Donaldson-Witten theory and indefinite theta functions

Moore

et al. 2019

Ann. Henri Poincaré

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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)

Section: Evaluation Of Correlation Functions Of Q-exact Observablesmentioning

confidence: 55%

“…where the first line is due to the product of the non-holomorphic Q-exact part of C 2 with G 2 , and the second line is the contribution of the product from the holomorpic 7 Note that we find a different sign of the ψ ∧ ψ term compared to [9,24].…”

Section: -Form Operatorsmentioning

confidence: 73%

Renormalization and BRST Symmetry in Donaldson–Witten Theory

Korpas

Moore

et al. 2019

Ann. Henri Poincaré

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“…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].…”

Section: Jhep06(2020)150mentioning

confidence: 99%

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

Moore

Zhang

2020

J. High Energ. Phys.

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The low energy effective couplings of a four-dimensional N = 2 supersymmetric gauge theory to topological invariants of the background gravitational field are described by two functions A and B. These two functions play an important role in the study of topologically twisted four-dimensional N = 2 supersymmetric gauge theories and in the computation of central charges of N = 2 superconformal theories. In this paper, we compute A and B from the partition function in the Ω-background for SU(2) gauge theories. Our results not only confirm the predicted expressions of the effective gravitational couplings, but also give the previously undetermined overall multiplicative factors. We also analyze A and B for the SU(N) super-Yang-Mills theory, and confirm all the previous predictions.

“…Such an integral is as in Equation (1.3), but with d c replaced by −τ . Besides their appearance in VW partition functions, mock modular forms have become an important element of the study of modular partition functions, such as in conformal field theory [13,14,15], Donaldson-Witten theory [16,17,18], AdS 3 gravity [19], black holes [20,21,22] and the moonshine phenomenon [23].…”

Section: Introductionmentioning

confidence: 99%

Vafa–Witten Theory and Iterated Integrals of Modular Forms

2019

Commun. Math. Phys.

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Vafa-Witten (VW) theory is a topologically twisted version of N = 4 supersymmetric Yang-Mills theory. S-duality suggests that the partition function of VW theory with gauge group SU (N ) transforms as a modular form under duality transformations. Interestingly, Vafa and Witten demonstrated the presence of a modular anomaly, when the theory has gauge group SU (2) and is considered on the complex projective plane P 2 . This modular anomaly could be expressed as an integral of a modular form, and also be traded for a holomorphic anomaly. We demonstrate that the modular anomaly for gauge group SU (3) involves an iterated integral of modular forms. Moreover, the modular anomaly for SU (3) can be traded for a holomorphic anomaly, which is shown to factor into a product of the partition functions for lower rank gauge groups. The SU (3) partition function is mathematically an example of a mock modular form of depth two.