nLab
Gromov-Witten invariants

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Quantum field theory

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).

References

Expositions

here are some seminar notes:

Some introductory notes:

  • Simon Rose, Introduction to Gromov-Witten theory, arXiv.

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

An exposition of GW theory as a TCFT is at

  • AJ Tolland, Gromov-Witten Invariants and Topological Field Theory (blog)

The origin of Gromov-Witten theory in and relation to string theory and other physics motivation is recalled and surveyed in

Via geometric quantization

Discussion in the context of geometric quantization is in

  • Emily Clader, Nathan Priddis, Mark Shoemaker, Geometric Quantization with Applications to Gromov-Witten Theory (arXiv:1309.1150)

As a TCFT

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:0904.4834)

  • Constantin Teleman, The structure of 2D semi-simple field theories (arXiv:0712.0160)

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

A generalization is discussed in

Expositions and summaries of this are in

In higher differential geometry / on orbifolds

GW theory of orbifolds (hence in higher differential geometry) has been introduced in

  • Weimin Chen, Yongbin Ruan, Orbifold Gromov-Witten Theory, in Orbifolds in mathematics and physics (Madison, WI, 2001), 25–85, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002 (arXiv:math/0103156)

A review with further pointers is in

In terms of motives

That the pull-push quantization of Gromov-Witten theory is naturally understood as a “motivic quantization” in terms of Chow motives of Deligne-Mumford stacks was suggested in

Further investigation of these stacky Chow motives then appears in

Revised on July 11, 2014 11:55:26 by Adeel Khan (46.114.6.77)