Quiver elliptic W-algebras

Abstract

We define elliptic generalization of W-algebras associated with arbitrary quiver using our construction (Kimura and Pestun in Quiver W-algebras, 2015. arXiv:1512.08533 [hep-th]) with six-dimensional gauge theory.

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Fig. 1

Notes

  1. 1.

    The Dirac convention is also often used in the literature, for example, [16, 41,42,43,44], while Refs. [31, 32, 45, 46] use the Dolbeault.

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Acknowledgements

The work of T.K. was supported in part by Keio Gijuku Academic Development Funds, JSPS Grant-in-Aid for Scientific Research (No. JP17K18090), the MEXT-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (No. S1511006), JSPS Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science” (No. JP15H05855), and “Discrete Geometric Analysis for Materials Design” (No. JP17H06462). VP acknowledges grant RFBR 15-01-04217 and RFBR 16-02-01021. The research of VP on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (QUASIFT grant agreement 677368).

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Correspondence to Vasily Pestun.

Appendix A: Proof of trace formula (3.19)

Appendix A: Proof of trace formula (3.19)

In this Appendix we prove the equivalence (3.19) using the coherent state basis.

Coherent state basis

The argument in this part is essentially parallel to the textbook [56].

For oscillator algebra generated by \((t, \partial _{t})\) with \([\partial _{t}, t]= 1\) we consider the coherent state basis in the Fock space

$$\begin{aligned} \left| n \right\rangle = \frac{t^n}{\sqrt{n!}} \left| 0 \right\rangle \,, \qquad \left\langle n \right| = \left\langle 0 \right| \frac{\partial ^n}{\sqrt{n!}} \,, \qquad |z) = e^{zt} \left| 0 \right\rangle \,, \qquad (z| = \left\langle 0 \right| e^{z^* \partial } \end{aligned}$$
(A.1)

The normalization is

$$\begin{aligned} \left\langle n|m \right\rangle = \delta _{n,m} \,, \qquad (z|w) = e^{z^* w} \,. \end{aligned}$$
(A.2)

The states in (A.1) are eigenstates of the filling number operator \(t \partial \left| n \right\rangle = n \left| n \right\rangle \) and the annihilation/creation operators \(\partial |z) = z |z)\), \((z| t = (z| z^*\). Notice that the operator \(a^{t \partial _t}\) acts on the states |z) and (z| as,

$$\begin{aligned} a^{t \partial _t} |z) = |az) \,, \qquad (z| a^{t \partial _t} = (a^* z| \,. \end{aligned}$$
(A.3)

The identity operator can be expressed in terms of the coherent state basis:

$$\begin{aligned} \mathbb {1} = \frac{1}{\pi } \int d^2 z \, |z) e^{-|z|^2} (z| \end{aligned}$$
(A.4)

where

$$\begin{aligned} \left\langle n | \mathbb {1} | m \right\rangle = \delta _{n,m} \,, \end{aligned}$$
(A.5)

so that the trace of an operator is

$$\begin{aligned} {{\mathrm{Tr}}}\mathcal {O} = \frac{1}{\pi } \int d^2 z \, e^{-|z|^2} (z| \mathcal {O} |z) \,. \end{aligned}$$
(A.6)

Then we find [57]

$$\begin{aligned} {{\mathrm{Tr}}}\left[ a^{t \partial } e^{b \partial } e^{c t} \right] = \frac{1}{1-a} \exp \left( \frac{abc}{1-a} \right) \end{aligned}$$
(A.7)

because

$$\begin{aligned} \frac{1}{\pi } \int d^2 z \, e^{-|z|^2} (z| a^{t\partial _t} e^{bt} e^{c\partial _t} |z) = \frac{1}{\pi } \int d^2 z \, e^{-(1-a)|z|^2 + ab z^* + cz} \,. \end{aligned}$$
(A.8)

Torus correlation function

Let us compute the torus correlation function (3.18). The product of the 5d screening currents is given by

$$\begin{aligned} S_{i,x}^\text {5d} S_{j,x'}^\text {5d} = \exp \left( - \sum _{m=1}^\infty \frac{1}{m} \frac{1-q_1^m}{1-q_2^{-m}} c_{ji}^{[m]} \frac{x'^m}{x^m} \right) : S_{i,x}^\text {5d} S_{j,x'}^\text {5d} : \,. \end{aligned}$$
(A.9)

Then we compute the trace part

$$\begin{aligned}&{{\mathrm{Tr}}}\left[ p^{L_0} : S_{i,x}^\text {5d} S_{j,x'}^\text {5d} : \right] \nonumber \\&= {{\mathrm{Tr}}}\Bigg [ \left( \prod _{i' \in \Gamma _0} \prod _{n = 1}^\infty p^{n t_{i',n} \partial _{i',n}} \right) \exp \left( \sum _{n=1}^\infty (1-q_1^n) \left( x^n t_{i,n} + x'^n t_{j,n}\right) \right) \nonumber \\&\qquad \times \exp \left( \sum _{n=1}^\infty - \frac{1}{n(1-q_2^{-n})} \left( x^{-n} c_{ki}^{[n]} \partial _{k,n} + x'^{-n} c_{lj}^{[n]} \partial _{l,n} \right) \right) \Bigg ] \nonumber \\&= \exp \left( \sum _{n=1}^\infty \left( - \frac{1-q_1^n}{n(1-q_2^{-n})} \frac{p^n}{1-p^n} c_{ji}^{[n]} \frac{x'^n}{x^n} + \frac{1-q_1^{-n}}{n(1-q_2^{n})} \frac{1}{1-p^{-n}} c_{ji}^{[-n]} \frac{x^{n}}{x'^{n}} \right) \right) \nonumber \\&\qquad \times \text {const} \end{aligned}$$
(A.10)

where we have used the formulas (A.7) and (2.20), and the constant term does not contain x nor \(x'\). Thus we obtain the torus correlator

$$\begin{aligned} {{\mathrm{Tr}}}\left[ p^{L_0} S_{i,x}^\text {5d} S_{j,x'}^\text {5d} \right] = \exp \left( - \sum _{n \ne 0} \frac{1-q_1^n}{n(1-q_2^{-n})(1-p^n)} c_{ji}^{[n]} \frac{x'^n}{x^n} \right) \,. \end{aligned}$$
(A.11)

This is equivalent to (3.17), and proves the relation (3.19).

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Kimura, T., Pestun, V. Quiver elliptic W-algebras. Lett Math Phys 108, 1383–1405 (2018). https://doi.org/10.1007/s11005-018-1073-0

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Keywords

  • Supersymmetric gauge theories
  • Conformal field theories
  • W-algebras
  • Quantum groups
  • Quiver
  • Instanton

Mathematics Subject Classification

  • 81T60
  • 81R10
  • 14D21
  • 81R50