Spin Gromov-Witten invariants.

*(English)*Zbl 1094.14042In 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.

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.

Reviewer: Edward Lee (Los Angeles)

##### MSC:

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

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\textit{T. J. Jarvis} et al., Commun. Math. Phys. 259, No. 3, 511--543 (2005; Zbl 1094.14042)

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