Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles

Resurgence theory implies that the non-perturbative (NP) and perturbative (P) data in a QFT are quantitatively related, and that detailed information about non-perturbative saddle point field configurations of path integrals can be extracted from perturbation theory. Traditionally, only stable NP saddle points are considered in QFT, and homotopy group considerations are used to classify them. However, in many QFTs the relevant homotopy groups are trivial, and even when they are non-trivial they leave many NP saddle points undetected. Resurgence provides a refined classification of NP-saddles, going beyond conventional topological considerations. To demonstrate some of these ideas, we study the $SU(N)$ principal chiral model (PCM), a two dimensional asymptotically free matrix field theory which has no instantons, because the relevant homotopy group is trivial. Adiabatic continuity is used to reach a weakly coupled regime where NP effects are calculable. We then use resurgence theory to uncover the existence and role of novel `fracton' saddle points, which turn out to be the fractionalized constituents of previously observed unstable `uniton' saddle points. The fractons play a crucial role in the physics of the PCM, and are responsible for the dynamically generated mass gap of the theory. Moreover, we show that the fracton-anti-fracton events are the weak coupling realization of 't Hooft's renormalons, and argue that the renormalon ambiguities are systematically cancelled in the semi-classical expansion. Our results motivate the conjecture that the semi-classical expansion of the path integral can be geometrized as a sum over Lefschetz thimbles.


Contents
where Z[λ] is the partition function. If λ can be kept small, 1 one expects to be able to evaluate the path integral using the saddle-point method, so that (schematically) Above p 0,n are the perturbative contributions to O, and encode an expansion in fluctuations around the trivial "perturbative" saddle-point U 0 of the path integral, which has zero action. There are also contributions from non-perturbative saddle point field configurations U c , which have finite actions S c measured in units of λ, and contributions from the perturbative fluctuations p c,n around U c . Eq. (1.2) is traditionally viewed as the semiclassical approximation to the original path integral. The reason is that in almost all interesting QFTs, and even in simple quantum mechanics or even simpler ordinary integrals, the perturbative series expansions around both the perturbative saddle U 0 as well as U c are actually divergent asymptotic expansions, with p 0,n , p c,n ∼ n! [1][2][3]. The standard way to give a meaning to such perturbative series is via Borel transform and resummation. After computing the Borel transform of an asymptotic series, and its analytic continuation, one obtains a function with singularities in the 'Borel plane'. The Borel sum of the perturbative series is defined as a Laplace transform of the analytic continuation of the Borel transform. The issue is that if p 0,n , p c,n ∼ n! then there will be singularities on the integration contour in the Borel sum, and the integral -and hence the sum -is not well-defined. Different choices of contour deformations to avoid the singularities give different results for the same physical observable. This is a reflection of the fact that λ ∈ R + is a Stokes line. As a result, the Borel sums of all of the perturbative series appearing in Eq. (1.2) are usually not well-defined.
Another (much less widely appreciated) fact about the semiclassical expansion is that the amplitudes associated with certain saddle points, for example, correlated instanton-antiinstanton [IĪ] events, are not well-defined either [4,5] along the λ ∈ R + Stokes line. That is, in addition to the ambiguities in the sum of the perturbative series, the sum over nonperturbative saddle points also suffers from ambiguities. But if every perturbative series and most of the non-perturbative factors appearing in our expansion are not well-defined, then in what sense, and to what extent, does the semiclassical expansion capture the physics encoded in the original path integral? How do we give a meaning to a saddle point expansion?
The standard perspective is that the semiclassical approximation has an inherent 'fuzziness' defined by the size of the resummation ambiguities, and Eq. (1.2) only approximates the value of the original integral up to semiclassically-incalculable corrections of the order of the ambiguities.
Although the inclusion of the contributions of the NP-saddles seems to make the problems in the semiclassical approximation even worse, we will argue that including the NP saddles is in fact the solution to defining our saddle point expansion for O [λ] in an ambiguity-free, meaningful way.

Resurgence theory
To see how the program of assigning unambiguous meaning to the semiclassical expansion for O[λ] might work, we note that it has been known for some time that there are special cases when an unambiguous meaning can be assigned to Eq. (1.2) by carefully including the contributions of the NP saddles. For example, in a double-well or periodic potential problem in quantum mechanics, it is known that the leading ambiguity in perturbation theory is cured by the ambiguity in the [IĪ] amplitude (and fluctuations around it), and the ambiguity in the perturbation theory around an instanton is cured by the ambiguity in in the [IIĪ] amplitude (and fluctuations around it), etc. [4][5][6], and see also [7,8].
Such cancellations of ambiguities may seem magical, but in fact underlying the cancellations there is a systematic mathematical framework called resurgence theory, a term coined in a different context by J. Ecalle in early 80s [9]. Applied to QFT, resurgence theory is a generalization of the venerable idea of Borel resummation of the perturbative expansion around the perturbative saddle which systematically incorporates Stokes phenomena [10][11][12][13][14]. As described above, in most interesting quantum mechanical systems and QFTs, Borel resummation does not work (i.e., gives ambiguities) due to singularities in the Borel plane. If the Borel transform of all perturbative series are endlessly continuable (i.e. the set of singularities in all Riemann sheets are discrete and there are no natural boundaries), then trans-series of the form of Eq. (1.2) can be viewed as expansions of resurgent functions. Ecalle's work [9] implies that for such trans-series, all would-be ambiguities of the semiclassical representation cancel, see also [15]. The key to these cancellations is that in the trans-series representation Eq. (1.2), there are also ambiguities associated with the non-perturbative factors e −Sc/λ , which exactly cancel the leading ambiguities in the perturbation theory, with further (more intricate) relations amongst the various terms in the trans-series leading to the cancellation of ambiguities at higher orders, in such a way that the trans-series representation is ambiguityfree to all orders 2 . If we conjecture that observables in QFT are resurgent functions, then resurgence theory implies that the expansions around any given saddle-point must contain exact information concerning the expansions around all other saddle-points of the theory. In particular, resurgence implies that encoded within the large order terms of perturbative series there is exact information about non-perturbative saddles. As a suggestive equation, one may call this idea "P-data = NP-data".
We should emphasize that resurgence suggests a major philosophical shift on the meaning of the semiclassical approximation. If the right hand side of Eq. (1.2) can indeed be systematically interpreted in an unambiguous way, then the semiclassical expansion should not be thought of as an approximation. Instead, when viewed as a resurgent trans-series, the saddle point expansion should be viewed as an exact coded representation of the observable O(λ) in the regime of the QFT which is smoothly connected to the small λ semiclassical limit.
In this work, we take resurgence as our guiding principle, and use it to find new saddles in certain QFTs. We are able to systematically test the predictions of resurgence theory by using the recently developing ideas of adiabatic continuity and weak coupling NP-calculability.

Beyond the topological classification of NP saddles
In the context of QFT, it has recently been proposed to use resurgence theory to provide evidence for a non-perturbative continuum definition in the semi-classical domain [16][17][18][19] by invoking the idea of adiabatic continuity [20,21]. This program provides a new insight into 't Hooft's mysterious renormalon problem [3,22]. In this context, resurgence theory has been applied to non-Abelian gauge theories on R 3 × S 1 and the CP N −1 non-linear sigma model on R × S 1 . In both cases, the theories involved have a non-trivial homotopy group classifying the stable NP saddle points, and consequently, they also have instantons, fractionalized instantons [23][24][25][26][27][28], and composite configurations made from some combination of correlated instanton and fractionalized instanton events. Using resurgence theory, it has recently been proposed that the ambiguities due to the most severe "infrared renormalon" sources of divergences in these asymptotically-free theories cancel against the contributions of the appropriate neutral bion (fractional instanton-anti-instanton) events with action 2 N in units where BPST instanton action is normalized to unity, in a semi-classical regime of the theory [16][17][18][19]. If it turns out that the cancellations of ambiguities persist to all orders, resurgence theory would yield a systematic non-perturbative semi-classical definition of asymptotically-free theories.
It is important to note that resurgence provides a classification of NP saddles which is more refined than the traditional topological classification of saddle points, based on π 3 [G] in 4D gauge theories with gauge group G, and on π 2 [T ] in 2D non-linear sigma models with target space T . If two saddles are in the same conjugacy class in these homotopy groups then they carry the same topological charge. So topology cannot be used to distinguish them. On the other hand, if these two topologically identical saddle points have different actions, then the non-analyticities in the coupling λ of their contributions to the path integral are different, and hence they are distinguishable using resurgence theory. For example, the perturbative saddle and the instanton saddle by definition constitute two different conjugacy classes according to homotopy, call them C 0 and C +1 . The elements of these conjugacy classes are The elements within each class are not distinguished by topological considerations. However, the elements of these conjugacy classes can be distinguished according to resurgence theory. This is the motivation of the "resurgence triangle" classification of saddle points discussed in [18,19].
In this work, we give a more dramatic realization of the idea of the resurgence triangle classification of saddle points. Some interesting QFTs have a trivial homotopy group. Relatedly, they do not possess any known topologically-stable finite-action field configurations like instantons, and hence cannot have fractionalized instantons either! So one might naively think that a semi-classical calculation of observables in such theories would include contributions only from the trivial perturbative saddle point.
However, high-order factorial divergences of perturbation theory are ubiquitous and are known to occur even in theories without instantons. If the resurgence formalism is the right way to think about the semiclassical representation of path integrals, it implies that there must always be finite-action configurations that contribute to the path integral whenever the sum of the perturbative series is ambiguous. This must be the case even when homotopy considerations leave no room for contributions from stable instantons or their constituents. Resurgence theory thus provides a more refined classification of the finite-action field configurations that can contribute to path integrals than conventional homotopy-theoretic methods. Understanding how this works in detail is a major focus of this paper. Previous works related to this question include [18,19,29] in the context of the CP N −1 model and [16,17] in QCD(adj) and deformed YM, which are theories with a non-trivial homotopy group, and [30] in a theory with trivial homotopy group. Indeed, the present paper is a detailed exposition of the results briefly announced in our joint work with G. V. Dunne [30].
We also note a related work [31] in the context of matrix models and topological string theory. Indeed, there is an important body of work applying resurgence theory to matrix models and string theory [32][33][34][35][36][37], where resurgent analysis is used to find new non-perturbative sectors which must also be taken into account in order to construct full non-perturbative solutions. For a recent review emphasizing resurgence in quantum mechanics and matrix models see [38].
The consequences of the application of resurgence theory to the PCM are rather striking. Resurgence tells us that if the PCM model exist as a quantum theory, item S6) in the similarity list implies that all the elements in the contrast list must be consequence of superficial reasoning. At a deeper level, the similarity of the large-order growth of perturbation theory in PCM and YM theory makes it impossible that the principal chiral model only has a trivial perturbative saddle point. In fact, it implies that it must possess a plethora of NP-saddles which is just as rich as in Yang-Mills theory. In this work, we confirm this resurgence theory expectation by explicit calculations.

Outline
The organization of this somewhat lengthy paper is as follows. In Section 2, we provide a zero dimensional toy example, related to the 2d theory via dimensional reduction, which exhibits Borel non-summability, Stokes phenomena and the cancellation of ambiguities upon Borel-Ecalle (BE)resummation. In combination with Appendix A, we hope that this provides a gentle introduction to some of the methods of resurgence theory. In Section 2, we also point out the relation between semiclassical expansions and Lefschetz thimbles, giving a geometric perspective on resurgence. Section 3 summarizes some basic facts and expectations about the dynamics of the PCM on R 2 . In Section 4 we explain the construction of the unique weakly-coupled small-L limit of the PCM on R × S 1 L which is continuously connected to the theory on R 2 . Our analysis is inspired by the one in [18,19] for the CP N −1 model. The weak-coupling parameter turns out to be N LΛ 2π 1, similarly to deformed YM and QCD(adj) [21] and 3d YM with adjoint matter or twisted boundary conditions [45,46]. At leading order in N LΛ, low-energy observables in the 2D PCM can be described by a simple onedimensional effective field theory, which is just quantum mechanics. In Section 5 we study large-order perturbative behavior of the weakly-coupled QM limit of the SU (N ) PCM. By using resurgence theory techniques, we identify the non-perturbative ambiguity in the Borel resummation and interpret it as pointing to the presence of new NP-saddles in the problem. In Section 6, we show that the model indeed has the predicted non-perturbative saddle points, the unitons and fractons, and describe their properties. In Section 7 we show that the amplitudes of correlated fracton-anti-fracton events (which we often refer to as neutral bions) have ambiguous parts on the Stokes line, and these ambiguous parts cancel the renormalon ambiguities of perturbation theory. Thus, we interpret the neutral bion as the semi-classical realization of the infrared renormalon. In Section 8 we show that the fractons are responsible for the generation of the mass gap in the bosonic PCM at small-L. We describe a plausible flow of the mass gap as the radius is dialed from small to large-L. Relatedly, we point out that the Borel plane singularities on R 2 are twice as dense as compared to the location of singularities on R × S 1 , and argue that there should exist a smooth flow in the location of the S-matrix using integrability -such as the presence of the mass gap -are certainly reasonable, they have yet to be demonstrated from first principles. Finally, the integrability-based approaches do not yield any information about the interpretation of the divergences of renormalized perturbation theory, which are a major focus of our work. singularities as the radius is dialed from small to large-L. We refer to this phenomenon as Borel flow. Understanding the exact nature of the Borel flow would amount to solution of the mass gap problem on R 2 , which is an open problem.

Zero dimensional prototype for resurgence and Lefschetz thimbles
In this section, we consider a zero dimensional integral using steepest descent methods. as a prototype of the semi-classical approach in path integrals. In fact, the zero dimensional model is related to the 2d QFT by dimensional reduction. Compactifying the 2d QFT on small R × S 1 with twisted boundary condition on S 1 , we land on a quantum mechanical problem with periodic potential. Further compactifying the QM problem and going to the small S 1 × S 1 regime, the integral over the zeroth Kaluza-Klein mode reduce to our 0d prototype.
Perturbative expansion of the finite-dimensional integral already exhibits non-Borel summability, Stokes phenomena and cancellation of ambiguities upon Borel-Ecalle (BE)resummation that also take place in path-integral of PCM, and hence provides a useful playground in which we can show many properties very explicitly. 4 Consider the zero dimensional partition function Z(λ) where I 0 is the modified Bessel function of the first kind. Z(λ) is an integral of a real function over a real domain on a finite interval I = − π 2 , π 2 , hence the result is manifestly real for real λ. In order to demonstrate the use of some of the resurgence technology that we will use in QFT, we would like to study this integral by using the steepest descent expansion, which is the counter-part of the semi-classical expansion in our QFT example. The fundamental idea of the analysis is that to understand the behavior of the Z(λ) for λ ∈ R + one should understand the behavior of the analytic continuation of Z(λ) when λ ∈ C.
Our analysis will proceed as follows: 1) Identify all critical points.
2) Allow λ to move off R + into C, and analytically continue Z(λ) by rewriting the original integration cycle as a sum over steepest descent paths, which are called Lefschetz thimbles in general.
3) Develop perturbation theory around the P and NP saddles, and derive the respective asymptotic expansions. This is the counterpart of the semi-classical approximation in QFT.

4)
Show that the action of the NP saddle governs the growth of late terms in the perturbative series around P-saddle, and that sub-leading corrections to the late terms in the perturbative series around the P-saddle are governed by early terms of the perturbative expansion around the NP-saddle and vice versa.

5)
Show the cancellation of ambiguities and the reality of the trans-series representation of Z(λ) on the λ ∈ R + Stokes line.
We first view the action as a meromorphic function S(z). This leads to a more natural description of steepest descent method and the semiclassical expansion both in the present zero-dimensional example and in QFT. It is also the natural way to study the properties of partition functions under analytic continuation. In fact, a judicious analysis of the semiclassical expansion urges us to view all actions as meromorphic functions of the fields as we will see very explicitly. So we now change perspective on the integration cycle I as where Σ has real dimension one for general arg(λ) 5 . We must now address the question of how the integration cycle in Z(λ) changes once θ = 0. There are two non-degenerate critical points, call them z 0 and z 1 , obtained by extremizing the action We call the first one the P-saddle (perturbative vacuum) since it has zero action, and call the latter the NP-saddle since it has a positive action: We have also defined the "relative action" S 10 (called the "singulant" by Dingle [10]), which plays an important role in asymptotic analysis. 6 Associated with each critical point z i , there is a unique integration cycle J i , called a Lefschetz thimble or a steepest descent path, along which the phase ImS remains stationary and Re S has a downward gradient flow. Strictly speaking, Lefschetz thimbles are the multidimensional generalizations of steepest descent paths. The J i cycle is determined by the equation Because of the Cauchy-Riemann equations, the minimization problem of Re S is same as the minimization of S(z). Equation (2.5) will in general have multiple solutions, but at a generic value of arg λ, only two of these pass through the saddle z i . One of these solutions correspond to a contour with downward gradient flow J i , and the other to a contour with upward gradient flow K i . So and there is a one-to-one correspondence between the critical points and Lefschetz thimbles. The set of the Lefschetz thimbles may be seen as forming a linearly independent and complete basis of integration cycles for integrals of e −S(z) . In general, the contours of integration deform smoothly as arg(λ) is varied, and pass through only the associated saddle. Exactly at the Stokes lines, these contours also pass through a subset of other saddles. Lefschetz thimbles are the natural geometric surfaces (lines in our example) which can be used to describe the analytic continuation of Z(λ) to complex λ. 5 More generally, we generalize I ⊂ R N −→ Σ ⊂ C N , where Σ has real dimension N . 6 One might naively think that a singulant is the equivalent of an instanton (which is a non-trivial saddle in the path integral formulation) in quantum mechanics or QFT, since both are nontrivial saddle points. However, in QM, or QFT, there is in general a charge (topological or perhaps emergent, as we will see here) associated with instantons, while the perturbative vacuum is neutral under this charge. Thus, the role that a singulant plays in the large-order behavior of perturbative series in ordinary integrals is actually played by instanton-anti-instanton [  The Lefschetz thimbles J i are generally unbounded, even when the original integration cycle is bounded, as illustrated in Fig.1. Therefore, we must address the issue of the convergence of the integration over a thimble. In doing so, we divide the complex z-plane into "good" and "bad" regions [48]. A good region corresponds to Re (S(z)) > 0 such that e −S(z) → 0 as |z| → ∞. A bad region is the complement, one in which Re (S(z)) < 0. An admissible contour for which the integral is finite by construction is the one which connects two good regions. In Fig. 1, the white regions are good and the red regions are bad. The J i cycles start and end in the good regions, while the K i cycles, which are not shown in the figure, start and end in the bad regions.
A general integration cycle Σ(θ) on which the integral converges can be written as a sum over the critical point cycles: The coefficients n i are piece-wise constant, but have jumps when θ crosses Stokes lines. To see an illustration of this, note that our original integration cycle I = − π 2 , π 2 can actually be written in two different ways, depending on how one approaches to θ = arg(λ) = 0 Stokes line: This is illustrated in Fig. 3.
The relation between the two choices of the integration cycles, the notion of of cancellation of ambiguities and BE-summability is discussed in Section 2.4. Note that despite the fact Figure 3. Stokes phenomenon (wall-crossing) at θ = 0: There is also a Stokes line at θ = π where J 1 jumps and J 0 does not.
that I is a finite interval in R, the critical point cycles J 0 and J 1 are infinite in C. Moreover, while J 0 (0 − ) and J 0 (0 + ) coincide on the real axis, their "tails" in the imaginary direction have a relative sign flip. There is also a flip of the sign of the coefficient of J 1 at θ = 0 (and J 0 at θ = π). These sign flips are realizations of the Stokes phenomenon [10]. They are responsible for the cancellation of the imaginary "tail" contributions to integrals running over either of the cycles in Eq. (2.8) 7 , so that the value of the integral for θ = 0 ± coincides with its value on the real integration cycle Σ(θ = 0) = I. This is an elementary but useful perspective on ambiguity cancellation, which is realized in Fig. 2 geometrically. More generally, note that as arg(λ) changes, the cycles J i deform in a smooth manner, except for Stokes lines, where they undergo jumps. The two Stokes lines are at θ = 0 and θ = π, and the respective jumps are: To see where these relations come from, note for example that when θ = 0 − , J 0 goes from one good domain to another good domain by passing through saddle z 0 and being arbitrarily close to saddle-z 1 . However, at θ = 0 + , there is no single path which can achieve this. To start and end in the same places when θ = 0 + as when θ = 0 − , one first needs to take the −J 1 thimble, then the J 0 thimble, then again go along −J 1 . So J 0 −→ J 0 − 2J 1 , as is illustrated in Fig. 3. Similar jumps in contours can be found in monodromies problems associated with certain Picard-Fuchs equations [49]. In summary, the expressions for the analytic continuation of the integration cycle from arg(λ) = 0 to arbitrary arg(λ) must take into account the Stokes phenomena Eq. (2.9) and it depends on the Stokes chamber: (2.10) This illustrates the reason that Lefschetz thimbles provide the natural basis for semi-classical expansions and the analytic continuation of Feynman path integral. Perturbative expansions around P and NP saddles: So far we have made no approximations in our analysis. If one could evaluate the integrals along the cycles Eq. (2.10) exactly, one would obtain an exact result for Z(λ). Usually, however, this is not possible, and the best one can do is evaluate the integrals using perturbative series. Hence we now find the perturbative expansion around each of the two saddles. The formal asymptotic expansion around the z 0 P-saddle is given by This is a non-alternating asymptotic series for θ = 0. The late terms grow as a n ∼ n! (S 10 ) n . The series is non-Borel-summable in the θ = 0 direction, but it is Borel-summable in the θ = π direction. This formal series is a perturbative representation of the contribution of the integral along the J 0 thimble.
The perturbative expansion around the NP-saddle (including the NP-factor) is given by This series is an alternating asymptotic series for θ = 0. The late terms grow as (−1) n n! (S 10 ) n , and the series is Borel-summable in the θ = 0 direction. On the other hand, it is not Borelsummable in the θ = π direction. This formal series is a perturbative representation of the contribution of the integral along the J 1 thimble.
The semiclassical expansion for Z(λ) can be written as a two term trans-series where σ i are trans-series parameters. Here the σ i parameters have exactly the same role as the n i coefficients of the Lefshetz thimbles in Eq. (2.7). The trans-series is an algebraic representation of the geometric information in Eq. (2.7). The analytic continuation of our original integral to complex λ involves different linear combinations of Z 0 (λ) and Z 1 (λ) in different Stokes wedges. The value of trans-series representation Z(λ, σ 0 , σ 1 ) of Z(λ) is that because it is an algebraic representation, it is well-suited for direct calculations using standard perturbative methods, while the value of the geometric Lefshetz thimble representation of Z(λ) is its ease of visualization.

Borel analysis and Stokes Phenomena
The Borel transforms of the formal series in Eq. (2.11) and Eq. (2.12) are given bŷ (−1) n a n t n n! , (2.15) and since the coefficients a n grow like n! bothΦ 0 (t) andΦ 1 (t) define two germs of analytic functions at t = 0. In this simple example we have the luxury of having closed-form expressions for all of the terms of the Borel transforms of the perturbative series, and indeed there are also closed-form expressions for the analytic continuations of these germs into C s in terms of hypergeometric functions: with singularity (branch point) at t = 1/2 for B[Φ 0 ](t) and at t = −1/2 for B[Φ 1 ](t). In more complicated examples, one might only have closed-form expressions for the low-order and high-order terms in perturbative series. However, this is still enough to determine the positions of the singularities of the Borel transform, since they are determined by the asymptotic behavior of the high-order terms of the perturbative series. The sectorial (directional) Borel resummations for Φ i , i = 0, 1 are given by These are well-defined holomorphic functions of λ in Re (e iθ /λ) > 0, where θ is now parametrizing a generic direction in the complex t-plane. For real coupling, arg λ = 0, we cannot directly work with S 0 due to the presence along the line of integration of a singularity of the Borel transform B[Φ 0 ](t). This is associated with the Stokes phenomenon in the complex λ-plane: the series Eq. (2.11) becomes non-alternating when θ = 0 and hence Φ 0 (λ) is not Borel summable when θ = 0. Φ 0 (λ) is, however, right and left Borel-summable. For the right summation one integrates along a contour which avoids the singularity in such a way that the singularity remains to the right of the contour (θ = 0 + in Eq. (2.17)), and analogously, for left summation, the singularity stays on the left of the contour (θ = 0 − in Eq. (2.17)).
As illustrated in Fig. 4, the difference of the right and left Borel resummation can be written as an integral over the Hankel contour γ which starts at ∞ below the imaginary axis, then circles the singular point at t = 1/2, and then goes back to ∞ above the imaginary axis: To obtain third line in Eq. (2.18) we used the known discontinuity property of the hypergeometric function [50]: valid for a + b = c, and the last line is the (unambiguous) Borel resummation of the Φ 1 (λ) series in the θ = 0 direction. If we only had access to the asymptotic expressions for the highorder behavior of the coefficients of the perturbative series, in the last line we would have obtained a perturbative series expression for S 0 Φ 1 (λ), rather than a closed-form expression for S 0 Φ 1 (λ) itself. The factor of 2i on the right-hand-side of Eq. (2.25) is called a Stokes constant (or "analytic invariant" inÉcalle's terminology): Stokes constants are non-vanishing only at singular points on Stokes line and are zero otherwise.
In fact, we can obtain the same result by changing the integration variable from the "field variable" z to the "action variable" u = S(z). Then the integrals over the J 0 (0 ± ) cycles are given by where ReS 0 Φ 0 is unambiguous. 8 More importantly, in the integral over the Lefschetz thimble associated with the P-saddle, J 0 (0 ∓ ), we immediately see the imprint of the NP physics! Moreover, we see all of the data associated with the NP saddle, both its NP weight and the perturbative fluctuations around it: it is encoded in the imaginary part of the integral along J 0 (0 ∓ ). Clearly, the imaginary part is ambiguous for θ exactly zero, since the result depends on how one approaches the Stokes line. This is a reflection of the non-Borel-summability of the perturbative series. In the geometric perspective we are following here, the flip in the imaginary part of the integral over J 0 (0 ∓ ) is due to the flip of the infinite "tail" of the integration cycle that takes place when crossing the Stokes line θ = 0. See Fig. 2 and Fig. 3. This is the geometric realization of non-Borel-summability. Happily, this is not the whole story, because the integration over the interval I is actually the linear combination of thimbles as seen in Eq. (2.8). We will come back to the story of how including the contribution from J 1 cures this problem in Section 2.4. First, however, it is useful to understand how this happens from the algebraic trans-series point of view.

Stokes automorphism and alien derivative
To understand how ambiguity cancellation works in the trans-series representation, it is useful to introduce the notions of Stokes automorphisms and alien derivatives from resurgence theory. To keep the presentation more streamlined, Appendix A summarizes some of the results and definitions that we are going to use in what follows.
Distinct sectorial solutions on two different sides of a Stokes line are "connected" through the Stokes automorphism, S θ : where Disc θ − denotes is the discontinuity arising on crossing the Stokes line, thus, In our example Eq. (2.18), the difference of the right and left summations in the θ = 0direction of Φ 0 (λ) is an exponentially small imaginary term given by The Stokes automorphisms connecting different sectorial Borel sums are non-trivial only at Stokes lines. The Stokes lines are θ = 0 for Φ 0 and θ = π for Φ 1 , and the resulting Stokes automorphisms are These equations encode the beautiful structure of resurgence: the P-saddle carries complete information about the NP saddle which is decoded using its Stokes automorphism in the θ = 0 direction. At the same time, the NP-saddle carries complete information about the P-saddle, which is decoded by its own Stokes automorphism, but in the θ = π direction. The set of all formal series appearing in our problem form a closed algebra under the action of the singularity derivative, also called alien derivative, which acts as The first one of these relations means that the action of the singularity derivative at ω = + 1 2 on Φ 0 is just Φ 1 times the Stokes constant Eq. (2.20). The action of the singularity derivative at any other point on Φ 0 just gives zero, because the Borel transform of Φ 0 does not have any other singularities, i.e., ∆ ω Φ 0 = 0 for ω = 1 2 .

Reality of resurgent trans-series for real λ and BE-summability
As stated earlier, for arg(λ) = 0, the partition function defined in Eq. (2.2) is manifestly real. On the other hand, the formal first sum Φ 0 (λ) in Eq. (2.28) is non-Borel summable in the singular direction θ = 0, and hence it has an ambiguity, which is of order ie − 1 2λ . On the other hand, for Φ 1 , the singular direction is θ = π, and hence it can be Borel resummed in the θ = 0 direction.
For example, for θ = 0 − (approaching the real line either from below), and we have similar cancellation for θ = 0 + , approaching the real line from above. The non-Borel summability of the perturbative expansion Φ 0 leads to two-fold purely imaginary ambiguity. But exactly at the Stokes line, the integration path is also two-fold ambiguous, J 0 ± J 1 , for θ = 0 ∓ . This maps to a two-fold ambiguity of the coefficient of the NP-term in the trans-series. The observable Z(λ) is the combination of the two contributions, and the ambiguities cancel in the appropriate combinations in any of the Stokes chambers, leading to the real (physical) result on positive real axis in coupling constant plane. This is an example of median resummation and Borel-Écalle summability. 9 Consequently, the trans-series expansions for the analytic continuation of the partition function Eq. (2.2) in different Stokes chambers are given by with a Stokes jump at arg(λ) = 0. Approaching the λ ∈ R + line from above or below, we observe that the real solution for a trans-series (2.13) is given by where s = 2i is once again the Stokes constant. This is a very simple example of the fact that cancellation of nonperturbative ambiguities leads to median resummation of trans-series [33]. This also seems to be valid for non-linear systems with infinitely many Borel plane singularities [15]. We comment on the generalization of this formula to QFT in Sec. 7.

Lefschetz thimbles and geometrization of ambiguity cancellation
We now return to the picture offered by the Lefshetz thimble decomposition of the integration cycle to see the geometrization of ambiguity cancellation on the Stokes line arg(λ) = 0 which we have already seen in the algebraic trans-series representation Eq. (2.27). The integral over the P-thimble J 0 at θ = 0 − and at θ = 0 + can be written schematically as where in the second step, we cut the segment [− π 2 + i∞, − π 2 ] and glued it to [ π 2 + i∞, π 2 ]. Because of the π periodicity of the integrand, the integral remains unchanged. Z is the original real valued partition function Eq. (2.2), and J 1 (0) is purely imaginary and is of order e − 1 2λ .
This formula makes it manifest that the integral over the P-thimble J 0 has an imaginary part and is not equivalent to the original partition function Z. For θ = 0 − the addition of + J 1 (0) kills the (undesired) imaginary part of the J 0 integral, and the combination J 0 (0 − ) + J 1 (0 − ) is the linear combination of thimbles associated with Z at θ = 0 − , namely Upon the Stokes jump at θ = 0 J 1 −→ J 1 , J 0 −→ J 0 − 2J 1 given in Eq. (2.9), we obtain where the combination of J 0 (0 + ), J 1 (0 + ) is simply the unique linear combination which cancels the imaginary part exactly at θ = 0 + . It is also important to note that except for arg(λ) = 0, (and arg(λ) = π), there is never an exact cancellation between the contribution of the two saddles, and both saddles contribute! In fact, at the anti-Stokes lines θ = ± π 2 , the modulus of the two contributions is the same, and there is an exchange of dominance.
Thus, the ambiguity in the imaginary part of the integration J 0 (0 ∓ ) is cancelled exactly by the ambiguity in the prefactor of the J 1 (0) integral. This is a simple geometric realization of the cancellation of ambiguities on the Stokes line. Stated another way, one can observe that on approaching the Stokes line from above θ = 0 + or from below θ = 0 − , the "amplitude" associated with the NP-saddle [z 1 ] θ=0 ± is given by a two-fold ambiguous result: This is the counter-part of the ambiguous structure of instanton-anti-instanton-type amplitudes in QM and QFT examples, where the associated amplitude (which may naively be expected to be real) actually possess an unambiguous real part and a two-fold ambiguous imaginary part: This is taking place in QFT for the same reason as in ordinary integration. Of course, in semi-classically calculable regimes of QFTs and in QM, there are infinitely many saddle points and Lefschetz thimbles. The thimbles are infinite dimensional algebraic varieties and when the theory is regularized on a finite lattice with a finite size, they become finite dimensional algebraic varieties. The saddle points are the perturbative vacuum and also instantons, biinstantons, topologically neutral instanton molecules, etc, with appropriate terminological modifications in theories without the topology to support instantons, as we explore in what follows. Let ρ 0 denote the perturbative vacuum and ρ n denote various NP-sectors, with action S n , that can communicate with the P-sector according to the structure of graded resurgence triangle explained in e.g. Sec 7. The evidence gathered so far suggests that the integration over J n , n = 0 yields both real and imaginary parts in QFT, while, in the 0d example in this section, J 1 yields only a purely imaginary contribution for arg(λ) = 0. The cancellation of the imaginary parts in path integral examples is essentially the same as for ordinary integrals. However, in QFT there are also real unambiguous contributions to observables from NPsaddles.

NP-data in late terms of P-expansion
Resurgence and the Stokes automorphism allow one to extract the structure of late terms in both P and NP sectors. The asymptotic large order behavior of the perturbative expansion can be deduced by using Cauchy's theorem. Taking z = 1 λ , we can write where summation over a is over the singular directions in Borel plane (or Stokes lines in the physical coupling plane), and C ∞ is a closed loop at infinity. In the present problem, the singular directions are θ = 0, π. For example, consider the large order behavior of Φ 0 . The only discontinuity of Φ 0 is in the θ = 0 direction and using Eq. (2.25) and Eq. (2.35), we obtain So the leading large order behavior of the asymptotic expansion around the P-saddle is determined by the relative action with respect to the NP-saddle. The corrections to the leading behavior are governed by the early terms in the perturbative expansion around the NP-saddle. This is a very explicit realization of the idea of resurgence stated just after Eq. (2.25): the information in the series expansion around the NP-saddle surges up, in a disguised form, in the expansion around the P-saddle and vice versa [9].

Review of the Principal Chiral Model on R 2
This section briefly summarizes some basic aspects of the principal chiral model (PCM) with and without fermions. The bosonic PCM in d = 2 dimensions is an asymptotically free matrix field theory. The classical action is given by where U (t, x) ∈ SU (N ), M is a two-dimensional manifold with µ running over t, x, and λ = g 2 N is a dimensionless coupling constant which must be held fixed when taking the large-N limit. This action is invariant under the global symmetry group SU (N ) L × SU (N ) R acting as U → g L U g R † , with g L ∈ SU (N ) L and g R ∈ SU (N ) R .
Classically, the theory is scale invariant, and the non-linear wave solutions for the U -field propagate at the speed of light. This means the classical theory has N 2 − 1 gapless degrees of freedom. This is similar to classical Yang-Mills theory, which also has N 2 − 1 massless (gapless) gluons.
In the quantum theory, the situation is believed to be both qualitatively and quantitatively different. In particular, a macroscopic observer in a hypothetical R 1,1 universe would not see the N 2 −1 non-linear U -field waves, just as we do not see non-linear Yang-Mills waves. Indeed, numerical lattice simulations indicate that the theory is gapped, see e.g. [51]. The mass gap is expected to be of the order of the dynamically generated strong scale Λ where µ is a cut-off scale, and β 0 is the leading coefficient of renormalization group betafunction. This model is interesting because it possesses a matrix-like large-N limit which is dominated by planar diagrams when N → ∞ with λ fixed, just like Yang-Mills theory. However, it has only received scant attention as a useful toy model for Yang-Mills. One of the primary reasons behind this is an apparent dissimilarity to Yang-Mills: the PCM does not have instantons while Yang-Mills theory does. We will come to the conclusion that this difference from Yang-Mills is only superficial, because under suitable conditions, there is an infinite class of the NP-saddles in PCM model as well.
As it happens, some NP saddles in the PCM, the unitons, have been discovered some time ago by Uhlenbeck [52]. Unitons are harmonic maps from S 2 to SU (N ) 10 . Unlike instantons, there is no homotopy argument for the stability of a uniton. However, (and sounding almost contrary to the previous statement), the uniton action is quantized in units of 8π g 2 [53][54][55][56][57] 11 . Thus the uniton amplitude is parametrically of the form: The structure of the moduli of a uniton, its zero and quasi-zero modes are so far not completely understood.
The bosonic principal chiral matrix theory is expected to posses the following properties: 1) Mass gap 10 Here, S 2 should be viewed as one point compactification of R 2 by including a point at infinity. 11 We are grateful to N. S. Manton for suggesting an elegant way to think about the quantization of the 2) Confinement: O(N 0 ) free energy in the low temperature regime.
3) Deconfinement: O(N 2 ) free energy in the high temperature regime.
A few remarks are in order about these properties: 1) In the integrability studies of the PCM where the model is viewed as solvable, the existence of the mass gap is an assumption. This assumption is something that we would like to derive in our framework. 2) We refer to the low temperature regime as a confined regime because there is no trace of the microscopic O(N 2 ) degree of freedom in the physical Hilbert space. 3) We refer to the high temperature regime as deconfined because of the liberation of the microscopic degrees of freedom.

Fermions
It will also be useful to add Lie algebra valued Majorana fermions to the bosonic Lagrangian, since that makes it easier to draw analogies with QCD with fermions in adjoint representation. The action Eq. (3.1) is replaced by and despite the asymmetric-looking form of the fermion transformations the theory still has an SU (N ) L × SU (N ) R symmetry for any N f .
The model with N f = 1 has N = (1, 1) supersymmetry, the minimal non-chiral supersymmetry in 2D, see e. g. [59,60]. On top of the three properties of the bosonic model mentioned above, this fermionic model is also believed to possess the property 4) Discrete chiral symmetry breaking and two isolated vacua (for any N ).
The N = (1, 1) model has a Z 2 discrete chiral symmetry: which is believed to be dynamically broken by the formation of fermion bilinear condensate, The existence of the two isolated vacua can also be backed up by the following independent argument, for the case of the N = (1, 1) SU (2) PCM. The group manifold for the SU (2) PCM is three-sphere S 3 , same as the O(4) sigma model. The supersymmetric index for the O(4) model (or generally for O(N ) models) was calculated a long time ago and it is equal to I S = 2 [61]. This is indeed compatible with the discrete chiral symmetry breaking and the existence of two isolated vacua. We claim that the supersymmetric Witten index for the PCM model for arbitrary N must also be equal to two: (3.6) N f > 1: The PCM model with N f Weyl Majorana fermions has an SU (N f ) continuous global symmetry and Z 2 discrete chiral symmetry. A continuous global symmetry in a finite N theory cannot be broken on R 2 due to the Mermin-Wagner-Coleman theorem. We expect that for sufficiently low N f the discrete chiral symmetry should be broken, and at large-N the SU (N f ) global symmetry may be broken as well.
The inclusion of an arbitrary number of fermion flavors does not modify the one loop beta function given in Eq. (3.2). Unlike in four dimensions, two-dimensional non-linear sigma models remain asymptotically free even at arbitrarily large-N f . Whether these theories are confining or exhibit IR conformal behavior should depend on the number of fermionic flavors, N f . The value N f at which these theories move from gapped (for U -fluctuations) behavior to gapless behavior is currently unknown. We expect that for N f < N f , the discrete chiral symmetry should be broken and these theories possess two isolated vacua.

Perturbation theory on R 2 and IR-renormalons
The structure of perturbation theory in PCM is similar to other asymptotically free matrix field theory. On general grounds, we expect two types of factorial divergences in the perturbative series associated with the P saddle point. These are 1. Combinatorial n! growth in the number of Feynman diagrams 2. Phase space n! contribution coming from the integration of high and low momenta from a fixed class of diagrams. These contributions are referred to as the UV and IR renormalons, respectively.
Usually, in theories with instantons, the combinatorial growth is associated with instantonanti-instanton [IĪ] pairs. That is, the ambiguity of Borel resummation of perturbation theory is cancelled against the ambiguity in the [IĪ]-amplitude. However, in the PCM, there are no instantons to begin with. In [30], we argued that the uniton saddle ought to substitute the [IĪ]-saddle and its ambiguity, yielding an ambiguity of order ±ie − 8π g 2 (Q) . On R 2 , Fateev, Kazakov, and Wiegmann [62,63] used integrability techniques to show that there exists a much larger ambiguity. It is guessed there that this is related to the leading renormalon ambiguity: where Q is a large Euclidean momentum. This is a sensible guess in the light of the operator product expansion (OPE). In the PCM, the leading non-trivial condensate is the dimension two operator Tr ∂ µ U ∂ µ U † . Since arg λ = 0 is a Stokes line in PCM, the expectation value of generic operators that can mix with perturbation theory are two-fold ambiguous [64]. The condensate evaluated at θ = 0 ± must give where c 1 and c 2 are pure numbers. Of course, one also expects more-suppressed IR renormalon ambiguities of the order Λ 2m /Q 2m , m > 1. On R 2 , there is no semi-classical interpretation for IR renormalons. However, at large Q 2 there is a useful hierarchy of non-perturbative scales associated with the IR renormalons which is exploited in the OPE approach. It is worth emphasizing that the leading IR renormalon ambiguity is parametrically much larger than the uniton ambiguity at large Euclidean momentum Q. It is actually the N th root of it.
In theories with fermions, if N f < N f , we expect only minor and relatively unimportant changes in the Borel plane structure. If N f > N f , one does not expect IR-renormalons to exist, since the coupling does not diverge in the infrared. This drastic change at N f is consistent with a speculation by 't Hooft, in the context of gauge theories, that IR-renormalons are connected with the mass gap and confinement [22].

Compactification to R × S 1 and adiabatic continuity
Usually, one can only hope to get analytic insights into the non-perturbative physics of a field theory when it is weakly coupled. This is definitely not the case for the PCM on R 2 , where the IR physics is strongly coupled in terms of λ. When the PCM is compactified on R × S 1 , however, at small enough circle size L the coupling will become set at the scale 1/L, and the theory is guaranteed to become weakly coupled thanks to asymptotic freedom.
In recent years it has become clear that in QCD-like gauge theories on R 3 × S 1 , the appearance of a weak-coupling regime at small-L can occur in two dramatically different ways: i) If the S 1 circle is thermal, as L is dialed from large to small, the gauge theory goes through a phase transition, or a rapid cross-over, in such a way that the physics in the small-L theory is not adiabatically connected to the physics in the large-L decompactification limit.
ii) If the S 1 circle is spatial in theories with e.g. massless adjoint fermions or if a centerstabilizing deformation is used, then as L is dialed from large to small, one finds that the small-L limit of the gauge theory is adiabatically connected to the large-L regime without any phase transition or rapid cross-over.
In gauge theory, thermal Yang-Mills theory and thermal N = 1 Super-Yang-Mills (SYM) yield examples of the first type of limit, as is well known from thermal field theory [65]. In this For small enough L, weak coupling guaranteed by asymptotic freedom Our approach is to put the theory on M = R time x S 1 (L) with physics adiabatically connected to original theory But with periodic boundary conditions, looks like a thermal circle! small L large L F ~ N 2 F ~ N 0 Large N phase transition, finite N cross-over! small L large L F ~ N 0 β , there is a rapid-crossover from an O(N 2 ) (deconfined) behavior of the free energy to O(N 0 ) confined regime, which becomes a genuine phase transition at N = ∞. Even at finite-N , however, the quantitative behavior of the theory changes dramatically between these two regimes, despite the fact that there is no sharp phase transition in a finite volume. Bottom: By using spatial compactification, we find a unique small-L limit in which "free energy" remains O(N 0 ). The behavior of the theory does not change dramatically from small-L to large-L. This is the idea of adiabatic continuity. Since the small-L theory is weakly coupled thanks to asymptotic freedom, it is NP-calculable, and the knowledge gained therein is continuously connected to the physics of decompactified theory on R 2 . case, the long distance 12 physics is strongly coupled and incalculable. Some examples of the second type are deformed Yang-Mills and spatially compactified N = 1 SYM [21,66]. In this case, the long distance physics remains weakly coupled and is non-perturbatively calculable.
Our goal in this section is to discuss the realization of both of these classes of small-L limits in the PCM by a careful analysis of the compactification procedure, as depicted in Defining the theory on an Euclidean base space manifold R × S 1 requires a choice of boundary conditions on the circle. We will consider a family of compactified theories labeled by a choice of boundary conditions where H L , H R ∈ su(N ). We refer to the (−) boundary conditions for the fermions as thermal, since they reduce to the standard anti-periodic thermal boundary conditions when H R = H L = 0, and we refer to (+) boundary conditions as spatial, since they become purely periodic BCs when H R = H L = 0. For generic H L , H R , these are non-thermal compactifications. For the supersymmetric N = (1, 1) theory, the spatial boundary conditions respect supersymmetry. The question is which theory, in this family, is in the same 'phase' at small-L as the theory on R 2 13 . With a thermal compactification, there is a rapid cross-over/phase transition at finite/large N as L is dialed from large to small values. Our goal is to construct a special small-L limit in which this does not happen, so that the changes between large-L and small-L are 'adiabatic'. One of the two observables we will use in our construction will sharpen into a proper order parameter in the large-N limit, where the Coleman-Mermin-Wagner theorem no longer forbids such a notion.
To proceed with the analysis we find it convenient to switch variables toŨ ,ψ: so thatŨ ,ψ are periodic on S 1 . In terms ofŨ ,ψ, the action becomes and H V,A = 1 2L (H L ±H R ). We can interpret 1 L H L and 1 L H R as background gauge fields for the global symmetry group SU (N ) L × SU (N ) R . The symmetry of the original action Eq.
which are effectively global 'gauge' transformations. A theory with the twisted boundary conditions (BCs) of Eq. (4.1) can be equivalently viewed as a theory with periodic BCs forŨ ,ψ with the constant background gauge fields 1 L H L/R 14 . From here onward we will work with the latter picture, and drop the tilde on U, ψ for simplicity. 13 We are working with a two-dimensional theory, for which the Coleman-Mermin-Wagner theorem implies the lack of any local order parameters which could distinguish distinct phases at finite N . Nevertheless, there can still be rapid cross-overs in the physical properties as a function of L at finite N (which become a sharp phase transition at N = ∞) and as a result small-L physics can be very different from large-L physics.
14 Note that these constant background fields HL,R are not standard chemical potentials for the conserved

Large-L expectations
To figure out the right choice for H V , H A , we have to decide what properties of the large-L theory we want to capture in the small-L limit. From our perspective, the most essential features of the bosonic PCM on R 2 and large S 1 × R are the existence of a mass gap and the order N 0 free energy density in the large-N limit. We want to find a small-L limit which allows both of these expected properties to persist. To translate these heuristic expectations into sharp conditions on the theory, we first observe that the existence of mass gap implies that when L → ∞, the dependence on the boundary conditions must vanish, since in a theory with a mass gap we expect finitevolume effects to vanish as e −∆L , where ∆ is the mass gap. Let Z(L; H L , H R ) be the partition function of the theory with the background fields turned on, and let us define as F(L; H L , H R ) = V −1 log Z(L; H L , H R ) as the "twist free energy", where V −1 is the volume of the space-time manifold. Then it is clear that in the large-L confined phase, the free energy must be independent of the boundary conditions at leading order in N . At small-L, it is not possible to have complete independence of the theory from L 15 . The highest degree of independence from L one can demand at small-L is , and the subscript on · H V ,H A is a reminder that the expectation value is to be taken with background fields turned on. Heuristically, this means that we demand that changing boundary conditions should not result in persistent currents in the compactified direction. Equation (4.9) automatically holds in the decompactification limit in the large-L theory, but it becomes a non-trivial constraint at small-L.
Next, to have any chance that the small-L theory is confined according to the count of the degrees of freedom contributing to the physical Hilbert space, the "twist free energy", i.e, the free energy of the system at a given value of the the boundary conditions must remain O(N 0 ). For instance, with purely 'thermal' boundary conditions, with H V = H A = 0, when L can be interpreted as an inverse temperature 1/T , at small-L the theory becomes weakly coupled, but one expects the theory to be 'deconfined', with all N 2 − 1 components of the matrix U liberated and contributing equally to free energy density, so that F ∼ O(N 2 ). This is in sharp contrast to what happens in the low temperature decompactification limit, where we expect the theory to be in a 'confining' phase with a free energy F ∼ O(N 0 ). At large-N , the value of 1 N 2 F becomes a bona-fide order parameter: it is zero in the decompactification limit, but may become order one at small-L, depending on the values of H V , H A . Hence we will demand that F (or more precisely the natural dimensionless quantities L 2 F or Λ −2 F associated to F) must scale as N 0 for any L. This expectation is believed to be automatically met in the decompactification limit based on previous studies using integrability [40,41] or lattice Monte Carlo simulations, see e.g. [51]. But demanding the relation L 2 F ∼ O(N 0 ) poses a non-trivial constraint at small-L.
To summarize, our adiabaticity conditions are 1. We demand an insensitivity to changes in boundary conditions, Eq.(4.9).
2. We demand a free energy that scales as O(N 0 ).
3. For the supersymmetric N f = 1 PCM, we impose susy preserving boundary conditions.
For the non-susy N f > 1 case we impose the same supersymmetric boundary conditions for all the fermions 16 .
We emphasize that we are not assuming that the theory is gapped at small-L. That is a dynamical question about the theory, which should be settled by a calculation at small-L. What makes the construction interesting is that the unique small-L theory selected by the conditions above does turn out to have a non-perturbatively-generated mass gap! This is consistent with the notion that the small-L limit we construct really is adiabatic.

Choosing the right small-L limit
Our goal for the rest of this section is to see which choice of H V , H A results in a Z(L; H L , H R ) which reproduces the large-L expectations encoded in Eq. (4.9) and L 2 F ∼ O(N 0 ) even at small-L. When LΛ is small enough, the theory will become weakly coupled in terms of U . The precise meaning of 'small enough' is subtle, and we address it below. If U is parametrized as we expect that at weak coupling the dominant contribution to Z will come from fluctuations which are quadratic in W . We first determine the form of H A consistent with Eq. (4.9). To do this, we perform a global axial 'gauge' transformation using g ∈ SU (N ) A , so that H A → g H A g † is diagonalized and lies within the Cartan subalgebra of su(N ) H A = a α t α , with {t α } being the Cartan generators normalized as Tr(t α t β ) = δ αβ /2. The small W (i.e., perturbative) action then becomes This constitutes a tree-level potential for the eigenvalues of H A , which is extremized when H A = 0. Hence to satisfy Eq. (4.9), we must set 17 H A = 0. So long as the theory is weakly 16 While it is clear that this is a sensible demand for the N f = 1 theory, the reason to demand this for general N f ≥ 1 is more subtle. The reason we do so is that we expect the large-N PCM with N f 'adjoint' fermions to have an emergent fermionic symmetry at large-N due to arguments similar to the ones recently given for QCD [Adj] in [67]. 17 Note that the partition function actually depends on the conjugacy classes of HL, HR: Hence once can trade a purely axial background HV = 0, HA = 0 (as in [62,63]) for a purely vectorial one, provided one modifies Eq. (4.10) accordingly. We thank V. Kazakov and Z. Bajnok for pointing this out to us. coupled, quantum effects cannot change the extremum of the potential energy as a function of H A , whether or not there are fermions in the theory. The reason is the presence of the non-vanishing tree-level potential above. Hence from here onward, we will set H A = 0. Now we can focus on working out the effective potential for H V . When H A = 0 we can use a global vectorial 'gauge' transformation to diagonalize H V , so that it lies in the Cartan subalgebra. The energy is independent of H V at tree-level, so now we must do a one-loop analysis to compute the first non-trivial contributions to the twist free energy.
Before presenting the result for the twist free energies with the two classes (thermal and spatial) of BCs we are considering, it is useful to introduce a Wilson loop operator associated to H V , which will allow us to write the one loop twist free energy in a more illuminating form: Note that the eigenvalues of Ω are SU (N ) V gauge-invariant, and Tr Ω transforms non-trivially under the Z N center symmetry of SU (N ) V acting as Ω → ωΩ, ω ∈ Z N . The expression for the twist free energy of the PCM in the presence of the background gauge field H V can now be rewritten in terms of the Ω 18 : One can now calculate the potential for the background holonomy Ω by integrating out the weakly coupled KK-modes at one-loop level. The result is 14) An intuitive way to derive Eq. (4.15) is as follows. (We only detail the derivation for the spatial case; the thermal analysis is very similar.) At sufficiently small-L, the Kaluza-Klein tower of modes in the principal chiral model can be viewed as a collection of simple harmonic bosonic and fermionic oscillators. The strategy of the derivation follows Lüscher and van Baal [68,69]. However, the idea of adiabatic continuity, which is instrumental for our purpose, did not appear in these earlier works.
In spatial compactification, i.e., with periodic boundary conditions for fermions, the KK modes of bosons and fermions are degenerate at the classical level. (This may or may not be lifted quantum mechanically, depending on the theory.) There are (N 2 − 1) physical bosonic fluctuations and N f (N 2 − 1) fermionic fluctuations for each Kaluza-Klein level k ∈ Z. The vacuum energy density associated with the background Eq. (4.13) is the sum of ground state energies of corresponding bosonic and fermionic harmonic oscillators, and it is equal to: Since the vacuum energy density Eq. (4.16) is periodic in µ ij ≡ µ i − µ j with period 2π, it can be Fourier transformed, which is actually the Poisson resummation of the original formula. Here, P n = (1−N f ) 2L 2 I n . The advantage of this form is that ij e inµ ij = | Tr Ω n | 2 , and we can express the result in terms of the background holonomy. The summation over KK modes k ∈ Z thus turns into a summation over the winding number of the line operator. The prefactor is: resulting in vacuum energy density which is identical to Eq. (4.15). Our adiabaticity conditions now tell us that we must find the extrema of the twist free energy, and compute the large-N scaling of Z = L 2 V at the extremum. Note that the expressions for the twist free energy is very similar to the effective potential for the eigenvalues of Polyakov loops in SU (N ) gauge theories on R 3 × S 1 [16,70] with N f adjoint Weyl fermions. The minimization problem for the potential is same as the one in QCD(Adj) on R 3 × S 1 19 . It turns out that there are two extrema that we must deal with in general.

Thermal compactification and non-adiabaticity
There are N 'thermal' extrema of the thermal or spatial twist free energy at where k labels the center-position of the lump of eigenvalues. For these extrema H V = 0, and if we use anti-periodic BCs for fermions this is precisely the standard Euclidean thermal compactification. Note that Tr Ω thermal transforms non-trivially under all non-trivial elements in the center subgroup In the thermal case, the free energy density is This gives a nice check of our calculations, because this is precisely the expected Stefan-Boltzmann law: (N 2 − 1) is the number of bosonic degrees of freedom and π 6 T 2 is the free energy per boson, while (N 2 − 1)N f is the number of fermionic degrees of freedom, with π 12 T 2 being the free energy per fermion. The factor of two difference follows from statistics, Bose-Einstein versus Fermi-Dirac. This is the deconfined regime shown in Fig. 5 where the O(N 2 ) degrees of freedom are liberated, not adiabatically connected to the large-L confined regime.

Spatial compactification and adiabaticity
The only other extremum of the spatial twist free energy for N f = 0, N f > 1 is This extremum of the twist free energy is unique, and Tr Ω is neutral under the Z N center symmetry Z N ⊂ SU (N ) V , with Tr Ω n vanishing for n mod N = 0. This is in sharp contrast with the thermal holonomy in Eq. (4.20), for which Tr Ω n = 0 for all n.
For N f = 1, the theory has N = (1, 1) supersymmetry. The fermion boundary conditions associated with V (+) also respect supersymmetry. Consequently, the vanishing of the one-loop contribution actually extends to all orders in perturbation theory. At the non-perturbative level, we expect to find non-vanishing contributions to V (+) , and for the background in Eq. (4.22) to be a non-trivial extremum, as is the case in 4D N = 1 SYM [71] and the 2D N = (2, 2) CP N −1 model. However, we leave an explicit verification of this for the PCM to future work. At the unique center-symmetric extremum characterized by Eq. (4.22), the twist free energy is (4.23) Note that it remains O(N 0 ) at arbitrarily small-L, as opposed to the thermal case, where the associated free energy is O(N 2 ), as anticipated in Fig. 5. Hence we refer to this centersymmetric small-L regime as confined. We also observe that only the theory associated to V + with Ω satisfying Eq. (4.22) satisfies all of our conditions of Section 4.1 for continuity between small-L and large-L regimes. Hence we have found a unique choice of boundary conditions for the principal chiral model for which one can expect that the physics at small-L should be smoothly connected to physics at large-L. We conjecture that with the Z N -symmetric background holonomy, the PCM on R × S 1 L has an adiabatic small-L limit. If L is small enough, the theory remains weakly coupled at long distances, as illustrated in Fig. 6. This is the counter-part of the adjoint-Higgs/Wilson line branch in gauge theories.
5 Structure of the perturbative series on small R × S 1 L Now that we have devised a weak-coupling limit on R × S 1 adiabatically connected to the theory on R 2 , we can expect long-distance observables to be calculable using semiclassical methods, with a trans-series representation of the form where λ = λ(1/N L). The reason for the inclusion of the log λ −1 terms will become apparent in Section 7.
As we mentioned in Section 3.1 the leading singularity, along the positive real axis of the Borel plane, for the R 2 theory is the IR-renormalon singularity and it is located at 1/N of the uniton action. On R 2 , the theory is strongly interacting, and there is no semiclassical interpretation of this singularity. Once we put the theory on R × S 1 and go to the adiabatic small-circle limit described in Sec. 4, the Borel singularity positions should move in a smooth way relative to their locations in the R 2 limit 20 . Since the adiabatic small-L limit we constructed in Sec. 4 is weakly coupled, however, we should be able to see renormalon ambiguities in the large-order behavior of perturbation theory in λ[1/N L], and reproduce at least the factor of N in the expression above. We now demonstrate that this is indeed the case.
The quickest route to see the semi-classical realization of the IR renormalons involves working out the perturbative series describing the contributions to the ground state energy E(λ) from fluctuations around the P saddle: where we have chosen to write the ground state energy in the natural units of the problem, which turn out to be ξ = 2π/(N L). The ground state energy will receive contributions from modes with momenta Q > 1/(N L), as well as modes with Q < 1/(N L).
The effective coupling constant for the UV modes with Q > 1/(N L) is λ(Q), and runs logarithmically, as can be seen in Fig. 6. Such modes will produce factorially-growing and sign-alternating contributions to the perturbative coefficients p n . More precisely, the signalternating part of the perturbative series is In the phase space integration over high momenta, the dominant contribution aries from the UV scale where Q is some large external momentum. These are associated with singularities on the negative real axis in the Borel plane, and are called the UV renormalons [3]. The UV renormalons can not be affected by compactification, because * UV ∼ LN e −n LN . In other words, UV renormalons probe only the high-momentum behavior of a theory and do not care if the space is compactified or not. So it is natural to see them appear unchanged in the compactified theory. Their presence is not related to the particular regularization used to compute the phase space integration over high momenta and the renormalons can be recovered also on the latticized theory [72][73][74].
The contributions from modes with Q < 1/(N L) are a different and more subtle matter. As illustrated in Fig. 6, the relevant coupling for these modes is effectively λ(1/N L), which ceases to run at energies lower than 1/LN . Naively, the absence of running would mean that the IR-renormalon should disappear. For example, if we take the PCM on R 2 with N f > N f , where N f is the value of N f above which the theory is conformal in the IR, the coupling stops running in the IR as well, and this was our reason for asserting in Section 3.1 that the IR-renormalons must disappear when N f > N f . So why is it that the freezing of the coupling at a small value in the infrared, which leads to the absence of IR-renormalon singularities when it happens on R 2 , does not lead to the same effect on R × S 1 ? Are we not running into a contradiction?
In fact, this is precisely at the heart of our adiabatic continuity idea. First, on R 2 , the dominant contribution to the renormalon singularities comes from the scale which is exactly the scale where the theory becomes strongly coupled and perturbation theory becomes unreliable. Studying the Borel resummation of perturbation theory, as briefly summarized in Section 3.1, one finds the leading-order ambiguity ±iΛ 2 Q −2 ∼ ±ie − 8π g 2 N (Q) for processes involving a large external momentum Q. Note that this is of order [e − 8π Because of the strong coupling problem, there is no semi-classical interpretation for the IR renormalons on R 2 . And indeed, until very recently it was widely believed that there is never a semi-classical interpretation for IR renormalons. However, the more useful/refined question to pose is actually the following.
• Start with an asymptotically free theory on R d which has IR renormalon singularities.
Compactify on R d−1 × S 1 and dial radius of S 1 to a small value. What happens to the renormalon singularities if we are able to work with a compactification which adiabatically connects the theory on R 2 to a regime where the long distance dynamics becomes weakly coupled? 21 21 It is conceivable that there may be alternative setups to compactification in which an analogous question can be posed.
Below, we show that in the adiabatic small-L limit of the PCM, we can describe the longdistance physics using a small-L effective field theory, which is a quantum mechanical system. This theory has ordinary 1d instantons whose actions are 1 N that of the unitons on R 2 . We refer to these 1d instantons as fractons, since they are the fractionalized constituents of the uniton saddle from a 2d point of view.
We then show that the IR renormalon singularities on R 2 which arise from phase space integration and which are located at ∼ S uniton N transmute into semi-classical correlated fractonanti-fracton saddles on small R × S 1 L which are also located at ∼ S uniton N . The crucial factor here is the appearance of 1 N , which is the characteristic of renormalons, but the exact location of the singularities will generally change mildly between the semi-classical and non-semiclassical regimes. This is what is guaranteed to hold via adiabaticity. So we have reached the same conclusion in PCM as was found in deformed YM theory, QCD(adj), and the CP N −1 model [16][17][18], but in a theory which has no topologically-stable stable points!

Covering SU (N ) with SU (2)s
Before discussing the construction of the leading term in the small-L effective field theory, we make some remarks on the organization of perturbation theory more generally. The most common way to set up a perturbative calculation would be to write U = e iW , W ∈ su(N ), expand the action in powers of W , and then do perturbation theory in terms of W . Indeed, this was how we organized our calculation in Section 4. From here onward, however, we find it advantageous to organize our perturbative calculations in a different way. We will examine the contributions to perturbative observables from field fluctuations within SU (2) subgroups of SU (N ), and then sum over the various SU (2) subgroups. 22 Consider the embedding of SU (2) into SU (N ) as two by two diagonal blocks: and

7)
22 This is the process by which which gauge configurations are generated in lattice Monte Carlo simulations for SU (N ) Yang-Mills theory [75], and is known to cover the whole SU (N ) manifold.
where z 1 , z 2 ∈ C, and |z 1 | 2 + |z 2 | 2 = 1. One can associated the affine root system of the associated su(N ) Lie algebra with each of these embeddings, U (1) j , j = 1, . . . , N , as the root associated with the corresponding SU (2). The affine root system consists of the N − 1 simple roots α i , i = 1, . . . , N − 1 along with the affine root α N : While the N distinguished SU (2) subgroups above will be the most important for our analysis, note that the generic SU (2) subgroup looks like (5.9) Each one of these SU (2) subgroups can be associated with sums of roots in the affine Lie algebra which are themselves roots.
To write down the actions describing the fluctuations within the U (k) i SU (2) sub manifolds, we need to choose a parametrization of SU (2) S 3 . We use the Hopf coordinate parametrization: and we take the angular variables to have the ranges θ ∈ [0, π], φ 1 ∈ [0, π], φ 2 ∈ [0, 2π]. For fixed θ, the variables φ 1 , φ 2 parametrize a torus. At the degeneration points, θ = 0, π/2, π, the torus shrinks to a circle. With this parametrization, for each one of our SU (2) embeddings, using the bosonic part of Eq. (4.5), we find that the fluctuations are described by where For the Z N -symmetric background, ξ i,k = 2πk/(N L) for all i = 1, ..., N (using affine roots when needed) 23 .

Large order behavior and Stokes phenomenon
To efficiently deduce the large-order behavior of the perturbative contributions to E when N LΛ/2π 1, we will use a small-L effective field theory. The IR properties of the theory for small enough L can be calculated from a theory where one integrates out modes with energies larger than the scale (N L) −1 , up to a subtlety which we comment on below. Therefore, the effective field theory is forgetful of UV-renormalon singularities, which can only be extracted from the microscopic theory. The leading term of the action of the small-L effective field theory is where ξ i,k is the same as Eq. (5.12). The expression in the second line comes from a restriction to a generic SU (2) subgroup of SU (N ), given in Eq. (5.9). This leading term of the small-L EFT is a 0 + 1 dimensional field theory, which is just quantum mechanics. An important subtlety with the derivation of Eq. (5.14) is that it is not quite true that all KK-momentum carrying states decouple at low energies. The issue is that states that carry non-zero winding number can contribute to the low-energy dynamics on the same footing as states that carry zero winding number. To see this, note that if we look at the contribution from U (N −1) i and allow φ 2 to carry −1 units of winding number, we find S = L g 2 dt θ 2 + cos 2 θφ 2 1 + sin 2 θφ 2 2 + −2π L + ξ(N − 1) 2 sin 2 θ (5.15) = L g 2 dt θ 2 + cos 2 θφ 2 1 + sin 2 θφ 2 2 + ξ 2 sin 2 θ , (5.16) where ξ = 2π/N L. This contribution is in fact associated precisely with the affine root of the su(N ) Lie algebra, and its relevance is due to the compactness of the Z N -symmetric background holonomy. So there are contributions to the low-energy physics from some configurations with winding number −1 which are of the same magnitude as contributions from field configurations that carry winding number 0. This is an illustration of the fact that the adiabatically-compactified theory "remembers" its two-dimensional nature even when L is very small. This subtlety will also be important in the analysis of non-perturbative saddle points in the next section.
To understand the high-order behavior of perturbation theory in g 2 [1/N L] using the leading order part of the EFT action -that is, leading order in an N LΛ/2π 1 expansion -it turns out to be useful to temporarily move back to Minkowski space and use Hamiltonian methods, because it lets us use some known results from the literature. At small-L the Hamiltonian associated with Eq. (5.14) is We emphasize that this Hamiltonian describes the contributions to the energies of the states from the N SU (2) subgroups U j , j = 1, . . . , N . In general, the way we have organized perturbation theory, we must sum over the contributions from all SU (2) subgroups to compute the energies of states in the full SU (N ) PCM. Fortunately, we will see that the U (1) j subgroups make the dominant contribution to e.g. the ground state energy at small-L, which simplifies our analysis. Also, to avoid confusion, we note that with the center-symmetric background gauge fields turned on, each of the U (1) j subgroups will make the same contribution to E. If the background fields were to be slightly perturbed away from the Z N symmetric points this would no longer be the case; the contributions are in principle distinguishable.
Having made the point about the importance of taking into account the various SU (2) subgroups, we present the subsequent calculations with ξ ≡ ξ 1 = 2π/(N L) to lessen the notational clutter. The dependence on ξ k always follows from the replacement ξ → ξ k .
The potential for θ has minima at θ = 0, π, and the action has a discrete symmetry P which acts as P : θ → π − θ. The states of the system will have well-defined eigenvalues ±1 under P . Moreover, since φ 1 , φ 2 are cyclic compact coordinates, associated to quantized conserved charges, a state with the quantum numbers of (±, n 1 , n 2 ) will have an energy of order E ±,n 1 ,n 2 ∼ g 2 ξ(n 2 1 + n 2 2 ) ∼ g 2 N L (n 2 1 + n 2 2 ) .

(5.18)
We expect the ground state and first excited states to have the quantum numbers (+, 0, 0), and (−, 0, 0), with an excitation energy non-perturbatively small compared to the scale 1/N L.
To compute the energy of the ground state as a function of g 2 , we use the Born-Oppenheimer (BO) approximation familiar from atomic physics. The Born-Oppenheimer approximation states that the lowest excitation energies of a coupled quantum system can be extracted by solving the Schrödinger equation for the 'light' degrees of freedom (in our case, the dynamics of θ field) with the 'heavy' degrees of freedom (the dynamics of the φ 1 , φ 2 fields) frozen to their energy-minimizing values. Essentially, one is exploiting a separation in energy scales between the two sets of degrees of freedom to solve for the behavior of the heavy variables classically, and then treating the light variables quantum-mechanically. In our case, we will see that the separation in energy scales between states with n 1 , n 2 = 0 and n 1 = n 2 = 0 is exponential in g 2 , so the use of the Born-Oppenheimer is parametrically well-justified when g 2 1, as is the case when N L is small compared to Λ −1 . So in the Bohr-Oppenheimer approximation, we only need to compute the ground state energy in the n 1 = n 2 = 0 sector of the theory, described by the Hamiltonian The Schrödinger equation can be written as where we have set L = 1. It is convenient to define a dimensionless HamiltonianH = Hξ −1 , and bring the kinetic term into canonical form by the change of variables θ = g 2 2ξθ . Then the Hamiltonian readsH This form is same as the Hamiltonian in Zinn-Justin's text, Section 41.2 [76] with the identification g ZJ = 2g 2 ξ . The energy of the ground state now follows from the P = + solution, and the large-order behavior of this solution was already determined by Stone and Reeve [77] using the methods developed by Bender and Wu [78,79]. From [77] we see that the contributions of each of the U 1 (i) in the SU (2) subgroups to the perturbative series for the ground state energy, E i , behave as with an asymptotic form given by We have confirmed the correctness of the asymptotic form (which includes sub-leading corrections to the leading asymptotic) by numerical analysis. See Fig. 7 for an illustration of the rapid convergence of the series coefficients to their asymptotic large-order form. The expansion parameter of perturbation theory is g 2 /(8ξ) = g 2 N/(16π). The perturbative series is non-alternating and has factorially-growing coefficients, of the form n! g 2 /(8ξ) n .
This means that the perturbative expansion is along a Stokes line, and its resummation is ambiguous. The analytic continuation of the Borel transform for the leading order n! divergence is given by (5.24) Hence the perturbative series will not be Borel summable along R + in Borel plane, due to the (leading) singularity located at t * = 8ξ = 16π N . If the integration contour defining the Borel-resummation of the series is deformed to pass either above or below the real axis near this singularity, so that it is avoided, one can obtain a finite result. However, the result of the resummation will depend on the choice of contour, leading to a two-fold ambiguity. An equivalent way to think about this which we find particularly useful in working with the NP saddles is to analytically continue g 2 → g 2 e i , ∈ (−π, π). Then so long as = 0, the Borel sum converges, but the analytic continuation back to = 0 depends on whether approaches 0 from above the real axis or below the real axis. This matches what one sees with the contour deformations.
Using either of these equivalent approaches we can define the lateral Eq. (2.17) Borel resummation as in the simple example discussed in Sec. 2.1. In particular, the right S 0 + E i and left S 0 − E i Borel resummations yield where P stands for the (unique) Cauchy principal value.

Stokes Jumps
We can now determine the action of the Stokes automorphism: subgroups. These N ambiguities are identical when the background holonomy is Z N -symmetric, so that the overall leading ambiguity in the SU (N ) theory is just N times the expression above 24 .
Of course, there are also ambiguities coming from other embeddings of SU (2) into SU (N ) such as the one shown in Eq. (5.9) which are associated with roots which can be written as positive linear combination of the simple roots. In this case, with the U (k) j embeddings in the center-symmetric background, the leading non-perturbative ambiguity is of order As advertised, this is exponentially small compared to the leading ambiguities coming from fluctuations living within U i .

NP-data from the late terms of the P-expansion
Energy eigenstates in a real periodic potential must be real. Yet we have just seen that the perturbative series generates a result which has an imaginary part upon Borel resummation. Furthermore, as seen in Eq. (5.25) the imaginary part is ambiguous. A non-perturbative completion of this result which gets rid of both of these problems is the resurgent transseries. See [15] for a very explicit discussion. In fact, the NP-completion of the perturbative results, even in the case of simplest single parameter trans-series, leads to infinitely many NP-saddles. If we assume that the semiclassical representation of observables in the PCM is resurgent, the appearance of the ambiguity in perturbation theory described above implies that there must exist some finite-action field configuration with action S = 16π N g 2 whose amplitude should be ambiguous in just the right way to cancel this leading ambiguity. Another way to say this is that resurgence theory predicts the existence of such non-perturbative saddle-points. However, as we have mentioned before, naive topological considerations seem to leave no room for stable finite-action solutions in the PCM. In the next section, we resolve this puzzle by showing the existence of such saddles, which in general are not classified according to the topological structure of the microscopic theory.
In the low energy effective field theory for the PCM on small R × S 1 , there is a sense in which topology is emergent. Indeed, the periodic potential Eq. (5.19) has one-dimensional instantons, which can be classified according to topology in quantum mechanics. However, in QM and QFT, it is well-known that instantons cannot mix with perturbation theory. The first NP-saddles which can mix with perturbation theory and fix its ambiguity are of the instanton-anti-instanton form. Indeed, we should expect a left/right [FF] ± = Re [FF] ± ie − 16π N g 2 correlated amplitude for some event with amplitude F: this is the only way the leading ambiguities of perturbation theory can be cancelled in the trans-series. So resurgence theory 24 The ambiguities would become split if the ZN symmetry were to be slightly broken. and perturbation theory in the semi-classically calculable regime of the 2d PCM compactified down to R × S 1 predicts that there must exist a NP saddle with amplitude (5.28) Note that the action of this NP saddle in the compactified theory is 1/N of the uniton action. Explicit field configurations with such actions are discussed in detail in the following Section.

Non-perturbative saddle points
We now begin our exploration of the non-perturbative features of the PCM at small S 1 , with the boundary conditions/background gauge field corresponding to Eq. (4.22). As we have mentioned in the introduction, there are no stable topological defects in the theory on R 2 . However, on R × S 1 with a Z N -symmetric background, the situation is different. The background holonomy amounts to a potential on the SU (N ) target space, effectively modifying it to U (1) N −1 , the maximal torus of SU (N ), at energies low compared to 1/(N L). This is the counterpart of adjoint Higgsing in gauge theories via a Wilson line, see e.g. [66,71].
As we briefly mentioned in Section 5.2, this potential has isolated minima on the group manifold. Consequently, there are tunneling events between these minima, which should admit a semiclassical Euclidean description as stable instanton-like field configurations when L is small enough and the theory is weakly coupled. This happens despite the fact that the microscopic theory has a trivial π 2 homotopy group and hence has no instantons. This is the sense in which there is an emergent topological classification in the low energy effective field theory.
We will give a careful classification of these new saddle points of the PCM path integral. Moreover, we will see that these small-L field configurations are the constituents of unitons, with the minimal uniton fractionalizing into N constituents at small-L. For this reason, we refer to these small-L stable saddle points as fractons. This terminology was originally introduced in [23] in the multi-flavor Schwinger model for fractionalized instantons. The fractons play a critical role in the non-perturbative physics of the PCM on small S 1 , and are responsible for the mass gap of the theory. Moreover, they lead us to the resolution of the deep puzzle about the interpretation of renormalon ambiguities in the PCM.

Unitons
We start with a description of the unitons, which are finite-action solutions of the second-order Euclidean equations of motion [52]. The reason that one should expect such solutions is that the CP N −1 manifold can be embedded as a totally geodesic submanifold into SU (N ) [80]. This means that solutions of the CP N −1 field equations can be lifted to solutions of the SU (N ) field equations. However, while finite-action solutions on CP N −1 carry a topological charge, and have no negative modes while staying within the CP N −1 manifold, once the solution can evolve in the full SU (N ) target space there is no longer any conserved topological charge.
Hence unitons are known to be "unstable", which just means that the fluctuation operator around a uniton has negative modes [81].
It turns out that in general, not all the solutions of the PCM are obtained directly from the CP N −1 embedding, and in [52] Uhlenbeck gave an exhaustive classifications of all possible solutions to the PCM equation of motion. However, our interest here will be in what happens to the minimal-action uniton, which can be obtained from a CP N −1 embedding, and has the action This minimal-action uniton solution on R × S 1 with the boundary conditions determined by Eq. (4.22) is associated with the minimal-action instanton of CP N −1 , and takes the form where P is a projector, P 2 = P. Here, we defined and and z = (x 1 + ix 2 )/L, with x 1,2 coordinates on Euclidean R × S 1 , while λ i ∈ C are moduli of the solution. We note that v(z) are precisely the twisted instanton configurations discussed in the context of the CP N −1 model in [26,27] and [18,19], with the correct twisted periodicity in x 2 . Once we work on R × S 1 with a center-symmetric background holonomy, we expect the emergence of some stable instanton-like field configurations, the fractons. The fastest way to see the emergence of the fractons is to look at some plots of the action densities associated to these solutions as a function of the moduli λ i shown above 25 . It turns out that 'small' unitons, which is to say ones whose characteristic size is small compared to L, resemble the profile of a uniton on R 2 . An example of such a small-uniton configuration is shown on the left in Fig. 8, where the minimal uniton looks like a single lump of Euclidean action density, L E , centered near x 1 = 0, x 2 = L/2. On the other hand, unitons which are large compared to L tend to fractionalize, in the background Eq. (4.22), into multiple lumps, the fractons. The locations of the different fractons are controlled by the λ i , with an example shown on the right of Fig. 8. The number of lumps is at most N . 26 More examples of fractionalization are shown for the SU (3) and SU (4) cases in Fig. 9 on the left and right respectively. From the plots we see that in the fractionalized limit, the lumps are constant along the compact direction x 2 , and seem to approach a vacuum configuration in between each other. This strongly suggests that the PCM should have isolated fracton solutions 27 for small-L, and this will be explicitly verified in the next section.
We also note that in the thermal compactification, at high temperatures, where the trivial 26 These type of plots, in the context of fractionalization of BPST instantons in the background of non-trivial holonomy, goes back to the work of van Baal et.al. [25] in QCD-like theories. For other works emphasizing similar effects in various QFTs, see e.g. [23,24,[26][27][28][82][83][84][85][86] The amusing feature illustrated by the PCM is that the idea of fractionalization even works in the absence of topologically stable instantons! A more direct route to study the constituents of instanton appears in [21,66,71] by studying the theory in a weakly coupled calculable regime, see e.g. [87][88][89] 27 In [90] it was shown that a direct extrapolation of classical field configurations from small-L to large-L, using only the leading term in the small-L EFT action, is quite subtle for non-BPS solutions. On the other hand we know already that the PCM, in the decompactified R 2 regime, becomes strongly coupled and one cannot recover the full quantum theory starting from classical field configurations. Indeed, even before thinking about the strongly-coupled L → ∞ limit, one can see that the small-L EFT action itself gets corrections as L is increased. In this paper, we confine our attention to the weakly-coupled small-L limit, in which the subtleties explored in [90] are not relevant.

SU (2) Fractons and KK Fractons
It is instructive to start our search for the fracton solutions by considering the SU (2) PCM, in which case the group manifold is S 3 . The discussion of the fractons becomes most transparent if we parametrize the group element using Hopf coordinates as we did in Section 5, see Eq. (5.10). The round metric on the 3-sphere in the Hopf coordinates is given by ds 2 = dθ 2 + cos 2 θdφ 2 1 + sin 2 θdφ 2 2 (6.5) and this fixes the kinetic term. From the action Eq. (4.5),we see that a non-trivial potential term is induced on the S 3 manifold due to the background field determined by Eq. (4.22). The associated Euclidean Lagrangian on R × S 1 is given by where ξ = 2π/N L = π/L. Given the fact that unitons which are large compared to L split into N configurations which appear to have flat profiles along the S 1 , it is tempting to start by setting all Kaluza-Klein (KK) momenta in Eq. (6.6) to zero, and thus obtain an effective one-dimensional theory. The usual reasoning behind such an approach is that field configurations carrying n units of KK momentum have an energy density n 2 /L 2 , so for studying physics on length scales large compared to L it is sufficient to focus on states with n = 0, since these are parametrically lighter. This would certainly be correct if ξ were set to zero. However, as we already saw in the perturbative context, this approach is too naive for the PCM in the Z N -symmetric small-L limit, since the small-L theory actually 'remembers' that it is microscopically a twodimensional theory, and neglecting all configurations with non-zero winding number (and hence KK momentum) is not quite correct. It will turn out to be important to consider finite-action field configurations also carrying non-zero winding number for φ 2 . SU(2) Fractons: Let us start by looking for instanton configurations carrying zero units of KK momentum in the compact direction x 2 for θ, φ 1 and φ 2 . By setting ∂ x 2 = 0, the euclidean action for the low-lying modes reduces to where dotted quantities are derived with respect to x 1 . As expected, the KK reduction lands us on a theory with a non-trivial potential on the target space T = S 3 due to presence of the Z 2 -symmetric background holonomy.
The Euclidean equations of motions associated with this action arë The existence of the stable fracton events is the crucial difference compared to R 2 , or to thermal compactification R × S 1 β associated with the trivial background holonomy. We can find the action of the instanton configuration by focusing only on the θ field, and using Bogomolny's method to rewrite the action in the simple form where we substituted the Z 2 symmetric value for ξ = π/L. Note that the action of a fracton S F is precisely 1/2 (1/N in the SU (N ) case) times the action of a uniton Eq. (6.1), supporting our claim that the fractons are constituents of unitons. The equality in Eq. (6.13) holds whenever the fracton/anti-fracton satisfy the first order BPS equationθ (x 1 ) − ξ sin θ = 0 , (6.14) θ(x 1 ) + ξ sinθ = 0. (6.15) The solutions to these equations take the form 1 is a position modulus. The fracton obeys the boundary conditions θ(x 1 → −∞) = 0, θ(x 1 → +∞) = π, while the anti-fracton goes from π to 0 and these solutions have precisely the BPS-saturated (from the small-L effective field theory point of view) action KK Fractons, and Unitons from Fractons: As we have mentioned above, discarding all KK non-zero modes misses some critical information, which is important even when L is small. In particular, consider doing the dimensional reduction to 1D with the periodic field φ 2 ∈ [0, 2π] winding n times as we move along the S 1 (i.e. φ 2 = 2π n x/L), rather than just being set to a constant, as we did above. We will shortly see that configurations with n = −1 have a distinguished role. Let us assume that 2π n x 2 L , (6.19) this means that φ 2 carries some KK momentum, by way of a non-trivial winding number n. The reduced action becomes The same arguments as before then show that this action gives rise to stable instanton configurations, which we will refer to as KK-fractons 28 . These configurations have action Now observe that when ξ takes on the Z 2 symmetric value ξ = π/L and n = −1, this becomes The action for a generic KK fracton with n = −1, will be higher than S fracton , and hence their contribution will be suppressed in the semiclassical expansion. But fractons and the n = −1 KK fractons enter the semiclassical expansion on the same footing! Despite having the same action as the fractons, it is important to emphasize that the n = −1 KK fractons and n = 0 fractons are distinct field configurations. Indeed, the explicit solutions for the field configurations of the n = −1 KK fractons are These KK fractons behave as θ( 1 , so that the combined field configurations goes from θ = 0 at x 1 = −∞ back to θ = 0 at x 1 = +∞. To see why the second object is a KK-fracton rather than an anti-fracton one can study the field φ 2 in the uniton solution on R × S 1 with the Z N -symmetric background holonomy. It can be shown that φ 2 interpolates between a configuration with zero winding in the x 2 direction at x 1 = −∞ and a configuration with −1 winding in the x 2 direction at x 1 = +∞: where for simplicity we fixed the moduli λ 1 = λ 2 = 1. Hence we see the precise sense in which the fractons are the constituents of unitons.

SU (N ) Fractons
We now discuss the generalization of the SU (2) fractons to the SU (N ) PCM. This can be done by embedding the SU (2) fractons into the SU (N ) model, by using Eq. (5.9). This construction permits us to classify all the fracton events with action is the action of the minimal fracton, and k may be seen as "level" of the associated action. Naive approach: Consider the root system of the Lie algebra su(N ). Denote the simple root system as We can use ∆ 1 to build the complete root system A N −1 . For each root α (simple or not), there is an associated su(2) sub-algebra. All fracton embeddings associated with simple roots are minimal action, level-1 events, with action: where we set µ i to their center-symmetric values. There are N − 1 simple roots, and hence there are N − 1 minimal action fracton events. The positive roots which can be written as a sum of just two adjacent simple roots correspond to level-2 fractons. There are N − 2 such positive roots. The actions of these events are (in the center-symmetric background) Similarly positive roots which can be written as a sum of just three (adjacent) simple roots correspond to level-3 fractons and so on. The general construction is that the root space A N −1 can be split into simple roots and roots that can be written as k-linear combination of simple roots: · · · · · · k = N − 2 : 2 positive roots : We can similarly define the negative roots, ∆ − j = −∆ j . Obviously, the root space A N −1 can be decomposed as This root space A N −1 contains N 2 −N roots, half positive and half negative, and together with the Cartan subalgebra, whose dimension is N − 1, they form the complete set of generators of SU (N ). As is well-known, there is an su(2) sub-algebra of su(N ) generated by E α , E −α , α · H where E ±α are raising and lowering operators, and H are Cartan sub-algebra matrices. We can embed an su(2) fracton into su(N ) by using any root α. Clearly, we have fracton embeddings associated with α, According to this rationale, there are N − 1 fractons at level-1, N − 2 fractons at level-2, so and so forth, and only one fracton at level N − 1. Clearly, the densities of events of level-1, level-2, etc are also hierarchical, and obeys More carefully, accounting for the compactness of holonomy: The previous discussion does not take into account the fact that the background holonomy is compact. For example, one might naively think that events in ∆ + N −1 are very rare tunneling paths, exponentially suppressed with respect to the minimal fractons, because their actions are S 1,N −1 = (N − 1) × S i,1 . Indeed, this is largely correct. However, if choose with a k = N − 1 configuration with winding number n = −1 for φ 2 , as was done in the SU (2) PCM in Sec. 6.2, and then apply the Kaluza-Klein-reduction to the action afterwards, we obtain This happens because µ i −µ j behaves as a fractional momentum in the compact direction, and combining (µ N −µ 1 ) with −1 unit of winding number results in a twisted KK-reduction. In the full 2D QFT, this tunneling path is exactly on the same footing with the elementary fractons corresponding to ∆ + 1 , and its action coincides with them (6.29) in the center-symmetric background of Eq. (4.22): We refer to this instanton-type event as the KK fracton, denoted F N . It is associated with the affine root of the su(N ) Lie algebra, defined by which is itself a negative root. The F N fracton is the counter-part of the so called KKmonopole-instanton (also called affine or twisted instanton) in gauge theories on R 3 × S 1 [24,25]. Similarly, by inserting −1 units of winding number into the two events living in k = N −2, we can turn them into tunneling events with action 2 N S uniton . This pattern continues for all levels: In particular, we learn that at level k, there are always N NP-saddles associated with the roots ∆ aff k 29 . This is actually a consequence of the cyclic Z N symmetry of the background holonomy. All N -saddles in ∆ aff k have actions S (k) = 8π g 2 N × k = S uniton N × k. The existence of the KK fracton events is due to the compactness of the SU (N ) target space and of the background holonomy Ω in Eq. (4.13). Equivalently, although the long distance effective field theory on small R × S 1 L may appear one-dimensional, the underlying microscopic theory is two-dimensional, and this is encoded in the structure of the effective field theory. As we saw above both in the perturbative and non-perturbative contexts, the details of dimensional reduction at small-L are quite subtle in the context of theories with adiabatic small-L limits.
The fractons associated with ∆ aff 1 are the leading (minimal action) topological configurations in the PCM. In the bosonic model, this will be of crucial importance in the determination of the mass gap. Furthermore, certain correlated fracton-anti-fracton events (neutral bions) will be crucial in resolving the renormalon ambiguity, similar to the CP N −1 model and YM theory [16][17][18][19].

Fracton amplitudes
The fracton amplitude is the same as the 1-d instanton amplitude associated with the quantum mechanical system in Eq. (5.19). Within the Born-Oppenheimer approximation in the low energy effective theory we find there is effectively only one bosonic zero mode, the position modulus of the fracton. The amplitude of the fracton event using dimensionless units 29 The advantage of ∆ aff k with respect to ∆ + k is that it makes the fact that the KK-fractons are on the same footing with the regular fractons manifest, along with the ZN cyclicity. Its main disadvantage is that one loses the notion of positivity associated with level-k. Recall that all roots living in ∆ + k were positive. Since and all roots (involving the affine root in some way) living in ∆ − N −k are negative, ∆ aff k is not comprised of positive roots only. Unfortunately, one cannot have both properties at once. The concept of positive roots will play some role when we discuss correlated fracton events, and hence, we work with both representation. introduced in Eq. (5.21), is given by is the quadratic fluctuation operator in the background of the fracton, det M indicates that the zero mode is dropped in evaluation of the determinant, and det M 0 is for normalization. The determinant can be evaluated in multiple different ways [38,76], for example, via the Gelfand-Yaglom method [91]. The Jacobian associated with the zero mode is The result of this calculation is given in Section 41.2 of [76], and is equal to I = 4 √ πg ZJ e −8/g ZJ .
Converting to our notation, see Eq. (5.21), g ZJ = 2g 2 ξ , we can write the minimal fracton amplitude (i = 1, . . . , N ) in the center-symmetric background as There are perturbative fluctuations around the fracton, and the amplitude incorporating those fluctuations is given by It should be noted that there is no non-perturbative ambiguity in the NP part of the amplitude, given by 16 On the other hand we expect Φ F (g 2 ) to be a non-Borel summable asymptotic series, similarly to perturbative expansion around the P-saddle Eq. (5.22).

Resurgence triangle and emergent topological structure
Now that we have found the fractons, the minimal-action non-perturbative field configurations at weak coupling, we are in a position to see how the renormalon ambiguities in the Borel resummation of the perturbative series, such as Eq. (5.25) are cancelled. We will show that they cancel against corresponding ambiguities in the contributions from the non-perturbative sector, leaving well-defined results order by order in an expansion in e −S F = e −8π/(g 2 N ) . The phenomenon we are describing is an example of a semiclassical expansion which is not Borel summable, but is Borel-Ecalle summable. This is a generalization of Eq. (2.27) for ordinary integrals.
We first emphasize again that there is no non-perturbative ambiguity in the fracton amplitude [F i ] itself 30 . Although there is no topological charge in the microscopic theory, there is an emergent topological structure in the low energy effective theory, and the fractons are 1d-instantons from that point of view, hence they cannot mix with perturbation theory. In theories with a topological Θ angle it is always true that events with non-zero instanton number cannot cure the ambiguity of perturbation theory [18] due to the Θ dependence of instanton amplitudes. In our case, despite the absence of a microscopic topological argument, at small-L we find that fractons cannot mix with perturbation theory either.
Second, it is now worthwhile to re-inspect perturbation theory around the perturbative saddle, and rewrite Eq. (5.22) and Eq. (5.23) as expansions in This emphasizes that the perturbative expansion parameter is the inverse of two times the fracton action. As already mentioned in Footnote 6, this is one of the major differences between ordinary integrals and path integrals. The leading ambiguity of perturbation theory is given by the right/left Borel resummation Eq. (5.25). Expressing these left/right resummations in terms of the action, makes it clear that the ambiguities can only be canceled by contributions which live in the fracton-anti-fracton sector. We show that this is the case in Section 7.2. The first step in seeing how this works is to understand the interactions of fractons and anti-fractons, which will be our task in Section 7.1.
Our main idea about emergent topology and resurgence is encoded in the resurgence triangle [18] classification of saddle points, which is illustrated for the PCM on R×S 1 in Fig. 10 in a schematic form, which does not take into account the "ramification" phenomenon we will describe later in this Section. The entries in the resurgence triangle are [FnF n ], which denote correlated events involving n-fracton andn anti-fractons. The vertical axis is the action of the events, while the horizontal axis tracks the "charge" q of the events, defined as q = n −n. Then the perturbative vacuum is the "primary" or "level zero" for the q = 0 sector, [F 1 ] is the "level zero" event for the q = 1 sector, [F 2 ] is the level zero event for the q = 2 sector, and so on. The cancellation of ambiguities happens between saddles in a given column, and different columns cannot mix in the cancellation of their respective ambiguities. Note that unlike theories in which different columns are classified according to different homotopy classes of the microscopic theory, in the present example where there is no microscopic topological classification, the classification of columns takes place according to the topology in the low energy effective theory, which is emergent. It should be emphasized that the resurgence triangle and the variety of NP-saddles for the PCM are just as rich as CP N −1 model [18] or deformed Yang-Mills theory or QCD(adj) [16]. In other words, the fact that the homotopy group π 2 is trivial in the microscopic PCM is only a superficial difference from Yang-Mills theory. At the deeper level of insight allowed by resurgence theory, it appears that the set of NP saddles in the PCM is in fact in one-to-one correspondence to the set of NP saddles in Yang-Mills theory, because in both cases the set of saddles are determined by the properties of the underlying su(N ) Lie algebra!

Fracton interactions and [F
To understand how the weak coupling realization of the renormalon ambiguity explained around Eq. (5.25) and Eq. (7.3) is cured, we must understand the structure of the semiclassical expansion at second and higher order. As we have described above, at first order in the semi-classical expansion, one has the minimal-action fractons, but their contributions (∼ e −S F ) are 1) exponentially larger than the renormalon ambiguity (∼ e −2S F ) and 2) are unambiguous. Hence the minimal-action fractons contributing at first order in the semiclassical expansion cannot contribute to resolving the issues of perturbation theory, which can only be cured at second order as shown by Eq. (7.3).
At second order in the semi-classical expansion, we encounter correlated fracton events, such as [F i F i ], [F iFi ], [F iFi ], as well as uncorrelated single fracton events associated with roots α ∈ ∆ aff 2 in Eq. (6.38). According to the emergent topological structure associated with the resurgence triangle of Fig. 10, [F i F i ], [F iFi ] are the leading saddles in their classes, and cannot cure the ambiguity of perturbation theory either. It is the fracton-anti-fracton configurations [F iFi ], i = 1, 2, . . . , N (which we refer to as "neutral bions" due to the universality of such configurations across a wide range of asymptotically-free theories) which can mix with perturbation theory. They are the first sub-leading saddles in the column associated with the perturbative saddle.
Below, we demonstrate that the non-perturbative amplitudes (without incorporating the perturbative fluctuations) associated with the correlated events are given by: Each of these representations is useful for slightly complementary reasons. Note that the correlated [F j F j ] event amplitude is not the same quantity as [F j ][F j ], and these are genuinely different (isolated, non-degenerate) saddles. In fact, in the semi-classical evaluation of partition function, we must sum over both types of saddles independently. For example, up to second order in NP trans-series expansion, we can write a grand canonical ensemble of all defects entering up to second order in the resurgence triangle: A pictorial description of this generalized grand canonical ensemble can be found in [8]. It is also implicit in the Zinn-Justin's exact quantization formula which incorporates all orders in non-perturbative effects [76] via a resurgent trans-series.

Computation of [F jFj ] and [F j F j ] and amplitude
Amplitudes for correlated two-events in the CP N −1 model on R × S 1 with Z N -twisted boundary conditions were calculated in [18], and as we have remarked there is a striking degree of similarity between the NP saddles of the CP N −1 model and the NP saddles of the PCM associated with the affine simple roots of su(N ). Moreover, it turns out that at small-L it is precisely (the interactions of) these N minimum-action saddle points of the PCM which are responsible for the cancellation of renormalon ambiguities. Hence our discussion below will amount to a review of the argument of [18], with a few additional clarifying comments. We suppress the index i in what follows to lessen the clutter, since we are looking for the correlated amplitudes for [F αFα ] and [F α F α ] for any α ∈ A N −1 . Indeed, α may be chosen to be α i ∈ ∆ aff 1 in the classification of Eq.
Here, t 0 is the center of action coordinate of the pair, while τ is the separation of the constituent fractons. It turns out that t 0 is an exact zero mode, while τ is a quasi-zero mode (with an 'energy' parametrically separated from the perturbative fluctuations). The contribution of [FF] events to the path integral is given by where [F][F] are the product of uncorrelated amplitudes. The explicit integration over τ , the separation between the constituent fractons, appears because it is a quasi zero mode, due to the presence of the "interaction potential" between the fractons. The integration over dΩ is an instruction to integrate over the 1d "solid angle", which simply counts for the two different (distinguishable) orderings of the events namely, [FF] and [FF]. One may evaluate the interactions between two fracton events separated by a distance τ by directly computing the action on the ansatz in Eq. (7.7). We find where V (τ ) characterize the interaction and are given by: In the latter expressions, we assumed the dilute fracton regime, where the separation τ between fractons is much larger than fracton size r F ∼ ξ −1 . In Euclidean space, where we map the proliferation of fractons into a dilute classical gas, these results can be interpreted as repulsive interactions between widely separated fracton-fracton events, and attractive interactions between widely separated fracton anti-fracton events.
According to the structure of the resurgence triangle of Fig. 10, the integral over the quasi-zero modes for [FF] should not yield an imaginary ambiguous part, while the one for [FF] should have an ambiguity. The quasi-zero mode integrals are of the form 11) where (−1) subtracts off the uncorrelated fracton-fracton events [4,18,19], and arises from the semi-classical expansion of the partition function automatically. In fact, not subtracting this factor would amount to double-counting the uncorrelated fracton events. For the I 1 (g 2 ) integral, using an integration by parts, we obtain This has its main support at the length scale τ * : 14) The integrand dies off for τ τ * because of the repulsion, and it dies off for τ τ * because of the subtraction of uncorrelated events. Thus, we identify τ * as the characteristic size of the correlated 2-events. Note that τ * is parametrically larger than the fracton size, but parametrically smaller than the inter-fracton separation, which is in turn much smaller than the typical separation between 2-events. So we have a hierarchy of scales: This hierarchy of scales means that the use of semi-classical methods for 1-events and 2-events is simultaneously justified. Now consider the I 2 (g 2 ) integral. The integral is dominated at separations where we cannot meaningfully talk about isolated fracton-anti-fracton events. This is neither a bug nor an accident. Rather, it is an important feature! Recall that the arg(g 2 ) > 0 line is a Stokes line, along which the Borel resummation of the perturbative series is ambiguous. The left or right Borel resummations are not ambiguous, but there is a Stokes jump associated with the crossing of the Stokes line. The [FF] configuration can mix with perturbation theory, and if perturbation theory is ambiguous with an ambiguity at order e −2S F , it is conceivable that configurations that can mix with perturbation theory may also have ambiguities of the same order (but not larger).
In fact, rather than trying to compute the [FF] directly on a Stokes line, we should instead calculate the right and left [FF] ± amplitudes. The simplest way to do this it to take g 2 → −g 2 , and then observe that I 2 (−g 2 ) = I 1 (g 2 ), an integral already performed. The analytic continuation back to +g 2 from −g 2 through the complex g 2 plane is two-fold ambiguous, with the result depending on whether we approach the positive real axis from above or from below. This method of evaluating the [FF] amplitude is called the Bogomolny-Zinn-Justin (BZJ) prescription [4,5]. Following the BZJ method, we find So the correlated [FF] ± events have an imaginary ambiguity.

Remark on analytic continuation
The reader may feel concerned by the following aspect of the previous derivation. It naively looks like we are taking g 2 → −g 2 for FF, evaluating the QZM integration there, and then we take −g 2 → +g 2 again either clock-wise or anti-clock-wise, producing a two-fold ambiguous result. But, naively, we are not performing any analytic continuation while calculating amplitudes for FF events. Does this mean that we are treating the theory inconsistently, by treating one sector differently from the other? As a matter of fact, we can (and should) move off the arg(g 2 ) = 0 line, the Stokes line, both for FF as well as FF. The point is that we can do so by just taking g 2 → g 2 e ±i , where 0 < < π, evaluate the integral , and then come back to the arg(g 2 ) = 0 line by taking → 0. Then, we find that  Fig. 10 is the consistent characterization of the semi-classical regime of the principle chiral model or other QFTs studied so far [16,18,30]. As we will demonstrate in the next section, the ambiguity associated with [FF] ± events cancels the ambiguity associated with the non-Borel-summability of the perturbative vacuum [0] which was calculated in Eqs. (5.25), (7.3). According to the emergent topological structure, [FF] is the lowest action configuration (the "level zero" or "primary") in the sector with "charge" +2. If [FF] were to have an ambiguity in its NP-part, that would imply that there must exist another configuration with lower action, and charge +2, which is impossible by construction.

SU(N)
We now discuss the generalization of this analysis to SU (N ), where the minimal action fractons are labeled by the roots α ∈ ∆ aff 1 of the su(N ) algebra. Based on some general arguments we expect that the interaction potential between the fractons are given by where α i · α j = 2δ i,j − δ i,j+1 − δ i,j−1 , which reduces to Eq. (7.10) for i = j.
The classification of 1-and 2-events in the bosonic SU (N ) PCM on R × S 1 is quite analogous to the classification of tunneling events in deformed Yang-Mills on R 3 × S 1 and CP N −1 on R × S 1 . There are tunneling 1-events in field space associated with the change of the field by α ∈ A N −1 , where α is any element of root space. These are uncorrelated single events which can be embedded as exact solution into SU (N ) PCM. At level-2 and above, there are a number of subtle issues which we address below.
• Neutral bions: These are correlated [F iFi ] ± = [B ii ] ± events, a tunneling occurring in an SU(2) subgroup associated with root α i followed by an anti-tunneling associated with −α i . In the parametrization of Eq. (4.10), we have These exist for all positive entries of the extended Cartan matrixÂ ii . ForÂ ii > 0, the interaction between the constituents is attractive. Since neutral bions have the quantum numbers of the perturbative vacuum, they can (and do) play a role in the cancellation of the ambiguities of perturbation theory. Furthermore, since α i − α i = 0, there is no single uncorrelated event associated with neutral bion. By its very nature, it is a correlated event. It also generates a non-perturbative contribution for the background holonomy potential (4.13) on top of the one-loop perturbative potential (4.15), and may play a role in the deconfinement phase transition (rapid crossover, for finite N ) in PCM.
• Charged bions: These are correlated [F iFj ] = B ij events that exist for all negative entries of the extended Cartan matrix,Â ij , thus j = i ± 1, which can be described as SinceÂ i,i±1 < 0, the interaction between the constituents is repulsive, and there is no ambiguity associated with these two-events. Furthermore, since α i −α i±1 is not a simple root, there there is no single uncorrelated event associated with charged bion. Like the neutral bion, by its very nature, the charged bions are fundamentally correlated events.
In theories with fermions, they play the leading role in the generation of the mass gap.
• Higher action elementary fractons: For all roots of the Lie algebra of su(N ), an exact solution can be embedded into an SU (2) subgroup of SU (N ). In particular this includes the roots α i + α i+1 ∈ ∆ aff 2 associated with the tunneling event Since this tunneling event is associated with a root, unlike the neutral and charged bions events, it should be considered as an elementary event, even though it carries two units of action 2S F . There is no ambiguity in the NP-part of the elementary events, while P-fluctuations around them will always have ambiguities.
• Higher action composite fracton-fracton pairs: Consider two roots α and β whose sum α + β is not a root itself, for example, 2α i . In all such cases, the tunneling event must be seen as composite. This is because one cannot embed a simple SU (2) fracton associated with the sum α+β into SU (N ). For all such correlated events, the interaction between the constituents is always repulsive, and so there is no ambiguity associated with the NP-part of such events: For example, for SU (N ), N ≥ 3, F α 1 +α 2 is a single event with action 2S F while [F α 1 F α 1 ] is a correlated event with action 2S F . This is because α 1 + α 2 is a root, while 2α 1 is not a root.
With these remarks in mind, we observe that the rows of the resurgence triangle of Fig. 10 get "ramified" according to a Lie algebraic structure. For example, the first three rows of the resurgence triangle in Fig. 10 should really have been written as : This carries rather refined information about the structure of the semi-classical expansion.
There are essentially six type of events at order e −2S F in the semiclassical expansion. This is the same order as the leading ambiguity in perturbation theory. Only one class out of the six, the neutral bion, participates in fixing the ambiguity of perturbation theory around the perturbative vacuum. The other classes of events have other roles in the rich inner life of the theory.
Despite the fact that all of these events appear at the same order in the semi-classical expansion, (e −2S F ), their amplitudes 31 differ is crucial ways: Even the two events which carry "charge" 2 under the emergent topological structure, The reason for this, as explained above, is that 2α i is not a root, and consequently the corresponding event is a correlated one. On the other hand, α i + α i+1 is a root, and is associated with a single tunneling within the corresponding SU (2) subgroup of SU (N ).
The expressions given in Eq. (7.24) do not include perturbative fluctuations around the NP saddles. Incorporating the perturbative fluctuations around the 2-events, we get n g 2n , (7.25) for the [FF] ± event, and similar expression for other events.

Reality of resurgent trans-series for real λ and BE-summability
The principal chiral model is a matrix field theory with a real action and a stable ground state. Hence its partition function must be real and unambiguous, similar to our toy example Eq. (2.2). In the preceding sections, however, we have calculated the leading perturbative and non-perturbative contributions to the the partition function, and have found that: 1. Perturbation theory is non-Borel resummable on the arg(g 2 ) = 0 Stokes line, meaning that the Borel sum of the perturbative series has a two-fold ambiguous imaginary part.
2. At second order in semi-classical expansion, we have calculated the neutral bion amplitude (via the BZJ prescription) and showed that it also has a two-fold ambiguous imaginary part.
Each of these ambiguities would by themselves be disastrous, indicating that the theory is not well-defined. The framework of resurgence suggests the resolution. Perturbation theory by itself is indeed not well-defined. The semi-classical expansion by itself is not well-defined either. However, neither is a direct physical observable, only their sum is. In fact, the ambiguities at order e −2S F cancel exactly to yield a result which is ambiguity free up to order e −4S F . Resurgence is the statement that these cancellations repeat order by order in the resurgent trans-series expansion for every physical observable.
Using Eq. (7.3) and Eq. (5.26), the right/left Borel resummation of the perturbative series can now be written in terms of fracton amplitudes as: where s = −4πi is the purely imaginary Stokes constant (analytic invariant) of the problem. Similarly, using Eq. (7.4), the right/left neutral bion amplitudes are given by Note that the same Stokes constant appears in the imaginary ambiguous part of the neutral bion amplitude. The ambiguities in both quantities are a manifestation of the fact that we are performing an expansion on a Stokes line. Consequently, the perturbative series is non-Borel summable, and exhibits a Stokes jump, which is mirrored by the jump in the neutral bion amplitude, leading to the cancellation of ambiguities: This is the counterpart of the cancellation of ambiguities we saw in d = 0 example Eq. (2.27), and it is an explicit realization of the median resummation [9,15] and BE-summability. The sum is ambiguity free up to higher order effects, and the non-canceling terms are of the form Physically, this quantity is the average of the ground state and first excited state. The difference of the first excited state and the ground state is the mass gap, and will be discussed in the next section.
Despite the fact that we have only shown Eq. (7.28) to be true at order O(e −2S F ), resurgence actually implies that This expression can be inverted, using Eq. (2.35), to obtain the non-alternating part of the late terms in the perturbative expansion around the perturbative vacuum: (2S F ) 2 n(n − 1) + . . . + . . . (7.31) which indeed agrees with Eq. (5.23). Note that this expression is very much in the same spirit as our zero dimensional example Eq. (2.36). Finally, although we do not attempt to derive it here (it is beyond the scope of our present work), we believe that all of the formal series appearing in our problem form an infinite dimensional algebra, the resurgence algebra, closed under the action of the singularity ("alien") derivative. For example, we expect to have the relations

Lefschetz thimbles and geometrization of ambiguity cancellation
In this section, we briefly sketch the geometric reason for the ambiguity cancellation, in connection with semi-classics and Lefschetz thimbles. The geometric explanation of the cancellations imaginary parts is very similar in the d = 0 and d = 1 examples. It is tied up with the steepest descent (or semi-classical) expansion. The lesson of resurgence theory is that whenever we consider a semi-classical expansion, we should in fact always work with a complexified version of the path integral. That this is the case becomes clear when one appreciates the nature of our approach in the preceding sections, which involved analytically continuing g, and hence to make sense of the ambiguities and use resurgence theory, we had to work with a complexified version of the path integral.
More specifically, in our example we first have to generalize where SL(N, C) is the complex special linear group. In a lattice version of the model, the infinite-dimensional path integral is regularized into a finite dimensional integral 32 . Let the two dimensional lattice have L 1 L 2 sites. Then the original partition function is an integral over [SU (N )] L 1 L 2 , an (N 2 − 1)L 1 L 2 dimensional space. When we complexify, the dimension of space is doubled, But since our goal is to find an analytic continuation of the original integral, we must pick special integration cycles within the complexified field space. Hence the integration runs over a sub-manifold Σ which is again (N 2 − 1)L 1 L 2 dimensional, same as original purely real cycle. For each saddle, there exists a unique Lefschetz thimble attached to it, J [saddle] (θ), whose structure depends on the phase arg(g 2 ) = θ. In fact, in the finite dimensional case, J [saddle] (θ) lives in [SL(N, C)] L 1 L 2 and its dimension is half of the complex space. The analytic continuation of the original integration cycle may be expressed as a linear combination of Lefschetz thimbles: and n i are piece-wise constant (in Stokes wedges) but jump at the Stokes lines. For example, consider the Lefschetz thimble attached to perturbative vacuum, J [0] (θ) at θ = 0 + and θ = 0 − . The integration cycle must have a dramatic change (upon crossing a Stokes line) which does not alter the real part of the integration, but leads to a jump in the imaginary part. Fig. 3 provides a cartoon of this phenomenon for an ordinary integral. In fact, Fig. 3 is related to our present problem via a dimensional reduction, in the one-site limit of the lattice model L 1 = L 2 = 1 with a twisted reduction in the L 2 direction. Fig. 3 shows that the real part of the cycle must remain unaltered upon a Stokes jump at θ = 0, while the The combination of the solvable d = 0 dimensionally reduced model and the non-Borel summability of the perturbative series Eq. (7.3) tells us that the integral over just the Pthimble J [0] (θ) will have a pathological θ = 0 limit, resulting in a two-fold ambiguous result, depending on the direction of the approach to θ = 0 in the complex g 2 plane. We have already seen that this pathology can be fixed by integration over other NP-thimbles. The integration over the fracton-anti-fracton thimble J [FF ] (0 ± ) contributes at the same order as the ambiguity of the P-thimble on the θ = 0 direction. In fact, we expect to be ambiguity free at order e −2S F , but to have some ambiguities at order e −4S F , which are cancelled thanks to the fact that the full integration contour includes thimbles which pass through the appropriate higher-action NP saddle points, and so on. It would be very interesting to understand the structure of all the thimbles and Stokes phenomena in this problem. We believe that this would provide a geometric understanding of the intricate relations between P and NP data which leads to the cancellation of ambiguities in resurgent trans-series, as described in e.g. [15].

Mass gap flow and Borel flow
In one of the first works on renormalons, 't Hooft speculated that they may be connected to the mass gap and confinement in gauge theories [22]. Using resurgence, we find a refinement and a confirmation of this idea in a semi-classical regime continuously connected to the strongly coupled regime of gauge theories and non-linear sigma models.
In every semi-classically calculable example studied so far, it turns out that the mass gap is due to half a renormalon in the semi-classical domain [16][17][18][19]30]. In deformed Yang-Mills on small R 3 × S 1 , the mass gap is due to monopole-instantons M [21], while [MM] ± yields the leading semi-classical realization of the renormalons [16,17]. In N =1 SYM and QCD(adj) on small R 3 ×S 1 , the mass gap is due to magnetic bions B [66], while the leading semi-classical realization of the renormalon is the neutral bion [BB] ± [16,17], etc.
In the SU(2) PCM, the evaluation of the mass gap in the small-L regime follows very closely the calculation of the mass gap in the CP 1 model. In Section 7.2, we already argued that the weak coupling realization of the renormalon is again the neutral bion, [FF] ± . Below, we show that the mass gap at leading order in semi-classical expansion in the small R × S 1 regime is due to fractons F. This is also true for the SU (N ) model in the N LΛ 2π 1 small circle limit adiabatically connected to R 2 .
The mass gap is defined as the energy required to excite the system from the ground state E (0) to the first excited state E (1) . As discussed in Section 5, in the small-L regime, we can work with the Hamiltonian Eq. (5.17) which describes the dynamics of a small-L EFT with zero KK-momentum, or of a small-L EFT with −1 units of winding number on the S 1 . In the Born-Oppenheimer approximation, we can further focus on Eq. (5.19), since the states which  Figure 11. Mass gap flow: The mass gap on the small R 1 × S 1 regime, corresponding to LN Λ 2π 1 is semi-classically calculable. At leading order, it is a one fracton effect. On R 2 or large R 1 × S 1 , a reliable analytical method which can address the mass gap question is at present unknown. Our small R 1 × S 1 theory is adiabatically connected to the theory on R 2 . carry non-zero P φ 1 and P φ 2 momentum acquire a gap of order g 2 /L, while (as we show below) the low lying states of Eq. (5.17) are split by a non-perturbatively small amount, justifying the use of the Born-Oppenheimer approximation.
In the SU (2) PCM the ground state is two-fold degenerate to all orders in perturbation theory. This degeneracy is lifted by non-perturbative fracton effects. The two lowest lying eigenstates described by Eq. (5.17) have the quantum numbers |±, n 1 = 0, n 2 = 0 , where ± are the eigenvalues of the parity operator P which acts on the polar coordinate θ in the Hopf parametrization as θ → π − θ. At leading non-perturbative order in the semiclassical expansion the SU (2) PCM mass gap is then given by For SU (N ), parametrically 33 , the mass gap is of order in the weak coupling semi-classical regime N LΛ 2π 1. 34 Note that the mass gap at small-L is N -independent in the large N limit, just as it is on R 2 , because when N LΛ 1 and N 1, L scales as L ∼ 1/N . This lends further support for the claim that our small-L limit is 33 At leading order in the SU (N ) PCM, the mass gap gets contributions from the N minimal-action fractons which describe tunneling between the ground state and the N directions in field space parametrized by of the N affine simple roots. To compute an exact expression for the mass gap in the SU (N ) case, one must diagonalize the resulting "tunneling matrix" and compute its smallest eigenvalue. The expression we show in Eq. (8.2) is the parametric form of the result which this calculation would give. 34 The appearance of the non-perturbative factor in the mass gap is the major difference with respect to thermal compactfication. In thermal theory at small S 1 β × R, usual KK-reduction works, and the gap is given ∆ thermal gap ∼ g 2 L . The thermal low energy theory does not remember its two dimensional origin, in contrast to the adiabatic small-L limit we have constructed.
adiabatic. At the boundary of the region of validity of the semi-classical regime N LΛ ∼ 1 where we can no longer rely on semiclassics, we observe that the mass gap acquires a strong scale value m g ∼ Λ. In the strong coupling regime, N LΛ 1, we expect the mass gap to be independent of the size of the circle. In fact, the onset of R 2 behavior at the compactification scale (Λ/N ) −1 rather than Λ −1 is a hallmark of large N volume independence, which can be shown to apply to the theory we are working with. Therefore, provided we are given a value m 0 for the gap at some N L 0 Λ 1, as N is varied m 0 can only change by order O(1/N 2 ) corrections in the SU (N ) model. In other words, we expect the mass gap to plateau and remain fixed in this regime as shown in Fig. 11, which shows the expected form of the mass gap as a function of L.
The connection with renormalons should now be clear. The field configurations F i that give rise to the mass gap at order e −S F then produce the leading renormalon singularities at the next order of the semi-classical expansion e −2S F . So the mass gap is tied to "half" of a renormalon. This is a concrete realization of 't Hooft's idea [22].

Borel flow
The idea of Borel flow is a more abstract version of the mass gap flow. Borel flow is tied up with all non-perturbative observables in the problem. The IR-singularities in the Borel plane on the small S 1 L × R regime and on R 2 are located at while the location of the UV-renormalon singularities remain unchanged no matter the value of L. See Fig. 12. The most dominant singularities (m = 1) lead to ambiguities of order where at small S 1 L × R, the 't Hooft coupling is evaluated at distance LN , while on R 2 , it is determined at a high (Euclidean) momentum scale Q (entering through an OPE with an external momentum insertion Q.) The crucial point is that Q Λ and 1 LN Λ so that the coupling is weak at the scale of Q and 1 LN , and in both cases, this gives a control parameter over the small NP-induced term. On S 1 L × R these ambiguities are cancelled respectively by the ambiguity in neutral bion events shown in Eq. (7.5), while on R 2 they are cancelled by the ambiguity in the condensate Eq. (3.8).
In the semiclassical regime, the mass gap is generated by half a renormalon, i.e., a fracton. The dimensionless mass gap is m g LN , and takes the form Borel plane structure on small R 1 × S 1 . The small-L effective field theory (in which small-L physics is integrated out) does not capture the UV renormalon structure by construction, but does capture the IR-renormalon singularities of the small-L theory.
demonstrating explicitly the relation between the mass gap and the leading renormalon. Compare this with the first line of Eq. (8.4). If we accept the behavior we have seen in the semi-classical regime as a rough guide to the behavior we should expect in the strongly coupled domain, we would deduce that the dimensionless mass gap (now measured in units of some external large momentum Q) behaves as which is a sensible result on R 2 . Thus, we are tempted to sharpen 't Hooft speculation.
In asymptotically free non-linear sigma models (including for instance CP N −1 , O(N ), Grassmannian, and principal chiral model-type matrix field theories) there may exist a quantitative relation between the mass gap of the theory and the location of the first renormalon singularity for all values of L.
As illustrated in Fig. 12 the IR-singularities in the Borel plane are twice as dense on R 2 with respect to R × S 1 L . The crucial point for adiabatic continuity is the fact that in both regimes the singularities are spaced by units of ∼ 1 N . Our framework strongly suggests that as we dial the radius from small to large, the singularities must exhibit a flow towards the origin rendering them twice as dense. The same phenomenon should also take place in deformed Yang-Mills, in which the dilution factor between the weak coupling regime and strong coupling regime is 11 3 [17]. Clearly, the flow of the singularities in the Borel plane as the radius is dialed, i.e, the Borel flow, and the flow of the mass gap as the theory is dialed from a weak coupling to strong coupling are intimately related. They are very likely manifestations of the same underlying dynamics. We believe that developing a thorough understanding of these flow equations would constitute a major step towards the solution of the mass gap problem in a large variety of asymptotically free theories.

Discussion and prospects
We have employed resurgence and adiabatic continuity to give a classification of the P-and NP-saddles in the principal chiral model. Due to insights from various techniques, such as lattice Monte Carlo calculations and integrability, these theories were believed to have highly non-trivial similarities to Yang-Mills theory. But the theory had no known NP-saddles, except for the uniton saddle discussed mostly in the mathematics literature [52], whose role in the quantum version of the PCM never became clear. So, from a semi-classical point of view, the only relevant saddle seemed to be the perturbative vacuum. This appeared to be a dramatic difference from Yang-Mills theory.
In this work we have constructed an infinite class of NP-saddles such as the fractons, as well as the zoo of correlated events, which turn out to play a crucial role in the dynamics of the PCM. The fractons lead to the generation of the mass gap of the theory, while neutral bions (correlated fracton-anti-fracton events) lead to the semi-classical realization of the IR renormalons in the weak coupling calculable regime N LΛ 2π 1. Our analysis is inspired by the recent treatment of the 2D in CP N −1 model [18,19], and has many parallels with the analysis of gauge theories on R 3 × S 1 initiated in [20,21,66], and recently revisited in the context of resurgence in [16,17].
In the present work, in the small-L regime, we were able to demonstrate the existence of fractons, whose action is S F = S uniton /N , by three independent methods: • Large order analysis of the small-L perturbative series describing fluctuations around the perturbative saddle point implies, via resurgence, that the non-perturbative completion of the problem involves saddles with action 2S F . Moreover, the notion of emergent topology encoded in the resurgence triangle leads to the conclusions that there must also be other saddles, with action S F , which are precisely the fractons.
• The effective field theory obtained via adiabatic continuity, which is just a particular quantum mechanical theory, allows a simple study of the NP-saddles. The NP saddles with the smallest action have actions S F , and they come in N different types.
• When the Z N symmetric background holonomy is turned on we have seen that the uniton splits into N lumps, each of which carries an action S F , as nicely shown by the plots in Fig. 9.
This last phenomenon is morally similar to the splitting of calorons (periodic instantons) into N -monopole instantons in gauge theories [24,25], but it is again worth emphasizing that the PCM does not have any instantons.

Future directions
There are a large number of interesting possible extensions of the study we have performed here. A few of them are: 1. WZW term and sign problem: Addition of a WZW-term to the action modifies the IR dynamics on R 2 and introduces a sign problem if one were to try to attack the system using Monte Carlo simulations. It would be interesting to study this system on the calculable small-L regime.
2. Borel and mass gap flows: An understanding of the non-perturbative dynamics on R 2 can perhaps be achieved by summing the resurgent trans-series at small-L, and extrapolating the sum to large-L. Less ambitiously, one may develop an understanding of an infinitesimal version of Borel flow from a more detailed analysis of the construction of the action of the small-L effective field theory, by exploring the effects of the higherderivative terms in the action induced by integrating out high-momentum modes.
3. PCM with fermions: The PCM model, like other non-linear sigma models, is asymptotically free for any number of fermion flavors (unlike QCD-like theories on R 4 .) It would be interesting to perform a detailed investigation of the impact of fermions on the dynamics, and to determine the boundary between the confining and IR-conformal regimes.

4.
Large-N reduced model: We can reduce the theory with N f ≥ 1 Majorana fermions to a one-site lattice theory, by imposing 't Hooft twisted boundary conditions. Let Φ represent the bosonic/fermionic degrees of freedom U, ψ i with e 1 , e 2 the two lattice vectors and then reduce the matrix model to one-site by imposing Φ(x + e 1 ) = Ω 1 Φ(x)Ω † 1 , Φ(x + e 2 ) = Ω 2 Φ(x)Ω † 2 , Ω 1 Ω 2 = e i 2π N Ω 2 Ω 1 , (9.1) where the last condition is ensures compatibility of the fields at x ∼ x + e 1 + e 2 . It would be useful to examine the dynamics of the associated reduced large-N model.

5.
Renormalons in the large-N reduced model: In the single-site reduction which is enabled by large-N volume independence, the space-time volume of the theory on R 2 is mapped to the matrix size (N = ∞) of the 1-site matrix model. Within planar perturbation theory there is an exact mapping between summation over spacetime momenta, and summation over the adjoint SU (N ) indices. This implies that there must exist a matrix field interpretation for both IR and UV renormalons. It would be interesting to understand this in detail.

A Resurgence Terminology
In this Appendix we briefly discuss some of the mathematics behind asymptotic series and resurgence methods. For further details, see e.g. [9,13,38,95]. As we already mentioned at the beginning of our paper, most perturbative series appearing in physics are not convergent. When we compute a generic physical quantity by means of perturbative expansion, generic observables take the form f (g) = ∞ n=0 a n g n , (A.1) with g is the coupling constant. When the constant term a 0 in (A.1) vanishes the asymptotic series is called a small power series. As noted long time ago by Dyson and Lipatov [1,2], a generic feature of quantum field theory is the factorial growth n! of the coefficients a n (i.e. combinatorics of Feynman diagrams or phase space integraltion of UV renormalons), which effectively makes the series Eq. (A.1) divergent for all non-zero g. Indeed, it turns out that series usually found in physics are only asymptotic series, meaning that the difference between the function f (g) and the partial sum tends to zero as lim g→0 g −N |f (g) − N n=0 a n g n | = 0 , (A.2) this for all N . Clearly this does not imply convergence, and a naive finite-order partial sum may differ enormously from the actual function f . We now define an important unitary subalgebra of the algebra of formal power series with coefficients in C, C[g]. Expansions of the form of Eq. (A.1) are examples of Gevrey order 1 formal power series with coefficients a n g n , which are defined by the property that |a n |/n! is growing at most as a geometric series. Some Gevrey-1 series can be assigned a meaningful sum by the method of Borel summation. To define a Borel sum of a Gevrey-1 series, we first insert the factor "1" into the series using the well known formula The formal series diverges for all g = 0. However, the coefficients a n = (−1) n−1 (n − 1)! alternate in sign, leading to a well defined and unique Borel sum. The Borel transform can be obtained from the definition Eq. (A.5) . 35 Note that usually Euler's equation is written in terms of x = 1/g. Consider now a simple modification of the formal series Eq. (A.6): This is a solution of the ordinary differential equation The coefficients a n = (n − 1)! are now non-alternating, and the Borel transform of F (g) is Due to the non-alternating nature of the coefficients a n we obtain a two-fold ambiguity in the Laplace transform, since the integration contour t ∈ [0, ∞) is a Stokes line. That is, it contains a pole for t = 1. If we allow the contour of integration to move into the complex t plane, also called Borel plane, we can avoid the singularity by either passing above it arg(t) > 0 or below it arg(t) < 0. Hence we can define two Laplace transform of B This exponential term is the hallmark of non-perturbative physics, and the example just presented is just a simplified version of what normally happens in generic asymptotically-free quantum field theories. As a result, the Borel sum is ambiguous since the integration line is then a Stokes line containing poles (or more generically branch cuts) of the Borel transform. For a generic asymptotic series f of Gevrey type 1 we will require its Borel transform to have only a "few" singularities in the Borel plane. More precisely we require the germ of analytic functions B[f ] to be endlessly continuable on C, meaning that for all L > 0 there exists only a finite set Ω L (B[f ]) ⊂ C, called the set of L-accessible singularities, such that B[f ] has an analytic continuation along every path whose length is less than L, while avoiding the set Ω L (B[f ]). This definition is slightly stronger than the original definition given by Ecalle. Both our previous example obtained from the E and F series satisfy this requirement. A conjecture consistent with all results available so far is that in fact the perturbative series arising from physical QFTs also satisfy this requirement, which is necessary for the technology of resurgence theory to be useful.
We will also say that B[f ] has only simple singularities if for all paths γ ending at a singular point t , the analytic continuation B[f ] γ of the germ B[f ], along the path γ, in a neighborhood of t takes the form where a γ ∈ C, while b γ and h γ are some analytic germs around the origin. The germs b γ and h γ are themselves endlessly continuable functions with simple singularities. With the concepts just introduced, we will define the formal power series F to be a simple resurgent function if it is of Gevrey order 1 and its Borel transform is an endlessly continuable function with only simple singularities. The set of simple resurgent function is actually a subalgebra of C[g] denoted by + R(1) as proven by Ecalle [9]. It is useful at this point to introduce the concept of directional Borel summation. where the contour of integration is the line, lying in the complex Borel plane, starting from the origin, t = 0, and going to infinity in the direction arg(t) = θ. This integral is convergent in the half-plane defined by P θ = {g ∈ C s.t. arg(t/g) > 0}, by Cauchy's theorem S θ [f ] and S φ [f ] will coincide on P θ ∩ P φ , so they are analytic continuation of one another. Furthermore they all have the same asymptotic expansion given by Eq. where the contour C θ comes from infinity in the direction arg(t) = θ, turns around the singular point t clockwise, and then goes back to infinity once again in the direction θ, as displayed in Fig. 4. Given the form of the singularity Eq. (A.13), one can show that where B −1 is the inverse of the Borel transform. As we can see from this last equation, the ambiguity in the Borel sum S θ [f ] is related to the presence of infinitesimal terms e −t /g (1 + O(g)). These terms cannot be captured by our perturbative series Eq. (A.1). On a Stokes line it is essential to take into account non-perturbative contributions in order to be able to assign a well-defined meaning to the sum of the perturbative series. Hence we have to replace our asymptotic series Eq. (A.1) with a trans-series of the schematic form This is precisely what we would expect from an observable computed using a saddle-point method, see Eq. (1.2). The e −tc/g factors are non-analytic for g → 0, so they have to be treated as objects external to the algebra of simple resurgent functions, and induce a grading on it. As shown above, when the direction θ is a singular one, the Borel summation jumps as we cross this Stokes line, and the full discontinuity across this direction plays a crucial role in linking perturbative and non-perturbative terms. The Stokes automorphism S θ is defined by where Disc θ encodes the full discontinuity across θ. When the Stokes automorphism in a particular direction θ acts as the identity operator, it means that the Borel transform of f (g) has no singularities along the θ direction and is given by a Borel-summable power series. Across a Stokes line S θ is non-trivial and it encodes the jump between the two lateral resummations. Passing from a standard asymptotic series Eq. (A.1) to a trans-series Eq. (A.17) with the inclusion of non-analytic (non-perturbative) terms of the form e −t/g is crucial. While these terms are exponentially suppressed for g ∼ 0 compared to terms of the form a n g n , when sitting on a Stokes line these terms are critical for making observables well-defined, and they must be taken into account.
By a contour deformation it is possible to show that the difference between the θ + and θ − deformation is nothing but a sum over Hankel's contours, and the discontinuity of S across θ is arising as an infinite sum of contribution coming from each one of the singular points. The logarithm of the Stokes automorphism defines the alien derivative where we denoted with Γ θ the set of singular points of the Borel transform along the θ direction. It is possible to show that this operator is a real derivation acting on the space of simple resurgent functions [13]. When the Borel transform of f has only one simple singularity Eq. (A.13) at t in the direction θ = arg(t ), the alien derivative takes the simpler form where the path γ is the line emanating from the origin in the direction θ = arg(t ). A more general definition is possible when there are multiple singular points along the chosen Stokes line but we will not need it for the present work [13].
In the particular case in which t is the only simple singularity for the asymptotic series Eq. (A.1) along θ, we can rewrite the Stokes automorphism using Eq. (A.20) S θ f (g) = 1 + e −t /g ∆ t + e −2 t /g 2 ∆ 2 t + ... f (g) . which is just a manifestation of Stokes phenomena written in the language of alien derivatives.