On matrix models and their q -deformations
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Springer
Received: September 3, 2020 Accepted: September 21, 2020 Published: October 20, 2020
Luca Cassia, Rebecca Lodin and Maxim Zabzine Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden
E-mail: [email protected], [email protected], [email protected] Abstract: Motivated by the BPS/CFT correspondence, we explore the similarities between the classical β-deformed Hermitean matrix model and the q-deformed matrix models associated to 3d N = 2 supersymmetric gauge theories on D2 ×q S 1 and Sb3 by matching parameters of the theories. The novel results that we obtain are the correlators for the models, together with an additional result in the classical case consisting of the W -algebra representation of the generating function. Furthermore, we also obtain surprisingly simple expressions for the expectation values of characters which generalize previously known results. Keywords: Conformal and W Symmetry, Matrix Models, Supersymmetric Gauge Theory ArXiv ePrint: 2007.10354
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)126
JHEP10(2020)126
On matrix models and their q-deformations
Contents 1 Introduction
1 4 4 8 9 9 12 13
3 Quantum models 3.1 Definitions 3.1.1 D2 ×q S 1 3.1.2 Sb3 3.2 q-Virasoro constraints 3.2.1 D2 ×q S 1 3.2.2 Sb3 3.3 Solution of the constraints 3.3.1 Nf = 1 3.3.2 Nf = 2 3.3.3 Comments on Nf ≥ 3
16 17 17 19 22 22 25 27 28 32 35
4 Conclusion
35
A Special functions
37
B Symmetric functions and characters
37
C Relating constraint generators to q-Virasoro generators
39
D Asymptotic analysis and convergence of Sb3 partition function
40
1
Introduction
The BPS/CFT correspondence [1, 2] has provided a mapping between the exact values of partition functions and certain BPS observables for supersymmetric theories on the one hand, to conformal field theories in two dimensions on the other hand. The computation of partition functions and BPS observables has been aided through the programme called localization (as reviewed in [3]), where the evaluation of infinite dimensional integrals
–1–
JHEP10(2020)126
2 Review of the classical models 2.1 Definitions 2.2 Virasoro constraints 2.3 Solving the constraints 2.3.1 p = 1 2.3.2 p = 2 2.3.3 Comments on p ≥ 3
–2–
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reduces to evaluation only at specific points. Such localization computations of partition functions sometimes result in the form of a finite dimensional matrix model, and it is such partition functions that will be of interest here. In the light of this BPS/CFT correspondence, we derive and solve the Virasoro constraints that generating functions (τ -function) for certain classical and “quantum” models satisfy. These constraints can be considered a type of Ward identities and were first studied in [4, 5]. Here, the notion of “quantum” will be related to the introduction of a special class of q-deformations with respect to an un-deformed model. In the classical case, we will be considering the β-deformed Her
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