Abstract
We discuss a class of vertex operator algebras {mathcal{W}}_{left.mright|nkern0.33em times kern0.33em infty } 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 = zmwn. We propose a free-field realization of such truncations generalizing the Miura transformation for {mathcal{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.
Highlights
It was observed in [27] that W1+∞ algebra contains a three-parameter family of truncations YN1,N2,N3 parametrized by non-negative integers Ni
We discuss a class of vertex operator algebras Wm|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 = zmwn
Relations in the ring of holomorphic functions lead to bosonization-like relations between different free-field realizations
Summary
According to the conjecture of [27], one can associate a vertex operator algebra to any toric divisor inside a toric Calabi-Yau three-fold. The special class of divisors at hand will be identified with configurations with zero shifts and leading to truncations of Wm|n×∞ itself. For N1, N2, N3, N4 non-negative integers, modulo the relation xy = zmwn in the coordinate ring of CYm3,n. Such functions scale with the power of ((n − m)h1 − h2)N3 + h2N2 − h1N4 + h1N1. Under the T 2 action described above This constant will play an important role in the discussion of vertex operator algebras below. This specialization is the reason for the above convention for the labels Ni in the exponents of (2.5)
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