An extension of the AGT relation from two to three dimensions begins from connecting the theory on domain wall between some two S-dual SYM models with the 3d Chern-Simons theory. The simplest kind of such a relation would presumably connect traces of the modular kernels in 2d conformal theory with knot invariants. Indeed, the both quantities are very similar, especially if represented as integrals of the products of quantum dilogarithm functions. However, there are also various differences, especially in the "conservation laws" for integration variables, which hold for the monodromy traces, but not for the knot invariants. We also discuss another possibility: interpretation of knot invariants as solutions to the Baxter equations for the relativistic Toda system. This implies another AGT like relation: between 3d Chern-Simons theory and the Nekrasov-Shatashvili limit of the 5d SYM.
Alday-Maldacena conjecture is that the area AΠ of the minimal surface in AdS5 space with a boundary Π, located in Euclidean space at infinity of AdS5, coincides with a double integral DΠ along Π, the Abelian Wilson average in an auxiliary dual model. The boundary Π is a polygon formed by momenta of n external light-like particles in N = 4 SYM theory, and in a certain n = ∞ limit it can be substituted by an arbitrary smooth curve (wavy circle). The Alday-Maldacena conjecture is known to be violated for n > 5, when it fails to be supported by the peculiar global conformal invariance, however, the structure of deviations remains obscure. The case of wavy lines can appear more convenient for analysis of these deviations due to the systematic method developed in [1] for (perturbative) evaluation of minimal areas, which is not yet available in the presence of angles at finite n. We correct a mistake in that paper and explicitly evaluate the h 2h2 terms, where the first deviation from the Alday-Maldacena duality arises for the wavy circle.
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