Quantum theory of compact and non-compact σ models

We discuss quantization of SO(N + 1) σ models and CP^N models, and of certain non-compact counterparts, SO(N, 1) and QP^N respectively, of these, both in canonical operator formalism and the covariant path integral formulation, showing the equivalence of the two approaches. We discuss also a class of supersymmetric σ models formulated in d ? 3 dimensions and apply the results to the SO(N + 1) and SO(N, 1) cases. This allows us to calculate the Witten index in each case. For SO(2l + 1,1) we thereby find supersymmetry breaking. However, for SO(2l, 1), we find supersymmetry is unbroken. Moreover, there is no unique ground state, invariant under SO(2l, 1), rather an infinite multiple of zero energy states, carrying a unitary irreducible representation of the non-compact SO(2l, 1) group. We discuss also field theoretic aspects of the models in d ? 2 dimensions, stressing differences of the non-compact to the compact cases. These include infrared instead of ultraviolet asymptotic freedom, lack of an energy gap, failure (in the QP^N case) of the auxiliary vector field to become dynamical. A further conclusion that is argued concerns the absence of a consistent particle interpretation for the QP^N model in exactly two dimensions. For d > 2 the non-compact symmetry of QP^N is broken down to the compact subgroup.     

Another supersymmetric extension of local Lorentz symmetry in the vierbein formalism of Einstein gravity

A supersymmetric extension of the vierbein formalism of Einstein gravity is investigated by takingSL(N, 2;C) superalgebra, which is shown to be the sole possibility other than the previously investigatedOSp(N 2;C). After general discussion on the ξ-field realization, a non-linear realization of linear representation introduced in a previous paper, it is applied to the fundamental representation ofSL(N, 2;C) t0 which the vierbein supermultiplet belongs. The chiral symmetry derived in the theory is shown to beSO(N) in the same way as in theOSp(N,2;C)-symmetric extension.     

Gradient property of low order covariants and truncated bifurcation equations for SO(N) symmetries

It is proved that the convariants of order p ≤ 3 of any irreducible representation of SO(N), and therefore the bifurcation equation truncated at order q ≤ 3 of any system with SO(N) symmetry, can be written in gradient form.     

Pseudo-Hermitian Systems,Involutive Symmetries and Pseudofermions

We have briefly analyzed the existence of the pseudofermionic structure of multilevel pseudo-Hermitian systems with odd time-reversal and higher order involutive symmetries. We have shown that 2N-level Hamiltonians with N- order eigenvalue degeneracy can be represented in the oscillator-like form in terms of pseudofermionic creation and annihilation operators for both real and complex eigenvalues. The example of most general four-level traceless Hamiltonian with odd time-reversal symmetry, which is an extension of the SO(5) Hermitian Hamiltonian, is considered in greater and explicit detail.     

Properties of SO(8) extended supergravity

The conjectured SU(8) invariance of the field equations of SO(8) extended supergravity is used to elucidate the general structure of the extended theories. Due to the representation content of the spinless fields this does not lead to a complete determination of the theory, as was the case for N = 4. The non-polynomial modifications by spinless fields are given in terms of a number of SU(8) covariant tensors, for which various identities are derived and discussed.     

Towards a finite N = 2 susy GUT

We consider finite, N = 2 supersymmetric GUTs based on gauge groups SU(n) and SO(n). As an example, we discuss a semirealistic model based on SO(12). We argue that in finite, N = 2 supersymmetric GUTs, gauge symmetry breaking should occur dynamically. We present a heuristic picture in which this is induced by soft, finiteness preserving SUSY breaking terms. The bound states formed cause a very rapid evolution of the SO(12) coupling constant and break SO(12) into SU(4)×SU(3)_C×U(1).     

Extended supergravity with local SO(5) invariance

We present a version of N = 5 supergravity with local SO(5) invariance and a lowest order calculation for N = 8 supergravity with local SO(8). The implications of these results and related aspects are discussed.     

Breaking of the SO(n) symmetry with vector and adjoint higgses

The spontaneous symmetry breaking of SO(n) is investigated by studying the most general quartic SO(n)-invariant Higgs potential with two multiplets of scalars belonging to a vector and to an adjoint representation. In the most general cases largest residual symmetry is found to be SU(l), SU(l-1), SO(l), SO(l-1) or SO(n-2) wherel is the rank of SO(n). In particular, the breaking of SO(n) into SU(n ₁)×SU(n ₂) is found to occur only in special cases.     

Twistor and polytope interpretations for subleading color one-loop amplitudes

We use the relation of the one-loop subleading-color amplitudes to the one-loop n-point leading color amplitudes in N=4 SYM, to derive a polytope interpretation for the former in the MHV case, and a representation in momentum twistor space for the general NkMHV case. These techniques are explored in detail for the 5-point and 6-point amplitudes. We briefly discuss the implications for IR divergences.     

N = 4 supergravity coupled to N = 4 matter and hidden symmetries

We present the N = 1 supergravity in 10 dimensions obtained by truncating the reduced N = 1 supergravity from 11 dimensions. This is further reduced to 4 dimensions to give SU(4) supergravity coupled to six SO(4) vector multiplets. As the reduction is from 10 dimensions, the theory is expected to have the symmetry SL(6R)_global×SO(6)_local, but we give a theoretical argument that this can be extended to SO(6,6)×SU(1,1)_global and SO(6)×SO(6)×U(1)_local.     

Bosonic construction of the Lie algebras of some non-compact groups appearing in supergravity theories and their oscillator-like unitary representations

We give a construction of the Lie algebras of the non-compact groups appearing in four dimensional supergravity theories in terms of boson operators. Our construction parallels very closely their emergence in supergravity and is an extension of the well-known construction of the Lie algebras of the non-compact groups $\text{SP}\text{(2n,}\text{R}$ and $\text{SO}\text{(2n)}{}^1$ from boson operators transforming like a fundamental representation of their maximal compact subgroup U(n). However this extension is non-trivial only for n?4 and stops at n = 8 leading to the Lei algebras of SU(4) × SU(1, 1), SU(1, 1), SU(5, 1), SO(12)¹ and E₇(7). We then give a general construction of an infinite class of unitary irreducible representations of the respective non-compact groups (except for E₇₍₇₎ and SO(12)¹ obtained from the extended construction). We illustrate our construction with the examples of SU(5, 1) and SO(12)¹.     

Antigravitating black hole solitons with scalar hair in N=4 supergravity

We present some new solutions of the equations of the N=4 supergravity theory which represent black holes with scalar, electric and magnetic charges. The solutions are parameterized by the mass and 6 electric and 6 magnetic charges which can be assembled into a complex 6-vector, Z^N. One can act on the solutions with SO(6)×U(1) to obtain new solutions with the same mass M but charges Z^N related by SO(6)×U(1) transformations, the U(1) factor corresponding to the duality subgroup of the hidden SU(1, 1) ssymetry of the N=4 model. In a certain limiting case the black holes have zero temperature and behave like solitons. In this case multisoliton solutions are exhibited which antigravitate, i.e. are in static equilibrium. We also present some solutions of the Kaluza-Klein theory which were anticipated by Scherk which also antigravitate. However, these latter solutions contain naked singularities. A discussion is also given of the relation of these solutions to dimensional reduction which has relevance for the black holes in the N=8 supergravity theory.     

Comments on a Minimal Quasiparticle Approach for the QGP and Its Large-N c Limits

A minimal quasiparticle approach for describing QGP at temperatures much higher than the critical one is discussed. It involves an ideal-gas framework in which quark and gluon masses depend on temperature. This model is able to reproduce the recent equations of state computed in lattice QCD for temperatures typically higher than 2 T _c , in a range in which it is reasonable to neglect interactions between quasiparticles. In addition, the equations of state for a generic gauge theory with gauge groups SU(N _c ) and quarks in an arbitrary representation are studied. The gauge independence in the pure glue sector and the large-N _c equivalence between the gauge groups SU(N _c ) and SO(2N _c ) in a full plasma is finally shown for normalized thermodynamic quantities.     

Moduli matrices of the vacua and walls on SO(2N)/U(N)

We construct parallel domain walls on the SO(2N)/U(N) manifold by using the moduli matrices, which were originally formulated on the Grassmann manifold. We propose a method to impose a quadratic constraint to the moduli matrices. This talk is based on [3]PRD83,125003(2011). This article is prepared for the proceedings of International Workshop on Supersymmetries and Quantum Symmetries-SQS 2011.     

The parallelizing S7 torsion in gauged N=8 supergravity

Using the parallelizing S⁷ torsion as an ansatz we investigate two solutions of gauged N=8 supergravity with SO(7) invariance. Supersymmetry is uniformly broken. We calculate the masses for these solutions which are both unstable. Certain apparent discrepancies with the results obtained by spontaneous compactification of d=11 supergravity are discussed. We establish that the compactification on the parallelized S⁷ has an SO(7) invariance and clarify the issue of supersymmetry breaking. The lack of stability in d=4 indicates that this d=11 solution is unstable.     

Multiplet shortening in Osp(N,4)

We describe a new type of multiplet shortening in Osp(N,4) which resolves a long-standing puzzle in Kaluza-Klein supergravities. Multiplet shortening implies quantization of mass eigenvalues in units of the inverse AdS radius. While detailed proofs are presented only for N = 3, we discuss implications for N = 8 supergravity and derive the SO(8) assignments of all higher modes on the round seven-sphere.     

A classification of compactifying solutions for d =11 supergravity

Supergravity in eleven dimensions is known to have classical solutions of the type (anti-de Sitter space-time) × (7-dimensional Einstein space). We give a list of all homogeneous 7-manifolds which admit an Einstein metric. Known solutions are reviewed, with some emphasis on the SU(3) × SU(2) × U(1) compactifications. Their topology is discussed in detail.The list includes three new solutions, with symmetry groups SU(3) × SU(2), SO(5) and SO(5) × U(1). The first solution has no supersymmetry, while the second and third yield respectively N = 1 and N = 2 supersymmetry in four dimensions. The last two solutions may be extended to solutions with nonzero internal photon curl, breaking all supersymmetry.The existence of a spin structure on homogeneous manifolds $\text{G}\text{H}$ is discussed and related to topological properties of $\text{G}\text{H}$. As an illustration, we treat the coset spaces SU(m + 1) × SU(n + 1)/SU(m) × SU(n) × U(1), which include the spaces with SU(3) × SU(2) × U(1) symmetry.     

Loop space Hamiltonians and numerical methods for large-N gauge theories

We consider large-N gauge theories in the hamiltonian, collective field approach. We derive an alternative collective representation which leads to significant reduction when translation invariance is invoked. It allows for a simplified computer simulation of loop rearrangements and the development of numerical techniques in the hamiltonian, loop space formalism. We proceed to give numerical evidence for validity of our representation and outline a general numerical approach for solving large-N QCD in terms of gauge-invariant Wilson loop variables.     

Gauged N = 8 supergravity in five dimensions

We construct gauged N = 8 supergravity theories in five dimensions. Instead of the twenty-seven vector fields of the ungauged theory, the gauged theories contain fifteen vector fields and twelve second-rank antisymmetric tensor fields satisfying self-dual field equations. The fifteen vector fields can be used to gauge any of the fifteen-dimensional semisimple subgroups of SL(6,R), specially SO(p, 6?p) for p = 0, 1, 2, 3. The gauged theories also have a physical global SU(1,1) symmetry which survives from the E₆₍₆₎ symmetry of the ungauged theory. This SU(1,1) for the SO(6) gauging is presumably related to that of the chiral N = 2 theory in ten dimensions. In our formalism we maintain a composite local USp(8) symmetry analogous to SU(8) in four dimensions.     

Gauge boson/Higgs boson unification: The Higgs bosons as superpartners of massive gauge bosons

The N = 4 supergravity theories with local SO(4) invariance are formulated in superspace. The gauged SO(4) theory with two coupling constants (g₁, g₂) is shown to reduce to three inequivalent models: g₁ = g₂ with negative cosmological constant, g₁ = ?g₂ with positive cosmological constant, and g₁ = 0, g₂ ≠ 0 which is a particular case of the Freedman-Schwarz gauged SU(2) ? SU(2) model. The Higgs effect in the vector-scalar sector of the gauged N = 5 supergravity is analyzed.