New Quiver-Like Varieties and Lie Superalgebras

Superspin chains from superstring theory

“…Further reduction of the trigonometric Rmatrix gives the rational R-matrix. The reduction of our elliptic sl(1|1) R-matrix down to the rational case matches with the previously computed rational sl(1|1) R-matrix from [30].…”

“…We present the explicit K-theoretic and cohomological stable envelopes for the quiver with a single node which is odd, see (2.84) and (2.87). The cohomological stable envelope (2.87) recovers the recent result of Rimányi and Rozansky [30,Theorem 7.3].…”

“…However, for more general 3d N = 2 theories, their Higgs branches do not have complex symplectic structure or polarization, even worse they can be singular. There have been efforts to construct stable envelopes for varieties without complex symplectic structure or polarization [30], the authors constructed cohomological stable envelopes using the super weight functions, which generalized their previous works on weight functions and stable envelopes on cotangent bundles of flag varieties [75,82]. They found the R-matrix of Y (sl(1|1)) as the geometric R-matrix of the total space of the bundle O(−1) ⊕2 P 1 .…”

“…The classical trigonometric solutions to YBE are constructed in [28] while the explicit trigonometric quantum R-matrices first appeared in [29]. Furthermore, the rational solution for g = sl(1|1) is constructed using the notion of stable envelopes in [30]. In this paper, we propose a general construction of elliptic R-matrices for Lie superalgebras of type A and we explicitly compute the elliptic R-matrix for g = sl(1|1).…”

“…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. Providing such a mathematical formulation for g being a Lie superalgebra of type A has been one of the central motivations of this work.…”