×

zbMATH — the first resource for mathematics

Spin Gromov-Witten invariants. (English) Zbl 1094.14042
In this paper, the authors construct the Gromov-Witten classes associated to the moduli space of maps of stable curves with a \(r\)-spin structure into a projective variety \(V\) and show that they satisfy axioms analogous to the famous M. Kontsevich, Y. Manin axioms [Commun. Math. Phys. 164, No.3, 525–562 (1994; Zbl 0853.14020)] for a cohomological field theory (CohFT).
The moduli space of \(n\)-pointed curves of genus \(g\) with a \(r\)-spin structure (that is, with a choice of \(r\)th root of the sheaf obtained from twisting the canonical sheaf by some negative linear combination of the marked points in order to make the degree divisible by \(r\)) was constructed by the first author in [Int. J. Math. 1, No. 5, 637–663 (2000; Zbl 1094.14504)]. To be more precise, note that this is a disjoint union of spaces indexed by \(\mathbf{m}\), where \(\mathbf{m}\) runs over all \(n\)-vectors of integers whose entries lie between 0 and \(r-1\). This is because the \(r\)-spin structure depends on a choice of twisting of the canonical sheaf of the underlying curve, but increasing the multiplicity of the twisting of the \(i\)th marked point by \(r\) leads to spaces which are canonically isomorphic. The \(r\)-spin virtual class \(\tilde{c}^{1/r}\) was constructed by the authors [Compos. Math. 126, No. 2, 157–212 (2001; Zbl 1015.14028)] in order to facilitate the construction of an intersection theory on the moduli space. There, the authors show that the \(r\)-spin virtual class plays the same role as the virtual fundamental class in the theory of moduli of stable maps [K. Behrend and B. Fantechi, Invent. Math. 128 No. 1, 45–88 (1997; Zbl 0909.14006)] and formulated and proved a generalization of the Witten conjecture. The first main task of the present paper is to construct the Deligne-Mumford stack \(\overline{\mathcal{M}}_{g,n}^{1/r}(V,\beta)\), again a disjoint union indexed by \(\mathbf{m}\). With this heavy task complete, the authors then define the virtual fundamental class \([\overline{\mathcal{M}}_{g,n}^{1/r}(V,\beta)]^{\text{virt}}\) in \(H_*(\overline{\mathcal{M}}_{g,n}^{1/r}(V,\beta)\) as the pullback of the usual fundamental class under the forgetful morphism \(\widetilde{p}: \overline{\mathcal{M}}_{g,n}^{1/r}(V,\beta) \rightarrow \overline{\mathcal{M}}_{g,n}(V,\beta)\). They also define the tautological cotangent line classes \(\psi_i\) coming from the data of the \(i\)th marked point. With all these ingredients in place, the authors then define the spin Gromov-Witten classes as the operations which take as input a collection of cohomology classes \(\gamma_i\) in \(H^*(V,\mathbb{C})\), pull them back via the evaluation maps ev\(_i\), take their common intersection with the spin virtual class, evaluate the resulting cohomology class in \(\overline{\mathcal{M}}_{g,n}^{1/r}(V,\beta)\) against the spin virtual fundamental class, and push forward the resulting homology class to \(\overline{\mathcal{M}}_{g,n}\). They then show that the spin Gromov-Witten classes satisfy a set of axioms analagous to the usual Kontsevich-Manin axioms for ordinary Gromov-Witten theory. In more condensed language, the CohFT of the spin Gromov-Witten classes is the tensor product of the CohFT associated to the usual Gromov-Witten theory of \(V\) and the \(r\)-spin CohFT. The authors also define spin gravitational correlators by inserting cotangent classes into the definition of the spin Gromov-Witten classes. When \(g = 0\), they show that different choices of \(\mathbf{m}\) which differ by a multiple of \(r\) yield spin virtual classes which are related by multiplication by the cotangent classes \(\psi_i\), as conjectured by the authors [in: Advances in algebraic geometry motivated by Physics. Proc. AMS sess. Lowell 2000, Contemp. Math. 276, 167–177 (2001; Zbl 0986.81105)]; this explains the appearance of the \(\psi\) classes in the definition of gravitational descendents. After restricting to genus zero, the authors also obtain the notion of \(r\)-spin quantum cohomology of \(V\), whose Frobenius structure is isomorphic to the tensor product of the Frobenius manifolds corresponding to the quantum cohomology of \(V\) and the \(r\)th Gelfand-Dickey hierarchy (or equivalently the \(A_{r-1}\) singularity). Finally, the authors show that the \(2\)-spin Gromov-Witten theory essentially reduces to the usual theory, and compute the spin gravitational correlators when \(g = 0, r = 3\) and \(V = \mathbb{P}^1\) by using the Kontsevich-Manin axioms to obtain a recursion relation.

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Abramovich D., Jarvis T.: Moduli of twisted spin curves. Proc. Amer. Math. Soc. 131, no. 3, 685–699 (2002) · Zbl 1037.14008
[2] Abramovich D., Vistoli A.: Compactifying the space of stable maps. J. Amer. Math. Soc. 15, no. 1. 27–75 (2001) · Zbl 0991.14007
[3] Behrend K.: Gromov-Witten invariants in algebraic geometry. Invent. Math. 127, 601–617 (1997) · Zbl 0909.14007
[4] Behrend K.: The product formula for Gromov-Witten invariants. J. Alg. Geom. 8, 529–541 (1999) · Zbl 0938.14032
[5] Behrend K., Manin Yu., Stacks of stable maps and Gromov-Witten invariants. Duke Math. J. 85, 1–60 (1996) · Zbl 0872.14019
[6] Chen W., Ruan Y.: A new cohomology theory for orbifold. Commun. Math. Phys. 248, no. 1, 1–31 (2004) · Zbl 1063.53091
[7] Dubrovin B.: Geometry of 2D topological field theories. In: ”Integrable Systems and Quantum Groups,” Lecture Notes in Math. 1620, Berlin: Springer-Verlag, 1996 · Zbl 0841.58065
[8] Grothendieck A., Dieudonné J.: Éléments de Géométrie Algébrique IV: Étude Locale des Schémas et des Morphismes de Schémas. Volume 28. Paris: Publications Mathématiques IHES, 1966
[9] Hartshorne R.: Algebraic Geometry. New York: Springer-Verlag, 1977 · Zbl 0367.14001
[10] Hitchin N.: Frobenius manifolds. In: ”Gauge Theory and Symplectic Geometry (Montreal, 1995),” J. Hurtubise et al. (eds.), NATO Adv. Sci. Inst. Series C 488, Dordrecht: Kluwer Publ., 1997, pp. 69–112. · Zbl 0867.53027
[11] Jarvis T. J.: Geometry of the moduli of higher spin curves. Internat. J. of Math. 11, 637–663 (2000) · Zbl 1094.14504
[12] Jarvis T. J.: Torsion-free sheaves and moduli of generalized spin curves. Compositio Math. 110, 291–333 (1998) · Zbl 0912.14010
[13] Jarvis T. J.: Picard group of the moduli of higher spin curves. New York J. Math. 7, 23–47 (2001) · Zbl 0977.14010
[14] Jarvis T. J.: Compactification of the universal Picard over the moduli of stable curves. Math. Zeitschrift 235, 123–149 (2000) · Zbl 0980.14020
[15] Jarvis T., Kimura T., Vaintrob A.: Gravitational descendants and the moduli space of higher spin curves. In: E. Previato (ed.), Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000), Contemporary Mathematics 276, Providence, RI: AMS, 2001, pp. 167–177 · Zbl 0986.81105
[16] Jarvis T., Kimura T., Vaintrob A.: Moduli spaces of higher spin curves and integrable hierarchies. Compositio Math. 126, no. 2, 157–212 (2001) · Zbl 1015.14028
[17] Jarvis T., Kimura T., Vaintrob A.: Tensor products of Frobenius manifolds and moduli spaces of higher spin curves. In: ”Conferénce de Moshé Flato 1999, Vol. 2,” G. Dito, D. Sternheimer (eds.), Dordrecht: Kluwer, 2000, pp. 145–166 · Zbl 0988.81120
[18] Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1–23 (1992) · Zbl 0756.35081
[19] Kontsevich M., Manin Yu. I.: Gromov-Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164, 525–562 (1994) · Zbl 0853.14020
[20] Kontsevich M., Manin Yu. I.: Relations between the correlators of the topological sigma-model coupled to gravity. Comm. Math. Phys. 196, no. 2, 385–398 (1998) · Zbl 0946.14032
[21] Kontsevich M., Manin Yu. I.: (with Appendix by R. Kaufmann): Quantum cohomology of a product. Invent. Math. 124, 313–340 (1996) · Zbl 0853.14021
[22] Manin Yu. I.: ”Frobenius manifolds, quantum cohomology, and moduli spaces.” Providence, RI: Amer. Math. Soc. 1999 · Zbl 0952.14032
[23] Manin Yu. I.: Three constructions of Frobenius manifolds: a comparative study. Asian J. Math. 3, 179–220 (1999) · Zbl 0992.53064
[24] Mochizuki T.: The virtual class of the moduli stack of r-spin curves. Preprint, December 2001 · Zbl 1136.14015
[25] Mumford D.: Towards an enumerative geometry of the moduli space of curves. In: ”Arithmetic and Geometry,” eds. M. Artin, J. Tate, Part II, Progress in Math., Vol. 36, Birkhäuser, 1983, pp. 271–328
[26] Polishchuk A.: Witten’s top Chern class on the moduli space of higher spin curves. Frobenius manifolds, Aspects Math., Vieweg, Wiesbaden, E36, 253–264 (2004) · Zbl 1105.14010
[27] Polishchuk A., Vaintrob A.: Algebraic construction of Witten’s top Chern class. In: E. Previato (ed.), Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000), Contemporary Mathematics 276, Providence, RI: AMS, 2001, pp. 229–249 · Zbl 1051.14007
[28] Vistoli A.: Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math. 97, 613–670, (1989) · Zbl 0694.14001
[29] Witten E.: Algebraic geometry associated with matrix models of two dimensional gravity. In: Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Houston: Publish or Perish, 1993, pp. 235–269 · Zbl 0812.14017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.