##
**Quantum cobordisms and formal group laws.**
*(English)*
Zbl 1103.14011

Etingof, Pavel (ed.) et al., The unity of mathematics. In honor of the ninetieth birthday of I. M. Gelfand. Papers from the conference held in Cambridge, MA, USA, August 31–September 4, 2003. Boston, MA: Birkhäuser (ISBN 0-8176-4076-2/hbk). Progress in Mathematics 244, 155-171 (2006).

From the introduction: The present paper is closely based on the lecture given by the second author at The Unity of Mathematics symposium and is based on our joint work in progress on Gromov-Witten invariants with values in complex cobordisms. We will mostly consider here only the simplest example, elucidating one of the key aspects of the theory. We refer the reader to [A. Givental, in: Frobenius Manifolds: Quantum Cohomology and Singularities, Aspects Math. E 36, 91–112 (2004; Zbl 1075.53091)] for a more comprehensive survey of the subject and to [T. Coates, Riemann-Roch Theorems in Gromov-Witten Theory, Ph.D. thesis, Berkeley, 2003; http://abel.math.harvard.edu/\(\sim\)tomc] for all further details.

Consider \(\overline{\mathcal M}_{0,n}\), \(n\geq 3\), the Deligne-Mumford compactification of the moduli space of configurations of \(n\) distinct ordered points on the Riemann sphere \(\mathbb{C}\mathbb{P}^1\). Obviously, \(\overline{\mathcal M}_{0,3}=PT\), \(\overline{\mathcal M}_{0,4}=\mathbb{C} \mathbb{P}^1\), while \(\overline{\mathcal M}_{0,5}\) is known to be isomorphic to \(\mathbb{C}\mathbb{P}^2\) blown up at four points. In general, \(\overline {\mathcal M}_{0,n}\) is a compact complex manifold of dimension \(n-3\), and it makes sense to ask what is the complex cobordism class of this manifold. The Thom complex cobordism ring, after tensoring with \(\mathbb{Q}\), is known to be isomorphic to \(U^*=\mathbb{Q}[\mathbb{C} \mathbb{P}^1,\mathbb{C}, \mathbb{P}^2,\dots]\), the polynomial algebra with generators \(\mathbb{C} \mathbb{P}^k\) of degree \(-2k\). Thus our question is to express \(\overline {\mathcal M}_{0,n}\), modulo the relation of complex cobordism, as a polynomial in complex projective spaces.

This problem can be generalized in the following three directions. First, one can develop intersection theory for complex cobordism classes from the complex cobordism ring \(U^*(\overline{\mathcal M}_{0,n})\). Such intersection numbers take values in the coefficient algebra \(U^*=U^* (\text{pt})\) of complex cobordism theory. Second, one can consider the Deligne-Mumford moduli spaces \(\overline{\mathcal M}_{g,n}\) of stable \(n\)-pointed genus-\(g\) complex curves. They are known to be compact complex orbifolds and for an orbifold, one can mimic (as explained below) cobordism-valued intersection theory using cohomology intersection theory over \(\mathbb{Q}\) against a certain characteristic class of the tangent orbibundle. Third, one can introduce more general moduli spaces \(\overline{\mathcal M}_{g,n} (X,d)\) of degree-\(d\) stable maps from \(n\)-pointed genus-\(g\) complex curves to a compact Kähler (or almost-Kähler) target manifold \(X\). One defines Gromov-Witten invariants of more, using virtual tangent bundles of the moduli spaces of stable maps and their characteristic classes, one can extend Gromov-Witten invariants to take values in the cobordism ring \(U^*\). The Quantum Hirzebruch-Riemann-Roch theorem expresses cobordism-valued Gromov-Witten invariant of \(X\) in terms of cohomological ones. Cobordism-valued intersection theory in Deligne-Mumford spaces is included as the special case \(X= \text{pt}\). In the notes, we will mostly be concerned with the special case and with curves of genus zero, i.e., with cobordism-valued intersection theory in the manifolds \(\overline{\mathcal M}_{0,n}\). The cobordism classes of \(\overline{\mathcal M}_{0,n}\) than we seek are then interpreted as the self-intersections of the fundamental classes.

For the entire collection see [Zbl 1083.00015].

Consider \(\overline{\mathcal M}_{0,n}\), \(n\geq 3\), the Deligne-Mumford compactification of the moduli space of configurations of \(n\) distinct ordered points on the Riemann sphere \(\mathbb{C}\mathbb{P}^1\). Obviously, \(\overline{\mathcal M}_{0,3}=PT\), \(\overline{\mathcal M}_{0,4}=\mathbb{C} \mathbb{P}^1\), while \(\overline{\mathcal M}_{0,5}\) is known to be isomorphic to \(\mathbb{C}\mathbb{P}^2\) blown up at four points. In general, \(\overline {\mathcal M}_{0,n}\) is a compact complex manifold of dimension \(n-3\), and it makes sense to ask what is the complex cobordism class of this manifold. The Thom complex cobordism ring, after tensoring with \(\mathbb{Q}\), is known to be isomorphic to \(U^*=\mathbb{Q}[\mathbb{C} \mathbb{P}^1,\mathbb{C}, \mathbb{P}^2,\dots]\), the polynomial algebra with generators \(\mathbb{C} \mathbb{P}^k\) of degree \(-2k\). Thus our question is to express \(\overline {\mathcal M}_{0,n}\), modulo the relation of complex cobordism, as a polynomial in complex projective spaces.

This problem can be generalized in the following three directions. First, one can develop intersection theory for complex cobordism classes from the complex cobordism ring \(U^*(\overline{\mathcal M}_{0,n})\). Such intersection numbers take values in the coefficient algebra \(U^*=U^* (\text{pt})\) of complex cobordism theory. Second, one can consider the Deligne-Mumford moduli spaces \(\overline{\mathcal M}_{g,n}\) of stable \(n\)-pointed genus-\(g\) complex curves. They are known to be compact complex orbifolds and for an orbifold, one can mimic (as explained below) cobordism-valued intersection theory using cohomology intersection theory over \(\mathbb{Q}\) against a certain characteristic class of the tangent orbibundle. Third, one can introduce more general moduli spaces \(\overline{\mathcal M}_{g,n} (X,d)\) of degree-\(d\) stable maps from \(n\)-pointed genus-\(g\) complex curves to a compact Kähler (or almost-Kähler) target manifold \(X\). One defines Gromov-Witten invariants of more, using virtual tangent bundles of the moduli spaces of stable maps and their characteristic classes, one can extend Gromov-Witten invariants to take values in the cobordism ring \(U^*\). The Quantum Hirzebruch-Riemann-Roch theorem expresses cobordism-valued Gromov-Witten invariant of \(X\) in terms of cohomological ones. Cobordism-valued intersection theory in Deligne-Mumford spaces is included as the special case \(X= \text{pt}\). In the notes, we will mostly be concerned with the special case and with curves of genus zero, i.e., with cobordism-valued intersection theory in the manifolds \(\overline{\mathcal M}_{0,n}\). The cobordism classes of \(\overline{\mathcal M}_{0,n}\) than we seek are then interpreted as the self-intersections of the fundamental classes.

For the entire collection see [Zbl 1083.00015].

### MSC:

14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |

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

14H10 | Families, moduli of curves (algebraic) |

55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |

14C40 | Riemann-Roch theorems |

14L05 | Formal groups, \(p\)-divisible groups |