Sanjaye Ramgoolam

Lectures on 2D yang-mills theory, equivariant cohomology and topological field theories

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.

Branes, fluxes and duality in M(atrix) theory

Abstract 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 (∞) N = 4 Super-Yang-Mills in (3 + 1) D . 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.

On spherical harmonics for fuzzy spheres in diverse dimensions

Abstract We construct spherical harmonics for fuzzy spheres of even and odd dimensions, generalizing the correspondence between finite Matrix algebras and fuzzy two-spheres. The finite Matrix algebras associated with the various fuzzy spheres have a natural basis which falls in correspondence with tensor constructions of irreducible representations of orthogonal groups SO ( n ). This basis is useful in describing fluctuations around various D-brane constructions of fuzzy spherical objects. The higher fuzzy spheres are non-associative algebras that appear as projections of associative algebras related to matrices. The non-associativity (as well as the non-commutativity) disappears in the leading large N limit, ensuring the correct classical limit. Some simple aspects of the combinatorics of the fuzzy four-sphere can be accounted by a heuristic picture of giant fractional instantons.

Diagonal multi-matrix correlators and BPS operators in N=4 SYM

We present a complete basis of multi-trace multi-matrix operators that has a diagonal two point function for the free matrix field theory at finite N. This generalises to multiple matrices the single matrix diagonalisation by Schur polynomials. Crucially, it involves intertwining the gauge group U(N) and the global symmetry group U(M) with Clebsch-Gordan coefficients of symmetric groups Sn. When applied to = 4 super Yang-Mills we consider the U(3) subgroup of the full symmetry group. The diagonalisation allows the description of a dual basis to multi-traces, which permits the characterisation of the metric on operators transforming in short representations at weak coupling. This gives a framework for the comparison of quarter and eighth-BPS giant gravitons of AdS5 × S5 spacetime to gauge invariant operators of the dual = 4 SYM.

LARGE N 2D YANG-MILLS THEORY AND TOPOLOGICAL STRING THEORY

Abstract: 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 Taylors “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.

Diagonal free field matrix correlators, global symmetries and giant gravitons

We obtain a basis of diagonal free field multi-matrix 2-point correlators in a theory with global symmetry group G. The operators fall into irreducible representations of G. This applies for gauge group U(N) at finite N. For composites made of n fundamental fields, this is expressed in terms of Clebsch-Gordan coefficients for the decomposition of the n-fold tensor products of the fundamental field representation in terms of G × Sn representations. We use this general construction in the case of the SL(2) sector of = 4 SYM. In this case, by using oscillator constructions, we reduce the computation of the relevant Clebsch-Gordans coupling infinite dimensional discrete series irreps of SL(2) to a problem in symmetric groups. Applying these constructions we write down gauge invariant operators with a Fock space structure similar to that arising in a large angular momentum limit of worldvolume excitations of giant gravitons. The Fock space structure emerges from Clebsch multiplicities of tensor products of symmetric group representations. We also give the action of the 1-loop dilatation operator of = 4 SYM on this basis of multi-matrix operators.

Finite factorization equations and sum rules for BPS correlators in N=4 SYM theory

Abstract A class of exact non-renormalized extremal correlators of half-BPS operators in N =4 SYM, with U ( N ) gauge group, is shown to satisfy finite factorization equations reminiscent of topological gauge theories. The finite factorization equations can be generalized, beyond the extremal case, to a class of correlators involving observables with a simple pattern of SO (6) charges. The simple group theoretic form of the correlators allows equalities between ratios of correlators in N =4 SYM and Wilson loops in Chern–Simons theories at k =∞, correlators of appropriate observables in topological G / G models and Wilson loops in two-dimensional Yang–Mills theories. The correlators also obey sum rules which can be generalized to off-extremal correlators. The simplest sum rules can be viewed as large k limits of the Verlinde formula using the Chern–Simons correspondence. For special classes of correlators, the saturation of the factorization equations by a small subset of the operators in the large N theory is related to the emergence of semiclassical objects like KK modes and giant gravitons in the dual AdS × S background. We comment on an intriguing symmetry between KK modes and giant gravitons.

Non commutative gravity from the ADS/CFT correspondence

The exclusion principle of Maldacena and Strominger is seen to follow from deformed Heisenberg algebras associated with the chiral rings of SN 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 AdS3 × S3 space-time of the associated six-dimensional supergravity theory acquires, when quantum effects are taken into account, a non-commutative structure given by SUq(1,1) × SUq(2). Both remarks imply that finite N effects are captured by quantum groups SLq(2) with q = eiπ/(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 SUq(1,1) at roots of unity.

Gravity from CFT on SN(X) : symmetries and interactions

The orbifold CFT dual to string theory on

Exact S-matrix for two-dimensional string theory

ADS_3 \times S^3