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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)].

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