GTM Seminar

Speaker: Alexis Roquefeuil (IPMU)
Title: Quantum K-theory of projective spaces and confluence of q-difference equations
Date: Thu, Jan 23, 2020, 15:30 - 17:00
Place: Seminar Room B
Blackboard: (Blackboard talk)
Abstract: Gromov--Witten invariants are numbers that, under certain conditions, can be understood as counting the number of complex curves on a projective variety that satisfy some incidence conditions. In algebraic geometry, these numbers are defined as the degree of some cohomological class constructed on Kontsevich's moduli space of stable maps. In the case of Fano varieties, these numbers can be encoded in an algebraic structure called quantum D-module, which an essential ingredient to produce computation of Gromov--Witten invariants through string theory's mirror symmetry.
In 2004, Y.P. Lee defined new invariants, called K-theoretic Gromov--Witten invariants, as the Euler characteristic of some sheaves living on the moduli space of stable maps. A natural question is to understand how these numbers are related to the usual Gromov--Witten invariants. A Riemann--Roch formula relating the two invariants was provided by A. Givental and V. Tonita in 2014, however this formula is quite technical and has not seen many applications so far. One of its consequences is that an important K-theoretical function expressed with K-theoretical Gromov--Witten invariants, called Givental's J-function, satisfy q-difference equations.

In the case of projective spaces, we suggest a new approach to compare the cohomological and K-theoretical Gromov--Witten theories. Using the q-difference equations satisfied by the J-function, we associate to K-theoretical Gromov--Witten invariants a q-difference module. In this talk, we will show that the confluence of q-difference equations provides a natural way to obtain the quantum D-module and the cohomological J-function as a limit of the q-difference module above and the K-theoretical J-function.