Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model
Belal Nazzal, Anton Nedelin, Shlomo S. Razamat
SciPost Phys. 12, 140 (2022) · published 26 April 2022
- doi: 10.21468/SciPostPhys.12.4.140
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Abstract
We consider supersymmetric surface defects in compactifications of the $6d$ minimal $(D_{N+3},D_{N+3})$ conformal matter theories on a punctured Riemann surface. For the case of $N=1$ such defects are introduced into the supersymmetric index computations by an action of the $BC_1\,(\sim A_1\sim C_1)$ van Diejen model. We (re)derive this fact using three different field theoretic descriptions of the four dimensional models. The three field theoretic descriptions are naturally associated with algebras $A_{N=1}$, $C_{N=1}$, and $(A_1)^{N=1}$. The indices of these $4d$ theories give rise to three different Kernel functions for the $BC_1$ van Diejen model. We then consider the generalizations with $N>1$. The operators introducing defects into the index computations are certain $A_{N}$, $C_N$, and $(A_1)^{N}$ generalizations of the van Diejen model. The three different generalizations are directly related to three different effective gauge theory descriptions one can obtain by compactifying the minimal $(D_{N+3},D_{N+3})$ conformal matter theories on a circle to five dimensions. We explicitly compute the operators for the $A_N$ case, and derive various properties these operators have to satisfy as a consequence of $4d$ dualities following from the geometric setup. In some cases we are able to verify these properties which in turn serve as checks of said dualities. As a by-product of our constructions we also discuss a simple Lagrangian description of a theory corresponding to compactification on a sphere with three maximal punctures of the minimal $(D_5,D_5)$ conformal matter and as consequence give explicit Lagrangian constructions of compactifications of this 6d SCFT on arbitrary Riemann surfaces.
Cited by 11
Authors / Affiliations: mappings to Contributors and Organizations
See all Organizations.- 1 Belal Nazzal,
- 1 Anton Nedelin,
- 1 2 Shlomo S. Razamat
- Ambrose Monell Foundation
- German-Israeli Foundation for Scientific Research and Development [GIF]
- Israel Science Foundation [ISF]
- United States - Israel Binational Science Foundation (through Organization: United States-Israel Binational Science Foundation [BSF])