Three-dimensional Sigma Model Research Articles

Note on Wess-Zumino-Witten models and quasiuniversality in 2+1 dimensions

We suggest the possibility that the two-dimensional SU(2)$_k$ Wess-Zumino-Witten (WZW) theory, which has global SO(4) symmetry, can be continued to $2+\epsilon$ dimensions by enlarging the symmetry to SO$(4+\epsilon)$. This is motivated by the three-dimensional sigma model with SO(5) symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\epsilon$ may be fixed by assuming analyticity in $\epsilon$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality $d_c>2$. We suggest that $d_c < 3$ for all $k$, and we compute $d_c$ in the limit of large $k$. The flows support the conjecture that the deconfined phase transition in SU(2) magnets is a ``pseudocritical'' point with approximate SO(5), controlled by a fixed point slightly outside the physical parameter space.

Bertotti-Robinson and soliton string solutions ofD=5minimal supergravity

We report on a series of new solutions to five-dimensional minimal supergravity. Our method applies to space-times with two commuting Killing symmetries and consists in combining dimensional reduction on two-spaces of constant curvature with reduction on a two-torus. The first gives rise to various generalized Bertotti-Robinson solutions supported by electric and magnetic fluxes, which presumably describe the near-horizon regions of black holes and black rings (strings). The second provides generating techniques based on U-duality of the corresponding three-dimensional sigma model. We identify duality transformations relating the above solutions to asymptotically flat ones and obtain new globally regular dyonic solitons. Some new extremal asymptotically flat multi-center solutions are constructed too. We also show that geodesic solutions of three-dimensional sigma models passing through the same target space point generically split into disjoint classes which cannot be related by the isotropy subgroup of U-duality.

Extremal multicenter black holes: nilpotent orbits and Tits Satake universality classes

Four dimensional supergravity theories whose scalar manifold is a symmetric coset manifold U[D=4]/Hc are arranged into a finite list of Tits Satake universality classes. Stationary solutions of these theories, spherically symmetric or not, are identified with those of an euclidian three-dimensional sigma-model, whose target manifold is a Lorentzian coset U[D=3]/H* and the extremal ones are associated with H* nilpotent orbits in the K* representation emerging from the orthogonal decomposition of the algebra U[D=3] with respect to H*. It is shown that the classification of such orbits can always be reduced to the Tits-Satake projection and it is a class property of the Tits Satake universality classes. The construction procedure of Bossard et al of extremal multicenter solutions by means of a triangular hierarchy of integrable equations is completed and converted into a closed algorithm by means of a general formula that provides the transition from the symmetric to the solvable gauge. The question of the relation between H* orbits and charge orbits W of the corresponding black holes is addressed and also reduced to the corresponding question within the Tits Satake projection. It is conjectured that on the vanishing locus of the Taub-NUT current the relation between H*-orbit and W-orbit is rigid and one-to-one. All black holes emerging from multicenter solutions associated with a given H* orbit have the same W-type. For the S^3 model we provide a complete survey of its multicenter solutions associated with all of the previously classified nilpotent orbits of sl(2) x sl(2) within g[2,2]. We find a new intrinsic classification of the W-orbits of this model that might provide a paradigm for the analogous classification in all the other Tits Satake universality classes.

Classification of BPS instantons in N = 4 D = 4 supergravity

This talk is based on the recent work in collaboration with M. Azreg-Aïnou and G. Clément [1] devoted to extremal instantons in the one-vector truncation of the Euclidean \U0001d4a9 = 4, D = 4 theory. Extremal solutions satisfying the no-force condition can be associated with null geodesic curves in the homogeneous target space of the three-dimensional sigma model arising in toroidal reduction of the four-dimensional theory. Here we (preliminarily) discuss the case of two vector fields sufficient to find all relevant metrics in the full \U0001d4a9 = 4, D = 4 theory. Classification of instanton solutions is given along the following lines. The first is their possible asymptotic structure: asymptotically locally flat (ALF), asymptotically locally Euclidean (ALE) and ALF or ALE with the dilaton growing at infinity. The second is the algebraic characterization of matrix generators according to their rank and the nature of the charge vectors in an associated Lorentzian space. Finally, solutions are distinguished by the number of independent harmonic functions with unequal charges (up to four).

The topology of U-duality (sub)groups

We discuss the topology of the symmetry groups appearing in compactified (super)gravity, and discuss two applications. First, we demonstrate that for three-dimensional sigma models on a symmetric space G/H with G non-compact and H the maximal compact subgroup of G, the possibility of oxidation to a higher dimensional theory can immediately be deduced from the topology of H. Second, by comparing the actual symmetry groups appearing in maximal supergravities with the subgroups of and Spin(32), we argue that these groups cannot serve as a local symmetry group for M-theory in a formulation of de Wit–Nicolai type.

Oxidation = group theory

Dimensional reduction of theories involving (super)gravity gives rise to sigma models on coset spaces of the form G/H, with G being a non-compact group, and H its maximal compact subgroup. The reverse process, called oxidation, is the reconstruction of the possible higher-dimensional theories, given the lower-dimensional theory. In three dimensions, all degrees of freedom can be dualized to scalars. Given the group G for a three-dimensional sigma model on the coset G/H, we demonstrate an efficient method for recovering the higher-dimensional theories, essentially by decomposition into subgroups. The equations of motion, Bianchi identities, Kaluza–Klein modifications and Chern–Simons terms are easily extracted from the root lattice of the group G. We briefly discuss some aspects of oxidation from the E8(8)/SO(16) coset, and demonstrate that our formalism reproduces the Chern–Simons term of 11D supergravity, knows about the T-duality of IIA and IIB theory, and easily deals with self-dual tensors, like the 5-tensor of IIB supergravity.

Ehlers-Harrison transformations and black holes in dilaton-axion gravity with multiple vector fields

Dilaton-axion gravity with $p U(1)$ vector fields is studied on space-times admitting a timelike Killing vector field. Three-dimensional sigma-model is derived in terms of K\"ahler geometry, and holomorphic representation of the SO(2,2+p) global symmetry is constructed. A general static black hole solution depending on $2p+5$ parameters is obtained via SO(2,2+p) covariantization of the Schwarzschild solution. The metric in the curvature coordinates looks as the variable mass Reissner-Nordstr\"om one and generically possesses two horizons. The inner horizon is pushed to the singularity if electric and magnetic SO(p) charge vectors are parallel. For non-parallel charges the inner horizon has a finite area except for an extremal limit when this property is preserved only for orthogonal charges. Extremal dyon configurations with orthogonal charges have finite horizon radii continuously varying from zero to the ADM mass. New general solution is endowed with a NUT parameter, asymptotic values of dilaton and axion, and a gauge parameter which can be used to ascribe any given value to one of scalar charges.