High-Temperature Asymptotics of the 4d Superconformal Index.

This dissertation contains a study of certain four-dimensional superconformal field theories (4d SCFTs). Any 4d SCFT has a spectrum of local operators. Some of these operators sit in short representations of the 4d N=1 superconformal group SU(2,2|1), and can be quantified using a partition function known as the 4d superconformal index. The superconformal index I(b,beta) is a function of two positive real parameters: the squashing parameter b, and the inverse temperature beta. Our study in the present dissertation is focused on the temperature- (or beta-) dependence of the superconformal index of 4d SCFTs. The superconformal index of a typical Lagrangian 4d SCFT is given by a complicated special function known as an elliptic hypergeometric integral (EHI). The high-temperature limit of the index corresponds to the hyperbolic limit of the EHI. The hyperbolic limit of certain special EHIs has been analyzed by Eric Rains around 2006; extending Rains’s techniques, we discover a surprisingly rich structure in the high-temperature limit of a (rather large) class of EHIs that arise as the superconformal index of unitary Lagrangian 4d SCFTs with non-chiral matter content. Our result has implications for N=1 dualities, the AdS/CFT correspondence, and supersymmetric gauge dynamics on R^3 x S^1. We also investigate the high-temperature asymptotics of the large-N limit of the superconformal index of a class of holographic 4d SCFTs (described by toric quiver gauge theories with SU(N) nodes). We show that from this study a rather general solution to the problem of holographic Weyl anomaly in AdS5/CFT4 at the subleading order (in the 1/N expansion) emerges. Most of this dissertation is based on published works by Jim Liu, Phil Szepietowski, and the author. We include here a few previously unpublished results though, one of which is the high-temperature asymptotics of the superconformal index of puncture-less SU(2) class-S theories.PhDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133352/1/ardehali_1.pd

On Exactly Marginal Deformations Dual to BB-Field Moduli of IIB Theory on SE5_5

The complex dimension of the space of exactly marginal deformations for quiver CFTs dual to IIB theory compactified on $Y^{p,q}$ is known to be generically three. Simple general formulas already exist for two of the exactly marginal directions in the space of couplings, one of which corresponds to the sum of the (inverse squared of) gauge couplings, and the other to the $\beta$-deformation. Here we identify the third exactly marginal direction, which is dual to the modulus $\int B_{2}$ on the gravity side. This identification leads to a relation between the field theory gauge couplings and the vacuum expectation value of the gravity modulus that we further support by a computation related to the chiral anomaly induced by added fractional branes. We also present a simple algorithm for finding similar exactly marginal directions in any CFT described by brane tiling, and demonstrate it for the quiver CFTs dual to IIB theory compactified on $L^{1,5,2}$ and the Suspended Pinch Point.Comment: 28 pages, JHEP style. v2: minor corrections, added references and acknowledgements. v3: a number of speculative comments regarding the application of the Konishi anomaly equation to our problem are removed. v4: the proposal in Eq. (2.4) added back as a conjectur

Central charges from the N=1\mathcal{N} = 1 superconformal index

We present prescriptions for obtaining the central charges, $a$ and $c$, of a four dimensional superconformal quantum field theory from the superconformal index. At infinite $N$, for holographic theories dual to Sasaki-Einstein 5-manifolds the prescriptions give the $\mathcal{O}(1)$ parts of the central charges. This allows us, among other things, to show the exact AdS/CFT matching of $a$ and $c$ for arbitrary toric quiver CFTs without adjoint matter that are dual to smooth Sasaki-Einstein 5-manifolds. In addition, we include evidence from non-holographic theories for the applicability of these results outside of a holographic setting and away from the large-$N$ limit.Comment: 4+1 pages. v2: minor revisions and added acknowledgements. v3: improved presentatio

Modified supersymmetric indices in AdS3_3/CFT2_2

We consider the $\mathcal{N}=(2,2)$ AdS$_3$/CFT$_2$ dualities proposed by Eberhardt, where the bulk geometry is AdS$_3\times(S^3\times T^4)/\mathbb{Z}_k$, and the CFT is a deformation of the symmetric orbifold of the supersymmetric sigma model $T^4/\mathbb{Z}_k$ (with $k=2,\ 3,\ 4,\ 6$). The elliptic genera of the two sides vanish due to fermionic zero modes, so for microstate counting applications one must consider modified supersymmetric indices. In an analysis similar to that of Maldacena, Moore, and Strominger for the standard $\mathcal{N}=(4,4)$ case of $T^4$, we study the appropriate helicity-trace index of the boundary CFTs. We encounter a strange phenomenon where a saddle-point analysis of our indices reproduces only a fraction (respectively $\frac{1}{2},\ \frac{2}{3},\ \frac{3}{4},\ \frac{5}{6}$) of the Bekenstein-Hawking entropy of the associated macroscopic black branes.Comment: v2: an outline added in the introductio

1/N^2 corrections to the holographic Weyl anomaly

We compute the O(1) contribution to holographic c-a for IIB supergravity on AdS_5 x S^5/Z_n and on AdS_5 x T^{1,1}/Z_n. In both cases, we find agreement with the dual field theory results, thus providing 1/N^2 checks of AdS/CFT with reduced supersymmetry. Since the holographic computation involves a sum over shortened multiplets in the KK tower, we provide some details on the S^5 and T^{1,1} spectra in a form that is convenient when considering their Z_n orbifolds. The computation for the even Z_n orbifolds of S^5 includes a sum over the multiplets in the twisted sector that is essential for obtaining agreement with the dual field theory.Comment: 23 pages, 2 figures. v2: minor corrections and references adde

c−ac-a from the N=1N=1 superconformal index

We present a prescription for obtaining the difference of the central charges, c-a, of a four dimensional superconformal quantum field theory from its single-trace index. The formula is derived from a one-loop holographic computation, but is expected to be valid independent of holography. We demonstrate the prescription with several holographic and non-holographic examples. As an application of our formula, we show the AdS/CFT matching of c-a for arbitrary toric quiver CFTs without adjoint matter that are dual to smooth Sasaki-Einstein 5-manifolds.Comment: 19 pages. v2: minor changes, reference added. v3: minor clarifications, published version. v4: a typo in Table 2 fixe

High-Temperature Expansion of Supersymmetric Partition Functions

Di Pietro and Komargodski have recently demonstrated a four-dimensional counterpart of Cardy's formula, which gives the leading high-temperature ($\beta\rightarrow{0}$) behavior of supersymmetric partition functions $Z^{SUSY}(\beta)$. Focusing on superconformal theories, we elaborate on the subleading contributions to their formula when applied to free chiral and U(1) vector multiplets. In particular, we see that the high-temperature expansion of $\ln Z^{SUSY}(\beta)$ terminates at order $\beta^0$. We also demonstrate how their formula must be modified when applied to SU($N$) toric quiver gauge theories in the planar ($N\rightarrow\infty$) limit. Our method for regularizing the one-loop determinants of chiral and vector multiplets helps to clarify the relation between the 4d $\mathcal{N} = 1$ superconformal index and its corresponding supersymmetric partition function obtained by path-integration.Comment: 15 pages plus appendices; v2: minor modifications and a "Note added"; v3: presentation improved and minor errors in app B correcte