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

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

Curve counting via stable pairs in the derived category

For a nonsingular projective 3-fold $X$, we define integer invariants virtually enumerating pairs $(C,D)$ where $C\subset X$ is an embedded curve and $D\subset C$ is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of $X$. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of $X$. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.Comment: Corrected typos and duality error in Proposition 4.6. 47 page

Cosmological Backreaction from Perturbations

We reformulate the averaged Einstein equations in a form suitable for use with Newtonian gauge linear perturbation theory and track the size of the modifications to standard Robertson-Walker evolution on the largest scales as a function of redshift for both Einstein de-Sitter and Lambda CDM cosmologies. In both cases the effective energy density arising from linear perturbations is of the order of 10^-5 the matter density, as would be expected, with an effective equation of state w ~ -1/19. Employing a modified Halofit code to extend our results to quasilinear scales, we find that, while larger, the deviations from Robertson-Walker behaviour remain of the order of 10^-5.Comment: 15 pages, 8 figures; replaced by version accepted by JCA

The charges of a twisted brane

The charges of the twisted D-branes of certain WZW models are determined. The twisted D-branes are labelled by twisted representations of the affine algebra, and their charge is simply the ground state multiplicity of the twisted representation. It is shown that the resulting charge group is isomorphic to the charge group of the untwisted branes, as had been anticipated from a K-theory calculation. Our arguments rely on a number of non-trivial Lie theoretic identities.Comment: 27 pages, 1 figure, harvmac (b

Conformal Field Theories, Graphs and Quantum Algebras

This article reviews some recent progress in our understanding of the structure of Rational Conformal Field Theories, based on ideas that originate for a large part in the work of A. Ocneanu. The consistency conditions that generalize modular invariance for a given RCFT in the presence of various types of boundary conditions --open, twisted-- are encoded in a system of integer multiplicities that form matrix representations of fusion-like algebras. These multiplicities are also the combinatorial data that enable one to construct an abstract ``quantum'' algebra, whose $6j$- and $3j$-symbols contain essential information on the Operator Product Algebra of the RCFT and are part of a cell system, subject to pentagonal identities. It looks quite plausible that the classification of a wide class of RCFT amounts to a classification of ``Weak $C^*$- Hopf algebras''.Comment: 23 pages, 12 figures, LateX. To appear in MATHPHYS ODYSSEY 2001 --Integrable Models and Beyond, ed. M. Kashiwara and T. Miwa, Progress in Math., Birkhauser. References and comments adde

Averaging Robertson-Walker Cosmologies

The cosmological backreaction arises when one directly averages the Einstein equations to recover an effective Robertson-Walker cosmology, rather than assuming a background a priori. While usually discussed in the context of dark energy, strictly speaking any cosmological model should be recovered from such a procedure. We apply the Buchert averaging formalism to linear Robertson-Walker universes containing matter, radiation and dark energy and evaluate numerically the discrepancies between the assumed and the averaged behaviour, finding the largest deviations for an Einstein-de Sitter universe, increasing rapidly with Hubble rate to a 0.01% effect for h=0.701. For the LCDM concordance model, the backreaction is of the order of Omega_eff~4x10^-6, with those for dark energy models being within a factor of two or three. The impacts at recombination are of the order of 10^-8 and those in deep radiation domination asymptote to a constant value. While the effective equations of state of the backreactions in Einstein-de Sitter, concordance and quintessence models are generally dust-like, a backreaction with an equation of state w_eff<-1/3 can be found for strongly phantom models.Comment: 18 pages, 11 figures, ReVTeX. Updated to version accepted by JCA

Exact solution and surface critical behaviour of open cyclic SOS lattice models

We consider the $L$-state cyclic solid-on-solid lattice models under a class of open boundary conditions. The integrable boundary face weights are obtained by solving the reflection equations. Functional relations for the fused transfer matrices are presented for both periodic and open boundary conditions. The eigen-spectra of the unfused transfer matrix is obtained from the functional relations using the analytic Bethe ansatz. For a special case of crossing parameter $\lambda=\pi/L$, the finite-size corrections to the eigen-spectra of the critical models are obtained, from which the corresponding conformal dimensions follow. The calculation of the surface free energy away from criticality yields two surface specific heat exponents, $\alpha_s=2-L/2\ell$ and $\alpha_1=1-L/\ell$, where $\ell=1,2,\cdots,L-1$ coprime to $L$. These results are in agreement with the scaling relations $\alpha_s=\alpha_b+\nu$ and $\alpha_1=\alpha_b-1$.Comment: 13 pages, LaTeX, to appear in J. Phys.

Theoretical study of incoherent phi photoproduction on a deuteron target

We study the photoproduction of phi mesons in deuteron, paying attention to the modification of the cross section from bound protons to the free ones with the aim of comparing with recent results at LEPS. For this purpose we take into account Fermi motion in single scattering and rescattering of the phi to account for phi absorption on a second nucleon as well as the rescattering of the proton. We find that the contribution of the double scattering is much smaller than the typical cross section of gamma p to phi p in free space, which implies a very small screening of the phi production in deuteron. The contribution from the proton rescattering, on the other hand, is found to be not negligible compared to the cross section of gamma p to phi p in free space, and leads to a moderate reduction of the phi photoproduction cross section on a deuteron at forward angles if LEPS set up is taken into account. The Fermi motion allows contribution of the single scattering in regions forbidden by phase space in the free case. In particular, we find that for momentum transferred squared close to the maximum value, the Fermi motion changes drastically the shape of d sigma / dt, to the point that the ratio of this cross section to the free one becomes very sensitive to the precise value of t chosen, or the size of the bin used in an experimental analysis. Hence, this particular region of t does not seem the most indicated to find effects of a possible phi absorption in the deuteron. This reaction is studied theoretically as a function of t and the effect of the experimental angular cuts at LEPS is also discussed, providing guidelines for future experimental analyses of the reaction.Comment: 17 pages, 16 figure

Orientifolds, Unoriented Instantons and Localization

We consider world-sheet instanton effects in N=1 string orientifolds of noncompact toric Calabi-Yau threefolds. We show that unoriented closed string topological amplitudes can be exactly computed using localization techniques for holomorphic maps with involution. Our results are in precise agreement with mirror symmetry and large N duality predictions.Comment: 25 pages, 10 figures, published version; v4: typos correcte