Superspin chains from superstring theory

Abstract

We present a correspondence between two-dimensional \mathcal{N} = (2,2)𝒩=(2,2) supersymmetric gauge theories and rational integrable \mathfrak{gl}(m|n)𝔤𝔩(m|n) spin chains with spin variables taking values in Verma modules. To explain this correspondence, we realize the gauge theories as configurations of branes in string theory and map them by dualities to brane configurations that realize line defects in four-dimensional Chern–Simons theory with gauge group GL(m|n)GL(m|n). The latter configurations embed the superspin chains into superstring theory. We also provide a string theory derivation of a similar correspondence, proposed by Nekrasov, for rational \mathfrak{gl}(m|n)𝔤𝔩(m|n) spin chains with spins valued in finite-dimensional representations.

Line Operators in 4d Chern-Simons Theory and Cherkis Bows

“…Known connections between spin chains and 3d gauge theories with Hanany-Witten type brane construction was used in [56] to establish bispectral dualities of integrable spin chains. Brane constructions similar to that of §5.1 was used in [40] to create spin chains with Verma modules and to illustrate fermionic dualities of superspin chains. We hope that our brane constructions and characterization of line operators using Cherkis bow varieties will further aid the study of non-trivial dualities of integrable spin chains and related gauge theories.…”

“…We shall find that the KW theory in a B-type Ω-background localizes to 2d BF theory. While this is certainly known in literature [38][39][40], some details have yet to appear, so for completion we go through the intermediate steps in this section. First we look at the bulk theory, disregarding the boundaries, and then we look at the boundary conditions in terms of the fields of the twisted theory.…”

“…The brane construction of the T-operator involves n D5, n NS5, and n D3 branes -one D3 brane suspended between each pair of five-branes. This setup was also used in [40] to create Wilson lines valued in Verma modules for 4d GL m|n CS theory with highest weights given by locations of NS5 branes in the holomorphic direction shifted by the Weyl vector. 16 It was also argued on general grounds in [46] that the quantized Coulomb branch algebras of 3d N = 4 theories act on Verma modules; they showed this by analyzing the action of the Coulomb branch operators -also called monopole operators -on vortex configurations.…”

5d AGT correspondence of supergroup gauge theories from quantum toroidal $$ \mathfrak{gl} $$1

“…S-duality of supergroup gauge theories with non-superalgebra quiver structure implies the existence of non-supergroup gauge theories with superalgebra quiver structure. For 3d supersymmetric gauge theories, a gauge theory with superquiver structure was discussed JHEP12(2022)157 in [54,[63][64][65]. However, for 5d gauge theories, to the author's knowledge, there is no known construction of supersymmetric supergroup superquiver gauge theories.…”

“…A detailed analysis is left for future work. A different construction of superquiver theories for three-dimensional supersymmetric gauge theories and their relations with Bethe/Gauge correspondence and supergroup gauge theories were studied in [37,54,[63][64][65]. Studying the relation with these papers is also left to be done.…”