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Hodge integrals and Gromov-Witten theory. (English) Zbl 0960.14031
Let \(\overline{M}_{g,n}\) be the non-singular moduli stack of a genus \(g\), \(n\)-pointed Deligne-Mumford stable curve \(C\). For each marking \(i\) there is an associated cotangent bundle \(\mathbb L_i\rightarrow\overline {M}_{g,n}\) with fiber \(T^*_{C,p_i}\) over the moduli point \([C,p_1,\ldots,p_n]\). Write \(\psi_i\) for the first Chern class \(c_1(\mathbb L_i)\in H^*(\overline{M}_{g,n},\mathbb Q)\). For a curve \(C\) let \(\omega_C\) denote its dualizing sheaf. Then the Hodge bundle \(\mathbb E\rightarrow\overline{M}_{g,n}\) is the rank \(g\) vector bundle with fiber \(H^0(C,\omega_C)\) over \([C,p_1,\ldots,p_n]\). Let \(\lambda_j=c_j(\mathbb E)\). A Hodge integral over \(\overline {M}_{g,n}\) is defined to be an integral of products of the \(\psi\) and \(\lambda\) classes.
Let \(X\) be a non-singular projective variety over \(\mathbb C\). Write \(\overline{M}=\overline{M}_{g,n}(X,\beta)\) for the moduli stack of stable maps to \(X\) representing the class \(\beta\in H_2(X,\mathbb Z)\). Let \([\overline{M}]^{\text{vir}}\in A_*(\overline{M})\) denote the virtual class (in the expected dimension). As a first result the following theorem (reconstruction theorem) is proven:
Theorem 1: The set of Hodge integrals over moduli stacks of maps to \(X\) may be uniquely reconstructed from the set of descendent integrals of the form \[ \int_{[\overline{M}_{g,n}(X,\beta)]^{\text{vir}}}\prod_{i=1}^n\psi_i^{a_i}\cup e_i^*(\gamma_i)\cup\prod_{j=1}^g\lambda_j^{b_j}. \] The proof of this result relies on an interpretation of Mumford’s calculation of Grothendieck-Riemann-Roch in Gromov-Witten theory.
The main result of the paper can be formulated as follows:
Theorem 2: Let \(F(t,k)\in{\mathbb Q}[k][[t]]\) be defined by \[ F(t,k)=1+\sum_{g\geq 1}\sum_{i=0}^gt^{2g}k^i\int_{\overline{M}_{g,1}}\psi_1^{2g-2+i}\lambda_{g-i}, \] then \(F(t,k)=\left({t/2\over\sin(t/2)}\right)^{k+1}.\)
Let \(C(g,d)=\int_{[\overline{M}_{g,0}(\mathbb P^1,d)]^{\text{vir}}}c_{\text{top}}(R^1\pi_*\mu^*N)\) denote the contribution to the genus \(g\) Gromov-Witten invariant of a Calabi-Yau \(3\)-fold of multiple covers of a fixed rational curve with normal bundle \(N=\mathcal O(-1)\oplus\mathcal O(-1)\). One knows that \(C(0,d)=1/d^3\) and \(C(1,d)=1/12d\). Here the general case is calculated:
Theorem 3: For \(g\geq 2\) one has \[ C(g,d)=|\chi(M_g)|\cdot{d^{2g-3}\over(2g-3)!}, \] where \(\chi(M_g)=B_{2g}/2g(2g-2)\) is the Harer-Zagier formula for the orbifold Euler characteristic of \(M_g\), and where \(B_{2g}\) is the \(2g\)-th Bernoulli number.
Theorem 4: For \(g\geq 2\) one has \[ \int_{\overline{M}_g}\lambda^3_{g-1}={|B_{2g}|\over 2g}{|B_{2g-2}|\over 2g-2}{1\over(2g-2)!}. \] Several methods to obtain relations between Hodge integrals are discussed in some detail: (i) via virtual localization, (ii) via classical curve theory, first via the canonical system, second via Weierstraß loci.
In the introduction the authors end with an interesting combinatorial conjecture relating Gromov-Witten theory to the intrinsic geometry of \(M_g\) via Hodge integrals. Let \(\mathcal R^*(M_g)\) be the ring of tautological Chow classes in \(M_g\). This ring is conjectured to be a Gorenstein ring with socle in degree \(g-2\). The top intersection pairings in \(\mathcal R^*(M_g)\) are determined by the Hodge integrals \(\int_{\overline{M}_{g,n}}\psi_1^{k_1}\ldots\psi_n^{k_n} \lambda_g\lambda_{g-1}\).
It was conjectured by C. Faber [in: Moduli of Curves and Abelian Varieties. The Dutch Intercity Seminar on Moduli, Aspects Math. E 33, 109-129 (1999; Zbl 0978.14029)] that \[ \int_{\overline{M}_{g,n}}\psi_1^{k_1}\ldots\psi_n^{k_n} \lambda_g\lambda_{g-1}={{(2g+n-3)!(2g-1)!!}\over{(2g-1)!\prod_{i=1}^n (2k_i-1)!!}}\int_{\overline{M}_{g,1}}\psi_1^{g-1}\lambda_g\lambda_{g-1}, \] where \(g\geq 2\) and \(k_i>0\). The conjecture has been shown to be implied by the so-called degree \(0\) Virasoro conjecture applied to \({\mathbb P}^2\) [cf. E. Getzler and R. Pandharipande, Nucl. Phys. B 530, No. 3, 701-714 (1998; Zbl 0957.14038)].

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14H10 Families, moduli of curves (algebraic)
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