The topologically twisted index of N=4\mathcal N=4 super-Yang-Mills on T2×S2T^2\times S^2 and the elliptic genus

We examine the topologically twisted index of $\mathcal N=4$ super-Yang-Mills with gauge group $SU(N)$ on $T^2\times S^2$, and demonstrate that it receives contributions from multiple sectors corresponding to the freely acting orbifolds $T^2/\mathbb Z_m\times\mathbb Z_n$ where $N=mn$. After summing over these sectors, the index can be expressed as the elliptic genus of a two-dimensional $\mathcal N=(0,2)$ theory resulting from Kaluza-Klein reduction on $S^2$. This provides an alternate path to the 'high-temperature' limit of the index, and confirms the connection to the right-moving central charge of the $\mathcal N=(0,2)$ theory.Comment: 29 pages, 1 figure; v2: restricted to real chemical potentials in section 4 and added a comment on the index where $c_r(\mathfrak n_a)<0$ in section

Subleading corrections to the S3S^3 free energy of necklace quiver theories dual to massive IIA

We investigate the $S^3$ free energy of $\mathcal N=3$ Chern-Simons-matter quiver gauge theories with gauge group $U(N)^r~(r\geq2)$ where the sum of Chern-Simons levels does not vanish, beyond the leading order in the large-$N$ expansion. We take two different approaches to explore the sub-leading structures of the free energy. First we evaluate the matrix integral for the partition function in the 't~Hooft limit using a saddle point approximation. Second we use an ideal Fermi-gas model to compute the same partition function, but in the limit of fixed Chern-Simons levels. The resulting expressions for the free energy $F=-\log Z$ are consistent with each other at the leading and first sub-leading order. The Fermi-gas approach also hints at a universal $\frac{1}{6}\log N$ correction to the free energy. Since the quiver gauge theories we consider are dual to massive Type IIA theory, we expect our results to match sub-leading corrections to the holographic dual free energy, which have not yet been fully investigated.Comment: v1: 50 pages, 1figure; v2: minor revision

Unsupervised Legendre-Galerkin Neural Network for Stiff Partial Differential Equations

Machine learning methods have been lately used to solve differential equations and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques such as deep neural networks and statistical learning are applied to classical problems of applied mathematics. Because neural networks provide an approximation capability, computational parameterization through machine learning and optimization methods achieve noticeable performance when solving various partial differential equations (PDEs). In this paper, we develop a novel numerical algorithm that incorporates machine learning and artificial intelligence to solve PDEs. In particular, we propose an unsupervised machine learning algorithm based on the Legendre-Galerkin neural network to find an accurate approximation to the solution of different types of PDEs. The proposed neural network is applied to the general 1D and 2D PDEs as well as singularly perturbed PDEs that possess boundary layer behavior.Comment: 29 pages, 8 figure

Holographic Thermal Observables and M2-branes

We use holography in conjunction with recent results from supersymmetric localization to compute certain thermal observables for 3d $\mathcal{N}=2$ holographic SCFTs arising on the worldvolume of $N$ M2-branes. We obtain results for the thermal free energy density on $S^1 \times \mathbb{R}^2$, the Casimir energy on $T^{2} \times \mathbb{R}$, and the three leading coefficients in the large temperature limit of the free energy on $S^1\times S^2$ valid to subleading order in the large $N$ limit. As a byproduct of our holographic analysis we also present a conjecture for the structure of the large temperature expansion of the thermal free energy of general 3d CFTs on $S^1\times S^2$.Comment: 22 page

Large NN Partition Functions of the ABJM Theory

We study the large $N$ limit of some supersymmetric partition functions of the $\mathrm{U}(N)_{k}\times \mathrm{U}(N)_{-k}$ ABJM theory computed by supersymmetric localization. We conjecture an explicit expression, valid to all orders in the large $N$ limit, for the partition function on the $\mathrm{U}(1)\times \mathrm{U}(1)$ invariant squashed sphere in the presence of real masses in terms of an Airy function. Several non-trivial tests of this conjecture are presented. In addition, we derive an explicit compact expression for the topologically twisted index of the ABJM theory valid at fixed $k$ to all orders in the $1/N$ expansion. We use these results to derive the topologically twisted index and the sphere partition function in the 't Hooft limit which correspond to genus $\tt g$ type IIA string theory free energies to all orders in the $\alpha'$ expansion. We discuss the implications of our results for holography and the physics of AdS$_4$ black holes.Comment: 56 pages, 4 figure

Large NN Partition Functions of 3d Holographic SCFTs

We study the $S^1\times\Sigma_{\mathfrak g}$ topologically twisted index and the squashed sphere partition function of various 3d $\mathcal N\geq2$ holographic superconformal field theories arising from M2-branes. Employing numerical techniques in combination with well-motivated conjectures we provide compact closed-form expressions valid to all orders in the perturbative $1/N$ expansion for these observables. We also discuss the holographic implications of our results for the topologically twisted index for the dual M-theory Euclidean path integral around asymptotically AdS$_4$ solutions of 11d supergravity. In Lorentzian signature this leads to a prediction for the corrections to the Bekenstein-Hawking entropy of a class of static asymptotically AdS$_4$ BPS black holes.Comment: v1: p.57; v2: minor revision

The N=4 SU(N) Super-Yang-Mills Index and Dual AdS Black Holes

The main subjects of this dissertation are indices of the N=4 SU(N) Super-Yang-Mills (SYM) theory, namely the topologically twisted index and the superconformal index. These indices have received a lot of attention since they provide microscopic understanding of AdS_5 black strings and black holes respectively through the AdS/CFT correspondence. In this dissertation, we focus on the field theory side and investigate these indices with a goal of improving the current microscopic understanding of AdS_5 black strings and black holes. As a result, we unveil interesting physics of the 4d indices such as modular properties and a relation to the S^3 partition function of effective Chern-Simons theory, and also make suggestions in the gravity side based on the structure of indices through the AdS/CFT correspondence. First, we study the topologically twisted index of the N=4 SU(N) SYM theory on T^2xS^2. We introduce the Bethe Ansatz (BA) formula that gives the twisted index as a sum over solutions to the Bethe Ansatz Equations (BAE) and categorize various solutions into two groups: standard ones that compose the SL(2,Z) family and non-standard ones that denote all the other BAE solutions including continuous families. Focusing on the contribution from standard BAE solutions, we confirm that it behaves as an elliptic genus with certain modular properties and further investigate its asymptotic behaviors in the Cardy-like limit. Lastly, we review how the twisted index counts the microstates associated with the dual AdS_5 black string entropy in the Cardy-like limit, and discuss missing steps that should be taken care of to validate the microstate counting. Next, we study the superconformal index of the N=4 SU(N) SYM theory. We compute the superconformal index first by saddle point evaluation and then by the BA formula. In due process, we establish a direct relation between the 4d superconformal index and the S^3 partition function of Chern-Simons theory. Then we investigate the phase structure of the superconformal index in the large-N after the Cardy-like limit, which contains partially-deconfined phases distinguished from the previously well-known fully-deconfined/confined phases. Finally, we discuss implications of a partially-deconfined phase, based on the counting of microstates associated with the dual AdS_5 black hole entropy by the superconformal index.PHDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/168069/1/junhoh_1.pd