Topological conformal field theories and Calabi–Yau categories

By K. Costello. 2004

375 citations recorded by the source corpus.

Abstract This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov–Witten invariants (at all genera). These Gromov–Witten type invariants depend on a Calabi–Yau A ∞ category, which plays the role of the target in ordinary Gromov–Witten theory. When we use an appropriate A ∞ version of the derived category of coherent sheaves on a Calabi–Yau variety, this constructs the B model at all genera. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the usual Gromov–Witten invariants are recovered. The assumptions are that open-closed Gromov–Witten theory can be constructed for X , and that the natural map from the Hochschild homology of the Fukaya category of X to the ordinary homology of X is an isomorphism.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/145216e3cabb83238703674e82dee295982a854c

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Citations in the corpus

Hochschild homology of structured algebras (2011, positive): “The proof of the main theorem, inspired by, though independent of, [11], uses simple properties of the double bar construction, and a quotiented version of it to take care of the equivariant version of the theorem under the action of the symmetric groups.”
D-branes and K-theory in 2D topological field theory (2006, methodology): “Our treatment of cochain-level theories and A∞ algebras has been developed considerably further by Costello [25].”
Higher string topology operations (2007, background): “In [7], Costello constructed a positiveboundary open-closed TQFT structure on the pair (A,HC∗A) where A is a Frobenius algebra and HC∗A are its Hochshild chains.”
Derived Categories (2016, neutral): “• Conformal field theories (see [42]).”
Hodge theoretic aspects of mirror symmetry (2008, background): “and some aspects of it were later developed in depth by K.Costello [ Cos07b , Cos05, Cos07c,”
Batalin-Vilkovisky algebra structures on Hochschild Cohomology (2007, positive): “This theorem has been reproved and extended by many people [24, 20, 28, 5, 21, 22, 18, 8] (in chronological order).”
On triangulated orbit categories (2005, background): “These are of considerable interest in mathematical physics, since, as Kontsevich shows [38], [37], cf. also [ 18 ], a topological quantum field theory is associated with each Calabi-Yau A∞-category satisfying some additional assumptions1.”
Homological mirror symmetry for Brieskorn–Pham singularities (2009, background): “Now let us discuss open string sector of the topological strings, which is the theory of cyclic A∞-categories from mathematical point of view (see [4] and references therein).”
The Mukai pairing, I: a categorical approach (2007, background): “Appropriate references for open-closed 2d topological field theories include Moore-Segal [14], Costello [7] and Lauda-Pfeiffer [9].”
Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations (2010, background): “3 Generalized abstract Hirzebruch-Riemann-Roch Theorem It is known that a Calabi-Yau dg-category gives rise to an open-closed 2d TQFT (see [5]).”