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The structure of 2D semi-simple field theories

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I classify the cohomological 2D field theories based on a semi-simple complex Frobenius algebra A. They are controlled by a linear combination of κ-classes and by an extension datum to the Deligne–Mumford boundary. Their effect on the Gromov–Witten potential is described by Givental’s Fock space formulae. This leads to the reconstruction of Gromov–Witten (ancestor) invariants from the quantum cup-product at a single semi-simple point and the first Chern class of the manifold, confirming Givental’s higher-genus reconstruction conjecture. This in turn implies the Virasoro conjecture for manifolds with semi-simple quantum cohomology. The classification uses the Mumford conjecture, proved by Madsen and Weiss (European Congress of Mathematics, pp. 283–303, 2005).

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  1. UC Department of Mathematics, 970 Evans Hall, Berkeley, CA, 94720, USA

    Constantin Teleman

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Teleman, C. The structure of 2D semi-simple field theories. Invent. math. 188, 525–588 (2012). https://doi.org/10.1007/s00222-011-0352-5

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