Tying Up Instantons with Anti-Instantons

Abstract

In quantizing classical mechanical systems one often sums over the classical trajectories as in localization formulas, but also takes into account the contributions of the "instanton gas": a set of approximate solutions of the equations of motion. This paper attempts to alleviate some of the frustrations of this 40+ years old approach by finding the honest solutions of equations of motion of the complexified classical mechanical system. These ideas originate in the Bethe/gauge correspondence. The examples include algebraic integrable systems, from the abstract Hitchin systems to the well-studied anharmonic oscillator. We also speculate on the applications to the black hole radiation. We elucidate the relation between Lefschetz thimbles and the Ω-deformed B-model. We propose the notion of the topological renormalization group.In memory of L. D. Faddeev 1 arXiv:1802.04202v2 [hep-th]

SCFT/VOA correspondence via Ω-deformation

“…Solutions to these equations are precisely the gradient trajectories on which two-dimensional B-model on R 2 localizes, as discussed in [10] with its full detail, generated by the function Re W ε where W is the holomorphic superpotential. For the case at hand, one could view the four-dimensional holomorphic-topological theory on C × C ⊥ as a two-dimensional B-twisted gauge theory on C ⊥ , as done in [28] for the six-dimensional holomorphic-topological theory to obtain the four-dimensional Chern-Simons theory.…”

“…Thus it can be written as a contribution from the boundary ∂C ⊥ . We can simply add a boundary term to the action to cancel this [10], yielding S hyp,cl,ε = 8i…”

“…The superpotential has to be chosen as W = C d 2 z q z Dzq to reproduce the four-dimensional holomorphic-topological theory in the ε → 0 limit. As we approch to infinity t → ∞, Az, q z , andq should end on the critical points {dW = 0} of the superpotential to guarantee that the action (3.28) does not diverge [10]. 3 We have to regard these flow equations as defined on the fields with complexified gauge group.…”

“…To obtain the two-dimensional chiral algebra, we have to perform supersymmetric localization of the Ω-deformed holomorphic-topological theory to produce a chiral CFT on C. The algebra of local operators of this CFT would provide our desired chiral algebra. It turns out that the localization procedure can be conducted in a very similar manner with [10], where the localization of the Ω-deformed two-dimensional Landau-Ginzburg model was discussed. In fact, our localization can be viewed as the gauge theory analogue of [10] on C ⊥ , which was discussed in [28] in its application of recovering four-dimensional Chern-Simons theory from six-dimensional supersymmetric gauge theory (see also [26,27] for the discussion of B-models on the compact disk where the localization locus was chosen to be constant maps), occuring at each point of C. The localization locus is given by solutions to certain gradient flow equations (emanating from the critical point of the superpotential as we take C ⊥ = R 2 ).…”

Instantons in the Hofstadter butterfly: difference equation, resurgence and quantum mirror curves

“…But there are several issues here. Firstly the semi-classics of path-integrals is to this day not a completely understood subject, but it has become clear recently that the correct interpretation of it is via the Picard-Lefschetz (PL) theory [19][20][21][22][23][24][25][26][27][28]. The PL theory analysis is by far not a straightforward matter, and requires the identification of saddles which contribute in the semi-classical expansion.…”