##### Department of Mathematics,

University of California San Diego

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### Math 292 - Topology Seminar

## Justin Roberts

#### UCSD

## Introduction to topological conformal field theories

##### Abstract:

A (1+1)-dimensional Topological Quantum Field Theory is a tensor functor from the category of 2-dimensional cobordisms to the category of vector spaces. It is easy to give a characterisation of such functors: they are determined by the vector space associated to a single circle together with the structure maps it inherits from the disc and the pair of pants, which make it into a finite-dimensional Frobenius algebra. A (1+1)-dimensional Conformal Field Theory is a much more subtle thing, being a functor from the category of Riemann surfaces (2-dimensional cobordisms equipped with complex structures or "moduli") to a category of Hilbert spaces. Somewhere between lies the idea of Topological Conformal Field Theory, which is a "chain level" version of a CFT. It is determined by a chain complex on which the spaces of chains of the (morphism spaces of the) category of Riemann surfaces act. Such a structure arises in several places in modern topology, most notably in the theory of Gromov-Witten invariants of symplectic manifolds and in the Sullivan-Chas string topology of a loop space. This term we aim to read Kevin Costello's paper "Topological Conformal Field Theories and Calabi-Yau categories" (math.QA/0412149), which gives an algebraic characterisation of TCFTs analogous to the "Frobenius algebra" classification of TQFTs. In the first talk in the series I will try to give an overview of what the paper says, and we will organise talks for the rest of the term.

### October 9, 2007

### 10:30 AM

### AP&M 7218

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