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Givental group action on topological field theories and homotopy Batalin-Vilkovisky algebras. (English) Zbl 1294.14019
Cohomological Field Theories (CohFT) are defined as certain algebras over homologies of the Deligne-Mumford compactifications of moduli spaces of punctured curves. They were introduced in order to study certain properties of Gromov-Witten invariants, and quickly proved to possess a very rich and interesting structure. In particular deep connections between CohFT and Givental’s theory have been found recently. One important challenge in this context is to understand the Givental group action on Cohomological Field Theories. In this paper the authors propose to study this action in terms of the so-called homotopy Batalin-Vilkovisky algebras, and characterize this action in genera zero and one. The main results of the paper are presented as two theorems that characterize Topological Field Theories (TFT), i.e. CohFT concentrated in degree zero. The first theorem states that in genus zero the stabilizers of Topological Field Theories are in one-to-one relation with commutative homotopy Batalin-Vilkovisky algebras. The second theorem asserts that in genera zero and one, the stabilizers of TFTs are in one-to-one correspondence with wheeled commutative homotopy Batalin-Vilkovisky algebras. These results, apart from being important in themselves, also open interesting directions for further studies – in particular, generalizing these results to arbitrary genera is an important problem.
The paper makes contact with several difficult topics and combines them in a very nice way. It is however not easy to make such a work fully self-contained, and readers unfamiliar with the subject should get acquainted with some background material first. The authors made an effort to facilitate this task, and in section one and in appendices a reader can find a concise summary of the most important notions necessary to understand the paper, such as: the intersection theory on the moduli spaces of curves, operads, Cohomological Field Theories, Givental group actions, some aspects of Batalin-Vilkovisky formalism, etc. The paper is nicely written, and is worth recommending to everyone interested in at least one of the above mentioned topics.

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
18D50 Operads (MSC2010)
57R56 Topological quantum field theories (aspects of differential topology)
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14H10 Families, moduli of curves (algebraic)
81T45 Topological field theories in quantum mechanics
13N10 Commutative rings of differential operators and their modules
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