PHYSICAL REVIEW D 97, 125013 (2018)

Holonomy saddles and supersymmetry

† ‡ Chiung Hwang,* Sungjay Lee, and Piljin Yi School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea

(Received 2 May 2018; published 14 June 2018)

In gauge theories on a spacetime equipped with a circle, the holonomy variables, living in the Cartan torus, play special roles. With their periodic nature properly taken into account, we find that a supersymmetric gauge theory in d dimensions tends to reduce in the small radius limit to a disjoint sum of multiple (d − 1) dimensional theories at distinct holonomies, called H-saddles. The phenomenon occurs regardless of the spacetime dimensions, and here we explore such H-saddles for d ¼ 4 N ¼ 1 2 2 theories on T fibered over Σg, in the limits of elongated T . This naturally generates novel relationships between 4d and 3d partition functions, including ones between 4d and 3d Witten indices, and also leads us to reexamine recent studies of the Cardy exponents and the Casimir energies and of their purported connections to the 4d anomalies.

DOI: 10.1103/PhysRevD.97.125013

I. GLUING GUAGE THEORIES holonomy. A more typical situation with minimal super- ACROSS DIMENSIONS symmetry is, on the other hand, that at generic vev the supersymmetry is spontaneously broken; one finds some Gauge theories in a spacetime with a circle admit discrete choices of the holonomy vev with the supersym- holonomy variables as special degrees of freedom. With metry intact. In either case, the process of the dimensional the spacetime sufficiently noncompact, the infrared proper- reduction, as the circle size is taken to zero, is typically ties of the theory are often characterized by the vacuum ambiguous until we specify at which holonomy vev this is expectation values (vev) of the Wilson line operator [1],or done. When the holonomy is nontrivial, the net effect is that the traced holonomy along the circle. of the Wilson line symmetry breaking. In many theories, the holonomy variables are not exactly When the space is compact or, more precisely, has no flat at the quantum level and the Wilson line often serves as more than two extended directions, on the other hand, the an order parameter. For example, 4d N ¼ 1 pure SUðNÞ special nature of the holonomy variables manifest some- Yang-Mills on a large circle, or on a circle with super- what differently, as they must be integrated over for the path symmetric boudnary condition, are known to admit N integral. For example, the localization for the twisted distinct vacua, whose confining nature is dictated by equally partition functions produces integration over gauge hol- spaced eigenvalues of the holonomy, hence a vanishing onomy variables at the end of the procedure. This means Wilson line expectation value. If we replace the circle by a that one must be rather careful in taking a small radius limit. sufficiently small thermal circle, with the antiperiodic If one naively replaces this integration over the holonomy, boundary condition on gauginos, the eigenvalues become living in the Cartan torus, by one over Rrank, the Cartan clustered at the origin, signaling a deconfined phase at high subalgebra, one ends up computing partition function of a temperature as evidenced by a nonvanishing Wilson line vev. dimensionally reduced theory in one fewer dimension, with If supersymmetry is extended enough to ensure that the vev of the holonomy variable naively frozen at the these variables correspond to genuine flat directions at identity. quantum level, compactification on the circle generates an As we commented already, however, dimensional reduc- infinite number of superselection sectors, labeled by the tion of a single supersymmetric gauge theory on a circle may produce distinct gauge theories in one fewer dimen- ’ *[email protected] sion, depending on what holonomy vev s are available and † [email protected] chosen. For partition function computations on a compact ‡ [email protected] spacetime with a circle, then, this ambiguity of the dimen- sional reduction must also manifest. How does this happen? Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Since the original integration range is over the Cartan torus Further distribution of this work must maintain attribution to rather than the Cartan subalgebra and since the periodic the author(s) and the published article’s title, journal citation, nature of the holonomy variables is not to be ignored so and DOI. Funded by SCOAP3. easily, the answer is quite clear: As we scan the holonomy

2470-0010=2018=97(12)=125013(28) 125013-1 Published by the American Physical Society CHIUNG HWANG, SUNGJAY LEE, and PILJIN YI PHYS. REV. D 97, 125013 (2018) along the Cartan torus, we often find special places where concretely, now armed with varieties of exact partition the Wilson line symmetry breaking leads to supersymmet- functions, and to consider other ramifications. Also related ric gauge theories in one fewer dimension. are Refs. [16,17] which found exceptions to the purported This translates to the supersymmetric partition function universal connection between the Cardy exponents and ΩG d of theory G in d-dimensions reducing, in an appropriate the anomaly coefficients [18]. What we find here is that scaling limit, to a discrete sum of (d − 1)-dimensional such a universal expression is often an artifact of ignoring ZH ’ H-saddles other than the naive one at u 0 and that when partition functions d−1 of theories H s sitting at special H ¼ holonomies uH, modulo some prefactors, as the theory comes with matter fields in gauge representa- X tions bigger than the defining ones, this “exception” tends ΩG → ∼ ZH ; 1:1 to occur generically for all acceptable spacetimes, including d d−1 ð Þ 1 3 uH S × S . Furthermore, we will find similar failures for the Casimir limit in general, although this side proves to be where these uH’s are distributed discretely along the more subtle. periodic Cartan torus. In the vanishing radius limit, distinct We wish to emphasize that this phenomenon is inherent uH’s are infinitely far from one another, so that taking the to the supersymmetric gauge theories themselves, rather naive limit of replacing the holonomies by scalars amounts than merely a property of the partition functions thereof. to concentrating on a small neighborhood near a single uH. Note that the latter quantities need compact spacetime for Since uH ≠ 0 would be infinitely far away from uH ¼ 0 their definition. When the spacetime has at least three from the perspective of dimensionally reduced theories, one noncompact directions, these special values of the holon- is often mislead to consider the theory at uH ¼ 0, tanta- omy give various superselection sectors where the theories mount to replacing the Cartan torus by the Cartan sub- in one less dimension are equipped with supersymmetry ΩG algeba, and ends up computing a wrong scaling limit of d . intact at quantum level. Nevertheless, the partition func- We will call these special holonomy values uH’s (and the tions in general and the Witten indices in particular offer supersymmetric theories sitting there) the holonomy sad- handy tools for classifying these special holonomies, which dles, or H-saddles. The authors of Ref. [2] had introduced is why we concentrate on computation of these quantities in this concept and thereby resolved a 15-year-old puzzle this paper. [3–5] on Witten indices of 1d pure Yang-Mills theories This paper is organized as follows. In the rest of this [6–11]; in retrospect, the puzzle had originated from a introductory section, which also serves as a rough sum- simple misconception that only the naive uH ¼ 0 saddle mary, we will overview supersymmetric twisted partition (and its images under the shift by the center) contributes to functions and give a broad characterization of H-saddle the right-hand side. Since the holonomy moduli space is phenomena. This phenomenon of H-saddles and their present universally for spacetimes with a circle, at least consequences will be studied in the subsequent sections classically, and since the holonomy must be integrated over for a large class of 4d N ¼ 1 theories defined on compact for compact enough space, it is clear that this H-saddle spacetimes which are T2 fiber bundles over smooth phenomenon will occur for twisted partition functions Riemannian surfaces. regardless of spacetime dimensions. Section II will review a recent construction of A-twisted For field theory Witten indices [12], for example, partition functions in such backgrounds, and recall the H-saddles dictate how the Witten indices of gauge theories detailed computational procedure. This is then extended to in adjacent dimensions could be related. Witten indices can the so-called “physical” backgrounds, one special case of easily differ in different dimensions despite the standard which is the superconformal index (SCI). Section III will rhetoric that compactification on torus does not change the classify the Bethe vacua in the small and the large τ limits. number of vacua. A well-known modern example of such The Bethe vacua are easily seen to be clustered into disparities is how the 1d wall-crossing phenomena does not subfamilies, each of which can be regarded as the Bethe manifest in 2d elliptic genera. H-saddles now give us a vacua of some 3d theories sitting at special value of the rather concrete way to relate such topologically protected holonomy. Although the latter viewpoint is physically quantities across dimensions, in a very definite manner. better motivated in the small τ limit, which we can really The importance of the holonomy in relating supersym- view as a compactification to 3d, the other limit of large τ metric theories between different dimensions has been follows the same pattern thanks to SLð2; ZÞ property of the noted elsewhere, if somewhat sporadically. Notable exam- fiber T2. Even when the SLð2; ZÞ is not available, such as ples are due to Aharony and collaborators [13,14] who in SCI’s, such clustering of Bethe vacua do occur as well, observed how a Seiberg-dual pair of 4d=3d theories may although, as we will see in Sec. IV. translate to multiple such in 3d=2d as well as an even earlier These limiting behaviors of Bethe vacua imply that a 4d work in Ref. [15] where, again, a 2d limit of a 3d mirror gauge theory typically decomposes into a disjoint sum of symmetry is explored. Our study can be viewed as an effort several, potentially distinct 3d theories: The 3d limit of 4d to explore such phenomena much more systematically and supersymmetric partition functions becomes a sum of

125013-2 HOLONOMY SADDLES AND SUPERSYMMETRY PHYS. REV. D 97, 125013 (2018) partition functions of these 3d theories, albeit with extra This is, however, not quite true in general. If the theory exponential factors. A special case of this is the 4d Witten admits a continuum spectrum whose energies are bounded 0 index, reexpressed as a sum of Witten indices of the below by Egap > , the trace (1.2) actually produces associated 3d theories at H-saddles, clearly without the F −βQ2 F −β extra exponential prefactors. In Sec. IV, we explore such −1 −1 Egap Trð Þ e ¼ TrkernelðQÞð Þ þ Oðe Þ: ð1:4Þ limits for various background geometries and spacetime. One noteworthy corollary here is that the Cardy exponents, This subtlety is relatively easy to handle since one may be and even the Casimir energies, to a lesser degree, would → ∞ able to scale Egap þ first, without affecting the ground generally deviate from the existing proposals [18–20], 0 state counting. When Egap ¼ , on the other hand, sepa- connected to various 4d anomalies. As we will see these 0 rating out the continuum contributions becomes something proposals are often tied to the naive uH ¼ saddle which of an art. One popular scheme in the face of such gapless may or may not be the dominant saddle. We should note, ν’ ’ asymptotic directions is to insert the chemical potentials s however, that the Casimir limit of SCI s is somewhat special for global symmetries F’s, in that the microscopic derivations in Refs. [21,22] and the anomaly connection thereof proved to be robust, despite the F ν −βQ2 Trð−1Þ e Fe ; ð1:5Þ presence of nontrivial H-saddles. We comment on this toward the very end of this paper. where ½F; Q¼0 is needed for this quantity to remain controllable. We may even have F involving an R-charge, A. Twisted partition functions and the Euclidean time as long as we choose one particular supercharge Q care- Twisted partition functions, to be denoted by Ω through- fully so that the two mutually commute. In many practical out this paper, are obtained by computing the partition examples, coming out of string theory, this option is function with an insertion of the chirality operator ð−1ÞF.A available and exploited. requisite for ð−1ÞF is that there is a notion of natural Although such an insertion of chemical potentials may Euclidean time coordinate, forming a circle S1. With the appear an innocent device to keep track of global charges of natural Z2 action of supersymmetry, say, Q, which anti- states, this is true only for theories suitably gapped to begin commutes with the chirality operator, this insertion allows with. With gapless theories, this chemical potential modifies generic bosonic states to cancel against fermionic states, the Lagrangian in such a way that asymptotic directions that and leaves behind a special subset of the Hilbert space. transform under F become massive. Since one gaps the When the theory is suitably gapped andthe space is taken to asymptotic dynamics artificially, one should not expect the be Td−1, this quantity would compute the Witten index [12], twisted partition function to behave nicely in the ν ¼ 0 limit. integral and enumerative of supersymmetric ground states. Recovering information about the original theory prior In recent years, many types of supersymmetric partition to turning on ν is hardly straightforward although well- functions have been proposed with the accompanying com- established routines exist for a handful of classes of putational tricks under the banner of the localization. The theories. The Atiyah-Patodi-Singer index theorem, appli- superconformal index [23,24] is one such class of well-known cable to non-linear sigma models onto manifolds with and much-computed objects, while the elliptic genera in 2d boundary, is one such classic example while a more recent [25,26] and the refined Witten indices in 1d have been one is the 1d gauged quiver quantum mechanics as developed to a very sophisticated level [27,28]. explained in Ref. [10]. Beyond these few, however, no The length of Euclidean time circle, β, may be inter- general prescription is known. Despite such difficulties, the preted as the inverse temperature. For the twisted version, twisted partition functions of such mass-deformed theories however, this parameter is often argued to disappear from proved to be very useful for some tasks, e.g., most notably, the end result, since supercharges Q act as a one-to-one checking Strong-Weak dualities [19,29–33]. map for positive energy bosonic and fermionic states. This What do we do to actually evaluate such objects? The disappearance is, of course, a desired feature of the index, popular trick of the localization naturally enters the story since the latter was designed, to begin with, to count Bose- when chemical potentials are turned on. The chemical Fermi asymmetry of the ground state sector. The twisted potentials ν tend to push the dynamics to a small subset of partition functions the configuration space or even to a small part of the spacetime; the localization method is then invoked to 2 Trð−1ÞF e−βQ ð1:2Þ amplify this effect maximally, whereby the path integral is reduced to that of a Gaussian path integral followed by a are thus argued to be projected to the ground state sector finite number of leftover zero-mode integrals from vector multiplets. −1 F An interesting fact about the localization routine is that, TrkernelðQÞð Þ ; ð1:3Þ 2 in the final expression, β as in e−βQ automatically drops which is necessarily integral and enumerative. out. This may happen because the system is fully gapped by

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ν so that the naive Bose-Fermi cancellation works perfectly. field content as the original theory. We will label this theory In fact, this is the case for main examples of this paper, on Md−1 by the same label G, whose partition function namely 4d N ¼ 1 theories which are maximally mass would also produce a localized path integral as deformed by ν’s. As such, we will work with Z ZG ν˜ rank ˜ ˜; ν˜ ð Þ¼ d ufGðu Þ: ð1:12Þ 2 ΩðνÞ ≡ Trð−1ÞF eνFe−βQ : ð1:6Þ localization Past experiences with such objects tell us that the limit is often equipped with an extra exponential factor, Sometimes this lift of the asymptotic flat direction by ν is incomplete, which tends to happen in odd spacetime rank SCardy=βþsubleading terms β g βu˜;βν˜ → e G f u˜;ν˜ as β → 0; dimensions. In such cases, the localization still removes Gð Þ Gð Þ β by computing, implicitly, a limit of β → 0, ð1:13Þ F νF −βQ2 F νF −βQ2 Cardy Trð−1Þ e e ¼ limTrð−1Þ e e : ð1:7Þ where SG is the Cardy exponent [34]. Then, the naive localization β→0 expectation is

This has been first noted for 1d systems [10] and further G SCardy=βþsubleading terms G Ω βν˜ → e G ×Z ν˜ as β → 0; 1:14 checked in Ref. [2]. ð Þ ð Þ ð Þ Although we have described how the Euclidean time where the two partition functions were computed for the span β naturally drops out in the localization computation, one and the same gauge theory, G, only in two different the resulting twisted partition function Ω can actually retain Cardy “ β indirectly, via the chemical potential ν understood as a dimensions. The exponent SG would dictate high ” holonomy associated with an external flavor gauge field, temperature behavior of the twisted partition function.

νF B. Holonomy saddles: An overview i∂t → i∂t þ : ð1:8Þ β However, comparing (1.10) and (1.12), one easily realizes that this is too rash. A limiting formula like In the small β limit, one has an option of keeping ν finite or (1.14) would hold if and only if the toroidal du integration keeping the alternate variable ν˜ finite with in (1.10) can be opened up to a planar integration in (1.12); since this is a discontinuous process, this may be justified ν ¼ βν˜: ð1:9Þ only if, in the small β limit, the infinitesimal region around u ¼ 0 contributes dominantly to the integral. The so-called “real” masses in 3d are, for example, nothing As a simple counterexample, which may look trivial but is but such finite ν˜. illuminating nevertheless, consider an SUð2Þ gauge theory We can think about something similar for the gauge with matter multiplets with integral isospins only. Suppose holonomy variables, u, which enter the localization for- that we choose the range of u suitable for the odd isospins, mulas for Ω as say [0, 1) in our convention where weight vectors are Z normalized to be integral and the holonomies are divided G rank by 2π. The integrand gðu; zÞ would then be invariant under Ω ðνÞ¼ d ugGðu; νÞ: ð1:10Þ the shift related by the center, u → u þ 1=2; the expansion of g around u ¼ 1=2 will look exactly the same as that around ν We introduced the label G to denote the theory and gGðu; Þ u ¼ 0, so that the integral near u ¼ 0 and that near u ¼ 1=2 is from the Gaussian integrals over nonzero modes. If we contribute exactly the same amount. Although this particular introduce the similarly rescaled variables u˜ ¼ u=β in the problem is easily countermanded by an overall factor 2, it small β limit, the following object where the integral is does warn us of a generic danger in confining ourselves to taken over u˜ instead of u, small regions near u ¼ 0 when u is a periodic variable. Z What happens generically is that the small β limit of Ω βν˜ Z’ ∼ dranku˜ lim βrankg ðβu˜; βν˜Þð1:11Þ ð Þ is actually a sum of s for several disjoint theories β→0 G on Md−1, such that the limit has the form X β G SCardy=βþsubleading terms H appears naturally. Since is taken to be arbitrarily small, Ω ðβν˜Þ → e H × Z ðν˜Þ as β → 0; the periodic nature of u is now lost. What would such an uH integral compute? ð1:15Þ To be precise, let us consider a spacetime of type 1 S × Md−1. In the small radius limit, the dimensional instead of (1.14). The summand is labeled by special 1 reduction on S produces a theory on Md−1 with the same holonomies values uH around which the dimensional

125013-4 HOLONOMY SADDLES AND SUPERSYMMETRY PHYS. REV. D 97, 125013 (2018) reduction gives a theory H, with potentially smaller field H-saddle with decoupled Uð1Þ’s or pure Yang-Mills content than the naive dimensional reduction of the original sectors, as long as appropriate Chern-Simons coefficients theory G. The integration over the toroidal u’s reduces to are generated from integrating out heavy modes. What ’ patches of planar integrations near such uH’s while con- remains unchanged, though, is that contributing uH s occur tributions from the rest become suppressed by e−1=β. discretely. See Sec. III for a complete characterization of The accompanying limit in the localization formulas should H-saddles. be similarly What we described above is a generic feature of gauge theories, due to the special roles played by the holonomy rank variables: The toroidal nature of the holonomy variables β gGðuH þ βu˜; βν˜Þ appears lost in the small radius limit, yet the periodic nature SCardy=βþsubleading terms → e H × f ðu˜; ν˜Þ as β → 0 Z H should not be ignored. Integrating over such holonomy variables, such as for gauge theories on compact spacetime ZH ν˜ rank ˜ ˜; ν˜ ð Þ¼ d ufHðu Þ: ð1:16Þ with a circle, we must remember to keep careful track of these holonomies. For supersymmetric partition functions, it so happens that there are multiple saddles which contribute The discrete locations uH are infinitely separated from one another, in the limit of β → 0, and thus cannot be captured to the total expression, each of which can be understood as a by the u˜ integration near the origin alone. partition function of some other theories in one fewer dimension. With at least one nontrivial uH, one must ask which of these saddles is dominant in the small β limit; one In this paper, we will consider implications of H-saddles 2 might have expected that the theory G at the naive saddle at in the context of 4d N ¼ 1 theories on T fibered over Riemannian surfaces of arbitrary genus Σ . General parti- uH ¼ 0 is the dominant one, given its largest light field g content, but it turns out this is generally false. In particular, tion functions of this class were given very recently via the d−1 G H when M −1 is T whereby Ω and Z would both so-called Bethe ansatz equation (BAE) [19]. In this d 2 compute the Witten indices, each of admissible uH generi- approach one first consider compactification on T reduc- cally contributes on equal footing. In other words, (1.15) ing the system to 2d, and vacua and partition functions are would reduce to found via the effective 2d twisted superpotential of X Coulombic variables. The vacua thus found is called G H Bethe vacua [35]. As such, this construction works for a I ðβν˜Þ → I −1ðν˜Þ as β → 0: ð1:17Þ d d restricted class of 4d theories, where, given the matter uH content, the superpotential is appropriately suppressed to allow maximal flavor symmetry. On the other hand, the This means that uH must be such that the theory H there must have supersymmetric vacua.1 construction is ideal for the investigation of the H-saddle Let us take d ¼ 4 N ¼ 1 theories, which will be our phenomena since the latter turns out to be quite manifest in 2 main examples. With generic holonomies, u, the 3d theory the classification of BAE vacuum solution. T fiber has two would be a product of free Uð1Þ’s whose vacuum manifolds circles, whose relative size is encoded in the complex are generically lifted by a combination of induced FI structure τ. In the large and the small τ limit, one of the two constants or Chern-Simon levels. Light charged multiplets, circles becomes very small relative to the other, and as such say with the charge λ with respect to the Cartan Uð1Þ’s, the phenomenon of H-saddles emerges on the smaller of would be needed for vacua with unbroken supersymmetry. the two circle directions. W This constrains the position u to quantized values, uH, The 2d twisted superpotential is naturally a function of the pair of the holonomies along T2, packaged into rank- many complex coordinates u. Bethe vacua are particular λ · uH ∈ Z ð1:18Þ 2πi∂W 1 holonomy values, u, where e ¼ . This vacuum for each such λ. In this manner, a contributing H-saddle is condition is periodic under integral shifts, u → u þ n þ mτ 2 equipped with a set of unbroken charges λ’s, which in turn where τ is the complex structure of T ,whichisagauge defines the 3d theory H at uH, modulo UV couplings in the equivalence. What we will discover is that these Bethe vacua 3d sense inherited and computable from the original 4d appear in clusters, scattered at discrete places in the unit cell, theory G. What we described here is a little simplified; it τ ≃ ˜ σ˜ ≃ σ turns out that when the matter content is not symmetric u= uH þ ;u uH þ ; ð1:19Þ under charge conjugation, one can actually have an where u˜ H ∼ 1=τ and uH ∼ τ with coefficients between 0 and þ 1 1forτ → i0 ;i∞ limits, respectively. Each such H-saddle When the theory possesses a gapless asymptotic sector, so σ˜ ’ σ ’ that the twisted partitions do not compute the true index, this would come with multiple and finite sand s, which condition should be relaxed since the twisted partition functions represents supersymmetric vacua in the reduced 3d theory at capture the so-called bulk part of the true index. such H-saddles. Depending on which circle is called the

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Euclidean time, the limit will also compute the Cardy Now let us see how field theory data enter this universal exponents or the Casimir energies at each of such H-saddles. formula. The basic variables are holonomies along T2. The The question of which saddle dominates becomes a non- gauge holonomies u1, u2, along these two circles trivial issue, generically compromising folklore on univer- Z Z sality of such exponents. We will revisit the asymptotics of 1 1 a1 ¼ A; a2 ¼ A; ð2:4Þ the partition functions in Sec. IV. 2π S1 2π S1 β1 β2 N II. 4D = 1 PARTITION FUNCTIONS AND BAE of 2π period each, are combined to Recently the supersymmetric partition function of a four- τ − dimensional N ¼ 1 theory on M4 was discussed [19] u ¼ a1 a2 ð2:5Þ where M4 is a torus bundle over a Riemannian surface Σg: and, similarly for the complexified flavor holonomy 2 T → M4 → Σ : ð2:1Þ g ðFÞ ðFÞ ν ¼ a1 τ − a2 : ð2:6Þ The partition function is obtained by considering an A-twisted theory on Σg via nontrivial background flux All of these holonomy variables obey n ¼ g − 1 for the Uð1Þ symmetry group. Here we give a R R ∼ 1 ∼ τ ν ∼ ν 1 ∼ ν τ quick summary of results in Ref. [19]. We will use G to ua ua þ ua þ ; A A þ A þ ; ð2:7Þ denote the N ¼ 1 theory itself, while G and G are the gauge group and the associated Lie algebra, respectively. under the respective large gauge transformations. Before we plunge into details, a cautionary remark is in The effective action of the theory is fully governed by order. Much of what follows will be phrased in terms of two holomorphic functions W and Ω, which are called the gauge holonomies on T2, valued in two copies of the Cartan effective twisted superpotential and the effective dilaton. torus. As was emphasized by Witten and others [36–40], Each component in (2.2) and (2.30) is then obtained from however, the space of connections on T2 and higher those two quantities, dimensional torii can in general admit disconnected com- ∂W ponents, even when the gauge group is connected. Such F 1ðu; ν; τÞ¼exp 2πi ; ð2:8Þ possibilities are not taken into account in the computations ∂τ outlined below, so we will confine ourselves, in this paper, to theories with G ¼ SUðr þ 1Þ, SpðrÞ. ∂W ∂W ∂W F 2ðu;ν;τÞ¼exp 2πi W − ua −νA −τ ; ∂ua ∂νA ∂τ A. A-twisted background ð2:9Þ 2 Compactifying on T , one has a two-dimensional N ¼ð2; 2Þ supersymmetric theory with infinite Kaluza- ∂2W u; ν; τ H ν; τ 2πiΩðu;ν;τÞ ð Þ Klein modes. Performing the path integral via localization, ðu; Þ¼e detab ; ð2:10Þ ∂ua∂ub summing over the magnetic flux sector, and then evaluating the resulting residue formulas, the partition function is ∂W Φ ðu; ν; τÞ¼exp 2πi ; written universally as a sum over the so-called Bethe vacua, a ∂u X a p1 p2 g−1 ∂W Ω ¼ F 1ðu ; ν; τÞ F 2ðu ; ν; τÞ Hðu ; ν; τÞ ΠAðu; ν; τÞ¼exp 2πi : ð2:11Þ ∈S ∂ν u YBE A Π ν; τ nα × Aðu; Þ : ð2:2Þ For semisimple G, the W-bosons and their superpartners do α not contribute to the effective twisted superpotential, and where p1, p2 ∈ Z are the two Chern numbers of the circle only charged chiral multiplets contribute. The contribution bundles: of a single chiral multiplet is given by Z Z 1 1 3 2 τ 1 W − u u − u p1 ¼ dA ;p2 ¼ dA : ð2:3Þ Φ ¼ þ þ 2π KK1 2π KK2 6τ 4 12 24 Σg Σg 1 X∞ k −1 kþ1 þ ðLi2ðxq Þ − Li2ðx q ÞÞ; ð2:12Þ τ ¼ τ1 þ iτ2 is the modular parameter of the torus with 2π 2 ð iÞ 0 β2 k¼ τ2 , where β1 and β2 are two radii of the torus. Note that ¼ β1 2 one can perform a modular transformation of T such where Li2 is a polylogarithm function. We have defined 2πiu 2πiτ that ðp1;p2Þ¼ðp; 0Þ. x ¼ e and q ¼ e .

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S Classification of the Bethe vacua BE starts with solving with

1 Φ ∞ ¼ a; ð2:13Þ Y 1 − t−1qnþ1 nþ1 Γ0ðw; τÞ¼ ; ð2:21Þ 1 − nþ1 which can be expressed more explicitly as n¼0 tq YY ρa Φ ν;τ Ψ ρ ν ;τ i aðu; Þ¼ ð i · u þ i Þ ; 1 ρ 2πiw 1 − 2πiw i i fðwÞ¼exp 2π Li2ðe Þþw logð e Þ : ð2:22Þ 2 i Ψðw;τÞ ≡ e−πiw =τθðw;τÞ−1; Y θðw;τÞ¼q1=12t−1=2 ð1 − tqkÞð1 − t−1qkþ1Þ; ð2:14Þ For the flux operator, the products are taken over every ρa k≥0 chiral multiplet and the weights of its representation. i is ρ ωα the ath Cartan charge of the gauge weight i while i is the 2πiw ν ≡ ν with t ¼ e , where i · Fi is the net sum of flavor αth Cartan charge of the flavor weight ωi. With nontrivial chemical potentials for the ith multiplet with flavor charges background flux nα for the flavor group, the flavor flux Fi. The product is over chiral multiplets, labeled by i for operator Π contributes to the partition function as in (2.2). ρ A each gauge multiplet and the charges i thereof with respect Before proceeding, however, we should mention a few to the Cartan. Then, SBE, which is nothing but the set of the caveats. The most obvious is the presence of nonanomalous 1 supersymmetric vacua for the 2d twisted superpotential, Uð ÞR symmetry. Since this symmetry is used for the may be defined as topological A-twisting [43], one must further require the Uð1Þ charges of chiral multiplets r be integral. As such, S Φ ν; τ 1 ∀ ≠ R i BE ¼fuj aðu; Þ¼ ; a; w · u u; neither for pure N ¼ 1 Yang-Mills theories nor for typical N 1 ∀w ∈ WGg=WG: ð2:15Þ ¼ superconformal theories would this methodology be applicable. Later, however, we will go to a slightly The additional constraint, that vacua invariant under any different class of spacetime, with the same geometry but 1 part of the Weyl group WG are to be ignored, has been noted different fluxes, so that the integrality of the Uð ÞR charges in the past literature, most notably Refs. [41,42]. can be relaxed. There, the partition function formula With the explicit form of W in (2.12), one can similarly should be applicable to N ¼ 1 superconformal theories compute the rest. The effective dilaton contribution is with a-maximized R-charges. given by A less obvious caveat, although quite rampant in the exact partition function computations, comes from the flavor Y Y 2π Ω −1 chemical potentials. As mentioned earlier, the latter means e i ¼ Ψðρ · u þ ν ; τÞri i i that the theory is artificially mass deformed, and that we may i ρi∈Ri Y not be able to recover physics of the original undeformed × ηðτÞ−2rankðGÞ Ψðα · u; τÞ ; ð2:16Þ theory easily. This danger is present in all exact twisted α∈g partition function computations via the localization, but perhaps a little more so in this class since this computation where the product in the second parentheses is taken over turns on a chemical potential for each and every chiral the roots of g ¼ LieðGÞ. From the effective dilaton (2.16) multiplet: One must always take extreme care in interpreting and the effective twisted superpotential (2.12), one can the results. H obtain the handle-gluing operator as well. The explicit With SUðNÞ theories with Nf fundamental flavors, for forms of the fibering operators F and the flavor flux example, this current computation would give a simple operators Πα are 4 numerical Witten index for all Nf if we take M4 ¼ T . Y Y However, such theories are often equipped with a manifold F ν; τ Ξ ρ ν ; τ 1;2ðu; Þ¼ 1;2ð i · u þ i Þ; ð2:17Þ of the vacuum moduli, which, since the number of ρ ∈ i i Ri spacetime dimensions is larger than two, should have made Y Y ωα the notion of the Witten index ill defined. One must really Π ν; τ Ψ ρ ν ; τ i Aðu; Þ¼ ð i · u þ i Þ ; ð2:18Þ regard these partition functions as probing theories that are ρ ∈ 2 i i Ri compactified on T with flavor holonomies necessarily turned on along the two circles. where we have defined

πi 3 πi 2w − 6 w Ξ1ðw; τÞ¼e3τ Γ0ðu; τÞ; ð2:19Þ B. Alternate backgrounds and superconformal index ∞ 3 2 Y τ So far we have considered an A-twisted theory, whose 2πiðw −w þwτþ 1 Þ fðw þ k Þ Ξ2ðw; τÞ¼e 6τ 4 12 24 ; ð2:20Þ supersymmetric background includes the nontrivial Uð1Þ fð−w þðk þ 1ÞτÞ R k¼0 gauge field of

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νR ¼ 0; nR ¼ g − 1: ð2:23Þ The authors of Ref. [19] proposed that the partition function in this background can be written in a similar On the other hand, there is another class of supersymmetric manner but with different operators H, F, etc., which will 1 backgrounds without the Uð ÞR flux, which is called be our working assumption, below. The flux operators are “physical gauge” in Ref. [19]: Y Y phys ρa Φa ðu;ν;νR;τÞ¼ Ψðρi · u þ νi þ νRðri −1Þ;τÞ i ; 1 − g i ρi∈Ri νR ¼ τ; nR ¼ 0; ð2:24Þ Y Y p phys ωα Π ν ν ;τ Ψ ρ ν ν −1 ;τ i A ðu; ; R Þ¼ ð i · u þ i þ Rðri Þ Þ ; ρ ∈ with g − 1 ≡ 0 mod p. This is achieved by starting with i i Ri p1 ¼ p and p2 ¼ 0, and taking a large gauge transforma- ð2:25Þ tion on R-symmetry that removes the R-flux in favor of the R-chemical potential. while the fibering operator for the circle 1 is

Y Y 1 Y lRðlRþ Þ phys 2 rankðGÞ 2l rankðGÞ F ðu;ν;νR;τÞ¼ Ξ1ðρi ·u þ νi þ νRðri − 1Þ;τÞ ð−1Þ ηðτÞ R Ξ1ðα · u þ νR;τÞ ; ð2:26Þ

i ρi∈Ri α∈G where where the pair of rotational chemical potentials that enter the usual superconformal index are identified. For the 1 − g superconformal index, the large and the small radius ν ¼ l τ;l¼ ∈ Z: ð2:27Þ R R R p limits are already discussed in literature quite extensively [16–18,20–22]. We reexamine these limits of the partition In addition, in the physical gauge, there is no effective function using the Bethe formalism. The subtlety with the dilaton contribution because nR ¼ 0. Thus, the handle- holonomy should be again present, and many results of the gluing operator is simply the Hessian determinant, previous section carry over to the new background verba- tim. Reference [19] initially motivated this construction via 1 ∂ log Φphys a large Uð1Þ transformation, as outlined above, from the phys ν ν ; τ a R H ðu; ; R Þ¼det : ð2:28Þ A-twisted cases with p ≠ 0. This cannot be really consid- ab 2πi ∂ub ered a derivation since the nonintegral values of ri ’s, As a result, the partition function in the physical gauge is inevitable for N ¼ 1 superconformal field theories, would given by be detrimental to such a process. On the other hand, an X alternate justification was also given by the same authors, 0 1 Ωphys ¼ F physðu ; ν; ν ; τÞpHðu ; ν; ν ; τÞg−1 where these BAE expressions for g ¼ , p ¼ are trans- 1 R R formed to the conventional form of the superconformal u ∈Sphys YBE index via contour manipulations. See Eq. (4.52). Πphys ν ν ; τ nα × A ðu; ; R Þ ; ð2:29Þ α III. BETHE VACUA FOR ELONGATED T2 where we restricted ourselves to the case p1 ¼ p and In this paper, for the sake of convenience, we will regard p2 ¼ 0, with the Bethe vacua circle 2 the Euclidean time. Then the large and the small Euclidean times correspond to, respectively, Sphys Φphys ν ν ; τ 1 ∀ BE ¼fuj a ðu; ; R Þ¼ ; a; β2 τ i → i∞ or i0þ: 3:1 ≠ ∀ ∈ ¼þ β ð Þ w · u u; w WGg=WG: ð2:30Þ 1 The small τ limit can be viewed as compactification along et al. Furthermore, Closset advocated that once we arrive at circle 2, while the large τ limit would be viewed as the “ ” this so-called physical gauge, the integral restriction on ri compactification along circle 1. This interpretation is can be lifted. possible as the size of the base Σg appears nowhere in the As such, this partition function is supposed to compute a partition functions, and also because only the ratio of the two 0 limit of the superconformal index [23,24] if we take g ¼ radii appears. For either compactification, the Kaluza-Klein 1 and p ¼ , towers will acquire a large mass shift at typical values of F 2 −β holonomy. This means that the low energy effective theory Ω 1 3 ; ≡ 3 −1 JþR GF 2H S ×S ðq xÞ TrS ½ð Þ q x e ; ð2:31Þ in the remaining three dimensions would be rank-many free

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Uð1Þ theories. At such a generic point we will find that the 3d m ≡ bw=τcð3:5Þ theory has supersymmetry spontaneously broken and thus cannot contribute solutions to the BAE. is an integer such that the real part of w=τ − m lies in [0, 1). What we will find is that solving 4d BAE will produce It follows that, for example, vacua clustered at some discrete and special choices of the holonomy. As τ → i∞, these special holonomies uH line up þ along circle 1, while for the other limit τ → i0 they line up fλ · uH=τg¼0; fρˆ · uH=τg ≠ 0; ð3:6Þ along circle 2. At such special places uH, i.e., at H-saddles, with the holonomy, the gauge charge set of the chiral in the large radius limit of A-twist cases (3.2), and multiplet will split,

fρg¼fλg ∪ fρˆg; ð3:2Þ ˜ ˆ ˜ fλ · uH=τ˜g¼0; fρ · uH=τ˜g ≠ 0; ð3:7Þ where those chirals associated with λ’s will produce light 3d chiral multiplets at the bottom of the KK tower, while those with u˜ ≡ u=τ and τ˜ ¼ −1=τ, in the small radius limit of associated with ρˆ will produce KK towers with no such light A-twisted cases. Note that we also use the same curly 3d field. The vector multiplets in the adjoint representation bracket f g as a symbol for sets, as is customary. would be also similarly decomposed, and a spontaneous Hopefully, the distinction between these two is self-evident. symmetry breaking by a Wilson line will occur G → H; ð3:3Þ A. H-saddles in the small τ˜ = − 1=τ limit τ → ∞ → 0 where the unbroken, 3d gauge group H has the same rank as We will start with i ,orq , although the the 4d gauge group G. The new 3d gauge theory H, typically discussion below may appear more natural in the other τ → 0þ τ with smaller light field content, both vectors and chirals, limit of i . In the small limit, the role of the two than the naive dimensional reduction of the theory G, circles will be exchanged, so that our finding here will carry over almost verbatim, via an SLð2; ZÞ action. appears. The holonomy uH’s and the new theories there, we will collectively call H-saddles [2]. We start by noting that there are two different types of This classification of H-saddles, to be explained in detail solutions to the 4d BAE equation in such a limit. The first below, could include some special cases. The case with type comes from assuming u remains finite under the scaling, and as such we find fλg¼fρg,forexampleatuH ¼ 0, would produce the 3d theory with the same field content as the naive dimensional reduction. Most of the related literature have assumed, q−1=12 effectively, that this type of H-saddle is either the only kind Ψðw; τÞ → t−1=2 − t1=2 or the dominant one. The other extreme λ ∅,ormore Y Y f g¼ 3 −ρ 2 −1=2 ρ 2 1=2 −ρa Φ ; τ → Φ d ≡ Λa i= − i= i generally the cases where λ do not span the charge vector aðu Þ a G ½x yi x yi ρ space, would produce 3d theory with a pure gauge sector. i i Also, H-saddles that include an Abelian subgroup in H ð3:8Þ require more care, since a large 3d Fayet-Iliopoulos constant can be generated even though the 4d theory had no such Abelian factor. For the latter types of H-saddles, a little more with care must be given, which we will go through in Secs. III C. P − ρa 12 ρ = and III D. Λa i; i i G ¼ q ; ð3:9Þ While most of this section is devoted to the A-twisted “ ” Q case, the physical version is really no different. The BAE 2π ν ρ ρa ≡ i i i ≡ i equations remain identical to those of the A-twisted case, where yi e and x axa . The resulting BAE 1 − 1 τ Φ3d 1 except the additional Uð ÞR chemical potential ðri ÞlR equations, a ¼ , look exactly the same as the 3d BAE for the chiral fields. As such, the large τ limit of the equations [30–33] of a dimensionally reduced theory with “physical” version requires extra care, which will be the same field content. The only unexpected feature is that addressed in Sec. III. E. 2πiτ now plays the role of a UV FI constant for the trace- A comment on a notation is in order, to avoid confusion. part UPð1Þ gauge field when the latter is present. ρa ≠ 0 For a holonomy variable w with the natural periods, τ and 1, If i , the locations of the solutions, xa, could be we will define its “fractional” part, fw=τg,as too far away and conflict with the above truncation. Thankfully, however, this never really happens for 4d fw=τg ≡ w=τ − m; ð3:4Þ theories with no Uð1Þ factor, as is necessary for the asymptotic freedom. With the gauge group G being at where most a product of semisimple Lie groups, we find

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Λa 1 λ ’ ρ ’ G ¼ ð3:10Þ for some subset i s of charge vectors i s and accompany- ing integers m’s. We can then invoke the language of 2 generally. Wilson-line symmetry breaking and consider uH as the On the other hand, a very different kind of solutions also point where λi is nearly massless while the others are exist. Suppose we consider a regime where x scales to zero, heavily gapped. The holonomy in question is along the along with q → 0. For example let us look for solutions direction 1, which was the fiber circle. na near xa ∼ q. At such points q → 0 might leave a factor In the neighborhood of uH, we write for the nearly ρi ð1 − q=x yiÞ as well. Note that we can still make use of the massless ones ρ infinite product formulas only if q=x i → 0 as well, so one ρ λ u ν mλ τ λ σ ν ; : must first shift the argument i · u by an integral multiple of i · þ i ¼ i þ i · þ i ð3 17Þ τ, using the identity with σ and νi understood to remain finite while τ → i∞.In 2 2 θðw; τÞ¼ð−1Þmtmqm = θðw þ mτ; wÞ; ð3:14Þ such a neighborhood of uH, then, we introduce new shifted σ ≡ − Cartan variables a ðu uHÞa such that bringing λa λa i m i 2πiσa xa ¼ q za ;za ≡ e : ð3:18Þ Ψðw; τÞ¼ð−1ÞmΨðfw=τgτ; τÞ 2 The shift of u to σ is integral in τ, so can at most change ¼ð−1Þme−πiτfw=τg þπiτfw=τgq−1=12 a a the sign of Ψ and Φa’s, and for these light fields, we can 1 proceed exactly the same way as the naive limit as in (3.8). × Q ; ð3:15Þ 1 − k 1 − kþ1 ρˆ’ k≥0ð ftgq Þð q =ftgÞ For heavy fields, from s, on the other hand, the infinite product for the relevant θ reduces to 1, and leaves behind a 2πiτfw=τg Φ where ftg ≡ e with fw=τg obtained by an additive prefactor only. Then, near uH, contributions to a from shift of w by −mτ for an integer m; the real part of fw=τg light and heavy modes accumulate to lies between 0 and 1. The aim is to make the infinite 3d;H product well defined in the limit of q → 0. At generic Φaðu; τÞ → Φa Y Y values of w, each of the infinite products reduces to 1, −λ 2 −1=2 λ 2 1=2 −λa ≡ ð−1ÞMΛ ½z i= y − z i= y i ð3:19Þ which shows that there is no solution to 4d BAE, Φ 1. H i i a ¼ i λ The new type of solutions will have to cluster around i ρ special places, called H-saddles, where the real part of ð i · for some integer M, with ν τ ρ ’ u þ iÞ= for i s are integral and not necessarily zero. In Y Y the large τ limit, with ν kept finite, such H-saddles, say u , a −πiτρˆaðfðρˆ ·u þρˆ ·σþν Þ=τg2−fðρˆ ·u þρˆ ·σþν Þ=τgÞ i H Λ ≡ e i i H i i i H i i ; are located at H i ρˆi λ ≃ τ ð3:20Þ i · uH mλi ð3:16Þ

where we used ΛG ¼ 1. 2This can be seen from the Weyl character formulas, with the As such, the solutions to the 4d BAE are neatly Cartan generators C’s in an irreducible representation R of a decomposed into union of the 3d BAE solutions at these G semisimple gauge group , distinct H-saddles, P λ ρ −1 jwjeu·wð Rþ W Þ χ u ≡ u·C Pwð Þ 3d;H Rðe Þ trRe ¼ ρ ; ð3:11Þ u ∶1 Φ u ∪ u σ ∶1 Φ σ ; −1 jwj u·wð W Þ f ¼ að Þgτ→i∞ ¼ uH f H þ ¼ a ð Þg wð Þ e P ð3:21Þ where λ is the highest weight, ρ ≡ α∈Δ α=2 is the Weyl R W þ vector, and the sums on the right-hand side are over the Weyl λ τ λ ’ group. It follows that χ eu χ ewðuÞ for any Weyl reflection where i · uH= are real and integral for some subset i sof Rð Þ¼ Rð Þ ρ ’ w and thus i s. For actual admissible BAE vacua, we must exclude X X those solutions that are fixed under some Weyl reflection but ρ · wðuÞ¼ ρ · u; ð3:12Þ it is clear that this does not interfere with this classification ρ∈R ρ∈R into clusters around H-saddles. There are two special Since u is arbitrary and since this holds for any Weyl reflection w, subclasses of H-saddles that are noteworthy. One is when it follows fλg¼fρg, which occurs when the gauge holonomy is X trivial. This of course corresponds to the naive dimensional ρa 0 3 ¼ ð3:13Þ reduction, where the 4d BAE reduces to 1 ¼ Φ d above. ρ∈ a R That is, we included the first type of solutions on equal for each irreducible representation R of G. footing as well, by assigning them to uH ¼ 0. The other

125013-10 HOLONOMY SADDLES AND SUPERSYMMETRY PHYS. REV. D 97, 125013 (2018) classes are when the unbroken matter charges λ ’s do not λ ≃− τλ σ˜ i i · u mλi þ i · : ð3:29Þ span the entire weight vector space. This means that the reduced 3d theory includes pure gauge sectors with no chiral As in the previous large τ case, the subset λi’s represents fields coupled. The latter deserves a more in-depth dis- light degrees of freedom among the matter fields. Shifting cussion which is postponed to the end of this section, as this the tilded variables similarly, vacua around such an Λ ’ requires a more explicit evaluation of H s. H-saddle solve YY 3 ; 0 2 −λ 2 −1=2 λ 2 1=2 −λa B. H-saddles in the small τ limit 1 Φ˜ d H ≡ −1 M = Λ˜ ˜ i= ˜ − ˜ i= ˜ i ¼ a ð Þ H ½z yi z yi λ We start with the identity i i : 2 ð3 30Þ θðw; τÞ¼ie−πiw =τθðw=τ; −1=τÞ; ð3:22Þ for some integer M0, with which implies that Y Y −π τ˜ρˆa ρˆ ˜ ρˆ σ˜ ν˜ τ˜ 2− ρˆ ˜ ρˆ σ˜ ν˜ τ˜ Λ˜ a ≡ i i ðfð i·uHþ i· þ iÞ= g fð i·uHþ i· þ iÞ= gÞ i i H e : Ψðw; τÞ¼ ¼ ; i ρˆ θðw=τ; −1=τÞ θðw;˜ τ˜Þ i 1 ð3:31Þ ˜ ≡ w τ˜ ≡ − w τ ; τ : ð3:23Þ As before, the latter ignores overall the phase factor. As such, the BAE in the small τ limit can be cast as The solutions to the 4d BAE can be again grouped into Y Y ˜ ρa ˜ 3d;H 1 Φ ν; τ ≡ Ψ ρ ˜ ν˜ ; τ˜ i ∶1 Φ þ ∪˜ ˜ σ˜ ∶1 Φ σ˜ ¼ aðu; Þ ð i · u þ i Þ ; ð3:24Þ fu ¼ aðuÞgτ→i0 ¼ uH fuH þ ¼ a ð Þg; ρ i i ð3:32Þ with where, again, λi · u˜ H=τ˜ are real and integral for some subset 1 λ ’ ρ ’ ˜ −π ˜ 2 τ˜ i sof i s. As before, the necessary exclusion of those Ψðw; τÞ¼ie iw = ; ð3:25Þ θðw;˜ τ˜Þ solutions fixed under some Weyl reflection should be performed, which does not interfere with this H-saddle since the quadratic exponents in the prefactor will cancel classification of solutions. The sum includes the special λ ρ out for Φa thanks to gauge and axial anomaly cancellation. case of f ig¼f ig, where the 3d BAE is nothing but This way, the small τ analysis will follow the large τ (3.27). Note that the locations of H-saddles are along analysis almost verbatim. direction 2 in this small τ limit while they were along Again, there are two types of solutions. The first class direction 1 in the large τ limit. It is reasonably clear that the comes with finite u˜’s, or equivalently, solutions to the BAE can be matched, between the large radius limit and the small radius limit, one on one and þ 0 u ∼ τ; as τ → i0 ; ð3:26Þ saddle by saddle. As before, the naive H-saddle at uH ¼ as well as those that involve a pure gauge sector should be which corresponds to a trivial holonomy along the time included. See Sec. III D. circle. In this obvious saddle, all chiral fields acquire finite ∼ν mass i and thus are in equal footing. The 4d BAE C. 3d UV couplings reduces to, as q˜ → 0, In both limits, the naive reduction on either circle gives Y Y 3d ˜ 3d ˜ 3d # −ρ 2 −1=2 ρ 2 1=2 −ρa 3d BAE, 1 ¼ Φ or 1 ¼ Φ , whereby one recovers vacua 1 ¼ Φ ≡ i ½x˜ i= y˜ − x˜ i= y˜ i ; ð3:27Þ a i i with negligible holonomies. This by itself does not count i ρ i all 4d BAE vacua, however. In order to account for all which is essentially the same equation as (3.8) of the large vacua, one must consider the possibility of turning on some 1 Φ3d;H radius limit, once we replace x, y, and q by x˜, y˜, and q˜. nontrivial holonomies, leading to ¼ near uH in the ˜ 3d;H Similarly to the large τ limit, more solutions appear as we large radius limit, or 1 ¼ Φ near u˜ H in the small radius allow u˜ to scale with the large τ˜ ¼ −1=τ, such that, limit. The holonomy in question is along circle 1 in the large radius limit, hence along the fiber circle, and along λ ˜ τ˜ λ σ˜ circle 2 in the small radius limit, hence along the time i · u ¼ mλi þ i · ; ð3:28Þ circle, respectively. Either way, the effective 3d theory at a again exactly as before, for some proper subset λi’sofρi ’s, given H-saddle comes with reduced field content: only or equivalently those associated with weights λi’s remain “light,” while the

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˜a rest acquire large masses of order β2=β1 or β1=β2, ζ þðσ˜ · κ˜ þ μ˜ · κ˜FÞa respectively. τ˜ X 2 2 a When we consider a particular H-saddle and 3d effective ≃ ðϵ¯ρ − ðϵ¯ρ Þ Þρ 2 i i i ρ theory sitting there, the effect of the heavy modes can i; i manifest via induced couplings.3 In 3d theories, the 1 X a þ ð1 − 2ϵ¯ρ Þρ Imðρ · σ˜ þ F · ν˜Þ; ð3:37Þ auxiliary D-term shows up as 2 i i i i i;ρi ðζ þ σ · κ þ μ · κFÞ · D ð3:33Þ ν˜ ν τ ϵ¯ ≡ ρ ˜ τ˜ while ¼ = as with others and ρi f i · uH= g. We use the common symbol ϵ¯ρ on the large and the small radius ζ κ with the FI constant , the gauge Chern-Simons level , and limits since these two sets of numbers are really identical. F the gauge-flavor-mixed Chern-Simons level κ . σ is the real In either expressions, we can infer the UV contributions μ scalar in the Cartan part of the 3d vector multiplet while is to these couplings in 3d sense, by expanding in 1=τ2 (1=τ˜2) the real masses associated with the flavor symmetries. and dropping λi contributions in the second sums for the This means that one-loop of chiral multiplet of charge Q Chern-Simons level, e.g., and an effective mass MðσÞ¼Q · σ þ M0 will induce a ζa σ κ μ κF a shift [29], UV þð · UV þ · UVÞ τ X 1X 2 2 a a 1 ≃ ðϵ¯ρˆ − ϵ¯ρˆ Þρˆ þ ð1 − 2ϵ¯ρˆ Þρˆ Imðρˆi · σ þ Fi · μÞ F 2 i i i 2 i i Δðζ þ σ · κ þ μ · κ Þ¼ QjQ · σ þ M0j: ð3:34Þ ρˆ ρˆ 2 i; i i; i ð3:38Þ In the τ ¼ τ1 þ iτ2 → i∞ limit, the heavy modes have masses M0 of order jτ2j ≫ jσj, and the leading terms will with μ ¼ ν. For the small τ limit, we take μ ¼ ν˜ and replace induce ζ ∼ jτj as σ in favor of σ˜. Although the KK mode sums associated λ ζ with the charge could have contributed to UV, they cancel ζ þ σ · κ þ μ · κF against the same contributionsP from ρˆ’s, thanks to the ρ 0 1 X X X observation we made earlier, ¼ , for each irreducible ρˆ τ ρˆ ρˆ σ ν representation for any semisimple group. With this, it is ¼ 2 i jImðn þ i · uH þ i · þ iÞj ∈Z i ρˆi n clear that these are precisely the couplings responsible for 1 X X X the prefactor Λ and Λ˜ , λa τ λ σ ν H H þ 2 i jImðn þ i · þ iÞj: ð3:35Þ i λ n∈Z −2π ζ σ κ a −2π ζ˜ σ˜ κ˜ a i Λa ≃ ð UVþ · UVÞ Λ˜ a ≃ ð UVþ · UVÞ H e ; H e ; ð3:39Þ τ → ∞ Regularizing the sum, we find, as i , provided that the left-hand sides are appropriately expanded in 1=τ2 (1=τ˜2) and truncated to the leading order. ζa σ κ μ κF a ζ þð · þ · Þ The FI constant UV must be present only for the Abelian τ X H 2 a part of the subgroup left unbroken by the holonomy uH, ≃ ρ ðϵ¯ρ þ Imðρ · σ þ ν Þ=τ2Þ 2 i i i i which we wish to confirm as a consistency check. Clearly ρ i; i α ζ 0 α this would hold if · UV ¼ for any root that belongs to − ϵ¯ ρ σ ν τ 2 H H ð ρi þ Imð i · þ iÞ= 2Þ Þ the unbroken groups . Since the symmetry breaking to τ X ≃ 2 ϵ¯ − ϵ¯2 ρa is due to the holonomy, this means that the irreducible ð ρ ρ Þ i 2 i i representation Ri of G will decompose into various spin s i;ρi representations under SUð2Þα ⊂ H ⊂ G, and that 1 X 1 − 2ϵ¯ ρa ρ σ ν þ ð ρi Þ i Imð i · þ Fi · Þ; ð3:36Þ 2 fϵ¯ρjρ ∈ Rig → fϵ¯ljRi ¼⊕l ½slg: ð3:40Þ i;ρi

At the holonomy such that ϵ¯α ¼ 0, therefore, we have where ϵ¯ρ ≡ fρ · u =τg so that ϵ¯λ ¼ 0. Repeating the i i H i X X X exercise for the small τ limit, we find α ρ ϵ¯ − ϵ¯2 ϵ¯ − ϵ¯2 α μ · ð ρ ρÞ¼ ð l l Þ · ð þÞ; ð3:41Þ ρ∈ Ri sl μ∈½sl 3Although we refer to induced 3d couplings at H-saddles here, the computation is straightforwardly extended to arbitrary hol- where ϵ¯l denotes the common value of those ϵ¯ρ’s that fall onomy values. At H-saddles, one typically considers split ρ’sto ˆ into the lth irreducible representation, say with spin sl,of λ’s for light modes and ρ’s for heavy modes, and the UV 2 μ’ contribution comes from the latter. At generic holonomy, how- SUð Þα. s are the weights of spin ½sl representation, ever, fλg¼∅ and the contribution comes from all charged chiral embedded into those of Ri, and the ellipsis denotes the part multiplets. invariant under SUð2Þα. We thus find

125013-12 HOLONOMY SADDLES AND SUPERSYMMETRY PHYS. REV. D 97, 125013 (2018) X X H ⊕ K ⊕ α ζ α ϵ¯ − ϵ¯2 μ 0 ¼ ; ð3:44Þ · UV ¼ · ð l l Þ ¼ ; ð3:42Þ l μ∈½s l where K is a semisimple Lie group with no light 3d chiral multiplet coupled. Where would such an unbroken group Pas expected, where in the last step we again used μ 0 be found? Recall that the location of the H-saddle was μ∈R ¼ for any irreducible representation R of a (semi)simple Lie group. Generalization to the entire set determined, so far, by the condition of α with ϵ¯α ¼ 0 is immediate. ϵ¯ 0 Also, the Chern-Simons coefficients should be appro- λ ¼ ð3:45Þ priately quantized. Indeed, we find the gauge Chern- Simons levels in the UV for some subset of matter charges, λ’s. In this current case, no such λ knows about K. Instead the location of the 1 X ab a b H-saddle is determined by the spontaneous symmetry κ ¼ ρˆ ρˆ ð1 − 2ϵ¯ρˆ Þ UV 2 i i i breaking as i;ρˆi 1 X 1 X a b a b ϵ¯ 0 α ∈ ¼ ρ ρ ð1 − 2ϵ¯ρ Þ − λ λ α ¼ ; K; ð3:46Þ 2 i i i 2 i i ρ i; i i;λi 1 X 1 X where K is the Lie algebra of K. When H contains no ρaρb 1 2 ρ τ − λaλb α ∪ λ ¼ 2 i i ð þ b i · uH= cÞ 2 i i ð3:43Þ Abelian subgroup, this combination of f g f g should ρ λ i; i i; i span the entire weight space of G, the Lie algebra of G, and again determine the acceptable positions u discretely. ϵ¯ 0 H holds since λP¼ and since the gauge anomaly cancella- In the absence of light matter fields coupled to K, and a b b tion demands ρ ρ ρ ρ ¼ 0. Recall that b c means the i; i i i i since no UV FI constant would exist for such non-Abelian real integral part, as in (3.5). All quantities in the sums are group, the 3d supersymmetric vacua in question are all manifestly integral, so the induced UV Chern-Simons “topological” types [29].ForK ¼ SpðrÞ, for example, 1 2 r coefficients are integral up to the overall factor = . one canP take a simple basis for the Cartan Uð1Þ such that κ˜ r Exactly the same applies to UV. σ ¼ 1 σ C with chiral fields in the defining representa- 1 2 s s The factor = , which some may find troublesome, is not tions have unit charges with respect to one and only one Cs. a problem at all. For many theories, such as the SQCD type As such, the reduced BAE will take the simple form, after where the fundamental chirals has to appear in pairs, we some rescaling expect that this is countermanded by the 4d spectrum. More to the point, the half-quantized Chern-Simons coefficient is 2κSpðrÞ 1 ¼ðz Þ UV ;s¼ 1; …;r: ð3:47Þ usually an indication that we must be more careful about s the effective action coming from integration out of massive We remove solutions with zs ¼1 for some s or those with fermions. The usual statement that this leads to Chern- ≠ Simons action is known to miss the global structure of the zs ¼ zt for some s t, and identify those related by Weyl Z r effective action; whenever the Chern-Simons coefficient transformations, W ¼ Sr × ð 2Þ . Then the vacua are πin=κSpðrÞ generated is half-integral, and thus apparently variant under labeled by unordered distinct r phases, e UV , with large gauge transformation, the effective action is actually 1 ≤ κSpðrÞ n< UV an eta-invariant with full gauge invariance [44,45]. κSpðrÞ − 1 j UV j D. Locating H-saddles with pure gauge sectors : ð3:48Þ r So far we have pretended that the H-saddle would come λ ϵ¯ 0 with charge vectors f j λ ¼ g, enough of them to span the Similarly,P forP K ¼ SUðr þ 1Þ, a simple choice is σ ¼ entire charge vector space. However, this needs not be the r σ C − σ C 1 s s ð s sÞ rþ1 in the redundant basis, whereby case in general, as one can have 3d theories with unbroken the reduced BAE equation becomes supersymmetry when the Chern-Simons is nontrivial. Also a further issue arises when the unbroken gauge group H κSUðrþ1Þ κSUðrþ1Þ 1 ¼ðz1=z 1Þ UV ¼¼ðz =z 1Þ UV ð3:49Þ contains a Uð1Þ factor with a UV FI constant generated. In rþ r rþ the latter cases, some topological vacua may appear shifted Q ≡ r −1 far away from σ, σ˜ ∼ Oð1Þ, potentially muddying the with zrþ1 ½ 1 zs understood. Each equation yields κSUðrþ1Þ − 1 ≠ 1 classification of the H-saddles. Here we wish to address j UV j acceptable solutions, zs=zrþ1 , upon ≠ issues related to such H-saddles. which we further impose zs=zrþ1 zt=zrþ1 for all pairs Consider the case where H contains no Abelian sector. s

125013-13 CHIUNG HWANG, SUNGJAY LEE, and PILJIN YI PHYS. REV. D 97, 125013 (2018) Q SUðrþ1Þ P P Q Q κ − 1 2 0 2 ðyz − 1Þ j UV j 1 ≃ ξ κ− Q =2þ ðQ Þ =2 QQ : : q z 0 0 ; ð3:54Þ ð3 50Þ Q − Q r Q0 ðz yÞ

These dovetail precisely with the 3d index computation by or N 2 Witten [46] once we extend the latter to ¼ ; the only Y P P Y N 2 κ0 κ − 0 0 κ− 2 2 0 2 2 new ingredient for ¼ is to take ¼ UV h, where h ðzQ − yÞQ ≃ qξz Q = þ ðQ Þ = ðyzQ − 1ÞQ; κ0 κ − 2 is the dual Coxeter number, instead of ¼ UV h= , for Q0 Q the bosonic theory in the end, since the adjoint fermion content is doubled between the two. In both cases, there- ð3:55Þ fore, a necessary condition for the existence of an H-saddle involving a pure SpðrÞ sector or a pure SUðr þ 1Þ sector where −Q and Q0 denote, collectively, the light charges of κ ≥ 1 ξ ≡ ζ = τ is UV r þ . negative and positive signs respectively, and UV Im κ κ ξ ≡ ζ˜ τˆ κ κ˜ Now let us allow K to include an Abelian factor. and ¼ UV or UV=Im and ¼ UV, in the large or Unbroken Uð1Þ’s can come with large 3d FI constants as in the small τ limits, respectively. Here, we will consider a we saw in the previous section, which will interfere with large τ limit, or q → 0, without loss of generality. reduction of 4d BAE to 3d BAE. If one started with 4d chiral Suppose that ξ > 0.Forξ < 0, we get the same result multiplets in real or pseudoreal representations, such FI after flipping Q ↔ Q0 and κ ↔ −κ.Ifξ ¼ 0, there is no constants would cancel out exactly, but of course this need issue, to begin with, as all solutions would be Oð1Þ and do not be the case. If one finds large FI constants, say at some not scale with q. Setting q ¼ 0 for the moment, we find Q0 1=Q0 0 uH, that scale as Imτ (or Imτ˜), the reduced 3d BAE of such finite solutions z ∼ y to (3.55), each of which are Q Uð1Þ ⊂ K cannot be solved for σ (or σ˜) kept finite. What this times degenerate. As we turn back on small q, these would really means is that H-saddles must be looked for, with the split but remain finite. To enumerate the other worrisome ζ 0 condition of UV ¼ imposed simultaneously, i.e., solutionsP that scaleP with q or 1=q,P it is usefulP to define X X k≡κ − Q2=2þ ðQ0Þ2=2, l≡κþ Q2=2− ðQ0Þ2=2. s s 2 s 2 ζ ∝ ρˆ ðϵ¯ρˆ − ϵ¯ Þ¼ ρ ðϵ¯ρ − ϵ¯ρ Þ → 0: ð3:51Þ i i ρˆi i i i We then find, ρˆ ρ i; i i; i (i) k ≥ 0, l>0, −ξ=l l large solutions z ∼ q ; P Recall that for this case, neither a matter charge λ nor a root α 0 2 a theP total numberP of solutions are ðQ Þ þ l ¼ can be invoked to fix the location of u . Instead, we have this 2 2 κ þ Q =2 þ ðQ0Þ =2; ζs 0 condition, again fixing the holonomy u to discrete ¼ H (ii) k ≥ 0, l ≤ 0, possibilities. no new solutions; Once this necessary condition is met and the candidate P the total number is ðQ0Þ2; location for the H-saddle is found, we must decide whether (iii) k<0, l ≤ 0, such an H-saddle will actually contribute, i.e., whether the − ∼ ξ=jkj reduced 3d BAE admits nontrivial vacua nearby. Because k small solutions z q P; 0 2 − −κ we are dealing with pure N ¼ 2 gauge theory in three Pthe totalP number is ðQ Þ k ¼ þ 2 2 0 2 2 dimensions, the latter is possible only if κ ≠ 0.Ata Q = þ ðQ Þ = ; UV 0 0 saddle with decoupled Uð1Þ unbroken group the actual (iv) k< , l> , a −ξ=l supersymmetric vacua are determined by the 3d BAE, l large solutions z ∼ q and −k small solutions ζ= k z ∼ q j j; P P κs 0 2 − 2 Cs ¼ðzsÞ UV ð3:52Þ the total number is ðQ Þ þ l k ¼ Q . Among these vacua, the large and the small ones z ∼ q# for some finite constant CUð1Þ, so the number of them is should be taken only as a qualitative indication that somewhere far away there exist supersymmetric and κs topological vacua of free U 1 Chern-Simons theory. j UVj: ð3:53Þ ð Þ The truncation to (3.54) is justified only at finite values Therefore, we have found that an H-saddle involving a of σ, and thus precise locations of these additional vacua 1 decoupled Uð Þs gauge sector is possible provided that should be worked out by going back to the 4d BAE. Why is ζs 0 κs ≠ 0 UV ¼ and UV . the reduction to 3d theory, which has worked flawlessly so The question of H-saddles with K ¼ Uð1Þ ⊂ H but now far, compromised? Simply because the dimensional reduc- with light 3d charged matter field, a general case of tion ends up with 3d FI constant which still remembers the (2) above is a little more involved. Suppose we located large value of τ and thus the fact that the purported 3d a candidate H-saddle using a condition of type ϵ¯λ ¼ 0 for theory came from 4d theory with the extremely elongated some Uð1Þ-coupled charge vector λ. A schematic form of T2. The number of vacua found for such Uð1Þ factor in the rank 1 3d BAE at such an H-saddle is the preceding analysis should still hold, but the precise

125013-14 HOLONOMY SADDLES AND SUPERSYMMETRY PHYS. REV. D 97, 125013 (2018) locations of those at z ∼ q# are not to be trusted. Rather, will become all massive. As such no FI constant would be one must really view this situation as a sum of distinct generated, as the 3d gauge group H ¼ G would have no H-saddles consisting of two types. One is Uð1Þ theory with Uð1Þ factor. For a qualitative understanding, we will charge matters, but with its topological vacua due to very confine our attention to theories with a single classical large FI constant excised. The others are free Uð1Þ theory Lie group G as the gauge group and further assume that elsewhere in the u-space, with no light matter field coupled 0

ð2Þ X ð2Þ H G Tadj T E. -saddles with large chemical potentials κ ¼ × i ð1=2 − r Þ UV γ ð2Þ ð2Þ i As we hinted already, the large radius limit of the GTdef i Tadj 1 3 “physical” S × S partition function deviates a little from ð2Þ P 2 T 3 Tð Þ the main story of this paper. Apart from why this has to be ¼ adj · 1 − · i i : ð3:60Þ γ ð2Þ 2 3 ð2Þ so from the viewpoint of how these objects are constructed, GTdef Tadj we can also trace the difference at a mathematical level to 1 − 1 τ the large Uð ÞR chemical potential ðr Þ . The latter Note that the second term inside the parentheses cannot shifts the argument of various operators by a large amount exceed 3=2, once we demand the asymptotic freedom. If in the large τ limit, common for each chiral multiplets in a the theory contains a single type of chiral multiplets, the single irreducible gauge representation. asymptotic freedom combined with 0

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κ 2 − UV ¼ Nc Nf ð3:62Þ IV. 4D THEORY AS A DISJOINT SUM OF 3D THEORIES 3 2 ≤ 3 for 4d conformal field theories, Nc= Nf < Nc. With These discussions lead us to a clear definition of the H-saddle for general supersymmetric gauge theories on a κ j UVj

Suppose that the matter content is symmetric under charge with the two circles ðp1;p2Þ-fibered over a genus g surface, conjugation, such that charge vectors always come in pairs we find that this decomposes, both in the large and in the ðρ; −ρÞ. Then, places where this happens generically are small τ limits, XX ρ · uˆ ∈ τZ=2; ð3:65Þ g;p1;p2 g−1 p1 p2 H Ω4 → H F 1 F 2 ; ð4:3Þ σ uH which allows ϵρˆ ¼ ϵ−ρˆ and thus pairwise cancellations in the sum (3.64). Assuming ri ≠ 1=2, the theory reduces to where the 4d BAE vacua are reorganized into sets of 3d pure Yang-Mills type and the Chern-Simons level is BAE vacua for mutually disjoint 3d theories at various H-saddles. As we already emphasized, we find such X τ κab δabγ ð2ÞκG ρaρb ρ ˆ τ decomposition even in the large limit, because, in effect, UV ¼ GTdef UV þ i i b i · uH= þ ric; ð3:66Þ ρ this is equivalent to a small radius limit of circle 1. i; i ≠ 1 σ For g , we can reorganize the sum over 3d vacua at ZH κG ˆ 0 each uH, in terms of the 3d BAE partition function, 3 and where UV is the Chern-Simons level at uH ¼ as in (3.60). a multiplicative factor cH. The latter will generically have Coming back to SUð2Þ theory with 2Nf fundamental 2 an exponential behavior, interpreted as the Cardy exponent flavors, we find that reduced theory is a pure SUð Þ in the small τ limit and as the Casimir energy in the large τ YMCS with limit. Previous estimates of such leading exponents have effectively considered only the naive saddle at u ¼ 0 κ −2 4 3 5 H UV ¼ ; ; for Nf ¼ ; ; ð3:67Þ [20,22]. The results from such computations, interpreted as being related to 4d anomaly polynomials, must be therefore implying 1 and 3 BAE vacua, respectively, which are questioned. consistent with the Witten index of the original 4d theories. 4 1 2 ˆ τ 2 Nf ¼ with r ¼ = at SCFTalso admit uH ¼ = as an A. 4d Witten index is a sum of 3d Witten indices H-saddle; the reduced theory is an SU 2 theory with ð Þ Before we plunge into quantitative studies, it is worth- 2Nf ¼ 8 fundamental chirals and vanishing UV Chern- while to consider the simplest case of p1;2 ¼ 0 and g ¼ 1. Simons level. The number of vacua for this 3d theory is For T4, the BAE computes the numerical Witten index, usually expected to be three. However, the actual theory at with the summand at each u equal to 1. This means that we this H-saddle, being a reduction from 4d where the have an intuitive relation 1 baryonic Uð Þ is anomalous, and, because this theory X cannot have UV FI constant, one of these potential vacua G H I4 ¼ I3 ; ð4:4Þ is pushed to the Coulombic infinity. The number of vacua uH at the uˆ H ¼ τ=2 saddle is actually 2 which is again consistent with the 4d Witten index. These suggest that whereby the 4d Witten is reconstructed from those for SQCD theories, uˆ H ¼ τ=2 is the only H-saddle in the of several 3d theories sitting at distinct holonomies. Casimir limit. Regardless of the details of computations to follow for

125013-16 HOLONOMY SADDLES AND SUPERSYMMETRY PHYS. REV. D 97, 125013 (2018) different g’s and p’s, this by itself tells us that the small dimensions if one is a dimensional reduction of the other. radius limit of 4d theory cannot be regarded as a single 3d However, we already know, via many examples, that this theory. If we are considering supersymmetric theories in is not quite correct. For instance, Witten pointed out compact spacetime, therefore, the 4d theory in the small how the index of 4d N ¼ 1 pure Yang-Mills is sensitive radius limit should be considered as a disjoint sum of 3d to disconnected sectors of mutually commuting holono- theories. mies along T3 [40]; such sectors would be dropped if one or ’ 3 Since the same set of uH s enters such decompositions more radii of T is taken to zero literally. g;p1;p2 for all M4 ’s, this also means that an H-saddle will Our relation is yet another reminder that such topological occur if and only if the reduced 3d theory there has a invariance argument should not be taken too far. The nontrivial Witten index. The latter condition can be problem with the zero radius limit of a spacetime circle considered as the most important single property of is that the compact space of holonomies becomes non- H-saddles. The class of theories we are considering in compact in a zero radius limit, and cannot be considered a this paper are maximally mass deformed by flavor holon- small deformation. The above formula, which is far more omies so that the partition functions and Witten indices general than the particular class of theories here and the are all integral. As such, an H-saddle would occur if and partition functions thereof, gives a neat way to relate Witten only if the Witten index of the reduced 3d theory at the indices of gauge theories in adjacent dimensions. candidate holonomy is nonvanishing. We close with two simple examples. The first is the On the other hand, the notion of an H-saddle clearly goes canonical SQCD, namely SUðNÞ theories with Nf funda- beyond the particular class of theories or background mental and Nf antifundamental chirals. For these, it is clear geometries we are considering in this paper. More generally that the only H-saddle is the one at uH ¼ 0, hence we have twisted partition functions are often not enumerative, resulting in nonintegral twisted partition functions. As IG IH 4 ¼ 3 ju =τ¼0; ð4:5Þ we recalled in the Introduction, a twisted partition function H would generally compute the analog of the “bulk index.” In where the reduced H theory at uH=τ ¼ 0 has the same such cases, the defining property of the H-saddle should be gauge group and the same chiral multiplet content as its 4d extended to allow a nonvanishing supersymmetric partition cousin G. Indeed, Closset et al. [19] found function of the reduced theory at the candidate holonomy. 1 R3 If we were considering the 4d theory on S × , the N − 2 G H f holonomies would label superselection sectors; the dimen- I 4 ¼ I 3 j τ 0 ¼ : ð4:6Þ uH= ¼ − 1 sional reduction process is ambiguous until we specify the N holonomy or compute the vacuum expectation value of the holonomy. The above relation tells us that there are discrete Since the matter representation is real collectively, neither choices of u whereby the dimensional reduction produces the FI constant nor the Chern-Simon level arise at UV. H 2 distinct 3d theories whose 3d supersymmetry is not The other, less trivial, example is an SUð Þ theory with spontaneously broken, and that the 4d Witten index is two fundamental chirals and two adjoint chirals. For this, a τ 1 2 reproduced only after we sum over the Witten indices of nontrivial H-saddle at uH= ¼ = is present as well as the naive one at uH ¼ 0. While we are formulating things via these 3d theories at distinct uH’s. Such a behavior of 4d theory on a circle, producing the large radius limit, the small radius limit is found, multiple 3d theories in the small radius limit, has been verbatim, by replacing the variables to the tilded ones with noted previously by Seiberg and collaborators while study- the identical result. As such we have ing how 4d dualities reduce to 3d dualities [13]. As the X G H above relation shows, a dual pair of 4d theories would I4 ¼ I 3 ð4:7Þ τ 0 1 2 produce, each, several 3d theories which must be collec- uH= ¼ ; = tively dual to each other. Whether or not this implies τ 1 2 individual 3d dualities, say, in our language at H-saddle where the 3d theory at uH= ¼ = has the two adjoint pairs, is in principle another matter. For 4d theory as a chirals only. Again no UV 3d coupling is generated at either starting point, however, the interpretation of uH as the saddle, and 3d BAE vacua can be counted straightfor- superselection sector label does suggest that the duality wardly. We find will hold for 3d theories pairwise, or in our terminology, IG IH IH 8 6 14: 4:8 H-saddle by H-saddle. 4 ¼ 3 juH=τ¼0 þ 3 juH=τ¼1=2 ¼ þ ¼ ð Þ The robust nature of the Witten index under small deformations is often invoked to simplify index computa- The main feature of the latter example, relative to the first, tions. One such would be the insensitivity to the size of the is a chiral multiplet with gauge representation beyond the circles in T4, but this, if used improperly, seems to imply fundamental one. In fact, the existence of a chiral multiplet that Witten indices agree between theories in different in a gauge representation larger than the defining one, for

125013-17 CHIUNG HWANG, SUNGJAY LEE, and PILJIN YI PHYS. REV. D 97, 125013 (2018) classical ones at least, is one obvious criterion for a of which the last piece can, at most, contribute logarithmic nontrivial H-saddle to exist. corrections in the exponent. The large and the small τ limit Note that, of these, the first example is not compatible of Ψ’s were already explored. These may be combined to, with a general A-twist background, since the anomaly-free for the gauge multiplet contributions, at each H-saddle, 1 Σ 2 Uð ÞR charge is not integral. For g ¼ T , however, the Y 1 η τ −2rankðGÞ Ψ α ; τ A-twist is null, so we do not need to restrict the Uð ÞR ð Þ ð · u Þjτ→i∞ charge to be integral. And as long as one can find non- α P 1 − G 12 πiτ ϵα 1−ϵα anomalous Uð ÞR, its chemical potential can be turned on. ∼ q dimð Þ= e α ð Þ; ð4:10Þ For this reason, this recursive computation of the Witten index can be used for more examples of theories than where ϵα ¼fα · ðuH þ σÞ=τg are real numbers between 0 generic geometries of this class would allow. Of course, this and 1, is up to the major caveat that theories being considered are Y all equipped with real masses in the 3d sense, as is a −2rankðGÞ ηðτÞ Ψðα · u; τÞjτ→ 0þ common downside of the BAE formulation. One must take i α P care, in general, not to confuse the Witten index computed − G 12 π τ˜ ϵ˜ 1−ϵ˜ ∼ ˜ dimð Þ= i α αð αÞ this way with those of the vanilla 4d N ¼ 1 theories on R4. q e ; ð4:11Þ We close this subsection with a caveat. In relating 4d ϵ˜ α ˜ σ˜ τ˜ theories to one or more 3d theories, obtained by dimen- where α ¼f · ðuH þ Þ= g are real numbers between 0 sional reduction at such saddles, we are always speaking of and 1. The chiral multiplet contributions can be written the small radius limit. This means that constraints from the similarly as 4d anomaly, for example, should be considered valid even Y Y −1 Ψðρ · u þ ν ; τÞri in the said 3d limit. One example is the SUðNcÞ SQCD, i i τ→ ∞ ρ i whose strict 3d form allows an extra Uð1Þ flavor symmetry i i P P P P − −1 12 π τ −1 ϵ 1−ϵ which would be anomalous in 4d. Our 3d theories at ρ ðri Þ= i ðri Þ ρˆ ρˆ ð ρˆ Þ ∼ q i i e i i i i ; ð4:12Þ H-saddles are the ones without such a global symmetry; this affects the allowed superpotential and hence the 3d and Witten index as well. Y Y −1 Ψðρ · u þ ν ; τÞri i i τ→ 0þ B. Asymptotics ρ i i i P P P P Now let us turn to other, more involved partition − −1 12 π τ˜ −1 ϵ˜ 1−ϵ˜ ðri Þ= i ðri Þ ˆ ρˆ ð ρˆ Þ ∼ ˜ i ρi i ρi i i functions. In the literature, some 4d partition functions q e ; ð4:13Þ have been discussed with a particular interest on their where ϵρ ¼fðρ ·ðu þσÞþν Þ=τg and ϵ˜ρ ¼fðρ ·ðu˜ þσ˜Þþ asymptotic behavior [16–18,20–22]. Especially, Ardehali i i H i i i ν˜ τ˜ first observed the influence of the holonomy on the Cardy iÞ= g are also real numbers between 0 and 1. limit of the superconformal index [16], i.e., the partition 3 1 2. Asymptotics of F function on S × S , which is later extended to more 1 general manifolds by Di Pietro and Honda [17]. The latter The first fibering operator is given by 1 discussed the Cardy limit of the M3 × S partition func- Y Y M 1 Σ 1 tion, with explicit examples 3 ¼ Lðn; Þ, g × S . F 1 ¼ Ξ1ðρi · u þ νi; τÞ ρ On the other hand, for the Casimir limit, the role of the i i 3 holonomy is rarely discussed as far as we are aware. In this Y Y ðρ ·uþν Þ ρ ·uþν 2πið i i − i iÞ 6τ2 12 Γ ρ ν ; τ section, we provide a unified way of examining both the ¼ e 0ð i · u þ i Þ; ð4:14Þ ρ Cardy limit and the Casimir limit of the partition function, i i which manifests itself in the H-saddle approach. where

∞ 1. Asymptotics of H Y 1 − x−1qnþ1 nþ1 Γ0ðu; τÞ¼ : ð4:15Þ The handle-gluing operator is, with r being the Uð1Þ 1 − nþ1 i R n¼0 xq charge of the ith chiral multiplet, Y Y Y To find the large radius limit of F 1, again we decompose −2 G −1 H ≡ ηðτÞ rankð Þ Ψðα · u; τÞ Ψðρ · u þ ν ; τÞri ρ ν ϵ τ ϵ i i i · u þ i into ð ρi þ mρi Þ where ρi belongs in the range α ρ i i 0 ≤ ϵ 1 ρi < and mρi is the integer part. Using ∂a log Φb × det ; ð4:9Þ −πiðm2þmÞ −m 2πi Ξ1ðu þ mτ; τÞ¼e 2 Ψðu; τÞ Ξ1ðu; τÞð4:16Þ

125013-18 HOLONOMY SADDLES AND SUPERSYMMETRY PHYS. REV. D 97, 125013 (2018) for an integer m, one can find the large radius limit of F 1 as vanish after summed over all the multiplets due to the follows: anomaly-free condition. fðuÞ comes from the 1-loop 3 3 determinant of a chiral multiplet on S and converges to Y Y ϵρ ϵρ mρ i 2 i i πiτ ϵρ mρ −ϵρ mρ − F ∼ ð 3 þ i i i i 6 þ 6 Þ 1 for large u, 1jτ→i∞ e ; ð4:17Þ i ρi fðuÞj → ∞ ¼ 1: ð4:24Þ ρ ν ϵ τ u i with i · u þ i ¼ð ρi þ mρi Þ . The identity (4.16) also resolves an apparent puzzle with Thus, F 1 has the following asymptotic behavior for this asymptotic formula. Note that under a large gauge τ → i0þ: transformation ua can be shifted to ua þ τ. This will induce 3 2 ϵ ’ ’ Y Y ϵ˜ ϵ˜ ϵ˜ shift of both ρ s and mρ s, under which the exponent of π τ˜2 ρi − ρi ρi F ∼ i ð 3 2 þ 6 Þ: (4.17) does not look particularly invariant. Let us first look 1jτ→i0þ e ð4:25Þ i ρi at how F 1 transforms. Since ρ · u will shift by ρaτ, the transformation is 3. Asymptotics of F Y Y 2 ρa −ρa −1 The second fibering operator F 2 is given by F 1 → ð−1Þ i Ψ i × F 1 ¼ Φa × F 1; ð4:18Þ i ρi Y Y P F 2 ¼ Ξ2ðρi · u þ νiÞð4:26Þ a ρ where we used ρρ ¼ 0 for each irreducible representa- i i tions, as was shown in Sec. III.2. What this formula tells us Y Y 3 2 F 2π ðρi·uþνiÞ −ðρi·uþνiÞ ðρi·uþνiÞτ 1 is that although the flux operator 1 is not invariant as a ¼ e ið 6τ 4 þ 12 þ24Þ function of u under such large gauge transformations, its i ρi values at supersymmetric vacua, where 1 ¼ Φ , are invari- Y∞ a fðρ · u þ ν þ kτÞ ant. Therefore, although the leading exponent in (4.17) may × i i : ð4:27Þ f −ρ · u − ν k 1 τ look odd, its values at H-saddles are really invariant under k¼0 ð i i þð þ Þ Þ u → u þ τ. The same kind of invariance will work for F 2 a a F under ua → ua þ 1, for the small τ limit, as the two are In the large radius limit, we find the following limit of 2: related by S-transformation. 3 2 Y Y ϵρ ϵρ ϵρ F π τ2 i − i i On the other hand, the small radius limit of 1 can be F i ð 3 2 þ 6 Þ 2jτ→i∞ ¼ e ; ð4:28Þ obtained using the S-transformation. First note that Ξ1 i ρi satisfies an identity [19] ρ ν ϵ τ 3 with i · u þ i ¼ð ρ þ mρ Þ . Similarly one can also find πiu u 1 i i Ξ ; τ τ2 3 Ξ ; − the small radius limit of F 2 using the S-transformation 1ðu Þ¼e 2 τ τ ; ð4:19Þ (4.19): where Ξ2 is defined by 3 Y Y ϵ˜ρ ϵ˜ρ ρ i 2 i m i −πiτ˜ ϵ˜ρ mρ −ϵ˜ρ m˜ ρ − F ð 3 þ i i i i 6 þ 6 Þ ∞ 2jτ→i0þ ¼ e ; ð4:29Þ 3 2 Y u u uτ 1 f u kτ ρ 2πið − þ þ Þ ð þ Þ i i Ξ2ðu; τÞ¼e 6τ 4 12 24 ; ð4:20Þ fð−u þðk þ 1ÞτÞ k¼0 ρ ˜ ν˜ ˜ ϵ˜ τ˜ ϵ˜ ∈ 0 1 where i · u þ i ¼ðmρi þ ρi Þ with ρi ½ ; Þ. Note that and the small and the large τ limits of F 2 mirror, under τ → −1=τ, the large and the small τ limits of F 1, faithfully 1 2πiu 2πiu and respectively. fðuÞ¼exp Li2ðe Þþu log ð1 − e Þ : ð4:21Þ 2πi 4. Asymptotics of F phys Ξ2 satisfies Unlike the A-twist gauge, the physical handle-gluing πim phys Ξ2ðu þ mτ; τÞ¼e 6 Ξ2ðu; τÞ; ð4:22Þ operator only contains Jacobian factor H , which does not contribute to the leading term of the partition function. and, as a result, Ξ1 can be written as The leading contribution then only comes from F phys. F phys 3 From (4.25) one can see that each component of has ϵ˜ρ ˜ ρ π ˜ ρ −π τ˜2ð i þm i Þ iðm i Þ i 3 þ 6 ˜ the following asymptotic behavior. Ξ1ðρi · u þ νi; τÞ¼e Ξ2ðϵρ τ˜; τ˜Þ; ð4:23Þ i In the small τ limit, with where ρi · u˜ þ ν˜i is decomposed into ðm˜ ρ þ ϵ˜ρ Þτ˜ with 0 ≤ i i ϵ˜0 ρ σ˜ ν˜ τ˜ − 1 τ˜ ϵ˜ 1 ˜ ρi ¼fð i · ðuH þ Þþ iÞ= þ lRðri Þ= g; ð4:30Þ ρi < and an integer mρi. The cubic phase term will

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5. H-saddles from the small radius limit of a fibered circle 0 ϵ˜α ¼fα · ðuH þ σ˜Þ=τ˜ þ lR=τ˜g; ð4:31Þ We close with a minor consistency check on the notion of we find similarly the H-saddle by considering the collapsing circle with nontrivial winding number p over the base. The Cardy limit Y Y β1 → 0 with p1 ≠ 0 would be the canonical example, while Ξ1ðρi · u þ νi þ lRτðri − 1Þ; τÞ τ→i0þ the Casimir limit β1 → ∞ with p2 ≠ 0 shares the same issue i ρ ∈R i i since as far as our partition functions goes this is equivalent 0 3 0 2 0 Y Y ðϵ˜ρ Þ ðϵ˜ρ Þ ϵ˜ρ π τ˜2 i − i i β → 0 ∼ e i ð 3 2 þ 6 Þ;; ð4:32Þ to the other Cardy limit 2 . The winding number p of the collapsing circle is not part of 3d spacetime data, so i ρi H should not enter the 3d partition functions Z3 ’s, since the and notion of the H-saddle relies on the existence of 3d theories Y that makes sense without referring to its 4d origin. It would l ðl þ1Þ R R rankðGÞ 2l rankðGÞ be allowed to enter the coefficients c ’s which serve as the ð−1Þ 2 ηðτÞ R Ξ1ðα · u þ l τ; τÞ H R τ→ 0þ α i glue between the 3d theories at H-saddles and the original 4d

Y 0 3 0 2 0 theory. π τ˜2 ðϵ˜αÞ −ðϵ˜αÞ ϵ˜α ∼ e i ð 3 2 þ 6 Þ;; ð4:33Þ In view of the lengthy discussions in Sec. III,itshouldbe p1 G relatively clear that the part of ðF 1Þ that could have H contributed to Z3 in the large τ limit resides entirely in Γ0 where the last product is taken over all the gauge gen- of Eq. (2.21). However the latter function reduces to 1 erators, again. It is important to note here that the set of universally as q → 0, regardless of the field content. The H-saddles and the subsequent values of ϵ¯’s to be used in the winding number p1 therefore contributes at most to cH’s, in subsequent expansion of the exponents are no different this limit, via the surviving exponential prefactors in front of τ from the preceding discussion of the small limit of Γ0’s, and does not interfere with the 3d theory at the ν A-twisted cases. This happens because the shift due to R is H-saddles. Then, SLð2; ZÞ automatically implies that the τ → 0þ negligible as i , as far as the values of uH are same happens for the β2 → 0 limit with p2 ≠ 0,asF 2 in concerned. Clearly this is not the case for the other the small τ limit is nothing but F 1 in the large τ limit modulo τ → ∞ limit i . exponential prefactors, which are again harmless for the As we noted already in Sec. III, the large radius limit issue here. τ → i∞ for “physical” cases follows a different pattern due For physical cases, one should look at how F phys ν τ 1−g τ τ → ∞ to the large shift R ¼ lR ¼ p . With behaves in the i limit. The relevant part of the latter fibering operator is made up of Ξ1 ’s, or Γ0’s therein, so ϵ0 ρ ˆ σ ν τ − 1 again, F phys reduces to a product of exponential functions: ρi ¼fð i · ðuH þ Þþ iÞ= þ lRðri Þg; ð4:34Þ The winding number p can contribute to cH’s at most, again 0 as promised. ϵα ¼fα · ðuˆ H þ σÞ=τ þ lRg; ð4:35Þ

0 0 C. Cardy and Casimir where mρi ;mα are the remaining integer parts, the expo- nential behavior goes as In the small and the large τ limits, we found exponential Y Y behaviors of the partition function which differ between different H-saddles. The partition function on A-twisted Ξ1ðρi · u þ νi þ lRτðri − 1Þ; τÞ τ→i∞ i ρ ∈R geometries, for example, always has an H-saddle at i i 0 0 3 0 0 u ¼ , and the exponential behavior there follows a Y Y ðϵρ Þ ϵρ mρ H i 0 2 0 0 0 i i πiτð þðϵρ Þ mρ −ϵρ mρ − þ Þ ∼ e 3 i i i i 6 6 ; ð4:36Þ universal form, i ρi −1 2π τ − 12 −1 2π τ˜ − 12 −1 Hg ∼ ½e i ·ð trfRÞ= g or ½e i ·ð trfRÞ= g ; 0 and uH¼ Y ð4:38Þ l ðl þ1Þ R R rankðGÞ 2l rankðGÞ ð−1Þ 2 ηðτÞ R Ξ1ðα · u þ l τ; τÞ R τ→ ∞ in the respective limits of large τ or τ˜ ¼ −1=τ:trfR means the α i trace of Uð1Þ charge over all 4d fermions. The exponents Y ϵ0 3 ϵ0 0 R π τ ð αÞ ϵ0 2 0 −ϵ0 0 − α mα ∼ e i ð 3 þð αÞ mα αmα 6 þ 6 Þ; ð4:37Þ inferred from this universal part have been identified in the G past and given interpretation of the Casimir energy and the Cardy exponent with respect to the large and the small radius where the last product is taken over all generators of the limit of β2 [19]. The same expression also appeared for the gauge group with ϵ ¼ 0 understood for the Cartan Cardy limit of SCI’s, which was then related to conformal generators. anomaly coefficients [18].

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Existence of H-saddles at uH ≠ 0 and the different which, combined with the overall factor 2πiτ (2πiτ˜), supply σ σ˜ leading exponents at such places, however, tell us that finite and ( ) dependent phases. Thus, cancellations the Cardy exponent and the Casimir energy may be rather between 3d BAE vacua in favor of smaller exponents at a different in general. In this last section, we will explore this given H-saddle cannot be ruled out in general. Although issue. For the sake of simplicity we will confine our such cancellations do not appear to be commonplace, we attention to pure imaginary τ ¼ iβ2=β1 and consider only will identify a few examples of this kind later. those 4d theories whose chiral field content is invariant For p1;2 ≠ 0, there is a further exponential contribution p1 p2 under the charge conjugation symmetry, ρ → −ρ. of the form, via F 1 F 2 in the sum. In the large τ limit, the additional terms, to be added to (4.39), are 1. A-twist X X 3 1 ϵρ ϵρ mρ We find, at each H-saddle at uH, the leading exponents i 2 i i −1 p1 × þ ϵρ mρ − ϵρ mρ − þ of Hg in the large τ limit is 2 3 i i i i 6 6 i ρ i X X X τ ϵ3 ϵ2 ϵ 1 1 ρi ρi ρi − 1 − ϵ 1 − ϵ þ p2 × − þ ; ð4:44Þ ðg Þ × ðtrfRÞþ αð αÞ 2 3 2 6 12 2 i ρ α i 1 X X þ ðr − 1Þ ϵρ ð1 − ϵρ Þ ð4:39Þ 2 i i i 2πiτ i ρ again modulo the large multiplicative factor . For the i small τ limit, we merely need to exchange the asymptotic forms of F 1 and of F 2 and replace ϵρ → ϵ˜ρ , mρ → m˜ ρ multiplied by 2πiτ, instead of the universal form i i i i τ → τ˜ and . 1 In particular, with the restriction of the matter content to − 1 − ðg Þ × 12 ðtrfRÞ ð4:40Þ be symmetric under the charge conjugation, all terms that involve ρ · u cancel away leaving behind those involving powers of ν ’s. The above then reduces to, e.g., for the large at uH ¼ 0. Clearly the H-saddle with the dominant con- i tribution and the exponent thereof may be identified only τ limit, after comparing this expression at different H-saddles. Furthermore, the actual exponent is given by this multiplied X X νi 1 2 1 g − 1 g 0 p1 × ϵ¯ρ þ 2ϵ¯ρ mρ − mρ − by ( ), so the dominant contributions for ¼ and the τ 2 i i i i 6 1 i ρ dominant contributions for g> will generically come i τ X X from different H-saddles. For the small limit, the same 1 2 1 þ p2 × ν ϵ¯ρ − ϵ¯ρ þ ; ð4:45Þ formulas work with ϵ’s and τ replaced by ϵ˜’s and τ˜. i 2 i i 6 ρ Note that, once we begin to identify 3d BAE vacua and i i evaluate the sum, ϵ’satagivenH-saddle would be really the latter of which contributes ν-dependent pieces to the ρ σ ν ρ σ˜ ν˜ i · þ i i · þ i Casimir energy, while the former 1=τ term contributes a ϵρ ¼ ϵ¯ρ þ or ϵ˜ρ ¼ ϵ¯ρ þ ð4:41Þ τ τ˜ finite imaginary piece to the exponent. σ ’ etc., for multiple s found by solving the 3d BAE at uH. Expanding (4.39), the leading term 2. Physical “ ” X For the physical case, the H-saddle behavior is differ- 1 1 ent between the large radius limit and the small radius limit, ðg − 1Þ × − ðtr RÞþ ϵ¯αð1 − ϵ¯αÞ 12 f 2 as we saw in the previous subsection. The small radius limit α 1 X X itself is on par with that of an A-twisted case, except that þ ðr − 1Þ ϵ¯ρ ð1 − ϵ¯ρ Þ ð4:42Þ only F phys contributes the leading exponential 2 i i i i ρi X X 3 2 τ˜ ϵ¯ρ ϵ¯ρ ϵ¯ρ 1 must be augmented by the subleading pieces i − i i ρ σ˜ ν˜ − 1 p × 2 3 2 þ 6 þ 2½ i · þ i þ lRðri Þ ρ ∈ X X X i i Ri ðg − 1Þ × ϵ¯αα þ ðr − 1Þ ϵ¯ρ ρ · σ ; 1 τ i i i 2 α ρ × ϵ¯ρ − ϵ¯ρ þ ; ð4:46Þ i i i i 6 X X X ðg − 1Þ − × ϵ¯αα þ ðr − 1Þ ϵ¯ρ ρ · σ˜ ; ð4:43Þ τ˜ i i i α ρ i i from matters, which reduces to

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X X ν˜i 2 1 the Cardy exponent between the previous approach and p × ϵ¯ρ − ϵ¯ρ þ 2 i i 6 the BAE. i ρ ∈ i Ri One can also deduce the large radius limit of the − 1 1 ri 2 “physical” case. Recall, for each BAE solution, the leading þð1 − gÞ × ϵ¯ρ − ϵ¯ρ þ ; ð4:47Þ 2 i i 6 exponent from F phys is given by X X 0 3 0 0 on theories with matter content which is symmetric under 1 ðϵρ Þ ϵρ mρ i 0 2 0 0 0 i i p × þðϵρ Þ mρ − ϵρ mρ − þ charge conjugation. The contribution from the vector is 2 3 i i i i 6 6 i ρ ∈R i i X 1 1 X 0 3 0 0 2 1 ðϵαÞ 2 ϵα mα ð1 − gÞ × ϵ¯α − ϵ¯α þ : ð4:48Þ ϵ0 0 −ϵ0 0 − 2 6 þ 2 3 þð αÞ mα αmα 6 þ 6 ; ð4:51Þ G G 2π τ˜ Both appear in the exponent with i multiplied. with 2πiτ multiplied. Since the partition function in total is ν˜ The expression (4.47) plus (4.48), with i set to zero, has obtained by summing up the contributions with those been isolated for the high-temperature limit of the 4d leading exponentials, the simplest guess would be that 1 0 superconformal index [16], i.e., p ¼ and g ¼ , and the Casimir energy equals the smallest exponent among the govern the asymptotic behavior of the integrand prior to the values of (4.51) evaluated at the BAE solutions. holonomy integration, called Veff as in Ref. [16] modulo a constant shift. One subtlety is that the expressions we found 3. Cancellations in the Casimir limit via the BAE are meant to be evaluated and used at discrete places, u ¼ uH’s, so agreement with Ref. [16] requires that However, we also encounter a large class of examples the maximum of Veff necessarily occurs at an H-saddle. In where the Casimir energies do not equal the smallest fact, this is very likely since Veff is a piece-wise linear exponents computed above. The primary examples are function, as a consequence of ABJ anomaly cancellation, found in the superconformal indices, i.e., the partition and the derivative changes only at points where one or more function in “physical” background with p ¼ 1, g ¼ 0. The charged fields become massless. Thus the local maximum Casimir energy of the resulting SCI’s turns out to be equal ∈ Z and minimum can only occur at places where q · u for to the value of (4.51)at uˆ H ¼ 0, despite the presence of some charge q, and for a full agreement we only need to nontrivial H-saddles. This holds, in many cases for SCI, α exclude places, u0 where a vector multiplet of charge even with the naive uˆ H ¼ 0 saddle absent. becomes massless and no chiral multiplets are. This surprising fact can be demonstrated by rewriting the Since the contributions of the chiral multiplets to Veff, partition function as a unit circle contour integral, after using the anomaly condition, cannot change abruptly X there and since contribution from the α-charged vector F phys ν; τ ν; τ −1 ðu; ÞHðu; Þ will make a sharp turn, it suffices to consider how the ∈ u SBE derivative of Z 1 dx ¼ F physðx; y; qÞ; ð4:52Þ 1 jW j x 1 2πix −ϵ2 ϵ − G j j¼ Vα ¼ α þ α 6 ð4:49Þ the leading factor of F phys, which would have generated the behaves at ϵα ¼ 0, i.e., where α · u becomes an integer. Let holonomy-dependent Casimir energy, us consider a small neighborhood around u0 parametrized −1 1 Y Y π 3 π by

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This cancellation is possible in part because the H-saddle computation. The previous observation by position of H-saddles is aligned along the real axis of Ardehali on Cardy exponents [16] has effectively cap- 2πiu in this case. This should be contrasted to the Cardy tured this H-phenomenon on such a unit circle version of limit, where the H-saddles are located along the unit the superconformal indices. In contrast, such a cancella- circle jxj¼1 so that H-saddle phenomena manifests tion does not happen in the Cardy limit. even in this alternate integral formula. When the Something similar happens for the Casimir limit of the H-saddle occurs along the unit circle jxj¼1, the A-twist case when the fibration is nontrivial. To see this, we cancellations due to the anomaly cancellation condition should keep the finite part of the leading terms of F and H. no longer works because the infinite product formula For simplicity, we focus on the rank-1 case with p1 ¼ p, must be rewritten in new shifted variables whenever p2 ¼ 0. For massive matter fields at a given H-saddle, uH, one crosses such an H-saddle; this was at the heart of the the contribution from F is

3 Y Y ϵ¯ ϵ¯ ρi 2 ρi mρi 2 1 πi 2 πiτð þϵ¯ρ mρ −ϵ¯ρ mρ − þ Þþπiðϵ¯ρ þ2ϵ¯ρ mρ −mρ − Þðρ σþν Þ− ðmρ þmρ Þ e 3 i i i i 6 6 i i i i 6 i i 2 i i : ð4:54Þ

i ρi ρ λ ϵ 0 For massless matter fields, i.e., for i ¼ i such that λi ¼ , we have an additional factor

λ mλ ×ð1 − z i yiÞ i ð4:55Þ

2πiσ λ τ with z ¼ e . Using mλi ¼ iuH= , the contribution from massless fields can be written as follows: Y Y −πi λ2 2 τ2 λ τ λ τ 12 −λ 12 −1=12 −λ 2 −1=2 λ 2 1=2 λ u =τ 2 ð i uH= þ iuH= Þ iuH= = i= i= − i= i H e q z yi ½z yi z yi i λ iY Y −πi λ2 2 τ2 λ τ−2 τ λ τ 12 −λ 12 −1=12 u =τ 3d;H −u =τ 2 ð i uH= þ iuH= MuH= Þ iuH= = i= Λ H Φ H ¼ e q z yi H ð a Þ ; ð4:56Þ i λi P P where M mρ ρ and Λ is defined in (3.20). Note that u =τ is a rational number in [0, 1). Thus, at σ σ , the last ¼ i ρi i i H H ¼ factor becomes a root of unity,

3d;H −u =τ −2πiku =τ ðΦ Þ H jσ σ ¼ e H : ð4:57Þ a ¼

H consists of two parts: e2πiΩ and H. For massive matter fields, the leading term of e2πiΩ is given by Y Y 2 1 πiτðr −1Þð−ϵ¯ρ þϵ¯ρ − Þþπiðr −1Þð−2ϵ¯ρ þ1Þðρ σþν Þþπiðr −1Þmρ e i i i 6 i i i i i i ; ð4:58Þ

i ρi while for massless matter fields, we have an additional factor

λi −ðri−1Þ ×ð1 − z yiÞ : ð4:59Þ

The same expansion can be made for vector fields, by replacing ρi → α and ri → 2. Moreover, for SUð2Þ, H is explicitly written as

∞ ρ ρ ∞ −ρ −ρ −1 1 X X 1 X i i k X i i kþ ρ 2 − ρ ˆ σ ν τ fxHz yigq − fxH z yi gq j ij fð iðuH þ Þþ iÞ= gþ ρ ρ −ρ −ρ −1 1 : ð4:60Þ 2 1 − fx i z i y gqk 1 − fx i z i y gqkþ i ρi k¼0 H i k¼0 H i

For massive fields, the first line is the leading contribution 4. Explicit examples with G =SUð2Þ: The Casimir limit of order q0 while for massless fields, there is an extra Oðq0Þ λ We now explore some explicit examples for the Casimir z i yi contribution þ λ . A similar expansion is made for limit; recall that this side is prone to further subtleties 1−z i yi physical gauge as well by replacing νi → νi þ νRðri − 1Þ. beyond H-saddles. Let us discuss the A-twist case first.

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Considering asymptotically free theories of SUð2Þ, allowed ð1 þ w1Þð1 þ w2Þ 1 3 representations are those with isospin ≤ s ≤ . For a 4 2 2 2 z ðw1 þ w2Þ − 2z ð1 þ w1w2Þþw1 þ w2 model with few number of matters, BAE tends to be trivial ¼ 2 2 : ð4:66Þ ð1 − z Þ due to a lack of enough flavor symmetry and cannot be discussed using the A-twist formalism. The Intrligator- The fibering operator and the handle-gluing operator at Seiberg-Shenker model is such an example [47]. It has no uH=τ ¼ 1=2 are expanded as follows: anomaly-free flavor symmetry and, as a result, has the fixed Q 3 5 πi ν μ 2 2 anomaly-free R-charge R ¼ = , which allows A-twist 2fð i; iÞ 1 − eτ j¼1ð z wjÞ 1 only on a manifold of genus g ∈ 5Z due to the Dirac F ¼ Q þ Oðq2Þ; 7=12 7=12 2 z2 − w quantization condition for R-charges. Thus, we relegate the w1 w2 j¼1ð jÞ πihðν ;μ Þ discussion of this model to the physical gauge case, and 7 4e τ i i ð1 − w1Þð1 − w2Þð1 − w1w2Þ H 12 here consider SUð2Þ with fundamentals and adjoints. ¼ q × 3=2 3=2 w w The RGG anomaly condition restricts R-charges of 1 2 4 2 fundamentals and adjoints such that z ðw1 þ w2Þ − 2z ð1 þ w1w2Þþw1 þ w2 13 × þ Oðq12Þ; ð1 − z2Þ2 XNf XNa ð4:67Þ ðri − 1Þþ4 ðr˜j − 1Þþ4 ¼ 0; ð4:61Þ i¼1 j¼1 where ˜ where ri and rj are the R-charges of fundamentals and 2 2 adjoints respectively. For simplicity, we take ν μ ν3 ν3 μ3 μ3 fð i; iÞ¼3 1 þ 3 2 þ 1 þ 2; ð4:68Þ 4 − 1 ðNa Þ ˜ ν μ −4ν2 − 4ν 3μ2 3μ2 ri ¼ 1 þ ; rj ¼ 0: ð4:62Þ hð i; iÞ¼ 1 2 þ 1 þ 2: ð4:69Þ Nf One immediately notes that the leading term of F is With one adjoint, the numbers of flavors allowed by the proportional to the square root of BAE. Thus, asymptotically free condition are Nf ¼ 2, 4, 6. In those πi cases, however, the exponent (4.39) is independent of u,so 2fðνi;μiÞ eτ 1 not very interesting in our discussion. Instead, we discuss Fj ¼ þ Oðq2Þ;z∈ S nf1g: ð4:70Þ z¼z 7=12 7=12 the SUð2Þ model with two fundamentals and two adjoints. w1 w2 Because of the adjoints, the H-saddles for this model are Similarly, the leading term of H is also simplified at each located at uH=τ ¼ 0 and uH=τ ¼ 1=2. BAE solution as follows: Take the H-saddle at uH=τ ¼ 1=2. At this H-saddle only the vector field and the adjoint matter fields are massless πi ν μ 2 2 3 ∓ 1 τ hð i; iÞ 1 − 1 − 1 − while the fundamental matter fields become massive. Thus, 7 ð Þe ð w1Þð w2Þð w1w2Þ Hj ¼ q12 × z¼z 3=2 3=2 the reduced BAE is given by w1 w2 Q 13 O 12 2 ðz2 − w Þ2 þ ðq Þ; ð4:71Þ Q j¼1 j 1 2 1 − 2 2 ¼ : ð4:63Þ 1ð z wjÞ ∈ 1 ∈ j¼ where z Snf g and we have used (4.66) for z S−. As a result, the generic leading term at uH=τ ¼ 1=2 is The equation has eight solutions, which are classified into given by two classes S satisfying X Q F pHg−1 2 ðz2 − w Þ j¼1 j ∈ 1 z¼z Q 1;z∈ S : : z Snf g 2 1 − 2 ¼ ð4 64Þ 1ð z w Þ πi πi j¼ j z¼z 7 −1 pfðν ;μ Þþ ðg−1Þhðν ;μ Þ ∼ q12ðg Þ × ½2g þð−1Þp4geτ2 i i τ i i 2 2 g−1 For positive sign, the equation reduces to ½ð1 − w1Þð1 − w2Þð1 − w1w2Þ × 7 3 7 3 : ð4:72Þ 12pþ2ðg−1Þ 12pþ2ðg−1Þ z4 ¼ 1; ð4:65Þ w1 w2 g 0 p which has solutions z ¼1, i. Among them, since Unless ¼ and is odd, this term does not vanish, and q z ¼1 are Weyl invariant, only z ¼i are relevant the leading -exponent is given by solutions. For negative sign, on the other hand, the equation 7 p g−1 ðg−1Þ F H j τ 1 2 ∼ q12 : ð4:73Þ can be reorganized into uH= ¼ =

125013-24 HOLONOMY SADDLES AND SUPERSYMMETRY PHYS. REV. D 97, 125013 (2018) X 0 3 0 0 On the other hand, if g ¼ 0 and p is odd, this naive leading 1 ðϵρÞ ϵρ mρ ϵ0 2 0 −ϵ0 0 − term cancels out. In such cases, we numerically find the E0 ¼ þð ρÞ mρ ρmρ þ 2 3 6 6 ϵ0 3 0 −1 5 ρ∈½3=2 ρ¼5;mρ¼ true leading term, which turns out to be of order q12. X 1 ϵ0 3 ϵ0 0 At uH=τ ¼ 0, on the other hand, the reduced BAE is ð αÞ 0 2 0 0 0 α mα þ þðϵαÞ mα −ϵαmα − þ given by 2 3 6 6 ϵ0 0 0 1 α∈½1 α¼ ;mα¼ Q Q 511 2 − 2 2 − 2 1ðz yiÞ 1ðz wjÞ ¼ : ð4:78Þ Q i¼ Qj¼ 1; : 1500 2 1 − 2 1 − 2 2 ¼ ð4 74Þ i¼1ð zyiÞ j¼1ð z wjÞ On the other hand, for each H-saddle, the reduced BAE and 4 4 F phys p g−1 with the anomaly-free condition y1y2w1w2 ¼ 1. Since it is the leading terms of ð Þ H are given by difficult to solve this equation analytically, instead, we tried 6 numerical analysis for a given random phase values of y , − 7 1 ˆ τ i z ¼ at uH= ¼ 35 ; wj and found −2 3 z ¼ 1 at uˆ H=τ ¼ ; −1 − 5 −1 10 F pHg ∼ q 12ðg Þ; ð4:75Þ τ 0 1 uH= ¼ 8 1 ˆ τ z ¼ at uH= ¼ 2 ; which shows the exact agreement with the leading −2 7 z ¼ 1 at uˆ H=τ ¼ ; exponent at uH=τ ¼ 0 predicted by (4.39). Thus, there is 10 no cancellation of the leading terms at uH=τ ¼ 0. 29 τ 0 −z7 ¼ 1 at uˆ =τ ¼ ; ð4:79Þ Combining these results at uH= ¼ and at H 35 uH=τ ¼ 1=2, the leading term of the total partition function is given by and ( − 7 − 23 1 6 q 12;g¼ 0;peven; q 10500 × at uˆ =τ ¼ ; Ω ∼ : 7 2 H 35 g;p − 5 −1 ð4 76Þ z q 12ðg Þ; otherwise 61 z 3 1500 ˆ τ q × 2 at uH= ¼ 10 ; for SUð2Þ theory with two fundamental chirals and two 2 2 211 ð1 − z Þ 1 adjoint chirals. 1500 ˆ τ q × 6 at uH= ¼ 2 ; Next, we move on to the physical gauge case. As 8z et al. 61 z 7 advocated by Closset [33], the integer quantization 1500 ˆ τ q × 2 at uH= ¼ 10 ; condition for R-charges can now be relaxed, and we can consider theories that flow to nontrivial superconformal − 23 1 29 q 10500 × − at uˆ =τ ¼ ; ð4:80Þ points. The superconformal R-charge is then determined 7z5 H 35 by the anomaly-free condition and the a maximization. 2πiσ For physical gauge, a canonical example with potential with z ¼ e . Note that z ¼1 at uˆ H=τ ¼ 1=2 are Weyl H-saddles is the ISS model, which is the SUð2Þ model with invariant and again excluded from the solution set. One can a single isospin-3=2 matter. From the condition in Sec. III D, see that those leading contributions all vanish for p ¼ 1 as one can determine the H-saddles in the large radius limit as expected. We also confirmed numerically that the sublead- ing terms with q-exponents less than 511 are all canceled 6 3 1 7 29 1500 uˆ =τ ¼ ; ; ; ; : ð4:77Þ out such that the true leading term of the total partition H 35 10 2 10 35 511 function is of order q1500. Note that the value happens to coincide with the would-be exponent at uˆ H=τ ¼ 0, even Note that for physical gauge in the large radius limit, BAE though the tower sits at the uˆ =τ ¼ 1=2 saddle. This is one 1−g H does depend on the manifold because it contains lR ¼ p . example of cancellations for SCI’s which was advertised For simplicity we stick to lR ¼ 1 cases. previously. As we mentioned, for p ¼ 1, g ¼ 0, i.e., the super- On the other hand, such cancellations in favor of the conformal index, the partition function can be written as the would-be exponent at uH ¼ 0 does not necessarily happen unit circle contour integral, which predicts the Casimir for general values of p.Forp ¼ −2, g ¼ 3, as the second energy example, the leading ðF physÞpHg−1 is given by

125013-25 CHIUNG HWANG, SUNGJAY LEE, and PILJIN YI PHYS. REV. D 97, 125013 (2018)

23 4 6 We also saw that something similar happens with the q5250 × 49z at uˆ =τ ¼ ; H 35 Casimir limit as well. We again find that the notion of

− 61 4 3 H-saddles would be valid even in the large radius limit q 750 × at uˆ =τ ¼ ; z2 H 10 provided that there are two circles in the spacetime, at least 12 for computation of the partition functions. This will gen- −211 64z 1 750 ˆ τ erally complicate the asymptotics of typical partition func- q × 2 4 at uH= ¼ 2 ; ð1 − z Þ tions, just as in the Cardy limit. For this Casimir energy side, − 61 4 7 however, the connection to the global anomaly [20,22] is a q 750 × at uˆ =τ ¼ ; z2 H 10 little more robust than the Cardy side, although somewhat dependent on the background geometry; SCI’s, in particular, 23 10 29 5250 49 ˆ τ q × z at uH= ¼ 35 : ð4:81Þ turned out to enjoy a rather special structure such that this naive Casimir energy, apparently from uH ¼ 0, stands The locations of H-saddles are the same as those of p ¼ 1, uncorrected even though nontrivial H-saddles exist, and, 0 g ¼ 0 because the two geometries share the same lR ¼ 1. more surprisingly, even when the naive uH ¼ saddle is Substituting the BAE solutions at each H-saddle, the absent, due to magical cancellations between BAE vacua or leading terms are evaluated as even between H-saddles. For general partition functions, for example with p>1, such cancellations are more scarce. 23 6 Much of this section explored such diverse forms of the q5250 0 uˆ =τ ; × at H ¼ 35 Casimir energies and the Cardy exponents, and gave

− 61 3 precise methods for isolating these, albeit with no obvious 750 8 ˆ τ q × at uH= ¼ 10 ; universal formula.

−211 1 750 72 ˆ τ q × at uH= ¼ 2 ; V. SUMMARY

− 61 7 We have introduced the notion of the holonomy saddle, 750 8 ˆ τ q × at uH= ¼ 10 ; or H-saddle, and explored how the phenomenon manifests 4 N 1 23 29 in d ¼ ¼ massive gauge theories. 5250 0 ˆ τ q × at uH= ¼ 35 : ð4:82Þ Certain discrete values of the gauge holonomy are found to support d ¼ 3 N ¼ 2 supersymmetric gauge theories. 211 Thus, the leading q-exponent − 750 persists in this case, When the space is taken to be noncompact, the existence of multiple H-saddles means that a theory G compactified on a meaning that the nontrivial uˆ H=τ ¼ 1=2 saddle is dominant and the leading exponent there suffers no cancellations, in small circle admits multiple superselection sectors at dis- ’ contrast to the p ¼ 1 case. crete holonomy values uH s, where supersymmetric vacua are clustered which are in turn attributable to an effective 3d 5. Anomaly or not theory H. Such an H-theory tends to have generally a smaller light field content than the naive dimensional reduction, due Several observations relating these asymptotic coeffi- to the symmetry breaking by the Wilson line, although one cients to the axial and to the conformal anomalies were also typically finds the naive saddle at u ¼ 0 as well. This – H made recently [18,20 22,48]. One well-known example is observation dovetails nicely against some of the existing a relation between the Cardy exponent and the sum of studies of 4d-to-3d and 3d-to-2d reduction of dualities 1 Uð ÞR charges of fermions, on par with (4.40), which, for [13,14], where one finds that a single dual pair typically “ − ” superconformal cases, translates to a c where a and c generates multiple dual pairs in the lower dimensions. are the usual conformal anomaly coefficients. Its manifestation in the compact spacetime equipped One main consequence of our investigation is that such a with a circle, on the other hand, implies precise relations connection cannot be trusted in general. Whenever the between the twisted partition function of the G theory and matter content involves gauge representation beyond the those of the subsequent H theories, to which we have ≠ 0 simplest ones, H-saddles will tend to appear at uH , devoted the bulk of the computations. As such, the Witten 0 some of which could dominate the naive one at uH ¼ index of the G theory would be generally a sum of Witten easily. The canonical example of SQCD escapes this, since indices of the H theories, which explains, in part, how the the chiral multiplets are all in the fundamental representa- number of the supersymmetric vacua differs between two tion. It probably explains why this rather generic phe- theories in the adjacent dimensions even with the same ’ nomenon has so far failed to be noticed. For SCI s, such a supermultiplet content. This also offers a definite method ð1−gÞ deviation from 12 ·trfR has been observed first by for reconstructing one from the others. Ardehali [16] and subsequently by Di Pietro and Honda We also investigated the consequences for supersym- [17] for a handful of examples, but, as we saw, this metry-preserving torus-fibered compact spacetimes by deviation is more of a rule than an exception. observing how the twisted partition functions behave in

125013-26 HOLONOMY SADDLES AND SUPERSYMMETRY PHYS. REV. D 97, 125013 (2018) a small radius limit of one of the two circle fibers. The 4d What we have not explored here is how this phenomenon twisted partition function reduces to a sum of 3d partition relates to and interacts with the matter of disconnected functions in those limits, modulo exponential prefactors holonomy sectors, well known in the context of the 4d which are interpreted either as the Cardy or as the Casimir Witten index computations of pure Yang-Mills theories. behavior, depending on which direction is taken to be the Since our H-saddles would occur already for the holonomy 1 Euclidean time. The results on such exponents are gen- on S and since the corresponding discrete choices uH arise erally different from the existing claims, as the latter tends from the dynamics rather than from the topology, it is clear to focus, effectively, on the naive saddle at uH ¼ 0. that the topological consideration must be separately In the current examples of partition functions and considered as well for a more general gauge group G. theories, which admit the BAE description, H-saddles An immediate question is how the discussion here should are located by asking which subset of chiral matter fields be generalized when G is not simply connected [40] or become light at which discrete values of the holonomy. FI when the so-called “triple” is relevant [36–39]. We suspect constants and Chern-Simons levels, generated by KK we will encounter more issues related to such holonomy modes, can further complicate the pattern, which we saddles and holonomy islands in the near future. also delineated in much detail. The characterization of H-saddles should be a bit more general, however: an ACKNOWLEDGMENTS H-saddle would appear in the holonomy space wherever the dimensionally reduced theory admits supersymmetric We would like to thank Cyril Closset, Richard Eager, vacua, normalizable or non-normalizable [2]. This general Heeyeon Kim, Nati Seiberg, and Edward Witten for useful criterion for H-saddles should be valid for any super- conversations. The research of S. L. is supported in part by ysmmetric gauge theories, as long as the gauge holonomy the National Research Foundation of Korea (NRF) Grant is not exactly flat at the quantum level. No. NRF-2017R1C1B1011440.

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