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Gamma classes and quantum cohomology of Fano manifolds: gamma conjectures. (English) Zbl 1350.14041
The authors propose Gamma conjectures for Fano manifolds which can be thought of as a square root of the index theorem. The quantum connection of a Fano manifold \(F\) has a regular singularity at \(z=\infty\) and an irregular singularity at \(z=0\). Flat sections near \(z=\infty\) are constructed by the Frobenius method and can be put into correspondence with cohomology classes in a natural way. Flat sections near \(z=0\) are classified by their exponential growth order (along a sector). Under the assumption of Property \(\mathcal O\), Gamma conjecture I claims that the flat section with the smallest asymptotics as \(z\to 0\) will transport to the flat section near \(z=\infty\) corresponding to the Gamma class \(\hat{\Gamma}_F\). Under further semi-simplicity assumption, Gamma conjecture II says that cohomology classes \(A_i\) that correspond to flat sections asymptotic to \(e^{-u_i/z}\) as \(z\to 0\), where \(u_i\) are eigenvalues of \(C_1(F)\star_0\), can be written as \(A_i=\hat{\Gamma}_F\mathrm{Ch}(E_i)\), where \(\{E_i\}\) form an exceptional collection of the bounded derived category of coherent sheaves on \(F\). Gamma conjecture II refines a part of Dubrovin’s conjecture which says that the Stokes matrix of the quantum connection equals the matrix formed by Euler pairings among the objects in an exceptional collection. The authors then prove the Gamma conjectures for projective spaces by elementary methods (without using mirror symmetry, instead they directly look at solutions to the quantum differential equation and its Laplace transform). They also prove Gamma conjectures for Grassmannians, which follow from the case of projective spaces and the quantum Satake principle, or abelian/non-abelian correspondence, which says that the quantum connection of \(\mathrm{Gr}(r, N)\) is the \(r\)-th wedge of the quantum connection of \(\mathbb P^{N-1}\).

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
11G42 Arithmetic mirror symmetry
14J45 Fano varieties
14J33 Mirror symmetry (algebro-geometric aspects)
14M15 Grassmannians, Schubert varieties, flag manifolds
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