arXiv subjects
Publications and source records attributed to Sanjaye Ramgoolam.
In this paper we point out that the spacetime uncertainty relation proposed for string theory has strong cosmological implications that can solve the flatness problem and the horizon problem without the need of inflation. We make minimal assumptions about the very early universe.
Projector equivalences used in the definition of the K-theory of operator algebras are shown to lead to generalizations of the solution generating technique for solitons in NC field theories, which has recently been used in the construction of branes from other branes in B-field backgrounds and in the construction of fluxon solutions of gauge theories. The generalizations involve families of static solutions as well as solutions which depend on euclidean time and interpolate between different configurations. We investigate the physics of these generalizations in the brane-construction as well as the fluxon context. These results can be interpreted in the light of recent discussions on the topology of the configuration space of string fields.
We study relations between different kinds of non-commutative spheres which have appeared in the context of ADS/CFT correspondences recently, emphasizing the connections between spaces that have manifest quantum group symmetry and spaces that have manifest classical symmetry. In particular we consider the quotient $SU_q(2)/U(1)$ at roots of unity, and find its relations with the fuzzy sphere with manifest classical SU(2) symmetry. Deformation maps between classical and quantum symmetry, the $U_q(SU(2))$ module structure of quantum spheres and the structure of indecomposable representations of $U_q(SU(2))$ at roots of unity conspire in an interesting way to allow the relation between manifestly $U_q(SU(2)$ symmetric spheres and manifestly U(SU(2)) symmetric spheres. The relation suggests that a subset of field theory actions on the q-sphere are equivalent to actions on the fuzzy sphere. The results here are compatible with the proposal that quantum spheres at roots of unity appear as effective geometries which account for finite N effects in the ADS/CFT correspondence.
T-duality (Fourier-Mukai duality) and properties of classical instanton moduli spaces can be used to deduce some properties of $α^{\prime}$-corrected moduli spaces of branes for Type IIA string theory compactified on $K3$ or $T^4$. Some interesting differences between the two compactifications are exhibited.
We present direct arguments for non-commutativity of spheres in the AdS/CFT correspondence. The discussion is based on results for the $S_N$ orbifold SCFT. Concentrating on three point correlations (at finite $N$) we exhibit a comparison with correlations on a non-commutative sphere. In this manner an essential signature of non-commutativity is identified giving further support for the original proposal of hep-th/9902059.
The counting of long strings in ADS3, in the context of Type IIB string theory on $ADS_3 \times S^3 \times T^4$, is used to exhibit the action of the duality group $O(5,5;Z)$, and in particular its Weyl Subgroup $S_5 \bowtie Z_2$, in the non-perturbative phenomena associated with continuous spectra of states in these backgrounds. The counting functions are related to states in Fock spaces. The symmetry groups also appear in the structure of compactifications of instanton moduli spaces on $T^4$.
The orbifold CFT dual to string theory on $ADS_3 \times S^3$ allows a construction of gravitational actions based on collective field techniques. We describe a fundamental role played by a Lie algebra constructed from chiral primaries and their CFT conjugates. The leading terms in the algebra at large $N$ are derived from the computation of chiral primary correlation functions. The algebra is argued to determine the dynamics of the theory, its representations provide free and interacting hamiltonians for chiral primaries. This dynamics is seen to be given by an effective one plus one dimensional field theory. The structure of the algebra and its representations shows qualitatively new features associated with thresholds at $L_0= N$, $ L_0 = N/2$ and $L_0=N/4$, which are related to the stringy exclusion principle and to black holes. We observe relations between fusion rules of $SU_q(2|1,1)$ for $q = e^{i π/ {N+1}} $, and the correlation functions, which provide further evidence for a non-commutative spacetime.
The truncation in the number of single-trace chiral primary operators of $\N=4$ SYM and its conjectured connection with gravity on quantum spacetimes are elaborated. The model of quantum spacetime we use is $AdS^5_q \times S^5_q$ for $q$ a root of unity. The quantum sphere is defined as a homogeneous space with manifest $SU_q(3)$ symmetry, but as anticipated from the field theory correspondence, we show that there is a hidden $SO_q(6)$ symmetry in the constrution. We also study some properties of quantum space quotients as candidate models for the quantum spacetime relevant for some $Z_n$ quiver quotients of the $\N=4$ theory which break SUSY to $\N=2$. We find various qualitative agreements between the proposed models and the properties of the corresponding finite $N$ gauge theories.
The exclusion principle of Maldacena and Strominger is seen to follow from deformed Heisenberg algebras associated with the chiral rings of S_N orbifold CFTs. These deformed algebras are related to quantum groups at roots of unity, and are interpreted as algebras of space-time field creation and annihilation operators. We also propose, as space-time origin of the stringy exclusion principle, that the $ADS_3 \times S^3$ space-time of the associated six-dimensional supergravity theory acquires, when quantum effects are taken into account, a non-commutative structure given by $SU_q(1,1) \times SU_q (2)$. Both remarks imply that finite N effects are captured by quantum groups $SL_q(2)$ with $q= e^{i π\over {N + 1}}$. This implies that a proper framework for the theories in question is given by gravity on a non-commutative spacetime with a q-deformation of field oscillators. An interesting consequence of this framework is a holographic interpretation for a product structure in the space of all unitary representations of the non-compact quantum group $SU_q(1,1)$ at roots of unity.
The non-commutative algebra which defines the theory of zero-branes on $T^4/Z_2$ allows a unified description of moduli spaces associated with zero-branes, two-branes and four-branes on the orbifold space. Bundles on a dual space $\hat T^4/Z_2$ play an important role in this description. We discuss these moduli spaces in the context of dualities of K3 compactifications, and in terms of properties of instantons on $T^4$. Zero-branes on the degenerate limits of the compact orbifold lead to fixed points with six-dimensional scale but not conformal invariance. We identify some of these in terms of the ADS dual of the $(0,2)$ theory at large $N$, giving evidence for an interesting picture of "where the branes live" in ADS.
We consider the scattering of zero branes off an elementary string in Matrix theory or equivalently gravitons off a longitudinally wrapped membrane. The leading supergravity result is recovered by a one-loop calculation in zero brane quantum mechanics. Simple scaling arguments are used to show that there are no further corrections at higher loops, to the leading term in the large impact parameter, low velocity expansion. The mechanism for this agreement is identified in terms of properties of a recently discovered boundary conformal field theory.
Scattering of zero branes off the fixed point in $R^8/Z_2$, as described by a super-quantum mechanics with eight supercharges, displays some novel effects relevant to Matrix theory in non-compact backgrounds. The leading long distance behaviour of the moduli space metric receives no correction at one loop in Matrix theory, but does receive a correction at two loops. There are no contributions at higher loops. We explicitly calculate the two-loop term, finding a non-zero result. We find a discrepancy with M(atrix)-theory. Although the result has the right dependence on $v$ and $b$ for the scattering of zero branes off the fixed point the factors of $N$ do not match. We also discuss scattering in the orbifolds, $R^5/Z_2$ and $R^9/Z_2$ where we find the predicted fractional charges.
We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realized as S-duality for $U(\infty)$ N=4 Super-Yang-Mills in 3+1D. We argue that Kaluza Klein states of gravitons correspond to electric fluxes, wrapped membranes become magnetic fluxes and instantonic membranes are related to Yang-Mills instantons. The T-duality transformation of gravitons into wrapped membranes is interpreted as the duality between electric and magnetic fluxes. The identification of M-theory T-duality as SYM S-duality provides a natural framework for studying the M-theory 5-brane as the S-dual object to the unwrapped membrane. Using the equivalence between compactified M(atrix) theory and SYM, we find a natural candidate for a description of the light-cone 5-brane of M-theory directly in terms of matrix variables, analogous to the known description of the M(atrix) theory membrane.
We describe a topological string theory which reproduces many aspects of the 1/N expansion of SU(N) Yang-Mills theory in two spacetime dimensions in the zero coupling (A=0) limit. The string theory is a modified version of topological gravity coupled to a topological sigma model with spacetime as target. The derivation of the string theory relies on a new interpretation of Gross and Taylor's ``Ω^{-1} points.'' We describe how inclusion of the area, coupling of chiral sectors, and Wilson loop expectation values can be incorporated in the topological string approach.
These are expository lectures reviewing (1) recent developments in two-dimensional Yang-Mills theory, and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinite-dimensional differential geometry. We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals.
We give a simple diagrammatic algorithm for writing the chiral large $N$ expansion of intersecting Wilson loops in $2D$ $SU(N)$ and $U(N) $Yang Mills theory in terms of symmetric groups, generalizing the result of Gross and Taylor for partition functions. We prove that these expansions compute Euler characters of a space of holomorphic maps from string worldsheets with boundaries. We prove that the Migdal-Makeenko equations hold for the chiral theory and show that they can be expressed as linear constraints on perturbations of the chiral $YM2$ partition functions. We briefly discuss finite $N$ , the non-chiral expansion, and higher dimensional lattice models.
We write down all orders large $N$ expansions for the dimensions of irreducible representations of $O(N)$ and $Sp(N)$. We interpret all the terms in these expansions as symmetry factors for singular worldsheet configurations, involving collapsed crosscaps and tubes. We use it to complete the interpretation of two dimensional Yang Mills Theories with these gauge groups, on arbitrary two dimensional manifolds, in terms of a String Theory of maps of the type considered by Gross and Taylor. We point out some intriguing similarities to the case of $U(N)$ and discuss their implications.
We show that the Hopf link invariants for an appropriate set of finite dimensional representations of $ U_q SL(2)$ are identical, up to overall normalisation, to the modular S matrix of Kac and Wakimoto for rational $k$ $\widehat {sl(2)}$ representations. We use this observation to construct new modular Hopf algebras, for any root of unity $q=e^{-iπm/r}$, obtained by taking appropriate quotients of $U_q SL(2)$, that give rise to 3-manifold invariants according to the approach of Reshetikin and Turaev. The phase factor correcting for the `framing anomaly' in these invariants is equal to $ e^{- {{i π} \over 4} ({ {3k} \over {k+2}})}$, an analytic continuation of the anomaly at integer $k$. As expected, the Verlinde formula gives fusion rule multiplicities in agreement with the modular Hopf algebras. This leads to a proposal, for $(k+2)=r/m$ rational with an odd denominator, for a set of $\widehat {sl(2)}$ representations obtained by dropping some of the highest weight representations in the Kac-Wakimoto set and replacing them with lowest weight representations. For this set of representations the Verlinde formula gives non-negative integer fusion rule multiplicities. We discuss the consistency of the truncation to highest and lowest weight representations in conformal field theory.