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Algebraic Geometry

arXiv:alg-geom/9612004 (alg-geom)
[Submitted on 6 Dec 1996 (v1), last revised 2 Jun 1997 (this version, v2)]

Title:Intersection theory on $\Mbar_{1,4}$ and elliptic Gromov-Witten invariants

Authors:Ezra Getzler (Northwestern University)
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Abstract: 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.)
Comments: 25 pages. amslatex-1.2. This is the revised version which will appear in J. Amer. Math. Soc
Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th)
Report number: MPI 96-161
Cite as: arXiv:alg-geom/9612004
  (or arXiv:alg-geom/9612004v2 for this version)
  https://doi.org/10.48550/arXiv.alg-geom/9612004

Submission history

From: Ezra Getzler [view email]
[v1] Fri, 6 Dec 1996 15:07:57 UTC (21 KB)
[v2] Mon, 2 Jun 1997 16:24:27 UTC (24 KB)
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