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.

We review our proposal for a constructive definition of superstring, type IIB matrix model. The IIB matrix model is a manifestly covariant model for space-time and matter which possesses N=2 supersymmetry in ten dimensions. We refine our arguments to reproduce string perturbation theory based on the loop equations. We emphasize that the space-time is dynamically determined from the eigenvalue distributions of the matrices. We also explain how matter, gauge fields and gravitation appear as fluctuations around dynamically determined space-time.Comment: 37 pages, LaTex with PTPTex.sty, 2 epsf figures. Proceedings of the 13th Nishinomiya Yukawa Memorial Symposium (November, 1998

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.Comment: 39 pages, Latex, uses epsf and axodra

Elaborating on our previous presentation, where the term dipolar quantization was introduced, we argue here that adopting L 0 − (L 1 + L −1 )/2 +L 0 − (L 1 +L −1 )/2 as the Hamiltonian instead of L 0 +L 0 yields an infinite circumference limit in two-dimensional conformal field theory. The new Hamiltonian leads to dipolar quantization instead of radial quantization. As a result, the new theory exhibits a continuous and strongly degenerated spectrum in addition to the Virasoro algebra with a continuous index. Its Hilbert space exhibits a different inner product than that obtained in the original theory. The idiosyncrasy of this particular Hamiltonian is its relation to the so-called sine-square deformation, which is found in the study of a certain class of quantum statistical systems. The appearance of the infinite circumference explains why the vacuum states of sinesquare deformed systems are coincident with those of the respective closed-boundary systems.

We study the ^-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 4K^/k~-^Oik ""*), where <7^* = 1. We also discuss the relation to the Euclidean Chern-Simons-Witten gravity in three dimensions.PACS numbers: 04.60.+n, 02.20.+b It is well known that three-dimensional gravity is perturbatively trivial in the sense that there are no local degrees of freedom. In three dimensions Ricci-flat spacetime means truly flat space-time, so there are no gravitational wave modes but only topological excitation. The topological nature of three-dimensional gravity has already been encountered in the old work of Ponzano and Regge on the semiclassical limit of the Racah coefficients of su(2) [iL In their seminal paper a triangulationindependent quantity was defined by utilizing a certain relation for 6j symbols, and was shown to be viewed as a path integral for three-dimensional quantum gravity in the semiclassical continuum limit. This work was developed further by several authors [2]. Their treatment has two advantages: First, the measure is naturally fixed, and second, the triangle inequality for each simplex is automatically satisfied by virtue of the property of 67 symbols. However, their expression diverges and needs some regularization.Recently, Turaev and Viro [3] constructed a new topological invariant from the ^-deformed su(2) spin network when ^ is a root of unity. Their construction strongly resembles that of Ref. [l], and, moreover, the Turaev-Viro (TV) invariant is naturally regularized and finite due to the restriction for the spin variables. Therefore the TV invariant is expected to be considered as a regularized path integral for three-dimensional quantum gravity. In this Letter we estimate the asymptotic behavior of the q-6J symbol by the WKB approximation along the lines of Ref. [4], and see how the path integral defined by the su(2) spin network receives "quantum" corrections from <7-deformation. We show that a contribution from the cosmological term is effectively included in the path integral from the TV invariant.In Ref. [1] a sum of the products of four (classical) su(2) Racah-Wigner 67 symbols was considered: X (2jc + l)(2/,-f 0(2/2+0(2/3+1)(-1)^ .v,/"/2,/3: allowed vulue J\ J2 J2 X l\ I2 J6 75 J\ /l /2 /3 74 72 76 M73 75 74 I2 h X JI/3 X l\ (1) where ;t'^^ + X/=iA+2f=i7/-By repeated application of the Biedenharn-Elliott identity and the orthogonal relation, (1) is reduced to a single 67 symbol: 2:(2x+l)^ x«0 71 72 73 74 75 76(2)To give meaning to the divergent expression (2) a large-angular-momentum cutoff L is introduced in the summation, so that we take the equality between (1) and (2) as the following renormalized identity: 71 72 73 7*4 7*5 7*6 3a = lim ^ X (2x + 0(2/, + 0(2/2+ 0(...