Solving q -Virasoro constraints
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Solving q-Virasoro constraints Rebecca Lodin1 · Aleksandr Popolitov1,4,5,6 · Shamil Shakirov2,3 · Maxim Zabzine1 Received: 28 March 2019 / Revised: 28 March 2019 / Accepted: 4 September 2019 © The Author(s) 2019
Abstract We show how q-Virasoro constraints can be derived for a large class of (q, t)-deformed eigenvalue matrix models by an elementary trick of inserting certain q-difference operators under the integral, in complete analogy with full-derivative insertions for βensembles. From free field point of view, the models considered have zero momentum of the highest weight, which leads to an extra constraint T−1 Z = 0. We then show how to solve these q-Virasoro constraints recursively and comment on the possible applications for gauge theories, for instance calculation of (supersymmetric) Wilson loop averages in gauge theories on D 2 × S 1 and S 3 . Keywords Ward identities · Matrix models · Difference equations · Virasoro constraints Mathematics Subject Classification 39-02
Shamil Shakirov: On leave from Harvard University.
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Rebecca Lodin [email protected] Aleksandr Popolitov [email protected] Shamil Shakirov [email protected] Maxim Zabzine [email protected]
1
Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden
2
Society of Fellows, Harvard University, Cambridge, MA 02138, USA
3
Mathematical Sciences Research Institute, Berkeley, CA 94720, USA
4
Moscow Institute for Physics and Technology, Dolgoprudny, Russia
5
ITEP, Moscow 117218, Russia
6
Institute for Information Transmission Problems, Moscow 127994, Russia
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R. Lodin et al.
Contents 1 2 3 4
Introduction . . . . . . . . . . . . . . . . . . . . . . Definition of the class of models . . . . . . . . . . . . (q, t)-deformed Virasoro matrix model and free fields Ward identities via q-difference operator insertion . . 4.1 The q-difference operator . . . . . . . . . . . . . 4.2 From full derivative to shift operators in times . . 4.2.1 The generic constraint . . . . . . . . . . . 4.2.2 The special additional constraint . . . . . . 4.2.3 Combining generic and special constraints . 5 Recursive solution . . . . . . . . . . . . . . . . . . . 6 Gauge theory applications . . . . . . . . . . . . . . . 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . A Special functions . . . . . . . . . . . . . . . . . . . . B Verification of starting relation . . . . . . . . . . . . . B.1 First starting relation . . . . . . . . . . . . . . . B.2 Second starting relation . . . . . . . . . . . . . . C Verification of algebraic identity . . . . . . . . . . . . D Contour change . . . . . . . . . . . . . . . . . . . . D.1 First constraint . . . . . . . . . . . . . . . . . . . D.2 Second constraint . . . . . . . . . . . . . . . . . E Action of T0 and T−1 from free field representation . References . . . . . . . . . . . . . . . . . . . . . . . . .
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