The gaugino condensate from asymmetric four-torus with twists

We calculate the gaugino condensate in $SU(2)$ super Yang-Mills theory on an asymmetric four-torus $\mathbb T^4$ with 't Hooft's twisted boundary conditions. The $\mathbb T^4$ asymmetry is controlled by a dimensionless detuning parameter $\Delta$, proportional to $L_3 L_4 - L_1 L_2$, with $L_i$ denoting the $\mathbb T^4$ periods. We perform our calculations via a path integral on a $\mathbb T^4$. Its size is taken much smaller than the inverse strong scale $\Lambda$ and the theory is well inside the semi-classical weak-coupling regime. The instanton background, constructed for $\Delta\ll 1$ in arXiv:hep-th/0007113, has fractional topological charge $Q=\frac{1}{2}$ and supports two gaugino zero modes, yielding a non-vanishing bilinear condensate, which we find to be $\Delta$-independent. Further, the theory has a mixed discrete chiral/$1$-form center anomaly leading to double degeneracy of the energy eigenstates on any size torus with 't Hooft twists. In particular, there are two vacua, $|0\rangle$ and $|1\rangle$, that are exchanged under chiral transformation. Using this information, the $\Delta$-independence of the condensate, and assuming further that the semi-classical theory is continuously connected to the strongly-coupled large-$\mathbb T^4$ regime, we determine the numerical coefficient of the gaugino condensate: $\langle 0| \mbox{tr}\lambda\lambda|0\rangle=|\langle 1| \mbox{tr}\lambda\lambda|1\rangle|=32\pi^2 \Lambda^3$, a result equal to twice the known $\mathbb R^4$ value. We discuss possible loopholes in the continuity approach that may lead to this discrepancy.


Introduction
Strongly coupled gauge theories have been under intense study over the past few years, thanks to the recent developments of generalized global symmetries [2]. These are operations that implement the group multiplication laws via topological constructions such that the symmetry operations are supported on topological surfaces that are robust under small deformations. An ordinary 0-form symmetry G, which acts on point-like particles, is implemented by operators supported on co-dimension 1 surfaces. These surfaces obey the group-multiplication laws via fusion rules and give rise to phases valued in G when they cross the charged objects. Likewise, a 1-form symmetry acts on 1-dimensional objects and is implemented via co-dimension 2 surfaces. For example, Wilson lines in 4-D SU (N ) pure or super Yang-Mills theories are charged under Z (1) N 1-form symmetry. The 1-form symmetry is implemented via topological 2-dimensional surfaces that obey the Z N group multiplication laws and give rise to Z N phases when they link with Wilson lines. Like ordinary 0-form symmetries, 1-form symmetries organize the spectrum of a theory into representations, satisfy Ward identities, and may become anomalous if one tries to gauge them. In particular, 't Hooft anomaly matching conditions (or 't Hooft anomalies for short), which impose stringent constraints on quantum field theory (QFT), can be generalized to include anomalies of 1-form symmetries. Detecting the anomaly in 4-D requires defining a given QFT on manifolds with nontrivial 2-cycles, the typical example being the 4-torus T 4 . Generalized 't Hooft anomalies provide a framework for classifying QFT and its phases. Many recent works entertained this generalized framework to shed light on a plethora of asymptotically free gauge theories, including vector-like [3][4][5][6][7][8][9][10] and chiral theories [11,12]. Assuming a gauge theory generates a mass gap in the infrared (IR), one way to match its 't Hooft anomalies is via the formation of condensates. A famous example, and the subject of this paper, is the formation of gaugino condensates in N = 1 SU (N ) super Yang-Mills theory (SYM), which are needed to match a generalized 't Hooft anomaly. This anomaly can be understood as follows. In addition to the Z (1) N 1-form symmetry that acts on the Wilson lines, SYM enjoys a Z dχ 2N 0-form global discrete chiral, or R, symmetry. We may gauge the 1-form symmetry by turning on a background gauge field of Z (1) N , which is done either by coupling SYM to a Z N TQFT [13] or by activating a 't Hooft flux [14,15]. The latter is a field configuration on T 4 that carries a fractional flux B ∈ H 2 (T 4 , Z N ) (i.e. the flux piercing 2-cycles obeys the quantization rule T 2 ⊂T 4 B ∈ 2πZ N ) and fractional topological charge Q = 1 8π 2 T 4 B ∧ B ∈ Z N . In mathematical language, we consider the P SU (N ) ≡ SU (N )/Z N bundle instead of the SU (N ) bundle. The former has a non-trivial Brauer class w ∈ H 2 (BP SU (N ), Z N ) that obstructs the lifting of P SU (N ) to SU (N ), and we set B = w. In practice, the P SU (N ) flux is turned on by imposing twisted boundary conditions on the fields that live on T 4 . Next, we examine the partition function Z[P SU (N )] of SYM in this background by performing a Z dχ 2N rotation. We find Z[P SU (N )] → e i 2π N Z[P SU (N )], and thus, the theory stops being invariant under the action of Z dχ 2N once we gauge the 1-form symmetry. This is the sought generalized 't Hooft anomaly-a mixed anomaly between the Z (1) N 1-form symmetry and the Z dχ 2N 0-form discrete chiral symmetry. The anomaly indicates that the IR theory cannot be trivially gapped. Assuming confinement, the theory breaks its Z dχ 2N symmetry and forms N vacua separated by domain walls. The order parameter of Z dχ 2N symmetry is the bilinear gaugino condensate trλ 2 or higherorder condensates (trλ 2 ) n , n > 1 and n mod N = 0. The existence of N vacua is also in accordance with the Witten index [16].

Gaugino condensate: a historical prelude
Generally speaking, the matching conditions can only provide kinematical constraints and do not, by themselves yield insights into the details of the IR dynamics of a gauge theory. One needs an extra guide if, at all, there is a hope to understand the dynamics, e.g., the condensates. Thanks to supersymmetry, such studies are possible in SYM. These date back to the early eighties [17][18][19][20][21] (see [22][23][24][25][26][27] for reviews). The condensate calculations are based on the Belavin-Polyakov-Schwartz-Tyupkin (BPST) instanton calculus [28]: BPST instantons are (anti) self-dual Yang-Mills configurations that violate the non-renormalization theorems of SYM and hence, give a non-zero vacuum expectation value to the condensates. On dimensional-analysis grounds one can write trλ 2 16π 2 = cΛ 3 , where Λ is the strong-coupling scale and c is a numerical factor. The exact value of c was a controversial issue that caused many debates in the past. Generally, there are two methods to compute the bilinear gaugino condensate in 4-D SYM: the strong-coupling and the weak-coupling instanton methods.
In the first method, we start directly from the 4-D SYM in its strong-coupling regime and do instanton calculus, as in [17,18]. A single SU (N ) BPST instanton carries integer topological charge Q ∈ Z, and the configuration with the lowest topological charge Q = 1 admits 2N gaugino zero modes. The saturation of the zero modes in the Q = 1 instanton background gives a nonzero value to the 2N -point function (trλ 2 ) N , from which one can naively extract the value of the 2-point function (trλ 2 ) = (trλ 2 ) N 1/N . A detailed calculation, keeping track of all numerical coefficients, gives (trλ 2 ) = 2((N − 1)!(3N − 1)) −1/N (16π 2 Λ 3 )e i 2πk N , with k = 0, 1, ..., N − 1. The complex phase results from taking the N th root of unity, in accordance with the expectation that the theory admits N distinct vacua needed to match the generalized 't Hooft anomaly.
In the weak-coupling instanton method, we consider super QCD with N − 1 fundamental flavors Φ i , i = 1, .., N − 1, where Φ i is a chiral superfield, and give all the flavors small masses m. We work in the limit |Φ i | Λ Q , where Λ Q is the strong scale in the presence of quarks. Since there are N − 1 flavors, the gauge group fully abelianizes and we are well inside the weak-coupling regime. The total superpotential of this theory takes the from W = m i jΦ j Φ i + Λ 2N +1 Q Det(ΦΦ) , where the second term is the Affleck-Dine-Seiberg (ADS) superpotential [20]. The ADS term is nonperturbative in nature and is based on holomorphy and the symmetry structure of super QCD. It also results from saturating the quarks' zero modes in the BPST instanton background (the numerical coefficient was obtained 1 in [29], and corrected in [30]). Since we are in a weak-coupling limit, the instanton calculations are reliable. Minimizing the energy, we obtain the su- Finally, we substitute this result back into W to find W = N Λ 2N +1 Q Detm 1/N . We then decouple the quarks by taking m Λ Q , thus, leaving the weak-coupling regime. Using holomorphy, we can write W eff = N Λ 3 , where Λ is the strong scale at the mass threshold, and it exactly coincides with SYM strong scale at the decoupling limit. Recalling that one can write the holomorphic strong scale as Λ = µe 2πiτ /3N , with τ = 4πi g 2 h (µ) (g h is the holomorphic gauge coupling, running at one loop only and µ is some arbitrary energy scale) and that trλ 2 = −8πi ∂W eff ∂τ , one obtains trλ 2 = 16π 2 Λ 3 e i 2πk N in the k-th vacuum. 2 Having two different methods that yield two different answers resulted in many debates in the literature about the validity of both methods. It was earlier understood that the strong-coupling instanton method is in tension with the cluster decomposition principle (CDP). Consider the correlator trλ 2 (x)trλ 2 (x ) . In the limit |x−x | → ∞ we expect trλ 2 (x)trλ 2 (x ) = trλ 2 2 . However, since a BPST instanton cannot saturate 2 gaugino zero modes, one finds trλ 2 = 0, contradicting CDP. A possible resolution of this puzzle was proposed in [32]. It was hypothesized that SYM admits an extra phase with vanishing bilinear condensate trλ 2 = 0 and that averaging over the chirally symmetric and non-symmetric phases gives the result of the strong-coupling instanton method. This hypothesis was carefully examined in [33] by considering a softly broken N = 2 Seiberg-Witten theory down to SYM. It was shown that the chirally-symmetric phase is absent, casting doubt on the averaging hypothesis. Further considerations in [34] excluded the symmetric vacuum. We also note that the anomaly-matching argument of [35] also precludes such a phase. It was further shown in [36] that strongcoupling multi-instanton calculations are inconsistent with CDP.
Extra support to the weak-coupling calculations came from studying SYM on L is a small spatial circle with circumference L (N Λ) −1 , and both gauge fields and gauginos are given periodic boundary conditions on S 1 L [31,37]. Compactification over a small circle causes this theory to fully abelianize and enter its weakly-coupled regime. 3 The theory on R 3 × S 1 L admits monopole-instantons, the microscopic constituents of calorons. 4 At the supersymmetric vacuum, the monopoles have topological charges of 1/N . Using the index theorem, we deduce that a single monopole can saturate 2 gaugino zero modes giving rise to the bilinear condensate. Detailed calculations give trλ 2 16π 2 = Λ 3 , the exact same result from 4-D weak-coupling instanton calculations. The advantage of the compactified theory over the 4-D theory at weak coupling is that in the former, one can understand the dynamics responsible for the condensate formation without relying on the miracle of holomorphy. As a bonus, the proliferation of the monopoles causes the compactified theory to generate a mass gap and confine, a result that is prohibitively difficult to understand in 4-D.
The continuity of confinement and condensate between the small and large L limits may imply that the fractional instantons are responsible for the dynamics on R 4 , even though one may not make analytical sense of them in a strongly-coupled setup. The 3 The 4-D coupling constant on R 3 × S 1 L ceases to run at scale ∼ 1/N L, roughly the W -boson mass. Taking L (N Λ) −1 , we stay in the weakly-coupled regime. 4 A SU (N ) caloron with a unit topological charge is composed of N monopole instantons. Calorons are Yang-Mills (anti) self-dual configurations defined on R 3 × S 1 with integral topological charges and non-trivial holonomy (the expectation value of the Polyakov's loop) along S 1 . A BPST instanton, in contrast, (or more precisely, the Harrington-Shepard solution [38] defined on R 3 × S 1 ) has a trivial holonomy. Calorons were discovered by Kraan and van Baal [39] using the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction [40] and independently by Lee and Yi [41] and Lee and Lu [42] in the context of D-branes. Let L and P be the size of S 1 and the holonomy (in units of 1/L). If LP 1, then the monopole constituents are well-separated in space, and one can make sense of them in a semi-classical treatment, as in the case of SYM on R 3 × S 1 L . In particular, in the supersymmetric vacuum, which preserves the 0-form Z N center symmetry of the theory, the constituent monopoles are of equal action S = 8π 2 N g 2 . In the opposite limit, LP 1, the monopoles hide inside the caloron core.
continuity conjecture was taken seriously over the past decade in supersymmetric and nonsupersymmetric theories; see [43,44] for reviews. The conjecture withstood many tests, but we are still far from a firm conclusion about the role of fractional microscopic objects in strongly-coupled phenomena. This continuity was mainly tested on R 3 × S 1 L , and one wishes to examine whether it holds in other geometries.
One such geometry is T 4 , very natural from the point of view of lattice practicalities. Right after 't Hooft presented his twisted solutions (solutions with twisted boundary conditions, i.e. P SU (N ) bundle solutions, with fractional topological charges) [15], van Baal studied their mathematical properties [45,46]. Later, it was argued in a series of works [47][48][49], see also the review [50], that (anti) self-dual 't Hooft fractional instantons can be seen in realistic simulations of 4-D pure Yang-Mills theory and that such configurations could be utilized to explain confinement. 5 Further, it was argued in [53], via extended lattice simulations, that 't Hooft fractional instantons on T 4 are ultimately connected to monopole instantons in the infinite 3-volume limit and finite time 6 direction. In particular, it was shown that an exact SU (2) caloron solution with a unit topological charge and equal-action constituent monopoles (see Footnote 4) can be constructed on the twisted T 4 by gluing two twisted solutions, each carrying Q = 1 2 charge, along the space directions. 7 1.2 Gaugino condensate on asymmetric T 4 : a summary of the procedure and results The possible connection between 't Hooft's solutions and monopole-instantons [53] calls for a serious examination of this finding. Since both the R 3 ×S 1 L monopole-instanton and 4-D weak-coupling instanton methods give the same gaugino condensate, one wonders whether the same result can be obtained in the 't Hooft flux background. 8 This is especially timely after the advent of the new generalized anomalies, which can be easily detected when a QFT is put on T 4 with a P SU (N ) bundle (or 't Hooft twisted boundary conditions). Recently,Ünsal advocated that a refinement of the instanton sum in the partition functions of SU (N ) theories has to be considered: while fractional instantons of P SU (N ) bundles contribute to observables like the gaugino condensate and vacuum energy, the sum over the fractional objects has to be constrained to yield the integer topological charges of the SU (N ) bundle [55] (see also [52] for an earlier assertion).
In this paper, we put this proposal under scrutiny and perform detailed calculations of the gaugino condensate on T 4 and a P SU (2) bundle. 9 Our conclusion is that the condensate in any of the two vacua is given by 0|trλ 2 |0 = | 1|trλ 2 |1 | = 32π 2 Λ 3 , with a coefficient that is twice that obtained via the weak-coupling instanton method on R 4 and the semi-classical calculations on R 3 × S 1 L . The extra factor of 2 is unexpected and calls for further serious examinations of the role of the P SU (N ) bundles in SU (N ) gauge theories and of the continuity conjecture.
The calculations that lead to this puzzling result are surprisingly rich. The simplest SU (2) pure Yang-Mills (i.e. Yang-Mills equations with zero source) fractional instanton with topological charge Q = 1 2 and action S 0 = 4π 2 g 2 was constructed by 't Hooft in his seminal work [15]. This solution is abelian in nature, i.e. the gauge field components are along the Cartan direction. The solution must be (anti) self-dual, otherwise, the fluctuations in this background will have negative modes signaling instability. Let L 1 , L 2 , L 3 , L 4 be the lengths of the periods in T 4 . Then, the self-duality of the abelian solution is guaranteed if and only if L 1 L 2 = L 3 L 4 . We dub the T 4 that obeys this relation as the self-dual torus, the simplest one being the symmetric T 4 with L 1 =L 2 =L 3 =L 4 . Given the simplicity of the abelian solution on the symmetric T 4 , one is tempted to use it as a source for the gaugino condensate. According to the index theorem, this background must saturate 2 gaugino zero modes. Nevertheless, the direct solution of the Dirac equation yields 6 zero modes (four "undotted" and two "dotted" ones). To make things worse, the extra zero modes source the super Yang-Mills equations of motion, hinting that such an abelian solution might not be a consistent background. The resolution of both puzzles could have been achieved had we been able to show that higher-order corrections (including loops) lift the extra gaugino zero modes. Although we believe that this should be the case, we were not able to show it in a satisfactory way.
In order to circumvent this difficulty, we chose to depart from the symmetric selfdual to the non-self-dual torus. This is both a blessing and a curse. The curse is the technical difficulty of the problem, while the blessing is that the asymmetric torus enables us to take various interesting limits, e.g. R × T 3 and R 2 × T 2 . If the relation L 1 L 2 = L 3 L 4 is violated, the abelian solution must be modified to include non-abelian pieces, ensuring that the solution persists to be (anti) self-dual. To date, there exists no analytical solution with Q = 1 2 on T 4 with arbitrary shape. However, a systematic procedure to deal with non-self-dual tori was developed in [1]. This method gives an approximate analytical self-dual solution, with charge |Q| = 1 2 and action S 0 = 4π 2 g 2 , as an expansion in a "detuning" parameter ∆ ≡ ( Solving the Dirac equation on a small asymmetric T 4 with ∆ 1 and limiting our explicit treatment to the lowest order in ∆, we find exactly 2 gaugino zero modes as per the index theorem. We also verified that the explicit solutions of the fermion zero modes are consistent with supersymmetry transformations. These modes do not source the Yang-Mills equations of motion, implying that our approximate solution is a consistent background. The gauge field fluctuations admit 4 bosonic zero modes (also in accordance with the index theorem), interpreted as translational moduli. Supersymmetry guarantees the cancelation between the bosonic and fermionic excited states, leaving only the bosonic and fermion zero mode integrals to deal with.
The contributions of zero modes to the path integral are taken into account using the method of collective coordinates, and therefore, we need to integrate over the moduli space M. To correctly identify the shape and size of M, we carefully examine all gauge invariant observables in the background of the fractional instanton on T 4 . This includes both local gauge-invariant densities and Wilson loops. While gauge-invariant densities are invariant under translations over a period on T 4 , a Wilson loop acquires a Z 2 phase. 10 Therefore, we find M ∼ T 4 with double the periods of the physical torus to account for the gauge inequivalent classes. This result is further supported by investigating the Hamiltonian formalism of the theory. Interestingly, the metric on M is found to be proportional to the background action and is, hence, independent of ∆. Putting the pieces together, we finally obtain trλ 2 = 64π 2 Λ 3 , with a coefficient four times the expected number on R 4 .
To understand the significance of this result, we need to interpret the expectation value trλ 2 using the Hamiltonian formalism. Due to the mixed discrete chiral/oneform center anomaly, the energy eigenstates are doubly degenerate in the background of a 't Hooft flux on any torus size: there are 2 vacua that are exchanged under the operation of chiral transformation, which explains the extra factor of 2, while the relative phase in the condensate in the two vacua is compensated by the 't Hooft twist in the Euclidean time direction [57]. Thus, restricting the condensate to one vacuum, upon taking the limit of large T 4 , we obtain 0|trλ 2 |0 = | 1|trλ 2 |1 | = 32π 2 Λ 3 . This coefficient is twice the known value on R 4 (using the weak-coupling instanton method) or on R 3 × S 1 L . We emphasize that our calculations are performed on a small T 4 , compared to the strong scale of the theory, and thus, we are well inside the semi-classical regime. The calculations in this regime are under analytic control, thanks to the smallness of the coupling constant. However, to make a connection to the result on R 4 , we made a few assumptions. The invalidation of any of these assumptions can explain the discrepancy between our result and the expected one. We now list the assumptions that led to our result: 1. We assume that there is a unique fractional instanton on the asymmetric T 4 with topological charge Q = 1 2 . This solution is nonabelian in nature and is obtained from 't Hooft's abelian solution on a symmetric T 4 as an expansion in the detuning parameter ∆ [1]. However, we were not able to prove the uniqueness 11 of the solution. Instead, we rely on the numerical evidence in [1].
2. Another lingering issue is the radius of convergence of the expansion, which is yet to be understood. Although our final result does not depend on ∆, one needs to be cautious when interpreting the result. In order to interpret the result and compare with the condensate on R 4 , we had to make use of the Hamiltonian formalism. Here, first, one puts the theory on a small spatial torus T 3 , with a 't Hooft twist, and with periods of length L Λ −1 , and then takes the limit of a large Euclidean time direction, L 4 Λ −1 . This is the limit ∆ ∼ L 4 L 1, well outside the small-∆ regime used to compute the condensate.
3. After taking ∆ large, one needs to take the size of the remaining T 3 periods beyond the inverse strong-coupling scale. Thus, we necessarily leave the semiclassical regime, yet we assume no extra instantons contribute to the condensate in this limit. (However, we can not help but note that this is similar, at least in spirit, to the extrapolation of the R 3 × S 1 L semiclassical calculation of the gaugino condensate to R 4 .) 4. In carrying out our calculations, we assume that quantum corrections due to bosonic, fermionic, and ghost fluctuations cancel exactly in our supersymmetric 11 To avoid possible confusion, we note that the series in ∆ determines a unique self-dual configuration for fixed values of the moduli (see the proof in Appendix A.1, near eqn. (A.31)). The possibility of nonuniqueness mentioned here refers to existence of a genuinely nonabelian Q = 1 2 solution not connected to 't Hooft's abelian solution. The numerical study of [1] appears to support uniqueness in this sense, but does not constitute a proof.
background, to all loop orders, and for arbitrary T 4 size. Although the cancellation of the determinants to any loop order is well-established in R 4 , one may need to examine this assumption more carefully in the fractional instanton background of T 4 .
In view of the above discussion, in the bulk of the paper, we present our calculations in sufficient detail to help the interested readers follow all numerical factors and enable them to dwell on the procedures used-and on improving the interpretation of our result.

Future directions
Here, in lieu of a conclusion, we point out that our study warrants a few expansion directions: 1. An immediate step would be generalizing the SU (2) result to SU (N ≥ 3). Selfdual instantons with topological charge 1 N , N ≥ 3, are necessarily non-abelian solutions of Yang-Mills equations [15]. This makes the treatment more involved, especially as we deviate from the self-dual torus. 12 Fortunately, a systematic analysis to deal with this problem, a generalization of the method in [1], appeared in [58]. One can follow the same line of thought in our work to compute the condensate in SU (N ).
2. It is tempting to study the condensates in other gauge groups, e.g., Sp(N ) and Spin(2N ) groups. These groups have small center groups: Sp(N ) has a Z 2 and Spin(2N ) has a Z 4 or a Z 2 × Z 2 center group depending on whether N is odd or even. Owing to the small centers, 't Hooft fluxes (or fractional instantons) of these groups will, in general, support more than 2 gaugino zero modes. This situation is different from the calculations on R 3 × S 1 L , where a monopole-instanton always supports 2 zero modes for all gauge groups, even those with no center symmetry. Therefore, at first, it is not clear how the twists and monopoles are connected. Investigating this problem is important for the continuity program.
3. It is also interesting to carry out our procedure to higher orders in ∆. The convergence of the series in ∆ may shed more light on the problem of continuity.
In this regard, we note that the closely related work [59] in the two-dimensional abelian Higgs model was able to carry the ∆-expansion to 51-st order, with the results indicating convergence to the infinite volume limit.
4. Another interesting geometry is to consider SYM on T 2 × R 2 with a 't Hooft flux turned on T 2 . This setup was considered previously in [60]. It was shown via dimensional reduction from T 2 × R 2 to R 2 that the theory admits Z N vortices and that the gaugino condensate forms in the 2-D effective theory trλ 2 ∼ Λ 3 , as expected. However, the authors did not attempt to compute the numerical coefficient of the condensate. Such a calculation would mandate a more careful treatment of the dimensionally reduced 2-D theory, presumably using the power of supersymmetry. We also note that T 2 × R 2 is the limiting geometry of our asymmetric T 4 after taking ∆ to infinity.

Outline
Our paper is organized as follows: In Section 2, we formulate the theory and spell out all the necessary ingredients to define the partition function and the condensate on T 4 with twisted boundary conditions. Further, we explain that a self-dual torus gives rise to extra, unexpected, zero modes. Then, we present the solution on the asymmetric T 4 in Section 3 and discuss both the fermion and bosonic zero modes. In Section 4, we calculate the measures of the bosonic and fermion zero modes. In Section 5, we deviate to the Hamiltonian formalism to discuss two important aspects. First, we argue that the moduli space of the bosonic zero modes is isomorphic to T 4 with a period size twice the size of the physical period. Next, we recall that the 0-form/1-form mixed anomaly implies that the energy eigenstates on T 3 are doubly degenerate and the theory admits 2 degenerate vacua. These features are important for the interpretation of our calculation of the gaugino bilinear. Finally, in Section 6, we put all the pieces together to obtain our result of the gaugino condensate.
Owing to the mismatch between our and both R 4 and R 3 ×S 1 L results for the gaugino condensate, and in order to offer the reader the opportunity to catch mistakes, if any, we present our rather detailed calculations of various quantities in two appendices.
In Appendix A, we review in great detail the construction of [1] of the fractional instanton to order ∆, with emphasis on the dependence on the collective coordinates. We also provide expressions of the field strengths and Wilson loops in the background of the leading-order solution.
In Appendix B, we construct the fermion zero modes by directly solving the Weyl equation as well as by using supersymmetry transformation, and we prove that both methods yield the same result. Then, we construct the bosonic zero modes to any order in ∆, first by employing the fermionic zero modes and then by taking derivatives of the classical background solution w.r.t. the collective coordinates modulo gauge trans-formation (we also determine these gauge transformations). Finally, we determine the Jacobian of the bosonic zero modes moduli space needed to complete the calculations.

Fractional instantons on the symmetric torus
We consider the SU (2) SYM theory on T 4 with periods of lengths L 1 , L 2 , L 3 , L 4 . The Euclidean action of the theory is given by 13 and λ is a left-handed adjoint Weyl fermion, the gaugino. D n = ∂ n + i[A n , ] is the covariant derivative, σ n ≡ (i σ, 1),σ n ≡ (−i σ, 1), σ are the Pauli matrices, and the Latin letters n, m run over 1, 2, 3, 4. The field strength is given by This action is invariant under the supersymmetry transformations where the spinors obey ξ 1 = ξ 2 , ξ 2 = −ξ 1 and likewise for the dotted ones. The equations of motion that result from the variation of S SY M are where A = 1, 2, 3 labels the color group generators T A = τ A /2 with τ A the Pauli matrices. We shall consider SYM with twisted boundary conditions on T 4 . Without loss of generality, we can use the following transition functions: whereê n is a unit vector in the x n direction, n 12 = n 34 = −n 21 = −n 43 = 1, and the rest of n ij are zeros. The periodicity condition on the gauge fields and gaugino in R 4 defines the T 4 fields: A n (x)dx n . Let the SU (2) non-abelian field Φ denote either the gauge field or the gaugino and expand Φ using the Cartan-Weyl basis: The Pauli matrices τ 3 , τ ± are the generators of SU (2) in this basis. Using the commutation relations [τ 3 , τ ± ] = ±τ 3 , [τ + , τ − ] = τ 3 along with the Baker-Campbell-Hausdorff formula, one finds that the boundary conditions on Φ are satisfied if and only if: where I is a nonhomogeneous term that contributes to the gauge field and appears upon shifting the x 2 coordinate: A 3 n (x +ê 2 L 2 ) = A 3 n (x) + 2π L 1 δ n,1 . Similar boundary conditions hold in the 3-4 plane. As we shall show, there is a classical solution to the equations of motion (2.3), which satisfies the boundary conditions (2.8) and has a topological charge Q = 1 2 . The Euclidean partition function of our system is given by the path integral: where we have emphasized that this path integral is to be computed with the twists in the 1-2 and 3-4 plane, as per the transition functions (2.4), and thus, all the fields need to satisfy the boundary conditions (2.8). Notice that one also needs to sum over the integer topological sectors ν ∈ Z in order for Z SY M to respect locality and unitarity. 14 The vacuum angle θ can be transformed away by applying an axial rotation on the gauginos. Our main aim is to find the bilinear gaugino condensate starting from (2.9).
The solution of Z SY M , as well as the condensate, will proceed by using the semiclassical techniques, which amount to computing the path integral as the sum of paths of small fluctuations in the background of instantons. The semiclassical approach is justified in the limit of small volume.
Notice, however, that this partition function vanishes identically, which can be understood in two different ways. First, as we shall show, the theory exhibits 2 fermion zero modes. These are the zero modes that are saturated in the twisted background and give rise to the fermion condensate. We can also argue that Z SY M = 0 because of the mixed anomaly between the 0-form discrete chiral and 1-form center symmetries, as was previously shown in [57].
The gaugino condensate calculations can proceed by inserting the gaugino bilinear tr λλ in the path integral (2.9), where the trace is taken in the color space. We define the expectation value of the gaugino condensate as: with normalization constant N . Only the ν = 0 sector (in the presence of the twists) can contribute to the bilinear condensate, and thus, we restrict our calculations to this sector. The standard way of normalizing the expectation value of an operator is to divide by the partition function. Since Z SY M = 0, we need to search for another appropriate way to normalize our physical observables. We choose to divide by the partition function with twists only in the 1-2 plane. Explicitly, This partition function is saturated by the ν = 0 term. In the small-T 4 limit, it can be evaluated using a semiclassical expansion around two zero-action classical saddle point configurations, related by the Z 2 center symmetry in the x 3 direction, generated by the "improper gauge transformation" T 3 . In a gauge with constant transition functions 15 in the x 1 -x 2 plane, these saddle points are A = 0 and A = −iT 3 dT −1 3 [16]. Each of these saddle points gives an identical contribution to N . There are no zero modes (see [61] for calculations of the spectrum) and all determinants, fermionic and bosonic, are the ones computed in the A = 0 background (we note that this is the normalization of instanton transition amplitudes already taken in [62]). The partition function (2.11), up to a normalization factor that is expected to cancel the numerical factor in the numerator of (2.10), is equal to the Witten index Tr[(−1) F ], which counts the number of the ground states. Thus, we take N = 2. See Section 5 for a Hamiltonian treatment.
As a first trial, let us find a self-consistent Q = 1/2 fractional-instanton solution to the equations of motion (2.3) with the boundary conditions (2.8). It is easy to see that the abelian gauge configuration: The bar is introduced here to serve a later convenience, and the constants z m are the collective coordinates, which will be set to zero in this section without loss of generality. This field configuration was first found by 't Hooft, and it solves the Pure Yang-Mills equations D m F mn = 0, owing to its abelian nature and constant field strength. It carries a topological charge Q = 1 2 and its action is half the action of a BPST instanton: S = S 0 ≡ 4π 2 g 2 . This solution must be self-dual or anti-self-dual, F mn = ± 1 2 mnpq F pq , which guarantees that the bosonic and fermionic determinants in this background do not yield negative zero modes, a sign of instability of the solution on T 4 . Self-duality of the Q = 1/2 solution (2.12) implies the condition (2.13) A torus that satisfies the relation (2.13) is said to be self-dual, and a simple choice that satisfies the condition (2.13) is the symmetric torus with L 1 = L 2 = L 3 = L 4 . The classical solution (2.12) will also hold in SYM provided that the right-hand side of the first equation in (2.3) vanishes. To check that, we first need to solve the Weyl equations (second and third equations in (2.3)) in the background (2.12), along with the boundary conditions (2.8). We find 6-independent zero modes: 4 modes for λ and 2 modes forλ: where ξ i are Grassmann numbers. The functions h 12 0 and h 34 0 are the ground eigenstates of the simple harmonic oscillator, of frequencies ω 12 = 2π L 1 L 2 and ω 34 = 2π L 3 L 4 , respectively; see Appendices A.1 and B.2.3. Owing to the abelian nature of the classical background, the Weyl equation yields the solutions in the τ 3 directions. Substituting the fermion zero mode solutions into the right-hand side of the bosonic equation of motion in (2.3), one easily finds that the source term is nonzero. This simple exercise shows that the abelian background (2.12) is either inconsistent or that some gaugino zero modes must be lifted by higher-order corrections in order to have trλσ n [T A , λ] = 0. Presumably, higher-order corrections will gap 4 of the fermion zero modes, and in the end, we will have a consistent story. We were not able to show that this is the case in a satisfactory way. 16 Instead, we chose to detune the self-dual torus, i.e., to relax the condition L 1 L 2 = L 3 L 4 , and modify the abelian background to include non-abelian pieces that are needed to guarantee the self-duality of the fractional instanton on an asymmetric T 4 . The detuned solution does not suffer from any of the above-mentioned problems. A further motivation for considering the asymmetric T 4 connects to interesting semiclassical limits that have been considered.

Fractional instantons and gaugino zero modes on the asymmetric torus
In order to circumvent the problems of the self-dual T 4 , we instead search for a selfdual instanton on an asymmetric torus using a perturbation technique introduced in [1]. The gauge field A n can be written in the general form where we have split the τ 3 component into two parts. The first partĀ 3 n (x, z) is the abelian background (2.12) that solves the sourceless Yang-Mills equations on the symmetric T 4 . The functions S n (x, z) and W n (x, z) will be determined perturbatively. We also introduce the dimensionless detuning parameter ∆ ≡ (L 3 L 4 − L 1 L 2 ) / √ L 1 L 2 L 3 L 4 , which measures the deviation from the self-dual torus, and take ∆ > 0. The solution of the sourceless Yang-Mills equations on the asymmetric T 4 is obtained by imposing the self-duality condition F mn = 1 2 mnpq F pq , where 1234 = 1, etc. In order to reduce the gauge redundancy, one also imposes the background gauge condition The details of the construction of the self-dual solution in an expansion in powers of ∆ is given in Appendix A.1. The presentation there follows [1] but for completeness we present it in detail using our notation, with an emphasis on the dependence of the solution of the z n variables. Here, we only give an idea of the construction of the fractional instanton and present its main features. To simplify the equations, it is convenient to use a quaternion notation. Thus, we introduce the matrices: where σ n = (i σ, 1 2×2 ),σ n = (σ n ) † and C = 0 −i i 0 . Self-duality and the background gauge conditions (see (A.12)) then yield the following equations: These equations are subject to the boundary conditions (2.8). The solutions to (3.3) is found as series expansions in √ ∆: The symmetry structure of the self-duality equations (3.3), as well as their solution to the leading order in ∆, is discussed in Appendix A.1. The final answer for the fractional instanton solution, giving the order √ ∆ terms in (3.1), reads: We stress that the O(∆) contribution to S n is determined by the O( √ ∆) terms shown above, in a manner described in Appendix A.1, but explicit expressions will not be needed. The arbitrary phase α is due to the gauge freedom to rotate around the τ 3 isospin direction, and the function F (x, z) is given by the expression: with the normalization are the same harmonic oscillator ground state wave functions appearing in eqn. (2.14) and described there.
It remains to check that this solution obeys the equations of motion (2.3), and in particular, that the fermionic source in the bosonic equation of motion vanishes identically. To this end, we need to find the fermion zero modes, which we do in Section 3.1. For this purpose, we will need to use the field strength, a computation carried out in detail in Appendix A.2. We find that to order O( √ ∆): The function h 1 is the first excited state of the simple harmonic oscillator, and just like F (x, z), G is dimensionless and similarly normalized To find the field strength to order O(∆) and establish the self-duality of the solution, one needs to solve for the functions S n , see (A.43). In Appendix A.1, we discuss the relations obeyed by the field strength of S n (explicit expressions will not be needed in our work). These are important for establishing the self-duality of the solution, and thus of the fact that its action S 0 saturates the BPS bound, S 0 = 4π 2 g 2 . Finally, we stress that the solutions, whose construction via a small-∆ expansion was described above, have been subjected to comparison with "exact" solutions obtained by numerically minimizing the lattice Yang-Mills theory action with an n 12 = n 34 = 1 twist. A good qualitative (and in some cases quantitative) agreement of the gauge invariants charactering the solution, computed analytically 17 to O(∆), with the numerical approximation of the exact solution was found, for ∆ in the range 0.02 − 0.09 for various lattice sizes, see [1] for a detailed discussion.
The existence of a consistent solution in the form (3.4) to all orders in ∆ has been shown in [1], but explicit calculations beyond the leading order have not yet been performed. The radius of convergence in ∆ is also not known. In what follows, we assume that the all-order ∆-expansion gives rise to a unique self-dual solution of action S 0 = 4π 2 g 2 , at least for sufficiently small ∆. This is supported by the consistency of (3.4) and the agreement with numerical tests [1].

Fermion zero modes
The fermion zero modes can be found by solving the Weyl equations from (2.3), D nσn λ = 0 , D n σ nλ = 0 in the background (3.1). We perform these explicit com- The fermion zero modes can also be obtained via the supersymmetry transformations (2.2), with the result agreeing with the direct solution of the Weyl equations. Consider the effect of the supersymmetry transformation (2.2) in the background of the bosonic solution (3.1), with fermions set to zero,λ = λ = 0. Since our solution is self-dual, i.e. obeysσ mn F mn = 0, the SUSY transformation only produces λ α variations: which is true to all orders in ∆. Computing δλ we obtain, see Appendix B.2.4, to O( √ ∆): where in going from the first to second line we used the definitions Thus, as expected, we find 2 fermion zero modes in accordance with the index theorem. One can easily see that trλσ n [T A , λ] = 0, and hence, the selfdual instanton (3.1) on the asymmetric T 4 solves the sourceless equations of motion D m F mn = 0 and is a consistent background for SYM.

Bosonic zero modes
For every fermionic zero mode there are two bosonic zero modes. Thus, in total, there are four independent bosonic zero modes. The advantage of the discussion that follows is that the bosonic zero modes automatically obey the background gauge condition and, furthermore, their construction holds to arbitrary orders in ∆. The four-vector expressions for the bosonic zero modes are denoted by Z (β) A n and Z (β ) A n , where β, β = 1, 2. These modes are determined as described in e.g. [25,26]: from each zero mode φ Using the expression of φ (β) α in terms of F mn , we can also express Z (β) n and Z (γ ) n in terms of F mn : It is a simple exercise to check D n Z and show in Appendix B.3.3 that There, we also explicitly find the expressions for the background-gauge restoring gauge transformations Λ (k) to order √ ∆ and show that they obey boundary conditions preserving the transition functions. For later convenience, we also define the inner product of the zero-mode wave functions (or moduli space metric) which in diagonal basis is simply U kl = δ lk u l . Using (3.12), see Appendix B.3.2 for details, we readily obtain This result is valid to all orders in ∆, since the action of a fractional instanton does not depend on the size or shape of T 4 .

The path integral: bosonic and fermionic measures
The contributions from the non-zero modes of fermions, bosons, and ghosts cancel in the path integral, thanks to supersymmetry. Thus, in our subsequent discussion, we only discuss the contribution from zero modes to the bosonic and fermionic measures.

Fermionic measure
The fermion zero modes measure will be inferred from the one for the non-zero modes.
We expand the fermions as eigenfunctions of the second order Hermitian operators where, in the second line, we used the self-duality of the background.
To discuss the measure, we begin by considering the contribution of a single nonzero eigenvalue ω to the fermion path integral. Let −(DD) β α φ i β = ω 2 φ i α , where i labels the different eigenfunctions, the commuting functions φ i α , with the same eigenvalue ω (we note that there are at least two of them). We expand the nonzero-mode part of the fermion field (for brevity, denoting it with the same letter λ,λ) where we used the fact that the nonzero eigenfunctions of DD andDD are related as shown and we attach the spinor index to the bosonic solution of the 2nd order equation and not to the Grassmann variable, χ i orχ i (the fact that there is more than a single solution for every ω is accounted by the index i). We also indicate that the λ andλ expansions have each their separate Grassmann variables χ i ,χ i .
Plugging (4.2) into the fermionic action (2.1), we obtain after integration by parts and using the fact that φ is an eigenvector of DD: where the fermion mode inner product matrix is Then we define the fermion nonzero mode path integral so that it produces ω (the minimal number of eigenfunctions with the same eigenvalue is two, i.e. i, j = 1, 2, with When all nonzero eigenvalues are taken into account, we obtain the square root of the product over all nonzero eigenvalues ofDD (or DD). The integrals over the fermion zero modes are defined in the same manner via the same mode normalization matrix, U ij F , defined in (4.4). Recalling that only the undotted spinors λ α have zero modes, we expand 18 where we use η i to denote the zero-mode Grassmann variable. The fermion zero-mode measure is then taken to be the "square root" of (4.5): 19 In Appendix B.3.1, we calculate the Pfaffian, using the zero-mode wave functions (3.11), and show that PfU Notice that this result is valid to all orders in ∆ (thus including the nonabelian part of the zero modes), 18 Here and below, we use φ i α to denote the zero-mode solutions of DD, obeying (DD) β α φ i β = 0. The reader should forgive us for using the same letter as in the non-zero mode discussion near (B.9). 19 This definition ensures that, upon perturbing with a zero-mode lifting mass term, δS m = m g 2 tr λ α λ α , one obtains m for the zero-mode contribution to the path integral.
since PfU F involves integrating the square of the field strength F mn over T 4 , which is proportional to the action, a ∆-independent quantity. Furthermore, it follows from (4.6) that tr δγ +(nonzero modes). Thus, combining with (4.7), one finally finds: We next note that all gauge invariant quantities in the fractional instanton background depend on the combinations of x i with the dimensionless collective coordinates z l , see eqn. (A.9) in Appendix A.1. Thus, to calculate the condensate, in eqn. (4.8) one should replace That all gauge invariant quantities depend on (4.9) follows from the actions of translations in our background and is explained in Appendix A.1 (see Footnote 30 there).

Bosonic measure
We express the bosonic field A n (x), to be integrated over in the path integral, as a sum of the classical solution A cl n (x, z) (3.1), the zero mode fluctuations normalized as in (B.66) of Appendix B.3.3, and the nonzero modes of the fluctuation operator denoted by Z q n (of eigenvalues ω q ) using the same notation as in Appendix B.1, see the discussion after eqn. (B.2): (4.10) Using the inner product defined in (3.16), the measure of the bosonic zero modes takes the form In Appendix B.3.4, we show how we change the variables ζ (0) k to the collective coordinates {z m } by inserting a unity,á la Faddeev-Popov gauge-fixing method. The resulting expression is

The Hamiltonian formalism, Wilson loops, and the moduli space
Up to this point, we have all pieces to compute the gaugino condensate except for the shape and size of the moduli space M, or in other words, the range of integration over the collective coordinates {z m }. Determining M will force us to deviate, for now, from the path integral to the Hamiltonian formalism. We refer the reader to [57] for a detailed description of the Hilbert space, while here, we only provide a synopsis needed to study the Wilson loops and moduli space.

Pure Yang-Mills theory
The moduli space is determined in the absence of fermions, and thus, we start by studying the Hamiltonian formalism of pure Yang-Mills theory with vacuum angle θ on T 3 with a unit 't Hooft magnetic flux. Consider L 1,2,3 of T 3 as space of volume V 3 and L 4 as Euclidean time, where V 3 is to be taken much smaller than the inverse of Λ 3 , the strong-coupling scale, while L 4 can be varied from small to large w.r.t. Λ −1 . Recall that we apply the twists m 3 ≡ n 12 = 1, this is a "magnetic flux" piercing the 1-2 plane, and k 3 ≡ n 34 = 1. The latter is related to a twist of the partition function by a centre symmetry transformation, to be described below. The physical Hilbert space lives along the constant time slices and, thus, is in the m 3 = 1 "magnetic flux" background. Let |ψ denote a state in the physical Hilbert space on T 3 with twisted boundary conditions, H m 3 =1 .
We introduce the operatorT 3 as the generator of the Z 2 1-form center symmetry in the L 3 direction. The generatorT 3 commutes with the Hamiltonian, [T 3 ,Ĥ] = 0, and thus, they can be diagonalized in the same basis. It can be shown that the action ofT 3 on its eigenstates, |ψ = |e 3 , in the physical Hilbert space isT 3 |e 3 = e iπe 3 −i θ 2 |e 3 , where e 3 ∈ {0, 1} is the Z 2 "electric flux" of the state. This terminology can be explained as follows. SinceT 3 is a center-symmetry generator, it acts on Wilson loops winding around the L 3 direction asT 3Ŵ3 = −Ŵ 3T3 . Then, one readily findsT 3 , the action ofŴ 3 on a state increases the Z 2 flux of this state by one unit. The fact thatŴ 3 can be thought of as creating a winding electric flux tube explains the terminology. We conclude thatT 3 measures the center flux of a given state in Hilbert space. The generators of the center symmetries in the other two spatial directions, x 1 and x 2 ,T , = 1, 2, act similarly on Wilson loops winding in the corresponding directions. They also commute with the Hamiltonian and thus all eigenstates ofĤ are also labeled by e 1 ∈ {0, 1} and e 2 ∈ {0, 1}. 20 20 The action ofT 1 on states of electric flux e 1 isT 1 |e 1 = e iπe1 |e 1 (and similar forT 2 ), without the The twisted partition function of pure Yang-Mills theory, the one we have been studying in the absence of fermions, is the one with an insertion of aT 3 twist: where in going from the first to the second line, we summed over a complete set of eigenstates in the physical Hilbert space H m 3 =1 that diagonalizeĤ andT 3 simultaneously (for brevity, omitting the summation over e 1,2 ). The expectation value of However, the insertion ofŴ p increases the electric flux by one unit in the L p direction and produces a sum over different energy eigenstates. Thus, its diagonal matrix element in the |E(e 1 , e 2 , e 3 ), e 1 , e 2 , e 3 state vanishes. We immediately conclude that a Wilson loop that wraps L 1 , L 2 , or L 3 must vanish on T 4 with twisted boundary conditions. The vanishing of Ŵ 4 follows by applying a 90-degree rotation to any spatial Wilson loop, i.e. upon considering a different T 4 direction as time. Thus, we conclude that for θ/2 factor in the action ofT 3 (which is due to the m 3 = 1 twist). 21 We ignore the normalization of Wilson loops. We will come back to normalization in calculating the gaugino condensate.
any winding Wilson loopŴ p , We spent so much time explaining the expected result (5.3) because its consistency with semiclassics is one of our main criteria used to determine the moduli space of the fractional instanton. Thus, we now contrast the general result (5.3) with the computation of the expectation value of a Wilson loop in the path integral formalism, in the semiclassical approximation. Consider a Wilson loop W (C n 1 ,n 2 ,n 3 ,n 4 ), with C beginning at an arbitrary point x in T 4 and winding n times around each direction L : where we inserted Ω n 2 2 (x) Ω n 4 4 (x) to enforce the gauge invariance of W (x, C n 1 ,n 2 ,n 3 ,n 4 ). Using the classical self-dual background (3.1),(3.5), we show in eqn. (A.58) in the Appendix, that W to order ∆ is: The O(1) and O(∆) contributions come from the abelian and nonabelian components of (3.1). The cosine function has 4π periodicity in {z i }, while the O(∆) piece F(x, z) is a periodic and even function of Now, using the results from the previous section and limiting our discussion to pure Yang-Mills theory and ignoring issues of normalization, the expectation value of a general Wilson loop is where in going from the first to the second line, we ignored the quantum loops and used the bosonic zero-mode measure (4.12). We limited the r.h.s. to the contributions of the ν = 0 sector, with total topological charge Q = 1 2 , and the ν = −1 sector, with total topological charge Q = − 1 2 . Ignoring higher-Q contributions is justified in the limit when semiclassics holds, i.e. for a small T 4 . Furthermore, in writing the above expression, we assumed that the fractional instantons (and antinstantons 22 ) we obtained are the only |Q| = 1 2 saddle points in the path integral, contributing to the first and second terms above, respectively. While we have no proof of uniqueness, the favourable comparison of "exact" (i.e. obtained by numerically minimizing the action) fractional instantons on an asymmetric T 4 with the solutions obtained by small-∆ expansion gives support in favor of this assumption, at least for small enough ∆. In this regard, we note that ref. [1] found good agreement for ∆ as large as 0.08.
Our point now is that the d 4 z integral in (5.6) vanishes, and is thus consistent with (5.3), for all values of θ and for all x, if and only if the limits of integration are taken z i ∈ [0, 4π). This leads us to conclude that the moduli space M ∼ T 4 with period 4π in every direction. 23 A further argument in favour of this identification of the moduli space is that the classical fractional instanton field configurations with z n differing by 2π are distinguished by the gauge invariant winding Wilson loop operators, while those differing by 2π are not distinguished. Thus one expects that z n and z n + 2π are not to be identified. In contrast, no local gauge invariant operators can distinguish between z n and z n + 2π, see Section A.3 in the Appendix for detail. 24

SYM theory
Before turning to the calculations of the gaugino condensate using the path integral method, we pause here to discuss SYM theory using the Hamiltonian formalism, akin to our discussion of pure Yang-Mills of Section 5.1. The partition function of SYM with twisted boundary conditions in the 1-2 and 3-4 planes, identical for bosons and 22W in the second term is the Wilson loop (5.4), but evaluated in the anti-instanton background, whose explicit form is similar to (5.5) and shall not be needed. 23 Moreover, it is easy to check that changing the limits of integration to any other values yields a position-dependent Wilson loop. For example, consider W (x, C n1=1,n2=0,n3=0,n4=0 ) and take the range of z i ∈ [0, 2π), which yields W (x, C n1=1,n2=0,n3=0,n4=0 ) ∼ − sin π x 2 L + O(∆). This result should not be expected on physical grounds since our background has constant field strength on T 4 (to leading order in ∆); the expectation value of the Wilson loop should, at most, be a constant (to leading order in ∆). 24 We note, without discussing the details, that a virtually identical argument for extending the limits of z n integration holds also in the SU (N ) case in the background of the Q = 1/N solutions found by 't Hooft [15], recently extended in the framework of the small-∆ expansion [58].

fermions, is given by
The insertion of the fermion number (−1) F guarantees that both fermions and bosons obey periodic boundary conditions in the time direction. LetX be the Z dχ 4 discrete chiral symmetry generator. Then, it can be shown that the symmetry operatorsT 3 and X obey the algebra: where the last relation is the result of the mixed anomaly between the 0-form discrete chiral and 1-form center symmetries [57]. 25 SinceĤ commutes withT 3 , we can as before, label the physical states in This equation provides one way to see that the twisted partition function of SYM vanishes, by noting that all states in Hilbert space are doubly degenerate and their contributions cancel each other in the twisted partition function (5.9). Now we insert the gaugino condensate, the tr λ 2 operator, where the trace is taken in the color space. The operator is inserted at x 4 = 0 and some position on T 3 : Here, we took the normalization constant N to be the Witten index: N = tr H m 3 =1 (−1) F = 2, given by the partition function on T 3 with an n 12 twist, but without aT 3 twist in the time direction (recall (2.11) in the path integral framework). Next, we know thatX|E, e 3 = |E, e 3 + 1 from the anomaly (5.8) and thatX acts on the condensate asX tr λ 2 = − tr λ 2X . Thus, because tr λ 2 has opposite expectation values in the two degenerate states e 3 = 0 and e 3 = 1. Thus, we find where the sum is over only the half of the Hilbert space with e 3 = 0 and we are reminded that the condensate is being computed in a small V 3 theory Hilbert space with boundary conditions twisted by n 12 . If L 4 is small compared to the energy scales in the problem, there is no reason to assume that any particular values of E dominate. Thus, eqn. (5.12) is the Hamiltonian expression of the expectation value we have computed for general L 4 . However, we can try take the L 4 → ∞ limit to obtain 26 tr λλ = 0, 0| tr λ 2 |0, 0 V 3 ,n 12 , where we used that supersymmetry is unbroken and the small-V 3 ground state is bosonic, (−1) F = 1. 27 In conclusion, eqn. (5.13) shows that in the large-L 4 limit, the calculation of tr λ 2 using theT 3 -twisted parition function and normalized by the Witten index, as in (5.10), yields the gaugino condensate in one of the two degenerate ground states of the small-V 3 theory with a unit 't Hooft flux n 12 . The state |0, 0 V 3 ,n 12 is further expected, as V 3 → ∞, to become one of the two degenerate ground states of the SU (2) SYM theory on R 4 .

The gaugino condensate
Finally, we put pieces together to read out the condensate. We use the fact that non-zero modes in (2.10) cancel between the bosons, fermions, and ghosts. Then, we combine the integral of tr λ 2 over the zero mode measure for the fermions from eqn. (4.8) with the boson zero mode measure (4.12). We also recall that we normalize by the Witten index N = 2 to obtain: ]. (6.1) To obtain the last line, we recalled the substitution of (4.9), the fact that all local gauge invariants are periodic functions ofẑ k (defined in (4.9)) of period 2π, and used shifts of the z n variables to set the x k -coordinates (of period L k ) to zero. Further, Pauli-Villars regularization has been utilized to renormalize the theory, and the scale M P V is the Pauli-Villars mass. The factor M 3 P V comes from regularizing the boson and fermion determinants, with each zero-mode contributing an appropriate factor of M P V : the bosonic determinant gives M 4 P V and the fermionic determinant gives M −1 P V (we do not describe this in detail, as it is a standard procedure in supersymmetric instanton calculations [25,26,31,37]). In going from the first to the second line of (6.1), we used the fact that Λ 3 = in the scheme often used for gaugino condensate calculations [25,26], notably, in the weakly-coupled R 3 × S 1 set up and its comparison with the weak-coupling calculation on R 4 [31].
Next, we rescale the z k variables in (6.1) by appropriate factors of 2π Lp , contributing an overall Jacobian (2π) 4 V . Taking the liberty to again call the rescaled variables x k , we obtain for the condensate As indicated, the integrand on the second line is the same as the one appearing in U 21 F from (4.4), safe for the absence of a factor of 1/g 2 . Thus, recalling that U 21 F = 16π 2 g 2 , as per the discussion after (4.7) (see also (B.33)), we finally obtain for (5.12): Further, assuming that the result extends to L 4 → ∞, as described in the previous section, we use (5.13) and (6.3) to conclude 0, 0| tr λ 2 |0, 0 V 3 ,n 12 = 2 × 16π 2 Λ 3 . (6.4) Thus, the condensate in one of the two degenerate vacua of the small-V 3 , n 12 = 1 theory has twice the value calculated on R 4 as described in the Introduction. 28 Acknowledgments: E.P. is grateful to Antonio Gonzalez-Arroyo for many conversations on fractional instantons and the ∆-expansion during the "

A Constructing instantons on the asymmetric T 4 with twists
Our SU (2) Lie algebra convention is as follows. We denote the generators in the Cartan as τ 3 2 and the non-Cartan as τ ± , where τ + = 0 1 0 0 = (τ − ) † . We expand the Hermitean gauge field as 28 In the spirit of Footnote 24, without discussing the details, we remark that a similar result is obtained, upon taking the large-L 4 limit of 't Hooft's SU (N ) solutions with Q = 1 N , constructed via the ∆ expansion in [58]. One finds, instead of (6.4), that the condensate in one of the N degenerate vacua on T 3 with an n 12 = 1 twist, 0, 0| tr λ 2 |0, 0 V3,n12 = N × 16π 2 Λ 3 , i.e. equals N times the R 4 result.
An identical expression (but without the reality condition) also holds for the adjoint representation fermions λ α ,λα (α,α = 1, 2), which are independent variables in Euclidean space, to be discussed in more detail later.
The fields are smooth and defined on R 4 , with coordinates x n , n = 1, 2, 3, 4. The restriction to a T 4 with periods L n , and with 't Hooft twists is accomplished by imposing periodicities defined by transition functions Ω n (x), x ∈ R 4 . As explained in the main text, we choose a gauge where the T 4 transition functions are trivial in the x 1 and x 3 directions and nontrivial in the x 2 and x 4 directions Thus, there are two nontrivial twists, n 12 = n 34 = 1, since (A.2) obey the cocycle conditions whereê n is a unit vector in the x n direction. The periodicity condition on the gauge fields in R 4 define the T 4 fields: where we denoted A(x) = For future use, we note that a fundamental-representation Wilson loop along a unit-winding loop C 1 (winding once along x n ) is (A.6) 29 Our convention for gauge transformations, for any adjoint φ. Adjoint fermions obey the same periodicity conditions as (A.4), but without the inhomogeneous term.

Here, the insertion of Ω n (x) ensures invariance under (A.5).
A particular field configuration obeying (A.4) is the constant field strength Abelian background, the "fractional instanton" introduced by 't Hooft, see [15,46,64]: Here, z n are constants whose significance as collective coordinates associated with the instanton will be discussed at length later. The field strength of the abelian background (A.7) is: The abelian background (A.7,A.8) has the following properties: 1. The field strength F (0) mn from (A.8) can be used to explicitly verify that the abelian background (A.7) has topological charge 1/2. This can be seen by recalling that the topological charge only depends on the transition functions. Its fractional nature is owing to the nonzero twists n 12 = n 34 = 1.
2. In addition, it also follows from (A.8), that for a "symmetric" T 4 -one where L 1 L 2 = L 3 L 4 -the background (A.7) is self-dual and hence stable, i.e. it has minimal action for the given topological charge. The action of the self-dual abelian solution is S 0 = 4π 2 g 2 , half that of the BPST instanton.

For use below, it is convenient to introduce the variables
Theẑ-variables are important since all gauge invariants characterizing the nonabelian instanton background depend onẑ n only. In the gauge we are using, the fact that the background depends onẑ 1 ,ẑ 3 is already evident in (A.7). The appearance of the combinationsẑ 2 ,ẑ 4 follows from the action of translations in x 1 and x 3 : in order to preserve the transition functions, an x 1 , x 3 translation is accompanied by a non-Ω-periodic gauge transformation which shifts z 2 and z 4 . This ensures that all gauge invariant quantities only depend onẑ n . 30 A.1 Constructing the self-dual fractional instanton for small ∆ As explained in the main text, there are issues regarding the abelian solution in the tuned T 4 that have not yet been addressed in full. Notably, they concern the lifting of the extra bosonic zero modes (found in [46]) present in the L 1 L 2 = L 3 L 4 limit. These issues, as well as our desire to probe the more interesting asymmetric T 4 limit 31 prompt us to introduce a "detuning" parameter ∆, which we define as follows We always take L 3 L 4 > L 1 L 2 , i.e. ∆ > 0. For small positive ∆, a nonabelian solution of the self-duality equations has been constructed as a series expansion in powers of √ ∆ [1], with the leading contribution being the abelian background (A.7).
In this Section, we exhibit this solution, to the leading nontrivial order in ∆, using our notation and carefully including the dependence on z n . To begin, consider the classical background A n (x, z) = (Ā 3 n (x, z) + S n (x, z)) where S, W are the deviations from the abelian background, to be determined in terms of an expansion in √ ∆. Let us momentarily denote them by a n ≡ S n τ 3 2 +W n τ + +W * n τ − . The construction of the self-dual solution on the asymmetric torus by means of an expansion in ∆ proceeds by imposing a self-duality condition on the field strength of (A.11) and solving the resulting equations in a series expansion in √ ∆ [1]. In order to 30 For example, a translation . At the same time, the g 1 action onĀ shifts is invariant under the combined action of g 1 and translation e 1 ∂1 . Our nonabelian solution also exhibits this property: considering, for example, its W 1 component of eqn. (A.40), we observe that it is invariant under the combined action of translation of x 1 , the shift of z 2 given above, and a gauge transformation by g 1 , which multiplies W 1 by a phase. Gauge invariants built from the solution will then only depend on theẑ n combinations. 31 This is because the asymmetric T 4 connects to various semiclassical limits that have been discussed in the literature, see the main text.
solve for a n , it is subjected to the background-Lorentz gauge condition: D n (Ā)a n = ∂ n a n + i[Ā n , a n ] = 0, or in components: (A.12) ∂ n S n = 0, Further, the boundary conditions (A.4) imply that S n is periodic in all x n , while W n is periodic in x 1 and x 3 , but not in x 2 and x 4 . Explicitly the boundary conditions are To proceed with the construction of the instanton on the twisted asymmetric T 4 , following [1], we calculate the field strength of (A.11), set its antiselfdual part to zero, and solve the equations imposing self-duality in a series expansion in powers of √ ∆. To write the subsequent equations, it is convenient to use a quaternion notation. Thus, we introduce the matrices 32 w = σ n W n , where σ n = (i σ, 1 2×2 ),σ n = (σ n ) † , (A.14) Here σ are the usual Pauli matrices, acting in spinor space, not to be confused with the τ 3 , τ ± group generators from earlier. In terms of the matrices (A.14), the condition that the field strength of (A.11) be self dual is F mnσ m σ n = 0. This, using (A.12), becomes . 32 The n = 1, 2, 3 components of the four-vector matrix σ n defined here should not be confused with σ = {σ a , a = 1, 2, 3} = (σ 1 , σ 2 , σ 3 ), the usual Pauli matrices. We use lower-case indices a = 1, 2, 3 to denote the usual Pauli matrices σ a and reserve τ a for the SU (2) generators.
Here, we shall study only the leading-order solution. To exhibit it explicitly, we substitute the expansions in (A.16) in (A.15) and keeping the leading term in each equation, we find that s 1 , w 1 obeȳ We begin with the equation for w 1 -the complex quaternion σ m W m , obeying the boundary conditions (A.13). Periodicity in x 1 , x 3 is obeyed by writing a Fourier series Before we continue, we note that (A.20) together with (A.18) implies that the norm of w 1 on T 4 can be expressed as an R 2 -integral Thus, finiteness of ||w 1 || implies that the elements of the quaternion u 1 (x 2 , x 4 ) should be square-normalizable functions on the R 2 -plane spanned by x 2 , x 4 . Any complete basis of such functions on R 2 can be used, but below, we shall argue that an especially convenient basis is given by appropriately chosen simple harmonic oscillator eigenfunctions. Continuing with our self-duality equations (A.17), now the second equation in (A.17) together with (A.20) implies that The holonomies in the partial derivatives can be absorbed by defining a new quaternioñ u 1 (x 2 − n 1 L 2 , x 4 − n 3 L 4 ) by means of the redefinition

Using this, equation (A.22) becomes
As already alluded to, we now recognize that the first order differential operators appearing above are the creation and annihilation operators of simple harmonic oscillators (SHO) of frequencies 2π/(L 1 L 2 ) and 2π/(L 3 L 4 ). Let h ij p (x) be the normalized p-th state of the SHO of frequency The relations that we need in what follows are the orthonormality relations where we have indicated that the differential operators in the middle and bottom line are the familiar lowering and raising operators A ij , A † ij of SHOs of frequencies ω ij (A.25). In terms of A 12 , A † 12 , A 34 , A † 34 , 33 we can rewrite (A.24) as √ It is then clear that this equation is satisfied byũ 1 given by the product of ground state wave functions of the ω 12 and ω 34 SHOs times a constant 2 × 2 matrix C 1 : where C 1 obeys, as a consequence of (A.28) with arbitrary complex numbers a, b.
Before continuing, let us make the following important remark: One begins by noticing that c 00 11 and c 00 12 do not appear above (this is easy to see by inspection recalling the range of p, q, which ensure that terms containing the nonexistent coefficients, like c −1, * * * , are always multiplied by 0). Then, recursively solving the equations, one finds that their only solution is that all c nm ij vanish except for the two that do not appear in equations (A.31), the undetermined c 00 11 and c 00 12 , called a and b in (A.29).
Let us flesh out the argument. We consider the first and third equations in (A.31); the second and fourth can be considered similarly and are left as an exercise. First notice that (A.31) imply that c 0,p 11 = c p,0 11 = 0 for p = 1, 2, 3.... Further, denote δ = ω 12 ω 34 and shift the p's and q's in (A.31) such that both equations involve c p,q 11 we find Next, we multiply the top equation by with the normalization We thus conclude that a = 0 while |b| 2 = 2π V 1/2 . Thus, our solution has the form with the phase α due to the gauge freedom to rotate around the τ 3 isospin direction. Before we continue, let us exhibit the self-dual solution we found, eqn. (A.39), w = ∆ 1/2 w 1 in terms of the 4-vector gauge potentials, W n , S n of (A.11): (A. 40) We note that the above implies that the only gauge-field zero modes in our background are the one due to translations (corresponding to shifting z n ) and constant gauge rotations around τ 3 (corresponding to shifts of α). The former are physical and gauge invariant quantities characterizing the solution will be seen to depend on z n , while the latter are gauge artifacts with no physical quantity exhibiting α-dependence. For calculating the action density and gauge-invariant "electric" and "magnetic" fields to leading nontrivial order, one also needs the expression for S 1 , which was not given in (A.40). Here we simply describe how this can be found. We have, with w 1 from (A.39) Hence, from the top equation in (A.17), we obtain the equation for s 1 : The consistency of this equation (the absence of a constant Fourier mode of the r.h.s.) discussed above is yet again seen to follow from (A.36). We also note that equation (A.42) allows adding a constant to s 1 ; this is a contribution to the constants z n , already included in our background (A.7). To solve (A.42), we multiply by σ m ∂ m to obtain This equation is solved by expanding both s 1 and the r.h.s. in Fourier modes in x 1 ...x 4 and equating the coefficients. This can be explicitly done and a unique s 1 can be written down, but we shall not need the explicit form here. As we mentioned in Section 1.3, understanding the higher orders in the ∆ expansion is an interesting problem for future studies.
A.2 The field strength tensor of the fractional instanton to order ∆ To find the gauge invariants of the solution to order ∆, we need the field strength of s 1 , namely 34 where for brevity we use S n to denote the coefficient of the O(∆) part of S n . The l.h.s. of (A.42) has the form .

(A.45)
Comparing with the r.h.s. of (A.42), using ∂ n S n = 0, we conclude that Note that these terms should be multiplied by ∆. Also note that these equations by themselves do not determine the individual F s 12 , F s 34 (etc.) but only their non-self-dual parts. To find the individual F s mn , we need to solve (A.43) to first find s 1 . The field strength of the O(1) abelian background (A.7) is F (0) mn , already given in (A.8). Combining this with the results for F s mn from (A.46), we find where F s 34 is understood to be expressed via F s 12 through the first equation in (A.46). To study the complete field strength tensor in the τ 3 direction to order ∆, we must add to (A.47) the O(∆) contribution to the τ 3 field strength coming from the commutator F For the background (A.40), this is only nonzero for F (1(w)) 12 . We find This non-selfdual contribution to F (0+1(s)) mn is cancelled by the O(∆) contribution of the W n shown in (A.48).
Displayed in full glory, the Cartan part of the field strength to order ∆ is given by the sum of (A.47) and (A.48): We stress again that F s 34 is understood to be expressed via F s 12 through (A.46), a substitution we have not explicitly done for lack of space. Further, the above expression is understood to be multiplied by τ 3 /2 and the undertermined F s mn are found by solving (A.43).
The non-Cartan part of the solution, the W n from (A.40) also produce a non-Cartan field strength. The O( ∆) field strength due to W n is F . It is self-dual by itself and the O( ∆) field strength of W n is the only non-Cartan contribution to order ∆ (the next non-Cartan contribution is of order ∆ 3/2 ): The nonzero entries of (A.51) are where, just like F (x, z), G is dimensionless.
In conclusion, the full field strength tensor of our solution, to order ∆, is given by the Cartan part (A.50) and the non-Cartan part (A.51). Owing to the self-duality, the action of the solution is S 0 = 4π 2 g 2 , as can also be explicitly inferred from the explicit form of the field strength. These properties of the solutions are useful in what follows.

A.3 The gauge invariants of the fractional instanton background
We now consider the z n -dependence of gauge invariants, both local and nonlocal (i.e. winding Wilson loops) characterizing the solution.
We begin with the z n dependence of local gauge invariants, formed of traces of powers the field strength tensor. Using the data given above, we now compute calling it the "i-th component of the electric field squared," averaged over x 3 and x 4 . 35 Omitting overall constants, for i = 1, we find for the leading contribution to (A.54), which is O(∆): The sum over m, n is rapidly converging and can be seen to produce localized bumps on T 4 when plotted, periodic on R 2 with period L 1 , L 2 (usually a limit of −4 ≤ m, n ≤ 4 suffices to make the errors tiny and produces a periodic picture). The point of this discussion is to illustrate two facts: 1. That all local gauge invariants characterizing our fractional instanton depend on the combinations z 2 − 2πx 1 L 1 and z 1 + 2πx 2 L 2 (this also holds for the dependence on z 4 − 2πx 3 L 2 and z 3 + 2πx 4 L 4 , not shown above).
2. That, furthermore, the local gauge invariants have 2π periodicity in these variables.
Next, we consider the z n dependence of winding Wilson loops. To order ∆ 0 , with the abelian background (A.7), this is a rather straightforward task. Consider a Wilson loop beginning at some x, going along the x 1 direction n 1 times (i.e. from x to x +ê 1 n 1 L 1 ), then n 2 times in the x 2 direction, n 3 times in the x 3 direction, and n 4 times in the x 4 direction, with the final point being x + 4 k=1ê k n k L k . The gauge invariant Wilson loop along the loop C n 1 ,n 2 ,n 3 ,n 4 , beginning at x and consisting of the winding straight segments described above, is given by W (C n 1 ,n 2 ,n 3 ,n 4 ) = tr Pe i Cn 1 ,n 2 ,n 3 ,n 4 where the nontrivial transition functions are inserted to ensure gauge invariance. The path-ordering can be disregarded for an abelian background (Ā of (A.7)) and one can conclude that .
We now observe that the O(∆ 0 ) Wilson loops W ∆ 0 n 1 ,n 2 ,n 3 ,n 4 are periodic functions of the However, in contrast with the local gauge invariants, like the earlier (A.55), the periodicity inẑ n is 4π, rather than 2π. Thus, while local gauge invariants can not distinguish values of z n differing by 2π, nonlocal gauge invariant observables distinguish such values. These values are, therefore, not physically equivalent.
We are led to conclude that the range of values ofẑ n which are distinguished by gauge invariant quantities-and are thus physically distinct-is given by T 4 . In our description, the T 4 has "circumference" 4π,ẑ n ≡ẑ n + 4π. This fact is important in the calculation of the gaugino condensate.
It should be clear that the O(∆) (and higher) contributions to the Wilson loops (A.56), which require taking the path ordering into account, are proportional to the same overall factor as (A.57)-since they come upon expanding the path-ordered exponential. Thus, eqn. (A.57), is multiplied by (1 + O(∆)). The O(∆) terms are 2π periodic functions ofẑ n , even with respect to their simultaneous reflection, not affecting our conclusion above.
As an explicit simple example, consider the O(∆) contribution to W (C 1,0,0,0 ), obtained by expanding the path-ordered exponential and using the explicit form of the solution (A.40). Omitting the overall constant, we obtain 36 The expression above illustrates the properties we mentioned earlier: as a function of z n , the O(∆) contribution multiplying the overall cosẑ 1 2 factor is 2π periodic.

B Zero modes of the fractional instanton B.1 Leading-order bosonic zero-modes and measure
Here, we study the leading order bosonic zero modes and construct the O(∆ 0 ) bosonic zero mode measure. In the following Sections, we shall argue that the measure remains the same to all orders in ∆.
The bosonic zero modes are related to the dependence of the solution on z p . Their leading-order wave functions are particularly simple . 36 We indicated that there is an additional contribution of the Cartan-direction O(∆) component s 1 , which we have not computed. However, since it is found by solving (A.43), whose r.h.s. is (schematically) ∼ F * F which has the same properties as the (also schematically) |W 1 | 2 contribution shown in (A.58). 37 It should be clear that, to order O(∆ 0 ), the zero modes (B.1) already obey the background gauge condition: the leading-order background is in the Cartan algebra and thus commutes with Z 0,(p) n and, furthermore, the leading-order zero modes (B.1) are constant.
We include the rather well-known details that follow in order to motivate the inner product of modes as well as to follow the factors of 2. Denote by Z k m the nonzeroeigenvalue eigenfunctions of O nm , i.e. O nm Z k m = ω k Z k n and expand the nonzero-mode part of the fluctuation a m (for brevity, using the same letter as the full fluctuation which includes the zero modes) as a m = k ζ k Z k m . Then we have that where the last equation defines the zero-mode norm matrix U kl , explicitly spelled out in (B.3) below. We diagonalize the matrix U kl with eigenvalues u k and define the measure of the path integral over the nonzero modes as k dζ k √ u k √ 2π , thus normalizing the path integral to simply produce the product k ω −1/2 k . The upshot is that we defined the inner product (or moduli space metric) which in a diagonal basis is simply U kl = δ lk u l . We use the same inner product for the zero modes Z k → z k , we note that to leading order in ∆ this is quite straightforward, since the dependence of A cl (z) on z k is linear hence one simply replaces ζ As described earlier, Section A.3, the z n are integrated over in the range from 0 to 4π modulo an overall reflection. We end this introductory discussion of bosonic zero modes with two comments: 1. Eqn. (B.5) was obtained by considering only the leading-order solution. Our next task is to show that the zero modes of the O( √ ∆) (and higher) solution enjoy the same measure. We need to ensure that the background gauge condition can be satisfied and that the change of variables from ζ (0) k to z n results in the same measure.
2. An additional question that needs to be discussed is the fact that for ∆ = 0, the self-dual abelian solution has extra non-Cartan zero modes, in addition to the constant modes, as was found long ago [46]. In contrast, the asymmetric T 4 self-dual solution has only the zero modes discussed above, as we show below.

B.2 Fermions and their zero-mode measure
It is well-known (see e.g. [25,26]) that there is a relation between the zero modes of the adjoint Dirac operator and the zero modes of the bosonic fluctuation operator O mn of eqn. (B.2), to be exploited later. To this end, as well as because we are interested in the theory with adjoint fermions, we now consider the adjoint fermions and their path integral in the fractional instanton background. The Euclidean space Lagrangian density with the fermions included is where λ andλ are independent variables. 38 For future use, we note that it is invariant under the supersymmetry δA n = ζ α σ n ααλα +ζασα α n λ α (B.7) where the σ's are the ones of [25]. As usual for spinors, ξ 1 = ξ 2 , ξ 2 = −ξ 1 and likewise for dotted.
The procedure for the fermions we shall follow is to again start from the nonzero modes. We expand the fermions as eigenfunctions of the second order Hermitean 38 All our notation regarding fermions is as in [25], save for the fact that ref. [25] uses antihermitean gauge fields, necessitating the replacement A that ref. = iA this paper . The four-vectors σ n ,σ n were already defined in (A.14). For futher use, notice, in particular, that σ mn = (σ mσn − σ nσm )/4, and that these matrices are, explicitly, σ 12 = σ 34 = i 2 σ 3 , σ 13 = −σ 24 = − i 2 σ 2 , σ 14 = σ 23 = i 2 σ 1 . As we already noted, we use σ a to denote the usual Pauli matrices, not to be confused with the components of the four-vector σ n of (A.14).

operators
where, in the second line, we used the self-duality of the background (A.15). To discuss the measure, begin by considering the contribution of a single (for brevity) nonzero eigenvalue ω to the fermion path integral. Let −(DD) β α φ i β = ω 2 φ i α , where i labels the different eigenfunctions, the commuting functions φ i α , with the same eigenvalue ω (we note that there are at least two of them). We expand the nonzero-mode part of the fermion field (for brevity, denoting it with the same letter λ,λ) where we used the fact that the nonzero eigenfunctions of DD andDD are related as shown and we attach the spinor index to the bosonic solution of the 2nd order equation and not to the Grassmann variable, χ i orχ i (the fact that there is more than a single solution for every ω is accounted by the index i). We also indicate that the λ andλ expansions have each their separate Grassmann variables χ i ,χ i . Plugging (B.9) into the fermionic action (B.6), we obtain after integration of parts and using the fact that φ is an eigenvector of DD: where the fermion mode inner product matrix is and again we remind ourselves that we are just looking at a single eigenvalue (one can imagine a sum over them). Then we define the fermion nonzero mode path integral so that it produces ω (the minimal number of eigenfunctions with the same eigenvalue is two, i.e. i, j = 1, 2, with U F generically a 2 × 2 matrix) When all nonzero eigenvalues are taken into account, we obtain the square root of the product over all nonzero eigenvalues ofDD (or DD).
The definition of the integrals over the fermion zero modes are done in the same manner using the same mode normalization matrix, U ij F defined in (B.11). We imagine (as we shall argue to be the case in our background) that only the undotted spinors λ α have zero modes, thus we expand 39 where we use η i to denote the zero-mode Grassman variable. The fermion zero-mode measure is then taken to be the "square root" of (B.12): 40 (B.14) B.2.1 No zero modes of D 2 = DD on the asymmetric T 4 The reason to include this Section is that in the ∆ = 0 abelian self-dual background, i.e. in the symmetric-T 4 case, the D 2 (=DD) operator has zero modes: a n = c n τ 3 , with arbitrary constant c n . These zero modes are also zero modes of O mn of (B.2), owing to the abelian nature of the background. The four Cartan zero modes of the ∆ = 0 self-dual solution appear in addition to the two non-Cartan (complex) zero modes, also obeying O mn a n = 0 but with a n in the τ ± directions, found in [46]. Thus the ∆ = 0 self-dual abelian background has 8, not 4 bosonic zero modes. While it is expected that interactions lift half of these zero modes, this has not been shown in any detail and is a problem for future studies.
In any case, since we find the ∆ = 0 detuned T 4 of greater physical interest, we proceed to show that in our self-dual background (A.40) the D 2 operator has no zero modes. Furst, suppose that there exists an adjoint field φ (generally complex) obeying D n D n φ = 0 in our background; φ is, of course, assumed to obey the boundary conditions appropriate to adjoints, i.e. (A.4) without the nonhomogeneous term. This implies that tr D n φ † D n φ = 0. This is only possible if D n φ = 0, 39 Here and below, we use φ i α to denote the zero-mode solutions of DD, obeying (DD) β α φ i β = 0. The reader should forgive us for using the same letter as in the non-zero mode discussion near (B.9). 40 This definition ensures that, upon perturbing with a zero-mode lifting mass term, δS m = m g 2 tr λ α λ α , one obtains m for the zero-mode contribution to the path integral.
or writing explicitly the components of this equation: where A 3,± n are to be substituted by the O( √ ∆) background fields (A.40). Eqns. (B.15) imply that the D 2 zero mode φ 0 of the abelian background (the ∼ ∆ 0 term with A ± = 0) has components φ 3 0 = c, φ ± 0 = 0. One can show that there are no other zero mode solutions obeying the right boundary conditions (see [46], or perform a SHO analysis of (B.15) in the background (A.7), similar to the analysis in Section A.1).
We now want to argue that the O( √ ∆) perturbations around the abelian background of eqn. (A.40) lift the zero eigenvalue of −D 2 . To this end, we use perturbation theory for the non-negative Hermitean operator −D 2 and compute its matrix element in the unperturbed "eigenstate" found above, φ 0 = c τ 3 2 . For the shift of the eigenvalue, this gives where only the n = 1, 2 components are nonzero in the order √ ∆ solution: A + 1 = W 1 and A + 2 = W 2 = iW 1 . The integral is positive definite, as it is easily seen to be proportional to T 4 |F | 2 , recall (A.36), showing that the zero eigenvalue is lifted in the detuned-T 4 self-dual background. This is a welcome feature of this background, compared to the one in the symmetric T 4 studied in [46].

B.2.2 The zero modes of D on the asymmetric T 4 via the Dirac equation
In this Section, we explicitly study the Dirac equation for the undotted fermions in the ∼ √ ∆ background and show that they have two zero modes. Since, as shown above, D 2 =DD has no zero modes, the presence of two zero modes of DD is guaranteed by the index theorem in the topological charge 1/2 background. Thus, we include this discussion only for completeness. In the later Section, we shall see that these zero modes can be obtained using supersymmetry.
The Dirac equation for the zero mode isDα α λ α = 0 yielding In our leading-order background (A.40), these equations give, for the λ 3 α components: On the other hand, the non-Cartan λ − α components satisfy the equations We now can follow exactly the same steps as in the study of the self-duality equation for w 1 , see discussion after (A.17): we introduce x 1 , x 3 Fourier modes for λ ± and the functions corresponding to (A.20); likewise, we can absorb the Wilson lines by a redefinition similar to (A.23). Proceeding thus, we can now solve the undotted fermion zero mode equations (B.17,B.18, B.19) in an expansion in √ ∆. Keeping in mind that γ ∼ √ ∆, it follows that the solution for λ 3 α is of order ∆ 0 and is simply given by a twocomponent constant Grassmann spinor η 3 α . This satisfies the λ 3 α equation to leading order, since the r.h.s. of (B.17) is of order ( √ ∆) 2 as the λ ± solutions are themselves of order √ ∆. Proceeding thus, we find the leading-order fermion zero-modes: That these are solutions of (B.17, B.18, B.19) with γ = 0 follows from observing that the annihilation operators of the ω 12 and ω 34 SHOs act on λ + 1 and λ − 2 components only, while the other components are acted upon by creation operators and are thus set to zero. These four zero modes ofDα α at ∆ = 0 combine with the two dotted zero modes, obeying D ααλα = 0, which exist due to the fact that D 2 has zero modes at ∆ = 0 (recall Section B. (B.24), and recalling Footnote 38, we obtain (1) 13 On the last line, we again used the small-∆ relation ω 34 2πV − 1 2 . Comparing (B.21) with (B.25), we see that they are identical provided the supersymmetry parameter in (B.25) is identified with the Grassmann coefficient in (B.21) as follows showing that all is consistent with SUSY.

B.3 The moduli space metric, to any order in ∆
Here we shall study the bosonic and fermionic moduli space to order ∆. It is well known that for every zero mode solution of the undotted Dirac equation φ (β) α , one can construct two zero mode eigenvalues of the operator O nm which obey the background gauge condition D m (A cl )a m = 0. We begin with our notation for the zero modes (B.24): we shall denote the corresponding commuting wave functions by φ β A α , where A = 1, 2, 3 denotes the SU (2) algebra index. These are the two solutions of the undotted Dirac equation considered above. Thus, with (β) = 1, 2 labeling the two zero modes, we have In the second equality, we used self-duality of the background and the explicit form of σ mn given in Footnote 38.

B.3.1 Fermion zero-mode measure to arbitrary order in ∆
In terms of the zero modes (B.27) the fermion zero-mode expansion (B.13), relabeling the index i (used there to label the different zero modes), i → β, we have , sum understood over β = 1, 2, where again A = 1, 2, 3 is the Lie algebra index, summed over the definition of U F . The fermion zero mode norm matrix U βγ F from (B.11) is antisymmetric and we have from (B.30) and (B.27) Consider now the matrix X ab = (σ a ) 1 2 (σ b ) 2 1 − (σ a ) 1 1 (σ b ) 2 2 implicitly defined above. We have, from the explicit form of the Pauli matrices: The Pfaffian of the fermion mode matrix is thus PfU F = −U 12 F = 4 × 4π 2 g 2 and the fermion zero-mode measure, defined via the Pfaffian in (B.14) is To obtain a nonzero result for the fermion zero mode integral, we insert the gaugino bilinear, tr λ α λ α = 1 2 η α η β φ (α)A γ φ (β)A δ δγ +(nonzero modes), and obtain where we used (B.33) and the explicit form of the Pfaffian. We stress that the fermion zero mode measure (B.14) as well as (B.35) hold to arbitrary order in ∆. Furthermore, the measure and the result (B.35) are independent on ∆.
B.3.2 Bosonic zero-modes and moduli-space metric to any order in ∆ Now to the wave functions of the bosonic zero modes obtained from the fermionic modes and automatically obeying the gauge condition. For every fermionic zero mode φ (β) α , β = 1, 2, there are two bosonic zero modes. Thus, in total there are four independent bosonic zero modes. The advantage of the discussion that follows is that the bosonic zero modes thus obtained automatically obey the gauge condition and, furthermore, that their construction holds to arbitrary orders in ∆.
The four-vector expressions for the bosonic zero modes thus obtained are denoted by Z (β) A n and Z (β ) A n , where β = 1, 2, A = 1, 2, 3. These modes are determined as follows (see e.g. [25,26]). First one forms the quaternions made out of the zero-mode solutions of the undotted Dirac equation, explicitly: Thus, each zero mode φ Now, knowing the four-vector components of the zero modes, we can use (B.36) and (B.27) to find det U kl and argue for its ∆-independence. We need to compute the 4 × 4 matrix of different overlaps (B.3) where the trace is now in the Lie-algebra generator space. The four-vector inner products are where momentarily we omitted the group index A (to be restored below). Thus, using (B.27), with the shorthand σ.F A ≡ σ mn F A mn , we find Let us now study the matrix (B.41) Using the notation of eqn. (B.27), we rewrite it as Thus, the 4 × 4 matrix U with matrix elements (B.40), expressed through A of (B.41) is of the form where we used the diagonal form of A form (B.42). Thus, we have for the norm matrix (B.3) of the bosonic zero modes (to any arbitrary order in ∆ to which the solution has been find): The point, so far, is to argue that the inner-product of the bosonic zero modes obeying the gauge condition-or the moduli space metric (B.44)-are ∆-independent. 42 The careful reader may remark that the fact that the bosonic zero mode metric is proportional to the classical action, as in (B.44), is well-known, hence the ∆independence follows from the fact that the action of the self-dual solution is ∆independent. However, we presented the steps outlined above, especially the explicit relation between the undotted Dirac equation zero modes and the bosonic zero modes of eqn. (B.37), in order to use them to verify that the derivatives of the O( √ ∆) classical solution we found with respect to z n are equal to the modes (B.37) obeying the background gauge condition, after an appropriate gauge transformation. This somewhat more involved procedure, compared to the similar task for the BPST instanton, is what we discuss next. 42 The reader may notice that while we use the same letter, the moduli space metric U kl of (B.44) does not equal the one constructed earlier from the leading-order derivatives of the classical solution w.r.t. z n , the metric U kl of eqns. (B.3, B.4). In fact, we have g 8 det U kl eq.(B.44) = (4π) 8 while g 8 det U kl eq.(B.4) = V 2 . This is accounted for by the difference in normalization of the respective zero modes; see Section B.3.3 for explicit expressions. This difference contributes an extra factor of ( 4π We relabelled the four zero modes Z  A cl n (x, z) = ( 2πn nm x m L n L m + z n L n ) τ 3 2 + τ + (δ n1 + iδ n2 )W + τ − (δ n1 − iδ n2 )W * , where for brevity we introduced the (not anti-symmetric!) tensor n nm where only n 12 = n 34 = 1 are nonzero. Consider now the derivative of the classical solution w.r.t. z p /L p L p ∂A cl n ∂z p = δ np τ 3 2 + τ + (δ n1 + iδ n2 ) L p ∂W ∂z p + τ − (δ n1 − iδ n2 ) L p ∂W * ∂z p ).
(B.48) 43 As these can be constructed from F cl. mn to any order in ∆, we shall call these "exact" zero modes.
The Cartan term is the one we considered before, see Section B. To find the zero mode wave functions, an explicit computation using the SHO h 0 and h 1 properties from (A.27) shows that Here, we defined, in addition to F (x, z) (A.35) and G(x, z) (A.53), a new functioñ G(x, z), similar to G(x, z) but where the 1st-excited state is in the ω 12 SHO instead: Using these expressions, we find the derivatives of the classical solution w.r.t. z p :

(B.51)
The expressions for the exact zero modes Y We shall see that on the T 4 with twists, this is slightly different from the usual BPST instanton. To find these gauge transformations, we now consider the derivatives of the classical solution w.r.t. each z k in turn.
Zero mode ∂ ∂z 1 vs Y (1) : Here, we compare Y (1) n to the derivative w.r.t. z 1 . From the above we find that the four-vectors of the exact zero mode and the derivative of the classical solution are Their difference is After some algebra, we find that the gauge transformation making the derivative of the classical solution obey the gauge condition is O( √ ∆): We again consider the difference Here, in contrast with (B.54), we find that the gauge transformation making the zero mode obey the background condition also has a Cartan-subalgebra piece ∼ ∆ 0 , in addition to an O( √ ∆) piece which is proportional to (B.54): Naturally, only terms of order √ ∆ are to be kept on the r.h.s. of (B.56).
Zero mode ∂ ∂z 3 vs Y (3) : Remarkably, here we find hence no compensating gauge transform is needed. This follows from the four-vector expressions for the zero modes  We shall next argue that the measure in terms of the z n variables remains the one we found earlier by studying the leading-order zero modes, eqn. (B.5). For use below, we also rewrite the zero modes (B.65) as

B.3.4 The Jacobian and the all-order bosonic measure
We begin by shifting the bosonic field A n (x), obeying (A.4), to be integrated over in the path integral by the classical solution A cl.
n (x, z) of (B.47). We can choose to expand On the other hand, evaluating the derivative of (B.69) , we find Next, we use (B.66) to replace ∂A cl. n ∂z l by Y (k) n . Using the fact thatΛ (k) is Ω-periodic allows us to integrate by parts on T 4 without a boundary term, i.e. set T 4 tr D nΛ (l) Y (k) n = 0, since Y (k) n obeys the background gauge condition. We thus find Thus, the unity insertion (B.70) becomes where we used (B.67) and took the liberty to set ζ (0) k = 0 in the determinant, due to integrating the delta function with the measure (B.68). The nonzero mode part of the fluctuations can be ignored to leading order. Thus, collecting everything, we find that the Jacobian factors from the delta function and the determinant in (B.73) cancel out and the bosonic measure (B.68) becomes, after inserting (B.73) and integrating over ζ The bosonic zero mode measure is thus equal to the leading-order measure constructed earlier in (B.5).