Wei Gu: Nonabelian mirrors and Gromov-Witten invariants

Abstract: We propose Picard-Fuchs equations for periods of nonabelian mirrors [1] in this paper. Thenumber of parameters in our Picard-Fuchs equations is the rank of the gauge group in the nonabelian GLSM, although we reduce them to the actual number of Kähler parameters ultimately. These Picard-Fuchs equations are concise but novel, so we reproduce existing mathematical Picard-Fuchs equations of Gr(k,N) and Calabi-Yau manifolds as complete intersections in Grassmannians [2] to justify our proposal. Furthermore, our approach can be applied to other nonabelian GLSMs, so we compute mathematical Picard-Fuchs equations of some other Fano-spaces, which were not calculated in the literature before. Finally, the cohomology-valued generating functions of mirrors can be read off from our Picard-Fuchs equations. Using these generating functions, we compute Gromov-Witten invariants of various Calabi-Yau manifolds including complete intersection Calabi-Yau manifolds in Grassmannians and non-complete intersection Calabi-Yau examples such as Pfaffian Calabi- Yau threefold and Gulliksen-Negård Calabi-Yau threefold, and find agreement with existing results in the literature. However, the generating functions we propose for the non-complete intersection Calabi-Yau manifolds are genuinely new.