Huai-Liang Chang: BCOV Feynman structure in A side

Abstract:  Gromov Witten invariants Fg encodes the numbers of genus g curves in Calabi Yau threefolds. They play a role in enumerative geometry and are not easy to be determined. 

In 1993 Bershadsky, Cecotti, Ooguri, Vafa solved holomorphic anomaly equations for Fg's in B side and their solution exhibited a hidden ``Feynman structure” governing all Fg’s at once. The method was via path integral while its counterpart in mathematic has been missing for decades.


In 2018, a large N topological string theory is developed in math and provides the wanted ``Feynman structure”. New features are (i) dynamically quantizing the Kaehler moduli parameter
in Witten’s GLSM, and (ii) enhancing the Calabi Yau target to a large N bulk with Calabi Yau boundary. Both are achieved within a Landau Ginzburg type construction (P fields and cosections in math terms).


This theory, called “N-Mixed Spin P field” in math, is thus responsible for BCOV theory in A side. In it the A model theory of the large N bulk encodes the B model propagators of the boundary’s mirror. This is a new phenomenon intertwining bulk-boundary correspondence, mirror symmetry, and Gauge-String(B-A) duality.

In this talk we will see genuine ideas behind these features, along a new angle to realize NMSP from people familiar with Givental theory.