Exact results and Schur expansions in quiver Chern-Simons-matter theories
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Springer
Received: August 6, 2020 Accepted: September 6, 2020 Published: October 5, 2020
Leonardo Santillia and Miguel Tierzb,a a
Grupo de F´ısica Matem´ atica, Departamento de Matem´ atica, Faculdade de Ciˆencias, Universidade de Lisboa, Campo Grande, Edif´ıcio C6, 1749-016 Lisboa, Portugal b Departamento de Matem´ atica, Faculdade de Ciˆencias, ISCTE - Instituto Universit´ ario de Lisboa, Avenida das For¸cas Armadas, 1649-026 Lisboa, Portugal
E-mail: [email protected], [email protected] Abstract: We study several quiver Chern-Simons-matter theories on the three-sphere, combining the matrix model formulation with a systematic use of Mordell’s integral, computing partition functions and checking dualities. We also consider Wilson loops in ABJ(M) theories, distinguishing between typical (long) and atypical (short) representations and focusing on the former. Using the Berele-Regev factorization of supersymmetric Schur polynomials, we express the expectation value of the Wilson loops in terms of sums of observables of two factorized copies of U(N ) pure Chern-Simons theory on the sphere. Then, we use the Cauchy identity to study the partition functions of a number of quiver Chern-Simons-matter models and the result is interpreted as a perturbative expansion in the parameters tj = −e2πmj , where mj are the masses. Through the paper, we incorporate different generalizations, such as deformations by real masses and/or Fayet-Iliopoulos parameters, the consideration of a Romans mass in the gravity dual, and adjoint matter. Keywords: Matrix Models, Field Theories in Lower Dimensions, Chern-Simons Theories, Supersymmetric Gauge Theory ArXiv ePrint: 2008.00465
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)022
JHEP10(2020)022
Exact results and Schur expansions in quiver Chern-Simons-matter theories
Contents 1 Introduction
1 3 3 5 5 6 6 7 8 8 10 11
3 Evaluation of partition functions 3.1 U(1)k Chern-Simons theory with a fundamental hypermultiplet 3.2 Single node quivers 3.2.1 Abelian A1 theory with two flavours 3.2.2 Abelian A1 theory with Nf flavours 3.2.3 Wilson loops from Mordell integrals: Abelian A1 theory 3.2.4 Non-Abelian A1 theory with Nf flavours 3.3 Lessons so far 3.3.1 U(1)k theory at rational k 3.4 Abelian quivers 3.4.1 Abelian A2 theory 3.4.2 Abelian A3 theory 3.4.3 Abelian ABJM 3.4.4 Non-Abelian A2 theory 3.5 Abelian quivers at k = ±1
11 11 15 15 16 18 18 19 20 20 21 22 24 26 28
4 Wilson loops in ABJ theory 4.1 On Lie superalgebras representations 4.2 Wilson loops in typical representations 4.3 Two Wilson loops 4.3.1 Inverting one of the two Wilson loops 4.4 Three or more Wilson loops 4.5 Necklace quivers
30 31 31 32 34 35 38
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JHEP10(2020)022
2 Physics background and mathematical setup 2.1 Chern-Simons theories on S3 2.1.1 ABJ(M) theories and CS levels 2.1.2 12 -BPS Wilson loops 2.1.3 Unknot invariant in pure Chern-Simons theory 2.2 Mordell integrals 2.3 Moments of the log-normal 2.4 Cauchy identities, Gauss sums and notation 2.4.1 Cauchy identity
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