Search

Not all planar algebras can encode the algebraic structure of a Yang--Baxter integrable model described in terms of a so-called homogeneous transfer operator. In the family of subfactor planar algebras, we focus on the ones known as singly generated and find that the only such planar algebras underlying homogeneous Yang--Baxter integrable models are the so-called Yang--Baxter relation planar algebras. According to a result of Liu, there are three such planar algebras: the well-known Fuss--Catalan and Birman--Wenzl--Murakami planar algebras, in addition to one more which we refer to as the Liu planar algebra. The Fuss--Catalan and Birman--Wenzl--Murakami algebras are known to underlie Yang--Baxter integrable models, and we show that the Liu algebra likewise admits a Baxterisation. We also show that the homogeneous transfer operator describing a model underlied by a singly generated Yang--Baxter relation planar algebra is polynomialisable, meaning that it is polynomial in a spectral-parameter-independent element of the algebra., Comment: 38 pages, v2: title changed, presentation restructured

The Quantum Inverse Scattering Method is a scheme for solving integrable models in $1+1$ dimensions, building on an $R$-matrix that satisfies the Yang--Baxter equation and in terms of which one constructs a commuting family of transfer matrices. In the standard formulation, this $R$-matrix acts on a tensor product of vector spaces. Here, we relax this tensorial property and develop a framework for describing and analysing integrable models based on planar algebras, allowing non-separable \textit{$R$-operators} satisfying \textit{generalised} Yang--Baxter equations. We also re-evaluate the notion of integrals of motion and characterise when an (algebraic) \textit{transfer operator} is polynomial in a single integral of motion. We refer to such models as {\em polynomially integrable}. In an eight-vertex model, we demonstrate that the corresponding transfer operator is polynomial in the natural hamiltonian. In the Temperley--Lieb loop model with loop fugacity $\beta\in\mathbb{C}$, we likewise find that, for all but finitely many $\beta$-values, the transfer operator is polynomial in the usual hamiltonian element of the Temperley--Lieb algebra $\mathrm{TL}_n(\beta)$, at least for $n\leq17$. Moreover, we find that this model admits a second canonical hamiltonian, and that this hamiltonian also acts as a polynomial integrability generator for small $n$ and all but finitely many $\beta$-values., Comment: 51 pages, v2: minor changes

The usual Galilean contraction procedure for generating new conformal symmetry algebras takes as input a number of symmetry algebras which are equivalent up to central charge. We demonstrate that the equivalence condition can be relaxed by inhomogeneously contracting the chiral algebras and present general results for the ensuing asymmetric Galilean algebras. Several examples relevant to conformal field theory are discussed in detail, including superconformal algebras and W-algebras. We also discuss how the Sugawara construction is modified in the asymmetric setting., Comment: 21 pages, comments welcome, v2: minor updates and notation changes, added references

The weights of finite-dimensional representations of simple Lie algebras are naturally associated with Weyl polytopes. Representation characters decompose into multiplicity-free sums over the weights in Weyl polytopes. The Brion formula for these Weyl polytope sums is remarkably similar to the Weyl character formula. Moreover, the same Lie characters are also expressible as Demazure character formulas. This motivates a search for new expressions for Weyl polytope sums, and we prove such a formula involving Demazure operators. It applies to the Weyl polytope sums of the simple Lie algebras $A_n$, for all dominant integrable highest weights and all ranks $n$., Comment: 8 pages

We introduce a dense and a dilute loop model on causal dynamical triangulations. Both models are characterised by a geometric coupling constant $g$ and a loop parameter $\alpha$ in such a way that the purely geometric causal triangulation model is recovered for $\alpha=1$. We show that the dense loop model can be mapped to a solvable planar tree model, whose partition function we compute explicitly and use to determine the critical behaviour of the loop model. The dilute loop model can likewise be mapped to a planar tree model; however, a closed-form expression for the corresponding partition function is not obtainable using the standard methods employed in the dense case. Instead, we derive bounds on the critical coupling $g_c$ and apply transfer matrix techniques to examine the critical behaviour for $\alpha$ small., Comment: 30 pages, v2: minor changes

We investigate a class of reducible yet indecomposable modules over the $N=2$ superconformal algebras. These so-called staggered modules exhibit a non-diagonalisable action of the Virasoro mode $L_{0}$. Using recent results on the coset construction of $N=2$ minimal models, we explicitly construct such modules for central charges $c = -1$ and $c = -6$. We also describe spectral-flow orbits and symmetries of the families of staggered modules which arise via the coset., Comment: 25 pages, 10 figures, comments are welcome.V2 corrected typos, improved arguments in section 5.2

Galilean conformal algebras can be constructed by contracting a finite number of conformal algebras, and enjoy truncated $\mathbb{Z}$-graded structures. Here, we present a generalisation of the Galilean contraction procedure, giving rise to Galilean conformal algebras with truncated $\mathbb{Z}^{\otimes\sigma}$-gradings, $\sigma\in\mathbb{N}$. Detailed examples of these multi-graded Galilean algebras are provided, including extensions of the Galilean Virasoro and affine Kac-Moody algebras. We also derive the associated Sugawara constructions and discuss how these examples relate to multivariable extensions of Takiff algebras. We likewise apply our generalised contraction prescription to tensor products of $W_3$ algebras and obtain new families of higher-order Galilean $W_3$ algebras., Comment: 17 pages

The dimer model on a strip is considered as a Yang-Baxter \mbox{integrable} six vertex model at the free-fermion point with crossing parameter $\lambda=\tfrac{\pi}{2}$ and quantum group invariant boundary conditions. A one-to-many mapping of vertex onto dimer configurations allows for the solution of the free-fermion model to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by $45\degree$ compared to their usual orientation. In a suitable gauge, the dimer model is described by the Temperley-Lieb algebra with loop fugacity $\beta=2\cos\lambda=0$. It follows that the model is exactly solvable in geometries of arbitrary finite size. We establish and solve transfer matrix inversion identities on the strip with arbitrary finite width $N$. In the continuum scaling limit, in sectors with magnetization $S_z$, we obtain the conformal weights $\Delta_{s}=\big((2-s)^2-1\big)/8$ where $s=|S_z|+1=1,2,3,\ldots$. We further show that the corresponding finitized characters $\chit_s^{(N)}(q)$ decompose into sums of $q$-Narayana numbers or, equivalently, skew $q$-binomials. In the particle representation, the local face tile operators give a representation of the fermion algebra and the fermion particle trajectories play the role of nonlocal degrees of freedom. We argue that, in the continuum scaling limit, there exist nontrivial Jordan blocks of rank 2 in the Virasoro dilatation operator $L_0$. This confirms that, with quantum group invariant boundary conditions, the dimer model gives rise to a {\em logarithmic} conformal field theory with central charge $c=-2$, minimal conformal weight $\Delta_{\text{min}}=-\frac{1}{8}$ and effective central charge $c_{\text{eff}}=1$.Our analysis of the structure of the ensuing rank 2 modules indicates that the familiar staggered $c=-2$ modules appear as submodules., Comment: 35 pages, 10 Figures. arXiv admin note: text overlap with arXiv:1612.09477

A Galilean contraction is a way to construct Galilean conformal algebras from a pair of infinite-dimensional conformal algebras, or equivalently, a method for contracting tensor products of vertex algebras. Here, we present a generalisation of the Galilean contraction prescription to allow for inputs of any finite number of conformal algebras, resulting in new classes of higher-order Galilean conformal algebras. We provide several detailed examples, including infinite hierarchies of higher-order Galilean Virasoro algebras, affine Kac-Moody algebras and the associated Sugawara constructions, and $W_{3}$ algebras., Comment: 15 pages

This work concerns the representation theory of the affine Lie algebra $A_1^{(1)}$ at fractional level and its links to the representation theory of the Virasoro algebra. We introduce affine Kac modules as certain finitely generated submodules of Wakimoto modules. We conjecture the existence of several classes of staggered $A_1^{(1)}$-modules and provide evidence in the form of detailed examples. We extend the applicability of the Goddard-Kent-Olive coset construction to include the affine Kac and staggered modules. We introduce an exact functor between the associated category of $A_1^{(1)}$-modules and the corresponding category of Virasoro modules. At the level of characters, its action generalises the Mukhi-Panda residue formula. We also obtain explicit expressions for all irreducible $A_1^{(1)}$-characters appearing in the decomposition of Verma modules, re-examine the construction of Malikov-Feigin-Fuchs vectors, and extend the Fuchs-Astashkevich theorem from the Virasoro algebra to $A_1^{(1)}$., Comment: 46 pages, v2: minor changes, references added

The family of $A^{(1)}_2$ models on the square lattice includes a dilute loop model, a $15$-vertex model and, at roots of unity, a family of RSOS models. The fused transfer matrices of the general loop and vertex models are shown to satisfy $s\ell(3)$-type fusion hierarchies. We use these to derive explicit $T$- and $Y$-systems of functional equations. At roots of unity, we further derive closure identities for the functional relations and show that the universal $Y$-system closes finitely. The $A^{(1)}_2$ RSOS models are shown to satisfy the same functional and closure identities but with finite truncation., Comment: 36 pages

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A layer sum is introduced as the sum of formal exponentials of the distinct weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that the character of every finite-dimensional irreducible $\mathfrak{g}$-module admits a decomposition in terms of layer sums, with only non-negative integer coefficients. Ensuing results include a new approach to the computation of Weyl characters and weight multiplicities, and a closed-form expression for the number of distinct weights in a finite-dimensional irreducible $\mathfrak{g}$-module. The latter is given by a polynomial in the Dynkin labels, of degree equal to the rank of $\mathfrak{g}$., Comment: 23 pages

Infinite-dimensional Galilean conformal algebras can be constructed by contracting pairs of symmetry algebras in conformal field theory, such as $W$-algebras. Known examples include contractions of pairs of the Virasoro algebra, its $N=1$ superconformal extension, or the $W_3$ algebra. Here, we introduce a contraction prescription of the corresponding operator-product algebras, or equivalently, a prescription for contracting tensor products of vertex algebras. With this, we work out the Galilean conformal algebras arising from contractions of $N=2$ and $N=4$ superconformal algebras as well as of the $W$-algebras $W(2,4)$, $W(2,6)$, $W_4$, and $W_5$. The latter results provide evidence for the existence of a whole new class of $W$-algebras which we call Galilean $W$-algebras. We also apply the contraction prescription to affine Lie algebras and find that the ensuing Galilean affine algebras admit a Sugawara construction. The corresponding central charge is level-independent and given by twice the dimension of the underlying finite-dimensional Lie algebra. Finally, applications of our results to the characterisation of structure constants in $W$-algebras are proposed., Comment: 45 pages, v2: minor changes, references added

The central charge of the dimer model on the square lattice is still being debated in the literature. In this paper, we provide evidence supporting the consistency of a $c=-2$ description. Using Lieb's transfer matrix and its description in terms of the Temperley-Lieb algebra $TL_n$ at $\beta = 0$, we provide a new solution of the dimer model in terms of the model of critical dense polymers on a tilted lattice and offer an understanding of the lattice integrability of the dimer model. The dimer transfer matrix is analysed in the scaling limit and the result for $L_0-\frac c{24}$ is expressed in terms of fermions. Higher Virasoro modes are likewise constructed as limits of elements of $TL_n$ and are found to yield a $c=-2$ realisation of the Virasoro algebra, familiar from fermionic $bc$ ghost systems. In this realisation, the dimer Fock spaces are shown to decompose, as Virasoro modules, into direct sums of Feigin-Fuchs modules, themselves exhibiting reducible yet indecomposable structures. In the scaling limit, the eigenvalues of the lattice integrals of motion are found to agree exactly with those of the $c=-2$ conformal integrals of motion. Consistent with the expression for $L_0-\frac c{24}$ obtained from the transfer matrix, we also construct higher Virasoro modes with $c=1$ and find that the dimer Fock space is completely reducible under their action. However, the transfer matrix is found not to be a generating function for the $c=1$ integrals of motion. Although this indicates that Lieb's transfer matrix description is incompatible with the $c=1$ interpretation, it does not rule out the existence of an alternative, $c=1$ compatible, transfer matrix description of the dimer model., Comment: 54 pages. v2: minor corrections

It is now well known that non-local observables in critical statistical lattice models, polymers and percolation for example, may be modelled in the continuum scaling limit by logarithmic conformal field theories. Fusion rules for such theories, sometimes referred to as logarithmic minimal models, have been intensively studied over the last ten years in order to explore the representation-theoretic structures relevant to non-local observables. Motivated by recent lattice conjectures, this work studies the fusion rules of the $N=1$ supersymmetric analogues of these logarithmic minimal models in the Neveu-Schwarz sector. Fusion rules involving Ramond representations will be addressed in a sequel., Comment: 37 pages, 6 figures, v2 added clarifying remarks

Virasoro Kac modules were initially introduced indirectly as representations whose characters arise in the continuum scaling limits of certain transfer matrices in logarithmic minimal models, described using Temperley-Lieb algebras. The lattice transfer operators include seams on the boundary that use Wenzl-Jones projectors. If the projectors are singular, the original prescription is to select a subspace of the Temperley-Lieb modules on which the action of the transfer operators is non-singular. However, this prescription does not, in general, yield representations of the Temperley-Lieb algebras and the Virasoro Kac modules have remained largely unidentified. Here, we introduce the appropriate algebraic framework for the lattice analysis as a quotient of the one-boundary Temperley-Lieb algebra. The corresponding standard modules are introduced and examined using invariant bilinear forms and their Gram determinants. The structures of the Virasoro Kac modules are inferred from these results and are found to be given by finitely generated submodules of Feigin-Fuchs modules. Additional evidence for this identification is obtained by comparing the formalism of lattice fusion with the fusion rules of the Virasoro Kac modules. These are obtained, at the character level, in complete generality by applying a Verlinde-like formula and, at the module level, in many explicit examples by applying the Nahm-Gaberdiel-Kausch fusion algorithm., Comment: 71 pages. v3: version published in Nucl. Phys. B

A new inner product is constructed on each standard module over the Temperley-Lieb algebra $\mathsf{TL}_n(\beta)$ for $\beta\in \mathbb R$ and $n \ge 2$. On these modules, the Hamiltonian $h = -\sum_i e_i$ is shown to be self-adjoint with respect to this inner product. This implies that its action on these modules is diagonalisable with real eigenvalues. A representation theoretic argument shows that the reality of spectra of the Hamiltonian extends to all other Temperley-Lieb representations. In particular, this result applies to the celebrated $U_q(sl_2)$-invariant XXZ Hamiltonian, for all $q+q^{-1}\in \mathbb R$., Comment: 30 pages

A new spin-chain representation of the Temperley-Lieb algebra $TL_n(\beta=0)$ is introduced and related to the dimer model. Unlike the usual XXZ spin-chain representations of dimension $2^n$, this dimer representation is of dimension $2^{n-1}$. A detailed analysis of its structure is presented and found to yield indecomposable zigzag modules., Comment: 23 pages. Comments and references added

For general Temperley-Lieb loop models, including the logarithmic minimal models ${\cal LM}(p,p')$ with $p,p'$ coprime integers, we construct an infinite family of Robin boundary conditions on the strip as linear combinations of Neumann and Dirichlet boundary conditions. These boundary conditions are Yang-Baxter integrable and allow loop segments to terminate on the boundary. Algebraically, the Robin boundary conditions are described by the one-boundary Temperley-Lieb algebra. Solvable critical dense polymers is the first member ${\cal LM}(1,2)$ of the family of logarithmic minimal models and has loop fugacity $\beta=0$ and central charge $c=-2$. Specializing to ${\cal LM}(1,2)$ with our Robin boundary conditions, we solve the model exactly on strips of arbitrary finite size $N$ and extract the finite-size conformal corrections using an Euler-Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double row transfer matrices. This inversion identity is established directly in the Temperley-Lieb algebra. We classify the eigenvalues of the double row transfer matrices using the physical combinatorics of the patterns of zeros in the complex spectral parameter plane and obtain finitized characters related to spaces of coinvariants of $\mathbb{Z}_4$ fermions. In the continuum scaling limit, the Robin boundary conditions are associated with irreducible Virasoro Verma modules with conformal weights $\Delta_{r,s-\frac{1}{2}}=\frac{1}{32}(L^2-4)$ where $L=2s-1-4r$, $r\in\mathbb{Z}$, $s\in\mathbb{N}$. These conformal weights populate a Kac table with half-integer Kac labels. Fusion of the corresponding modules with the generators of the Kac fusion algebra is examined and general fusion rules are proposed., Comment: 53 pages, v2: comments and references added

A Temperley-Lieb (TL) loop model is a Yang-Baxter integrable lattice model with nonlocal degrees of freedom. On a strip of width N, the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TL_N(beta) with loop fugacity beta=2cos(lambda). Similarly on a cylinder, the single-row transfer tangle T(u) is an element of the enlarged periodic TL algebra. The logarithmic minimal models LM(p,p') comprise a subfamily of the TL loop models for which the crossing parameter lambda=(p'-p)pi/p' is parameterised by coprime integers 02 takes the form of functional relations for D(u) and T(u) of polynomial degree p'. These derive from fusion hierarchies of commuting transfer tangles D^{m,n}(u) and T^{m,n}(u) where D(u)=D^{1,1}(u) and T(u)=T^{1,1}(u). The fused transfer tangles are constructed from (m,n)-fused face operators involving Wenzl-Jones projectors P_k on k=m or k=n nodes. Some projectors P_k are singular for k>p'-1, but we argue that D^{m,n}(u) and T^{m,n}(u) are well defined for all m,n. For generic lambda, we derive the fusion hierarchies and the associated T- and Y-systems. For the logarithmic theories, the closure of the fusion hierarchies at n=p' translates into functional relations of polynomial degree p' for D^{m,1}(u) and T^{m,1}(u). We also derive the closure of the Y-systems for the logarithmic theories. The T- and Y-systems are the key to exact integrability and we observe that the underlying structure of these functional equations relate to Dynkin diagrams of affine Lie algebras., Comment: 77 pages

The higher fusion level logarithmic minimal models LM(P,P';n) have recently been constructed as the diagonal GKO cosets (A_1^{(1)})_k oplus (A_1^{(1)})_n / (A_1^{(1)})_{k+n} where n>0 is an integer fusion level and k=nP/(P'-P)-2 is a fractional level. For n=1, these are the logarithmic minimal models LM(P,P'). For n>1, we argue that these critical theories are realized on the lattice by n x n fusion of the n=1 models. For n=2, we call them logarithmic superconformal minimal models LSM(p,p') where P=|2p-p'|, P'=p' and p,p' are coprime, and they share the central charges of the rational superconformal minimal models SM(P,P'). Their mathematical description entails the fused planar Temperley-Lieb algebra which is a spin-1 BMW tangle algebra with loop fugacity beta_2=x^2+1+x^{-2} and twist omega=x^4 where x=e^{i(p'-p)pi/p'}. Examples are superconformal dense polymers LSM(2,3) with c=-5/2, beta_2=0 and superconformal percolation LSM(3,4) with c=0, beta_2=1. We calculate the free energies analytically. By numerically studying finite-size spectra on the strip with appropriate boundary conditions in Neveu-Schwarz and Ramond sectors, we argue that, in the continuum scaling limit, these lattice models are associated with the logarithmic superconformal models LM(P,P';2). For system size N, we propose finitized Kac character formulas whose P,P' dependence only enters in the fractional power of q in a prefactor. These characters involve Motzkin and Riordan polynomials defined in terms of q-trinomial coefficients. Using the Hamiltonian limit, we argue that there exist reducible yet indecomposable representations for which the Virasoro dilatation operator L_0 exhibits rank-2 Jordan blocks confirming that these theories are indeed logarithmic. We relate these results to the N=1 superconformal representation theory., Comment: 55 pages, v2: comments and references added

Working in the Virasoro picture, it is argued that the logarithmic minimal models LM(p,p')=LM(p,p';1) can be extended to an infinite hierarchy of logarithmic conformal field theories LM(p,p';n) at higher fusion levels n=1,2,3,.... From the lattice, these theories are constructed by fusing together n x n elementary faces of the appropriate LM(p,p') models. It is further argued that all of these logarithmic theories are realized as diagonal cosets (A_1^{(1)})_k \oplus (A_1^{(1)})_n / (A_1^{(1)})_{k+n} where n is the integer fusion level and k=np/(p'-p)-2 is a fractional level. These cosets mirror the cosets of the higher fusion level minimal models of the form M(m,m';n), but are associated with certain reducible representations. We present explicit branching rules for characters in the form of multiplication formulas arising in the logarithmic limit of the usual Goddard-Kent-Olive coset construction of the non-unitary minimal models M(m,m';n). The limiting branching functions play the role of Kac characters for the LM(p,p';n) theories., Comment: 16 pages, v2: footnote added

A lattice model of critical dense polymers is solved exactly for arbitrary system size on the torus. More generally, an infinite family of lattice loop models is studied on the torus and related to the corresponding Fortuin-Kasteleyn random cluster models. Starting with a cylinder, the commuting periodic single-row transfer matrices are built from the periodic Temperley-Lieb algebra extended by the shift operators Omega and Omega^{-1}. In this enlarged algebra, the non-contractible loop fugacity is alpha and the contractible loop fugacity is beta. The torus is formed by gluing the top and bottom of the cylinder. This gives rise to a variety of non-contractible loops winding around the torus. Because of their nonlocal nature, the standard matrix trace does not produce the proper geometric torus. Instead, we introduce a modified matrix trace for this purpose. This is achieved by using a representation of the enlarged periodic Temperley-Lieb algebra with a parameter v that keeps track of the winding of defects on the cylinder. The transfer matrix representatives and their eigenvalues thus depend on v. The modified trace is constructed as a linear functional on planar connectivity diagrams in terms of matrix traces Tr_d (with a fixed number of defects d) and Chebyshev polynomials of the first kind. For critical dense polymers, where beta=0, the transfer matrix eigenvalues are obtained by solving a functional equation in the form of an inversion identity. The solution depends on d and is subject to selection rules which we prove. Simplifications occur if all non-contractible loop fugacities are set to alpha=2 in which case the traces are evaluated at v=1. In the continuum scaling limit, the corresponding conformal torus partition function obtained from finite-size corrections agrees with the known modular invariant partition function of symplectic fermions., Comment: 51 pages, v2: added comments and references

Solvable critical dense polymers is a Yang-Baxter integrable model of polymers on the square lattice. It is the first member LM(1,2) of the family of logarithmic minimal models LM(p,p'). The associated logarithmic conformal field theory admits an infinite family of Kac representations labelled by the Kac labels r,s=1,2,.... In this paper, we explicitly construct the conjugate boundary conditions on the strip. The boundary operators are labelled by the Kac fusion labels (r,s)=(r,1) x (1,s) and involve a boundary field xi. Tuning the field xi appropriately, we solve exactly for the transfer matrix eigenvalues on arbitrary finite-width strips and obtain the conformal spectra using the Euler-Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double-row transfer matrices. The transfer matrix eigenvalues are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity which takes the form of a decomposition into irreducible blocks corresponding combinatorially to finitized characters given by generalized q-Catalan polynomials. This decomposition is in accord with the decomposition of the Kac characters into irreducible characters. In the scaling limit, we confirm the central charge c=-2 and the Kac formula for the conformal weights Delta_{r,s}=((2r-s)^2-1)/8 for r,s=1,2,3,... in the infinitely extended Kac table., Comment: 40 pages. arXiv admin note: substantial text overlap with arXiv:hep-th/0610273. v2: Added comments and references

We consider boundary conditions compatible with discrete holomorphicity for the dilute O(n) and C_2^(1) loop models. In each model, for a general set of boundary plaquettes, multiple types of loops can appear. A generalisation of Smirnov's parafermionic observable is therefore required in order to maintain the discrete holomorphicity property in the bulk. We show that there exist natural boundary conditions for this observable which are consistent with integrability, that is to say that, by imposing certain boundary conditions, we obtain a set of linear equations whose solutions also satisfy the corresponding reflection equation. In both loop models, several new sets of integrable weights are found using this approach., Comment: 23 pages, 27 figures; Minor changes to text, figures improved

Working with Lieb's transfer matrix for the dimer model, we point out that the full set of dimer configurations may be partitioned into disjoint subsets (sectors) closed under the action of the transfer matrix. These sectors are labelled by an integer or half-integer quantum number we call the variation index. In the continuum scaling limit, each sector gives rise to a representation of the Virasoro algebra. We determine the corresponding conformal partition functions and their finitizations, and observe an intriguing link to the Ramond and Neveu-Schwarz sectors of the critical dense polymer model as described by a conformal field theory with central charge c=-2., Comment: 44 pages

The fractal dimensions of the hull, the external perimeter and of the red bonds are measured through Monte Carlo simulations for dilute minimal models, and compared with predictions from conformal field theory and SLE methods. The dilute models used are those first introduced by Nienhuis. Their loop fugacity is beta = -2cos(pi/barkappa}) where the parameter barkappa is linked to their description through conformal loop ensembles. It is also linked to conformal field theories through their central charges c = 13 - 6(barkappa + barkappa^{-1}) and, for the minimal models of interest here, barkappa = p/p' where p and p' are two coprime integers. The geometric exponents of the hull and external perimeter are studied for the pairs (p,p') = (1,1), (2,3), (3,4), (4,5), (5,6), (5,7), and that of the red bonds for (p,p') = (3,4). Monte Carlo upgrades are proposed for these models as well as several techniques to improve their speeds. The measured fractal dimensions are obtained by extrapolation on the lattice size H,V -> infinity. The extrapolating curves have large slopes; despite these, the measured dimensions coincide with theoretical predictions up to three or four digits. In some cases, the theoretical values lie slightly outside the confidence intervals; explanations of these small discrepancies are proposed., Comment: 41 pages, 32 figures, added references

We construct new Yang-Baxter integrable boundary conditions in the lattice approach to the logarithmic minimal model WLM(1,p) giving rise to reducible yet indecomposable representations of rank 1 in the continuum scaling limit. We interpret these W-extended Kac representations as finitely-generated W-extended Feigin-Fuchs modules over the triplet W-algebra W(p). The W-extended fusion rules of these representations are inferred from the recently conjectured Virasoro fusion rules of the Kac representations in the underlying logarithmic minimal model LM(1,p). We also introduce the modules contragredient to the W-extended Kac modules and work out the correspondingly-extended fusion algebra. Our results are in accordance with the Kazhdan-Lusztig dual of tensor products of modules over the restricted quantum universal enveloping algebra $\bar{U}_q(sl_2)$ at $q=e^{\pi i/p}$. Finally, polynomial fusion rings isomorphic with the various fusion algebras are determined, and the corresponding Grothendieck ring of characters is identified., Comment: 28 pages

For each pair of positive integers r,s, there is a so-called Kac representation (r,s) associated with a Yang-Baxter integrable boundary condition in the lattice approach to the logarithmic minimal model LM(1,p). We propose a classification of these representations as finitely-generated submodules of Feigin-Fuchs modules, and present a conjecture for their fusion algebra which we call the Kac fusion algebra. The proposals are tested using a combination of the lattice approach and applications of the Nahm-Gaberdiel-Kausch algorithm. We also discuss how the fusion algebra may be extended by inclusion of the modules contragredient to the Kac representations, and determine polynomial fusion rings isomorphic to the conjectured Kac fusion algebra and its contragredient extension., Comment: 31 pages, v2: comments, subsection and references added

The logarithmic minimal models are not rational but, in the W-extended picture, they resemble rational conformal field theories. We argue that the W-projective representations are fundamental building blocks in both the boundary and bulk description of these theories. In the boundary theory, each W-projective representation arising from fundamental fusion is associated with a boundary condition. Multiplication in the associated Grothendieck ring leads to a Verlinde-like formula involving A-type twisted affine graphs A^{(2)}_{p} and their coset graphs A^{(2)}_{p,p'}=A^{(2)}_{p} x A^{(2)}_{p'}/Z_2. This provides compact formulas for the conformal partition functions with W-projective boundary conditions. On the torus, we propose modular invariant partition functions as sesquilinear forms in W-projective and rational minimal characters and observe that they are encoded by the same coset fusion graphs., Comment: 33 pages, v2: comments and references added, v3: minor corrections

It is well known that the Kerr-NUT-AdS-dS black hole admits two linearly independent Killing vectors and possesses a hidden symmetry generated by a rank-2 Killing tensor. The near-horizon geometry of an extremal Kerr-NUT-AdS-dS black hole admits four linearly independent Killing vectors, and we show how the hidden symmetry of the black hole itself is carried over by means of a modified Killing-Yano potential which is given explicitly. We demonstrate that the corresponding Killing tensor of the near-horizon geometry is reducible as it can be expressed in terms of the Casimir operators formed by the four Killing vectors., Comment: 7 pages

We consider Kerr-Newman-AdS-dS black holes near extremality and work out the near-horizon geometry of these near-extremal black holes. We identify the exact U(1)_L x U(1)_R isometries of the near-horizon geometry and provide boundary conditions enhancing them to a pair of commuting Virasoro algebras. The conserved charges of the corresponding asymptotic symmetries are found to be well defined and non-vanishing and to yield central charges c_L\neq0 and c_R=0. The Cardy formula subsequently reproduces the Bekenstein-Hawking entropy of the black hole. This suggests that the near-extremal Kerr-Newman-AdS-dS black hole is holographically dual to a non-chiral two-dimensional conformal field theory., Comment: 11 pages

We consider excitations around the recently introduced near-NHEK metric describing the near-horizon geometry of the near-extremal four-dimensional Kerr black hole. This geometry has a U(1)_L x U(1)_R isometry group which can be enhanced to a pair of commuting Virasoro algebras. We present boundary conditions for which the conserved charges of the corresponding asymptotic symmetries are well defined and non-vanishing and find the central charges c_L=12J/hbar and c_R=0 where J is the angular momentum of the black hole. Applying the Cardy formula reproduces the Bekenstein-Hawking entropy of the black hole. This suggests that the near-extremal Kerr black hole is holographically dual to a non-chiral two-dimensional conformal field theory., Comment: 11 pages, v2: references updated, added

We consider the W-extended logarithmic minimal model WLM(p,p'). As in the rational minimal models, the so-called fundamental fusion algebra of WLM(p,p') is described by a simple graph fusion algebra. The fusion matrices in the regular representation thereof are mutually commuting, but in general not diagonalizable. Nevertheless, we show that they can be brought simultaneously to block-diagonal forms whose blocks are upper-triangular matrices of dimension 1, 3, 5 or 9. The directed graphs associated with the two fundamental modules are described in detail. The corresponding adjacency matrices share a complete set of common generalized eigenvectors organized as a web constructed by interlacing the Jordan chains of the two matrices. This web is here called a Jordan web and it consists of connected subwebs with 1, 3, 5 or 9 generalized eigenvectors. The similarity matrix, formed by concatenating these vectors, simultaneously brings the two fundamental adjacency matrices to Jordan canonical form modulo permutation similarity. The ranks of the participating Jordan blocks are 1 or 3, and the corresponding eigenvalues are given by 2cos(j\pi/n) where j=0,...,n and n=p,p'. For p>1, only some of the modules in the fundamental fusion algebra of WLM(p,p') are associated with boundary conditions within our lattice approach. The regular representation of the corresponding fusion subalgebra has features similar to the ones in the regular representation of the fundamental fusion algebra, but with dimensions of the upper-triangular blocks and connected Jordan-web components given by 1, 2, 3 or 8. Some of the key results are illustrated for W-extended critical percolation WLM(2,3)., Comment: 48 pages, v2: minor changes

A lattice model of critical dense polymers is solved exactly on a cylinder with finite circumference. The model is the first member LM(1,2) of the Yang-Baxter integrable series of logarithmic minimal models. The cylinder topology allows for non-contractible loops with fugacity alpha that wind around the cylinder or for an arbitrary number ell of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley-Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even N, this involves a non-diagonalizable braid operator J and an involution R=-(J^3-12J)/16=(-1)^{F} with eigenvalues R=(-1)^{ell/2}. The number of defects ell separates the theory into sectors. For the case of loop fugacity alpha=2, the inversion identity is solved exactly for the eigenvalues in finite geometry. The eigenvalues are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane yielding selection rules. The finite-size corrections are obtained from Euler-Maclaurin formulas. In the scaling limit, we obtain the conformal partition functions and confirm the central charge c=-2 and conformal weights Delta_t=(t^2-1)/8. Here t=ell/2 and t=2r-s in the ell even sectors with Kac labels r=1,2,3,...; s=1,2 while t is half-integer in the ell odd sectors. Strikingly, the ell/2 odd sectors exhibit a W-extended symmetry but the ell/2 even sectors do not. Moreover, the naive trace summing over all ell even sectors does not yield a modular invariant., Comment: 44 pages, v3: minor corrections

The near-horizon geometry of the extremal four-dimensional Kerr black hole and certain generalizations thereof has an SL(2,R) x U(1) isometry group. Excitations around this geometry can be controlled by imposing appropriate boundary conditions. For certain boundary conditions, the U(1) isometry is enhanced to a Virasoro algebra. Here, we propose a free-field construction of this Virasoro algebra., Comment: 10 pages, v2: comments and references added

The infinite series of logarithmic minimal models LM(1,p) is considered in the W-extended picture where they are denoted by WLM(1,p). As in the rational models, the fusion algebra of WLM(1,p) is described by a simple graph fusion algebra. The corresponding fusion matrices are mutually commuting, but in general not diagonalizable. Nevertheless, they can be simultaneously brought to Jordan form by a similarity transformation. The spectral decomposition of the fusion matrices is completed by a set of refined similarity matrices converting the fusion matrices into Jordan canonical form consisting of Jordan blocks of rank 1, 2 or 3. The various similarity transformations and Jordan forms are determined from the modular data. This gives rise to a generalized Verlinde formula for the fusion matrices. Its relation to the partition functions in the model is discussed in a general framework. By application of a particular structure matrix and its Moore-Penrose inverse, this Verlinde formula reduces to the generalized Verlinde formula for the associated Grothendieck ring., Comment: 28 pages, v2: section, comments and references added

The near-horizon geometries of the extremal Kerr black hole and certain generalizations thereof are considered. Their isometry groups are all given by SL(2,R) x U(1). The usual boundary conditions of the Kerr/CFT correspondence enhance the U(1) isometry to a Virasoro algebra. Various alternatives to these boundary conditions are explored. Partial classifications are provided of the boundary conditions enhancing the SL(2,R) isometries or separately the U(1) isometry to a Virasoro algebra. In the case of SL(2,R)-enhancing boundary conditions of a near-horizon geometry of the type considered, the conserved charges associated to the generators of the asymptotic Virasoro symmetry form a centreless Virasoro algebra., Comment: 16 pages, v3: comments and references added

We consider the Grothendieck ring of the fusion algebra of the W-extended logarithmic minimal model WLM(1,p). Informally, this is the fusion ring of W-irreducible characters so it is blind to the Jordan block structures associated with reducible yet indecomposable representations. As in the rational models, the Grothendieck ring is described by a simple graph fusion algebra. The 2p-dimensional matrices of the regular representation are mutually commuting but not diagonalizable. They are brought simultaneously to Jordan form by the modular data coming from the full (3p-1)-dimensional S-matrix which includes transformations of the p-1 pseudo-characters. The spectral decomposition yields a Verlinde-like formula that is manifestly independent of the modular parameter $\tau$ but is, in fact, equivalent to the Verlinde-like formula recently proposed by Gaberdiel and Runkel involving a $\tau$-dependent S-matrix., Comment: 13 pages, v2: example, comments and references added

Based on symmetry principles, we derive a fusion algebra generated from repeated fusions of the irreducible modules appearing in the W-extended logarithmic minimal model WLM(p,p'). In addition to the irreducible modules themselves, closure of the commutative and associative fusion algebra requires the participation of a variety of reducible yet indecomposable modules. We conjecture that this fusion algebra is the same as the one obtained by application of the Nahm-Gaberdiel-Kausch algorithm and find that it reproduces the known such results for WLM(1,p') and WLM(2,3). For p>1, this fusion algebra does not contain a unit. Requiring that the spectrum of modules is invariant under a natural notion of conjugation, however, introduces an additional (p-1)(p'-1) reducible yet indecomposable rank-1 modules, among which the identity is found, still yielding a well-defined fusion algebra. In this greater fusion algebra, the aforementioned symmetries are generated by fusions with the three irreducible modules of conformal weights Delta_{kp-1,1}, k=1,2,3. We also identify polynomial fusion rings associated with our fusion algebras., Comment: 25 pages, v3: new section, comments and a reference added

The countably infinite number of Virasoro representations of the logarithmic minimal model LM(p,p') can be reorganized into a finite number of W-representations with respect to the extended Virasoro algebra symmetry W. Using a lattice implementation of fusion, we recently determined the fusion algebra of these representations and found that it closes, albeit without an identity for p>1. Here, we provide a fusion-matrix realization of this fusion algebra and identify a fusion ring isomorphic to it. We also consider various extensions of it and quotients thereof, and introduce and analyze commutative diagrams with morphisms between the involved fusion algebras and the corresponding quotient polynomial fusion rings. One particular extension is reminiscent of the fundamental fusion algebra of LM(p,p') and offers a natural way of introducing the missing identity for p>1. Working out explicit fusion matrices is facilitated by a further enlargement based on a pair of mutual Moore-Penrose inverses intertwining between the W-fundamental and enlarged fusion algebras., Comment: 48 pages

In statistical mechanics, observables are usually related to local degrees of freedom such as the Q < 4 distinct states of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuum scaling limit, these models are described by rational conformal field theories, namely the minimal models M(p,p') for suitable p, p'. More generally, as in stochastic Loewner evolution (SLE_kappa), one can consider observables related to nonlocal degrees of freedom such as paths or boundaries of clusters. This leads to fractal dimensions or geometric exponents related to values of conformal dimensions not found among the finite sets of values allowed by the rational minimal models. Working in the context of a loop gas with loop fugacity beta = -2 cos(4 pi/kappa), we use Monte Carlo simulations to measure the fractal dimensions of various geometric objects such as paths and the generalizations of cluster mass, cluster hull, external perimeter and red bonds. Specializing to the case where the SLE parameter kappa = 4p'/p is rational with p < p', we argue that the geometric exponents are related to conformal dimensions found in the infinitely extended Kac tables of the logarithmic minimal models LM(p,p'). These theories describe lattice systems with nonlocal degrees of freedom. We present results for critical dense polymers LM(1,2), critical percolation LM(2,3), the logarithmic Ising model LM(3,4), the logarithmic tricritical Ising model LM(4,5) as well as LM(3,5). Our results are compared with rigourous results from SLE_kappa, with predictions from theoretical physics and with other numerical experiments. Throughout, we emphasize the relationships between SLE_kappa, geometric exponents and the conformal dimensions of the underlying CFTs., Comment: Added references

We consider the continuum scaling limit of the infinite series of Yang-Baxter integrable logarithmic minimal models LM(p,p') as `rational' logarithmic conformal field theories with extended W symmetry. The representation content is found to consist of 6pp'-2p-2p' W-indecomposable representations of which 2p+2p'-2 are of rank 1, 4pp'-2p-2p' are of rank 2, while the remaining 2(p-1)(p'-1) are of rank 3. We identify these representations with suitable limits of Yang-Baxter integrable boundary conditions on the lattice. The W-indecomposable rank-1 representations are all W-irreducible while we present a conjecture for the embedding patterns of the W-indecomposable rank-2 and -3 representations. The associated W-extended characters are all given explicitly and decompose as finite non-negative sums of W-irreducible characters. The latter correspond to W-irreducible subfactors and we find that there are 2pp'+(p-1)(p'-1)/2 of them. We present fermionic character expressions for some of the rank-2 and all of the rank-3 W-indecomposable representations. To distinguish between inequivalent W-indecomposable representations of identical characters, we introduce `refined' characters carrying information also about the Jordan-cell content of a representation. Using a lattice implementation of fusion on a strip, we study the fusion rules for the W-indecomposable representations and find that they generate a closed fusion algebra, albeit one without identity for p>1. We present the complete set of fusion rules and interpret the closure of this fusion algebra as confirmation of the proposed extended symmetry. Finally, 2pp' of the W-indecomposable representations are in fact W-projective representations and they generate a closed fusion subalgebra., Comment: 34 pages

Two-dimensional critical percolation is the member LM(2,3) of the infinite series of Yang-Baxter integrable logarithmic minimal models LM(p,p'). We consider the continuum scaling limit of this lattice model as a `rational' logarithmic conformal field theory with extended W=W_{2,3} symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture. We find that the representation content of the ensuing closed fusion algebra contains 26 W-indecomposable representations with 8 rank-1 representations, 14 rank-2 representations and 4 rank-3 representations. We identify these representations with suitable limits of Yang-Baxter integrable boundary conditions on the lattice and obtain their associated W-extended characters. The latter decompose as finite non-negative sums of W-irreducible characters of which 13 are required. Implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit Cayley table. The closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry., Comment: 30 pages

We consider the logarithmic minimal models LM(1,p) as `rational' logarithmic conformal field theories with extended W symmetry. To make contact with the extended picture starting from the lattice, we identify 4p-2 boundary conditions as specific limits of integrable boundary conditions of the underlying Yang-Baxter integrable lattice models. Specifically, we identify 2p integrable boundary conditions to match the 2p known irreducible W-representations. These 2p extended representations naturally decompose into infinite sums of the irreducible Virasoro representations (r,s). A further 2p-2 reducible yet indecomposable W-representations of rank 2 are generated by fusion and these decompose as infinite sums of indecomposable rank-2 Virasoro representations. The fusion rules in the extended picture are deduced from the known fusion rules for the Virasoro representations of LM(1,p) and are found to be in agreement with previous works. The closure of the fusion algebra on a finite number of representations in the extended picture is remarkable confirmation of the consistency of the lattice approach., Comment: 15 pages

We identify quotient polynomial rings isomorphic to the recently found fundamental fusion algebras of logarithmic minimal models., Comment: 18 pages