### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 048, 21 pages      arXiv:1407.1370      https://doi.org/10.3842/SIGMA.2017.048
Contribution to the Special Issue on Combinatorics of Moduli Spaces: Integrability, Cohomology, Quantisation, and Beyond

### Equivariant Gromov-Witten Invariants of Algebraic GKM Manifolds

Chiu-Chu Melissa Liu a and Artan Sheshmani bc
a) Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA
b) Harvard University, Department of Mathematics (CMSA), 20 Garden Street, Cambridge, MA, 02138, USA
c) Aarhus University, Department of Mathematics, QGM, Ny Munkegade 118, 8000 Aarhus, Denmark

Received January 16, 2017, in final form June 21, 2017; Published online July 01, 2017

Abstract
An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use virtual localization to express equivariant Gromov-Witten invariants of any algebraic GKM manifold (which is not necessarily compact) in terms of Hodge integrals over moduli stacks of stable curves and the GKM graph of the GKM manifold.

Key words: Gromov-Witten theory; GKM manifold; moduli space; equivariant cohomology; localization.

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