1
$\begingroup$

Quantization of of an elliptic curve can be done in different ways. In C^*-algebraic version, one can start with the C^*-algebra of continuous functions on ordinary torus and by inserting a deformation parameter \theta into the product obtain a deformed non-commutaive C^*-algebra of functions on the quantum torus.

My question is:

Is there any natural way for deformation quantization of closed Riemann surfaces with higher genus in the above sense?

$\endgroup$
3
$\begingroup$

See the paper Quantization of Multiply Connected Manifolds, by Eli Hawkins. arXiv link.

$\endgroup$
3
$\begingroup$

One should definitely take a look at the work of Bordemann, Meinrenken, and Schlichenmaier: they provide a Berezin-Toeplitz inspired deformation quantization for all compact quantizable (i.e. the Kähler form is integral) Kähler manifolds. The asymptotics of this was discussed before by Cahen, Gutt, and Rawnsley in their 4 papers of quantization on Kähler manifolds. Later, Schlichenmaier and Karabegov determined the characteristic classes of the resulting formal star products explicitly, relating the BT-approach also to other versions.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.