nLab Gromov-Witten invariants

Idea

Gromov-Witten theory studies symplectic manifolds via maps of Riemann surfaces into it. In contrast to the fundamental group, where one studies maps from the circle into the space, there is no composition law for Riemann surfaces. Instead, one considers pseudoholomorphic curves in a fixed homology class and with fixed boundary conditions such that only a finite number of such maps exists. These numbers are symplectic invariants. Their computation is difficult and uses enumerative and analytic techniques from Floer homology.

Gromov-Witten invariants are used to deform the cup product of cohomology classes (using only maps from the Riemann sphere into the manifold), leading to quantum cohomology.

Gromov-Witten theory originates in physics from the A-model. A powerful tool to compute Gromov-Witten invariants is mirror symmetry.

The precise mathematical definition uses the notion of the moduli space of stable maps.

Definition

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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

• stable map

• orbifold cohomology

• quantum sheaf cohomology

• Donaldson theory

• A-model

• GW/DT correspondence

• enumerative geometry

References

Expositions

Introductory notes:

• Simon Rose, Introduction to Gromov-Witten theory (arXiv:1407.1260)

• Albrecht Bertram, Stable Maps and Gromov-Witten Invariants, School and Conference on Intersection Theory and Moduli Trieste, 9-27 September 2002 (pdf, pdf)

• Sheldon Katz, Enumerative Geometry and String Theory, Student Mathematical Library 32, AMS 2006 (ISBN:978-1-4704-2143-4, spire:739788)

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

• 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

• Daniel Grunberg, Gromov-Witten Theory and Threshold Corrections (arXiv:hep-th/0605087)

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

• 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)

• Maxim Kontsevich, Yuri 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 Cruz 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

• 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)

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

• Dan Abramovich, Lectures on Gromov-Witten invariants of orbifolds (arXiv:math/0512372)

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

• Kai Behrend, Yuri Manin, Stacks of Stable Maps and Gromov-Witten Invariants (arXiv:alg-geom/9506023)

Further investigation of these stacky Chow motives then appears in

• Bertrand Toën, On motives for Deligne-Mumford stacks, International Mathematics Research Notices 2000, 17 (2000) 909-928 (arXiv:math/0006160, web, pdf)

• Utsav Choudhury, Motives of Deligne-Mumford Stacks (arXiv:1109.5288)

On its relation to tropical geometry:

• Grigory Mikhalkin?. Enumerative Tropical Algebraic Geometry in $\mathbb{R}^2$. Journal of the American Mathematical Society, Vol. 18, No. 2 (Apr., 2005), pp. 313-377. (doi)

• Emil Albrychiewicz, Kai-Isaak Ellers, Andrés Franco Valiente, Petr Hořava. Tropological Sigma Models. (2023). (arXiv:2311.00745)