A journey to 3d: exact relations for adjoint SQCD from dimensional reduction

In this note we elaborate on the reduction of four dimensional Seiberg duality with adjoint matter to three dimensions. We use the exact formulation of the superconformal index and of the partition function as instruments to test this reduction. We translate the identity between indices of the dual 4d theories to 3d. This produces various new identities between partition functions of 3d dual phases.Comment: 18 pages, appendix added on the large mass scaling of the partition function, version published in JHE

ABCD of Beta Ensembles and Topological Strings

We study beta-ensembles with Bn, Cn, and Dn eigenvalue measure and their relation with refined topological strings. Our results generalize the familiar connections between local topological strings and matrix models leading to An measure, and illustrate that all those classical eigenvalue ensembles, and their topological string counterparts, are related one to another via various deformations and specializations, quantum shifts and discrete quotients. We review the solution of the Gaussian models via Macdonald identities, and interpret them as conifold theories. The interpolation between the various models is plainly apparent in this case. For general polynomial potential, we calculate the partition function in the multi-cut phase in a perturbative fashion, beyond tree-level in the large-N limit. The relation to refined topological string orientifolds on the corresponding local geometry is discussed along the way.Comment: 33 pages, 1 figur

Multiply-refined enumeration of alternating sign matrices

Four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices (ASMs). Specifically, these statistics are the positions of the 1's in the first and last rows and columns of an ASM, and the numbers of generalized inversions and -1's in an ASM. Previously-known and related results for the exact enumeration of ASMs with prescribed values of some of these statistics are discussed in detail. A quadratic relation which recursively determines the generating function associated with all six statistics is then obtained. This relation also leads to various new identities satisfied by generating functions associated with fewer than six of the statistics. The derivation of the relation involves combining the Desnanot-Jacobi determinant identity with the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions.Comment: 62 pages; v3 slightly updated relative to published versio

String Field Theory Vertices, Integrability and Boundary States

We study Neumann coefficients of the various vertices in the Witten's open string field theory (SFT). We show that they are not independent, but satisfy an infinite set of algebraic relations. These relations are identified as so-called Hirota identities. Therefore, Neumann coefficients are equal to the second derivatives of tau-function of dispersionless Toda Lattice hierarchy (this tau-function is just the partition sum of normal matrix model). As a result, certain two-vertices of SFT are identified with the boundary states, corresponding to boundary conditions on an arbitrary curve. Such two-vertices can be obtained by the contraction of special surface states with Witten's three vertex. We analyze a class of SFT surface states,which give rise to boundary states under this procedure. We conjecture that these special states can be considered as describing D-branes and other non-perturbative objects as "solitons" in SFT. We consider some explicit examples, one of them is a surface states corresponding to orientifold.Comment: 28pages plus appendices, acknowledgments adde

2D Fermions and Statistical Mechanics: Critical Dimers and Dirac Fermions in a background gauge field

In the limit of the lattice spacing going to zero, we consider the dimer model on isoradial graphs in the presence of singular $SL(N,\mathbb{C})$ gauge fields flat away from a set of punctures. We consider the cluster expansion of this twisted dimer partition function show it matches an analogous cluster expansion of the 2D Dirac partition function in the presence of this gauge field. The latter is often referred to as a tau function. This reproduces and generalizes various computations of Dub\'edat (J. Eur. Math. Soc. 21 (2019), no. 1, pp. 1-54). In particular, both sides' cluster expansion are matched up term-by-term and each term is shown to equal a sum of a particular holomorphic integral and its conjugate. On the dimer side, we evaluate the terms in the expansion using various exact lattice-level identities of discrete exponential functions and the inverse Kasteleyn matrix. On the fermion side, the cluster expansion leads us to two novel series expansions of tau functions, one involving the Fuschian representation and one involving the monodromy representation.Comment: 45+26 pages, 26 figure

Arithmetic properties of overpartition functions with combinatorial explorations of partition inequalities and partition configurations

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2017.In this thesis, various partition functions with respect to `-regular overpartitions, a special partition inequality and partition con gurations are studied. We explore new combinatorial properties of overpartitions which are natural generalizations of integer partitions. Building on recent work, we state general combinatorial identities between standard partition, overpartition and `-regular partition functions. We provide both generating function and bijective proofs. We then establish an in nite set of Ramanujan-type congruences for the `-regular overpartitions. This signi cantly extends the recent work of Shen which focused solely on 3{regular overpartitions and 4{regular overpartitions. We also prove some of the congruences for `-regular overpartition functions combinatorially. We then provide a combinatorial proof of the inequality p(a)p(b) > p(a+b), where p(n) is the partition function and a; b are positive integers satisfying a+b > 9, a > 1 and b > 1. This problem was posed by Bessenrodt and Ono who used the inequality to study a maximal multiplicative property of an extended partition function. Finally, we consider partition con gurations introduced recently by Andrews and Deutsch in connection with the Stanley-Elder theorems. Using a variation of Stanley's original technique, we give a combinatorial proof of the equality of the number of times an integer k appears in all partitions and the number of partition con- gurations of length k. Then we establish new generalizations of the Elder and con guration theorems. We also consider a related result asserting the equality of the number of 2k's in partitions and the number of unrepeated multiples of k, providing a new proof and a generalization.MT201