Cohomological field theories and BKP integrability: Omega classes times Witten-classes

Abstract: There is a deep interaction between Cohomological field theories (CohFTs), introduced by Kontsevich and Manin, and integrable hierarchies. For instance, the celebrated Witten-Kontsevich result shows that the trivial CohFT gives rise to a solution of the KdV integrable hierarchy. As another example, Kazarian’s theorem shows that the Hodge CohFT gives rise to a solution of the KP hierarchy, and so do Hurwitz numbers, which by ELSV formula are descendant integrals of the Hodge CohFT. The change of variable which carries the partition function of Hurwitz numbers into the partition function of pure descendant Hodge integrals is triangular and KP-preserving, it is in fact essentially given by the Topological Recursion spectral curve in the sense of Eynard and Orantin.

We study spin-Hurwitz numbers (not to be confused with completed cycles Hurwitz numbers) enumerating branches Riemann covers weighted by the parity of theta characteristics. They obey the BKP integrable hierarchy. We prove that the Topological Recursion conjecture for these numbers is equivalent to their underlying CohFT to be an explicit product between Witten’s class and Omega-classes computed by Chiodo. The Topological Recursion conjecture has recently been proved by Alexandrov and Shadrin in a more general framework for BKP integrability.

HEP - theorymathematical physicsalgebraic geometrycombinatoricsexactly solvable and integrable systems

Audience: researchers in the discipline

IBS-CGP Mathematical Physics Seminar

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