From 4d Yang-Mills to 2d $\mathbb{CP}^{N-1}$ model: IR problem and confinement at weak coupling

We study four-dimensional $\mathrm{SU}(N)$ Yang-Mills theory on $\mathbb{R} \times \mathbb{T}^3=\mathbb{R} \times S^1_A \times S^1_B \times S^1_C$, with a twisted boundary condition by a $\mathbb{Z}_N$ center symmetry imposed on $S^1_B \times S^1_C$. This setup has no IR zero modes and hence is free from IR divergences which could spoil trans-series expansion for physical observables. Moreover, we show that the center symmetry is preserved at weak coupling regime. This is shown by first reducing the theory on $\mathbb{T}^2=S_A \times S_B$, to connect the model to the two-dimensional $\mathbb{CP}^{N-1}$-model. Then, we prove that the twisted boundary condition by the center symmetry for the Yang-Mills is reduced to the twisted boundary condition by the $\mathbb{Z}_N$ global symmetry of $\mathbb{CP}^{N-1}$. There are $N$ classical vacua, and fractional instantons connecting those $N$ vacua dynamically restore the center symmetry. We also point out the presence of singularities on the Borel plane which depend on the shape of the compactification manifold, and comment on its implications.


Contents
1 Introduction

Confinement and resurgence
It has been a long-standing problem to show that a non-trivial quantum Yang-Mills theory exists on the flat space R 4 , and that it has a mass gap. This is a challenging problem since Yang-Mills theory, while asymptotic free in the UV, is intrinsically strongly coupled in the IR.
One possible approach is to start with the perturbative expansion of the theory: this is the best-understood part of the theory, can also be formulated mathematically [1]. One might then hope that a suitable re-summation of such a perturbative series might lead to a complete theory [2].
Of course, Yang-Mills theory on R 4 is strongly-coupled in the IR and hence the perturbative expansion breaks down. One might nevertheless try to avoid this problem by compactifying the theory onto e.g. a circle S 1 of radius L. When the radius is smaller than the dynamical scale Λ of the theory (L Λ −1 ), the renormalization-group-running of the gauge coupling constant stops at the scale 1/L, at which the gauge coupling is small. One then hopes to do a reliable weak coupling computation, perturbatively in the gauge coupling constant. Once we have done this, one might hope to adiabatically continue back to the flat space.
The hope is that such an adiabatic continuation can be achieved by the theory of Borel-Écalle re-summation and resurgence [3,4] (see e.g. [5,6] for recent exposition). This states that the perturbative series (say of the vacuum expectation value of an operator O) should be thought of as part of the so-called trans-series expansion, containing both perturbative and non-perturbative corrections: c I,k g k + . . . , (1.1) where I is a label for the saddle points of the action S, and S I denotes the value of the action at the I-th saddle point; the coefficients c I,k represent the perturbative expansions around the I-th saddle point. Assuming that we know all the saddle points contributing to the path integral, all the coefficients c 0,k and c I,k can be computed by perturbative methods, and we obtain a trans-series as in (1.1). Now one hopes to turn the trans-series into a well-defined function O (g) as a function of the coupling constant with the help of the resurgence theory, 1 and if that function is indeed resurgent (as might be expected from resurgence theory), i.e. endlessly continuable, we can adiabatically continue back to the large value of the coupling constant along a suitable choice of path in the complex plane.
Challenges in resurgence There are problems with such an optimistic scenario. First, such a compactification of the theory can dramatically change the physics; for example, if we compactify the theory on a temporal circle we expect to encounter the confinement-deconfinement transition, and thus the weak coupling computation is possible only in the deconfined phase. One way to avoid such a phase transition is to ensure the existence of the center symmetry (more on this later in this paper). This can be achieved, for example, by twisted boundary conditions or by suitable deformation of the Lagrangian. Such a continuity from small to large size of the compactified direction is 1 The word resurgence in the physics literature sometimes refers to a stronger statement. Namely, largeorder asymptotic growth (as k large) of the perturbative coefficients c 0,k around the trivial saddle point contains the information of the non-perturbative saddle points (inside the same topological sector). This has been discussed in the literature since long ago [7,8]. See e.g., [9][10][11][12][13][14] for various points of view on this issue. called adiabatic continuity. It was considered in physics independently and later combined with the mathematical idea of resurgence: see e.g., [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. (See also literature on large N twisted Eguchi-Kawai model [35][36][37]).
Second, in order to cancel the ambiguity of Borel re-summation of perturbative series, we need to have a corresponding semi-classical saddle point for a singularity of the Borel plane on the positive real axis. In particular, it was argued that there is the renormalon pole [2], whose action is of order 1/N of that of the instanton in the large N limit. Recently there has been huge progress in this respect (together with the adiabatic continuity), in the context of recent connection with resurgence. The activities in this field has been triggered by the works of [23,24] on 2d CP N −1 -model and [21,22] for 4d adjoint QCD. 2 While these points are of relevance to this paper, our main focus here is yet another, although related, source of trouble for the resurgence program, associated with IR divergences. The problem of IR divergences is already stressed in the context of resurgence in [23], but we discuss this problem again by emphasizing the point that they spoil the formal trans-series expansion of physical observables.

The problem of the infrared divergences
In this paper we point out and study the subtlety which spoils the resurgence program: the issue of IR divergence. In short, the problem is that the perturbative (or more generally trans-series) expansion (1.1) in itself is ill-defined if the observable suffers from IR divergences.
Instanton computation of vacuum energy. To illustrate the issue of IR divergence, let us study the theta-angle dependence of the vacuum energy of the Yang-Mills theory on R 4 . Instanton computation gives the formula for the θ-dependence of vacuum energy E(θ) as [39] where µ is an arbitrary renormalization scale, b 1 is the coefficient of the one-loop beta function, ρ is the size modulus of the instanton, and g(µ) is the running gauge coupling constant at the scale µ. This integral is divergent in the IR region ρ → ∞. Still, one might hope that the formula gives qualitatively right answer by introducing naive IR cutoff, which we may take to be the dynamical scale Λ of the theory. This gives the answer 3 the large N limit, there are infinitely-many metastable vacua labelled by an integer e ∈ Z, and the vacuum energy of each of the metastable vacua are given by Note that each E e (θ) is not even a periodic function of θ! We can recover the periodicity by considering the energy of the true vacuum, namely minimum of all the E e 's: however this is discontinuous at θ = ±π. (See [46] and references therein for the physics at θ = ±π.) The actual situation is that each vacuum labeled by e ∈ Z is metastable (in the large N limit), and the true stable vacuum e = 0 at θ = 0 is adiabatically continued to a metastable vacuum under the monodromy θ → θ + 2π. This is very different from the naive answers expected from the instanton calculus (1.2), even at the qualitative level.
Linde's problem. Let us next consider the thermal free energy F defined by where β = T −1 is the inverse temperature and Vol. is the space-time volume, which we regularize to be a finite number by introducing some IR cutoff. After the compactification on the thermal circle S 1 β , the 4d Yang-Mills is reduced to 3d Yang-Mills up to Kaluza-Klein (KK) modes with the 3d gauge coupling given by g 2 3 = T g 2 . The contribution of 3d Yang-Mills to the free energy is expanded by g 2 3 = T g 2 , but it has a mass dimension. Thus the expansion is of the form where [KK contribution] means the contributions from nonzero modes on S 1 β , and the expansion by T g 2 comes from 3d Yang-Mills. In naive perturbation theory, m IR = 0 and the above expansion has severe infrared problem. In improved perturbation theory, m IR may be taken to be the thermal mass of the fields. However, for the 3d gauge fields, the thermal mass is known to be at most m thermal < ∼ T g 2 . Therefore, in the above expansion, higher loops do not give higher powers of the coupling g and hence all the higher loops give the fixed order O(g 6 ) of the coupling expansion (in the most optimistic case of m thermal ∼ T g 2 ). The perturbative expansion, and hence the (trans)series expansion, is therefore not welldefined [47].
The naive cutoff prescription of the IR divergence does not work in each of the two cases discussed above. Of course, some observables (IR safe observables) do not depend on the details of IR regularizations, but many observables of physical interest are sensitive to such subtleties.
We stress that the IR problems mentioned above are conceptually different from (although related to) the problem of whether the coupling g is large or small and we can numerically trust the weak coupling results or not. 4 The issue is whether or not the coefficient in the perturbative (or trans-series) expansion is well-defined. Even if g is very large, the resurgence might give a sensible answer as long as the trans-series expansion is well-defined. But if there is IR divergences, there is no way to do resurgence from the beginning.
The question is then to find a concrete setup in which the trans-series expansion is well-defined. Doing this for the pure SU(N ) Yang-Mills theory 5 is the motivation for the next section.

Our setup
In this paper we study the pure Yang-Mills theory with gauge group SU(N ), with gauge coupling constant g and theta angle θ. Our setup is summarized in Figure 1.
We include a unit 't Hooft magnetic flux (i.e., twisting by the center symmetry) along S 1 B × S 1 C , which plays rather crucial roles in this paper.

Yang-Mills theory in a box
In this paper the pure Yang-Mills theory is put into a box, namely compactified on a spatial three-torus where the non-compact direction R is the temporal direction. Here S 1 A,B,C are spatial directions. We will denote their circumferences of three spatial directions by L A,B,C , respectively. These circumferences play the role of the IR cutoff, and the limit to the flat space R 4 is achieved by L A,B,C → ∞ with their ratios kept finite.
In this paper we will primarily study the parameter region

2)
4 For example, even for weakly coupled QED, there are IR divergences due to soft photons. Therefore the problem of IR divergences are conceptually different from the strong dynamics. However, in the case of soft photons, physical treatment of IR divergences are not so difficult. 5 One can also add adjoint fermions, and also fundamental fermions whose flavor number is a multiple of N .
with the relative size of L A and L B kept finite. Note this breaks the symmetries between S 1 A,B and S 1 C . The hierarchy in scales (2.2) means that below energy scales 1/L A and 1/L B the theory is effectively described by the two-dimensional theory on R × S 1 C . We show that the resulting two-dimensional theory is given by the two-dimensional sigma model whose target space is CP N −1 . This connection with the two-dimensional physics will be rather useful for understanding the physics of four-dimensional Yang-Mills theory.
Notice that the parameter region (2.2) is different from the decompactification limit of R × T 3 into R 3 × S 1 , where one might encounter a version of Linde's problem: However, one can hope that after a suitable resurgence analysis in the region (2.2) we may extrapolate the answer to the region (2.3). We will give some comments on this point in section 6.

Twist by 1-form Z N symmetry
As another crucial ingredient for our construction, we consider twisted boundary condition along S 1 B × S 1 C . The four-dimensional pure Yang-Mills theory has a center symmetry, or in modern language a global "electric" one-form Z N -symmetry [48]. In the Language of [48], the one-form symmetry twist along S 1 B × S 1 C used in this paper is generated by the surface operator supported on (pt) (See appendix A for quick summary of one-form symmetry). As we will see later, this global symmetry acts on the Wilson lines along the cycle S 1 B . This twist is crucial for the following reason. First, we will see that this ensures that • gauge symmetry is completely broken at classical vacua around which we perform perturbation, hence there is no massless modes in the IR.
We therefore do not encounter a subtle issue of IR divergences. Second, we will see later in this paper (section 5.3) that while the one-form center symmetry is broken classically, it is restored after quantum tunneling effects are taken into account: • center symmetry is preserved.
Since the presence of the one-form center symmetry is proposed as a characterization of the confinement phase [48], we conclude that our system confines, at least in the parameter regions (2.2). We expect that this is true for all L A , L B , and L C as long as their ratios are kept finite. (The case of taking L A,B /L C → ∞ is commented in section 6.) Such a dynamical restoration of center symmetry is shown by reducing the theory onto the R × S 1 C -directions, and matching the resulting theory with the two-dimensional CP N −1 -model with Z N -twisted boundary condition studied recently in connection with resurgence [23,24,34]. Namely, we show that the following two twists are the same after the reduction from four to two dimensions: • the twist in the SU(N ) Yang-Mills theory by the one-form symmetry on S 1 • the twist in the CP N −1 model by the Z N global symmetry on S 1 C .
For this reason we will spend the next two sections exploring the connection between four-dimensional Yang-Mills theory and the two-dimensional CP N −1 -model. 3 From SU(N ) Yang-Mills to CP N −1 Let us first consider the compactification of the pure SU(N ) Yang-Mills theory on the two-dimensional torus (elliptic curve) E = T 2 = S 1 A × S 1 B down to two dimensions. When the Yang-Mills theory is compactified on E and the Kaluza-Klein modes are integrated out, the light degrees of freedom on E are the zero-energy configuration of gauge fields, i.e. flat connections on E, except near singular points of the moduli space of flat connections where there is an enhanced symmetry and W-bosons become light. The moduli space of flat SU(N ) connections on E is known to be the projective space CP N −1 at the holomorphic level [49][50][51]. Here we review and spell out some details of this relation between flat connections and CP N −1 .

Yang-Mills flat connections on a torus
We choose a complex coordinate of the torus E = T 2 to be where τ is a complex parameter (modulus of torus) with Im(τ ) > 0. The torus E has the flat metric where L E is the size of the torus. In the notation of the previous section, these parameters are given as τ = i(L B /L A ) and L E = L A . But it is not difficult to take τ to be more general complex numbers. We call the cycles on E corresponding to z → z + 1 and z → z + τ as the A-cycle and B-cycle, respectively. In our previous notation these are S 1 A and S 1 B , respectively. The gauge field on E can be represented as A = Azdz +A z dz in the complex coordinate z. We take A to be anti-hermitian so that the field strength is given by F = dA + A ∧ A. For flat connections, thez-component Az is expanded as where φ i are complex scalars which are independent of z and satisfy N i=1 φ i = 0. The constant prefactor −2πi/(τ − τ * ) is chosen for later convenience. Wilson lines in the Aand B-cycles are given by (3.4) The scalars φ i have periodicities and thus can be regarded as a point on E with the constraint i φ i = 0. Furthermore, φ i 's are permuted by the Weyl group of SU(N ), namely the N -th symmetric group S N : an element of the Weyl group σ ∈ S N acts as φ i → φ σ(i) . Therefore, the moduli space of flat connections is given by The case of N = 2 is particularly simple, since then we have One can then see as in Figure 2 that this is indeed CP 1 = S 2 as an algebraic variety. What has to be kept in mind is that the manifold has four singular points, where extra massless W-bosons appear. The case of N > 2 is more complicated, but there is a convenient way to describe such N points φ i on E. Consider a space of meromorphic functions F which are allowed to have N -th order pole at z = 0. Namely, F is allowed to behave as F (z) ∼ z −N (z ∼ 0). We denote such a space of meromorphic functions on E as H 0 (E, O(N p)), where p represents the point p = {z = 0 ∈ E}, and O(N p) indicates that the functions are allowed to have Nth order pole at p. 6 When F ∈ H 0 (E, O(N p)), the Cauchy theorem for dF/F = ∂ z F/F dz on E implies that there must be N -points q i (i = 1, · · · , N ) at which F (z) has a zero. This is because by the Cauchy theorem, the sum of residues of dF/F must be zero and it has the residue +1 at a simple zero of F and −1 at a simple pole, with obvious generalization when zero or pole have multiplicity. If q i = p for some i, that means that the degree of the pole at p = {z = 0} is reduced by the number of q i for which q i = p. If q i = q j , that means that the function F has double (or more generally multiple) zero at that point.
Then the claim is that the set of unordered points {φ i } 1≤i≤N on the torus E, that is This can be shown by well-known techniques in algebraic geometry (see e.g. [51]). Here we would like to give more or less elementary explanation.
Let φ i be the coordinates corresponding to q i associated to the zero points of a function F . Then, first we want to show that they satisfy i φ i = 0 so that they define a point on E defined in (3.7). One way to see this is as follows. Corresponding to each q i , take a path γ i from p to q i such that γ i and γ j do not intersect other than at the point p. See Figure 3. By eliminating γ i from E, the log F becomes well defined which has branch cuts at γ i . Consider the integration of log F dz on a loop of the form ABA −1 B −1 , where A and B are the A-cycle and B-cycle of the torus E. Then we get a value of the form 2πi(Z + τ Z). This is because the contributions from A and A −1 add up to give 2πin A dz where 2πin ∈ 2πiZ comes from the difference of log F before and after going around the loop B. Similarly, the contributions from B and B −1 add to give 2πim B dz. On the hand, applying the Cauchy theorem to this integral, we get − i 2πi γ i dz. By using γ i dz = (φ i − 0), we get the desired result ( i φ i ) ∈ (Z + τ Z).
From the results obtained above, we see that a meromorphic function F ∈ H 0 (E, O(N p)) gives a data {φ i }/S N needed for a flat connection. Suppose that two meromorphic functions F and G have the same set of zero points {φ i }. Then, their ratio F/G does not have either a zero or pole. Such a holomorphic function must be a constant, F/G = const. This means the following. First, notice that the space of functions H 0 (E, O(N p)) is a complex vector space. Let PH 0 (E, O(N p)) be the projective space associated to this vector space. Namely, we introduce equivalence relation F ∼ cF for constants c ∈ C − {0}, and divide the space It turns out that the map above is also surjective. Indeed, given a set of points {φ i } satisfying i φ i = 0, we can explicitly construct a function F ∈ H 0 (E, O(N p)) which has zeros at these points and an N -th order pole at p = {z = 0}: The infinite product is convergent if This establish that the moduli space of flat connections is given by where we have used the fact that H 0 (E, O(N p)) is an N -dimensional vector space, as can be seen by counting the number of parameters to specify F ∈ H 0 (E, O(N p)) (i.e., φ i with i φ i = 0 gives N − 1 parameters and the overall scale gives one parameter.) We remark that the equivalence (3.10) should be interpreted as equivalence as algebraic varieties, and at the classical level, the metric of the moduli space M flat is not the Fubini-Study metric. However, this point does not affect our qualitative discussions in this paper.

Explicit map from
The above discussion was rather abstract. We now make the map M flat = E/S N → CP N −1 more explicit.
We denote the root lattice of SU(N ) by L The weight lattice is spanned by the fundamental weights We define theta functions as where φ = (φ 1 , · · · , φ N ) and the inner product between vectors is defined as φ · = i φ i i . These theta functions are invariant under the Weyl symmetry S N acting on φ, because each set is Weyl invariant, and the Weyl symmetry preserves the inner product. Furthermore, under the shift they transform as Note that the factor e −πiτ m 2 −2πi m· φ is independent of k.
We denote points of CP N −1 by using homogeneous coordinates as [Z 1 , · · · , Z N ]. Then, if we define then the above properties imply that this is a well-defined map from M flat to CP N −1 . 7 We claim that this is an isomorphism between M flat and CP N −1 .
In the above discussion, we have given θ k explicitly. In fact, there is an algebraic geometric reason for the existence of such a map. (The reader who is not interested in algebraic geometry can skip this paragraph.) Let E be the space of flat connections before dividing by the Weyl group as defined in (3.7). In the previous subsection, we established that E/S N ∼ = CP N −1 . Thus there is a natural projection map On the CP N −1 , there is a line bundle O(1) (which is also written as O(H) for a hyperplane H in CP N −1 and is called the hyperplane line bundle). It has the properties that its holomorphic sections are one-to-one correspondence with linear functions of the homogeneous coordinates Z k (k = 1, · · · , N ) of the CP N −1 . With this identification, there are N linearly independent sections Z k of O(1). Now, we can pull-back this line bundle by π to define a line bundle L = π * O(1) on E. Then we also get N linearly independent sections θ k = π * Z k of the line bundle L by the pull-back. By using these sections, we can define a map By construction, this map coincides with the original projection map π. On the other hand, E is an Abelian variety because it is equivalent to E N −1 as is clear from (3.7). Also, L is a positive line bundle in the sense that its curvature can be positive definite by choosing an appropriate metric on L. There is a general classification and explicit construction of such line bundles and their sections on Abelian varieties. See the section on "Theta Functions" in [52]. These are precisely the theta functions θ k ( φ) given in (3.13). In fact, the nontrivial transformation (3.16) implies that θ k are not functions, but sections of a line bundle L.
By compactification of space-time on S 1 , a 1-form symmetry splits to a 1-form symmetry and a 0-form symmetry. Here, "0-form" symmetry means a usual global symmetry. So, we get a usual global symmetry for each S 1 .
In the context of reducing four dimensional Yang-Mills to two dimensional CP N −1 , we are interested in the two 0-form symmetries Z where the minus signs are introduced just for later convenience. Notice that the two Z N 's, where τ Z + Z means an arbitrary constant of the form τ m i − n i such that i φ i = 0 is preserved. By defining a vector the above action is written as for arbitrary j. The reason that j can be arbitrary is because c j − c j ∈ L for any j and j . Now we can study the action of them on the theta functions θ k defined in Sec. 3.2. First, one can check that for (p, q) = (0, 1), the change of the theta functions is given as (3.24) Next, for (p, q) = (1, 0), we get Therefore we get Note that the overall factor e −πiτ c 2 j −2πi c j · φ is independent of k, and hence is irrelevant when θ k are mapped to homogeneous coordinates of CP N −1 .
In summary, in terms of the homogeneous coordinates, the symmetries Z 29) or more generally instantons are discussed in more mathematically precise way by Atiyah [53] motivated by the analogy between the Yang-Mills and the CP N −1 model.

The topological theta term
We explain the equivalence of the topological terms between the CP N −1 and Yang-Mills. Actually, we discuss more details of the actions. We follow the explanation in section 4 of [54] adapted to our case.

4d Yang-Mills as 2d gauge theory with infinite gauge group
Let E be a Riemann surface which for our application is T 2 . We consider a 4d spacetime , where x i (i = 0, 1) are the coordinates of R 2 and y a (a = 2, 3) are the coordinates of E. They are related to the complex coordinate z on E as z = y 2 + iy 3 . The metric on E is denoted as ds 2 = g ab dy a dy b = 2g zz |dz| 2 . The following discussion can be easily generalized even if we replace R 2 × E by more general X 2 × Σ for 2d space-time X 2 and a Riemann surface Σ. The 4d Yang-Mills theory with a compact, simple and simply connected gauge group G on R 2 × E can be regarded as a 2d gauge theory on R 2 with gauge group G, where G is the group of all gauge transformations on E. Here we are regarding the space E as an internal manifold, and considering a gauge theory on the 2d space-time R 2 . Notice that this G is an infinite dimensional group, with the obvious group structure. In the following, all the Lie algebra generators are taken to be anti-hermitian which is different from the standard physics convention. The total gauge field A is split as A = a + b, where a = i=0,1 a i dx i is the gauge field of G and b = a=2,3 b a dy a is now regarded as "matter fields" from the 2-dimensional point of view. Regarding b as matter fields may be intuitively understood if we perform Kaluza-Klein reduction, since in the 2d space-time R 2 they are just scalar fields. However, we can proceed in an abstract way without any Kaluza-Klein reduction.
A gauge transformation g(x, y) of G on R 2 × E can be regarded as a gauge transformation of G on R 2 , and it acts as where d x and d y are exterior derivatives on R 2 and E, respectively. The matter field b takes values in the space of connections on E which we denote as A. The space A is an infinite dimensional Kähler manifold, once we fix a complex structure on E. The holomorphic coordinates on A are given by bz, i.e., the anti-holomorphic component of the connection b = bzdz + b z dz on E. The Kähler form ω is given by 8 where δ is the exterior derivative on A, or more intuitively, δb means infinitesimal fluctuations of b. The trace is normalized in such a way that the usual instanton number is given by integration of 1 8π 2 tr(F ∧ F ). (For the case of G = SU(N ), the trace is in the defining N -dimensional representation.) Even if we forget about the complex structure of E, the ω still defines a natural symplectic structure on A. Under the gauge transformation, the δb transforms as δb → g −1 (δb)g, and hence the gauge transformation preserves the Kähler form.
Because G acts on A preserving the Kähler structure, we can define the moment map of G as follows. An element of the Lie algebra of G is an infinitesimal gauge transformation on E. It is described by a vector field on A given by where [b, ] + d y is the infinitesimal gauge transformation of b. Let ι V be the contraction operator of a vector field V acting on differential forms. We get where D y = d y + [b, ·] is the covariant exterior derivative. Then we can write where µ( ) is the moment map associated to , and is given by We define µ by This µ takes values in the Lie algebra of G. Now we can rewrite the 4d Yang-Mills action as a G gauge theory on the 2d spacetime R 2 . First, let us introduce a notation for an inner product on the internal space. For elements of the Lie algebra of G, r and s, we define The minus sign comes from the fact that we take the Lie algebra to be anti-hermitian, and hence this inner product is positive definite. Now the kinetic term can be written as The Lagrangian L 2d,kin is given by where Therefore, the 4d Yang-Mills is now rewritten as a 2d gauge theory with the gauge kinetic term f a ij , f a ij , the matter kinetic term D i bz, gz z D i b z and the potential energy µ, µ . The topological term of 4d Yang-Mills is given as where where f a = 1 2 f a ij dx i ∧ dx j and D x bz = D i bzdx i .

CP N −1 as Kähler quotient A//G
Having derived the 2d Lagrangian L 2d = L 2d,kin + L 2d,top , let us look at the potential minima of this Lagrangian. The potential energy is proportional to µ, µ , and hence the potential minima at the classical level is given by 2πµ = f b 23 = 0. This is just the flat connections on E. If we divide the space of the potential minima by the gauge group G, we get the CP N −1 for the case G = SU(N ) as discussed before. On the other hand, taking µ = 0 and dividing by the group G is precisely the Kähler (or symplectic) quotient of the space A by G denoted as A//G. Therefore we get The Kähler form (4.3) descends to this space A//G. Now let us look at the topological term in this subspace µ = 0. We get However, this is precisely the pull-back of the Kähler form given by (4.3). Let Φ be the map Φ : Then, by the pull-back we get Φ * δbz = D x bz, and hence Therefore, we finally get The ω is a Kähler form on CP N −1 = A//G. Furthermore, it must be integrally quantized on A//G and can take the value 1 as is clear from the fact that Φ * ω comes from the 4d topological term 1 8π 2 tr(F ∧ F ). Therefore ω is the generator of H 2 (CP N −1 , Z), and the instanton number of CP N −1 corresponds to the instanton number of 4d Yang-Mills.

Moduli space of instantons
Next let us compare the moduli spaces of instantons of Yang-Mills and those of CP N −1model. To simplify the problem, we compactify the 2d space-time R 2 to S 2 .
For the Yang-Mills theory, the total 4d space-time is S 2 ×E. Then, the Atiyah-Hitchin-Singer theorem [55] states that the number of (virtual 9 ) dimensions of the instanton moduli space for SU(N ) with topological charge Q is given by where χ and σ are the Euler number and the signature of the 4-manifold. 10 For the 4-manifold S 2 × E, we have χ = σ = 0, and thus the dimensions is 4N Q. Let us next consider the instantons of the CP N −1 -model on S 2 . If we regard S 2 = CP 1 , then the instantons of CP N −1 -model is given by a holomorphic map CP 1 → CP N −1 , and the instanton number is given by the winding number of this map. The standard explanation is as follows. Consider a general sigma model whose target space M is Kähler. Let ξ I be holomorphic coordinates of the sigma model, and let g IJ be the Kähler metric such that the Kähler form ω = ig IJ dξ I ∧ dξJ is integrally quantized, i.e., the generator of H 2 (M, Z). Also let x i be the coordinates of the spacetime S 2 = CP 1 with w = x 0 + ix 1 its complex coordinate. Then we get where Q is the instanton number defined in terms of the Kähler form ω. The equality is saturated if and only if ∂ i ξ I ± ij ∂ j ξ I = 0, which implies ∂wξ I = 0 (instanton) or ∂ w ξ I = 0 (anti-instanton) in terms of w = x 0 + ix 1 . The equation ∂wξ I = 0 means that the map from the space-time CP 1 to the target space M is holomorphic.
Here we consider the case where the instanton number is integer. The case of fractional instantons is discussed in section 5.
Let [Z k ] (k = 1, · · · , N ) be the homogeneous coordinates of CP N −1 , and let w ∈ C ∪ {∞} = CP 1 be the usual coordinate of CP 1 . Then, holomorphic maps are given by where a k, are constants. The fact that the degree of the polynomial Q corresponds to the instanton number is easy to see by considering simple cases. Let us consider the case N = 2 and [Z 1 , Z 2 ] = [1, w]. This map is the identity map from the spacetime CP 1 to the target space CP N −1=1 . Therefore it gives the instanton number Q = 1. In the case [Z 1 , Z 2 ] = [1, w Q ], the spacetime CP 1 wraps Q times around the target CP N −1 , so the instanton number is Q. The general case is seen by using the invariance of the topological charge under continuous deformation, and also by using the embedding  When Q > 1, the number of the parameters of instantons of CP N −1 -model are much smaller than those of the Yang-Mills theory. The missing moduli parameters can still be described in algebraic geometry by using spectral cover construction [51] because CP 1 × E can be regarded as a (trivial) elliptic manifold. We do not discuss it in this paper, however.

Fractional instantons, vacuum structure, and center symmetry
Having studied the relation between instantons in 4d and 2d theories, we now study fractional instantons. In particular we show that such instantons restore the center symmetry of the system. Let (t, x) be the coordinates of R × S 1 C with x ∼ x + 1. The metric may then be taken to be where L C and L E are some length parameters. The multiplication of the term dt 2 by the same L 2 C as the one multiplying dx 2 is introduced just for later convenience. For the twisted compactification, we choose to use the symmetry Z N on the homogeneous coordinates is given by [· · · , Z k (t, x + 1), · · · ] = [· · · , e 2πik N Z k (t, x), · · · ] .
(5.2) This is precisely the boundary condition which played an important role in the analysis of the CP N −1 model [23,24,34]. For this reason let us start our analysis in the CP N −1 -model, and then subquently discuss the four-dimensional theory.

Fractional instantons and the vacuum structure in CP N −1 -model
Here we discuss the vacuum structure of the CP N −1 -model. First of all, let us discuss the candidates for the vacuum at the classical level in the presence of the twist (5.2). A minimal energy configuration should be constant on 2d space-time (t, x), and also invariant under the above twist. This requires e 2πik/N Z k = cZ k (k = 1, · · · , N ), where c is a constant independent of k which takes care of the fact that Z k are homogeneous coordinates and hence an overall constant is irrelevant. There are N solutions of these equations, which are given by These points are the fixed points of the twisting (5.2). These are the classical vacua. Thanks to the twisting, the fields have energy gap. For example, near the point P N , the fields are described by inhomogeneous coordinates U k := Z k /Z N with the boundary condition U k (t, x+1) = e 2πik N U k (t, x). This boundary condition forces U k to have a nonzero momentum in the direction S 1 C . Therefore, the vacua have energy gap already at the classical level, so there are no IR divergences. This makes straightforward perturbation theory possible around these vacua.
There are transitions among these classical vacua P k by fractional instantons and antiinstantons, as we now describe. First, let us identify R × S 1 C as S 2 = CP 1 by regarding the infinite future t = ∞ and infinite past t = −∞ as the north pole and south pole of S 2 = CP 1 , respectively. More explicitly, we take the coordinate w = e 2π(t+ix) ∈ C + {0} = CP 1 . (5.4) The point w = 0 corresponds to the infinite past t = −∞ and the point w = ∞ corresponds to the infinite future t = +∞. Then, as stated in the previous section, instantons are given by holomorphic maps from CP 1 to CP N −1 . However, we have to take into account the boundary condition (5.2) appropriately when we consider instanton solutions.
In the present case of twisted compactification, there exist fractional instantons (see [56][57][58][59][60] for early works). There are 1/N instantons connecting P k and P k+1 which we denote as I k→k+1 . They are shown in Figure 4 and 5 and given by I k→k+1 =(P k → P k+1 ) : [· · · , Z k (w), Z k+1 (w), · · · ] = [0, · · · , 0, a, w 1 N , 0, · · · , 0] , (5.5) where a is a constant complex moduli parameter of the 1/N instantons. Notice that the twisted boundary condition (5.2) is satisfied by these fractional instanton solutions. The loop x → x + 1 corresponds to w → e 2πi w, under which w 1/N → e 2πi/N w 1/N . By using the fact that homogeneous coordinates Z k are defined only up to overall multiplication by constants, one can check that (5.2) is satisfied. This really has 1/N topological charge, because by going to the N -covering space of CP 1 as w = w 1/N , the above solutions are [· · · , 0, a, w , 0, · · · ] which is the standard one instanton of CP N −1 . There are also −1/N anti-instantonsĪ k→k+1 which are obtained by replacing w → 1/w.
Note that the Atiyah-Hitchin-Singer theorem [55] predicts that there are 4N Q = 4 moduli fields for Q = 1/N instantons. Two of the four real parameters are provided by a. The other two may be provided by the moduli associated to translations along E as discussed earlier in the case of Q = 1. Actually, the 1 2πi log a also gives translation moduli along the direction R × S 1 C . Therefore, all the four moduli parameters of the Yang-Mills 1/N instantons are the translations on R × T 3 . 11 11 For some gauge groups G other than SU(N ), there exist fractional instantons on R × T 3 even without twisting. In those cases, the fact that the four moduli parameters are given by translations on R × T 3 can be seen by using string duality. See section 3.1 of [61] for these discussions. Figure 4. A fractional instanton for CP 1 -model interpolates between two different vacua, 0 and ∞ (the black dots represents singular points, as discussed in Figure 2). In cylindrical coordinate (so that the geometry is R × S 1 ) with fixed x, this moves along the R-direction from past to future infinities, as we vary the parameter t. The full trajectory, with both t and x varied, is a hemisphere, and when the two such hemispheres are combined we obtain a full sphere. This is a manifestation of the fact that two fractional instantons combine into a single instanton. Figure 5. The geometry is more complicated for N > 2 case than the N = 2 case in Figure 4. However, the idea is the same: the classical moduli space CP N −1 is lifted by twisted boundary condition along S 1 C into N different points P 1 , . . . P N , however tunneling between different these points, as given by fractional instantons, lifts the N -fold degeneracy, restoring the center symmetry dynamically. Now the true vacuum structure of the theory at the quantum level is described as follows. First of all, note that in the present case, the 1/N instantons do not have size modulus because all the four moduli are associated to translations on space-time R × T 3 . The absence of the size modulus is related to the fact that there are no zero modes in the twisted compactification used in this paper. Then, at least if the length scale of the compactification is small, the dilute gas approximation of fractional instanton/antiinstanton is valid. This is the crucial difference from the instanton computation on R 4 discussed in the Introduction.
Let H be the Hamiltonian of the system, and |P k be the classical vacuum staying at the point P k . Then the usual dilute gas computation of instantons would give the transition amplitudes P k+n |H|P k = C n (g, e iθ , L A , L B , L C , µ) e inθ N , (5.6) where C n (g, e iθ , L A , L B , L C , µ) is a function of the coupling g, the exponential of the theta angle e iθ , the lengths L A,B,C along S 1 A,B,C and the renormalization scale µ. The k and n should be considered mod N by regarding C n+N := C n e −iθ . This C n (g, e iθ , L A , L B , L C , µ) has perturbative as well as non-perturbative terms, and is expected to have the form of a resurgent function. The dependence of C n (g, e iθ , L A , L B , L C , µ) on the theta angle θ through e iθ is because one-instanton N k=1 I k→k+1 is a transition from one vacuum to itself. In the dilute gas approximation, we roughly get where M > 0 is given by a combination of the parameters (g, e iθ , L A , L B , L C , µ) with mass dimension one, while we neglect other C n (g, e iθ , L A , L B , L C , µ) for |n| ≥ 2. The minus sign was put to make the later results consistent with the general argument of Vafa and Witten [40], as was already mentioned around (1.3). By diagonalizing the Hamiltonian, we get the energy eigenstates and eigenvalues as There are N states, and the true vacuum is the one which minimizes this energy while the others are metastable. Let us notice several important properties of the above vacuum structure. First, the energies E e depend on e iθ/N , while the θ is supposed to be a periodic variable θ ∼ θ + 2π. They are consistent due to the existence of N (meta)stable vacua. For each e ∈ Z N , we have one vacuum. When the θ is adiabatically changed to θ + 2π, the vacuum |e goes to the vacuum |e − 1 , mapping one vacuum to another. This turns a metastable vacuum to a true vacuum, or a true vacuum into a metastable vacuum. This is a manifestation of the Witten effect [62], because our twisted boundary condition by the center symmetry is equivalent to including a unit 't Hooft magnetic flux m = 1 on S 1 B × S 1 C [63], and the e ∈ Z N can be identified with the 't Hooft electric flux, as will be seen in later discussions. In this way, the dependence on θ as e iθ/N is consistent with the periodicity. See [46] for recent related discussion of this type of Witten effect.
Next, in the large N limit and with finite e, the energy is proportional to These properties are exactly as expected from the large N analysis of CP N −1 -model and Yang-Mills theory, as mentioned already in section 1.2.
The results above are obtained in a small volume region where weak coupling analysis is reliable. However, the above vacuum structure such as E e ∝ cos[(θ − 2πe)/N ] was also obtained in a different regime where the volume of the space-time is infinite while there is a light adjoint Majorana fermion [45]. The dependence of E e on θ can have important implications for cosmology of (non-QCD) axions such as string axions, and hence it would be very interesting to investigate it further.

4d situation
The discussion of the previous subsection was for the CP N −1 model. Let us see what is happening in the 4d Yang-Mills theory. We introduce coordinates (x A , x B , x C ) related to the previously introduced coordinates as z = x A + τ x B and x = x C (where τ = iL B /L A ), and call the corresponding directions as A, B and C cycles, respectively.
We would like to identify the points P k given in (5.3) in terms of the 4d Wilson lines U A , U B , and U C , where these Wilson lines are computed along the A, B and C cycles. In the following, it may be convenient to consider the covering space R 3 of S 1 A × S 1 B × S 1 C and regard the gauge fields as living on this R 3 with certain periodic properties. The Wilson lines U A,B,C are defined by integration over 0 ≤ x A,B,C ≤ 1 as and similarly for U B and U C . Implementation of the twist is a little more complicated in the Yang-Mills [63]. (See [64] for a detailed explanation of mathematical background behind the following discussion.) We introduce a gauge transformation h(x B ) which has the property that Then, the gauge field is assumed to have the periodic boundary condition where we suppressed x A because it is irrelevant for the discussion of the nontrivial part of the periodic boundary condition. This periodicity property means in particular that This corresponds to the twisted boundary condition Z k (x C + 1) = e 2πik/N Z k (x C ) of the CP N −1 as can be seen from (3.20) and (3.29).
Another point which requires a care is the following. Under gauge transformation g on the space-time, the U C transforms as g −1 (x B , x C = 1)U C g(x B , x C = 0). However, the g also satisfies the boundary condition g( . This requires us to modify the definition of U C as Then the gauge invariant Wilson loops can be defined as tr U C . This new U C has a boundary condition as The reason for this is as follows. As mentioned in section 2.2 and reviewed in appendix A, the symmetry operators of the 1-form symmetry C(Y ) is defined on some codimension-2 surface Y . One way to implement the twisted boundary condition (which might look different from the implementation by (5.14) but is actually equivalent) is by inserting such an operator. In the present case, the Y extends in the direction R × S A while it is localized on a point pt ∈ S B × S C , so Y = pt × R × S A . Schematically we write this twist as C(R × S A ). Then the directions S B and S C are on the same footing in this implementation of the 1-form twist. In particular, when we move U C from x B to x B + 1, this Wilson line is crossed across the operator C(R × S A ). This crossing produces the phase factor e 2πi/N in the boundary condition of U C . Similarly U B gets the phase U B (x C + 1) = e − 2πi N U B (x C ) by crossing C(R × S A ). However, instead of C(R × S A ), we just use the boundary conditions (5.14) in this subsection. 12 Now let us restrict our attention to flat connections F = dA + A 2 = 0. In this case we have and setting U B := U B (0) and U C := U C (0), we get 63,64]. Up to conjugation (i.e., gauge transformation), there is a unique solution to the equation (5.19). This is because U C can be regarded as the "raising operator" of the eigenvalues of U B . The solution up to conjugation is given by (5.20) 12 Mathematically speaking, any realization of the twist is described by a nontrivial topology of the SU(N )/ZN bundle over S 1 B × S 1 C . The (5.14) is one way to realize such a nontrivial bundle. Another way to realize it is to consider a small patch near pt ∈ S 1 B × S 1 C , and take a nontrivial transition function around this point by using an element of π1(SU(N )/ZN ). These two are equivalent.
On the other hand, U A = U A (0) satisfies because the boundary condition in the direction x A is trivial (i.e., there is no twist in the S A direction). Then, there are N solutions for U A given by Notice that the gauge symmetry is completely broken by the Wilson lines U B and U C , because the subset of SU(N ) whose elements commute with both U B and U C is discrete. This makes all fields massive and there is no infrared divergence, and we are away from the singular points of the moduli space of flat connections on the two-torus E because W-bosons are massive.
The above Wilson lines correspond to the coordinates of CP N −1 as follows. In terms of φ introduced in (3.4), the above (U A , U B ) are given by where c j was introduced in (3.22), and ρ is the Weyl vector (i.e., half the sum of all positive roots) of SU(N ) with respect to the root lattice L (3.11): Now let us map the above φ to CP N −1 . First, by a computation as in Sec. 2.2 we get Next, let W ∈ S N be an element of the Weyl group such that it sends the i-th component to i + 1-th component. Then, one can see that W ( ρ) = ρ − N c 1 . Noticing the fact that the lattices e k + L are invariant under the Weyl group, we get and hence θ k ( ρ/N ) = 0 for k = 0 mod N . Therefore, we conclude that By the identification Z k = θ k , this is precisely the point P K introduced in (5.3); (5.28)

Dynamical restoration of the center symmetry
Now we can show that the center symmetry (i.e. dimensional reduction of 1-form symmetry) is unbroken. Because we are compactifying the theory on S 1 A ×S 1 B ×S 1 C , we get three center symmetries Z Thus, at each classical vacuum on P k , the center symmetry is broken by the VEV (vacuum expectation value) of the Wilson loop tr U A . However, we found that the true quantum vacua are linear combinations of |P k given in (5.8). For them, we get Therefore, the center symmetry is dynamically restored by 1/N fractional instanton/antiinstanton effects! The fact that the symmetry is restored itself is not so surprising because all the spatial directions are compact T 3 = S 1 A × S 1 B × S 1 C and a symmetry breaking does not usually happen in quantum mechanics. However, we stress that the "strong dynamics" properties of confinement such as the center symmetry restoration and the dependence of the vacuum energy on θ are realized completely in weak coupling regime. Because of this, we conjecture that the vacua discussed in the previous subsection are adiabatically continued when we go from small to large volume limit.
Although the center symmetry is unbroken, the vacua have charges under Z (A) N . The vacuum states go to itself under the action of the symmetry and hence all operators having nonzero charge under it have vanishing VEVs. But there can be a possible phase factor on the action of the symmetry operator on the vacua. To see this, let C A be the generator of the center symmetry Z (A) acting on the Hilbert space. 13 From (3.28), it acts as C A |P k = |P k−1 . 14 Then we find that the basis |e diagonalizes the electric one-form center symmetry C A (hence the notation e for "electric"): (5.32) 13 In the description C(Y ) in appendix A, CA is defined by choosing Y which is extending in the direction SB × SC while localized in the direction R × SA. So schematically we can write CA = C(SB × SC ). 14 In principle, we can have a nontrivial phase factors here as CA|P k = η k |P k−1 for |η k | = 1. By successive redefinitions of states |P k , we can assume η k = 1 for k = 0 mod N . Furthermore, the fact that the symmetry is ZN implies that (CA) N = 1, and then we get N k=1 η k = 1. From these facts, we can take all η k to be 1. Once we fix the phase factors of the states |P k in this way, there is no more freedom to change them except for the overall factor. Then the phase e iθ/N appearing in (5.6) is well-defined, and hence θ has a physical meaning. However, there is a freedom to change the definition of CA as C A = e 2πi/N CA and |P k = e −2πik/N |P k . They still satisfy C A |P k = |P k−1 , but we get the change θ → θ = θ + 2π. Of course, this is the usual periodicity of θ. Therefore, the different vacua has different charges under C A . The e ∈ Z N which determines the eigenvalues of C A is the 't Hooft electric flux [63,64].
We stress again that the center symmetry is still preserved, because C A |e ∝ |e and hence they define the same ray in the Hilbert space (i.e., the same physical state). So, the center symmetry is unbroken at each vacuum. By using (5.31), one can see that we can move between these vacua by the action of the operator tr U A , This is consistent because U A is charged under C A .

Borel plane of QFTs and Linde's problem
Having constructed a setup where IR divergence is properly taken into account and the center symmetry ensured, we now have a setup where the resurgence program can be applicable, at least in principle. While the detailed analysis of such an analysis is left for future work, let us here point out an important subtlety: we demonstrate that there can exists singularities in Borel plane in QFT which depend on the shapes of manifolds. Such manifold-dependent singularities could in some cases dominate over renormalons, and hence dramatically change the singularity structure of the Borel plane of a compactified theory. This dependence is on the shapes of manifolds, so they survive even in the large volume limit. This might suggest that there is no possibilities for resurgence directly on R 4 for Yang-Mills theory, and we always have to start from a compactified geometry without IR problems.

Quantum mechanics
That we have manifold-dependent singularities in the Borel plane is a general phenomenon.
To explain this, let us start with 1d quantum mechanics with target space M. The Lagrangian is where h IJ is the metric on the target space M, and g 2 QM is the coupling constant (which is usually called ). The canonical momentum is where ∇ I is the covariant derivative on the target space M with respect to the metric h IJ . The Hamiltonian is therefore given by where ∆ = −∇ I ∇ J is the Laplacian.
Let λ be the -th eingenvalue of the Laplacian ∆, and let n be the multiplicity of that eigenvalue. Then the partition function of the theory on S 1 with circumference 2πR is given by Asymptotic expansion. Now we restrict our attention to the case M = CP 1 = S 2 which has the unit radius. We want to perform perturbative expansion of Z in terms of the coupling constant. In this case, we know that the eigenfunctions of the Laplacian ∆ are given by spherical harmonics and the eigenvalues and their degeneracies are given by where = 0 is the ground state and ≥ 1 are excited states. Now we can expand the partition function Z by using the Euler-Maclaurin formula. If f (x) is an analytic function, we have asymptotic expansion given by where f (2m−1) means (2m−1)-th order differential of the function f . The B 2m are Bernoulli numbers and can be expressed as Note that ζ(s) → 1 as s → ∞. Thus these formulas give the asymptotic behavior of B 2m . Now we can rewrite the partition function of CP 1 = S 2 quantum mechanics. We set Then we get and Thus we get Let us study the asymptotic behavior. The coefficient grows as where C m → 1/ √ 4πm. Therefore, a crude asymptotic behavior of Z is given as This has a singularity on the positive real axis of the Borel plane with the associated ambiguity of order exp − π Rg 2 QM . (6.14) Assuming that resurgence works, this implies that the path integral of this quantum mechanics has unstable saddle points with the classical action π/Rg 2 QM . Interestingly, we can indeed find such a saddle point.
Classical saddle points. In the path integral with the Lagrangian (6.1), there are saddle points which are given by geodesics on the target space M. Let us consider the simple case M = CP 1 = S 2 . Let (θ, φ) be the polar coordinates of M = S 2 . Then, classical solutions of equations of motion are geodesics wrapping great circles of S 2 . For example, we have where n ∈ Z. The action can be easily evaluated. The n = 0 is the lowest stable solutions. For n = ±1 it is given as This is exactly as expected from the above analysis of the asymptotic expansion and the singularities of the Borel plane. Notice that this action explicitly depends on the parameter R which specify the manifold S 1 .
Twist. We can also introduce a twist in the partition function above. Explicitly, we can impose the boundary condition such as (θ, φ) (t+2πR) = (θ, φ + π) (t) . (6.17) Then the classical solutions (6.15) still exists, but now with the condition that n ∈ 1 2 + Z. The stable lowest action solutions are θ = 0, π. Therefore, the above solutions are unstable. The order of the classical action is the same up to numerical constants. The analysis of the asymptotic behavior of the perturbative expansion should also be straightforward, but we do not perform that explicitly.

2d sigma models
Having established that for 1d quantum mechanics the asymptotic behavior is reproduced from corresponding unstable saddle points, we can easily see that there are singularities on the Borel plane of 2d sigma models.
The Lagrangian for the 2d theory is Now let us consider partition function of this theory on the spacetime T 2 = S 1 × S 1 , where one S 1 has radius R 1 and the other S 1 has radius R 2 ; in this section we compactify the temporal direction of the 2d sigma models. If R 1 R 2 , we may first dimensionally reduce the theory to the quantum mechanics discussed above. The coupling constants are related as The radiative corrections from the Kaluza-Klein modes are higher orders of g 2 2d . Then, in the dimensionally reduced theory, we can use the above results about quantum mechanics.
Let us consider the case M = CP 1 = S 2 . The classical action obtained above is given by (6.20) From this, we conclude that the Borel plane contains a singularity which depends on the shape R 1 /R 2 of the manifold. Notice that when R 1 /R 2 1, this pole is very close to the origin than renormalons and instanton-anti-instanton pairs.
In the large volume limit we expect to recover the theory on flat space R 2 . In particular, we expect that it is sensitive only to the volume R 1 R 2 of spacetime, and not to the shape R 1 /R 2 if the theory has a mass gap. 15 Nevertheless, the above-mentioned singularity of the Borel plane with the corresponding action (6.20) still exists. This suggests that the structure of the Borel plane is more complicated than is usually thought in this IR regulated situation.
On the other hand, if we take then the partition function gives the thermal free energy with temperature T = (2πR 1 ) −1 .
In this limit, the action (6.20) becomes extremely small and hence the singularity on the Borel plane merges into the origin. This is a manifestation of the Linde's problem in thermal free energy in the present case of 2d CP 1 sigma model. However, we stress that if we can complete the resurgence program of QFT, we can still get a sensible answer. Let us demonstrate this point by using a toy function. Consider This has an asymptotic expansion given roughly as Naively from this perturbative expansion, it seems hopeless to get a sensible answer in the limit R 2 → ∞. In the notation of the subsection 1.2, we identify β = T −1 → 2πR 1 and m IR → R −1 2 , so the IR cutoff m IR goes to zero as R 2 → ∞. However, the integral in (6.23) makes perfect sense in the limit R 2 → ∞ with a finite value for the toy thermal free energy. Of course, the actual partition functions of QFT's are more complicated than the above toy function. But the point here is that Linde's problem can in principle be overcome by resurgence.
Twist. Suppose that we introduce the twisted boundary condition on the S 1 with radius R 2 . If R 1 R 2 , the result is qualitatively similar to the one above; we have a singularity on the Borel plane which depends on R 1 /R 2 . However, in the region R 1 R 2 , the theory has a mass gap of order R −1 2 which is large, and hence there is no IR divergences even in the limit R 1 → ∞. Thus we expect that there is no problem in resurgence in this limit. This is the situation studied in e.g., [23,24,31,[65][66][67][68] and also in this paper.

Speculation on 4d Yang-Mills
We would like to make speculative comments on the structure of the Borel plane and Linde's problem in 4d Yang-Mills. By a compactification on small T 2 and considering only flat connections, we have seen that the 4d SU(N ) Yang-Mills is reduced to 2d CP N −1 at a qualitative level. 16 Then, we expect that the 4d Yang-Mills encounters all the phenomena discussed above for CP 1 .
For example, we can find a classical solution of the Yang-Mills equation on a torus. We consider the 4-manifold T 4 = S 1 T × S 1 A × S 1 B × S 1 C with the circumference of each circle given by L T , L A , L B and L C . We take the metric as Thus it satisfies the Yang-Mills equations. Notice that this configuration a = 2πx C dx B is similar to the classical solutions in CP N −1 given above via the relations between Yang-Mills and CP N −1 model given in (3.4) and (3.13).
With the twist on S 1 B × S 1 C , we still have solutions such as iA = iA µ dx µ = diag(a, , 0, · · · , 0) − 1 N diag(a, a, · · · , a) (6.29) with the above a. These classical solutions have actions of order up to order 1 factor. These actions explicitly depend on the shape of the 4-manifold even in the large volume limit. We have not analyzed the asymptotic behavior of the perturbative expansion of the free energy, but the existence of these unstable solutions suggests that the structure of the Borel plane is more complicated that is usually thought. If L T L A L B L C , this classical action becomes arbitrary small. On the other hand, if L T L A L B L C , it depends on whether there is the twist or not. Without the twist, we can have a similar solution a = 2πx T dx A with the action proportional to (L B L C )/(L T L A ). So the situation is parallel to the above case. With twist, such a solution is forbidden basically because the twist generates a mass gap determined by L B,C and there is no IR 16 Quantitatively, the classical action of the 4d Yang-Mills gives a metric of CP N −1 which is different from the standard Fubini-Study metric. Also, there are radiative corrections from W-bosons. It would be interesting to perform detailed computations on these points. divergences in the limit L T,A → ∞. Thus there is no classical saddle points with arbitrarily small action.
The value of the action (6.30) is very large in the parameter region L T → ∞ which we have studied in this paper, and hence the above unstable saddle point is unimportant for the resurgence analysis at zero temperature L T → ∞. By contrast, when we decompactify the spatial regions L A,B,C → ∞ with fixed ratios and constant finite L T , then the action becomes arbitrarily small. This is a manifestation of Linde's problem of 4d Yang-Mills and this problem should be overcome by resurgence as discussed above.

A One-form global symmetry
Let us quickly review the concept of the one-form global symmetry [48,69]. The usual (i.e., 0-form) symmetries act on local operators, while 1-form symmetries act on line operators such as Wilson loop operators. More explicitly, they can be described as follows. The usual 0-form symmetry for a continuous symmetry can be described by picking up a codimension-1 submanifold Y inside a spacetime X. By integrating a current j = j µ dx µ on Y we get a charge as Q(Y ) = Y * j. For discrete symmetries, there is no current operator, but we can still consider charge operators Q(Y ) defined on Y . Conservation of the charge is translated to the statement that this operator Q(Y ) only depends on the topology of Y outside the locus where other operators are inserted. If we cross a local operator across Y , then that operator is transformed by Q(Y ). We can do similar things for p-form symmetries. Pick up a codimension-(p + 1) submanifold Y and define a charge operator Q(Y ). Operators charged under them are extended in p dimensional manifold Z. We denote that operator as O(X). If the Z is crossed across Y , the operator under consideration is changed by the charge operator Q(Y ).
For example, if Z ∼ = R p is a flat subspace inside the total spacetime X = R d , and Y ∼ = S d−p−1 is a sphere surrounding Z where d = dim X, then we get where L(Y, Z) is the linking number of Y and Z. For example, in the case discussed above, namely Z ∼ = R p and Y ∼ = S d−p−1 is a sphere surrounding Z, we get L(Y, Z) = 1.