Contents

Definition

…

Relation to TQFT

Gromov-Witten invariants may be understood (and have originally been found) as arising from a particular TQFT, or actually a TCFT, called the A-model.

For a useful exposition of this see (Tolland).

Related concepts

• quantum sheaf cohomology

References

Expositions

here are some seminar notes:

• basic ideas of moduli stacks of curves and Gromov-Witten theory

And this introductory bit on the moduli stack of elliptic curves:

• A Survey of Elliptic Cohomology - elliptic curves.

An exposition of GW theory as a TCFT is at

• A. J. Tolland, Gromov-Witten Invariants and Topological Field Theory (blog)

As a TCFT

• Kevin Costello, The Gromov-Witten potential associated to a TCFT (arXiv:0509264)

See also the references at A-model.

General

A discussion by quantization of quadratic Hamiltonians is in

• Alexander Givental?, Gromov-Witten invariants and quantization of quadratic Hamiltonians (pdf)

• M. Kontsevich, Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562 (euclid).

• Yuri Manin, Frobenius manifolds, quantum cohomology and moduli spaces, Amer. Math. Soc., Providence, RI, 1999,

• W. Fulton, R. Pandharipande, Notes on stable maps and quantum cohomology, in: Algebraic Geometry- Santa uz 1995 ed. Kollar, Lazersfeld, Morrison. Proc. Symp. Pure Math. 62, 45–96 (1997)

• J Robbin, D A Salamon, A construction of the Deligne-Mumford orbifold, J. Eur. Math. Soc. (JEMS) 8 (2006), no. 4, 611–699 (arxiv; pdf at JEMS); corrigendum J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 901–905 (pdf at JEMS).

• J Robbin, Y Ruan, D A Salamon, The moduli space of regular stable maps, Math. Z. 259 (2008), no. 3, 525–574 (doi).

• Martin A. Guest, From quantum cohomology to integrable systems, Oxford Graduate Texts in Mathematics, 15. Oxford University Press, Oxford, 2008. xxx+305 pp.

• Joachim Kock, Israel Vainsencher, An invitation to quantum cohomology. Kontsevich’s formula for rational plane curves, Progress in Mathematics, 249. Birkhäuser Boston, Inc., Boston, MA, 2007. xiv+159 pp.

• Dusa McDuff, Dietmar Salamon, Introduction to symplectic topology, 2 ed. Oxford Mathematical Monographs 1998. x+486 pp.

• Sheldon Katz, Enumerative geometry and string theory, Student Math. Library 32. IAS/Park City AMS & IAS 2006. xiv+206 pp.

• Eleny-Nicoleta Ionel, Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96 (doi).

• Edward Frenkel, Constantin Teleman, AJ Tolland, Gromov-Witten Gauge Theory I (arXiv)

• Constantin Teleman, The structure of 2D semi-simple field theories (arXiv)

• Oliver Fabert, Floer theory, Frobenius manifolds and integrable systems arxiv/1206.1564

A generalization is discussed in

• Edward Frenkel, A. Losev, Nikita Nekrasov, Instantons beyond topological theory I (arXiv:hep-th/0610149)

• Edward Frenkel, A. Losev, Nikita Nekrasov, Instantons beyond topological theory II (arXiv:hep-th/0610149)

Expositions and summaries of this are in

• Edward Frenkel, A. Losev, Nikita Nekrasov, Notes on instantons in topological field theory and beyond (arXiv:hep-th/0702137)

• Jacques Distler, Localized (2006) (blog)