On extensions of $$ \mathfrak{gl}\widehat{\left(\left.m\right|n\right)} $$ Kac-Moody algebras and Calabi-Yau singularities

Abstract

We discuss a class of vertex operator algebras W m|n×∞ generated by a supermatrix of fields for each integral spin 1, 2, 3,. .. . The algebras admit a large family of truncations that are in correspondence with holomorphic functions on the Calabi-Yau singularity given by solutions to xy = z m w n. We propose a free-field realization of such truncations generalizing the Miura transformation for W N algebras. Relations in the ring of holomorphic functions lead to bosonization-like relations between different free-field realizations. The discussion provides a concrete example of a non-trivial interplay between vertex operator algebras, algebraic geometry and gauge theory.

Quantum integrable systems from supergroup gauge theories

“…However the gauge theoretical interpretation of such a parameter remains rather unclear, and we merely regarded it as the rescaling of Ω-background parameter . We can consider realizing such a θ-deformation through the intersecting D-brane configuration at an angle [25,53], in particular in the gauge origami setup for the folded instanton configuration [15], θ-deformation parameter naturally equals to k by having more -parameters.…”

The matrix-extended $$ {\mathcal{W}}_{1+\infty } $$ algebra

“…This algebra and its cousins were studied before in [22][23][24][25][26][27], where it was sometimes referred to as 'rectangular W-algebra'. Moreover, its geometric realizations are described in [28].…”

“…Since nothing in the operator product expansion depends on the rank m of matrices or the structure of matrix multiplication (and this is true even for Jacobi identities), one expects that there should exist various generalizations of this construction. For super Lie algebras this is worked out explicitly in [28].…”

“…The second one is obtained from it by applying duality automorphism as already mentioned. The last two elementary Miura factors are studied in greater detail in [28]. We should also mention that the latter three curves in parameter space degenerate for n = 1 to the singular fields of gl(m) κ Kac-Moody algebra (provided that m ≥ 3).…”

“…We thank Miroslav Rapčák for useful discussions as well as for sharing the draft [28] prior to publication. We would also like to thank Thomas Creutzig, Branislav Jurčo, Yasuaki Hikida, Wei Li, Andy Linshaw, Shashank Kanade, Luca Matiello, Cheng Peng and Ivo Sachs for useful discussions.…”