Quantum states to brane geometries via fuzzy moduli spaces of giant gravitons

Eighth-BPS local operators in N=4 SYM are dual to quantum states arising from the quantization of a moduli space of giant gravitons in AdS5xS5. Earlier results on the quantization of this moduli space give a Hilbert space of multiple harmonic oscillators in 3 dimensions. We use these results, along with techniques from fuzzy geometry, to develop a map between quantum states and brane geometries. In particular there is a map between the oscillator states and points in a discretization of the base space in the toric fibration of the moduli space. We obtain a geometrical decomposition of the space of BPS states with labels consisting of U(3) representations along with U(N) Young diagrams and associated group theoretic multiplicities. Factorization properties in the counting of BPS states lead to predictions for BPS world-volume excitations of specific brane geometries. Some of our results suggest an intriguing complementarity between localisation in the moduli space of branes and localisation in space-time.Comment: 69 pages, 6 figures. v2: references adde

A remark on T-duality and quantum volumes of zero-brane moduli spaces

T-duality (Fourier-Mukai duality) and properties of classical instanton moduli spaces can be used to deduce some properties of $\alpha^{\prime}$-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.Comment: 6-pages, Harvmac big, 2 figures; version 2 : ref added v3 : final JHEP version - minor clarification + ref adde

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.Comment: 247 pages (280 pages "l" mode), 60 figures. Lectures presented at the 1994 Les Houches Summer School ``Fluctuating Geometries in Statistical Mechanics and Field Theory.'' (also available at http://xxx.lanl.gov/lh94/ ). replaced to correct inessential typo

Zero-Branes on a Compact Orbifold

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.Comment: 34 pages (harvmac big); version to appear in JHE

Non-associative gauge theory and higher spin interactions

We give a framework to describe gauge theory on a certain class of commutative but non-associative fuzzy spaces. Our description is in terms of an Abelian gauge connection valued in the algebra of functions on the cotangent bundle of the fuzzy space. The structure of such a gauge theory has many formal similarities with that of Yang-Mills theory. The components of the gauge connection are functions on the fuzzy space which transform in higher spin representations of the Lorentz group. In component form, the gauge theory describes an interacting theory of higher spin fields, which remains non-trivial in the limit where the fuzzy space becomes associative. In this limit, the theory can be viewed as a projection of an ordinary non-commutative Yang-Mills theory. We describe the embedding of Maxwell theory in this extended framework which follows the standard unfolding procedure for higher spin gauge theories.Comment: 1+49 pages, LaTeX; references and clarifying remarks adde

Large N 2D Yang-Mills Theory and Topological String Theory

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 ``\Omega^{-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.Comment: 95 pages, 15 Postscript figures, uses harvmac (Please use the "large" print option.) Extensive revisions of the sections on topological field theory. Added a compact synopsis of topological field theory. Minor typos corrected. References adde

D-Branes and Physical States

States obtained by projecting boundary states, associated with D-branes, to fixed mass-level and momentum generically define non-trivial cohomology classes. For on-shell states the cohomology is the standard one, but when the states are off-shell the relevant cohomology is defined using a BRST operator with ghost zero modes removed. The zero momentum cohomology falls naturally into multiplets of $SO(D-1,1)$. At the massless level, a simple set of D-brane configurations generates the full set of zero-momentum states of standard ghost number, including the discrete states. We give a general construction of off-shell cohomology classes, which exhibits a non-trivial interaction between left and right movers that is not seen in on-shell cohomology. This includes, at higher mass levels, states obtained from typical D-brane boundary states as well as states with more intricate ghost dependence.Comment: 22 pages, harvmac, no figure

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 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 $q= e^{{i \pi \over {N + 1}}}$. 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.Comment: 28 pages in harvmac big ; v2: Minor corrections, ref adde