….
Gromov-Witten invariants may be understood (and have originally been found) as arising from a particular TQFT, or actually a TCFT?.
For a useful exposition of this see
here are some seminar notes:
And this introductory bit on the moduli stack of elliptic curves:
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)
An explicit FQFT point of view is developed in