BRST-Invariant Deformations of Geometric Structures in Topological Field Theories

Abstract

We study a Lie algebra of formal vector fields $W_n$ with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi-Yau manifold with boundaries in the B-model. A relevant concept in the vertex operator algebra and the BRST cohomology is that of the elliptic genera (the one-loop string partition function). We show that the elliptic genera can be written in terms of spectral functions of the hyperbolic three-geometry (which inherits the cohomology structure of BRST-like operator). We show that equivalence classes of deformations are described by a Hochschild cohomology theory of the DG-algebra $\mathfrak{A}=(A,Q)$, $Q=\bar{\partial }+{\partial }_{\mathrm{deform}}$, which is defined to be the cohomology of $(-1{)}^{n}Q+d_{\mathrm{Hoch}}$. Here $\bar{\partial }$ is the initial non-deformed BRST operator while ${\partial }_{\mathrm{deform}}$ is the deformed part whose algebra is a Lie algebra of linear vector fields ${\mathrm{gl}}_n$. We discuss the identification of the harmonic structure $(H{T}^{\bullet }(X);H{\Omega }_{\bullet }(X))$ of affine space $X$ and the group ${\mathrm{Ext}}_X^n({\mathcal{O}}_{△},{\mathcal{O}}_{△})$ (the HKR isomorphism), and bulk-boundary deformation pairing.

44 pages. arXiv admin note: text overlap with arXiv:0909.3643, arXiv:hep-th/0210057, arXiv:math/0308080, arXiv:hep-th/0405232 by other authors

References in corpus (2)

• A-twisted heterotic Landau-Ginzburg models
• String Partition Functions, Hilbert Schemes, and Affine Lie Algebra Representations on Homology Groups