On extensions of $$ \mathfrak{gl}\widehat{\left(\left.m\right|n\right)} $$ Kac-Moody algebras and Calabi-Yau singularities
Miroslav Rapčák
Abstract
A bstract We discuss a class of vertex operator algebras $$ {\mathcal{W}}_{\left.m\right|n\kern0.33em \times \kern0.33em \infty } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mfenced> <mml:mi>m</mml:mi> </mml:mfenced> <mml:mi>n</mml:mi> <mml:mspace/> <mml:mo>×</mml:mo> <mml:mspace/> <mml:mo>∞</mml:mo> </mml:mrow> </mml:msub> </mml:math> generated by a super- matrix 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 $$ {\mathcal{W}}_N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> 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.