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Department of Geometry and Topology, Steklov Mathematical Institute, Gubkina ul. 8, 117966, Moscow GSP-1, Russia and Independent University of Moscow, Bol'shoi Vlasievskii per. 11, Moscow 121002, Russia oleg AT sheinman.mccme.ru
The second order casimirs for affine Krichever-Novikov algebras $\widehat{\frak gl}_{g,2}$ and $\widehat{\frak sl}_{g,2}$ are described. More general operators which we call semi-casimirs are introduced. It is proved that semi-casimirs induce well-defined operators on conformal blocks and there is a natural projection of $T_\Sigma {\Cal M}_{g,2}^{(1)}$ onto the space of these operators where ${\Cal M}_{g,2}^{(1)}$ is the moduli space of Riemann surfaces with two marked points and fixed 1-jets of local coordinates at those points, $\Sigma\in {\Cal M}_{g,2}^{(1)}$ is the Riemann surface involved into the definition of Krichever-Novikov algebras in question.
Let $\Sigma$ be a compact algebraic curve over ${\Bbb C}$ of genus $g$ with two marked points $P_{\pm}$, ${\Cal A}$ be the algebra of meromorphic functions on $\Sigma$ which are regular exept at the points $P_\pm$, ${\g}$ be a complex semi-simple or reductive Lie algebra. Then
$${\gh}=\g\otimes_{\Bbb C}{\Cal A}\oplus{\Bbb C}c
$$
is called the {\it affine Krichever-Novikov algebra}
(first introduced in \cite{1}). The bracket on $\gh$ is given by the formula
$$[x\otimes A,y\otimes B]=[x,y]\otimes
AB+\gamma(x\otimes A,y\otimes B)c , \,\,
[x\otimes A,c]=0,
$$
where
$$ \gamma (x\otimes A,y\otimes B)=
\text{tr}(\text{ad}\,x\circ \text{ad}\,y)
\text{res}_{P_+}(AdB).
$$
Let also $\L$ be the algebra of meromorphic vector
fields on $\Sigma$ with the same analitic properties and $\D$ be the
semi-direct sum of $\gh$ and $\L$.
In [2] the fermion (or wedge) representations of the affine Krichever-Novikov algebras were introduced. They are parametrized by the triples consisting of a holomorphic vector bundle on $\Sigma$, a dominant weight of $\g$ and of an integer ({\it the charge}). There are two actions of $\L$ in the space of a fermion representation. The first is due to the existence of a meromorphic (therefore flat) logarithmic conection on the corresponding bundle (for $e\in\L$ we denote this action $\eh$). The second is the Sugawara representation of $\L$ (for $e\in\L$ we denote it $T(e)$). Consider operators of the form $\Delta(e)=\eh-T(e)$. All these actions can be continued on a certain completion $\overline{\L}$ of $\L$. Observe that $[\Delta(e),\widehat{\frak sl}_{g,2}]=0$ for any $e\in\overline{\L}$.
Definition 1 $\Delta(e)$ is called a (second order) casimir for $\gh$ iff $[\Delta(e),{\Cal A}]=0$.
Theorem 1 For a generic fermion representation there is exactly one (up to a scalar factor) casimir. The corresponding vector field has a simple zero at the point $P_+$.
Definition 2 $\Delta(e)$ is called a semi-casimir for $\gh$ iff $[\Delta(e),\tilde{\Cal A}_-]=0$ where $\tilde{\Cal A}_-$ denotes the subspace of ``generating'' operators in $\Cal A$.
Theorem 2 Semi-casimirs induce well-defined operators on the space of conformal blocks. For a generic fermion representation there is a natural projection of $T_\Sigma {\Cal M}_{g,2}^{(1)}$ onto the space of operators induced by semi-casimirs on conformal blocks over $\Sigma$.
