Presentations of Wess-Zumino-Witten Fusion Rings

The fusion rings of Wess-Zumino-Witten models are re-examined. Attention is drawn to the difference between fusion rings over Z (which are often of greater importance in applications) and fusion algebras over C. Complete proofs are given characterising the fusion algebras (over C) of the SU(r+1) and Sp(2r) models in terms of the fusion potentials, and it is shown that the analagous potentials cannot describe the fusion algebras of the other models. This explains why no other representation-theoretic fusion potentials have been found. Instead, explicit generators are then constructed for general WZW fusion rings (over Z). The Jacobi-Trudy identity and its Sp(2r) analogue are used to derive the known fusion potentials. This formalism is then extended to the WZW models over the spin groups of odd rank, and explicit presentations of the corresponding fusion rings are given. The analogues of the Jacobi-Trudy identity for the spinor representations (for all ranks) are derived for this purpose, and may be of independent interest.Comment: 32 pages, 3 figures, added references, minor additions to text. To be published in Rev. Math. Phy

Hadronic Density of States from String Theory

Exactly soluble string theories describing a particular hadronic sector of certain confining gauge theories have been obtained recently as Penrose-Gueven limits of the dual supergravity backgrounds. The effect of taking the Penrose-Gueven limit on the gravity side translates, in the gauge theory side, into an effective truncation to hadrons of large U(1) charge (annulons). We present an exact calculation of the finite temperature partition function for the hadronic states corresponding to a Penrose-Gueven limit of the Maldacena-Nunez embedding of N=1 SYM into string theory. It is established that the theory exhibits a Hagedorn density of states. Motivated by this exact calculation we propose a semiclassical string approximation to the finite temperature partition function for confining gauge theories admitting a supergravity dual, by performing an expansion around classical solutions characterized by temporal windings. This semiclassical approximation reveals a hadronic energy density of states of Hagedorn type, with the coefficient determined by the gauge theory string tension as expected for confining theories. We argue that our proposal captures primarily information about states of pure N=1 SYM, given that this semiclassical approximation does not entail a projection onto states of large U(1) charge.Comment: 15 page

Conformal and Superconformal Mechanics

We investigate the conformal and superconformal properties of a non-relativistic spinning particle propagating in a curved background coupled to a magnetic field and with a scalar potential. We derive the conditions on the couplings for a large class of such systems which are necessary in order their actions admit conformal and superconformal symmetry. We find that some of these conditions can be encoded in the conformal and holomorphic geometry of the background. Several new examples of conformal and superconformal models are also given.Comment: 46 pages, Phyzzx.te

D-brane charges on non-simply connected groups

The maximally symmetric D-branes of string theory on the non-simply connected Lie group SU(n)/Z_d are analysed using conformal field theory methods, and their charges are determined. Unlike the well understood case for simply connected groups, the charge equations do not determine the charges uniquely, and the charge group associated to these D-branes is therefore in general not cyclic. The precise structure of the charge group depends on some number theoretic properties of n, d, and the level of the underlying affine algebra k. The examples of SO(3)=SU(2)/Z_2 and SU(3)/Z_3 are worked out in detail, and the charge groups for SU(n)/Z_d at most levels k are determined explicitly.Comment: 31 pages, 1 figure. 2 refs added. Added the observation: the charge group for each su(2) theory equals the centre of corresponding A-D-E grou

Topics on the geometry of D-brane charges and Ramond-Ramond fields

In this paper we discuss some topics on the geometry of type II superstring backgrounds with D-branes, in particular on the geometrical meaning of the D-brane charge, the Ramond-Ramond fields and the Wess-Zumino action. We see that, depending on the behaviour of the D-brane on the four non-compact space-time directions, we need different notions of homology and cohomology to discuss the associated fields and charge: we give a mathematical definition of such notions and show their physical applications. We then discuss the problem of corretly defining Wess-Zumino action using the theory of p-gerbes. Finally, we recall the so-called *-problem and make some brief remarks about it.Comment: 29 pages, no figure

Towards reduction of type II theories on SU(3) structure manifolds

We revisit the reduction of type II supergravity on SU(3) structure manifolds, conjectured to lead to gauged N=2 supergravity in 4 dimensions. The reduction proceeds by expanding the invariant 2- and 3-forms of the SU(3) structure as well as the gauge potentials of the type II theory in the same set of forms, the analogues of harmonic forms in the case of Calabi-Yau reductions. By focussing on the metric sector, we arrive at a list of constraints these expansion forms should satisfy to yield a base point independent reduction. Identifying these constraints is a first step towards a first-principles reduction of type II on SU(3) structure manifolds.Comment: 20 pages; v2: condition (2.13old) on expansion forms weakened, replaced by (2.13new), (2.14new

Dynamic SU(2) Structure from Seven-branes

We obtain a family of supersymmetric solutions of type IIB supergravity with dynamic SU(2) structure, which describe the local geometry near a stack of four D7-branes and one O7-plane wrapping a rigid four-cycle. The deformation to a generalized complex geometry is interpreted as a consequence of nonperturbative effects in the seven-brane gauge theory. We formulate the problem for seven-branes wrapping the base of an appropriate del Pezzo cone, and in the near-stack limit in which the four-cycle is flat, we obtain an exact solution in closed form. Our solutions serve to characterize the local geometry of nonperturbatively-stabilized flux compactifications.Comment: 49 pages, 2 figures; v2: minor corrections, references adde

D-branes, KK-theory and duality on noncommutative spaces

We present a new categorical classification framework for D-brane charges on noncommutative manifolds using methods of bivariant K-theory. We describe several applications including an explicit formula for D-brane charge in cyclic homology, a refinement of open string T-duality, and a general criterion for cancellation of global worldsheet anomalies

On general flux backgrounds with localized sources

We derive new consistency conditions for string compactifications with generic fluxes (RR, NSNS, geometrical) and localized sources (D-branes, NS-branes, KK-monopoles). The constraints are all related by string dualities and share a common origin in M-theory. We also find new sources of instabilities. We discuss the importance of these conditions for the consistency of the effective action and for the study of interpolating solutions between vacua.Comment: 29 pages, 2 figures, v2: published versio

Homology and K--Theory Methods for Classes of Branes Wrapping Nontrivial Cycles

We apply some methods of homology and K-theory to special classes of branes wrapping homologically nontrivial cycles. We treat the classification of four-geometries in terms of compact stabilizers (by analogy with Thurston's classification of three-geometries) and derive the K-amenability of Lie groups associated with locally symmetric spaces listed in this case. More complicated examples of T-duality and topology change from fluxes are also considered. We analyse D-branes and fluxes in type II string theory on ${\mathbb C}P^3\times \Sigma_g \times {\mathbb T}^2$ with torsion $H-$flux and demonstrate in details the conjectured T-duality to ${\mathbb R}P^7\times X^3$ with no flux. In the simple case of $X^3 = {\mathbb T}^3$, T-dualizing the circles reduces to duality between ${\mathbb C}P^3\times {\mathbb T}^2 \times {\mathbb T}^2$ with $H-$flux and ${\mathbb R}P^7\times {\mathbb T}^3$ with no flux.Comment: 27 pages, tex file, no figure