Quantum statistical mechanics in arithmetic topology

Abstract

This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere replacing abelian extensions of the field of rational numbers. The operator algebraic properties of this system differ significantly from the Bost-Connes case, due to the properties of the action of the semigroup of knots on a direct limit of knot groups. The resulting algebra of observables is a noncommutative Bernoulli product. We describe the main properties of the associated quantum statistical mechanical system and of the relevant partition functions, which are obtained from simple knot invariants like genus and crossing number.

Quantum Statistical Mechanics of the Absolute Galois Group

“…The definition of a quantum statistical mechanical system associated to this construction, relies on the general setting for a quantum statistical mechanics for the group Q * + described in [57], which we recall briefly here, adapted to our environment.…”

“…The smallest sub-C * -algebra A(Q * + ) of B 2 (Q * + ) that contains C * r (Q * + ) and is preserved by the time evolution σ t (X) = e −itH Xe itH , with H as above, is generated by C * r (Q * + ) and the projections Π k, with Π k, a/b = a/b when k|a and |b, for (a, b) = 1, and zero otherwise. The spectral projections of H belong to this C * -algebra, hence the time evolution acts by inner automorphisms, see [57] for a more detailed discussion. Proof .…”

“…We now consider the enveloping group Υ = Q * + Z/2Z. We proceed as in the quantum statistical mechanics of Q * + described in [57] and recalled above. For a positive rational number r ∈ Q * + with prime decomposition r = p a 1 1 • • • p a where p i a pairwise distinct primes and a i ∈ Z,…”

“…This shows that the sub-C * -algebra of B( 2 (Υ)) generated by C * r (Υ) and by the projections Π m, is stable under the time evolution σ t . As in the case of Q * + discussed in [57] the spectral projections of H are in the algebra A Υ hence the time evolution is acting by inner automorphisms on A Υ .…”

“…We discuss here some functions associated to the evaluation of low temperature KMS states (Gibbs states) of the time evolutions defined earlier in this section. We start with a simpler case based on the quantum statistical mechanics of Q * + of [57], and then we move to the dynamics considered in Lemma 3.16.…”