group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
…
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).
here are some seminar notes:
Some introductory notes:
And this introductory bit on the moduli stack of elliptic curves:
An exposition of GW theory as a TCFT is at
The origin of Gromov-Witten theory in and relation to string theory and other physics motivation is recalled and surveyed in
Discussion in the context of geometric quantization is in
See also the references at A-model.
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
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)
GW theory of orbifolds (hence in higher differential geometry) has been introduced in
A review with further pointers is in
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