报告题目：Topological recursion and W-constraints

报告人：Gaetan Borot,德国波恩马普数学所

时间：11月06日（周二）下午14:30—16:00

地点：东区第二教学楼 2403教室

摘要：The topological recursion of Eynard and Orantin has many applications in enumerative geometry (Hurwitz theory, intersection theory on the moduli space of curves, ...). Its starting point is a spectral curve (i.e a branched cover of curves) to which the topological recursion associates a collection F_{g,n} of generating series, which encode enumerative information.

It has been recently related to deformation quantization by Konsevich and Soibelman, who introduced the notion of quantum Airy structures. 

A quantum Airy structure is a collection of differential operators, which among other conditions, must form a graded Lie ideal. To any quantum Airy structure one can associate the (essentially unique) function annihilated by these differential operators, which is always computed by a kind of topological recursion. For instance in the Gromov-Witten theory of a point or more generally for semi-simple cohomological field theories, the quantum Airy structure encode the Virasoro constraints for the so-called ancestor potential, which is itself computed by the topological recursion.I will explain that W(gl_r)-algebras naturally give rise to a family of quantum Airy structures, for which the topological recursion F_{g,n} can be expressed in terms of the ancestor potential of the Witten r-spin class. The latter was studied in Witten's conjecture/Faber-Shadrin-Zvonkine theorem, and by Milanov from the perspective of W-algebra constraints. Conversely, any spectral curve with arbitrary ramification give rise to quantum Airy structures in this family.

The talk is based on joint work with Andersen-Chekhov-Orantin and Bouchard-Chidambaram-Noshchenko.

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