Affine Gaudin models and hypergeometric functions on affine opers

Abstract

We conjecture that quantum Gaudin models in affine types admit families of higher Hamiltonians, labelled by the (countably infinite set of) exponents, whose eigenvalues are given by functions on a space of meromorphic opers associated with the Langlands dual Lie algebra. This is in direct analogy with the situation in finite types. However, in stark contrast to finite types, we prove that in affine types such functions take the form of hypergeometric integrals, over cycles of a twisted homology defined by the levels of the modules at the marked points. That result prompts the further conjecture that the Hamiltonians themselves are naturally expressed as such integrals.We go on to describe the space of meromorphic affine opers on an arbitrary Riemann surface. We prove that it fibres over the space of meromorphic connections on the canonical line bundle Ω. Each fibre is isomorphic to the direct product of the space of sections of the square of Ω with the direct product, over the exponents j not equal to 1, of the twisted cohomology of the j th tensor power of Ω. 32 7. Twisted homology and the integral pairing 39 8. Discussion 41 Appendix A. Hyperplane arrangements and quadratic Hamiltonians 47 References 51

Cubic hypergeometric integrals of motion in affine Gaudin models

“…According to the conjecture in [LVY], the HamiltoniansQ γ i can be diagonalized by means of the Bethe ansatz and have certain specific eigenvalues encoded by affine opers. ForQ γ 1 this is known; see [LVY,§5]. In the next two sections we perform a simple Bethe ansatz calculation to check that the conjecture is correct also forQ γ 2 at least for zero and one Bethe roots.…”

ODE/IQFT correspondence for the generalized affine $$ \mathfrak{sl} $$(2) Gaudin model

“…Note that the spectrum of the first few lowest integrals of motion for the affine Gaudin model associated to sl(N ) was studied in the work [13], while for the case of an arbitrary Lie algebra some conjectures are formulated in ref. [12]. Also, the classical limit of H…”

Affine opers and conformal affine Toda

“…The space of g-opers is by definition the quotient of the set of connections op g by the gauge action ofB + (A): Recall from Section 5.3 the definition of the exponents j ∈ E of g and of the generators p j ∈ a + . The following result was established in [33], following [12,29]. (The proof in [33] is in the meromorphic setting, but the same proof goes through the case of the disc.…”