3d N $$ \mathcal{N} $$ = 4 OPE coefficients from Fermi gas
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Springer
Received: June 2, 2020 Accepted: June 17, 2020 Published: July 7, 2020
Shai M. Chester, Rohit R. Kalloor and Adar Sharon Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot, Israel
E-mail: [email protected], [email protected], [email protected] Abstract: The partition function of a 3d N = 4 gauge theory with rank N can be computed using supersymmetric localization in terms of a matrix model, which often can be formulated as an ideal Fermi gas with a non-trivial one-particle Hamiltonian. We show how OPE coefficients of protected operators correspond in this formalism to averages of n-body operators in the Fermi gas, which can be computed to all orders in 1/N using the WKB expansion. We use this formalism to compute OPE coefficients in the U(N )k × U(N )−k ABJM theory as well as the U(N ) theory with one adjoint and Nf fundamental hypermultiplets, both of which have weakly coupled M-theory duals in the large N and finite k or Nf regimes. For ABJM we reproduce known results, while for the Nf theory we compute the all orders in 1/N dependence at finite Nf for the coefficient cT of the stress tensor two-point function. Keywords: 1/N Expansion, Supersymmetric Gauge Theory, Conformal Field Theory ArXiv ePrint: 2004.13603
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)041
JHEP07(2020)041
3d N = 4 OPE coefficients from Fermi gas
Contents 1 Introduction
1
2 Fermi gas for n-body operators
4 7 8 11 12 13
4 Nf matrix model 4.1 Matrix model 4.2 ∂b2 Z b=1
13 14 16
5 Conclusion
17
A The double sine function
19
B Phase space integrals B.1 ABJM B.1.1 Phase space integrals for A B.1.2 Phase space integrals for B B.2 Nf matrix model
20 20 20 22 23
C ~ corrections in Fermi gas
24
−
1
m=0
Introduction
Supersymmetric localization is one of the few available tools for analytically computing nontrivial physical observables non-perturbatively in interacting quantum field theories. In the original case of 4d N = 4 super-Yang-Mills (SYM) with gauge group SU(N ), Pestun used localization to compute the sphere partition function and the expectation value of Wilson loops in terms of (N − 1)-dimensional matrix model integrals [1]. Localization was then applied by [2] to 3d N = 2 Chern-Simons-matter theories, such as the N = 6 ABJM theory with gauge group U(N )k ×U(N )−k and Chern-Simons level k [3], to compute matrix model observables as N -dimensional integrals. For low N , these integrals can be computed by hand, but this becomes infeasible for large N . The large N regime is of particular interest for conformal field theories (CFTs) with holographic duals, such as 4d N = 4 SYM dual to Type IIB string theory on AdS5 × S 5 , and 3d ABJM dual to Type IIA
–1–
JHEP07(2020)041
3 ABJM matrix model 3.1 Matrix model 2 Z 3.2 ∂m − m=0 2 Z 3.3 ∂m + m=0 4 Z 3.4 ∂m
To study the M-theory limit of ABJM, [10] showed that the ABJM matrix model could be written in the form
Z=
Z N O 1 X (−1)|σ| dN xhx1 . . . xN | ρˆ|xσ(1
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