Shlomo S. Razamat

We elaborate on the relation between the generalized Schur index of {\cal N}=2 SCFTs in four dimensions and the non-relativistic limit of the elliptic Ruijsenaars-Schneider model. In particular we discuss explicitly how to express generalized Schur indices of theories of class {\cal S} in terms of elliptic Jack functions. For example, in the A_1 A 1 case the indices are given naturally in terms of eigenfunctions of the Lamé equation. We use the expression in terms of eigenfunctions to further check the recent observation that the generalized Schur indices of different theories in the Deligne-Cvitanović series can be mapped onto each other. This mapping implies non trivial identities on unrefined sums of eigenfunctions of non-relativistic elliptic Calogero-Moser models associated to different root systems. We claim then that the non-relativistic limits of various integrable models give rise naturally to generalized Schur-like limits of classes of {\cal N}=1 SCFTs. As an example we discuss the relation of the Inozemtsev model, the non relativistic limit of the van Diejen model, and compactifications of the rank Q Q E-string theory. We argue that in general the “Schur index” of {\cal N}=1 4d 4 d SCFTs can be understood as being related to the free fermionic limit of a non-relativistic integrable model.

We revisit a double-scaled limit of the superconformal index of N = 2 superconformal field theories (SCFTs) which generalizes the Schur index. The resulting partition function, Z ^ ( q , α ) , has a standard q -expansion with coefficients depending on a continuous parameter α . The Schur index is a special case with α = 1 . Through explicit computations we argue that this partition function is an invariant of certain mass deformations and vacuum expectation value (vev) deformations of the SCFT. In particular, two SCFTs residing in different corners of the same Coulomb branch, satisfying certain restrictive conditions, have the same partition function with a nontrivial map of the parameters, Z ^ 1 ( q , α 1 ) = Z ^ 2 ( q , α 2 ( α 1 ) ) . For example, we show that the Schur index of the SCFTs in the Deligne-Cvitanović series is given by special values of α of the partition function of the SU ( 2 ) N f = 4 N = 2 SQCD.

We consider supersymmetric conformal quantum field theories (SCFTs) with degrees of freedom labeled by lattice data. We will assume that in terms of the corresponding lattice the interactions are nearest neighbor and exactly marginal. For example, one can construct such theories by coupling many copies of a single SCFT with exactly marginal deformations. In particular, we discuss the interplay between conformal manifolds of such theories and their global, on-site and lattice, symmetries. We show that one can interpret certain current non-conservation equations for symmetries broken by the interactions as conservation equations including the lattice directions. Moreover, we discuss a class of exactly marginal deformations which are labeled by lattice holonomies that are topological on the lattice. We discuss concrete examples of such constructions and comment on their relevance to compactifications of SCFTs.

We consider supersymmetric conformal quantum field theories (SCFTs) with degrees of freedom labeled by lattice data. We will assume that in terms of the corresponding lattice the interactions are nearest neighbor and exactly marginal. For example, one can construct such theories by coupling many copies of a single SCFT with exactly marginal deformations. In particular, we discuss the interplay between conformal manifolds of such theories and their global, on-site and lattice, symmetries. We show that one can interpret certain current non-conservation equations for symmetries broken by the interactions as conservation equations including the lattice directions. Moreover, we discuss a class of exactly marginal deformations which are labeled by lattice holonomies that are topological on the lattice. We discuss concrete examples of such constructions and comment on their relevance to compactifications of SCFTs.

We identify a categorical structure of the set of all conformal field theories (CFTs). In particular, we show that the set of all CFTs has a natural monoidal strict 2-category structure with the 1-morphisms being sequences of deformations and 2-morphisms determined by 0-form symmetries of the CFTs. Published by the American Physical Society 2024

In this set of five Lectures we present a basic toolbox to discuss the dynamics of four dimensional supersymmetric quantum field theories. In particular we overview the program of geometrically engineering the four dimensional supersymmetric models as compactifications of six dimensional SCFTs. We discuss how strong coupling phenomena in four dimensions, such as duality and emergence of symmetry, can be naturally imbedded in the geometric constructions. The Lectures mostly review results which previously appeared in the literature but also contain some unpublished derivations.

We derive ground state eigenfunctions and eigenvalues of various relativistic elliptic integrable models. The models we discuss appear in computations of superconformal indices of four-dimensional theories obtained by compactifying six-dimensional models on Riemann surfaces. These include, among others, the Ruijsenaars-Schneider model and the van Diejen model. The derivation of the eigenfunctions builds on physical inputs, such as conjectured Lagrangian across dimensions IR dualities and assumptions about the behavior of the indices in the limit of compactifications on surfaces with large genus/number of punctures/flux.

We discuss 4D Lagrangian descriptions, across dimensions IR duals, of compactifications of the 6D (D, D) minimal conformal matter theory on a sphere with arbitrary number of punctures and a particular value of flux as a gauge theory with a simple gauge group. The Lagrangian has the form of a “star shaped quiver” with the rank of the central node depending on the 6D theory and the number and type of punctures. Using this Lagrangian one can construct across dimensions duals for arbitrary compactifications (any, genus, any number and type of USp punctures, and any flux) of the (D, D) minimal conformal matter gauging only symmetries which are manifest in the ultraviolet.