Cohomological Field Theories and Four-Manifold Invariants

Abstract

Four-dimensional cohomological quantum field theories possess topological sectors of correlation functions that can be analyzed non-perturbatively on a general four-manifold. In this thesis, we explore various aspects of these topological models and their implications for smooth structure invariants of four-manifolds. Cohomological field theories emerge when one considers topological twisting of ordinary quantum field theories with extended (N = 2 in the context of this thesis) supersymmetry. The global scalar supersymmetry of these theories allows one to use integrals/sums over their quantum vacua as a tool for their exact analysis. In the case of pure SU(2) N = 2 gauge theory this has lead to remarkable success of Witten’s field theory formulation of Donaldson invariants and discovery of Seiberg-Witten invariants which are the best presently available tool for distinguishing smooth structures on four-manifolds with fixed topological type. In chapter 3 of this thesis we analyze a new prescription for defining the integral over a Coulomb branch of vacua in Donaldson-Witten theory as well as discuss possible treatment of IR divergences associated with certain BRST-exact operators. Chapiiter 3 of the thesis is based on the work reported in [20] (arXiv:1901.03540 [hep-th]) and partly has been extracted from that paper. On general grounds one expects that topological twisting of any N = 2 supersymmetric theory defines a smooth structure invariant. However, examples of Lagrangian theories strongly suggest that topological partition functions of Lagrangian theories are expressible through the classical cohomological invariants and Seiberg-Witten invariants. Therefore, the search for new 4-manifold invariants has to be restricted to so-called N = 2 theories. Though full non-Lagrangian theories are, at present, difficult to analyze due to their strongly-coupled nature and the lack of action principle, in chapter 4 we show how one can derive the topological partition function of a simplest non-trivial non-Lagrangian theory discovered by Argyres and Douglas and known as AD3 theory. We obtain a formula for the partition function of topologically twisted version of the AD3 theory on any compact, oriented, simply connected, four-manifolds without boundary and with b_2^+ > 0. The result can be, once again, expressed in terms of classical cohomological invariants and Seiberg-Witten invariants. We argue that our results hint at the existence of four-manifolds of new, presently unknown, type as well as narrow the search for new field theory invariants of four-manifolds to Non-Lagrangian superconformal points that admit Higgs branches. Chapter 4 of this thesis is based on the work reported in [40] (arXiv:1711.09257 [hep-th]) and partly has been extracted from that paper. Finally, in chapter 5 we derive a twisted (0,2) two-dimensional model by putting the abelian low energy theory of single M5 brane described by the PST action on a direct product of a Riemann surface and a four-manifold. The resulting two dimensional topological model can potentially be used as a tool refining the u-plane integral to study topologically twisted N = 2 theories of class S .