Fluctuating Commutative Geometry

We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can also fluctuate. The model describes the geometry of spaces with a countable number $n$ of points. The spectral principle of Connes and Chamseddine is used to define dynamics.We show that this simple model has two phases. The expectation value , the average number of points in the universe, is finite in one phase and diverges in the other. Moreover, the dimension $\delta$ is a dynamical observable in our model, and plays the role of an order parameter. The computation of is discussed and an upper bound is found, $< 2$. We also address another discrete model defined on a fixed $d=1$ dimension, where topology fluctuates. We comment on a possible spontaneous localization of topology.Comment: 7 pages. Talk at the conference "Spacetime and Fundamental Interactions: Quantum Aspects" (Vietri sul Mare, Italy, 26-31 May 2003), in honour of A. P. Balachandran's 65th birthda

Quasi-Topological Quantum Field Theories and Z2Z_2 Lattice Gauge Theories

We consider a two parameter family of $Z_2$ gauge theories on a lattice discretization $T(M)$ of a 3-manifold $M$ and its relation to topological field theories. Familiar models such as the spin-gauge model are curves on a parameter space $\Gamma$. We show that there is a region $\Gamma_0$ of $\Gamma$ where the partition function and the expectation value of the Wilson loop for a curve $\gamma$ can be exactly computed. Depending on the point of $\Gamma_0$, the model behaves as topological or quasi-topological. The partition function is, up to a scaling factor, a topological number of $M$. The Wilson loop on the other hand, does not depend on the topology of $\gamma$. However, for a subset of $\Gamma_0$, depends on the size of $\gamma$ and follows a discrete version of an area law. At the zero temperature limit, the spin-gauge model approaches the topological and the quasi-topological regions depending on the sign of the coupling constant.Comment: 19 pages, 13 figure

Quasi-Topological Field Theories in Two Dimensions as Soluble Models

We study a class of lattice field theories in two dimensions that includes gauge theories. Given a two dimensional orientable surface of genus $g$, the partition function $Z$ is defined for a triangulation consisting of $n$ triangles of area $\epsilon$. The reason these models are called quasi-topological is that $Z$ depends on $g$, $n$ and $\epsilon$ but not on the details of the triangulation. They are also soluble in the sense that the computation of their partition functions can be reduced to a soluble one dimensional problem. We show that the continuum limit is well defined if the model approaches a topological field theory in the zero area limit, i.e., $\epsilon \to 0$ with finite $n$. We also show that the universality classes of such quasi-topological lattice field theories can be easily classified. Yang-Mills and generalized Yang-Mills theories appear as particular examples of such continuum limits.Comment: 23 pages, 16 figures, uses psbox.te

A Recipe for Constructing Frustration-Free Hamiltonians with Gauge and Matter Fields in One and Two Dimensions

State sum constructions, such as Kuperberg's algorithm, give partition functions of physical systems, like lattice gauge theories, in various dimensions by associating local tensors or weights, to different parts of a closed triangulated manifold. Here we extend this construction by including matter fields to build partition functions in both two and three space-time dimensions. The matter fields introduces new weights to the vertices and they correspond to Potts spin configurations described by an $\mathcal{A}$-module with an inner product. Performing this construction on a triangulated manifold with a boundary we obtain the transfer matrices which are decomposed into a product of local operators acting on vertices, links and plaquettes. The vertex and plaquette operators are similar to the ones appearing in the quantum double models (QDM) of Kitaev. The link operator couples the gauge and the matter fields, and it reduces to the usual interaction terms in known models such as $\mathbb{Z}_2$ gauge theory with matter fields. The transfer matrices lead to Hamiltonians that are frustration-free and are exactly solvable. According to the choice of the initial input, that of the gauge group and a matter module, we obtain interesting models which have a new kind of ground state degeneracy that depends on the number of equivalence classes in the matter module under gauge action. Some of the models have confined flux excitations in the bulk which become deconfined at the surface. These edge modes are protected by an energy gap provided by the link operator. These properties also appear in "confined Walker-Wang" models which are 3D models having interesting surface states. Apart from the gauge excitations there are also excitations in the matter sector which are immobile and can be thought of as defects like in the Ising model. We only consider bosonic matter fields in this paper.Comment: 52 pages, 58 figures. This paper is an extension of arXiv:1310.8483 [cond-mat.str-el] with the inclusion of matter fields. This version includes substantial changes with a connection made to confined Walker-Wang models along the lines of arXiv:1208.5128 and subsequent works. Accepted for publication in JPhys