Integrable Boundaries, Conformal Boundary Conditions and A-D-E Fusion Rules

The $sl(2)$ minimal theories are labelled by a Lie algebra pair $(A,G)$ where $G$ is of $A$-$D$-$E$ type. For these theories on a cylinder we conjecture a complete set of conformal boundary conditions labelled by the nodes of the tensor product graph $A\otimes G$. The cylinder partition functions are given by fusion rules arising from the graph fusion algebra of $A\otimes G$. We further conjecture that, for each conformal boundary condition, an integrable boundary condition exists as a solution of the boundary Yang-Baxter equation for the associated lattice model. The theory is illustrated using the $(A_4,D_4)$ or 3-state Potts model.Comment: 4 pages, REVTe

Virasoro constraints and the Chern classes of the Hodge bundle

We analyse the consequences of the Virasoro conjecture of Eguchi, Hori and Xiong for Gromov-Witten invariants, in the case of zero degree maps to the manifolds CP^1 and CP^2 (or more generally, smooth projective curves and smooth simply-connected projective surfaces). We obtain predictions involving intersections of psi and lambda classes on the compactification of M_{g,n}. In particular, we show that the Virasoro conjecture for CP^2 implies the numerical part of Faber's conjecture on the tautological Chow ring of M_g.Comment: 12 pages, latex2

Integrable and Conformal Boundary Conditions for sl(2) A-D-E Lattice Models and Unitary Minimal Conformal Field Theories

Integrable boundary conditions are constructed for the critical A-D-E lattice models of statistical mechanics. In particular, using techniques associated with the Temperley-Lieb algebra and fusion, a set of explicit boundary Boltzmann weights which satisfies the boundary Yang-Baxter equation is obtained for each boundary condition. When appropriately specialised, these boundary weights, each of which depends on three spins, decompose into more natural two-spin edge weights. The specialised boundary conditions are also naturally in one-to-one correspondence with the conformal boundary conditions of sl(2) unitary minimal conformal field theories. Supported by this and further evidence, we conclude that, in the continuum scaling limit, the integrable boundary conditions provide realisations of the complete set of conformal boundary conditions in the corresponding field theories

Gauges and Cosmological Backreaction

We present a formalism for spatial averaging in cosmology applicable to general spacetimes and coordinates, and allowing the easy incorporation of a wide variety of matter sources. We apply this formalism to a Friedmann-LeMaitre-Robertson-Walker universe perturbed to second-order and present the corrections to the background in an unfixed gauge. We then present the corrections that arise in uniform curvature and conformal Newtonian gauges.Comment: 13 pages. Updated: reference added, typos corrected, exposition clarified. Version 3: Replaced with version published by JCA

Solution of the dual reflection equation for An−1(1)A^{(1)}_{n-1} SOS model

We obtain a diagonal solution of the dual reflection equation for elliptic $A^{(1)}_{n-1}$ SOS model. The isomorphism between the solutions of the reflection equation and its dual is studied.Comment: Latex file 12 pages, added reference

Donaldson-Thomas invariants and wall-crossing formulas

Notes from the report at the Fields institute in Toronto. We introduce the Donaldson-Thomas invariants and describe the wall-crossing formulas for numerical Donaldson-Thomas invariants.Comment: 18 pages. To appear in the Fields Institute Monograph Serie

A Construction of Solutions to Reflection Equations for Interaction-Round-a-Face Models

We present a procedure in which known solutions to reflection equations for interaction-round-a-face lattice models are used to construct new solutions. The procedure is particularly well-suited to models which have a known fusion hierarchy and which are based on graphs containing a node of valency $1$. Among such models are the Andrews-Baxter-Forrester models, for which we construct reflection equation solutions for fixed and free boundary conditions.Comment: 9 pages, LaTe