Kyoto Journal of Mathematics

A Fock sheaf for Givental quantization

Abstract

We give a global, intrinsic, and coordinate-free quantization formalism for Gromov–Witten invariants and their B-model counterparts, which simultaneously generalizes the quantization formalisms described by Witten, Givental, and Aganagic–Bouchard–Klemm. Descendant potentials live in a Fock sheaf, consisting of local functions on Givental’s Lagrangian cone that satisfy the $(3g-2)$-jet condition of Eguchi–Xiong; they also satisfy a certain anomaly equation, which generalizes the holomorphic anomaly equation of Bershadsky–Cecotti–Ooguri–Vafa. We interpret Givental’s formula for the higher-genus potentials associated to a semisimple Frobenius manifold in this setting, showing that, in the semisimple case, there is a canonical global section of the Fock sheaf. This canonical section automatically has certain modularity properties. When $X$ is a variety with semisimple quantum cohomology, a theorem of Teleman implies that the canonical section coincides with the geometric descendant potential defined by Gromov–Witten invariants of $X$. We use our formalism to prove a higher-genus version of Ruan’s crepant transformation conjecture for compact toric orbifolds. When combined with our earlier joint work with Jiang, this shows that the total descendant potential for a compact toric orbifold $X$ is a modular function for a certain group of autoequivalences of the derived category of $X$.

Article information

Source
Kyoto J. Math., Volume 58, Number 4 (2018), 695-864.

Dates
Revised: 17 November 2015
Accepted: 12 December 2016
First available in Project Euclid: 27 July 2018

https://projecteuclid.org/euclid.kjm/1532656825

Digital Object Identifier
doi:10.1215/21562261-2017-0036

Mathematical Reviews number (MathSciNet)
MR3880240

Zentralblatt MATH identifier
07000589

Citation

Coates, Tom; Iritani, Hiroshi. A Fock sheaf for Givental quantization. Kyoto J. Math. 58 (2018), no. 4, 695--864. doi:10.1215/21562261-2017-0036. https://projecteuclid.org/euclid.kjm/1532656825

References

• [1] D. Abramovich, T. Graber, and A. Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), 1337–1398.
• [2] M. Aganagic, V. Bouchard, and A. Klemm, Topological strings and (almost) modular forms, Comm. Math. Phys. 277 (2008), 771–819.
• [3] M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño, and C. Vafa, Topological strings and integrable hierarchies, Comm. Math. Phys. 261 (2006), 451–516.
• [4] M. Alim, E. Scheidegger, S.-T. Yau, and J. Zhou, Special polynomial rings, quasi modular forms and duality of topological strings, Adv. Theor. Math. Phys. 18 (2014), 401–467.
• [5] J. A. Ball and M. W. Raney, Discrete-time dichotomous well-posed linear systems and generalized Schur-Nevanlinna-Pick interpolation, Complex Anal. Oper. Theory 1 (2007), 1–54.
• [6] S. Barannikov, Quantum periods, I: Semi-infinite variations of Hodge structures, Int. Math. Res. Not. IMRN 2001, no. 23, 1243–1264.
• [7] S. Barannikov, Semi-infinite Hodge structures and mirror symmetry for projective spaces, preprint, arXiv:math/0010157v2 [math.AG].
• [8] V. V. Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993), 349–409.
• [9] V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), 493–535.
• [10] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), 45–88.
• [11] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Holomorphic anomalies in topological field theories, Nuclear Phys. B 405 (1993), 279–304.
• [12] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Comm. Math. Phys. 165 (1994), 311–427.
• [13] A. Bertram, I. Ciocan-Fontanine, and B. Kim, Gromov-Witten invariants for abelian and nonabelian quotients, J. Algebraic Geom. 17 (2008), 275–294.
• [14] L. A. Borisov, L. Chen, and G. G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005), 193–215.
• [15] L. A. Borisov and R. P. Horja, Mellin-Barnes integrals as Fourier-Mukai transforms, Adv. Math. 207 (2006), 876–927.
• [16] A. Brini and R. Cavalieri, Crepant resolutions and open strings, II, preprint, arXiv:1407.2571v3 [math.AG].
• [17] A. Brini, R. Cavalieri, and D. Ross, Crepant resolutions and open strings, J. Reine Angew. Math., published electronically 8 April 2017.
• [18] A. Brini and A. Tanzini, Exact results for topological strings on resolved $Y^{p,q}$ singularities, Comm. Math. Phys. 289 (2009), 205–252.
• [19] J. Bryan and T. Graber, “The crepant resolution conjecture” in Algebraic Geometry—Seattle 2005, Part 1, Proc. Sympos. Pure Math. 80, Amer. Math. Soc., Providence, 2009, 23–42.
• [20] S. Cecotti and C. Vafa, Topological–anti-topological fusion, Nuclear Phys. B 367 (1991), 359–461.
• [21] S. Cecotti and C. Vafa, On classification of $N=2$ supersymmetric theories, Comm. Math. Phys. 158 (1993), 569–644.
• [22] S. B. Chae, Holomorphy and Calculus in Normed Spaces, with an appendix by A. E. Taylor, Monogr. Textb. Pure Appl. Math. 92, Dekker, New York, 1985.
• [23] W. Chen and Y. Ruan, “Orbifold Gromov-Witten theory” in Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemp. Math. 310, Amer. Math. Soc., Providence, 2002, 25–85.
• [24] D. Cheong, I. Ciocan-Fontanine, and B. Kim, Orbifold quasimap theory, Math. Ann. 363 (2015), 777–816.
• [25] A. Chiodo, H. Iritani, and Y. Ruan, Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence, Publ. Math. Inst. Hautes Études Sci. 119 (2014), 127–216.
• [26] A. Chiodo and Y. Ruan, Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations, Invent. Math. 182 (2010), 117–165.
• [27] I. Ciocan-Fontanine and B. Kim, Moduli stacks of stable toric quasimaps, Adv. Math. 225 (2010), 3022–3051.
• [28] I. Ciocan-Fontanine and B. Kim, Higher genus quasimap wall-crossing for semipositive targets, J. Eur. Math. Soc. (JEMS) 19 (2017), 2051–2102.
• [29] I. Ciocan-Fontanine and B. Kim, Wall-crossing in genus zero quasimap theory and mirror maps, Algebr. Geom. 1 (2014), 400–448.
• [30] I. Ciocan-Fontanine, B. Kim, and C. Sabbah, The abelian/nonabelian correspondence and Frobenius manifolds, Invent. Math. 171 (2008), 301–343.
• [31] T. Coates, A. Corti, H. Iritani, and H.-H. Tseng, Computing genus-zero twisted Gromov-Witten invariants, Duke Math. J. 147 (2009), 377–438.
• [32] T. Coates, A. Corti, H. Iritani, and H.-H. Tseng, A mirror theorem for toric stacks, Compos. Math. 151 (2015), 1878–1912.
• [33] T. Coates and A. Givental, Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), 15–53.
• [34] T. Coates and H. Iritani, On the convergence of Gromov-Witten potentials and Givental’s formula, Michigan Math. J. 64 (2015), 587–631.
• [35] T. Coates and H. Iritani, On the existence of a global neighbourhood, Glasg. Math. J. 58 (2016), 717–726.
• [36] T. Coates, H. Iritani, and Y. Jiang, The crepant transformation conjecture for toric complete intersections, Adv. Math. 329 (2018), 1002–1087.
• [37] T. Coates, H. Iritani, Y. Jiang, and E. Segal, $K$-theoretic and categorical properties of toric Deligne-Mumford stacks, Pure Appl. Math. Q. 11 (2015), 239–266.
• [38] T. Coates, H. Iritani, and H.-H. Tseng, Wall-crossings in toric Gromov-Witten theory, I: Crepant examples, Geom. Topol. 13 (2009), 2675–2744.
• [39] T. Coates, Y.-P. Lee, A. Corti, and H.-H. Tseng, The quantum orbifold cohomology of weighted projective spaces, Acta Math. 202 (2009), 139–193.
• [40] T. Coates and Y. Ruan, Quantum cohomology and crepant resolutions: A conjecture, Ann. Inst. Fourier (Grenoble) 63 (2013), 431–478.
• [41] K. Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007), 165–214.
• [42] K. Costello, The partition function of a topological field theory, J. Topol. 2 (2009), 779–822.
• [43] K. Costello and S. Li, Quantum BCOV theory on Calabi–Yau manifolds and the higher genus B-model, preprint, arXiv:1201.4501v1 [math.QA].
• [44] L. David and I. A. B. Strachan, $tt^{*}$-geometry on the big phase space, Comm. Math. Phys. 329 (2014), 295–323.
• [45] R. Dijkgraaf and E. Witten, Mean field theory, topological field theory, and multi-matrix models, Nuclear Phys. B 342 (1990), 486–522.
• [46] A. Douai and C. Sabbah, Gauss-Manin systems, Brieskorn lattices and Frobenius structures, I, Ann. Inst. Fourier (Grenoble) 53 (2003), 1055–1116.
• [47] A. Douai and C. Sabbah, “Gauss-Manin systems, Brieskorn lattices and Frobenius structures, II” in Frobenius Manifolds, Aspects Math. E36, Vieweg, Wiesbaden, 2004, 1–18.
• [48] B. Dubrovin, Geometry and integrability of topological-antitopological fusion, Comm. Math. Phys. 152 (1993), 539–564.
• [49] B. Dubrovin, “Geometry of $2$D topological field theories” in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math. 1620, Springer, Berlin, 1996, 120–348.
• [50] B. Dubrovin, “Painlevé transcendents in two-dimensional topological field theory” in The Painlevé Property, CRM Ser. Math. Phys., Springer, New York, 1999, 287–412.
• [51] B. Dubrovin and Y. Zhang, Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants, preprint, arXiv:math/0108160v1 [math.DG].
• [52] T. Eguchi and C.-S. Xiong, Quantum cohomology at higher genus: Topological recursion relations and Virasoro conditions, Adv. Theor. Math. Phys. 2 (1998), 219–229.
• [53] C. Faber, S. Shadrin, and D. Zvonkine, Tautological relations and the $r$-spin Witten conjecture, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 621–658.
• [54] M. Florig and S. J. Summers, Further representations of the canonical commutation relations, Proc. London Math. Soc. (3) 80 (2000), 451–490.
• [55] L. Gårding and A. Wightman, Representations of the commutation relations, Proc. Natl. Acad. Sci. USA 40 (1954), 622–626.
• [56] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Mod. Birkhäuser Class., Birkhäuser, Boston, 2008.
• [57] E. Getzler, “The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants” in Frobenius Manifolds, Aspects Math. E36, Vieweg, Wiesbaden, 2004, 45–89.
• [58] A. B. Givental, “Homological geometry and mirror symmetry” in Proceedings of the International Congress of Mathematicians, Vols. 1–2 (Zürich, 1994), Birkhäuser, Basel, 1995, 472–480.
• [59] A. B. Givental, Homological geometry, I: Projective hypersurfaces, Selecta Math. (N.S.) 1 (1995), 325–345.
• [60] A. B. Givental, “A mirror theorem for toric complete intersections” in Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progr. Math. 160, Birkhäuser Boston, Boston, 1998, 141–175.
• [61] A. B. Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001), 551–568, 645.
• [62] A. B. Givental, Semisimple Frobenius structures at higher genus, Int. Math. Res. Not. IMRN 2001, no. 23, 1265–1286.
• [63] A. B. Givental, $A_{n-1}$ singularities and $n$KdV hierarchies, Mosc. Math. J. 3 (2003), 475–505, 743.
• [64] A. B. Givental, “Symplectic geometry of Frobenius structures” in Frobenius Manifolds, Aspects Math. E36, Vieweg, Wiesbaden, 2004, 91–112.
• [65] P. R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102–112.
• [66] C. Hertling, $tt^{*}$ geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math. 555 (2003), 77–161.
• [67] C. Hertling and Y. Manin, Weak Frobenius manifolds, Int. Math. Res. Not. 1999, no. 6, 277–286.
• [68] C. Hertling and Y. Manin, “Unfoldings of meromorphic connections and a construction of Frobenius manifolds” in Frobenius Manifolds, Aspects Math. E36, Vieweg, Wiesbaden, 2004, 113–144.
• [69] K. Hori and C. Vafa, Mirror symmetry, preprint, arXiv:hep-th/0002222v3.
• [70] H. Iritani, Quantum $D$-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (2006), 577–622.
• [71] H. Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), 1016–1079.
• [72] H. Iritani, “Ruan’s conjecture and integral structures in quantum cohomology” in New Developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry (RIMS, Kyoto, 2008), Adv. Stud. Pure Math. 59, Math. Soc. Japan, Tokyo, 2010, 111–166.
• [73] H. Iritani, Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), 2909–2958.
• [74] H. Iritani, $tt^{*}$-geometry in quantum cohomology, preprint, arXiv:0906.1307v1 [math.DG].
• [75] L. Katzarkov, M. Kontsevich, and T. Pantev, “Hodge theoretic aspects of mirror symmetry” in From Hodge Theory to Integrability and TQFT $tt^{*}$-Geometry, Proc. Sympos. Pure Math. 78, Amer. Math. Soc., Providence, 2008, 87–174.
• [76] L. Katzarkov, M. Kontsevich, and T. Pantev, personal communication, March 2012.
• [77] Y. Kawamata, Log crepant birational maps and derived categories, J. Math. Sci. Univ. Tokyo 12 (2005), 211–231.
• [78] A. A. Kirillov, “Geometric quantization” in Dynamical Systems, IV, Encyclopaedia Math. Sci. 4, Springer, Berlin, 2001, 139–176.
• [79] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1–23.
• [80] M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525–562.
• [81] M. Kontsevich and Y. Manin, Relations between the correlators of the topological sigma-model coupled to gravity, Comm. Math. Phys. 196 (1998), 385–398.
• [82] M. Kontsevich and Y. Soibelman, “Notes on $A_{\infty}$-algebras, $A_{\infty}$-categories and non-commutative geometry” in Homological Mirror Symmetry, Lecture Notes in Phys. 757, Springer, Berlin, 2009, 153–219.
• [83] A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1–31.
• [84] M. Krawitz and Y. Shen, Landau-Ginzburg/Calabi-Yau correspondence of all genera for elliptic orbifold $\mathbb{P}^{1}$, preprint, arXiv:1106.6270v1 [math.AG].
• [85] Y.-P. Lee, Invariance of tautological equations, I: Conjectures and applications, J. Eur. Math. Soc. (JEMS) 10 (2008), 399–413.
• [86] Y.-P. Lee, Invariance of tautological equations, II: Gromov-Witten theory, with an appendix by Y. Iwao and Y.-P. Lee, J. Amer. Math. Soc. 22 (2009), 331–352.
• [87] C. Li, S. Li, K. Saito, and Y. Shen, Mirror symmetry for exceptional unimodular singularities, J. Eur. Math. Soc. (JEMS) 19 (2017), 1189–1229.
• [88] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), 119–174.
• [89] S. Li, BCOV theory on the elliptic curve and higher genus mirror symmetry, preprint, arXiv:1112.4063v1 [math.QA].
• [90] S. Li, Variation of Hodge structures, Frobenius manifolds, and gauge theory, preprint, arXiv:1303.2782v1 [math.QA].
• [91] Y. I. Manin, Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, Amer. Math. Soc. Colloq. Publ. 47, Amer. Math. Soc., Providence, 1999.
• [92] T. Milanov, The equivariant Gromov-Witten theory of $\mathbb{C}\mathrm{P}^{1}$ and integrable hierarchies, Int. Math. Res. Not. IMRN 2008, no. 21, art. ID rnn 073.
• [93] T. Milanov, Analyticity of the total ancestor potential in singularity theory, Adv. Math. 255 (2014), 217–241.
• [94] T. Milanov and Y. Ruan, Gromov–Witten theory of elliptic orbifold $\mathbb{P}^{1}$ and quasi-modular forms, preprint, arXiv:1106.2321v1 [math.AG].
• [95] T. Milanov, Y. Ruan, and Y. Shen, Gromov-Witten theory and cycle-valued modular forms, J. Reine Angew. Math. 735 (2018), 287–315.
• [96] J. Morava, “Heisenberg groups and algebraic topology” in Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press, Cambridge, 2004, 235–246.
• [97] R. Pandharipande, Rational curves on hypersurfaces (after A. Givental), Astérisque 252 (1998), 307–340, Séminaire Bourbaki 1997/1998, no. 848.
• [98] R. Pandharipande, A. Pixton, and D. Zvonkine, Relations of $\overline{M}_{g,n}$ via $3$-spin structures, J. Amer. Math. Soc. 28 (2015), 279–309.
• [99] T. Reichelt, A construction of Frobenius manifolds with logarithmic poles and applications, Comm. Math. Phys. 287 (2009), 1145–1187.
• [100] T. Reichelt and C. Sevenheck, Logarithmic Frobenius manifolds, hypergeometric systems and quantum $\mathcal{D}$-modules, J. Algebraic Geom. 24 (2015), 201–281.
• [101] C. Sabbah, Hypergeometric period for a tame polynomial, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 603–608.
• [102] C. Sabbah, Isomonodromic Deformations and Frobenius Manifolds, Universitext, Springer, London, 2007.
• [103] C. Sabbah, Fourier-Laplace transform of a variation of polarized complex Hodge structure, J. Reine Angew. Math. 621 (2008), 123–158.
• [104] K. Saito, Period mapping associated to a primitive form, Publ. Res. Inst. Math. Sci. 19 (1983), 1231–1264.
• [105] M. Saito, On the structure of Brieskorn lattice, Ann. Inst. Fourier (Grenoble) 39 (1989), 27–72.
• [106] M. Sato and Y. Sato, “Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold” in Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), North-Holland Math. Stud. 81, North-Holland, Amsterdam, 1983, 259–271.
• [107] G. Segal and G. Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5–65.
• [108] T. P. Srinivasan, Doubly invariant subspaces, Pacific J. Math. 14 (1964), 701–707.
• [109] C. Teleman, The structure of 2D semi-simple field theories, Invent. Math. 188 (2012), 525–588.
• [110] H.-H. Tseng, Orbifold quantum Riemann-Roch, Lefschetz and Serre, Geom. Topol. 14 (2010), 1–81.
• [111] E. Witten, “Two-dimensional gravity and intersection theory on moduli space” in Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, Penn., 1991, 243–310.
• [112] E. Witten, Quantum background independence in string theory, preprint, arXiv:hep-th/9306122v1.
• [113] N. M. J. Woodhouse, Geometric Quantization, 2nd ed., Oxford Math. Monogr., Clarendon Press, Oxford Univ. Press, New York, 1992.