# Gromov-Witten invariants

I'm a mathematician studying Schubert calculus, and I'm out to compute the Gromov-Witten invariants of the complete flag manifold. Well, I actually already know how to compute them, but only in a way that involves cancellation, and the goal is to masterfully cancel everything and have only a formula that describes what's left.

I've heard this has some relationship to quantum field theory. I know very little physics other than what an enthusiast would read in a popular science book, but I do have some interest in the relationship of the problem I will spend my life failing to solve (far easier problems have been unsolved for 136 years) to reality. If anyone could give a brief overview of why physicists care about Gromov-Witten invariants I would appreciate it.

• One can think of Gromov-Witten invariants as scattering amplitudes in a 'topological' version of a string theory, known as the topological A-model. This theory is a subsector of a physical string theory known as type IIA string theory. Intuitively, Gromov-Witten invariants of a space X study pseudoholomorphic maps from a Riemann surface (of a certain genus and possibly with punctures) to X; and in string theory, maps from a punctured Riemann surface to a manifold representing spacetime is just a possible string scattering process, with the punctures representing incoming/outgoing strings. – Meer Ashwinkumar Nov 19 '15 at 7:39
• The pseudoholomorphic condition comes about since it gives the classical configurations of the A-model about which quantum perturbations manifest. – Meer Ashwinkumar Nov 19 '15 at 7:44
• I believe the definition used in physics is the former. In any case perhaps you should take a look at Witten's papers 'Topological Sigma Models' and 'Mirror Manifolds and Topological Field Theory'. The monograph 'Mirror Symmetry' (which is available online) is also helpful, take a look at chapter 16 for a treatment of the topological A-model. The math section of the book does discuss Gromov-Witten invariants, see chapter 26. As you can see the definition there is given as an integral. – Meer Ashwinkumar Nov 20 '15 at 3:13
• That is interesting, I did not know that the Gromov-Witten invariants were the coefficients of a polynomial. I should mention a very nice essay by Mina Aganagic, 'String theory and Math: Why this Marriage May Last', section 2 introduces Gromov-Witten invariants, and 2.1 is an answer to your original question. I believe other parts of the essay may also be helpful for you. – Meer Ashwinkumar Nov 20 '15 at 4:50
• I think you may be able to find a physics paper which discusses Gromov-Witten invariants for complete flag manifolds, this could be it arxiv.org/pdf/0704.1761. – Meer Ashwinkumar Nov 20 '15 at 5:04