BCFG Drinfeld-Sokolov hierarchies and FJRW-theory. (English) Zbl 1333.14053
E. Witten [in: Surveys in differential geometry. Vol. I: Proceedings of the conference on geometry and topology, held at Harvard University, Cambridge, MA, USA, April 27-29, 1990. Providence, RI: American Mathematical Society; Bethlehem, PA: Lehigh University. 243–310 (1991; Zbl 0757.53049)] gave a remarkable conjecture relating the intersection theory of the (Deligne-Mumford compactified) moduli space \(\overline{\mathcal{M}}_{g,k}\) of genus \(g\) Riemann surfaces with \(k\) punctures and integrable systems in KdV hieracrchies. This was later proved by M. Kontsevich [Commun. Math. Phys. 147, No. 1, 1–23 (1992; Zbl 0756.35081)]. More precisely, the generating (partition) function \(\exp\left(\mathcal{F}\right)\) where \( \mathcal{F} = \sum\limits_{g=0}^\infty \hbar^{g-1} \sum\limits_{k=0}^\infty \frac{t_{l_1} \ldots t_{l_k}}{k!} I_g \) with intersection numbers \(I_g = \left< \tau_{l_1}, \ldots, \tau_{l_k} \right>_g = \int_{\overline{\mathcal{M}}_{g,k}} \prod_i \psi_i^{l_i}\) for \(\psi_i \in H^{2i}(\overline{\mathcal{M}}_{g,k})\) being the first Chern class of certain line-bundles, should be a tau-function of the KdV hierarchy.
A host of activities ensued in the past 2 decades, in various generalization relating the intersection theory, especially in light of Gromov-Witten invariants and other integrable hierarchies, by A. Okounkov and R. Pandharipande [Ann. Math. (2) 163, No. 2, 561–605 (2006; Zbl 1105.14077)], B. Dubrovin and Y. Zhang [Commun. Math. Phys. 198, No. 2, 311–361 (1998; Zbl 0923.58060)], E. Getzler [in: Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14–18, 2000. Singapore: World Scientific. 51–79 (2001; Zbl 1047.37046)], Fan-Jarvis-Ruan (FJRW) [H. Fan et al., Ann. Math. (2) 178, No. 1, 1–106 (2013; Zbl 1310.32032)] et al. This correspondence can be thought of a manifestation of mirror symmetry with the geometry side playing the role of the A-model and the integrable side, the B-model.
Of particular note is the beautiful result of FJRW that the Drinfeld-Sokolov hierarchy for affine simply-laced Lie algebras ADE corresponds to geometric orbifolds of ADE-type. However, the Saito-Givental-Dubronvin-Zhang partition functions of the non-simply-laced BCGF singularities have been shown not to be tau-functions of the corresponding Drinfeld-Sokolov hierarchy. The purpose of the current paper is to nicely complete this story (cf. Theorems 1.3 and 1.4): a correction is needed. In particular, the partition function of the \(\Gamma\)-invariant sectors of \(A,D^T,E_6\) (where the \(D^T\) is a mirror version of the D-type singularity) FJRW theory with maximal diagonal symmetry group is a tau-function of the corresponding \(B,C,F_4\) Drinfeld-Sokolov hierarchy (note the foldings of the respective Dynkin diagrams). Moreover, the partition function of the \(\mathbb{Z}_3\)-invariant sector of a \(D_4\) FJRW theory gives that of the \(G_2\) case.
A host of activities ensued in the past 2 decades, in various generalization relating the intersection theory, especially in light of Gromov-Witten invariants and other integrable hierarchies, by A. Okounkov and R. Pandharipande [Ann. Math. (2) 163, No. 2, 561–605 (2006; Zbl 1105.14077)], B. Dubrovin and Y. Zhang [Commun. Math. Phys. 198, No. 2, 311–361 (1998; Zbl 0923.58060)], E. Getzler [in: Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14–18, 2000. Singapore: World Scientific. 51–79 (2001; Zbl 1047.37046)], Fan-Jarvis-Ruan (FJRW) [H. Fan et al., Ann. Math. (2) 178, No. 1, 1–106 (2013; Zbl 1310.32032)] et al. This correspondence can be thought of a manifestation of mirror symmetry with the geometry side playing the role of the A-model and the integrable side, the B-model.
Of particular note is the beautiful result of FJRW that the Drinfeld-Sokolov hierarchy for affine simply-laced Lie algebras ADE corresponds to geometric orbifolds of ADE-type. However, the Saito-Givental-Dubronvin-Zhang partition functions of the non-simply-laced BCGF singularities have been shown not to be tau-functions of the corresponding Drinfeld-Sokolov hierarchy. The purpose of the current paper is to nicely complete this story (cf. Theorems 1.3 and 1.4): a correction is needed. In particular, the partition function of the \(\Gamma\)-invariant sectors of \(A,D^T,E_6\) (where the \(D^T\) is a mirror version of the D-type singularity) FJRW theory with maximal diagonal symmetry group is a tau-function of the corresponding \(B,C,F_4\) Drinfeld-Sokolov hierarchy (note the foldings of the respective Dynkin diagrams). Moreover, the partition function of the \(\mathbb{Z}_3\)-invariant sector of a \(D_4\) FJRW theory gives that of the \(G_2\) case.
Reviewer: Yang-Hui He (London)
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 81T45 | Topological field theories in quantum mechanics |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
| 14H10 | Families, moduli of curves (algebraic) |
Citations:
Zbl 0757.53049; Zbl 0756.35081; Zbl 1105.14077; Zbl 0923.58060; Zbl 1047.37046; Zbl 1310.32032References:
| [1] | Balog, J., Fehér, L., O’Raifeartaigh, L., Forgács, P., Wipf, A.: Toda theory and \[WW\]-algebra from a gauged WZNW point of view. Ann. Phys. 203(1990), 76-136 (1990) · Zbl 0725.58046 · doi:10.1016/0003-4916(90)90029-N |
| [2] | Buryak, A., Posthuma, H., Shadrin, S.: On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket. J. Geom. Phys. 62, 1639-1651 (2012) · Zbl 1242.53113 · doi:10.1016/j.geomphys.2012.03.006 |
| [3] | Buryak, A., Posthuma, H., Shadrin, S.: A polynomial bracket for the Dubrovin-Zhang hierarchies. J. Diff. Geom. 92, 153-185 (2012) · Zbl 1259.53079 |
| [4] | Chiodo, A., Iritani, H., Ruan, Y.: Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence. Publ. Math. Inst. Hautes tudes Sci. 119, 127-216 (2014) · Zbl 1298.14042 · doi:10.1007/s10240-013-0056-z |
| [5] | Drinfeld, V., Sokolov, V.: Lie algebras and equations of Korteweg-de Vries type. J. Soviet Math. 30, 1975-2036 (1985). Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya) 24, 81-180 (1984) · Zbl 1221.37132 |
| [6] | Dubrovin, B.: Integrable systems and classification of 2-dimensional topological field theories. In: Integrable Systems (Luminy, 1991), pp. 313-359. Progress in Mathematics, vol. 115, Birkhäuser, Boston (1993) · Zbl 0824.58029 |
| [7] | Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable systems and Quantum Groups (Montecatini Terme, 1993), pp. 120-348. Lecture Notes in Mathematics, vol. 1620, Springer, Berlin (1996) · Zbl 0841.58065 |
| [8] | Dubrovin, B., Liu, S.-Q., Zhang, Y.: On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasi-triviality of bi-Hamiltonian perturbations. Comm. Pure Appl. Math. 59, 559-615 (2006) · Zbl 1108.35112 |
| [9] | Dubrovin, B., Liu, S.-Q., Zhang, Y.: Frobenius manifolds and central invariants for the Drinfeld-Sokolov bihamiltonian structures. Adv. Math. 219, 780-837 (2008) · Zbl 1153.37032 · doi:10.1016/j.aim.2008.06.009 |
| [10] | Dubrovin, B., Zhang, Y.: Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation. Comm. Math. Phys. 198, 311-361 (1998) · Zbl 0923.58060 · doi:10.1007/s002200050480 |
| [11] | Dubrovin, B., Zhang, Y.: Frobenius manifolds and Virasoro constraints. Selecta Math. (N.S.) 5, 423-466 (1999) · Zbl 0963.81066 |
| [12] | Dubrovin, B., Zhang, Y.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. eprint arXiv:math/0108160 · Zbl 1221.35458 |
| [13] | Dubrovin, B., Zhang, Y.: Virasoro Symmetries of the Extended Toda Hierarchy. Comm. Math. Phys. 250, 161-193 (2004) · Zbl 1071.37054 · doi:10.1007/s00220-004-1084-9 |
| [14] | Faber, C., Shadrin, S., Zvonkine, D.: Tautological relations and the r-spin Witten conjecture. Ann. Sci. Éc. Norm. Supér. 43, 621-658 (2010) · Zbl 1203.53090 |
| [15] | Fan, H., Jarvis, T., Ruan, Y.: The Witten equation and its virtual fundamental cycle. eprint arXiv:math/0712.4025 · Zbl 0756.35081 |
| [16] | Fan, H., Jarvis, T., Ruan, Y.: The Witten equation, mirror symmetry and quantum singularity theory. Ann. Math. 178, 1-106 (2013) · Zbl 1310.32032 |
| [17] | Fan, H., Francis, A., Jarvis, T., Merrell, E., Ruan, Y.: Witten’s \[D_4\] D4 Integrable hierarchies conjecture. eprint arXiv:1008.0927 · Zbl 1342.14110 |
| [18] | Frenkel, E., Givental, A., Milanov, T.: Soliton equations, vertex operators, and simple singularities. Funct. Anal. Other Math. 3, 47-63 (2010) · Zbl 1203.37108 · doi:10.1007/s11853-010-0035-6 |
| [19] | Getzler, E.: The Toda conjecture. In: Symplectic Geometry and Mirror Symmetry (Seoul, 2000), pp. 51-79. World Scientific Publishing, River Edge (2001) · Zbl 1047.37046 |
| [20] | Givental, A.: Semisimple Frobenius structures at higher genus. Internat. Math. Res. Notices 2001(23), 1265-1286 (2001) · Zbl 1074.14532 · doi:10.1155/S1073792801000605 |
| [21] | Givental, A.: Gromov-Witten invariants and quantization of quadratic Hamiltonians. Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. Mosc. Math. J. 1, 551-568, 645 (2001) · Zbl 1008.53072 |
| [22] | Givental, A., Milanov, T.: Simple singularities and integrable hierarchies. In: The Breadth of Symplectic and Poisson Geometry. Progress in Mathematics, vol. 232, pp. 173-201. Birkhauser Boston, Boston (2005) · Zbl 1075.37025 |
| [23] | Hollowood, T., Miramontes, J.: Tau-functions and generalized integrable hierarchies. Comm. Math. Phys. 157, 99-117 (1993) · Zbl 0796.35144 · doi:10.1007/BF02098021 |
| [24] | Johnson, P.: Equivariant Gromov-Witten theory of one dimensional stacks. eprint arXiv:0903.1068 · Zbl 0853.14020 |
| [25] | Kac, V.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990) · Zbl 0716.17022 · doi:10.1017/CBO9780511626234 |
| [26] | Kac, V., Wakimoto, M.: Exceptional hierarchies of soliton equations. In: Theta Functions-Bowdoin 1987, Part 1 (Brunswick, ME, 1987). Proceedings of Symposia in Pure Mathematics, vol. 49, pp. 191-237. American Mathematical Society, Providence (1989) · Zbl 0691.17014 |
| [27] | Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys. 147, 1-23 (1992) · Zbl 0756.35081 · doi:10.1007/BF02099526 |
| [28] | Kontsevich, M., Manin, Y.: Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys. 164, 525-562 (1994) · Zbl 0853.14020 · doi:10.1007/BF02101490 |
| [29] | Krawitz, M.: FJRW rings and Landau-Ginzburg mirror symmetry, eprint arXiv:0906.0796 · Zbl 1250.81087 |
| [30] | Liu, S.-Q., Wu, C.-Z., Zhang, Y.: On the Drinfeld-Sokolov hierarchies of D type. Int. Math. Res. Notices 2011, 1952-1996 (2011) · Zbl 1221.35458 |
| [31] | Liu, S.-Q., Yang, D., Zhang, Y.: Uniqueness theorem of \[{\cal{W}}W\]-constraints for simple singularities. Lett. Math. Phys. 103, 1329-1345 (2013) · Zbl 1285.53080 · doi:10.1007/s11005-013-0643-4 |
| [32] | Liu, S.-Q., Zhang, Y.: Deformations of semisimple bihamiltonian structures of hydrodynamic type. J. Geom. Phys. 54, 427-453 (2005) · Zbl 1079.37058 · doi:10.1016/j.geomphys.2004.11.003 |
| [33] | Liu, S.-Q., Zhang, Y.: Jacobi structures of evolutionary partial differential equations. Adv. Math. 227, 73-130 (2011) · Zbl 1221.37132 · doi:10.1016/j.aim.2011.01.015 |
| [34] | Milanov, T., Tseng, H.-H.: Equivariant orbifold structures on the projective line and integrable hierarchies. Adv. Math. 226, 641-672 (2011) · Zbl 1222.14121 · doi:10.1016/j.aim.2010.07.004 |
| [35] | Milanov, T.: Analyticity of the total ancestor potential in singularity theory. Adv. Math. 255, 217-241 (2014) · Zbl 1295.14051 · doi:10.1016/j.aim.2014.01.009 |
| [36] | Okounkov, A., Pandharipande, R.: The equivariant Gromov-Witten theory of \[P^1\] P1. Ann. Math. 163, 561-605 (2006) · Zbl 1105.14077 · doi:10.4007/annals.2006.163.561 |
| [37] | Rossi, P.: Gromov-Witten theory of orbicurves, the space of tri-polynomials and symplectic field theory of Seifert fibrations. Math. Ann. 348, 265-287 (2010) · Zbl 1235.14053 · doi:10.1007/s00208-009-0471-0 |
| [38] | Saito, K.: On a linear structure of the quotient variety by a finite reflexion group. Publ. Res. Inst. Math. Sci. 29, 535-579 (1993) · Zbl 0828.15002 · doi:10.2977/prims/1195166742 |
| [39] | Saito, K.: Primitive forms for a universal unfolding of a function with an isolated critical point. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3) (1981), 775-792 (1982) · Zbl 0523.32015 |
| [40] | Teleman, C.: The structure of 2D semi-simple field theories. Invent. Math. 188, 525-588 (2012) · Zbl 1248.53074 · doi:10.1007/s00222-011-0352-5 |
| [41] | Witten, E.: Two-dimensional gravity and intersection theory on the moduli space. In: Surveys in Differential Geometry (Cambridge, MA, 1990), pp. 243-310. Lehigh University, Bethlehem (1991) · Zbl 0757.53049 |
| [42] | Witten, E.: Algebraic geometry associated with matrix models of two-dimensional gravity. In: Topological Models in Modern Mathematics (Stony Brook, NY, 1991) pp. 235-269. Publish or Perish, Houston (1993) · Zbl 0812.14017 |
| [43] | Witten, E.: Private communication |
| [44] | Wu, C.-Z.: A remark on Kac-Wakimoto hierarchies of D-type. J. Phys. A 43, 035201, 8 pp (2010) · Zbl 1182.35195 |
| [45] | Wu, C.-Z.: Tau functions and Virasoro symmetries for Drinfeld-Sokolov hierarchie. eprint arXiv:1203.5750 · Zbl 1369.37074 |
| [46] | Zhang, Y.: On the \[CP^1\] CP1 topological sigma model and the Toda lattice hierarchy. J. Geom. Phys. 40, 215-232 (2002) · Zbl 1001.37066 · doi:10.1016/S0393-0440(01)00036-5 |
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