Renormalization and BRST Symmetry in Donaldson–Witten Theory

Abstract

The presence of a BRST symmetry in topologically twisted gauge theories makes a precise analysis of these theories feasible. While the global BRST symmetry suggests that correlation functions of BRST exact observables vanish, this decoupling might be obstructed due to a contribution from the boundary of field space. Motivated by divergent BRST exact observables on the Coulomb branch of Donaldson-Witten theory, we put forward a new prescription for the renormalization of correlation functions on the Coulomb branch. This renormalization is based on the relation between Coulomb branch integrals and integrals over a modular fundamental domain, and establishes that BRST exact observables indeed decouple in Donaldson-Witten theory. arXiv:1901.03540v2 [hep-th] 9 Aug 2019 5. Evaluation of correlation functions of Q-exact observables 30 6. Discussion and conclusion 31 A. Modular forms and theta functions 33 B. Siegel-Narain theta function 34 C. The self-dual twisted operator 35 -1 -and whose restriction to H 2 (M, Z)×H 2 (M, Z) is an integral bilinear form with signature (1, b 2 − 1). The bilinear form provides the quadratic form Q(k) := B(k, k) ≡ k 2 , which can be brought to a simple standard form [15, Section 1.1.3]. We denote the period point by J, i.e. the harmonic two-form, satisfying * J = J ∈ H 2 (M, R), J 2 = 1, (3.2)

Duality and mock modularity

“…Now we mention some previous and current work on related problems. Mock modularity arising from Coulomb branch integrals in gauge theories with N = 2 supersymmetry has been systematically explored in [19], extending previous calculations that had been done by more special methods [5,[20][21][22][23][24]. Moreover, in forthcoming work, Manschot and Moore have analyzed the Coulomb branch integral and the associated mock modularity in the N = 2 * theory, which of course is closely related to N = 4 super Yang-Mills, which we study in the present paper.…”

Effective gravitational couplings of four-dimensional $$ \mathcal{N} $$ = 2 supersymmetric gauge theories

“…With the understanding of the low energy effective dynamics of the theory, an alternative formulation of the Donaldson invariants was conjectured in terms of the Seiberg-Witten invariants [5]. Subsequently, a physical derivation of the conjecture was given in [6] and later extended and clarified in [7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

“…The term Ψ [K] comes essentially from the evaluation of the photon partition function of the low energy effective abelian gauge theory, and takes the form of a Siegel-Narain theta function with kernel K depending on the inserted observables [19,20]. It was found by Shapere and Tachikawa [21] that the functions A and B appearing in the topologically twisted theory can be used to compute the central charges of the physical N = 2 superconformal theory that corresponds to a superconformal point in the Coulomb moduli space of an N = 2 supersymmetric gauge theory.…”

“…where Y is the total number of boxes in N Young diagrams. We introduce the conversion operator which maps characters into weights, 19) and the dual operator ∨,…”

“…In order to compare our results with the prediction (1.10), we take the Seiberg-Witten curve and the Seiberg-Witten differential to be [30][31][32] t 2 =ũ − νm 2 ℘ (z; τ UV ) , λ = tdz, (4.18) where the parameterũ in the curve is the same as u up to an additive constant, 19) and ℘ (z; τ UV ) is Weierstrass's elliptic function (see appendix D for its basic properties). We see from the curve (4.18) thatũ is modular under the ultraviolet S-duality transformation.…”