It has been known that D = 5 simple supergravity resembles D = 11 supergravity in many respects. We present their further resemblances in (1) the duality groups upon dimensional reduction, and (2) the worldsheet structure of the solitonic string of the D = 5 supergravity. We show that the D = 3, G 2(+2) /SO(4) (bosonic) nonlinear sigma model is obtained by using Freudenthal's construction in parallel to the derivation of the D = 3, E 8(+8) /SO(16) sigma model from D = 11 supergravity. The zero modes of the string solution with unbroken (4,0) supersymmetry consist of three (non-chiral) scalars, four Majorana-Weyl spinors of the same chirality and one chiral scalar, which suggests a duality to a certain six-dimensional chiral string theory. The worldsheet gravitational anomaly indicates a quantum correction to the Bianchi identity for the dualized two-form gauge field in the bulk just like the M5-brane case.

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(...

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Extending the seminal work of Bilal and Gervais, we construct a tachyon-free, modular invariant partition function for critical superstrings on four-dimensional Minkowski × two-dimensional black hole. This model may be thought of as an SL(2, R)/U(1) version of Gepner models and corresponds to a conifold point on the moduli space of CalabiYau compactifications. We directly deal with N = 2, c = 9 unitary superconformal characters. Modular invariance is achieved by requiring the string to have a momentum along an extra noncompact direction, in agreement with the picture of singular CFTs advocated by Witten. The four-dimensional massless spectrum coincides with that of the tensionless strings, suggesting a possible dual description of type II strings on a conifold in terms of two intersecting NS5-branes. An interesting relation to D = 6, N = 4 gauged supergravity is also discussed.

We give a complete set of generators for the discrete exceptional U-duality groups of toroidal compactified type II theory and M-theory in d 3. For this, we use the DSZ quantization in d = 4 as originally proposed by Hull and Townsend, and determine the discrete group inducing integer shifts on the charge lattice. It is generated by fundamental unipotents, which are constructed by exponentiating the Chevalley generators of the corresponding Lie algebra. We then extend a method suggested by the above authors and used by Sen for the heterotic string to obtain the discrete U-duality group in d = 3, thereby obtaining a quantized symmetry in d = 3 from a d = 4 quantization condition. This is studied first in a toy model, corresponding to d = 5 simple supergravity, and then applied to M-theory. It turns out that, in the toy model, the resulting U-duality group in d = 3 is strictly smaller than the one generated by the fundamental unipotents corresponding to all Chevalley generators. However, for M-theory, both groups agree. We illustrate the compactification to d = 3 by an embedding of d = 4 particle multiplets into the d = 3 theory.