Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

We study a quantum (non-commutative) representation of the affine Weyl group mainly of type E (1) 8 , where the representation is given by birational actions on two variables x, y with q-commutation relations. Using the tau variables, we also construct quantum “fundamental” polynomials F (x, y) which completely control the Weyl group actions. The geometric properties of the polynomials F (x, y) for the commutative case is lifted distinctively in the quantum case to certain singularity structures as the q-difference operators. This property is further utilized as the characterization of the quantum polynomials F (x, y). As an application, the quantum curve associated with topological strings proposed recently by the first named author is rederived by the Weyl group symmetry. The cases of type D (1) 5 , E (1) 6 , E (1) 7 are also discussed.

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