Noncommutative Spectral Invariants and Black Hole Entropy

Abstract

We consider an intrinsic entropy associated with a local conformal net A by the coefficients in the expansion of the logarithm of the trace of the "heat kernel" semigroup. In analogy with Weyl theorem on the asymptotic density distribution of the Laplacian eigenvalues, passing to a quantum system with infinitely many degrees of freedom, we regard these coefficients as noncommutative geometric invariants. Under a natural modularity assumption, the leading term of the entropy (noncommutative area) is proportional to the central charge c, the first order correction (noncommutative Euler characteristic) is proportional to log µ A , where µ A is the global index of A, and the second spectral invariant is again proportional to c.We give a further general method to define a mean entropy by considering conformal symmetries that preserve a discretization of S 1 and we get the same value proportional to c.We then make the corresponding analysis with the proper Hamiltonian associated to an interval. We find here, in complete generality, a proper mean entropy proportional to log µ A with a first order correction defined by means of the relative entropy associated with canonical states. * Supported in part by JSPS. † Supported in part by GNAMPA and MIUR. 1By considering a class of black holes with an associated conformal quantum field theory on the horizon, and relying on arguments in the literature, we indicate a possible way to link the noncommutative area with the BekensteinHawking classical area description of entropy.

Structure and Classification of Superconformal Nets

“…If A b is modular, we can express d(ρ) by the Kac-Peterson Verlinde matrix S as this is equal to the Rehren matrix S. By combining all this information and an argument in [37], we then get the formula…”

“…We recall here a discussion made in [37]. Let B be a completely rational local conformal net of von Neumann algebras on S 1 .…”

“…We expect all local conformal completely rational nets to be modular (see [31] for results of similar kind). Furthermore, if B is a modular local conformal net and C an irreducible extension of B, then C is also modular [37], that allows to check the modularity property in several cases.…”

“…Now, if B is a modular net, then B is two-dimensional log-elliptic with noncommutative area a 0 = 2πc/24 [37], indeed the following asymptotic formula holds:…”

Nuclearity and Thermal States in Conformal Field Theory

“…Note that log-ellipticity naturally holds with α = 1 in rational conformal field theory (modular nets) [21].…”