The 4d superconformal index near roots of unity and 3d Chern-Simons theory

We consider the $S^3\times S^1$ superconformal index $\mathcal{I}(\tau)$ of 4d $\mathcal{N}=1$ gauge theories. The Hamiltonian index is defined in a standard manner as the Witten index with a chemical potential $\tau$ coupled to a combination of angular momenta on $S^3$ and the $U(1)$ R-charge. We develop the all-order asymptotic expansion of the index as $q = e^{2 \pi i \tau}$ approaches a root of unity, i.e. as $\widetilde \tau \equiv m \tau + n \to 0$, with $m,n$ relatively prime integers. The asymptotic expansion of $\log\mathcal{I}(\tau)$ has terms of the form $\widetilde \tau^k$, $k = -2, -1, 0, 1$. We determine the coefficients of the $k=-2,-1,1$ terms from the gauge theory data, and provide evidence that the $k=0$ term is determined by the Chern-Simons partition function on $S^3/\mathbb{Z}_m$. We explain these findings from the point of view of the 3d theory obtained by reducing the 4d gauge theory on the $S^1$. The supersymmetric functional integral of the 3d theory takes the form of a matrix integral over the dynamical 3d fields, with an effective action given by supersymmetrized Chern-Simons couplings of background and dynamical gauge fields. The singular terms in the $\widetilde \tau \to 0$ expansion (dictating the growth of the 4d index) are governed by the background Chern-Simons couplings. The constant term has a background piece as well as a piece given by the localized functional integral over the dynamical 3d gauge multiplet. The linear term arises from the supersymmetric Casimir energy factor needed to go between the functional integral and the Hamiltonian index.


Introduction
In the last few years there has been a renewed interest in the study of the superconformal index of 4d N = 1 superconformal field theories (SCFTs) and, in particular, N = 4 super Yang-Mills (SYM). The index in question is the supersymmetric partition function of the SCFT on S 3 ×S 1 which receives contributions from BPS states that preserve two supercharges (Q, Q). In the large-N limit, the expectation from AdS/CFT is that the index should account for the entropy of the BPS black holes (BH) that preserve the same two supercharges in the dual supergravity on AdS 5 . This question was introduced in [1][2][3], and the work of the last few years has shown that the index indeed captures the BH entropy in different asymptotic limits .
The focus of the present paper is the Cardy-like limit in which the BH entropy becomes very large. In the canonical ensemble, this translates to the study of the exponential growth of the index as τ → 0, where the parameter τ is the chemical potential dual to the charge. As pointed out in [13], the τ → 0 limit is in fact one of an infinite number of inequivalent Cardylike limits in which the index is expected to grow exponentially. These limits correspond to τ approaching a rational number or, equivalently, q = e 2πiτ approaching a root of unity. In this paper we analyze the 4d superconformal index near a general root of unity, and find interesting relations to three-dimensional Chern-Simons (CS) theory. The main statement is that the asymptotics of the index near a rational point −n/m is equal (to all orders in perturbation theory in deviations τ = mτ + n from the rational point) to the partition function of a certain 3d N = 2 gauge theory with Chern-Simons couplings that involve background as well as dynamical fields on an S 3 /Z m orbifold. The background couplings give rise to singular terms at O(1/ τ 2 ) and O(1/ τ ) that govern the growth of the index, while the constant O(1) term receives contributions from both background fields and the dynamical Chern-Simons theory.
We demonstrate this statement from two points of view-by direct asymptotic analysis of the index near rational points, and from an analysis of the reduced three-dimensional theory and calculating the various couplings using high-temperature effective-field theory (EFT) techniques. The latter method, based on [25,26], relates the high-temperature asymptotics of the index to a low-energy effective field theory, in the spirit of the Cardy formula. 1 The four-dimensional superconformal index and its asymptotic growth In this paper we study N = 1 gauge theories with a Lagrangian description and a U (1) R symmetry, with a focus on N = 4 SYM which we use to illustrate some statements in detail. The symmetry algebra of N = 1 SCFT on S 1 × S 3 is SU (2, 2|1), which includes the energy E which generates translations around S 1 , the angular momenta J 1 , J 2 on S 3 , and the U (1) R-charge Q. One can pick a complex supercharge obeying the following algebra, (1.1) The most general index built out of the N = 1 superconformal algebra is an extension of the Witten index of Q and is defined as the following trace over the physical Hilbert space, I(σ, τ ) = Tr H (−1) F e −γ{Q,Q}+2πiσ(J 1 + 1 2 Q)+2πiτ (J 2 + 1 2 Q) . (1. 2) The trace (1.2) only receives contributions from states annihilated by the supercharges ( 1 4 -BPS states) so that the right-hand side of (1.1) vanishes for these states. This index I(σ, τ ) can be calculated from either Hamiltonian or functional integral methods and reduces to a unitary matrix integral [3,[32][33][34], which can be written as an integral over the space of gauge holonomies around the S 1 of certain infinite products, as written in Equation (2.1).
Our focus in this paper is the analog, in the present context, of the high-temperature Cardy limit of 2d CFT. This means fixing the rank and taking the charges (J i , Q) to be 1 In the high-temperature picture the (Euclidean) time direction is taken along the S 1 , while in the lowenergy picture time is a fiber inside the S 3 . Relating the two pictures involves swapping time and space as in the derivation of the 2d Cardy formula [27]. Unlike in the two-dimensional context where one uses SL(2, Z) automorphy to relate the swapped problem to the original one, here we do not have an a priori understanding of the automorphic properties of the 4d index I(τ ). Aspects of this question are being addressed in [28]. See also [29,30] for related work on modular-type transformation properties relating different indices, and [31] for a discussion of the automorphic behavior of a different index in N = 2 SCFTs. larger than any other scale in the theory. In the canonical ensemble this translates to taking Im σ, Im τ → 0 at fixed rank. In order to calculate the asymptotic growth of states along a certain direction in the charge lattice, one needs to fix the relation between σ and τ . We study 2 the slice σ = τ − n 0 with n 0 an integer, as in [12,13,19]. Setting 2J = J 1 + J 2 , the resulting canonical index I is given by I(τ ; n 0 ) = Tr H (−1) F e −γ{Q,Q}−2πin 0 (J 1 + 1 2 Q)+2πiτ (2J+Q) . (1. 3) The large-charge asymptotics then implies Im τ → 0, while Re τ is not fixed a priori by the limit. We consider asymptotic limits as τ approaches a rational number τ → −n/m with gcd(m, n) = 1, introduced in the present context as new Cardy-like limits in [13]. The index I clearly depends on the gauge group G. We generally suppress it in our notation, but sometimes use the notation I N to emphasize the dependence on N for U (N ) or SU (N ) N = 4 SYM theory (which should be clear from the context). Our motivation to consider these rational points comes from the study of the index I N (τ ) of N = 4 SYM in the large-N limit. 3 In this limit one considers charges scaling as N 2 as N → ∞, which translates to N → ∞ at fixed τ in the canonical ensemble [20]. In this large-N limit one expects the field theory index I N (τ ) to be written as a sum over saddles. This picture has been partially realized in the last few years using two different approachesthe Bethe-ansatz-like approach developed in [7,35,36], and the direct study of large-N saddle points using an elliptic extension of the action [13,19]. In particular, the large-N approach in [13] found a class of saddles labelled by rational numbers −n/m, where the perturbation expansion around each saddle is given by the asymptotic limit τ → −n/m. 4 Setting n 0 = −1, we have log I N (τ ) ∼ −S eff (m, n; τ ) , τ → −n/m , (1.4) where the effective action at each saddle is given by S eff (m, n; τ ) = N 2 πi 27 m 2 τ + χ 1 (m + n) 3 τ 2 , τ := mτ + n . (1.5) where χ 1 (n) is the Dirichlet character equal to 0, ±1 when n ≡ 0, ±1 (mod 3), respectively. There was one caveat in the above result, which was stressed in [13,19], namely that the pure-imaginary τ -independent term could not be fixed by the methods used in those papers. The constant term in the effective action (1.4), therefore, was a convenient choice made using inputs coming from outside the field-theory analysis. 2 Our methods can be generalized to study the case where σ and τ are linearly dependent over the rationals, but we shall not develop this in the present paper. 3 Another motivation comes from the mathematical literature on q-series, where it is also natural to consider expansions around roots of unity. We thank D. Zagier for emphasizing this point to us. 4 These saddles map to residues of the Bethe-ansatz type approach-see [22] for a recent discussion of the connections between the two approaches. A larger set of saddles have been classified in [19], but the full set of important/contributing saddles is not understood in either approach. In particular, interesting continuum configurations of the Bethe-ansatz equations have been recently discovered in [37][38][39] whose role in the large-N limit is not fully understood.
complete asymptotic expansion in τ terminates at O(τ )-i.e. the perturbation theory only contains 1/τ 2 , 1/τ , τ 0 and τ up to exponentially suppressed corrections. (This is, in fact, more generally true when the index is dominated by isolated saddles, and not true when there are flat directions; see [48].) Interestingly, it was found in [41] that the constant term in the expansion contains the partition function of SU(N ) pure Chern-Simons theory on S 3 at level ±N .
In this paper we find that the same structure persists at all rational points. We see that the constant term in the expansion as τ → −n/m involves Chern-Simons theory whose action is 1/m times the action as τ → 0. We present evidence that this corresponds to CS theory on an orbifold space S 3 /Z m (with the action of Z m depending on n such that the orbifold coincides with the lens space L(m, −1) when n = 1) at level ±N [49,50]. In other words, the 4d SYM index appears to play the role of a master index which governs the partition function of three-dimensional CS theory on an infinite family of S 3 orbifolds.
The appearance of 3d Chern-Simons theory from the 4d superconformal index is intriguing, and gives rise to two related questions: (a) is there a direct three-dimensional physics explanation of the appearance of Chern-Simons theory? (b) can we also understand the singular terms in the asymptotic expansions around rational points as being related to 3d Chern-Simons theory? The answers to both these questions are positive, as we now explain.

The asymptotics of the 4d index from supersymmetric Chern-Simons theory
The natural idea is that the reduction of the four-dimensional theory on S 1 gives rise to a three-dimensional theory on S 3 in a "high-temperature" expansion in powers of the circumference β of the shrinking circle. If we calculate the functional integral of the threedimensional theory, we should recover the four-dimensional functional integral as β → 0. The three-dimensional effective field theory is known to have a derivative expansion, where the most relevant terms are Chern-Simons terms [51,52]. This EFT approach was developed in the supersymmetric context in [25,26] who presented supersymmetrized CS actions involving the dynamical as well as background fields, which are necessary for preserving supersymmetry on S 3 ×S 1 . In particular, the 1/β 2 and 1/β effective actions derived this way in [26] reproduced the asymptotics of the the index as found in [48] for n 0 = 0 and arg(τ ) = π/2. (Note that when the metric on S 3 × S 1 has a direct product form with S 3 the unit round three-sphere, a real value of β determines a purely imaginary τ = iβ 2π .) The coefficient of the leading 1/β 2 term in these works is pure imaginary, and also does not grow as N 2 (it is in fact zero for non-chiral theories), therefore the exponential growth of states corresponding to the BH is not captured there.
One of the motivations for the current paper is to explain the exponential growth associated to the bulk black holes from the three-dimensional point of view, which requires arg(τ ) = π/2 and n 0 = 0. 5 (Note, in particular, that the (m, n) = (1, 0) saddle in (1.5) given for n 0 = −1, would have its leading piece a pure phase if arg(τ ) = π/2.) For this purpose we consider, as in [5], a background geometry of the form S 3 × Ω S 1 , with S 3 the unit round three-sphere, γ the circumference of the circle, and Ω a twist parameter 6 controlling the deviation of the metric from a direct product form (Equation (3.1)). The imaginary part of the twist parameter determines a non-zero real part of τ via (Equation (3.4)) As shown in [5], the integer n 0 in (1.3) controls the periodicity of the fermions in this background, and n 0 = ±1 (which is naturally dual to the BH) corresponds to anti-periodic fermions, i.e., as in a Scherck-Schwarz reduction. In the present context we insist on supersymmetry being preserved-and that necessitates the turning on of other background fields under which the fermions are charged. In the three-dimensional background supergravity, we have a non-zero graviphoton from the fibration as well as non-zero auxiliary background gauge and scalar fields. As we explain in Section 3, the resulting configuration is effectively described by a circle of radius R, which in the limit γ → 0, Ω → ∞ with τ fixed obeys R → τ . Now, what is the actual calculation? There are two types of fields in the three-dimensional functional integral-background fields which take constant values, and dynamical modes which fluctuate in the integral. The latter is further made up of light modes (with zero momentum around S 1 ) and heavy (Kaluza-Klein) modes. The first step is to integrate out the heavy modes in order to obtain an effective action for the light modes. The integration over heavy modes also generates corrections to the coefficients of the supersymmetric Chern-Simons terms of the non-zero background fields, see e.g. [25,54]. In these calculations, we need to include, in addition to the couplings discussed in [26], the supersymmetrized RR and gravitational CS actions which were discussed in [55]. The effective actions of the background gauge fields turn out to produce precisely the singular pieces 1/τ 2 and 1/τ in the asymptotic expansion of the index, as well as a constant piece. The remaining functional integral is described by an N = 2 SYM theory with a certain one-loop induced CS coupling on S 3 , whose partition function is known to agree, up to a sign, with that of pure Chern-Simons theory [56]. This explains the appearance of the dynamical Chern-Simons theory in the constant term of the asymptotic expansion.
Two technical remarks are in order. Firstly, recall that supersymmetry implies that the 4d superconformal index should not depend on γ and Ω separately, but only on their combination τ as in (1.6). In [5] this was shown to be true in 5d gravitational variables, as well as through a localization computation in 4d field theory. In this paper we verify this also in 3d effective field theory. Secondly, the order of limits is important to have a smooth 5 The leading order 1/τ 2 behavior was found from similar considerations in [6] for N = 4 SYM with flavor chemical potentials, and in [11] for more general gauge theories in a setting similar to ours. In this paper we follow a systematic, manifestly supersymmetric approach developed in [25,26], which allows us to obtain all-order results for general gauge theories around generic rational points. 6 Similar twists had been described in slightly different contexts in [33,53].
calculational set up. We first send γ → 0 keeping Ω fixed, so that the three-dimensional geometry is smooth and finite. Then we take Ω → ∞ at fixed τ and express the result in terms of τ using (1.6). We find there are no singularities generated in the latter step and thus the limiting procedure is perfectly smooth. Finally, we repeat the same analysis as τ approaches rational points. The dimensional reduction in this case naturally leads us to considering orbifolds of S 3 ×S 1 , which, as far as we understand, are related to the orbifolds discussed in [40]. The three-dimensional calculation then leads to the 1/ τ 2 and 1/ τ terms as well as a constant piece from the background fields, and we provide evidence that the remaining dynamical piece is the partition function of N = 2 SYM with a one-loop induced CS coupling on S 3 /Z m .
Notation. We have σ, τ ∈ H and z, u ∈ C, and we set p = e 2πiσ , q = e 2πiτ , ζ = e 2πiz . We use to mean an all-order asymptotic equality of the logarithms of the two sides.

The 4d superconformal index and its asymptotic expansion
We consider a four-dimensional N = 1 gauge theory which flows to a superconformal fixed point. The theory has gauge group G (which we take to be semi-simple, and separately comment on the U (N ) case), and a number of chiral multiplets labelled by I with R-charge r I and in the representation R I of the gauge group. We assume 0 < r I < 2 for all chiral multiplets. The superconformal index for these theories on S 3 × S 1 has been calculated in the Hamiltonian as well as functional integral formalism [3,[32][33][34], and the answer is expressed as an integral over the Cartan torus which we parameterize by the vector of gauge holonomies u = (u 1 , . . . , u rk(G) ), with u i ∈ R/Z. The index is given by the following expression [57][58][59] I(σ, τ ) = [Du] Z vec (u; σ, τ ) Z chi (u; σ, τ ) .
The factors Z vec , Z chi denote the vector multiplet and chiral multiplet contribution respectively given by Here α + runs over the set of positive roots of the gauge group G, I runs over all the chiral multiplets of the theory, and ρ is the weight of the gauge representation R I . The Pochhammer symbol is defined by and the elliptic gamma function [44,60] is defined by the infinite product formula Γ e (z; σ, τ ) = j,k≥0 From now on in this section we set σ = τ − n 0 , and use the notation Γ e (z) = Γ e (z; τ, τ ). We have where the action S ind (u) = S ind (u; τ ) is given by Our goal now is to calculate the asymptotics of the function I(τ, n 0 ) as τ approaches a rational number or, equivalently, q = e 2πiτ approaches a root of unity. For N = 4 SYM we have for U (N ) and a similar expression for SU (N ) with N replaced by N − 1. Using the product expression (2.4) we see that for N = 4 SYM the index I N (τ ; n 0 ) has the symmetry τ → τ + 3 for fixed n 0 , so that we can restrict our attention to, say, τ ∈ [0, 3]. Relatedly, the independent values of n 0 are 0, ±1. More generally, the periodicity of τ depends on the quantization of R-charge in the theory. Before analyzing these asymptotic limits we briefly discuss a slightly independent motivation to study these new limits and, relatedly, the origin of the number n 0 in (2.5), (2.7). One of the motivations in the recent developments in this subject has been to "find the dual black hole" in the superconformal index. In terms of the microcanonical Fourier coefficients the problem in the context of the Cardy-like limit is to check if |d N (n; n 0 )| ∼ N 2 s(n/N 2 ) as n → ∞ [20]. In the canonical setting this is reflected by a corresponding asymptotic growth of the function I(τ ) as τ approaches the real axis or, equivalently, as q = e 2πiτ approaches the unit circle. The leading asymptotics of the growth of microcanonical degeneracies is governed by the dominant singularity of I. As it turns out, the index I N (τ ; 0) of N = 4 SYM does not have any exponential growth as τ → 0 (the growth is power-law [48]). It is the asymptotic growth of log I N (τ ; 0) as τ → ±1 instead that matches the on-shell action of the AdS 5 BH (the two points giving growth of equal magnitude and opposite phases). From a numerical study of the microcanonical degeneracies one can deduce that this is, in fact, the leading growth of the index [20]. In this case, noting that I N (τ ± 1; n 0 ) = I N (τ ; n 0 ∓ 1), we see that the leading growth can be stated as coming from the growth of the function I N (τ, ∓1) as τ → 0. Actually, one finds that the growth of states at n 0 = ±1 matches the BH growth of states for very large classes of N = 1 SCFTs [11,12]. Once we understand that the growth can come from a region with Im τ → 0 but Re τ = 0, it is perhaps more natural to set n 0 = 0 (for which the two regions of leading growth have a symmetric placement around τ = 0). We will, nevertheless, keep n 0 as an explicit parameter in the following to make contact with related literature.
It is clear from the above discussion that one should equally well explore other points on the unit circle in q. 7 As it turns out there is exponential growth near any root of unity consistent with (1.4), (1.5), i.e. partial deconfinement in the sense of asymptotic growth [37]. In the following subsections we proceed to analyze the asymptotic behavior of the index as τ → 0 and then as τ approaches any rational number.

Asymptotics of the index as τ → 0
In this subsection we perform an all-order asymptotic analysis of the integral (2.5) as τ → 0. This calculation was done for N = 4 SYM recently in [41] using a saddle-point analysis. Here we find the asymptotics for the general class of theories discussed in the introduction, using the rigorous method of [46,48] (see in particular Section 3.1 of [48]). The application in [48] was restricted to real τ and n 0 = 0, but the method is more general and we apply it to the case of complex τ and general n 0 .
We first calculate the all-order asymptotic expansion of the integrand. In order to do this we need the the asymptotic behavior of the elements in (2.6), namely the Pocchammer symbol and the elliptic gamma function, which we review in Equations (A.1), (A.2), (A.13). Using these estimates we find that in the range α + · u ∈ (−1 + δ, 1 − δ) (for fixed small δ) the integrand of (2.5) can be written, up to exponentially suppressed corrections, as The all-order effective potential as τ → 0 is given by The superconformal index as a function of q is defined on a branched cover of the complex plane and one should explore the full covering space. For N = 4 SYM one has a three-sheeted cover and the leading growth occurs on two of the three sheets [11,12,37]. with where α runs over all the roots of G including the rk(G) zero roots, I runs over all the chiral multiplets, and ρ I runs over all the weights of the representation R I . Note that in (2.9) we have separated the supersymmetric Casimir energy given by [61] E susy = 1 6 TrR 3 − 1 12 TrR . (2.12) We make a brief comparison to [12] in which the singular pieces were studied. The potential V 2 in (2.11) coincides (up a multiplicative −iπ/6 factor) with the V 2 studied in [12]. At finite u, the sinh 2 ( πα + ·u −iτ ) factors in (2.9) also contribute to O(1/τ ). Including this piece in V 1 renormalizes it to V r 1 as 8 The potential V r 1 coincides (up to a multiplicative −iπ factor) with the V 1 studied in [12]. In our treatment below we keep the sinh 2 factors separate and place them in the "dynamical measure" (2.14) Compared to [12], here we also include the O(τ 0 ) piece corresponding to exp(−V 0 ). Finally, the O(τ ) piece of the exponent is determined by the supersymmetric Casimir energy and, notably, there are no O(τ 2 ) or higher corrections in the perturbative effective potential.
We now investigate the local behavior of the potential near u = 0. The potentials V 2 , V 1 , V 0 are piecewise polynomials, and using B j = jB j−1 we obtain their Taylor expansion near u = 0 8 Instead of using (A.13) for vector multiplet gammas and then simplifying the sinh term for finite u as above, we could alternatively use (A.2) for vector multiplet gammas (assuming α+ · u / ∈ Z) and get V r 1 directly as in [12].
(2.15) Importantly, the second lines of V 2 , V 1 , V 0 above vanish due to gauge 3 , U (1) R -gauge 2 , and U (1) 2 R -gauge and gravity 2 -gauge anomaly cancellations. Therefore V 2 is actually piecewise quadratic in u, while V 1 is piecewise linear and V 0 is piecewise constant. This is similar to Section 3.1 of [48].
The leading asymptotic behavior of V as τ → 0 is determined by V 2 . In order to obtain a local minimum of Re(V ) at u = 0, we want (i) the linear term in V 2 to vanish, and (ii) the quadratic term to be on the negative (respectively positive) imaginary axis for arg(τ ) − π 2 > 0 (respectively arg(τ ) − π 2 < 0). As found in [12], we can achieve both of these requirements in any theory in which 0 < r I < 2 by specializing to n 0 = ±1. Explicitly, for n 0 = ±1 we can use the fact that for |x| < 1 we have B 2 (x) = x 2 − |x| + 1 6 to deduce that the linear term in V 2 is equal to which vanishes thanks to the U (1) 2 R -gauge and gravity 2 -gauge anomaly cancellations. Similarly we can use the fact that for 0 < |x| < 1 we have where the equality follows from U (1) R -gauge 2 anomaly cancellation, and we have used that sign(n 0 ) = n 0 for n 0 = ±1. This quadratic piece is on the positive (respectively negative) imaginary axis for n 0 = +1 (respectively n 0 = −1). In this manner we see that u = 0 is a local minimum of Re(V ). Therefore in the rest of this subsection we focus on n 0 = ±1, and take arg(τ ) − π 2 to have the opposite sign to n 0 , i.e. n 0 (arg(τ ) − π 2 ) < 0.
Using the explicit expressions of B 1,2,3 (x) in the range 0 < |x| < 1, and using the anomaly cancellation conditions, the potentials V 2 , V 1 , V 0 simplify, for n 0 = ±1, to Note that, as a bonus, V 1 also becomes independent of u for n 0 = ±1 and small enough u. 9 We now consider a small neighborhood h cl around u = 0, defined by the cutoff |u j | < , whose contribution to the index is From the above discussion we have that where the background piece is and the dynamical piece is Here we suppress the dependence of these functions on the gauge group and the matter content.
To simplify Z dyn further, we first define x j = u j −iτ , so that the integral becomes along straight contours from x j = − −iτ to x j = + −iτ . With our choice of n 0 and arg(τ ), the 9 We will shortly interpret the quadratic term in V2 as inducing a Chern-Simons type coupling in the integrand. If the linear term in V1 were present, it would similarly induce an FI parameter in the integrand. While this is impossible for semi-simple gauge theories near u = 0 which we are focussing on in this section, there are cases of semi-simple gauge theories in which one must expand around u = 0 and as a result one finds the measure of an abelian gauge theory in the integrand, where such induced FI parameters do arise. See Section 3.3.1 of [48] where the ISS model displaying an SU(2)→ U (1) breaking pattern with an induced FI parameter in the τ → 0 limit is discussed, and see Appendix A of [62] where that induced FI parameter is given an effective field theory explanation.
integrand is locally exponentially suppressed away from u = 0, so we can complete the tails of the contours along straight lines to infinity (i.e. send → +∞) introducing only exponentially small error. The contours make an angle π 2 − arg(τ ) with the positive real axis. However, observing that (i) the integrand is exponentially suppressed as |x j | → ∞, and (ii) there are no poles between the contour of x j and the real axis, we can deform the contours back to the real axis. We thus obtain, with x = (x 1 , . . . , x n ) As noted in [41] for N = 4 SYM, and in [42,43] for more general groups, Z dyn is related to the partition function of pure Chern-Simons theory [63] on S 3 as with the gauge group implicit and the same on both sides, and with Chern-Simons coupling given by Notice that Z dyn (n 0 ) is independent of τ . The tails completion (i.e. sending → ∞) and contour deformation mentioned above removed the τ -dependence of Z dyn at the cost of an exponentially small error.
The considerations of three-dimensional effective field theory in the next section show that Z dyn arises naturally in fact as the partition function of three-dimensional N = 2 gauge theory on S 3 with the same gauge group and the same CS coupling. (The latter is well-known to be related to Z CS exactly as in (2.24).) The above analysis was local around u = 0. We now focus on SU(N ) N = 4 SYM for which we know that u = 0 is a global minimum of the leading potential V 2 /τ 2 for n 0 = ±1 and n 0 (arg(τ ) − π 2 ) < 0. 10 However, this is not the only global minimum-there are N isolated global mimima labelled by k = 1, . . . , N which are related to u j = 0 by center symmetry, namely the points u j = (k − 1)/N , k = 1, . . . , N [10]. Upon summing over these minima we obtain The factor of N arises from the sum over N minima as explained above. The other three factors can be calculated by specializing our general discussion to this case: In this case the matrix of Chern-Simons couplings reduces to a single level (k ij = kδ ij ), and we have For n 0 = ±1 the SU(N ) Chern-Simons partition was found in [41] to simplify to The analogous result for the case with U (N ) gauge group is obtained by adding the contribution of a decoupled U (1) N = 4 multiplet to that of the SU(N ) theory: This finishes the discussion of our methods illustrated in the special case τ → 0. Before moving on to the more general case of rational points, we make a few technical remarks.
Firstly, since we are analyzing the index by estimating its integrand, we need uniform estimates. For n 0 = ±1, the estimate (A.2) when applied to the chiral multiplet gamma functions gives uniformly valid asymptotics near u = 0, because the −n 0 r I /2 shift in the argument takes us safely into the domain of validity of (A.2). For the vector multiplet gamma functions, however, there is no finite shift in the argument, so (A.2) does not apply uniformly around u = 0. We had to use instead (A.13) to obtain uniform asymptotics near u = 0 for the vector multiplet gammas.
Secondly, we emphasize that our asymptotic analysis is essentially real-analytic (as in [46,48]). We only appeal to complex-analytic tools (specifically, contour deformation), after having done the asymptotic analysis, to simplify the final answer for Z dyn in (2.22) to the more familiar form (2.23). 11 For comparison with [16], we set ξ there a = −n here 0 /3. The result in that paper contains the number η ∈ {−1, +1}. This is related to our n0 as η = 6B1(− n 0 3 ). For n0 = ±1, a simple calculation shows that η = n0.
Thirdly, we note that when actually doing the saddle-point analysis, one finds that the dominant holonomy configurations spread into the complex plane, as in the analysis of [7,13,41]. Upon taking the τ → 0 limit the spreading shrinks, and the answers from those approaches indeed agree with our results.

Asymptotics of the index as τ → Q
We now study the index (2.5) in the limit with m, n relatively prime, keeping arg( τ ) away from integer multiples of π/2. As in the previous subsection we first calculate the all-order asymptotics of the integrand. The required small-τ estimates for the Pocchammer symbol and the elliptic gamma function are given in Equations (A.17), (A.23), (A.26). Using these estimates we find that in the range α + · u ∈ (− 1 m + δ, 1 m − δ), for some fixed small δ, the integrand of (2.5) can be written up to exponentially suppressed corrections as The all-order effective potential as τ → 0 is given by where α runs over all the roots of G including the rk(G) zero roots, I runs over all the chiral multiplets, and ρ I runs over all the weights of the representation R I . Here s(n, m) is the Dedekind sum defined in (A.18) and the function C(m, n, r, z) is defined in (A.24). We have defined to emphasize an analogy with the analysis in [10,37] of the index with flavor chemical potential ξ, although we do not have flavor fugacities in our problem. Note also that we have separated the supersymmetric Casimir energy in (2.35) as in the τ → 0 case.
Next, as in the previous subsection we expand the potentials near u = 0. Anomaly cancellations again lead to simplifications, but here we further assume the theory is nonchiral (i.e. that ρ I come in pairs of opposite sign) so that the answer takes a particularly simple form. Analogously to (2.18) we obtain (2.39) Next we focus on a small neighborhood h cl around u = 0, defined by the cutoff |u j | < , whose contribution to the index as τ → 0 is Upon putting the above discussion together we obtain where the background piece is 42) and the dynamical piece is (2.43) Upon defining the rescaled variable x j = u j −i τ , we recognize Z dyn (τ ; m, n, n 0 ) as the CS partition function on S 3 with gauge group G and level k ij = − 1 m I,ρ I B 1 mξ I ρ i I ρ j I . We will see momentarily that it is more natural to define the rescaled variable as x j = mu j −i τ . Upon tails completion and deforming the integration contour we obtain Up to the overall m −rk(G) factor, this coincides [50] with the topologically trivial sector of the S 3 /Z m partition function of N = 2 SYM with Chern-Simons coupling While the explicit expression for the dominant potential V 2 in (2.37) was derived in a neighborhood (− 1 m + δ, 1 m − δ) of u = 0, it is actually correct more generally, because it follows from (A.23) which we can use as long as mρ I · u + mξ I / ∈ Z. Moreover, since ρ j I are integers and u j appears in V 2 in the combination m ρ I · u, the 1-periodicity of V 2 implies that any holonomy configuration with u j a multiple of 1/m gives the same leading asymptotics as the u = 0 configuration. In the SU(N ) case these non-trivial holonomy configurations correspond to with m j ∈ Z/mZ, and N j=1 m j = 0 (mod m). For n = 1, we can use the estimate (A.31) for the vector multiplet gamma functions to compute the contribution of the saddles (2.46). The result is similar to (2.40), with the same V 2,1 and the same SUSY Casimir piece, but with the dynamical piece modified to (modulo an overall constant factor) where m indicates that we are considering the contribution from a neighborhood |u j − m j /m| < , and the re-scaled integration variable arises as . This coincides (again up to an overall constant factor) with the topologically non-trivial sector of the partition function of SU(N ) Chern-Simons theory with coupling (2.45) on the lens space L(m, −1) [50].
We expect that similarly for general n, including the contribution of the non-trivial saddles (2.46) to the index would complete Z dyn 0 (m, n, n 0 ) to the full S 3 /Z m partition function, including all the topologically non-trivial sectors. We motivate this expectation further from an EFT perspective in the next section where we also present the (n-dependent) action of Z m on the S 3 . The explicit demonstration, which we leave to future work, requires generalizing the estimate (A.31) to arbitrary n, and improving it to include the overall constant.
The above analysis was local in nature: we considered the contribution to the index from only a small neighborhood of u = 0. We now study the specific case of SU(N ) N = 4 SYM for which we present a global picture of the dominant holonomy configurations. Note that in the previous subsection rather than performing the global analysis from scratch we had borrowed the result of [11,12] that in a certain domain of parameters (n 0 (arg(τ ) − π/2) < 0) the u = 0 configuration is globally dominant (see Section 2.3 for the complementary domain).
The global structure of the leading potential for N = 4 SYM For SU(N ) N = 4 theory the potential V 2 reads where the factor of 3 comes from the sum over three chiral multiplets, and with As mentioned below (2.45) the expression (2.48) applies as long as u ij + ξ I avoid Z m . We now have to minimize the real part of V 2 (u)/ τ 2 as | τ | → 0. Since the u ij -independent piece and the real positive overall multiplicative constants are irrelevant in finding the dominant holonomy configurations, our problem boils down to minimizing the potential which is analogous to the pairwise holonomy potential in [10]. As in that work, we first consider the qualitative behavior of V Q . We assume arg( τ ) − π 2 > 0, and comment below on what happens for the opposite sign. We find that the potential is (see Figure 1) We also see from Equation (2.49) that we have {mξ} ∈ {0, 1 3 , 2 3 }. (2.51) For arg( τ ) − π 2 < 0, the M-and W-regions switch places. So in order to have u ij = 0 as the dominant saddle we must assume m, n are such that {mξ} = { −mn 0 −2n (2.53) In this case we have that B 2 (2/3) = +B 2 (1/3) = −1/18, which leads to exp − 2πi 9 for arg( τ ) > π 2 as well as arg( τ ) < π 2 . Note that since B 2 (0) = 1/6, we see from (2.53) that there is no O(1/ τ ) exponent for {mξ} = 0, as alluded to above.

The Chern-Simons coupling
Specializing the Chern-Simons coupling (2.45) to SU(N ) N = 4 theory we find with We emphasize that all the topologically nontrivial sectors necessary for agreement with an S 3 /Z m partition function are present in our analysis, but we leave the investigation of their explicit contributions to future work.
The linear (in τ ) exponent can be read from (2.40) to be −2πi τ E susy /m. Note again that while (2.40) was derived near u = 0, as the estimate (A.31) shows the O( τ ) exponent remains valid near u ∈ Z m as well (at least for n = 1, and we expect more generally as well). Since for SU(N ) N = 4 theory E susy = 4 Summary: the smallτ asymptotics for N = 4 SYM We can summarize the asymptotics of the SU(N ) N = 4 SYM index analyzed above as follows for τ near any rational point −n/m, with and with C N (n 0 , m, n) an overall constant. Note that we have used η 3 = η = ±1 to simplify the final expression. Also, by completing the cube inside the exponent we have introduced an O( τ 0 ) factor at the cost of redefining C N (n 0 , m, n).
We have only demonstrated that there is a contribution to Z CS S 3 /Zm (k) from near u = 0 that coincides with the topologically trivial sector of the S 3 /Z m partition function of Chern-Simons theory with coupling k. As mentioned below (2.46) we expect that summing over the contributions from neighborhoods of the non-trivial configurations u j = m j /m would lead to the complete orbifold partition function.
We can include the contribution of a decoupled U (1) N = 4 multiplet in a straightforward manner. This effectively changes the dimension of the group in the exponent to N 2 , introduces a prefactor 1/ τ , and change the constant from C N (n 0 , m, n) to a new constant C N (n 0 , m, n), so that we have We see that the background (and the SUSY Casimir) piece in (2.60) matches the effective action (1.5) and, in addition, we have a dynamical Chern-Simons term. In the following section we explain both these pieces from the point of view of 3d N = 2 field theory.

C-center phases
Focussing on SU(N ) N = 4 theory, we now move on to studying the τ → 0 limit of the index in the W region, which as shown in Figure 1 for arg τ > π/2 corresponds to 1/2 < {mξ} < 1. As before we assume arg τ is in compact domains avoiding integer multiples of π/2 as | τ | → 0. Recall from (2.49) that only the values {mξ} = 0, 1/3, 2/3 are realized in our problem. But to highlight the parallels with the analysis of partially-deconfined phases in the W regions of the (flavored) 4d N = 4 index in [37], we will study the phase structure for arbitrary {mξ} ∈ ( 1 2 , 1) below, and only at the end specialize our result to the single "physical" point {mξ} = 2/3 in that interval.
Asymptotic analysis of the index for arbitrary {mξ} ∈ ( 1 2 , 1) is difficult for general N , because finding the dominant holonomy configurations is not possible analytically in the W regions. Analogously to [37] we consider now the large-N limit (on top of the τ → 0 limit), and conjecture that the C-center phases suffice for extremizing the potential in the W region. Also, similarly to [37] we consider only the leading (here O(1/ τ 2 )) exponent of the index in the W region.
A C-center holonomy configuration consists of C packs of N/C holonomies uniformly distributed on the circle such that the SU(N ) constraint is satisfied. While at finite N it is possible to have such configurations only for C a divisor of N , in the large-N limit any integer C ≥ 1 provides an acceptable C-center configuration [37]. For such a distribution the "on-shell" value of the potential V 2 in (2.48) becomes (2.61) where d := N/C. The second term above is the contribution from pairs in the same pack, and the third term is from pairs with each end on a different pack. To simplify the above expression further, we use the following identity which can be proven from (A.6) and (A.21): where g := gcd(m, C) and C := C/g. Keeping only the O(N 2 ) terms we hence end up with Since the leading asymptotics of the index is given as exp(− V 2 / τ 2 ), we then find the analog of the main result of [37] (Equation (3.19) of that work) for our case to be with mξ = − mn 0 +2n 3 as before.
The competition between various terms in (2.64) can be visualized by comparing the exponents as in Figure 2, which shows the range of ∆ := {mξ} for which a given phase dominates when arg τ − π/2 > 0. The figure implies that for the "physical" values {mξ} = 1/3, 2/3, the index is respectively in the 1-center, and 2-center phase when arg τ − π/2 > 0, and vice versa for arg τ − π/2 < 0. As mentioned above, for {mξ} = 0 the index is in a confined phase and does not yield exponential O(N 2 ) growth. Therefore up to an o(N 2 / τ 2 ) error in the exponents we have the following simplification of (2.64) by restricting to C = 1, 2: .

(2.65)
This is the analog of Conjecture 1 in [37]. Since B 3 (2/3) = −B 3 (1/3), we see from (2.65) that the action of the 2-center saddle has the opposite sign and is smaller in absolute value by a factor of 8 compared to that of the 1-center saddle. Figure 2: The functions C −3 B 3 (C ∆) with C = 1, · · · , 13, for 0 ≤ ∆ ≤ 1. For 0 < ∆ < 1/2 the blue curve corresponding to the fully-deconfined phase takes over. The take-over of the orange curve signifies the partially-deconfined 2-center phase in the corresponding region (1/2 < ∆ .72), and so on.

Asymptotics of the 4d index from 3d field theory
In this section we consider the dimensional reduction of the four-dimensional N = 1 gauge theory on a Hopf surface. This surface is topologically S 3 × S 1 and we reduce along the S 1 fiber. The dimensionally reduced theory describes a three-dimensional dynamical gauge supermultiplet coupled to background three-dimensional supergravity on S 3 . The Wilsonian effective action of the gauge multiplet can be calculated by integrating out the tower of massive Kaluza-Klein modes, and the resulting theory is described by a functional integral over the gauge multiplet fields with this effective action. We find that the functional integral of the three-dimensional theory can be written as a perturbative expansion in τ . The singular terms in the expansion behave as O(1/τ 2 ) and O(1/τ ), and are captured by three-dimensional effective field theory. In particular, these terms are independent of the dynamical fields, and are completely accounted by the (supersymmetrized) Chern-Simons couplings of the background supergravity. The result agrees with the corresponding singular terms in the microscopic expansion (2.26), (2.28).
The all-order asymptotic formula from the microscopic index includes, in addition to these singular terms, constant and linear terms in τ . Using a localization argument we show that the constant term in τ , besides a background part, has a dynamical piece captured by the integral over the fluctuations of the dynamical fields in three-dimensional path integral, which is essentially the partition function of N = 2 supersymmetric CS theory at level ±N . Finally, the linear term in the microscopic formula is precisely the supersymmetric Casimir energy which is needed to translate between the microscopic Hamiltonian index and the macroscopic functional integral. 12 In this manner the full asymptotic formula for the four-dimensional index is explained by three-dimensional physics. The fact that the asymptotic formula does not contain any higher order terms in τ implies a non-renormalization theorem, namely that there are no corrections to the three-dimensional effective action at any polynomial order in τ . We leave the explanation of this interesting point to future work. Finally, we show that corresponding statements also hold near rational points when τ → −n/m. Here we present evidence that the relevant three-dimensional manifold is a Z m orbifold of S 3 and the results agree with the microscopic asymptotic expansion given in (2.58).
We begin by recalling the functional integral definition of the N = 1 superconformal index on S 3 × S 1 . In the Hamiltonian trace definition (1.2) we have two chemical potentials that couple to linear combinations of the two angular momenta J 1 , J 2 on S 3 and the U (1) R-charge Q. This is equal to the supersymmetric functional integral of the theory on S 3 × S 1 with twisted boundary conditions on the fields as we go around the S 1 . Equivalently, one can explicitly introduce a background gauge field (for the R charge) and background off-diagonal terms in the metric (for the angular momenta) in a manner, so as to preserve supersymmetry. As explained in [65], such background configurations can be obtained as solutions to the condition of vanishing gravitino variations of off-shell supergravity (and then taking a rigid limit so as to decouple the fluctuations of gravity).
The relevant background configuration for the calculation of the 4d superconformal index for complex τ and nonzero n 0 was studied in [5] in the context of 4d new minimal supergravity [66,67]. Recall that the bosonic fields of new minimal supergravity are the metric, a gauge field A nm , and another vector field V nm which is covariantly conserved. The background con- 12 The supersymmetric Casimir energy that appears in our asymptotic formulas is the one given in [61]. Note in particular that (unlike in [5]) this is independent of n0. We can understand this in the path-integral picture by appealing to the result in Section 4 of [10] (based on the regularization method of [64]) which demonstrated that the supersymmetric Casimir energy is independent of flavor fugacities when they are on the unit circle, and by noting that e 2πi(−n 0 r I /2) is effectively a flavor fugacity in our problem.
figuration [5] preserving the supercharges (Q, Q) is 13 Here θ ∈ [0, π/2], the angles φ 1 , φ 2 are 2π-periodic, and the Euclidean time coordinate has the independent periodicity condition 14 This configuration admits the following Killing spinor which is identified with Q, The twist parameters Ω i , Φ are related to the chemical potentials σ, τ in the index as follows 15 Ω with In this section for ease of presentation we focus on the case with Ω 1 = Ω 2 = Ω, which implies σ = τ = iγ 2π (1 − Ω). The partition function on the above background is related to the index I(σ − n 0 , τ ), which for σ = τ coincides with the index I(τ ; n 0 ) in (1.3). In Appendix D we comment on the more general case with Ω 1 = Ω 2 and hence σ = τ .
The four-dimensional supersymmetric partition function of the theory corresponding to the Hamiltonian index (1.2) can then be expressed as a functional integral of the gauge theory with 4d N = 1 chiral and vector multiplets on the background (3.1). 16 As discussed in [5], this functional integral localizes to an integral over flat connections of the gauge field on the KK circle, The Wilson loop (3.20) maps to the scalar in the three-dimensional vector multiplet in the KK reduction. We now proceed to derive an expression for the supersymmetric partition function of the three-dimensional gauge theory. 13 A real metric corresponds to pure imaginary Ωi. General complex Ωi correspond to analytic continuation in the background metric. 14 In [5] the parameter γ was called β. 15 Here (Ω * 1 , Ω * 2 , Φ * ) = (1, 1, 3 2 ) are the values of the potentials on the supersymmetric BH solution. 16 More precisely the Hamiltonian index equals the functional integral for the supersymmetric partition function up to the supersymmetric Casimir energy factor [61,64].

Dimensional reduction to three dimensions
We first consider the reduction of the above four-dimensional background as a configuration in three-dimensional supergravity. In three dimensions we use the off-shell supergravity formalism [68][69][70][71], and follow the treatment [34,[72][73][74] for the reduction from four to three dimensions. The bosonic fields in the off-shell three-dimensional supergravity are the metric, the KK gauge field (the graviphoton) written as a one-form c, a two-form B, and the R-symmetry gauge field one-form A R . The equations are often presented in terms of the dual one-form v = −i * dc and the dual scalar H = i * dB.
We begin by writing the background in (3.1) as a Kaluza-Klein (KK) compactification to three dimensions, i.e. a circle fibration on a 3-manifold M 3 . We define the rescaled S 1 coordinate which obeys the periodicity condition Writing the metric (3.1) in the KK form, we find that the graviphoton field is 10) and the metric on the 3-manifold M 3 is The three-dimensional metric obeys We see that we effectively have a KK reduction on a circle of radius R.
In order to study the effective theory in three dimensions, we consider the limit R → 0. From the relation (3.8) we see that this is implemented by taking the original circle size γ → 0. Our eventual interest is in the limit τ → 0. The question is how to correlate these two limits of γ and τ . If we take γ → 0 first, then we see from the relation (3.4) that Ω → ∞ and from (3.12) that M 3 shrinks to zero size. Although the local Lagrangian involves background fields and terms such as the Ricci scalar which diverge in this limit, the three-dimensional effective action turns out to be finite. We can understand this in a cleaner manner as follows. We first scale τ and γ to zero at the same rate keeping Ω finite and fixed, i.e. take γ = ετ with fixed ε = 2πi/(Ω − 1), and only take ε → 0 at the end of all calculations. In particular, the three-dimensional calculations are all performed at finite ε, i.e. on smooth backgrounds. The action turns out to have two pieces, one of which stays finite and the other vanishing in the limit ε → 0, and, in particular, there are no diverging terms in this limit. Thus we can safely take the limit Ω → ∞ at the end of calculations. In this limit we have that R → τ , so that the effective field theory answers are effectively written as a perturbative series in τ .
In the treatment of three-dimensional background supergravity we need the Hodge dual of the graviphoton, v = v µ dx µ = −i * dc , (3.13) whose value in the above background is v = 2 Ω 1 − Ω 2 sin 2 θ dφ 1 + cos 2 θ dφ 2 , (3.14) so that v µ = 2 Ω(1, 1, 0). The associated Chern-Simons action is The identification between the four-dimensional and the three-dimensional gauge fields is made by comparing the respective Killing spinor equations. As shown in [34,74], one has 17 The background gauge fields in (3.1) are given by so that the auxiliary fields in the background supergravity multiplet are We now discuss the Kaluza-Klein reduction of the dynamical gauge multiplet. The N = 1 gauge multiplet in four dimensions reduces to an N = 2 gauge multiplet in three dimensions, whose bosonic field content is a vector A µ , a scalar σ, and the auxiliary D field. These are related to the four-dimensional fields as follows,

19)
17 In [74] it is assumed that V nm Y = A nm Y , which is not satisfied in our background. Therefore we follow more closely the treatment of [34]. and the three-dimensional fermions are the reduction of the corresponding four-dimensional fermions. As discussed above, the theory localizes on the BPS configurations given by with vanishing values of all other fields in the off-shell gauge and chiral multiplets. In the three-dimensional theory the non-zero fields on the BPS locus are

Effective action and functional integral of the three-dimensional theory
We now turn to the calculation of the partition function of the three-dimensional supersymmetric theory that we just discussed. Our strategy is to first calculate the three-dimensional Wilsonian effective action of u i , and then use this to calculate the three-dimensional partition function. The tree-level action (coming from a mode expansion of the four-dimensional theory) consists of matter-coupled super Yang-Mills theory. The full quantum effective action of the three-dimensional theory is obtained by integrating out the tower of massive KK modes on the circle. In order to calculate this action, we draw from known results in the effective field theory in three dimensions. The effective field theory on backgrounds of the type M 3 × S 1 R was studied in a general context in [51,52], and in the special context of supersymmetry in [25,26]. The resulting three-dimensional action begins with a term proportional to 1/R 2 , and continues as a perturbation expansion as the radius R → 0. At each order in R one has a combination of three-dimensional actions of the background and the dynamical fields, which are all related by supersymmetry to a certain Chern-Simons term. The Chern-Simons terms are of the form M 3 A x ∧ dA y , where A x and A y represent the various gauge fields. As discussed in the previous subsection, these are the dynamical gauge field, the background graviphoton, the background R-gauge field, and the spin connection. We follow, and review in Appendix B, the treatment of [26] for the supersymmetrized Chern-Simons action of all the background and the dynamical gauge fields up to O(R 0 ). The full effective action also includes RR and gravitational supersymmetrized CS terms discussed in [55], which turn out to be crucial for our purposes.
It follows from the above discussion that the overall coefficient at each order in R can be fixed by calculating the coefficient of the Chern-Simons terms themselves. These coefficients, in turn, can be obtained by integrating out all the fermions coupling to the corresponding gauge fields. The resulting induced Chern-Simons coefficient is one-loop exact. Thus the strategy is to integrate out the fermions in each KK mode, write the resulting Chern-Simons action, and sum over all the fermions in the theory. The KK momenta of the fermions take the values p Y = k Y /R, with k Y = n + n 0 2 , n ∈ Z. The shift n 0 /2 appears because of the gauge fields in the background (3.1). (Recall, for example, that the four-dimensional Killing spinor (3.3) has momentum n 0 /2.) The result for the complete action obtained by integrating out a fermion f of R-charge r f and transforming in a representation of weight ρ f under the gauge group is given in Appendix B and take the following form, The terms in (3.22) depend on the real mass m f (related to the central charge appearing in the three-dimensional algebra). The first two terms depend on the dynamical gauge field. On the configuration (3.21) they take the following values, where A M 3 and L M 3 are functions of the three-dimensional background given in (B.11). The last two terms in (3.22) do not depend on the dynamical gauge field, and given by where R M 3 and G M 3 are functions of the three-dimensional background given in (B.3).
In Appendix C we calculate the values of these background actions. 18 As explained above, we perform the calculations keeping R, Ω finite so that the three-dimensional physics is manifestly smooth. The result is that there is a smooth limit as γ → 0 keeping fixed τ . The limiting values of the actions are as follows, (3.25) Using these values, we obtain the total effective action of the fermion f to be (3.26) 18 We note that there is a subtlety with the gravitational CS term in (B.3), concerning the dependence of the term on the frame [63]. There should be a choice of frame which is consistent with the supersymmetry and the 4d to 3d reduction. We do not work out the details of this issue in this paper, and instead rely on consistency with [55] where this term is obtained indirectly by considering integrating out chiral multiplets. We thank Cyril Closset for a discussion on this point. Now we turn to the sum over all the fermions in the theory. The value of the real mass is given in (B.7) to be, as R → 0, 19

28)
In order to obtain the full effective action we now have to sum over all the fermions. For the chiral multiplets, this implies summing over all the weights in representations ρ f ∈ R f , as well as over all momenta labelled by n ∈ Z. The summation over KK modes can be evaluated using with x = ρ f · u − 1 2 n 0 r I , for j = 1, 2, 3 (cf. Section 4 of [10]). 20 Here we have used the relation r f = r I − 1 between the R-charge of the fermion and that of the bottom component of the multiplet I to which the fermion belongs.
For the vector multiplet contribution the analysis is quite similar: there is a tower of massive KK gaugino modes that are integrated out. These generate CS actions whose supersymmetrization yields the vector multiplet contribution to δS 1-loop . In the present context there is an important difference with the chiral multiplet analysis however. Near u = 0 there is a single gaugino mode in the tower that has real mass of order α · u/R, and is therefore considered a "light" mode for small enough |α · u|. Therefore we do not integrate out this mode and, instead, keep it as a dynamical mode in the path integral of the three-dimensional theory.
More precisely, recall that the nth KK gaugino mode associated to a root α of the gauge group has p Y = (n + n 0 )/R and hence a real mass (α · u − n − n 0 )/R. Therefore the mode corresponding to n = −n 0 is light near α · u = 0. We now describe how removing this term from the sum over the KK tower modifies the result compared to the chiral multiplet computation. The vector multiplet contributions is a sum over roots α that come in pairs ±α + , as a result of which they give vanishing contributions to the quadratic and constant terms in u in the action of a single KK mode. We therefore focus on the contribution to the linear term in u , which is proportional to 1/R. The calculation is similar to the corresponding chiral multiplet calculation. Upon summing over all the KK modes, we obtain the vector multiplet 19 In fact the first three terms in (3.26) sum up to and, using (3.29) with x = ρ f · u − n − 1 2 n0 rI + (rI − 1)R to perform the sum over the KK modes, we obtain an effective potential which reproduces the chiral multiplet contributions in (3.33). Essentially the same comment can be made in the microscopic analysis of Section 2. 20 Here, the sgn function is interpreted as applying to m f with R a real positive number (which therefore scales out of the formula so as to give sgn(ρ f · u − 1 2 n0 rI )). Note that in the subsequent formulas R is taken to be complex.
contribution from a root α to be where the prime indicates that we are not including the light mode corresponding to n = −n 0 . Upon adding and subtracting the n = −n 0 contribution, we obtain, using (3.29), Now, since we are interested in the proximity of u = 0, we use the fact that for |x| < 1 we have B 2 (x) = x 2 − |x| + 1 6 , to simplify the result to Upon putting all the pieces together, we obtain the total one-loop correction to the Wilsonian action of the three-dimensional theory, which we call V eff (u) (we justify this name below). We have We now localize the path integral of the light gauge multiplet mode that was excluded from the sum (3.30), using its Wilsonian effective action, which consists of the tree-level action coming from the light n = −n 0 mode in 4d, as well as the one-loop action δS 1-loop derived above (in the bosonic sector, which is relevant for the localization calculation) from integrating out the heavy modes. It is useful to keep in mind the different but related problem of calculating the partition function of superconformal CS theory coupled to matter on M 3 [56], [75]. In that case the theory localizes onto arbitrary constant values of the scalar σ and is supported by the auxiliary scalar H. The measure including the one-loop determinant of the localizing action in the non-BPS directions is 21 (3.34) 21 Compare with Section 5 of [76], noting that for squashed S 3 with squashing parameter b one has ω thf 1 = ib, ω thf 2 = ib −1 . We leave the derivation of (3.35) from the metric (3.11) to future work.
with ω thf 1,2 the moduli of the transversely holomorphic foliation (THF) [77] of M 3 , which we expect to be Recalling from (3.21) that σ i = u i /R, and adding the contribution from δS 1-loop in (3.33) (which although arises at one-loop in high-temperature EFT, contributes as a "classical" piece in the localization computation), we obtain the final result for the three-dimensional partition function Noting that the supersymmetric partition function and the Hamiltonian index are related as [61,64] Z(τ ) = e 2πiτ Esusy I(τ ) , (3.37) we see that the result (3.36) agrees precisely with the microscopic result (2.9)-(2.12). We emphasize that while the above derivation of V eff in (3.33) applies to u near 0, it can be easily extended to generic finite u by modifying the vector multiplet discussion. For generic u, the non-Cartan components of the n = −n 0 mode of the vector multiplet are also heavy, and ought to be integrated out. Consequently the sum in (3.30) would no longer have a prime, and we end up with V r 1 as in (2.13) rather than V 1 in (3.33). This is the EFT derivation of the finite-u potentials V 1,2 found microscopically in [12].
On the other hand, when n 0 = 0, the small-u discussion leading up to (3.33) needs to be modified because now the chiral multiplets have light modes (corresponding to n = 0). As in the discussion around (3.30) the light mode should be removed from the KK sum and instead be included in the dynamical part (to be localized). Indeed, it is well-known that for n 0 = 0 the constant piece of the small-τ expansion coming from the u = 0 saddle contains the (localized) S 3 partition function of the dimensionally reduced chiral as well as vector multiplets [48] (see [78][79][80][81][82] for earlier work on the connection between 4d indices and S 3 partition functions).
A technical remark is in order regarding our EFT derivation of (3.36). To reproduce the desired asymptotics, we have sent ε(= γ τ = 2πi Ω−1 ) → 0, and hence Ω → ∞, when evaluating the CS actions in Appendix C. It would be interesting to have a formula of the type (3.36) for γ → 0 at finite ε, which would imply that Ω and the resulting 3d geometry would be finite-sized. 22 Finally, as discussed in Appendix D, we find that the effective potential for τ and σ not necessarily equal is given by making the replacement The recent work [83] presents such a derivation, although using a background different from ours. The precise relation between the two backgrounds is not clear to us at the moment.
in the effective potential (3.33). The singular pieces are indeed in agreement with the microscopic calculations reported in [11,12].

Rational points
We now turn our attention to the limit of τ approaching a rational point. In the discussion of the previous subsection we used the fact that the radius of the circle R equals τ which becomes small in the limit, so that we could use an effective three-dimensional description. Now we are interested in τ = mτ + n → 0, with n, m ∈ Z (with no common factor) as in [13].
In terms of the variable τ we have that ω = 2πiτ = 2πi( τ − n)/m so that (3.39) and the four-dimensional metric background (3.1) is now (3.40) In terms of the following new coordinates and new parameters, the above metric is with φ 1 , φ 2 being 2π-periodic as before, and the periodic identification going around the time circle is The metric configuration (3.42) with the identifications (3.43) is simply a global identification, or orbifold, of the configuration considered in the previous subsection with the new parameters ( γ, τ , Ω) replacing (γ, τ, Ω). 23 On the covering space, going around the time circle shifts t E → t E + γ and φ i → φ i +2πn. The latter identification can be trivialized by using the independent 2π-periodicity of φ i , so that we have the identification t E , φ 1 , φ 2 ∼ t E + γ , φ 1 , φ 2 . On this configuration we can perform the dimensional reduction to three dimensions. The relevant considerations of the previous subsection go through exactly as before with the replacement (γ, τ, Ω) → ( γ, τ , Ω). Actually, because the gauge holonomies on the cover wrap a circle m times larger than the original S 1 , we also get a replacement u j → mu j . Moreover, since ξ I (which equals −n 0 r I /2 23 We learned about these orbifolds from a talk by O. Aharony at the Stony Brook seminar series in November 2020 [40]. for (m, n) = (1, 0)) effectively plays the role of a flavor chemical potential in our problem as mentioned around (2.38), we expect a similar replacement −n 0 r I /2 → mξ I . We can see this replacement arise more directly as follows.
We multiply the first term in (3.26) by m 2 m 2 , and the second term by m m . This amounts to γ → mγ and u j → mu j as mentioned above, but also k Y → mk Y (which corresponds to keeping only the singlet modes under the Z m quotient) as well as n 0 → mn 0 . On the other hand, writing A nm Y in (3.17) in terms of τ instead of τ amounts to yet another replacement n 0 → n 0 + 2n m . Combining these two effects yields the desired −n 0 r I /2 → mξ I replacement.
With the preceding substitutions in the results of the previous subsection, we thus arrive at the potentials V 2,1 in (2.37). We then take the Z m quotient which has two effects as usual. Firstly it reduces the volume of the three-dimensional space, and secondly it introduces new topologically non-trivial sectors in the path integral over the gauge-field configurations. The change in calculations involving local gauge-invariant Lagrangians will therefore be only a reduction in the action by a factor of m. This explains the reduction of the effective potential by a factor of m as in (2.35).
Finally we discuss the constant terms (in τ ) arising from the functional integral over the dynamical gauge multiplet. There are a few subtleties. Firstly the actions like the gravitational CS action will depend on the global properties of the orbifold. Then we need to calculate the partition function of the orbifold space with a background graviphoton. Assuming as in the previous subsection that the expected THF moduli arise, and that by rescaling and contour deformation (as discussed around (2.44)) the THF moduli can be replaced with those of round S 3 , the calculation presumably reduces to an S 3 /Z m partition function as in [50,75,84,85], with the Z m action following from (3.43) to be which for n = 1 coincides with that of the lens space L(m, −1). Here one has to be careful about how the measure on the space of constant scalars σ i is affected by the four-dimensional orbifold (3.43). We leave these interesting questions to future work, noting that the result of these considerations indeed agrees with the microscopic answer (2.58), with the O( τ ) piece explained by the supersymmetric Casimir energy factor as before.
Note Added. The paper [83], which appeared on the arXiv the same day as the first version of this paper, has some overlap with our section 3. The paper [86], which appeared on the arXiv soon after, has some overlap with our section 2.
N. 681908, "Quantum black holes: A macroscopic window into the microstructure of gravity", and by the STFC grant ST/P000258/1. AAA would like to especially thank Junho Hong for several helpful discussions and collaboration on a related project.
A Asymptotic estimates of the special functions A.1 τ → 0 We first consider the limit τ → 0. More precisely, in the rest of this subsection we assume arg(τ ) is in compact domains avoiding integer multiples of π 2 as |τ | → 0. For the Pochhammer symbol (q; q) the small-τ asymptotics is standard: Recall that the symbol means that logarithms (on appropriate branches) of the two sides (assumed to be non-zero) are equal to all orders in the small parameter (here in |τ |).
For the chiral multiplet elliptic gamma functions we have the following estimate, valid for any r ∈ R, uniformly in z over compact subsets of R \ Z (see Proposition 2.11 of [46] or Equation (3.53) of [48]): as |τ | → 0. Here B j (z) are the periodic Bernoulli polynomials defined, for z ∈ R through their Fourier series expansion, The prime in the above formula means that k = 0 has to be omitted, and that in the j = 1 case-where the series is not absolutely convergent-the sum is in the sense of Cauchy principal value. For x ∈ R \ Z, we have B j (x) = B j ({x}) with {·} := · − · the fractional-part function. When j > 1 this also holds for x ∈ Z (and so B j (Z) = B j (0)). When j = 1 on the other hand The Bernoulli polynomials are uniquely characterized by and the first three non-trivial ones are explicitly The connection between B j and the Bernoulli polynomials can be verified by first noting that for j = 1 the left-hand side of (A.3) is essentially the Taylor expansion of the logarithm function, and then observing that B j are uniquely characterized by With the aid of (A.3) one can easily prove relations such as and n−1 valid for m, n ∈ Z >0 relatively prime and x ∈ R, by using the Fourier expansion of the Bernoulli functions, and swapping the sum over Fourier modes with the sum over . 24 The estimate (A.2) is particularly useful for the chiral multiplet gamma functions in (2.6) when the integral is dominated by the 1-center holonomy configurations with z i − z j = 0. This is because the complex phase 2πn 0 /3 shifts the argument of the chiral multiplet gamma functions safely into the interior of the domain z ∈ R \ Z where the estimate is uniformly valid. On the other hand, since the vector multiplet gamma functions in (2.6) lack such phase shifts in their arguments, the estimate (A.2) is not appropriate for them near the 1-center holonomy configurations when τ → 0.
The estimate (A.2) is not uniformly valid, with respect to z, over intervals containing Z. There is a well-known improvement of it around z = 0 however, which is valid uniformly over compact subsets of (−1, 1), and we will use for vector multiplet elliptic gamma functions in the index. It reads (see Proposition 2.10 of [46] or Equation (2.16) of [48]) Γ e (rτ + z) e 2πiR 0 (rτ +z;τ ) Γ h (rτ + z; τ, τ ), and Γ h (x; ω 1 , ω 2 ) is the hyperbolic gamma function. 24 Note that similar operations with B1 are not allowed, because its Fourier expansion is not absolutely convergent. This is the source of sophistication of the Dedekind sum defined below in terms of B1-or more specifically the source of the nontrivial dependence of (A.18) on n. (Readers familiar with Eisenstein series might recall similar "anomalous" behavior from E2 and its associated elliptic functions.) A closely related fact is that Bj>1 are continuous, but B1 has discontinuities on Z.
To obtain the asymptotics of the Pochhammer symbol we note that for integer a, b, c, d satisfying ad − bc = 1 with c > 0, we have for r ∈ R. This is the analog of (A.2) for τ → 0. The explicit expression for C(m, n, z, r) is (A.24) The estimate (A.23) is important to derive our results for the asymptotic expansion of the index near the roots of unity. It is valid uniformly over compact subsets of z ∈ R \ Z m , because using (A.2) on the right-hand side of (A. 19) is allowed only if z − n l m / ∈ Z for = 0, . . . , m − 1. We can also use (A.19) for z near 0. More precisely, on the RHS of (A.19), for fixed z ∈ (− 1 m , 1 m ), we can use (A.13) for the = 0, m terms, and use (A.2) for all other . With the aid of the "reflection formula" Γ h (x + ω 1 + ω 2 2 ; ω 1 , ω 2 )Γ h (−x + ω 1 + ω 2 2 ; ω 1 , ω 2 ) = 1. (A. 25) which gets rid of the hyperbolic gammas arising from = m, and using the "product formula" (A.10) to trade the hyperbolic gammas arising from = 0 for hyperbolic sines, we obtain This is the analog of (A.13)-(A.12) for τ → 0, and is similarly useful (i.e. uniformly valid) in a neighborhood of z = 0. An estimate similar to (A.26) for z near general nonzero Z m can be obtained as well. We focus for simplicity on the n = 1 case (i.e. τ → − 1 m ). We write z = 0 /m + z and appeal to (A.19). We have to use the estimate (A.8) for = 0 , 0 + m, and the estimate (A.2) for all other in the product (A. 19) for Γ e (z). Similarly we have to use the estimate (A.8) for = − 0 + m, − 0 + 2m, and the estimate (A.2) for all other in the product for Γ e (−z). The result is (up to a constant phase that we suppress)

B Supersymmetric three-dimensional Chern-Simons actions
In this appendix we present the bosonic part of supersymmetrized three-dimensional Chern-Simons actions. We work in the context of three-dimensional N = 2 supersymmetric gauge theory coupled to off-shell three-dimensional supergravity. We first collect all allowed Chern-Simons terms including background and dynamical gauge fields, and then write the corresponding supersymmetrizations, following the presentation of Appendix A of [26]. We then evaluate the actions for the field configurations that we consider in Section 3. The CS terms have the form where x and y run over all possible gauge fields and with a coupling that we discuss below. Below we present the bosonic parts of the supersymmetric completions of the various cases 26 This relation can be proven using the reflection formula (A.25) together with Γ h (x + τ, τ, τ ) = 2 sin πx τ Γ h (x, τ, τ ). and the graviphoton and KK scalar are e 2φ = 1 − Ω 2 1 sin 2 θ − Ω 2 2 cos 2 θ , c = c µ dx µ = −i e −2φ Ω 1 sin 2 θ dφ 1 + Ω 2 cos 2 θ dφ 2 . (D.3) For the case Ω 1 = Ω 2 = Ω, we have that e 2φ = 1 − Ω 2 , so that the graviphoton c defined in (3.9) is related to c as c = e φ c. The magnitude of the volume form in three dimensions is . (D.6) For the identification between the four-dimensional and the three-dimensional fields, we follow the treatment of [34] applied to the metric (D.1). The result is The values of these fields are We can now calculate the various actions as in Appendix C, and we find that, in the γ → 0, Ω → ∞ limit we have the effective replacement 10) in the effective potential (3.33).