On Razamat's $A_2$ and $A_3$ kernel identities

Abstract

In recent work on superconformal quantum field theories, Razamat arrived at elliptic kernel identities of a new and striking character: They relate solely to the root systems A 2 and A 3 and have no coupling type parameters [Ra18]. The pertinent 2and 3-variable Hamiltonians are analytic difference operators and the kernel functions are built from the elliptic gamma function. Razamat presented compelling evidence for the validity of these identities, and checked them to a certain order in a power series expansion. This paper is mainly concerned with analytical proofs of these identities. More specifically, we furnish a complete proof of the identities for the A 2 elliptic case and for the A 3 hyperbolic case, and consider several specializations. We also discuss the implications the kernel identities might have for a Hilbert space scenario involving common eigenvectors of the Hamiltonians and the integral operators associated with the kernel functions.

Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model

“…But in some cases operators were previously unknown as in the case of the minimal 6d SCFTs compactifications. Study of such novel operators constitutes an interesting field of research with some initial steps already taken in this direction [14,15].…”

“…x I(x) ∼ W (C 2 ;ha;1,0) (x)W (C 2 ;h b ;1,0) (x) − W (C 2 ;h b ;1,0) (x)W (C 2 ;ha;1,0) (x) I(x) = 0 . (E. 14) This term is obviously zero since it does not involve any shift. 6) Terms with ∆ −1 q (x i )∆ −1 q (x j )I(x).…”

A Panorama Of Physical Mathematics c. 2022

“…For example, various dualities lead to non-trivial identities of elliptic-hypergeometric integrals and limits thereof, Painlevé tau-functions, etc. A set of pairs of concrete examples of the intimate relation between the mathematical works and the supersymmetric physics is [1139,1214], [1215,1216], [1155,1217], [1218,1219], and [1220,1221]. Many times these quantities can be related to topological, conformal, and integrable QFTs in even lower dimensions (not necessarily supersymmetric).…”

“…See [1226] for a comprehensive review and list of references. Another example is the association of an integrable quantum mechanical system to supersymmetric theories with two dimensional N = 2 super-Poincare invariance, the Bethe/gauge correspondence [336], including those originating as four or six dimensional theories [1176,1182,1215,1216,1221,[1227][1228][1229][1230][1231][1232][1233]. In the geometric program of relating compactifications of six dimensional theories to four dimensional ones the quantum mechanical integrable models appear for example as operators acting on supersymmetric indices [668][669][670] of the four dimensional theories introducing surface defects in to the index computation [1227].…”