Bethe/Gauge correspondence for ABCDEFG-type 3d gauge theories

Abstract

In this paper, we give a new effective superpotential that makes clear Bethe/Gauge correspondence between 2d (and 3d) SO/Sp gauge theories and open XXX (and XXZ) spin chains with diagonal boundary conditions, and also works in the case of 2d (and 3d) BCN-type gauge theories which is not previously discussed in the literature. Especially, for exceptional Lie algebras F4, G2, we give the effective superpotential and vacuum equations. For E6,7,8, we only give theirs effective superpotential for convenience.

Superspin chains from superstring theory

“…Part of their conjecture is an action of the spectrum-generating algebra of a quantum integrable system, Yangian for example, on the cohomology (or generalized cohomologies) of the Higgs branch of the corresponding gauge theory. The correspondence and its string-theory origin, when g is a bosonic Lie algebra, have been extensively studied [31,[33][34][35][36][37][38][39][40][41][42][43][44][45][46][47], generalized to higher dimensions [32,35,[48][49][50][51][52][53], and the mathematical formulation has been established [54][55][56][57][58][59]. For g being a Lie superalgebra, the correspondence and its string-theory origin were studied recently [60][61][62][63] but despite progress [30,64], a full mathematical treatment was lacking.…”

“…ab = α ab = e a − e b , a = b, (C 41). where α ab denotes the ab th root of u(N ), associated to the generator E ab , and {e 1 , • • • , e N } is the standard basis for R N 53. ThereforeW I (σ C ) = ρ ab (σ C ) = σ C a − σ C b , a = b, W I (n) = ρ ab (n) = n a − n b , a = b.…”

“…t a is the image of the I th u(1) generator of the center of g * under the embedding c * ֒ h * 53. For α ab = ea − e b for a = b, we consider the one-dimensional vector space generated by E ab .…”