We extend a result about the gauge action on noncommutative solitons by showing that a family of functions can be gauged away to a Gaussian using the quantification condition given in On a gauge action on sigma model solitons IDAQP(2018).
We consider noncommutative theory of a compact scalar field. The recently discovered projector solitons are interpreted as classical vacua in the model considered. Localized solutions to the projector equation are pointed out and their brane interpretation is discussed. An example of the noncommutative soliton interpolating between such vacua is given. No strong noncommutativity limit is assumed.
Given a positive function on the set of edges of an arbitrary directed graph $E=(E^0,E^1)$, we define a one-parameter group of automorphisms on the C*-algebra of the graph $C^*(E)$, and study the problem of finding KMS states for this action. We prove that there are bijective correspondences between KMS states on $C^*(E)$, a certain class of states on its core, and a certain class of tracial states on $C_0(E^0)$. We also find the ground states for this action and give some examples.
We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as morphisms and a suitable metric category of spectral triples over commutative pre-C*-algebras. We also construct an embedding of a quotient of the category of spectral triples introduced in arXiv:math/0502583v1 into the latter metric category. Finally we discuss a further related duality in the case of orientation and spin-preserving maps between manifolds of fixed dimension.
We construct exact solitons on noncommutative tori for the type of actions arising from open string field theory. Given any projector that describes an extremum of the tachyon potential, we interpret the remaining gauge degrees of freedom as a gauge theory on the projective module determined by the tachyon. Whenever this module admits a constant curvature connection, it solves exactly the equations of motion of the effective string field theory. We describe in detail such a construction on the noncommutative tori. Whereas our exact solution relies on the coupling to a gauge theory, we comment on the construction of approximate solutions in the absence of gauge fields.
We consider the Type 1 and Type 2 noncommutative Borsuk-Ulam conjectures of Baum, D$k{a}$browski, and Hajac: there are no equivariant morphisms $A to A circledast_delta H$ or $H to A circledast_delta H$, respectively, when $H$ is a nontrivial compact quantum group acting freely on a unital $C^*$-algebra $A$. Here $A circledast_delta H$ denotes the equivariant noncommutative join of $A$ and $H$; this join procedure is a modification of the topological join that allows a free action of $H$ on $A$ to produce a free action of $H$ on $A circledast_delta H$. For the classical case $H = mathcal{C}(G)$, $G$ a compact group, we present a reduction of the Type 1 conjecture and counterexamples to the Type 2 conjecture. We also present some examples and conditions under which the Type 2 conjecture does hold.