Three-dimensional extensions of the~Alday--Gaiotto--Tachikawa relation
Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 1, pp. 73-99
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An extension of the two-dimensional (2d) Alday–Gaiotto–Tachikawa (AGT) relation to three dimensions starts from relating the theory on the domain wall between some two $S$-dual supersymmetric Yang–Mills (SYM) models to the 3d Chern–Simons (CS) theory. The simplest case of such a relation would presumably connect traces of the modular kernels in 2d conformal theory with knot invariants. Indeed, the two quantities are very similar, especially if represented as integrals of quantum dilogarithms. But there are also various differences, especially in the “conservation laws” for the integration variables holding for the monodromy traces but not for the knot invariants. We also consider another possibility: interpreting knot invariants as solutions of the Baxter equations for the relativistic Toda system. This implies another AGT-like relation: between the 3d CS theory and the Nekrasov–Shatashvili limit of the 5d SYM theory.
Keywords:
Alday–Gaiotto–Tachikawa relation, Chern–Simons theory, knot invariant.
@article{TMF_2012_172_1_a5,
author = {D. V. Galakhov and A. D. Mironov and A. Yu. Morozov and A. V. Smirnov},
title = {Three-dimensional extensions of {the~Alday--Gaiotto--Tachikawa} relation},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {73--99},
publisher = {mathdoc},
volume = {172},
number = {1},
year = {2012},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2012_172_1_a5/}
}
TY - JOUR AU - D. V. Galakhov AU - A. D. Mironov AU - A. Yu. Morozov AU - A. V. Smirnov TI - Three-dimensional extensions of the~Alday--Gaiotto--Tachikawa relation JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2012 SP - 73 EP - 99 VL - 172 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2012_172_1_a5/ LA - ru ID - TMF_2012_172_1_a5 ER -
%0 Journal Article %A D. V. Galakhov %A A. D. Mironov %A A. Yu. Morozov %A A. V. Smirnov %T Three-dimensional extensions of the~Alday--Gaiotto--Tachikawa relation %J Teoretičeskaâ i matematičeskaâ fizika %D 2012 %P 73-99 %V 172 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2012_172_1_a5/ %G ru %F TMF_2012_172_1_a5
D. V. Galakhov; A. D. Mironov; A. Yu. Morozov; A. V. Smirnov. Three-dimensional extensions of the~Alday--Gaiotto--Tachikawa relation. Teoretičeskaâ i matematičeskaâ fizika, Tome 172 (2012) no. 1, pp. 73-99. http://geodesic.mathdoc.fr/item/TMF_2012_172_1_a5/