Topology Change for Fuzzy Physics: Fuzzy Spaces as Hopf Algebras

Fuzzy spaces are obtained by quantizing adjoint orbits of compact semi-simple Lie groups. Fuzzy spheres emerge from quantizing S^2 and are associated with the group SU(2) in this manner. They are useful for regularizing quantum field theories and modeling spacetimes by non-commutative manifolds. We show that fuzzy spaces are Hopf algebras and in fact have more structure than the latter. They are thus candidates for quantum symmetries. Using their generalized Hopf algebraic structures, we can also model processes where one fuzzy space splits into several fuzzy spaces. For example we can discuss the quantum transition where the fuzzy sphere for angular momentum J splits into fuzzy spheres for angular momenta K and L.Comment: LaTeX, 13 pages, v3: minor additions, added references, v4: corrected typos, to appear in IJMP

Chaos from Massive Deformations of Yang-Mills Matrix Models

We focus on an $SU(N)$ Yang-Mills gauge theory in $0+1$-dimensions with the same matrix content as the bosonic part of the BFSS matrix model, but with mass deformation terms breaking the global $SO(9)$ symmetry of the latter to $SO(5) \times SO(3) \times {\mathbb Z}_2$. Introducing an ansatz configuration involving fuzzy four and two spheres with collective time dependence, we examine the chaotic dynamics in a family of effective Lagrangians obtained by tracing over the aforementioned ansatz configurations at the matrix levels $N = \frac{1}{6}(n+1)(n+2)(n+3)$, for $n=1,2,\cdots\,,7$. Through numerical work, we determine the Lyapunov spectrum and analyze how the largest Lyapunov exponents(LLE) change as a function of the energy, and discuss how our results can be used to model the temperature dependence of the LLEs and put upper bounds on the temperature above which LLE values comply with the Maldacena-Shenker-Stanford (MSS) bound $2 \pi T$ , and below which it will eventually be violated.Comment: 32+1 pages, 7 tables, 6 figures. Expanded discussion in section 3 establishing upper bound on temperature above which largest Lyapunov exponent complies with the MSS bound, published versio

Duality in Fuzzy Sigma Models

Nonlinear `sigma' models in two dimensions have BPS solitons which are solutions of self- and anti-self-duality constraints. In this paper, we find their analogues for fuzzy sigma models on fuzzy spheres which were treated in detail by us in earlier work. We show that fuzzy BPS solitons are quantized versions of `Bott projectors', and construct them explicitly. Their supersymmetric versions follow from the work of S. Kurkcuoglu.Comment: Latex, 9 pages; misprints correcte

Nonlocal regularisation of noncommutative field theories

We study noncommutative field theories, which are inherently nonlocal, using a Poincar\'e-invariant regularisation scheme which yields an effective, nonlocal theory for energies below a cut-off scale. After discussing the general features and the peculiar advantages of this regularisation scheme for theories defined in noncommutative spaces, we focus our attention onto the particular case when the noncommutativity parameter is inversely proportional to the square of the cut-off, via a dimensionless parameter $\eta$. We work out the perturbative corrections at one-loop order for a scalar theory with quartic interactions, where the signature of noncommutativity appears in $\eta$-dependent terms. The implications of this approach, which avoids the problems related to UV-IR mixing, are discussed from the perspective of the Wilson renormalisation program. Finally, we remark about the generality of the method, arguing that it may lead to phenomenologically relevant predictions, when applied to realistic field theories.Comment: 1+11 pages, 6 figures; v2: references added, typos corrected, conclusions unchange