The quantum Hilbert space of a chiral two-form in d = 5 + 1 dimensions

We consider the quantum theory of a two-form gauge field on a space-time which is a direct product of time and a spatial manifold, taken to be a compact five-manifold with no torsion in its cohomology. We show that the Hilbert space of this non-chiral theory is a certain subspace of a tensor product of two spaces, that are naturally interpreted as the Hilbert spaces of a chiral and anti-chiral two-form theory respectively. We also study the observable operators in the non-chiral theory that correspond to the electric and magnetic field strengths, the Hamiltonian, and the exponentiated holonomy of the gauge-field around a spatial two-cycle. All these operators can be decomposed into contributions pertaining to the chiral and anti-chiral sectors of the theory.Comment: 15 page

BPS states in (2,0) theory on R x T5

We consider $(2, 0)$ theory on a space-time of the form $R \times T^5$, where the first factor denotes time, and the second factor is a flat spatial five-torus. In addition to their energy, quantum states are characterized by their spatial momentum, 't Hooft flux, and $Sp (4)$ $R$-symmetry representation. The momentum obeys a shifted quantization law determined by the 't Hooft flux. By supersymmetry, the energy is bounded from below by the magnitude of the momentum. This bound is saturated by BPS states, that are annihilated by half of the supercharges. The spectrum of such states is invariant under smooth deformations of the theory, and can thus be studied by exploiting the interpretation of $(2, 0)$ theory as an ultra-violet completion of maximally supersymmetric Yang-Mills theory on $R \times T^4$. Our main example is the $A$-series of $(2,0)$ theories, where such methods allow us to study the spectrum of BPS states for many values of the momentum and the 't Hooft flux. In particular, we can describe the $R$-symmetry transformation properties of these states by determining the image of their $Sp (4)$ representation in a certain quotient of the $Sp (4)$ representation ring.Comment: 22 page

The partition bundle of type A_{N-1} (2, 0) theory

Six-dimensional (2, 0) theory can be defined on a large class of six-manifolds endowed with some additional topological and geometric data (i.e. an orientation, a spin structure, a conformal structure, and an R-symmetry bundle with connection). We discuss the nature of the object that generalizes the partition function of a more conventional quantum theory. This object takes its values in a certain complex vector space, which fits together into the total space of a complex vector bundle (the `partition bundle') as the data on the six-manifold is varied in its infinite-dimensional parameter space. In this context, an important role is played by the middle-dimensional intermediate Jacobian of the six-manifold endowed with some additional data (i.e. a symplectic structure, a quadratic form, and a complex structure). We define a certain hermitian vector bundle over this finite-dimensional parameter space. The partition bundle is then given by the pullback of the latter bundle by the map from the parameter space related to the six-manifold to the parameter space related to the intermediate Jacobian.Comment: 15 pages. Minor changes, added reference

Analyticity Properties of Graham-Witten Anomalies

Analytic properties of Graham-Witten anomalies are considered. Weyl anomalies according to their analytic properties are of type A (coming from $\delta$-singularities in correlators of several energy-momentum tensors) or of type B (originating in counterterms which depend logarithmically on a mass scale). It is argued that all Graham-Witten anomalies can be divided into 2 groups: internal and external, and that all external anomalies are of type B, whereas among internal anomalies there is one term of type A and all the rest are of type B. This argument is checked explicitly for the case of a free scalar field in a 6-dimensional space with a 2-dimensional submanifold.Comment: 2 typos correcte

Rigid surface operators and S-duality: some proposals

We study surface operators in the N=4 supersymmetric Yang-Mills theories with gauge groups SO(n) and Sp(2n). As recently shown by Gukov and Witten these theories have a class of rigid surface operators which are expected to be related by S-duality. The rigid surface operators are of two types, unipotent and semisimple. We make explicit proposals for how the S-duality map should act on unipotent surface operators. We also discuss semisimple surface operators and make some proposals for certain subclasses of such operators.Comment: 27 pages. v2: minor changes, added referenc

Extended superspace, higher derivatives and SL(2,Z) duality

We consider the low-energy effective action for the Coulomb phase of an N=2 supersymmetric gauge theory with a rank one gauge group. The N = 2 superspace formalism is naturally invariant under an SL(2, {\bf Z}) group of duality transformations, regardless of the form of the action. The leading and next to leading terms in the long distance expansion of the action are given by the holomorphic prepotential and a real analytic function respectively. The latter is shown to be modular invariant with respect to SL(2, {\bf Z})

BPS surface observables in six-dimensional (2,0) theory

The supergroup OSp(8*|4), which is the superconformal group of (2,0) theory in six dimensions, is broken to the subgroup OSp(4|2)xOSp(4|2) by demanding the invariance of a certain product in a superspace with eight bosonic and four fermionic dimensions. We show that this is consistent with the symmetry breaking induced by the presence of a flat two-dimensional BPS surface in the usual (2,0) superspace, which has six bosonic and sixteen fermionic dimensions.Comment: 9 pages, LaTeX. v2: reference adde

Bound states in N = 4 SYM on T^3: Spin(2n) and the exceptional groups

The low energy spectrum of (3+1)-dimensional N=4 supersymmetric Yang-Mills theory on a spatial three-torus contains a certain number of bound states, characterized by their discrete abelian magnetic and electric 't Hooft fluxes. At weak coupling, the wave-functions of these states are supported near points in the moduli space of flat connections where the unbroken gauge group is semi-simple. The number of such states is related to the number of normalizable bound states at threshold in the supersymmetric matrix quantum mechanics with 16 supercharges based on this unbroken group. Mathematically, the determination of the spectrum relies on the classification of almost commuting triples with semi-simple centralizers. We complete the work begun in a previous paper, by computing the spectrum of bound states in theories based on the even-dimensional spin groups and the exceptional groups. The results satisfy the constraints of S-duality in a rather non-trivial way.Comment: 20 page

First Law, Counterterms and Kerr-AdS_5 Black Holes

We apply the counterterm subtraction technique to calculate the action and other quantities for the Kerr--AdS black hole in five dimensions using two boundary metrics; the Einstein universe and rotating Einstein universe with arbitrary angular velocity. In both cases, the resulting thermodynamic quantities satisfy the first law of thermodynamics. We point out that the reason for the violation of the first law in previous calculations is that the rotating Einstein universe, used as a boundary metric, was rotating with an angular velocity that depends on the black hole rotation parameter. Using a new coordinate system with a boundary metric that has an arbitrary angular velocity, one can show that the resulting physical quantities satisfy the first law.Comment: 19 pages, 1 figur