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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)].

MSC:
 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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