TY - JOUR

T1 - Toward U (N| M) knot invariant from ABJM theory

AU - Eynard, Bertrand

AU - Kimura, Taro

N1 - Funding Information:
We would like to thank G. Borot and M. Mariño for fruitful discussions. We are also grateful to A. Brini for carefully reading the manuscript and giving useful comments. BE thanks Centre de Recherches Mathématiques de Montréal, the FQRNT grant from the Québec government, Piotr Sułkowski and the ERC starting grant Fields-Knots. The work of TK is supported in part by Keio Gijuku Academic Development Funds, MEXT-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (No. S1511006), and JSPS Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science” (No. JP15H05855).
Publisher Copyright:
© 2017, Springer Science+Business Media Dordrecht.

PY - 2017/6/1

Y1 - 2017/6/1

N2 - We study U (N| M) character expectation value with the supermatrix Chern–Simons theory, known as the ABJM matrix model, with emphasis on its connection to the knot invariant. This average just gives the half-BPS circular Wilson loop expectation value in ABJM theory, which shall correspond to the unknot invariant. We derive the determinantal formula, which gives U (N| M) character expectation values in terms of U (1 | 1) averages for a particular type of character representations. This means that the U (1 | 1) character expectation value is a building block for the U (N| M) averages and also, by an appropriate limit, for the U (N) invariants. In addition to the original model, we introduce another supermatrix model obtained through the symplectic transform, which is motivated by the torus knot Chern–Simons matrix model. We obtain the Rosso–Jones-type formula and the spectral curve for this case.

AB - We study U (N| M) character expectation value with the supermatrix Chern–Simons theory, known as the ABJM matrix model, with emphasis on its connection to the knot invariant. This average just gives the half-BPS circular Wilson loop expectation value in ABJM theory, which shall correspond to the unknot invariant. We derive the determinantal formula, which gives U (N| M) character expectation values in terms of U (1 | 1) averages for a particular type of character representations. This means that the U (1 | 1) character expectation value is a building block for the U (N| M) averages and also, by an appropriate limit, for the U (N) invariants. In addition to the original model, we introduce another supermatrix model obtained through the symplectic transform, which is motivated by the torus knot Chern–Simons matrix model. We obtain the Rosso–Jones-type formula and the spectral curve for this case.

KW - ABJM theory

KW - Chern–Simons theory

KW - Knot invariant

KW - Matrix model

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U2 - 10.1007/s11005-017-0936-0

DO - 10.1007/s11005-017-0936-0

M3 - Article

AN - SCOPUS:85011265970

SN - 0377-9017

VL - 107

SP - 1027

EP - 1063

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

IS - 6

ER -