arXiv subjects
Publications and source records attributed to Ezra Getzler.
Operads may be represented as symmetric monoidal functors on a small symmetric monoidal category. We discuss the axioms which must be imposed on a symmetric monoidal functor in order that it give rise to a theory similar to the theory of operads: we call such functors patterns. We also develop the enriched version of the theory, and show how it may be applied to axiomatize topological field theory.
Let X be a projective manifold, and let H be the associated small phase space; this is a Frobenius manifold with canonical choice of fundamental solution for the Dubrovin connection. The large phase space of X may be identified with the jet-space of curves in H. In this paper, we formulate the differential equations satisfied by the higher-genus potentials F_g of X, such as topological recursion relations and the Virasoro constraints, in an intrinsic fashion on the jet space of H, that is, in such a way that the equations do not depend on the choice of fundamental solution. This effort is rewarded by a closer relationship between the resulting theory and the geometry of moduli spaces of stable curves. A consequence of our analysis is the proof of a conjecture of Eguchi and Xiong: F_g is a function on the jet space of jets of order 3g-2.
We study a reduction of the Toda lattice in the limit of infintesimal lattice spacing. Using this reduction, we formulate a conjecture for the equivariant Gromov-Witten invariants of the sphere, which we prove in genus 0. This conjecture follows from recent work of Okounkov and Pandharipande (math.AG/0207233), combined with results from the sequel to this paper.
Applying recent ideas of Carlet, Dubrovin and Zhang (to appear), who, following a suggestion of Eguchi and Yang (hep-th/9407134), study the logarithm of the Lax operator of the Toda lattice, we show that the equivariant Toda lattice introduced in math.AG/0207025 is a Hamiltonian integrable system.
We give explicit formulas for the multipoint series of Gromov-Witten invariants of CP^1. We use the recursions of the Toda hierarchy to calculate these in degree 0, and the Toda equation and the degree 0 formulas to extend the calculation to higher degrees. We also obtain formulas for the Hodge series for the Hodge classes lambda_g and lambda_{g-1}.
We study the Toda conjecture of Eguchi and Yang for the Gromov-Witten invariants of CP^1,using the bihamiltonian method of the formal calculus of variations. We also study its relationship to the Virasoro conjecture for CP^1, recently proved by Givental (math.AG/0108100).
We prove a Darboux theorem for formal deformations of Hamiltonian operators of hydrodynamic type (Dubrovin-Novikov). Not all deformations are equivalent to the original operator: there is a moduli 2-stack of normal forms. The paper utilizes three main concepts: 1) dg Lie algebras concentrated in degrees [-1,\infty) such as the Schouten algebra - these give a convenient language for describing deformation problems; 2) the Deligne 2-groupoid associated to such a dg Lie algebra, which represents the moduli of formal deformations; 3) a refined version of the Schouten bracket in the formal calculus of variations, due to V. O. Soloviev (hep-th/9305133).
We calculate the genus 2 correlation functions of two-dimensional topological gravity, in a background with two primary fields, using the genus 2 topological recursion relations.
We calculate the Euler characteristics of the local systems S^k(V) \otimes S^l\Wedge^2(V) on the moduli space M_2 of curves of genus 2, where V is the rank 4 local system R^1π_*C.
The Virasoro conjecture is a conjectured sequence of relations among the descendent Gromov-Witten invariants of a smooth projective variety in all genera; the only varieties for which it is known to hold are a point (Kontsevich) and Calabi-Yau threefolds (S. Katz). We review the statement of the conjecture and its proof in genus 0, following Eguchi, Hori and Xiong. This version incorporates many changes and improvements compared with the first, and a small number of misprints from the second. It will appear in the AMS volume "Hirzebruch 70", edited by P. Pragacz et al.
In Part 1 of this paper, we study gravitational descendents of Gromov-Witten invariants for general projective manifolds, applying the Behrend-Fantechi construction of the virtual fundamental classes. In Part 2, we calculate the topological recursion relations in genus 2. There are two of these, one for the second descendent of a field, and one for the first descendents of two fields. The proof uses the results of Part 1 together with a thorough study of intersection theory on the moduli space $\bar{M}_{2,2}$.
Given a mixed Hodge module E on a scheme X over the complex numbers, and a quasi-projective morphism f:X->S, we construct in this paper a natural resolution of the nth exterior tensor power of E restricted to the nth configuration space of f. The construction is reminiscent of techniques from the theory of hyperplane arrangements, and relies on Arnold's calculation of the cohomology of the configuration space of the complex line. This resolution is S_n-equivariant. We apply it to the universal elliptic curve with complete level structure of level N>=3 over the modular curve Y(N), obtaining a formula for the S_n-equivariant Serre polynomial (Euler characteristic of H^*_c(V,Q) in the Grothendieck group of the category of mixed Hodge structures) of the moduli space M_{1,n}. In a sequel to this paper, this is applied in the calculation of the S_n-equivariant Hodge polynomial of the compactication \bar{M}_{1,n}.
The WDVV equation is satisfied by the genus 0 correlation functions of any topological field theory in two dimensions coupled to topological gravity, and may be used to determine the genus 0 (rational) Gromov-Witten invariants of many projective varieties (as was done for projective spaces by Kontsevich). In this paper, we present an equation of a similar universal nature for genus 1 (elliptic) Gromov-Witten invariants -- however, it is much more complicated than the WDVV equation, and its geometric significance is unclear to us. (Our prove is rather indirect.) Nevertheless, we show that this equation suffices to determine the elliptic Gromov-Witten invariants of projective spaces. In a sequel to this paper, we will prove that this equation is the only one other than the WDVV equation which relates elliptic and rational correlation functions for two-dimensional topological field theories coupled to topological gravity. It is unclear if there are any further equations of this type on the small phase space in higher genus, but we think it unlikely. (The genus 0 and 1 cases are special, since the correlation functions on the small phase space determine those on the large phase space.)
We present two explicit recursions which determine the elliptic Gromov-Witten invariants of CP^3 in terms of the rational ones, and give a table up to degree 5. Unlike the rational Gromov-Witten invariants, the coefficients are negative and fractional. In a further paper, we will prove that N^1_ab + (2n-1)N^0_ab/12 is the number of elliptic space curves through a generic lines and b generic points.
The semi-classical approximation is an explicit formula of mathematical physics for the sum of Feynman diagrams with a single circuit.In this paper, we study the same problem in the setting of modular operads (see dg-ga/9408003); instead of being a number, the interaction at a vertex of valence n is an S_n-module. As an application, we calculate the S_n-equivariant Hodge polynomials of the moduli spaces \Mbar_{1,n}.
The symmetric group S_n acts freely on the configuration space of n distinct points in a quasi-projective variety. In this paper, we study the induced action of the symmetric group S_n on the de Rham cohomology of this space, using mixed Hodge theory, combined with methods from the theory of symmetric functions. (We prove a motivic version of this as well.) As an application of our results, we calculate the S_n-equivariant Hodge polynomial of the Fulton-MacPherson compactification X[n] of the configuration space.
We study a pair of dual operads which arise in the study of moduli spaces of pointed genus 0 curves (this duality is similar to that between commutative and Lie algebras). These operads are both quadratic, and even Koszul, and arise in the theory of quantum cohomology.
This paper provides some background to the theory of operads, used in the first author's papers on 2d topological field theory (hep-th/921204, CMP 159 (1994), 265-285; hep-th/9305013). It is intended for specialists.