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Two proofs of a conjecture of Hori and Vafa. (English) Zbl 1082.14055
One of the incarnations of mirror symmetry is the equivalence between the nonlinear sigma model for a Fano manifold \(X\) and the Landau-Ginzburg theory with a suitable potential \(W_X\) [K. Hori and C. Vafa, hep-th/0002222; E. Witten, Nucl. Phys., B 403, No. 1–2, 159–222 (1993; Zbl 0910.14020)]. Hori and Vafa conjecture in [K. Hori and C. Vafa, loc. cit.] that the Landau-Ginzburg mirror of the Grassmannian \({\mathbf G}=G(r,n)\) can be obtained by a symmetrization procedure from the Landau-Ginzburg mirror of the product of projective spaces \({\mathbf P}=\prod_{i=1}^r{\mathbb P}^{n-1}\). Translating this on the sigma model side, one obtains a conjectural relation between the \(J\)-function of \({\mathbf G}\) and the \(J\)-function of \({\mathbf P}\). Although the sigma/Landau-Ginzburg model correspondence has not been given a rigorous mathematical treatment, yet, the conjectural relation between \(J^{\mathbf G}\) and \(J^{\mathbf P}\) that one obtains is a precise mathematical statement, which the authors are able to give two different proofs in the present paper.
The first proof uses an explicit computation of the \(J\)-functions of \({\mathbf G}\) and \({\mathbf P}\), obtained by Givental’s localization formula for equivariant \(J\)-functions [A. B. Givental, Int. Math. Res. Not. 1996, No. 13, 613–663 (1996; Zbl 0881.55006)]. More precisely, the \(J\)-function \(J^{\mathbf G}\) can be expressed in terms of the 1-point Gromov-Witten invariants of \({\mathbf G}\), i.e. involves the geometry of the moduli space \(\overline{M}_{0,1}({\mathbb G},d)\) of 1-pointed degree \(d\) maps \({\mathbb P}^1\to {\mathbf G}\). This space can be identified with a component of the fixed point locus for the natural \({\mathbb C}^*\)-action on \(\overline{M}_{0,0}({\mathbf G}\times {\mathbb P}^1, (d,1))\) induced by the action of \({\mathbb C}^*\) on \({\mathbb P}^1\). The moduli space \(\overline{M}_{0,0}({\mathbf G}\times {\mathbb P}^1, (d,1))\) is a compactification of the Hilbert scheme of degree \(d\) maps \({\mathbb P}^1\to {\mathbf G}\). This Hilbert scheme has another natural compactification: the quot scheme \(\text{Quot}_{{\mathbb P}^1,d}({\mathbb C}^n,n-r)\), and also this space carries a \({\mathbb C}^*\)-action induced by the \({\mathbb C}^*\)-action on \({\mathbb P}^1\). The two spaces \(\overline{M}_{0,0}({\mathbf G}\times {\mathbb P}^1, (d,1))\) and \(\text{Quot}_{{\mathbb P}^1,d}({\mathbb C}^n,n-r)\) can be related via equivariant maps to \({\mathbb P}(\text{Sym}^d(\mathbb{C}^2)^\vee\otimes {\mathbb C}^n)\); using these maps the authors are able to explicitly compute the \(J\)-function of \({\mathbf G}\) in terms of geometrical data of the quot scheme; this formula specializes to Givental’s formula for \(J^{{\mathbb P}^{n-1}}=J^{G(1,n)}\). One also knows the \(J\)-function of \({\mathbf P}\), since \(J^{\mathbf P}=\prod_{i=1}^r J^{{\mathbb P}^{n-1}}\). Comparing the explicit expressions for \(J^{\mathbf G}\) and \(J^{\mathbf P}\) one proves the Hori-Vafa conjecture.
The second proof is bases on quantum cohomology: Givental’s \(J\)-function encodes the data of flat sections for the connection \(\nabla_{\hbar}=d+\frac{1}{\hbar}*\), where \(*\) is the quantum product in the small quantum cohomology ring. One has well-known presentations for the small quantum cohomology rings of \({\mathbf G}\) and \({\mathbf P}\) and also an explicit relation between the cup product of Schubert classes in the Grassmannian and the cup product of their images in \(H^*({\mathbf P})\) via the Martin-Ellingsrud-Strømme map \(\theta\colon H^*({\mathbf G})\to H^*({\mathbf P})\) [S. Martin, math.SG/0001002; G. Ellingsrud and S. A. Strømme, Ann. Math. (2) 130, No. 1, 159–187 (1989; Zbl 0716.14002)]. Elaborating on this, the authors are able to relate the quantum product in \(QH^*({\mathbf G})\) with the quantum product in \(QH^*({\mathbf P})\), and to consequently deduce the conjectured relation between \(J^{\mathbf G}\) and \(J^{\mathbf P}\).
Moreover, as a consequence of the Hori-Vafa relation, the authors prove Givental’s \(R\)-conjecture, and consequently the Virasoro conjecture, for Grassmannians [A. B. Givental, Mosc. Math. J. 1, No. 4, 551–568 (2001; Zbl 1008.53072)]. In the final part of the paper, the authors with Dennis Stanton give a proof of a formula for (a part of) the \(J\)-function for \(G(2,n)\) which had been conjectured by V. V. Batyrev, I. Ciocan-Fontanine, B. Kim and D. Van Straten [Nucl. Phys., B 514, No. 3, 640–666 (1998; Zbl 0896.14025)].

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14M15 Grassmannians, Schubert varieties, flag manifolds
Full Text: DOI arXiv
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