Quantum Statistical Mechanical Systems Associated to Riemann Surfaces
Mentor:Matilde Marcolli, Professor of Mathematics, California Institute of Technology
Given a compact hyperbolic Riemann surface, we construct a quantum statistical mechanical system that encodes its conformal isomorphism class. Building from the spectral triple construction of Cornelissen and Marcolli (J. Geom. Phys. 58, no. 5, pp. 619-632, 2008), the C*-dynamical system is defined using a Hamiltonian constructed from the Dirac operator. The induced time evolution gives a noncommutative C*-algebra of observables that extends the C*-algebra of the aforementioned spectral triple. The partition function and equilibrium states of the quantum statistical mechanical system are found to be equivalent to zeta functions of the spectral triple, and again from the results of Cornelissen and Marcolli, characterization of conformal isomorphism class is established. Generalization of the construction is discussed at the end.