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## Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations

Cornell University
,
2014.

We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard
(KZB) equations with $\tilde N$ punctures by deformation of the corresponding
quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit
of the first one brings the model to the ordinary rational KZ equation. Another
one is $\tau$. At the level of classical mechanics the deformation parameter
$\tau$ allows to extend the previously obtained modified Gaudin models to the
modified Schlesinger systems. Next, we notice that the identities underlying
generic (elliptic) KZB equations follow from some additional relations for the
properly normalized $R$-matrices. The relations are noncommutative analogues of
identities for (scalar) elliptic functions. The simplest one is the unitarity
condition. The quadratic (in $R$ matrices) relations are generated by
noncommutative Fay identities. In particular, one can derive the quantum
Yang-Baxter equations from the Fay identities. The cubic relations provide
identities for the KZB equations as well as quadratic relations for the
classical $r$-matrices which can be halves of the classical Yang-Baxter
equation. At last we discuss the $R$-matrix valued linear problems which
provide ${\rm gl}_{\tilde N}$ Calogero-Moser (CM) models and Painleve equations
via the above mentioned identities. The role of the spectral parameter plays
the Planck constant of the quantum $R$-matrix. When the quantum ${\rm gl}_N$
$R$-matrix is scalar ($N=1$) the linear problem reproduces the Krichever's
ansatz for the Lax matrices with spectral parameter for the ${\rm gl}_{\tilde
N}$ CM models. The linear problems for the quantum CM models generalize the KZ
equations in the same way as the Lax pairs with spectral parameter generalize
those without it.