412 research outputs found
Harmonic Superspace, Minimal Unitary Representations and Quasiconformal Groups
We show that there is a remarkable connection between the harmonic superspace
(HSS) formulation of N=2, d=4 supersymmetric quaternionic Kaehler sigma models
that couple to N=2 supergravity and the minimal unitary representations of
their isometry groups. In particular, for N=2 sigma models with quaternionic
symmetric target spaces of the form G/HXSU(2) we establish a one-to-one mapping
between the Killing potentials that generate the isometry group G under Poisson
brackets in the HSS formulation and the generators of the minimal unitary
representation of G obtained by quantization of its geometric realization as a
quasiconformal group. Quasiconformal extensions of U-duality groups of four
dimensional N=2, d=4 Maxwell-Einstein supergravity theories (MESGT) had been
proposed as spectrum generating symmetry groups earlier. We discuss some of the
implications of our results, in particular, for the BPS black hole spectra of
4d, N=2 MESGTs.Comment: 20 pages; Latex file: references added; minor cosmetic change
Gauging the Full R-Symmetry Group in Five-dimensional, N=2 Yang-Mills/Einstein/tensor Supergravity
We show that certain five dimensional, N=2 Yang-Mills/Einstein supergravity
theories admit the gauging of the full R-symmetry group, SU(2)_R, of the
underlying N=2 Poincare superalgebra. This generalizes the previously studied
Abelian gaugings of U(1)_R subgroup of SU(2)_R and completes the construction
of the most general vector and tensor field coupled five dimensional N=2
supergravity theories with gauge interactions. The gauging of SU(2)_R turns out
to be possible only in special cases, and leads to a new type of scalar
potential. For a large class of these theories the potential does not have any
critical points.Comment: Latex file, 15 pages ; section two is split in two and the discussion
of the critical points is moved into the new section. Version to appear in
Physical Review
Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups
We study the symmetries of generalized spacetimes and corresponding phase
spaces defined by Jordan algebras of degree three. The generic Jordan family of
formally real Jordan algebras of degree three describe extensions of the
Minkowskian spacetimes by an extra "dilatonic" coordinate, whose rotation,
Lorentz and conformal groups are SO(d-1), SO(d-1,1) XSO(1,1) and
SO(d,2)XSO(2,1), respectively. The generalized spacetimes described by simple
Jordan algebras of degree three correspond to extensions of Minkowskian
spacetimes in the critical dimensions (d=3,4,6,10) by a dilatonic and extra
(2,4,8,16) commuting spinorial coordinates, respectively. The Freudenthal
triple systems defined over these Jordan algebras describe conformally
covariant phase spaces. Following hep-th/0008063, we give a unified geometric
realization of the quasiconformal groups that act on their conformal phase
spaces extended by an extra "cocycle" coordinate. For the generic Jordan family
the quasiconformal groups are SO(d+2,4), whose minimal unitary realizations are
given. The minimal unitary representations of the quasiconformal groups F_4(4),
E_6(2), E_7(-5) and E_8(-24) of the simple Jordan family were given in our
earlier work hep-th/0409272.Comment: A typo in equation (37) corrected and missing titles of some
references added. Version to be published in JHEP. 38 pages, latex fil
Spectrum Generating Conformal and Quasiconformal U-Duality Groups, Supergravity and Spherical Vectors
After reviewing the algebraic structures that underlie the geometries of N=2
Maxwell-Einstein supergravity theories (MESGT) in five and four dimensions with
symmetric scalar manifolds, we give a unified realization of their three
dimensional U-duality groups as spectrum generating quasiconformal groups. They
are F_{4(4)}, E_{6(2)}, E_{7(-5)}, E_{8(-24)} and SO(n+2,4). Our formulation is
covariant with respect to U-duality symmetry groups of corresponding five
dimensional supergravity theories, which are SL(3,R), SL(3,C), SU*(6), E_{6(6)}
and SO(n-1,1)X SO(1,1), respectively. We determine the spherical vectors of
quasiconformal realizations of all these groups twisted by a unitary character.
We also give their quadratic Casimir operators and determine their values. Our
work lays the algebraic groundwork for constructing the unitary representations
of these groups induced by their geometric quasiconformal actions, which
include the quaternionic discrete series. For rank 2 cases, SU(2,1) and
G_{2(2)}, corresponding to simple N=2 supergravity in four and five dimensions,
this program was carried out in arXiv:0707.1669. We also discuss the
corresponding algebraic structures underlying symmetries of matter coupled N=4
and N>4 supergravity theories. They lead to quasiconformal realizations of
split real forms of U-duality groups as a straightforward extension of the
quaternionic real forms.Comment: Section 4 is split with the addition of a subsection on quadratic
Casimir operators; references added; typos corrected. Latex file; 53 page
Lectures on Spectrum Generating Symmetries and U-duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace
We review the underlying algebraic structures of supergravity theories with
symmetric scalar manifolds in five and four dimensions, orbits of their
extremal black hole solutions and the spectrum generating extensions of their
U-duality groups. For 5D, N=2 Maxwell-Einstein supergravity theories (MESGT)
defined by Euclidean Jordan algebras, J, the spectrum generating symmetry
groups are the conformal groups Conf(J) of J which are isomorphic to their
U-duality groups in four dimensions. Similarly, the spectrum generating
symmetry groups of 4D, N=2 MESGTs are the quasiconformal groups QConf(J)
associated with J that are isomorphic to their U-duality groups in three
dimensions. We then review the work on spectrum generating symmetries of
spherically symmetric stationary 4D BPS black holes, based on the equivalence
of their attractor equations and the equations for geodesic motion of a
fiducial particle on the target spaces of corresponding 3D supergravity
theories obtained by timelike reduction. We also discuss the connection between
harmonic superspace formulation of 4D, N=2 sigma models coupled to supergravity
and the minimal unitary representations of their isometry groups obtained by
quantizing their quasiconformal realizations. We discuss the relevance of this
connection to spectrum generating symmetries and conclude with a brief summary
of more recent results.Comment: 55 pages; Latex fil
EXTENDED SUPERCONFORMAL SYMMETRY, FREUDENTHAL TRIPLE SYSTEMS AND GAUGED WZW MODELS
We review the construction of extended ( N=2 and N=4 ) superconformal
algebras over triple systems and the gauged WZW models invariant under them.
The N=2 superconformal algebras (SCA) realized over Freudenthal triple systems
(FTS) admit extension to ``maximal'' N=4 SCA's with SU(2)XSU(2)XU(1) symmetry.
A detailed study of the construction and classification of N=2 and N=4 SCA's
over Freudenthal triple systems is given. We conclude with a study and
classification of gauged WZW models with N=4 superconformal symmetry.Comment: Invited talk presented at the Gursey Memorial Conference I in
Istanbul, Turkiye (June 6-10, 1994). To appear in the proceedings of the
conference. (21 pages. Latex document.
Triple Products and Yang-Baxter Equation (II): Orthogonal and Symplectic Ternary Systems
We generalize the result of the preceeding paper and solve the Yang-Baxter
equation in terms of triple systems called orthogonal and symplectic ternary
systems. In this way, we found several other new solutions.Comment: 38 page
Quaternion Octonion Reformulation of Quantum Chromodynamics
We have made an attempt to develop the quaternionic formulation of Yang -
Mill's field equations and octonion reformulation of quantum chromo dynamics
(QCD). Starting with the Lagrangian density, we have discussed the field
equations of SU(2) and SU(3) gauge fields for both cases of global and local
gauge symmetries. It has been shown that the three quaternion units explain the
structure of Yang- Mill's field while the seven octonion units provide the
consistent structure of SU(3)_{C} gauge symmetry of quantum chromo dynamics
Topological wave functions and heat equations
It is generally known that the holomorphic anomaly equations in topological
string theory reflect the quantum mechanical nature of the topological string
partition function. We present two new results which make this assertion more
precise: (i) we give a new, purely holomorphic version of the holomorphic
anomaly equations, clarifying their relation to the heat equation satisfied by
the Jacobi theta series; (ii) in cases where the moduli space is a Hermitian
symmetric tube domain , we show that the general solution of the anomaly
equations is a matrix element \IP{\Psi | g | \Omega} of the
Schr\"odinger-Weil representation of a Heisenberg extension of , between an
arbitrary state and a particular vacuum state .
Based on these results, we speculate on the existence of a one-parameter
generalization of the usual topological amplitude, which in symmetric cases
transforms in the smallest unitary representation of the duality group in
three dimensions, and on its relations to hypermultiplet couplings, nonabelian
Donaldson-Thomas theory and black hole degeneracies.Comment: 50 pages; v2: small typos fixed, references added; v3: cosmetic
changes, published version; v4: typos fixed, small clarification adde
Orbifolds and Flows from Gauged Supergravity
We examine orbifolds of the IIB string via gauged supergravity. For the
gravity duals of the A_{n-1} quiver gauge theories, we extract the massless
degrees of freedom and assemble them into multiplets of N=4 gauged supergravity
in five dimensions. We examine the embedding of the gauge group into the
isometry group of the scalar manifold, as well as the symmetries of the scalar
potential. From this we find that there is a large SU(1,n) symmetry group which
relates different RG flows in the dual quiver gauge theory. We find that this
symmetry implies an extension of the usual duality between ten-dimensional IIB
solutions which involves exchanging geometric moduli with background fluxes.Comment: 37 pages, harvma
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