Tensors, !-graphs, and non-commutative quantum structures

Categorical quantum mechanics (CQM) and the theory of quantum groups rely heavily on the use of structures that have both an algebraic and co-algebraic component, making them well-suited for manipulation using diagrammatic techniques. Diagrams allow us to easily form complex compositions of (co)algebraic structures, and prove their equality via graph rewriting. One of the biggest challenges in going beyond simple rewriting-based proofs is designing a graphical language that is expressive enough to prove interesting properties (e.g. normal form results) about not just single diagrams, but entire families of diagrams. One candidate is the language of !-graphs, which consist of graphs with certain subgraphs marked with boxes (called !-boxes) that can be repeated any number of times. New !-graph equations can then be proved using a powerful technique called !-box induction. However, previously this technique only applied to commutative (or cocommutative) algebraic structures, severely limiting its applications in some parts of CQM and (especially) quantum groups. In this paper, we fix this shortcoming by offering a new semantics for non-commutative !-graphs using an enriched version of Penrose's abstract tensor notation.Comment: In Proceedings QPL 2014, arXiv:1412.810

Open Graphs and Monoidal Theories

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph. While the category of open-graphs is not itself adhesive, we introduce the notion of a selective adhesive functor, and show that such a functor embeds the category of open-graphs into the ambient adhesive category of typed graphs. Using this functor, the category of open-graphs inherits "enough adhesivity" from the category of typed graphs to perform double-pushout (DPO) graph rewriting. A salient feature of our theory is that it ensures rewrite systems are "type-safe" in the sense that rewriting respects the inputs and outputs. This formalism lets us safely encode the interesting structure of a computational model, such as evaluation dynamics, with succinct, explicit rewrite rules, while the graphical representation absorbs many of the tedious details. Although topological formalisms exist for string diagrams, our construction is discreet, finitary, and enjoys decidable algorithms for composition and rewriting. We also show how open-graphs can be parametrised by graphical signatures, similar to the monoidal signatures of Joyal and Street, which define types for vertices in the diagrammatic language and constraints on how they can be connected. Using typed open-graphs, we can construct free symmetric monoidal categories, PROPs, and more general monoidal theories. Thus open-graphs give us a handle for mechanised reasoning in monoidal categories.Comment: 31 pages, currently technical report, submitted to MSCS, waiting review

Towards Quantum Dielectric Branes: Curvature Corrections in Abelian Beta Function and Nonabelian Born-Infeld Action

We initiate a programme to compute curvature corrections to the nonabelian BI action. This is based on the calculation of derivative corrections to the abelian BI action, describing a maximal brane, to all orders in F. An exact calculation in F allows us to apply the SW map, reducing the maximal abelian point of view to a minimal nonabelian point of view (replacing 1/F with [X,X] at large F), resulting in matrix model equations of motion. We first study derivative corrections to the abelian BI action and compute the 2-loop beta function for an open string in a WZW (parallelizable) background. This beta function is the first step in the process of computing string equations of motion, which can be later obtained by computing the Weyl anomaly coefficients or the partition function. The beta function is exact in F and computed to orders O(H,H^2,H^3) (H=dB and curvature is R ~ H^2) and O(DF,D^2F,D^3F). In order to carry out this calculation we develop a new regularization method for 2-loop graphs. We then relate perturbative results for abelian and nonabelian BI actions, by showing how abelian derivative corrections yield nonabelian commutator corrections, at large F. We begin the construction of a matrix model describing \a' corrections to Myers' dielectric effect. This construction is carried out by setting up a perturbative classification of the relevant nonabelian tensor structures, which can be considerably narrowed down by the constraint of translation invariance in the action and the possibility for generic field redefinitions. The final matrix action is not uniquely determined and depends upon two free parameters. These parameters could be computed via further calculations in the abelian theory.Comment: JHEP3.cls, 64 pages, 3 figures; v2: added references; v3: more references, final version for NP

Dynamics of Higher Spin Fields and Tensorial Space

The structure and the dynamics of massless higher spin fields in various dimensions are reviewed with an emphasis on conformally invariant higher spin fields. We show that in D=3,4,6 and 10 dimensional space-time the conformal higher spin fields constitute the quantum spectrum of a twistor-like particle propagating in tensorial spaces of corresponding dimensions. We give a detailed analysis of the field equations of the model and establish their relation with known formulations of free higher spin field theory.Comment: JHEP3 style, 40 pages; v2 typos corrected, comments and references added; v3 published versio

Infrared Consistency and the Weak Gravity Conjecture

The weak gravity conjecture (WGC) asserts that an Abelian gauge theory coupled to gravity is inconsistent unless it contains a particle of charge $q$ and mass $m$ such that $q \geq m/m_{\rm Pl}$. This criterion is obeyed by all known ultraviolet completions and is needed to evade pathologies from stable black hole remnants. In this paper, we explore the WGC from the perspective of low-energy effective field theory. Below the charged particle threshold, the effective action describes a photon and graviton interacting via higher-dimension operators. We derive infrared consistency conditions on the parameters of the effective action using i) analyticity of light-by-light scattering, ii) unitarity of the dynamics of an arbitrary ultraviolet completion, and iii) absence of superluminality and causality violation in certain non-trivial backgrounds. For convenience, we begin our analysis in three spacetime dimensions, where gravity is non-dynamical but has a physical effect on photon-photon interactions. We then consider four dimensions, where propagating gravity substantially complicates all of our arguments, but bounds can still be derived. Operators in the effective action arise from two types of diagrams: those that involve electromagnetic interactions (parameterized by a charge-to-mass ratio $q/m$) and those that do not (parameterized by a coefficient $\gamma$). Infrared consistency implies that $q/m$ is bounded from below for small $\gamma$.Comment: 37 pages, 5 figures. Minor typos fixed and equation numbers changed to match journal. Published in JHE

Geometric spin foams, Yang-Mills theory and background-independent models

We review the dual transformation from pure lattice gauge theory to spin foam models with an emphasis on a geometric viewpoint. This allows us to give a simple dual formulation of SU(N) Yang-Mills theory, where spin foam surfaces are weighted with the exponentiated area. In the case of gravity, we introduce a symmetry condition which demands that the amplitude of an individual spin foam depends only on its geometric properties and not on the lattice on which it is defined. For models that have this property, we define a new sum over abstract spin foams that is independent of any choice of lattice or triangulation. We show that a version of the Barrett-Crane model satisfies our symmetry requirement.Comment: 28 pages, 27 diagrams, typos correcte