The toroidal block and the genus expansion

We study the correspondence between four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories in the case of N=2* gauge theory. We emphasize the genus expansion on the gauge theory side, as obtained via geometric engineering from the topological string. This point of view uncovers modular properties of the one-point conformal block on a torus with complexified intermediate momenta: in the large intermediate weight limit, it is a power series whose coefficients are quasi-modular forms. The all-genus viewpoint that the conformal field theory approach lends to the topological string yields insight into the analytic structure of the topological string partition function in the field theory limit.Comment: 36 page

Algebra diagrams: a HANDi introduction

A diagrammatic notation for algebra is presented – Hierarchical Al- gebra Network Diagrams, HANDi. The notation uses a 2D network notation with systematically designed icons to explicitly and coherently encode the fun- damental concepts of algebra. The structure of the diagrams is described and the rules for making derivations are presented. The key design features of HANDi are discussed and compared with the conventional formula notation in order demonstrate that the new notation is a more logical codification of intro- ductory algebra

A Universal Approach to Vertex Algebras

We characterize vertex algebras (in a suitable sense) as algebras over a certain graded co-operad. We also discuss some examples and categorical implications of this characterization.Comment: To appear in the Journal of Algebr

Non-renormalization theorems without supergraphs: The Wess-Zumino model

The non-renormalization theorems of chiral vertex functions are derived on the basis of an algebraic analysis. The property, that the interaction vertex is a second supersymmetry variation of a lower dimensional field monomial, is used to relate chiral Green functions to superficially convergent Green functions by extracting the two supersymmetry variations from an internal vertex and transforming them to derivatives acting on external legs. The analysis is valid in the massive as well as in the massless model and can be performed irrespective of properties of the superpotential at vanishing momentum.Comment: 20 pages, Latex, added acknowledgment

Determination of the anomalous dimension of gluonic operators in deep inelastic scattering at O(1/N_f)

Using large N_f methods we compute the anomalous dimension of the predominantly gluonic flavour singlet twist-2 composite operator which arises in the operator product expansion used in deep inelastic scattering. We obtain a d-dimensional expression for it which depends on the operator moment n. Its expansion in powers of epsilon = (4-d)/2 agrees with the explicit exact three loop MSbar results available for n less than or equal to 8 and allows us to determine some new information on the explicit n-dependence of the three and higher order coefficients. In particular the n-dependence of the three loop anomalous dimension gamma_{gg}(a) is determined in the C_2(G) sector at O(1/N_f).Comment: 26 latex pages, 7 postscript figure

Space-time S-matrix and Flux tube S-matrix II. Extracting and Matching Data

We elaborate on a non-perturbative formulation of scattering amplitudes/null polygonal Wilson loops in planar N=4 Super-Yang-Mills theory. The construction is based on a decomposition of the Wilson loop into elementary building blocks named pentagon transitions. Our discussion expands on a previous letter of the authors where these transitions were introduced and analyzed for the so-called gluonic excitations. In this paper we revisit these transitions and extend the analysis to the sector of scalar excitations. We restrict ourselves to the single particle transitions and bootstrap their finite coupling expressions using a set of axioms. Besides these considerations, the main focus of the paper is on the extraction of perturbative data from scattering amplitudes at weak coupling and its comparison against the proposed pentagon transitions. We present several tests for both the hexagon and heptagon (MHV and NMHV) amplitudes up to two- and three-loop orders. In attached notebooks we provide explicit higher-loop predictions obtained from our method.Comment: 73 pages, 42 figures. v2,v3: typos correcte

The super-correlator/super-amplitude duality: Part I

We extend the recently discovered duality between MHV amplitudes and the light-cone limit of correlation functions of a particular type of local scalar operators to generic non-MHV amplitudes in planar N=4 SYM theory. We consider the natural generalization of the bosonic correlators to super-correlators of stress-tensor multiplets and show, in a number of examples, that their light-cone limit exactly reproduces the square of the matching super-amplitudes. Our correlators are computed at Born level. If all of their points form a light-like polygon, the correlator is dual to the tree-level amplitude. If a subset of points are not on the polygon but are integrated over, they become Lagrangian insertions generating the loop corrections to the correlator. In this case the duality with amplitudes holds at the level of the integrand. We build up the superspace formalism needed to formulate the duality and present the explicit example of the n-point NMHV tree amplitude as the dual of the lowest nilpotent level in the correlator.Comment: 56 page

Adler-Bardeen theorem and manifest anomaly cancellation to all orders in gauge theories

We reconsider the Adler-Bardeen theorem for the cancellation of gauge anomalies to all orders, when they vanish at one loop. Using the Batalin-Vilkovisky formalism and combining the dimensional-regularization technique with the higher-derivative gauge invariant regularization, we prove the theorem in the most general perturbatively unitary renormalizable gauge theories coupled to matter in four dimensions, and identify the subtraction scheme where anomaly cancellation to all orders is manifest, namely no subtractions of finite local counterterms are required from two loops onwards. Our approach is based on an order-by-order analysis of renormalization, and, differently from most derivations existing in the literature, does not make use of arguments based on the properties of the renormalization group. As a consequence, the proof we give also applies to conformal field theories and finite theories.Comment: 43 pages; EPJ