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We propose a discrete model whose continuum limit reproduces the string susceptibility and the scaling dimensions of (2, 4m) minimal superconformal models coupled to 2D supergravity. The basic assumption in our presentation is a set of super-Virasoro constraints imposed on the partition function. We recover the Neveu-Schwarz and Ramond sectors of the theory, and we are also able to evaluate all planar loop correlation functions in the continuum limit. We find evidence to identify the integrable hierarchy of nonlinear equations describing the double scaling limit as a supersymmetric generalization of KP studied by Rabin.

Explicit answer is given for the HOMFLY polynomial of the figure eight knot 4 1 in arbitrary symmetric representation R = [p ]. It generalizes the old answers for p = 1 and 2 and the recently derived results for p = 3, 4, which are fully consistent with the Ooguri-Vafa conjecture. The answer can be considered as a quantization of the H R = H |R|[1] identity for the "special" polynomials (they define the leading asymptotics of HOMFLY at q = 1), and arises in a form, convenient for comparison with the representation of the Jones polynomials as sums of dilogarithm ratios. In particular, we construct a difference equation ("non-commutative A-polynomial") in the representation variable p. Simple symmetry transformation provides also a formula for arbitrary antisymmetric (fundamental) representation R = [1 p ], which also passes some obvious checks. Also straightforward is a deformation from HOMFLY to superpolynomials. Further generalizations seem possible to arbitrary Young diagrams R, but these expressions are harder to test because of the lack of alternative results, even partial.

Interpretation of exact results on the low-energy limit of 4d N = 2 SUSY YM in the language of 1d integrability theory is reviewed. The case of elliptic Calogero system, associated with the flow between N = 4 and N = 2 SUSY in 4d, is considered in some detail.

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