Spacetime Diffeomorphisms and Topological W-Infinity Symmetry in Two Dimensional Topological String Theory

TL;DR

This paper explores the rich structure of spacetime symmetries in two-dimensional topological string theory, revealing infinite-dimensional algebras and their deformations within the BRST cohomology framework.

Contribution

It identifies the algebraic structure of spacetime symmetries in topological string theory, including the emergence of $w_$ superalgebras and their behavior under model deformations.

Findings

Spacetime symmetries form an infinite algebra in absolute BRST cohomology.

In semirelative cohomology, symmetries reduce to $w_$ and topologically twisted $w_$ superalgebra.

Deformations can lead to different $w_$ superalgebras in the cohomology.

Abstract

This paper analyzes spacetime symmetries of topological string theory on a two dimensional torus, and points out that the spacetime geometry of the model is that of the Batalin-Vilkovisky formalism. Previously I found an infinite symmetry algebra in the absolute BRST cohomology of the model. Here I find an analog of the BV $\Delta$ operator, and show that it defines a natural semirelative BRST cohomology. In the absolute cohomology, the ghost-number-zero symmetries form the algebra of all infinitesimal spacetime diffeomorphisms, extended at non-zero ghost numbers to the algebra of all odd-symplectic diffeomorphisms on a spacetime supermanifold. In the semirelative cohomology, the symmetries are reduced to $w_\infty$ at ghost number zero, and to a topologically twisted N=2 $w_\infty$ superalgebra when all ghost numbers are included. I discuss deformations of the model that break parts of the spacetime symmetries while preserving the topological BRST symmetry on the worldsheet. In the absolute cohomology of the deformed model, another topological $w_\infty$ superalgebra may emerge, while the semirelative cohomology leads to a bosonic $w_\infty$ symmetry.