In Levin, Olshanetsky and Zotov (JHEP 10 (2014) 109), Lax pairs and linear problems for Calogero–Moser-type systems were built from quantum R-matrices with the Planck constant ℏ\hbar playing the role of the spectral parameter, and structural analogies with the Knizhnik–Zamolodchikov–Bernard (KZB) equations were established, while the precise correspondence between the two structures was explicitly deferred to future work. The companion conjecture of that paper — the trace formula (4.21) asserting that traces of powers of the R-matrix-valued Lax matrix reproduce the glN~\mathfrak{gl}_{\tilde N} Calogero–Moser Hamiltonians — was subsequently proven by Zotov (2016, 2018), and the surviving open core is to construct the exact correspondence between the R-matrix-valued linear problems and the KZB connections.

Background

A Lax pair is a reformulation of a Hamiltonian evolution equation L˙=[L,M]\dot L=[L,M] as a compatibility condition of two linear problems; when the matrix LL depends on an auxiliary complex variable, the spectral parameter, the eigenvalues of L(z)L(z) are conserved quantities, which makes integrability manifest. The classical many-body Calogero–Moser (CM) model — particles on a line interacting through elliptic pair potentials — admits such a representation with an elliptic spectral parameter (Krichever's construction), whose matrix entries are values of the Kronecker function ϕ(ℏ,z)\phi(\hbar,z), the basic genus-one elliptic function satisfying the Fay three-term identity. A quantum R-matrix is a matrix R12(z)R_{12}(z) solving the quantum Yang–Baxter equation R12R13R23=R23R13R12R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}; the Baxter–Belavin R-matrix is its elliptic, glN\mathfrak{gl}_N fundamental-representation solution, and it can be viewed as a matrix generalization of the Kronecker function, with ℏ\hbar entering as the Planck constant of the underlying quantum theory. Besides the Yang–Baxter equation it satisfies unitarity (R12(z)R21(−z)R_{12}(z)R_{21}(-z) proportional to the identity) and the associative Yang–Baxter equation (AYBE) R12R23=R13R12+R23R13R_{12}R_{23}=R_{13}R_{12}+R_{23}R_{13} with shifted arguments, the R-matrix analogue of the scalar Fay identity. The Knizhnik–Zamolodchikov (KZ) equations are the differential equations satisfied by conformal blocks of two-dimensional conformal field theory: first-order linear equations ℏ∂ziψ=∑j≠irijψ\hbar\partial_{z_i}\psi=\sum_{j\ne i}r_{ij}\psi whose connections are built from a classical r-matrix (here ℏ\hbar plays the role of the KZ level parameter; in the constructions described below the same symbol ℏ\hbar instead serves as the spectral parameter of the classical problem); their elliptic-curve version, augmented by an additional modular connection ∇τ\nabla_\tau (acting by differentiation with respect to the elliptic modulus τ\tau), is due to Bernard (KZB equations). In the source paper (Levin–Olshanetsky–Zotov, JHEP 10 (2014) 109), Levin, Olshanetsky and Zotov reversed the usual roles: they used the quantum R-matrix in place of the scalar Kronecker function inside Krichever-type Lax matrices, so that the Planck constant ℏ\hbar of the quantum R-matrix serves as the spectral parameter of a classical glN~\mathfrak{gl}_{\tilde N} Calogero–Moser system (the integers NN and N~\tilde N label the quantum and classical matrix sizes, and ν\nu is a coupling constant). They proved that the Yang–Baxter, unitarity and associative Yang–Baxter identities imply compatibility of these R-matrix-valued Lax pairs, constructed rational KZB equations from a τ\tau-deformed family of R-matrices (a deformation in the elliptic modulus τ\tau of the curve), and observed that the associated quantum linear problem resembles the KZ/KZB connections — generalizing them in the same way as Lax pairs with spectral parameter generalize those without. They stated the trace formula (4.21) — traces of powers of the Lax matrix reproduce the glN~\mathfrak{gl}_{\tilde N} Calogero–Moser Hamiltonians — as a conjecture with the proof deferred, and closed with: 'We hope to clarify exact relations between R-matrix valued linear problems and KZB equations in our future papers.' The trace-formula conjecture has since been proven (Zotov, Theoret. Math. Phys. 189 (2016) 1554; 197 (2018) 1755), and several structural bridges are known — the Matsuo–Cherednik construction relates KZ-type equations to Calogero models, and the R-matrix-valued Lax pairs were identified with Hitchin systems (integrable systems on moduli spaces of bundles) on elliptic curves with non-trivial characteristic classes. What no later work supplies, however, is the precise map the authors deferred: an explicit correspondence under which the R-matrix-valued linear problems with ℏ\hbar-as-spectral-parameter and the KZB connections become identifiable objects. Flatness of the KZB connections and the Calogero–Moser dynamics generated by traces of the R-matrix-valued Lax matrix are already theorems of the source; the missing exact relation, made explicit by Grekov–Zotov (2018), is a precise relation between the KZB conformal blocks and the Baker–Akhiezer (wave) functions of the corresponding R-matrix-valued Lax pair.

Problem Statement

Construct the exact relation between the R-matrix-valued linear problems of Levin–Olshanetsky–Zotov (JHEP 10 (2014) 109) and the Knizhnik–Zamolodchikov–Bernard (KZB) equations, which the source explicitly deferred ('We hope to clarify exact relations between R-matrix valued linear problems and KZB equations in our future papers'). Concretely: for the quantum R-matrices actually used in the source — the elliptic Baxter–Belavin R-matrix and the rational τ\tau-deformed R-matrix obtained by deformation of the quantum glN\mathfrak{gl}_N rational R-matrix, from which the rational KZB equations are constructed (both solutions of the quantum Yang–Baxter equation, the unitarity condition, and the associative Yang–Baxter equation) — establish a precise, mathematically identified correspondence between (a) the linear problems whose Lax matrices are built from the R-matrix with the Planck constant ℏ\hbar in the role of the spectral parameter — the classical linear problems (∂t+M)Ψ=0(\partial_t+M)\Psi=0, (∂ℏ+L)Ψ=0(\partial_\hbar+L)\Psi=0 and the quantum linear problem L^Ψ=ΨΛ\hat L\Psi=\Psi\Lambda — and (b) the KZB connections constructed in the same paper: the elliptic KZB equations and the rational τ\tau-deformed KZB equations on the punctured elliptic curve. The correspondence must be exact rather than a structural association: as Grekov–Zotov (J. Phys. A 51 (2018) 315202) states at the end of its §3, what is missing is a precise relation between the conformal blocks of the KZB equations and the Baker–Akhiezer (wave) functions Ψ\Psi of the corresponding R-matrix-valued linear problems, identifying the two structures rather than associating KZB connections built from the classical-limit rr- and mm-matrices to the Lax pair. The trigonometric degeneration is explicitly deferred by the source to separate publications and lies outside this target; the coupling constant ν\nu enters as a parameter of the R-matrix-valued Lax matrices.

References

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