From Exact Results to Gauge Dynamics on $\mathbb{R}^3\times S^1$

We revisit the vacuum structure of the $\mathcal{N}=1$ Intriligator-Seiberg-Shenker model on $\mathbb{R}^3\times S^1$. Guided by the Cardy-like asymptotics of its Romelsberger index, and building on earlier semi-classical results by Poppitz and \"{U}nsal, we argue that previously overlooked non-perturbative effects generate a Higgs-type potential on the classical Coulomb branch of the low-energy effective 3d $\mathcal{N}=2$ theory. In particular, on part of the Coulomb branch we encounter the first instance of a dynamically-generated quintic monopole superpotential.


Introduction
Some twenty-five years ago, Intriligator, Seiberg, and Shenker (ISS) [1] published a study of a fascinating N = 1 model which has continued to display surprising features ever since. The model is an SU(2) gauge theory, which besides the gauge multiplet only has a single chiral multiplet Q I=3/2 transforming in the spin-3/2 representation of the gauge group. It is asymptotically free (with the one-loop beta-function coefficient b = 6 − 5 = 1), and has a U(1) R symmetry under which Q I=3/2 has charge 3/5. Using holomorphy arguments it was shown that the model, placed on R 4 , admits a moduli space of SUSY vacua parametrized by the basic gauge-singlet superfield u = Q 4 I=3/2 (with a totally-symmetric contraction of the gauge indices). As for the most interesting point u = 0 on the moduli space, there used to be two main proposals for its IR behavior: confinement versus non-trivial superconformal fixed-point. The original study [1] deemed the confinement scenario more likely because of the remarkable TrR and TrR 3 't Hooft anomaly matchings between the UV constituents and u as the IR superfield. Under the confinement hypothesis, it was shown that dynamical SUSY breaking can be realized by adding a tree-level superpotential.
However, the confinement proposal was subsequently questioned [2,3], and after several years of suspense eventually ruled out by Vartanov in [4]. Vartanov's work is an outstanding example of use of exact results to settle questions in N = 1 gauge dynamics. The exact observable employed in [4] was the Romelsberger index [5], which up to a Casimir-energy factor can be identified with the supersymmetric partition function on S 3 × S 1 [6]. The index is known to be RG-invariant. Therefore under the confinement scenario of [1] it should have matched between the IR superfield u and the UV fields of the ISS model; explicit computation found otherwise [4].
Closely related is the dynamics of the ISS model on R 3 × S 1 , with periodic spin structure on the circle. It has been studied by Poppitz andÜnsal [7] with the motivation to approach the R 4 behavior via an adiabatic argument by varying the radius r S 1 of the circle. Their results include a description of the vacua on R 3 × S 1 . In particular, it was suggested in [7] that the moduli space of vacua on R 3 × S 1 contains a new flat direction, namely a compact Coulomb branch (for the low-energy effective 3d N = 2 theory living on R 3 ) parameterized by a periodic holonomy variable related to the component of the 4d gauge field along the circle.
In the present paper we employ the Romelsberger index to clarify the low-energy dynamics of the ISS model on R 3 × S 1 . Specifically, our study is motivated by the finding in [8] that the asymptotic behavior of the Romelsberger index of the model, as r S 3 → ∞, exhibits a non-trivial potential for the holonomy variable. In light of this result, and building on earlier semi-classical developments due to Poppitz andÜnsal, we re-examine the arguments and conclusions of [7] regarding the compactified ISS model.
Below, by the "low-energy" regime on R 3 × S 1 we mean the regime where energies are much smaller than 1/r S 1 . Whenever performing semi-classical analysis, we moreover assume that r S 1 Λ ISS 1, with Λ ISS the dynamical scale of the model. In such low-energy, semiclassical regimes, a Kaluza-Klein (KK) reduction on the circle gives a finite number of fields that are massless on R 3 and constitute the dynamical degrees of freedom, together with infinite towers of massive KK modes that ought to be integrated out.
Our main results are as follows. Firstly, at the classical level, we carefully identify the fields that are massless in the three-dimensional sense. We find that the mixture of KK charge and gauge charge leads to extra massless fields on the classical Coulomb branch. This leads to a new picture of the classical moduli space. Secondly, at the semi-classical level, we find a monopole-induced potential on the classical Coulomb branch which has the same qualitative profile as the holonomy potential arising from the r S 3 → ∞ limit of the Romelsberger index! This monopole-induced potential has the effect of lifting the parts of the classical Coulomb branch that are away from the loci supporting charged massless 3d fields.
We emphasize that our use of the index in addressing gauge dynamics on R 3 × S 1 is not as direct as Vartanov's use of it in [4]. Rather, we utilize (the asymptotics of) the index as a guide for a subsequent careful semi-classical analysis, and it is the latter that yields our main results. While it is intuitively expected that the r S 3 → ∞ limit of the index might encode semi-classical dynamics on R 3 × S 1 , the details of the connection are somewhat mysterious at the moment; in particular, the asymptotics of the index yields a potential for the holonomy which is piecewise linear, whereas the semi-classical (multi-)monopole potential on R 3 × S 1 is piecewise exponential, and the relation between the two is far from clear. We will comment more on this point in section 5.
The rest of this paper is organized as follows. In section 2 we explain how the leading asymptotics of the Romelsberger index might be used as a guide for studying semi-classical gauge dynamics on R 3 × S 1 . In section 3 we perform a precise semi-classical analysis of the low-energy dynamics of the compactified ISS model. In section 4 we make a few preliminary remarks on the IR phase of the low-energy effective 3d N = 2 theory. In section 5 we put our results in a wider context and also comment on some related open problems. Appendix A discusses the physical content of the subleading asymptotics of the index, appendix B elaborates on the IR phase of the low-energy 3d N = 2 theory, and appendix C discusses an alternative perspective on the piecewise-linear potential arising in the leading asymptotics of the index.

The index perspective
In this section we present intuitive arguments (as in section 5.4 of [8]) which will guide our semi-classical analysis in the next section.
We approach R 3 × S 1 by taking the large-r S 3 limit of S 3 × S 1 . The partition function on the latter space is called the Romelsberger index [5], and can be computed exactly for any N = 1 gauge theory with a U(1) R symmetry. 1 Focusing on non-chiral theories with semi-simple gauge group for simplicity, the asymptotics of the index as the radius of the S 3 is sent to infinity takes the form [8][9][10] where β = 2πr S 1 /r S 3 , and E DK 0 (β) = π 2 3β TrR is the piece studied in [11]. The symbol x stands for the collection x 1 , . . . , x r G parametrizing the unit hypercube in the Cartan subalgebra of the gauge group, whose rank is denoted r G ; the unit hypercube is denoted by h cl . The exponential function z i = e 2πix i maps h cl to the moduli space of the eigenvalues of the holonomy matrix P exp(i S 1 A 4 ), with A 4 the component of the gauge field A along the S 1 . The integral in (2.1) is thus over the "classical" moduli space of the holonomies around the circle; hence the subscript cl in h cl . When S 3 decompactifies to R 3 , this moduli space becomes a middledimensional section of the classical Coulomb branch of the low-energy effective 3d N = 2 theory living on R 3 ; the rest of the classical Coulomb branch is parametrized by the dual photons (see e.g. [12]), which the index does not see.
The integral in (2.1) localizes in the "Cardy-like limit" β → 0 to the minima of the Rains function which, as in [8], we would like to intuitively interpret as a quantum effective potential on the classical Coulomb branch of the low-energy effective 3d N = 2 theory on R 3 . In Eq. (2.2) the set of weights of the gauge-group representation of the chiral multiplet χ is denoted by ∆ χ , and the positive roots of the gauge group by α + . The function ϑ(x) in (2.2) is defined following [9] as with {x} := x − x . Note that while ϑ(x) is piecewise quadratic, the Rains function L h (x) is piecewise linear thanks to the U(1) R ABJ anomaly cancelation.
With the above reasoning we are led to expect that the minima of the Rains function encode the vacua of the gauge theory on R 3 × S 1 . Of course any non-trivial vacuum structure due to the dual photons would be invisible in this approach, as the index is blind to them. Moreover, any non-trivial Higgs branch of the 3d N = 2 theory would also escape this approach, because for any finite r S 3 the curvature couplings lift them.
Some words of caution are due at this point. Even modulo the limitations spelled out in the previous paragraph, strictly speaking there is no guarantee that the Rains function is a reliable guide to the dynamics on R 3 × S 1 . Our arguments regarding the decompactification limit of S 3 have been intuitive, and suggestive at best. In fact there seem to be some clear counter-examples, such as 4d N = 2 SCFTs with non-vanishing Rains function (the Z 2 orbifold theory discussed in [8] being a specific instance); the extended supersymmetrywhich the index is blind to-is expected to protect the classical Coulomb branch on R 3 × S 1 , so the Rains function cannot be interpreted as a quantum effective potential in such cases. Besides, the index is insensitive also to various "global" subtleties (such as the spectrum of the line operators [13]), so might not capture the dynamics on R 3 ×S 1 when such global issues become crucial in determination of vacua. Nevertheless, the ISS model which is our focus in the present paper is expected to be free of various "accidents" (such as SUSY enhancement) or global subtleties, so we are going to take its Rains function seriously as a guide to R 3 × S 1 dynamics, and see where it leads us.
For the ISS model, in terms of x := x 1 , the Rains function reads and looks like a Mexican hat as in Figure 1.
Continuing the intuitive line of argument advocated above, Figure 1 would suggest that the quantum Coulomb branch of the ISS model on R 3 × S 1 consists of a single point (x = 1/3, which is Weyl-equivalent to x = −1/3). As pointed out in [8], this appears to be in conflict with an earlier semi-classical analysis of the compactified ISS model by Poppitz and Unsal [7]. We now proceed to argue that the analysis of Poppitz andÜnsal, once extended slightly further, actually reveals non-perturbative effects generating a Higgs-type potential qualitatively compatible with Figure 1.

Revisiting the semi-classical analysis
We begin our semi-classical analysis with a brief discussion of the perturbative vacuum moduli space of the compactified ISS model. We assume 1/r S 1 Λ ISS , so that a semi-classical analysis is reliable in appropriate regions of the classical moduli space.
At energies well below the compactification scale 1/r S 1 , semi-classically we have an effective 3d N = 2 theory. The scalar in the 3d N = 2 vector multiplet comes from the holonomy of the 4d gauge field as follows. Since the gauge group is SU(2), the two eigenvalues of P exp(i S 1 A 4 ) are of the form z ±1 , and picking one of them as z we can define a periodic scalar x through z = e 2πix ; of course there is a Weyl redundancy in picking one of the two eigenvalues as z, and throughout the rest of this paper we fix this redundancy in our scalar by taking x ∈ [0, 1/2]. There is no tree-level potential for this scalar, so it parametrizes what is called the classical Coulomb branch of the 3d N = 2 theory. Actually, for any x strictly inside the [0, 1/2] interval, the super-Higgs mechanism breaks the SU(2) gauge group down to U(1), and the 3d photon can be Hodge-dualized to a compact scalar a, which also does not have a tree-level potential, and hence combines with x to parametrize the full classical Coulomb branch.
The Coulomb-branch parameter x also determines the real mass of the 3d descendants of the 4d fields that are charged under the Cartan of the gauge group. The chiral multiplet Q I=3/2 contains fields with charges −3, −1, +1, +3 under the Cartan of the SU(2). To find the real masses of their 3d descendants we first note that the mth KK mode of a 4d field with charge n under the Cartan has real mass (nx + m)/r S 1 ; picking the m ∈ Z that minimizes the absolute value of this real mass, we get the real mass of the "lightest" among all the KK modes-which we refer to as the 3d descendant.
At x = 0 all fields yield massless 3d descendants with KK charge m = 0 of course. Since the descendants q I=3/2 of the chiral multiplet have no tree-level potential at x = 0, we have a classical Higgs branch there parametrized by the gauge-invariant combination u ∼ q 4 I=3/2 . More interestingly, we see that the x = 1/3 point also supports massless 3d fields; these are the descendants of the fields with charges +3, −3 inside the chiral multiplet, and we denote them by q +3 , q −3 . (Note that q ±3 have KK charges m = ∓1.) These 3d fields also lack a tree-level potential, and thus yield a classical Higgs branch at x = 1/3, which can be parametrized by the gauge-invariant combination M ∼ q +3 q −3 .
Finally, at x = 1/2 only the 4d vector multiplet yields massless descendants. The descendants of the 4d gauge multiplet (which have gauge charges ±2 and KK charges ∓1) restore the SU(2) gauge group at this point, so we have a 3d pure N = 2 SYM sitting there.
Note that both at x = 0 and at x = 1/2 the enhanced SU(2) is recovered. At these points the "dual photon" a ceases to make sense. In fact even in proximity of these pointsin neighborhoods that we suspect to be more precisely of size O(r S 1 Λ ISS ) and O(r S 1 Λ SYM ) = O((r S 1 Λ ISS ) 1/6 ) respectively-we expect our semi-classical techniques to lose reliability. 2 We have now almost completely explained Figure 2. The one remaining feature, namely the pinches at x = 0, 1/3, 1/2, are due to the perturbative running of the gauge coupling, which sets the radius of the dual photon; this radius shrinks near points where charged fields become massless-see [15] for a clear exposition.
No further perturbative effects are expected to modify the vacuum structure (because of the perturbative non-renormalization theorems in particular), so we now turn to the nonperturbative effects.
There are known semi-classical, non-perturbative effects on R 3 × S 1 due to BPS (anti-) 2 The KK reduction at scale 1/r S 1 ought to be valid irrespective of x as long as r S 1 ΛISS 1; it is the semi-classical Higgs picture for x = 0, 1/2 that might lose reliability in proximity of the x = 0, 1/2 points. Our emphasis on these subtleties is motivated by the observation that the Rains function also receives correction near x = 0, 1/2; see footnote 3 of [8]. The "perils of compactification" discussed in [14] seem related to these subtleties as well.
For SU(2) gauge group, inside the classical Coulomb branch parametrized by x(= x 1 ) ∈ (0, 1/2), and away from pinch-points and the end-points of the interval, the (lowest component of the) monopole superfield takes the form [19] Y ∼ exp 2x  (2) instanton is often thought of as a composite configuration made out of a BPS and a KK monopole: 1 Y × ηY = η.) For such multi-monopoles to actually contribute to the low-energy superpotential though, it is necessary that they have precisely two fermionic zero-modes [19]. The BPS and the KK monopole both have precisely two gaugino zero-modes, so in absence of chiral multiplets (i.e. in pure SU(2) SYM) they both contribute to the superpotential, which hence reads 1/Y +ηY . In the ISS model we have to take into account the extra zero-modes arising from the chiral multiplet in the spin-3/2 representation of the gauge group. These zero-modes are counted by the following index theorem, given by Poppitz andÜnsal in [22] (see also [23]): As indicated in [22] (below its Eq. (3.10)), the above index theorem implies that for 0 < x < 1/3 there are 4 BPS zero-modes-thus 6 KK zero-modes, while for 1/3 < x < 1/2 there are 10 BPS zero-modes-hence no KK zero-modes.
It appears like in their later paper [7], Poppitz andÜnsal did not consider the range 1/3 < x < 1/2. Since in this range the chiral multiplet of the ISS model has no fermionic zeromodes on a KK-monopole background, there should be an ηY superpotential generated on this part of the classical Coulomb branch of the model. This superpotential yields a potential which increases with increasing Y , and hence with increasing x; therefore its qualitative behavior is compatible with the Rains function of the ISS model in that range-c.f. Figure 1! What about the range 0 < x < 1/3? As the index theorem shows, both the BPS and the KK monopole have too many zero-modes to generate a superpotential there. At this point, a result of Poppitz andÜnsal can crucially guide us further [7]: they found that in the range 0 < x < 1/3 the monopole superfield Y has U(1) R charge −2/5. Thus the natural candidate for a superpotential term would be 1/Y 5 , signaling a five-BPS-monopole effect. At first glance it seems like such a term is forbidden because a five-BPS-monopole background would have 5×4 = 20 chiral-multiplet zero-modes and 5×2 = 10 gaugino zero-modes-far too many. However, we now argue that the extra zero-modes are not protected by any symmetry, so are expected to be lifted by quantum effects! In particular, because the U(1) R charge of the chiral-multiplet fermions is 3/5 − 1 = −2/5 and that of the gauginos is +1, the 20 chiral-multiplet zero-modes can combine with 8 of the gauginos and be lifted without violating U(1) R ; the corresponding vertex ψ 20 λ 8 has U(1) R -charge 20 × (−2/5) + 8 × (+1) = 0, so is not prohibited by any symmetry, and should therefore naturally-in the technical sensebe generated. We expect that the low-energy supersymmetric sigma model on the moduli space of a five-BPS-monopole configuration would have such a vertex (or actually the more fundamental ψ 10 λ 4 vertex which would also do the job), soaking up the extra zero-modes of the five-monopole vertex.
Although our proposed mechanism for lifting of the extra zero-modes might appear contrived, in fact several effects of similar nature have already been observed both in three and four dimensions, with various amounts of supersymmetry-see e.g. [21,24,25]. For the sceptic readers we point out that the non-perturbative two-instanton vertex (c.f. [7]) also does the job of lifting the extra fermion zero-modes; this would of course suppress the five-monopole effect by an η 2 factor, but that would not significantly modify our discussion below. We conclude that a quintic monopole-superpotential of the form 1/Y 5 is indeed generated over the range 0 < x < 1/3 of the classical Coulomb branch. This potential decreases with increasing x; therefore its qualitative behavior is compatible with the Rains function of Figure 1 in that range! (Some similar higher-power monopole superpotentials have been recently used in [26][27][28]-though only up to quadratic degree.) It is useful-for holomorphy arguments in particular-to employ different Coulombbranch operators on different sides of the pinch-points. Thus for x > 1/3 instead of Y we may use Y + ∼ exp 2x ĝ 2 , with x := x − 1/3. At the quantum-mechanical level we expect Y + to have the nice property that Y + → 0 as x → 0 + [20]. On the other hand, on a neighborhood to the left of x = 1/3, the Coulomb-branch operator Y − ∼ exp − 2x ĝ 2 , proportional to 1/Y , becomes useful as we expect that quantum-mechanically Y − → 0 as x → 0 − . Therefore we can summarize our findings in this section by writing the low-energy superpotential on the classical Coulomb branch of the ISS model on R 3 × S 1 as This superpotential-previously thought to be zero-is the main result of the present paper.
Over the range 0 < x < 1/3 we argued that a vertex of the form ψ 20 λ 8 (or two vertices of the form ψ 10 λ 4 ) should naturally lift the extra fermionic zero-modes of a five-BPS-monopole background. For 1/3 < x < 1/2 we simply used an earlier result of Poppitz andÜnsal [22] that the KK-monopole background does not have extra fermionic zero-modes. In particular, the semi-classical vacuum structure implied by the low-energy superpotential (3.4) is compatible with that suggested by the Rains function (2.4) shown in Figure 1.
Note that the superpotential (3.4) is useful only outside a small neighborhood of x = 1/3 wherein new light states associated to q ±3 appear. However, at least near x = 1/3 the superpotential seems to stabilize the theory to a vacuum. The nature of this vacuum is not entirely clear to us, and we will only present some preliminary comments on it in the next section and in appendix B.
It might be possible that a perspective from Seiberg-Witten theory [29] (see also [20]) can shed further light on the above picture once the ISS model is embedded in the N = 2 I * 1 theory of [30,31]. This approach is currently under investigation.

The low-energy phase
In this section we make some rudimentary remarks concerning the low-energy phase. A more careful study is left for future work.
First, following the conventional wisdom we assume that the red neighborhoods of the x = 0, 1/2 points, although not really amenable to semi-classical analysis, presumably contain no acceptable vacua because in their semi-classical vicinity the low-energy effective potential is repulsive-c.f. the case of 3d pure N = 2 SYM for instance [20]. Hence below we discuss only the vacuum structure near x = 1/3.
The SU(2) gauge group is of course broken down to U(1). Moreover, of the four components of the chiral multiplet Q I=3/2 , having U(1) charges −3, −1, +1, +3, the two with charges −1, +1 do not yield massless descendants in 3d. In the resulting low-energy theory we are thus left with a 3d N = 2 SQED with chiral multiplets q −3 , q +3 of charge −3, +3 coming from the KK modes ±1 of their 4d parents. This is in fact compatible with the subleading Cardy-like asymptotics of the Romelsberger index of the model as described in appendix A.
In this SQED theory near x = 1/3, there are no FI or Chern-Simons terms induced through one-loop effects, because the massive fields come in pairs of opposite charge under the low-energy U(1) gauge group. Non-perturbative effects presumably cannot induce a properly quantized Chern-Simons level either, but since FI parameters need not be quantized we leave open the possibility of an instanton-induced FI parameter ζ. (As we will discuss in appendix A, the subleading β → 0 asymptotics of the S 3 × S 1 partition function signals an induced 'imaginary FI parameter', which we will explain from the r S 1 → 0 perspective via "high-temperature" perturbative effects; it is the possibility of an induced FI parameter from the R 3 × S 1 perspective via "low-energy" non-perturbative effects that we are leaving open.) We expect that the near-pinch SQED inherits a superpotential of the form (3.4), which would lift its Coulomb branch. In fact one might argue that an extra term of the form 3 M Y 1/3 − Y 1/3 + is also generated dynamically in the theory, since it has the right R-charge: recalling that q ±3 have R-charge 3/5, the R-charge of M ∼ q +3 q −3 becomes 6/5, while those of Y −,+ follow from (3.4) to be 2/5, 2, therefore M Y wherein Y + , Y − , M ought to be considered independent operators. We present some further plausibility arguments for this proposal in appendix B.
Whether the above description is accurate, and in case it is what is the fate of the remaining classical Higgs branch, are important questions requiring a careful treatment. Here we only note that assuming the above description, for nonzero ζ we expect our SQED would have a Higgs branch parametrized by vevs of the two chiral-multiplet scalars subject to |q +3 | 2 − |q −3 | 2 = ζ/6π (c.f. section 2 of [32]); if ζ = 0 on the other hand, the M = 0 point would become a viable vacuum as well. In either case, the low-energy phase appears to be gapless.
A discussion of gapped phases of somewhat similar models can be found in [33].
5 Discussion: what are exact results good for?
It is often said that exact results are valuable because they reach beyond semi-classical techniques. One of the main points of the present article has been that exact results can be useful also to shed further light on semi-classical regimes; they might inform us of subtle effects that we may have neglected in our earlier semi-classical analyses. With this vision in mind, one can begin comparing the semi-classical results on arbitrary N = 1 gauge theories with a U(1) R symmetry with the behavior expected from the Rains-function perspective. In many cases the two perspectives are compatible, at least as far as the dimension of the implied quantum Coulomb branch is concerned; see subsection 5.4 of [8]. In the special case of N = 2 SCFTs the Rains function is not useful for this purpose of course, since it is blind to the extended supersymmetry protecting the classical Coulomb branch. (However, at least in some simple cases-e.g. the Z 2 orbifold theory discussed in [8]-it seems to correctly account for the vacuum structure of the softly-broken theories, which have proportional Rains function.) More seriously perhaps, even in a couple of models without enhanced supersymmetry (such as the SO(5) BCI model [2]) our preliminary investigations actually indicate mismatch, and we are not sure how best to interpret the incompatibility; it could be that various other subtleties which the index is blind to undermine the connection in those cases after all. The precise connection between the Rains function and the low-energy superpotentials on R 3 × S 1 is certainly far from clear at this point. So far a satisfactory semi-classical understanding of the Rains function exists only in the "direct channel" where the β → 0 limit is interpreted as shrinking the S 1 [34] rather than decompactifying the S 3 ; there, it originates from perturbative effects in high-temperature effective field theory. Here, as in [8], we have suggested that the Rains function might also encode non-perturbative effects in the "crossed channel", i.e. on R 3 × S 1 . Such a remarkable connection between perturbative effects in the direct channel and non-perturbative effects in the crossed channel would be reminiscent of strong-weak duality, and a more systematic investigation of it would be quite worthwhile.
An alternative "crossed-channel" perspective, due to Shaghoulian [35], is given on the Rains function in appendix C. It involves a rather heuristic argument relating S 1 r S 1 →0 × S 3 to S 1 × S 3 /Z p→∞ , and does not appear to bear implications for our main discussions in this paper. We nevertheless find it an interesting complementary angle worth further exploration.
Finally, what (if any) implications our results might have for gauge dynamics of the ISS model on R 4 remains to be understood.
where E ISS susy = 511/12, and with Γ h the hyperbolic gamma function. (One can of course numerically evaluate Y ISS S 3 ≈ 0.423 [10], but here we are interested in the physical interpretation of the integral representation of Y ISS S 3 .) The precise meaning of the symbol in (A.2) is that after taking logarithms of the two sides we get an equality to all orders in β.
As alluded to in the main text, Di Pietro and Honda [34] have given a semi-classical explanation of the leading asymptotics-the first factor on the RHS of (A.2)-in the "direct channel" where the Cardy limit is interpreted as shrinking the circle of S 3 × S 1 . This was done through high-temperature effective field theory, building on Di Pietro and Komargodski's work [11].
A high-temperature effective field theory explanation for the E susy piece-i.e. the third factor on the RHS of (A.2)-follows from the more recent work [39].
The purpose of this appendix is to provide a similar "direct-channel" explanation for the Y ISS S 3 piece. Of course Y ISS S 3 looks like the S 3 partition function of the SQED arising after the SU(2)→U(1) Higgsing driven by the Rains function-compare for instance with the expressions in section 5 of [12]. The only part of it deserving further explanation is the factor e −2πi(− 4 5 i)x in the integrand, which as noted in section 5 of [8] appears to signal an induced FI parameter. The semi-classical explanation of this 'imaginary FI parameter' 4 ζ = − 4 5 i is as follows. (See section 3 of [40] for a related discussion).
On the Coulomb branch of an SU(2) theory-in the Weyl chamber with x > 0 to be concrete-massive fermions can be integrated out. Upon integrating out each of these massive fermions, a one-loop mixed gauge-U(1) R Chern-Simons term is generated with coefficient where R is the U(1) R charge, and n is the U(1) gauge charge of the fermion. More precisely, each such fermion is accompanied by a tower of KK modes which modify the above coefficient to Here we have used the regularizations m∈Z sign(m+T ) = 1−2{T } valid for T / ∈ Z. Note that in the case nx ∈ Z where there is a massless descendant, the regularization m∈Z sign(m + T ) = 0 valid for T ∈ Z guarantees that the massive modes in its KK tower do not contribute.
The mixed Chern-Simons coefficient (A.5) appears as an FI term in the (high-temperature) effective Lagrangian: Here ω is the background gauge field that couples the U(1) R current. For S 3 it is ω = i. (Setting ω = i(b + b −1 )/2 instead, generalizes the story to squashed S 3 with squashing parameter b.) The expression (A.6) in fact matches the FI coefficient arising from the Cardy limit of the index of a single chiral multiplet-c.f. the estimate (3.53) in [8]. To see specifically how the FI parameter ζ = − 4 5 i in (A.3) is generated, we note that (A.6) in this case implies with the first line on the RHS coming from the gauginos which have gauge charges n = ±2 and R-charge R = 1, and the second line coming from the two chiral-multiplet fermions with n = ±1 and R = −2/5. The other two chiral-multiplet fermions with n = ±3 have nx = ±3 · 1 3 ∈ Z, and hence (according to the remarks below (A.5)) do not contribute to the FI parameter.

B On the effective SQED near the pinch
In this appendix we present some further intuitive arguments enforcing plausibility of the superpotential (4.1).
First of all, we consider the 3d SQED of section 4 as an effective field theory valid only up to a cut-off scale (or "threshold") That is to say we devise this theory as an effective description of only an O( ) neighborhood of the pinch 5 at x (:= x − 1/3) = 0.
The cut-off in (B.1) is much below the cut-off scale of the effective 3d N = 2 U(1) gauge theory that is normally used (as in section 3 for instance) to describe generic points of O( 0 ) distance away from pinches.
The main significance of the cut-off (B.1) for the present discussion is as follows. Recall that the chiral multiplet Q I=3/2 yields states of charge −3, −1, +1, +3 on the Coulomb branch. However, because the states of charge ±1 have masses ≥ 1/3r S 1 , they lie far beyond the SQED cut-off /r S 1 . Therefore we will not worry about their quantization in the near-pinch effective theory.
The charge-±1 states should of course be included in the N = 2 U(1) theory away from the pinch, since their masses do not lie far beyond the U(1) theory cut-off 1/r S 1 ; so indeed the minimal-flux operators for generic distances |x | = O( 0 ) away from the pinch would be [12] with a the dual photon which we restore in this appendix. On the other hand, when the 3d SQED Coulomb-branch parameterx (↔ x /r S 1 ) is below the threshold /r S 1 , the minimal-flux operators can be taken to be Here the factors of 3 in the denominators of the exponent indicate that V ± create magnetic monopoles of charge ±1/3, which are the minimal allowed by Dirac quantization with the electric sources of charge ±3 present in the near-pinch effective SQED.
Recalling thatĝ 2 = r S 1ĝ 2 3 , the forms of (B.3) and (B.4) suggest that at the threshold (If the SQED had charges ±n, the exponent of the RHS would be n.) Now, matching at the threshold with the superpotential (3.4) of the U(1) theory away from the pinch suggests taking However, we might expect from the SQED/XYZ duality [20] that in the IR our SQED is alternatively described via an XYZ model. We propose that the natural identification would be between X, Y and the minimal-flux operators V ± , as well as between Z and M per usual [20]. The XYZ superpotential would then read Interpreting the XY Z term in (B.7) as a dynamically generated M V − V + term in the SQED picture would then justify (4.1).
Finally, note that the superpotential (B.7) (or alternatively (4.1) when written in terms of V ± , M ) is stationary at X = Y = 0 (alternatively at V ± = 0). Therefore the low-energy phase appears to be supersymmetric.

C Rains function from Shaghoulian's perspective
Shaghoulian has conjectured that the leading asymptotics of the supersymmetric partition functions on S 1 r S 1 →0 × S 3 and S 1 × S 3 /Z p→∞ are equal upon the "modular" identificatioñ with the tilded parameters those of the latter space [35]. The (rather heuristic) reasoning behind the conjecture is essentially as follows: both S 1 r S 1 × S 3 and S 1 × S 3 /Z p are torusbundles over S 2 ; the limit r S 1 → 0 shrinks one cycle of the torus, while p → ∞ shrinks the other, and modularity relates the two limits; effects of non-trivial fibration undermine the asymptotic equality of the partition functions at subleading orders, but (conjecturally) not at the leading order.
A Cardy-like [41] argument then gives the leading small-β asymptotics of the S 1 × S 3 partition function in terms of the supersymmetric Casimir energy E susy,p on S 1 × S 3 /Z p . Shaghoulian produced the e −E DK 0 factor in (2.1) by appealing to the supersymmetric Casimir energy in the zero-holonomy sector of S 1 × S 3 /Z p [35]. Below we show how the e − 4π 2 β L h piece in (2.1), and hence the Rains function, arise when the dependence of E susy,p on the "spatial" holonomies around the non-trivial cycle of S 3 /Z p is taken into account. 6 The starting point is Eq. (5.6) of Martelli and Sparks [42] for the supersymmetric Casimir energy on S 1 × S 3 /Z p . We are interested in the p → ∞ limit, and in the round S 3 case corresponding to b 1 = b 2 = 1 in [42]. We begin by considering a chiral multiplet χ with Rcharge r χ . Below we denote m/p in that work byx, and hence ν/p in that work by {ρ χ ·x}. Eq. (5.6) of [42] then immediately gives (from its linear term in u = r χ − 1) E χ susy,p (x) p p→∞ −→ r χ − 1 12 + 1 2 (1 − r χ )ϑ(ρ χ ·x).

(C.2)
It is clear that by summing over all the chiral multiplets, and also including the vector multiplets the above relation becomes The "vacuum" energy is of course obtained by minimizing E susy,p (x) over the moduli space of the "spatial" holonomiesx. For this minimized value, E min susy,p , we get in complete accord with (2.1). When L min h = 0 we of course recover the Di Pietro-Komargodski formula; this happens when E susy,p (x) is minimized in the zero-holonomy sector, since L h (0) = 0. But in the case of the ISS model, Higgs vacua withx = 0 minimize E susy,p (x), thereby modifying the Di Pietro-Komargodski asymptotics.