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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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References:
[1] L. Abrams, The quantum Euler class and the quantum cohomology of the Grassmannians , Israel J. Math 117 (2000), 335–352. · Zbl 0954.53048 · doi:10.1007/BF02773576
[2] G. E. Andrews, q-Series: Their Development and Applications in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra , CBMS Reg. Conf. Ser. Math. 66 , Amer. Math. Soc., Providence, 1986. · Zbl 0594.33001
[3] V. V. Batyrev, I. Ciocan-Fontanine, B. Kim, and D. van Straten, Conifold transitions and mirror symmetry for Calabi-Yau complete intersections in Grassmannians , Nuclear Phys. B 514 (1998), 640–666. · Zbl 0896.14025 · doi:10.1016/S0550-3213(98)00020-0
[4] –. –. –. –., Mirror symmetry and toric degenerations of partial flag manifolds , Acta Math. 184 (2000), 1–39. · Zbl 1022.14014 · doi:10.1007/BF02392780
[5] K. Behrend and B. Fantechi, The intrinsic normal cone , Invent. Math. 128 (1997), 45–88. · Zbl 0909.14006 · doi:10.1007/s002220050136
[6] A. Bertram, “Computing Schubert’s calculus with Severi residues: An introduction to quantum cohomology” in Moduli of Vector Bundles (Sanda and Kyoto, Japan, 1994) , Lecture Notes in Pure and Appl. Math. 179 , Dekker, New York, 1996, 1–10. · Zbl 0885.14023
[7] –. –. –. –., Quantum Schubert calculus , Adv. Math. 128 (1997), 289–305. · Zbl 0945.14031 · doi:10.1006/aima.1997.1627
[8] –. –. –. –., Another way to enumerate rational curves with torus actions , Invent. Math. 142 (2000), 487–512. · Zbl 1031.14027 · doi:10.1007/s002220000094
[9] A. Bertram, I. Ciocan-Fontanine, and W. Fulton, Quantum multiplication of Schur polynomials , J. Algebra 219 (1999), 728–746. · Zbl 0936.05086 · doi:10.1006/jabr.1999.7960
[10] A. Bertram, G. Daskalopoulos, and R. Wentworth, Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians , J. Amer. Math. Soc. 9 (1996), 529–571. JSTOR: · Zbl 0865.14017 · doi:10.1090/S0894-0347-96-00190-7 · links.jstor.org
[11] M. Brion, “The push-forward and Todd class of flag bundles” in Parameter Spaces (Warsaw, 1994) , Banach Center Publ. 36 , Polish Acad. Sci., Warsaw, 1996, 45–50. · Zbl 0873.14004 · eudml:208581
[12] B. Dubrovin, “Geometry of 2D topological field theories” in Integrable Systems and Quantum Groups (Montecatini Terme, Italy, 1993) , Lecture Notes in Math. 1620 , Springer, Berlin, 1996, 120–348. · Zbl 0841.58065 · doi:10.1007/BFb0094793
[13] T. Eguchi, K. Hori, and C.-S. Xiong, Gravitational quantum cohomology , Internat. J. Modern Phys. A. 12 (1997), 1743–1782. · Zbl 1072.32500 · doi:10.1142/S0217751X97001146
[14] –. –. –. –., Quantum cohomology and Virasoro algebra , Phys. Lett. B 402 (1997), 71–80. · Zbl 0933.81050 · doi:10.1016/S0370-2693(97)00401-2
[15] G. Ellingsrud and S. A. Strømme, On the Chow ring of a geometric quotient , Ann. of Math. (2) 130 (1989), 159–187. JSTOR: · Zbl 0716.14002 · doi:10.2307/1971479 · links.jstor.org
[16] A. B. Givental, Equivariant Gromov-Witten invariants , Internat. Math. Res. Notices 1996 , no. 13, 613–663. · Zbl 0881.55006 · doi:10.1155/S1073792896000414
[17] –. –. –. –., Gromov-Witten invariants and quantization of quadratic Hamiltonians , Mosc. Math. J. 1 (2001), 551–568. · Zbl 1008.53072 · www.ams.org
[18] –. –. –. –., Semisimple Frobenius structures at higher genus , Internat. Math. Res. Notices 2001 , no. 23, 1265–1286. · Zbl 1074.14532 · doi:10.1155/S1073792801000605
[19] K. Hori and C. Vafa, Mirror symmetry , · Zbl 1044.14018 · arxiv.org
[20] D. Joe and B. Kim, Equivariant mirrors and the Virasoro conjecture for flag manifolds , Internat. Math. Res. Notices 2003 , no. 15, 859–882. · Zbl 1146.14302 · doi:10.1155/S1073792803201148
[21] B. Kim, On equivariant quantum cohomology , Internat. Math. Res. Notices 1996 , no. 17, 841–851. · Zbl 0881.55007 · doi:10.1155/S1073792896000517
[22] ——–, Quot schemes for flags and Gromov invariants for flag varieties , · arxiv.org
[23] B. Lian, C.-H. Liu, K. Liu, and S.-T. Yau, “The \(S^1\) fixed points in Quot-schemes and mirror principle computations” in Vector Bundles and Representation Theory (Columbia, Mo., 2002) , Contemp. Math. 322 , Amer. Math. Soc., Providence, 2003. · Zbl 1067.14056
[24] S. Martin, Symplectic quotients by a nonabelian group and by its maximal torus , · arxiv.org
[25] R. Pandharipande, Rational curves on hypersurfaces (after A. Givental) , Astérisque 252 (1998), 307–340., Séminaire Bourbaki 1997/98, no. 848. · Zbl 0932.14029 · numdam:SB_1997-1998__40__307_0 · eudml:110250
[26] B. Siebert and G. Tian, On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator , Asian J. Math. 1 (1997), 679–695. · Zbl 0974.14040
[27] E. Witten,“The Verlinde algebra and the cohomology of the Grassmannian” in Geometry, Topology, and Physics , Conf. Proc. Lecture Notes Geom. Topology 4 , Internat. Press, Cambridge, Mass., 1995, 357–422. · Zbl 0863.53054
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