We quantize the superstring on the AdS 2 × S 2 background with Ramond-Ramond flux using a P SU (1, 1|2)/U (1) × U (1) sigma model with a WZ term. One-loop conformal invariance of the model is guaranteed by a general mechanism which holds for coset spaces G/H where G is Ricci-flat and H is the invariant locus of a Z 4 automorphism of G. This mechanism gives conformal theories for the P SU (1, 1|2) × P SU (2|2)/SU (2) × SU (2) and P SU (2, 2|4)/SO(4, 1) × SO(5) coset spaces, suggesting our results might be useful for quantizing the superstring on AdS 3 × S 3 and AdS 5 × S 5 backgrounds.

We begin a classification of the symmetry algebras arising on configurations of type IIB [p, q] 7-branes. These include not just the Kodaira symmetries that occur when branes coalesce into a singularity, but also algebras associated to other physically interesting brane configurations that cannot be collapsed. We demonstrate how the monodromy around the 7-branes essentially determines the algebra, and thus 7-brane gauge symmetries are classified by conjugacy classes of the modular group SX(2,Z). Through a classic map between the modular group and binary quadratic forms, the monodromy fixes the asymptotic charge form which determines the representations of the various (p, q) dyons in probe D3-brane theories. This quadratic form also controls the change in the algebra during transitions between different brane configurations. We give a unified description of the brane configurations extending the DJV? EN an d Argyres-Douglas H^ series beyond the Kodaira cases. We anticipate the appearance of affine and indefinite infinite-dimensional algebras, which we explore in a sequel paper.

It is confirmed that geodesic string junctions are necessary to describe the gauge vectors of symmetry groups that arise in the context of IIB superstrings compactified in the presence of nonlocal 7-branes. By examining the moduli space of 7-brane backgrounds for which the dilaton and axion fields are constant, we are able to describe explicitly and geometrically how open string geodesics can fail to be smooth, and how geodesic string junctions then become the relevant BPS representatives of the gauge bosons. The mechanisms that guarantee the existence and uniqueness of the BPS representative of any gauge vector are also shown to generalize to the case where the dilaton and axion fields are not constant. *

In a previous paper we explored how conjugacy classes of the modular group classify the symmetry algebras that arise on type IIB [p,q] 7-branes. The Kodaira list of finite Lie algebras completely fills the elliptic classes as well as some parabolic classes. Loop algebras of E N fill additional parabolic classes, and exotic finite algebras, hyperbolic extensions of E N and more general indefinite Lie algebras fill the hyperbolic classes. Since they correspond to brane configurations that cannot be made into strict singularities, these non-Kodaira algebras are spectrum generating and organize towers of massive BPS states into representations. The smallest brane configuration with unit monodromy gives rise to the loop algebra E 9 which plays a central role in the theory. We elucidate the patterns of enhancement relating E 8 , E 9 , E 9 and E 10 . We examine configurations of 24 7-branes relevant to type IIB compactifications on a two-sphere, or F-theory on K3. A particularly symmetric configuration separates the 7-branes into two groups of twelve branes and the massive BPS spectrum is organized by E 10 ⊕ E 10 . Review of 7-branes and Lie AlgebrasWe shall introduce the infinite-dimensional algebras arising on 7-brane configurations by exploring how they arise as enhancements of more familiar, finite algebras. Let us first review the 7-brane configurations realizing finite algebras, and their supported junctions.Consider a configuration G = X z 1 . . . X zn of 7-branes X z i , where we use a charge-vector notation z i = [p i , q i ] to label the charges of each brane. Each 7-brane has an associated monodromy K z = 1 1 + zz T S = 1+pq −p 2 q 2 1−pq , (2.1)

BPS states of N = 2, D = 4 Super Yang-Mills theories with ADE flavor symmetry arise as junctions joining a D3-brane to a set of 7-branes defining the enhanced flavor algebra. We show that the familiar BPS spectrum of SU(2) theories with N f ≤ 4 is simply given by the set of junctions whose self-intersection is bounded below as required by supersymmetry. This constraint, together with the relations between junction and weight lattices, is used to establish the appearance of arbitrarily large flavor representations for the case of D n≥5 and E n symmetries. Such representations are required by consistency with decoupling down to smaller flavor symmetries.

A map between string junctions in the affine 7-brane backgrounds and vector bundles on del Pezzo surfaces is constructed using mirror symmetry. It is shown that the lattice of string junctions with support on an affine 7-brane configuration is isomorphic to the K-theory group of the corresponding del Pezzo surface. This isomorphism allows us to construct a map between the states of the N = 2, D=4 theories with E N global symmetry realized in two different ways in Type IIB and Type IIA string theory. A subgroup of the SL(2, Z Z) symmetry of the E 9 7-brane background appears as the Fourier-Mukai transform acting on the D-brane configurations realizing vector bundles on elliptically fibered B 9 .

We study the general structure of Smirnov's axioms on form factors of local operators in integrable models. We find various consistency conditions that the form factor functions have to satisfy. For the special case of the O(3) σ-model we construct simple polynomial solutions for the operators of the spin-field, current, energy-momentum tensor and topological charge density.

We study the algebra of SU (n)-currents in the many-flavor chiral Gross-Neveu model. The general structure of the current-current OPE leading to non-local quantum conserved charges is reviewed. We calculate the OPE in the one-flavor and the many-flavor models perturbatively and use renormalization group invariance to prove that our results are not altered by higher-order corrections. We conclude that in these models the non-local quantum charge exists which is the first step towards the proof of the absence of particle production and factorization.