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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)
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