Quantum field theory on noncommutative spaces

2D gravity and random matrices

We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization approach that removes local entanglement and produces a coarse-grained lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixedpoint tensors Tinv plus the symmetry group Gsym of the tensors i.e., the symmetry group of the Lagrangian characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by two-dimensional 2D statistical Ising model, 2D statistical loop-gas model, and 1+1D quantum spin-1/2 and spin-1 models. In particular, using such a Gsym,Tinv characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry-protected topological phase. The Gsym,Tinv characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.

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Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order

U duality and M theory

Here we examine the noncommutative counterpart of QED, which is called as noncommutative QED. The theory is obtained by examining the consistent minimal coupling to noncommutative U(1) gauge field. The *-product admits the coupling of the matter with only three varieties of charges, i.e., 0, +1 and -1. Ultraviolet divergence can be absorbed into the rescaling of the fields and the parameters at least at one loop level. To examine the infrared aspect of the theory the anomalous magnetic dipole moment is calculated. The dependence on the direction of photon momentum reflects the Lorentz symmetry violation of the system. The explicit calculation of the finite part of the photon vacuum polarization shows the singularity ln({q C^TC q}) (C^{\mu\nu} is a noncommutative parameter.) in the infrared side which also exists in noncommutative Yang-Mills theory. It is associated with the ultraviolet behavior of the theory. We also consider the extension to chiral gauge theory in the present context, but the requirement of anomaly cancellation allows only noncommutative QED.

Perturbative analysis on infrared aspects of noncommutative QED on R^4

Non-commutative Yang–Mills in IIB matrix model

We derive a long distance effective action for space-time coordinates from a IIB matrix model. It provides us an effective tool to study the structures of space-time. We prove the finiteness of the theory for finite $N$ to all orders of the perturbation theory. Space-time is shown to be inseparable and its dimensionality is dynamically determined. The IIB matrix model contains a mechanism to ensure the vanishing cosmological constant which does not rely on the manifest supersymmetry. We discuss possible mechanisms to obtain realistic dimensionality and gauge groups from the IIB matrix model.

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Space-time structures from iib matrix model

We show that twisted reduced models can be interpreted as noncommutative Yang-Mills theory. Based upon this correspondence, we obtain noncommutative Yang-Mills theory with D-brane backgrounds in IIB matrix model. We propose that IIB matrix model with D-brane backgrounds serve as a concrete definition of noncommutative Yang-Mills. We investigate D-instanton solutions as local excitations on D3-branes. When instantons overlap, their interaction can be well described in gauge theory and AdS/CFT correspondence. We show that IIB matrix model gives us the consistent potential with IIB supergravity when they are well separated.

Noncommutative Yang-Mills in IIB Matrix Model

IIB Matrix Model

We study the q-deformed su(2) spin network as a three-dimensional quantum gravity model. We show that in the semiclassical continuum limit the Turaev-Viro invariant obtained recently defines a naturally regularized path integral a` la Ponzano and Regge, in which a contribution from the cosmological term is effectively included. The regularization-dependent cosmological constant is found to be 4${\mathrm{\ensuremath{\pi}}}^{2}$/${\mathit{k}}^{2}$+O(${\mathit{k}}^{\mathrm{\ensuremath{-}}4}$), where ${\mathit{q}}^{2\mathit{k}}$=1. We also discuss the relation to the Euclidean Chern-Simons-Witten gravity in three dimensions.

Tsukasa Tada

Papers

Non-commutative Yang–Mills in IIB matrix model

Space-time structures from iib matrix model

Noncommutative Yang-Mills in IIB Matrix Model

IIB Matrix Model

Three-dimensional gravity from the Turaev-Viro invariant.