Orbifold Gromov-Witten theory.

*(English)*Zbl 1091.53058
Adem, Alejandro (ed.) et al., Orbifolds in mathematics and physics. Proceedings of a conference on mathematical aspects of orbifold string theory, Madison, WI, USA, May 4–8, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2990-4/pbk). Contemp. Math. 310, 25-85 (2002).

Inspired by orbifold string theory [L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261, No. 4, 678–687 (1985) and Nucl. Phys. B 274, No. 2, 285–314 (1986)], the authors show how the Gromov-Witten theory can be extended to the context of orbifolds. The main obstacle to such an extension is the fact that the pull-back of an orbifold bundle \(E\to X\) along an orbifold map \(f: X\to Y\) is not an orbifold bundle in general. Moreover, even when \(f^*E\) admits an orbifold bundle structure, this needs not be unique. To overcome this problems, Chen and Ruan introduce the notions of compatible system and of good orbifold map.

More precisely, a good map is an orbifold map \(f: X\to Y\) such that the pull-back bundles \(f^*E\) admit at least an orbifold bundle structure; the additional datum needed to fix an isomorphism class of orbifold bundle structures among the possible ones on \(f^*E\) is precisely what Chen and Ruan call a compatible system of \(f\). If \((X,J)\) is an almost complex orbifold, orbifold stable maps from a (nodal) Riemann surface \(\Sigma\) to \(X\) are defined as pairs \((\tilde\varphi,\xi)\) where \(\tilde\varphi\) is a good map whose underlying map \(\varphi: \Sigma\to X\) is a pseudo-holomorphic map satisfying the usual stability conditions, and \(\xi\) is a compatible system of \(\tilde\varphi\). What makes this a good definition is the following finiteness result: let \(\Sigma\) be a genus \(g\) Riemann surface with \(k\) marked points; then, for any pseudo-holomorphic map \(\varphi: \Sigma\to X\), there are finitely many orbifold structures on \(\Sigma\) whose singular set is contained in the set of marked points of \(\Sigma\), and for each of these orbifold structures there are finitely many pairs \((\tilde\varphi,\xi)\) where \(\tilde\varphi\) is a good map whose underlying map is \(\varphi\), and \(\xi\) is an isomorphism class of compatible systems of \(\tilde\varphi\). The total number is bounded from above by a constant depending only on \(X\), \(g\) and \(k\).

Evaluation of a stable map \(\varphi\) at a point \(z\) of \(\Sigma\) determines a point \(\varphi(z)\) in \(X\) together with a conjugacy class in the local group \(G_{\varphi(z)}\). That is, evaluation at \(z\) actually takes its values in the space \(\tilde{X}=\{(p,(g)^{}_{G_p})\,\mid p\in X, g\in G_p\}\). The connected components of \(\tilde{X}\) are called the twisted sectors of the orbifold \(X\). If \({\mathbf x}\) is a connected component of \(\tilde{X}^k\), then a \(k\)-pointed stable map \(\varphi: \Sigma\to X\) is said to be of type \({\mathbf x}\) if the evaluation of \(\varphi\) at the marked points of \(\Sigma\) lies in \({\mathbf x}\).

Let now \(X\) be a symplectic orbifold with a compatible almost complex structure \(J\) or a projective orbifold with an integrable complex structure \(J\). In these hypothesis Chen and Ruan show that the moduli space \(\overline{\mathcal M}_{g,k}(X,J,A,\mathbf{x})\) of \(k\)-pointed genus \(g\) stable curves \(\varphi: \Sigma\to X\) of type \({\mathbf x}\) and such that \(\varphi_*([\Sigma])=A\in H_2(X;{\mathbb Z})\) is a compact metrizable space, which comes equipped with a map \(\overline{\mathcal M}_{g,k}(X,J,A,{\mathbf x})\to \overline{\mathcal M}_{g,k}\) defined by forgetting the stable map and contracting the unstable components of the domain. Orbifold Gromov-Witten invariants are then defined by means of the moduli spaces \(\overline{\mathcal M}_{g,k}(X,J,A,{\mathbf x})\) as in the classical Gromov-Witten theory and is shown that the orbifold GW-invariants satisfy the same axioms as the ordinary GW-invariants, except the divisor axiom which only holds if one restricts to a divisor class in the nontwisted sector. As in the smooth case, orbifold GW-invariants are independent of the almost complex structure \(J\) and are therefore invariants of the symplectic structure. Moreover, genus zero orbifold GW-invariants define an orbifold quantum cohomology, which is an associative deformation of the orbifold cohomology constructed in [W. Chen and Y. Ruan, Commun. Math. Phys. 248, No. 1, 1–31 (2001; Zbl 1063.53091)].

The appendix to the paper provides an excellent introduction to the general theory of orbifolds, with rigorous definitions, detailed proofs and clear examples.

The notion of orbifold stable maps in the context of algebraic stacks has been independently introduced and studied by [D. Abramovich and A. Vistoli [J. Am. Math. Soc. 15, No. 1, 27–75 (2002; Zbl 0991.14007)].

For the entire collection see [Zbl 1003.00015].

More precisely, a good map is an orbifold map \(f: X\to Y\) such that the pull-back bundles \(f^*E\) admit at least an orbifold bundle structure; the additional datum needed to fix an isomorphism class of orbifold bundle structures among the possible ones on \(f^*E\) is precisely what Chen and Ruan call a compatible system of \(f\). If \((X,J)\) is an almost complex orbifold, orbifold stable maps from a (nodal) Riemann surface \(\Sigma\) to \(X\) are defined as pairs \((\tilde\varphi,\xi)\) where \(\tilde\varphi\) is a good map whose underlying map \(\varphi: \Sigma\to X\) is a pseudo-holomorphic map satisfying the usual stability conditions, and \(\xi\) is a compatible system of \(\tilde\varphi\). What makes this a good definition is the following finiteness result: let \(\Sigma\) be a genus \(g\) Riemann surface with \(k\) marked points; then, for any pseudo-holomorphic map \(\varphi: \Sigma\to X\), there are finitely many orbifold structures on \(\Sigma\) whose singular set is contained in the set of marked points of \(\Sigma\), and for each of these orbifold structures there are finitely many pairs \((\tilde\varphi,\xi)\) where \(\tilde\varphi\) is a good map whose underlying map is \(\varphi\), and \(\xi\) is an isomorphism class of compatible systems of \(\tilde\varphi\). The total number is bounded from above by a constant depending only on \(X\), \(g\) and \(k\).

Evaluation of a stable map \(\varphi\) at a point \(z\) of \(\Sigma\) determines a point \(\varphi(z)\) in \(X\) together with a conjugacy class in the local group \(G_{\varphi(z)}\). That is, evaluation at \(z\) actually takes its values in the space \(\tilde{X}=\{(p,(g)^{}_{G_p})\,\mid p\in X, g\in G_p\}\). The connected components of \(\tilde{X}\) are called the twisted sectors of the orbifold \(X\). If \({\mathbf x}\) is a connected component of \(\tilde{X}^k\), then a \(k\)-pointed stable map \(\varphi: \Sigma\to X\) is said to be of type \({\mathbf x}\) if the evaluation of \(\varphi\) at the marked points of \(\Sigma\) lies in \({\mathbf x}\).

Let now \(X\) be a symplectic orbifold with a compatible almost complex structure \(J\) or a projective orbifold with an integrable complex structure \(J\). In these hypothesis Chen and Ruan show that the moduli space \(\overline{\mathcal M}_{g,k}(X,J,A,\mathbf{x})\) of \(k\)-pointed genus \(g\) stable curves \(\varphi: \Sigma\to X\) of type \({\mathbf x}\) and such that \(\varphi_*([\Sigma])=A\in H_2(X;{\mathbb Z})\) is a compact metrizable space, which comes equipped with a map \(\overline{\mathcal M}_{g,k}(X,J,A,{\mathbf x})\to \overline{\mathcal M}_{g,k}\) defined by forgetting the stable map and contracting the unstable components of the domain. Orbifold Gromov-Witten invariants are then defined by means of the moduli spaces \(\overline{\mathcal M}_{g,k}(X,J,A,{\mathbf x})\) as in the classical Gromov-Witten theory and is shown that the orbifold GW-invariants satisfy the same axioms as the ordinary GW-invariants, except the divisor axiom which only holds if one restricts to a divisor class in the nontwisted sector. As in the smooth case, orbifold GW-invariants are independent of the almost complex structure \(J\) and are therefore invariants of the symplectic structure. Moreover, genus zero orbifold GW-invariants define an orbifold quantum cohomology, which is an associative deformation of the orbifold cohomology constructed in [W. Chen and Y. Ruan, Commun. Math. Phys. 248, No. 1, 1–31 (2001; Zbl 1063.53091)].

The appendix to the paper provides an excellent introduction to the general theory of orbifolds, with rigorous definitions, detailed proofs and clear examples.

The notion of orbifold stable maps in the context of algebraic stacks has been independently introduced and studied by [D. Abramovich and A. Vistoli [J. Am. Math. Soc. 15, No. 1, 27–75 (2002; Zbl 0991.14007)].

For the entire collection see [Zbl 1003.00015].

Reviewer: Domenico Fiorenza (Roma)