arXiv subjects

C. Fronsdal

Publications and source records attributed to C. Fronsdal.

8 recordsLinked to original sources

arXiv2001

Harrison Cohomology and Abelian Deformation Quantization on Algebraic Varieties

Abelian deformations of ordinary algebras of functions are studied. The role of Harrison cohomology in classifying such deformations is illustrated in the context of simple examples chosen for their relevance to physics. It is well known that Harrison cohomology is trivial on smooth manifolds and that, consequently, abelian *-products on such manifolds are trivial to first order in the deformation parameter. The subject is nevertheless interesting; first because varieties with singularities appear in the physical context and secondly, because deformations that are trivial to first order are not always (indeed not usually) trivial as exact deformations. We investigate cones, to illustrate the situation on algebraic varieties, and we point out that the coordinate algebra on (anti-) de Sitter space is a nontrivial deformation of the coordinate algebra on Minkowski space -- although both spaces are smooth manifolds.

hep-th↗

arXiv2000

Conformal Fields in Higher Dimensions

We generalize, to any space-time dimension, the unitarity bounds of highest weight UIR's of the conformal groups with Lie algebras $so(2,d)$. We classify gauge theories invariant under $so(2,d)$, both integral and half-integral spins. A similar analysis is carried out for the algebras $so^*(2n)$. We study new unitary modules of the conformal algebra in $d>4$, that have no analogue for $d\leq 4$ as they cannot be obtained by "squaring" singletons. This may suggest the interpretation of higher dimensional non-trivial conformal field theories as theories of "tensionless" $p$-branes of which tensionless strings in $d=6$ are just particular examples.

hep-th↗

arXiv1999

The Ideals of Free Differential Algebras

We consider the free ${\bf C}$-algebra ${\cal B}_q$ with $N$ generators $\{ξ_i\}_{i = 1,...,N}$, together with a set of $N$ differential operators $\{\partial_i\}_{i = 1,...,N}$ that act as twisted derivations on ${\cal B}_q$ according to the rule $\partial_iξ_j = δ_{ij} + q_{ij}ξ_j\partial_i$; that is, $\forall x \in {\cal B}_q, \partial_i(ξ_jx) = δ_{ij}x + q_{ij}ξ_j\partial_i x,$ and $\partial_i{\bf C} = 0$. The suffix $q$ on ${\cal B}_q$ stands for $\{q_{ij}\}_{i,j \in \{1,...,N\}}$ and is interpreted as a point in parameter space, $q = \{q_{ij}\}\in {\bf C}^{N^2}$. A constant $C \in {\cal B}_q$ is a nontrivial element with the property $\partial_iC = 0, i = 1,...,N$. To each point in parameter space there correponds a unique set of constants and a differential complex. There are no constants when the parameters $q_{ij}$ are in general position. We obtain some precise results concerning the algebraic surfaces in parameter space on which constants exist. Let ${\cal I}_q$ denote the ideal generated by the constants. We relate the quotient algebras ${\cal B}_q' = {\cal B}_q/{\cal I}_q$ to Yang-Baxter algebras and, in particular, to quantized Kac-Moody algebras. The differential complex is a generalization of that of a quantized Kac-Moody algebra described in terms of Serre generators. Integrability conditions for $q$-differential equations are related to Hochschild cohomology. It is shown that $H^p({\cal B}_q',{\cal B}_q') = 0$ for $p \geq 1$. The intimate relationship to generalized, quantized Kac-Moody algebras suggests an approach to the problem of classification of these algebras.

math.QA↗

arXiv1998

Gauge Fields and Singletons of $AdS_{2p+1}$

We show that $p$-forms on $AdS_{2p+1}$ describe both singletons and massless particles. On the $2p$-dimensional boundary the singleton $p$-form Lagrangian reduces to the conformally invariant functional $\int F^2$. All the representations, singletons as well as massless, are zero center modules and involve a vacuum mode. Two- and three-form singleton fields are required by supersymmetry in AdS$_5$ and AdS$_7$ supergravity respectively.

hep-th↗

arXiv1998

Gauge fields as composite boundary excitations

We investigate representations of the conformal group that describe "massless" particles in the interior and at the boundary of anti-de Sitter space. It turns out that massless gauge excitations in anti-de Sitter are gauge "current" operators at the boundary. Conversely, massless excitations at the boundary are topological singletons in the interior. These representations lie at the threshold of two "unitary bounds" that apply to any conformally invariant field theory. Gravity and Yang-Mills gauge symmetry in anti-De Sitter is translated to global translational symmetry and continuous $R$-symmetry of the boundary superconformal field theory.

hep-th↗

arXiv1998

Conformal Maxwell theory as a singleton field theory on AdS_5, IIB three-branes and duality

We examine the boundary conditions associated with extended supersymmetric Maxwell theory in 5-dimensional anti-De Sitter space. Excitations on the boundary are identical to those of ordinary 4-dimensional conformal invariant super electrodynammics. Extrapolations of these excitations give rise to a 5-dimensional topological gauge theory of the singleton type. The possibility of a connection of this phenomenon to the world volume theory of 3-branes in IIB string theory is discussed.

hep-th↗

arXiv1997

8-Vertex Correlation Functions and Twist Covariance of q-KZ Equation

We study the vertex operators $Φ(z)$ associated with standard quantum groups. The element $Z = RR^{t}$ is a "Casimir operator" for quantized Kac-Moody algebras and the quantum Knizhnik-Zamolodchikov (q-KZ) equation is interpreted as the statement $:ZΦ(z): = Φ(z)$. We study the covariance of the q-KZ equation under twisting, first within the category of Hopf algebras, and then in the wider context of quasi Hopf algebras. We obtain the intertwining operators associated with the elliptic R-matrix and calculate the two-point correlation function for the eight-vertex model.

q-alg↗

arXiv1995

Deformations of Multiparameter Quantum gl(N)

Multiparameter quantum gl(N) is not a rigid structure. This paper defines an essential deformation as one that cannot be interpreted in terms of a similarity transformation, nor as a perturbation of the parameters. All the equivalence classes of first order essential deformations are found, as well as a class of exact deformations. This work provides quantization of all the classical Lie bialgebra structures (constant r-matrices) found by Belavin and Drinfeld for sl(n). A special case, that requires the Hecke parameter to be a cubic root of unity, stands out.

q-alg↗