Characters of the W3 algebra

Traces of powers of the zero mode in the W3 Algebra have recently been found to be of interest, for example in relation to Black Hole thermodynamics, and arise as the terms in an expansion of the full characters of the algebra. We calculate the first few such powers in two cases. Firstly, we find the traces in the 3-state Potts model by using null vectors to derive modular differential equations for the traces. Secondly, we calculate the exact results for Verma module representations. We compare our two methods with each other and the result of brute-force diagonalisation for low levels and find complete agreement.Comment: v2: Numerous small changes, version to appear in JHEP, 22 pages. v3: Typos corrected, matches published version, 22 page

Decision making dynamics in corporate boards

Members of boards of directors of large corporations who also serve together on an outside board, form the so called interlock graph of the board and are assumed to have a strong influence on each others' opinion. We here study how the size and the topology of the interlock graph affect the probability that the board approves a strategy proposed by the Chief Executive Officer. We propose a measure of the impact of the interlock on the decision making, which is found to be a good predictor of the decision dynamics outcome. We present two models of decision making dynamics, and we apply them to the data of the boards of the largest US corporations in 1999.Comment: 20 pages, 10 figures, submitte

Modular properties of characters of the W3 algebra

In a previous work, exact formulae and differential equations were found for traces of powers of the zero mode in the W3 algebra. In this paper we investigate their modular properties, in particular we find the exact result for the modular transformations of traces of $W_0^n$ for n = 1, 2, 3, solving exactly the problem studied approximately by Gaberdiel, Hartman and Jin. We also find modular differential equations satisfied by traces with a single $W_0$ inserted, and relate them to differential equations studied by Mathur et al. We find that, remarkably, these all seem to be related to weight 0 modular forms with expansions with non-negative integer coefficients.Comment: 20 pages. v2: 23 pages. v3: 23 pages, published in JHE

Signature Characters for A_2 and B_2

The signatures of the inner product matrices on a Lie algebra's highest weight representation are encoded in the representation's signature character. We show that the signature characters of a finite-dimensional Lie algebra's highest weight representations obey simple difference equations that have a unique solution once appropriate boundary conditions are imposed. We use these results to derive the signature characters of all $A_2$ and $B_2$ highest weight representations. Our results extend, and explain, signature patterns analogous to those observed by Friedan, Qiu and Shenker in the Virasoro algebra's representation theory.Comment: 22 p

Parity and Spin CFT with boundaries and defects

This paper is a follow-up to [arXiv:2001.05055] in which two-dimensional conformal field theories in the presence of spin structures are studied. In the present paper we define four types of CFTs, distinguished by whether they need a spin structure or not in order to be well-defined, and whether their fields have parity or not. The cases of spin dependence without parity, and of parity without the need of a spin structure, have not, to our knowledge, been investigated in detail so far. We analyse these theories by extending the description of CFT correlators via three-dimensional topological field theory developed in [arXiv:hep-th/0204148] to include parity and spin. In each of the four cases, the defining data are a special Frobenius algebra $F$ in a suitable ribbon fusion category, such that the Nakayama automorphism of $F$ is the identity (oriented case) or squares to the identity (spin case). We use the TFT to define correlators in terms of $F$ and we show that these satisfy the relevant factorisation and single-valuedness conditions. We allow for world sheets with boundaries and topological line defects, and we specify the categories of boundary labels and the fusion categories of line defect labels for each of the four types. The construction can be understood in terms of topological line defects as gauging a possibly non-invertible symmetry. We analyse the case of a $\mathbb{Z}_2$-symmetry in some detail and provide examples of all four types of CFT, with Bershadsky-Polyakov models illustrating the two new types.Comment: v2 - expanded some discussions in the introduction and in the examples section - 112 page

Adjustment and social choice

We discuss the influence of information contagion on the dynamics of choices in social networks of heterogeneous buyers. Starting from an inhomogeneous cellular automata model of buyers dynamics, we show that when agents try to adjust their reservation price, the tatonement process does not converge to equilibrium at some intermediate market share and that large amplitude fluctuations are actually observed. When the tatonnement dynamics is slow with respect to the contagion dynamics, large periodic oscillations reminiscent of business cycles appear.Comment: 13 pages, 6 figure

Non-unitarity in quantum affine Toda theory and perturbed conformal field theory

There has been some debate about the validity of quantum affine Toda field theory at imaginary coupling, owing to the non-unitarity of the action, and consequently of its usefulness as a model of perturbed conformal field theory. Drawing on our recent work, we investigate the two simplest affine Toda theories for which this is an issue - a2(1) and a2(2). By investigating the S-matrices of these theories before RSOS restriction, we show that quantum Toda theory, (with or without RSOS restriction), indeed has some fundamental problems, but that these problems are of two different sorts. For a2(1), the scattering of solitons and breathers is flawed in both classical and quantum theories, and RSOS restriction cannot solve this problem. For a2(2) however, while there are no problems with breather-soliton scattering there are instead difficulties with soliton-excited soliton scattering in the unrestricted theory. After RSOS restriction, the problems with kink-excited kink may be cured or may remain, depending in part on the choice of gradation, as we found in [12]. We comment on the importance of regradations, and also on the survival of R-matrix unitarity and the S-matrix bootstrap in these circumstances.Comment: 29 pp, LaTex2e, 6 eps and 1 ps figure

On the renormalisation group for the boundary Truncated Conformal Space Approach

In this paper we continue the study of the truncated conformal space approach to perturbed boundary conformal field theories. This approach to perturbation theory suffers from a renormalisation of the coupling constant and a multiplicative renormalisation of the Hamiltonian. We show how these two effects can be predicted by both physical and mathematical arguments and prove that they are correct to leading order for all states in the TCSA system. We check these results using the TCSA applied to the tri-critical Ising model and the Yang-Lee model. We also study the TCSA of an irrelevant (non-renormalisable) perturbation and find that, while the convergence of the coupling constant and energy scales are problematic, the renormalised and rescaled spectrum remain a very good fit to the exact result, and we find a numerical relationship between the IR and UV couplings describing a particular flow. Finally we study the large coupling behaviour of TCSA and show that it accurately encompasses several different fixed points.Comment: 27 pages, 19 figure

Defect flows in minimal models

In this paper we study a simple example of a two-parameter space of renormalisation group flows of defects in Virasoro minimal models. We use a combination of exact results, perturbation theory and the truncated conformal space approach to search for fixed points and investigate their nature. For the Ising model, we confirm the recent results of Fendley et al. In the case of central charge close to one, we find six fixed points, five of which we can identify in terms of known defects and one of which we conjecture is a new non-trivial conformal defect. We also include several new results on exact properties of perturbed defects and on the renormalisation group in the truncated conformal space approach.Comment: 35 pages, 21 figures. 1 reference adde