School of Mathematics and Statistics - Theses
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ItemSingular vectors for the WN algebras and the BRST cohomology for relaxed highest-weight Lk(sl(2)) modules( 2019)This thesis presents the computation of singular vectors of the W_n algebras and the BRST cohomology of modules of the simple vertex operator algebra L_k(sl2) associated to the affine Lie algebra of sl2 in the relaxed category We will first recall some general theory on vertex operator algebras. We will then introduce the module categories that are relevant for conformal field theory. They are the category O of highest-weight modules and the relaxed category which contains O as well as the relaxed highest-weight modules with spectral flow and non-split extensions. We will then introduce the W_n algebras and the simple vertex operator algebra L_k(sl2). Properties of the Heisenberg algebra, the bosonic and the fermionic ghosts will be discussed as they are required in the free field realisations of W_n and L_k(sl2) as well as the construction of the BRST complex. We will then compute explicitly the singular vectors of W_n algebras in their Fock representations. In particular, singular vectors can be realised as the image of screening operators of the W_n algebras. One can then realise screening operators in terms of Jack functions when acting on a highest-weight state, thereby obtaining explicit formulae of the singular vectors in terms of symmetric functions. We will then discuss the BRST construction and the BRST cohomology for modules in category O. Lastly we compute the BRST cohomology for L_k(sl2) modules in the relaxed category. In particular, we compute the BRST cohomology for the highest-weight modules with positive spectral flow for all degrees and the BRST cohomology for the highest-weight modules with negative spectral flow for one degree.
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ItemYang-Baxter integrable dimers and fused restricted-solid-on-solid lattice models( 2019)The main objects of investigation in this thesis are two Yang-Baxter integrable lattice models of statistical mechanics in two dimensions: nonunitary RSOS models and dimers. At criticality they admit continuum descriptions with nonunitary conformal field theories (CFTs) in (1+1) dimensions. CFTs are quantum field theory invariant under conformal transformations. They play a major role in the theory of phase transition and critical phenomena. In quantum field theory unitarity is the requirement that the probability is conserved, hence realistic physical problems are associated with unitary quantum field theories. Nevertheless, in statistical mechanics this property loses a physical meaning and statistical systems like polymers and percolations, which model physical problems with long-range interactions, in the continuum scaling limit give rise to nonunitary conformal field theories. Both the nonunitary RSOS models and dimers are defined on a two-dimensional square lattice. Restricted solid-on-solid (RSOS) models are so called because their degrees of freedom are in the form of a finite (therefore restricted) set of heights which live on the sites of the lattice and their interactions take place between the four sites around each face of the lattice (solid-on-solid). Each allowed configuration of heights maps to a specific Boltzmann weight. RSOS are integrable in the sense that their Boltzmann weights and transfer matrices satisfy the Yang-Baxter equation. The CFTs associated to critical RSOS models are minimal models, the simplest family of rational conformal field theories. The process of fusion on elementary RSOS models has a different outcome on the CFT side depending on both the level of fusion and the value of their crossing parameter λ. Precisely, in the interval 0 < λ < π/2, the 2x2 fused RSOS models correspond to higher-level conformal cosets with integer level of fusion equal to two. Instead in the complementary interval π/n < λ < π the 2x2 fused RSOS models are related to minimal models with integer level of fusion equal to one. To prove this conjecture one-dimensional sums, deriving from the well-known Yang-Baxter corner transfer matrix method, have been calculated, extended in the continuum limit and ultimately compared to the conformal characters of the related minimal models. The dimer model has been for a long time object of various scientific studies, firstly as simple prototype of diatomic molecules and then as equivalent formulation to the well-known domino tilings problem of statistical mechanics. However, only more recently it has attracted attention as conformal field theory thanks to its relation with another famous integrable lattice model, the six-vertex model. What is particularly interesting of dimers is the property of being a free-fermion model and at the same time showing non-local properties due to the long-range steric effects propagating from the boundaries. This non-locality translates then in the dependance of their bulk free energy on the boundary conditions. We formulate the dimer model as a Yang-Baxter integrable free-fermion six-vertex model. This model is integrable in different geometries (cylinder, torus and strip) and with a variety of different integrable boundary conditions. The exact solution for the partition function arises from the complete spectra of eigenvalues of the transfer matrix. This is obtained by solving some functional equations, in the form of inversion identities, usually associated to the transfer matrix of the free-fermion six-vertex model, and using the physical combinatorics of the pattern of zeros of the transfer matrix eigenvalues to classify the eigenvalues according to their degeneracies. In the case of the cylinder and torus, the transfer matrix can be diagonalized, while, in the other cases, we observe that in a certain representation the double row transfer matrix exhibits non trivial Jordan-cells. Remarkably, the spectrum of eigenvalues of dimers and critical dense polymers agree sectors by sectors. The similarity with critical dense polymers, which is a logarithmic field theory, raises the question whether also the free-fermion dimer model manifests a logarithmic behaviour in the continuum scaling limit. The debate is still open. However, in our papers we provide a final answer and argue that the type of conformal field theory which best describe dimers is a logarithmic field theory, as it results by looking at the numerically estimate of the finite size corrections to the critical free energy of the free-fermion six-vertex-equivalent dimer model. The thesis is organized as follows. The first chapter is an introduction which has the purpose to inform the reader about the basics of statistical mechanics, from one side, and CFTs, on the other side, with a specific focus on the two lattice models that have been studied (nonunitary RSOS and dimers) and the theories associated to the their continuum description at criticality (minimal models and logarithmic CFTs). The second chapter considers the family of non-unitary RSOS models with π/n < λ < π and brings forward the discussion around the one-dimensional sums of the elementary and fused models, and the associated conformal characters in the continuum scaling limit. The third and fourth chapters are dedicated to dimers, starting with periodic conditions on a cylinder and torus, and then more general integrable boundary conditions on a strip. In each case, a combinatorial analysis of the pattern of zeros of the transfer matrix eigenvalues is presented and extensively treated. It follows then the analysis of the finite-size corrections to the critical free energy. Finally, the central charges and minimal conformal dimensions of critical dimers are discussed in depth with concluding remarks about the logarithmic hypothesis. Next, there is a conclusion where the main results of these studies are summarized and put into perspective with possible future research goals.
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ItemCoset construction for the N=2 and osp(1|2) minimal models( 2019)The thesis presents the study of the N=2 and osp(1|2) minimal models at admissible levels using the method of coset constructions. These sophisticated minimal models are rich in mathematical structure and come with various interesting features for us to investigate. First, some general principles of conformal field theory are reviewed, notations used throughout the thesis are established. The ideas are then illustrated with three examples of bosonic conformal field theories, namely, the free boson, the Virasoro minimal models, and the admissible-level Wess-Zumino-Witten models of affine sl(2). The concept of supersymmetry is then introduced, and examples of fermionic conformal field theories are discussed. Of the two minimal models of interest, the N=2 minimal model, tensored with a free boson, can be extended into an sl(2) minimal model tensored with a pair of fermionic ghosts, whereas an osp(1|2) minimal model is an extension of the tensor product of certain Virasoro and sl(2) minimal models. We can therefore induce the known structures of the representations of the coset components and get a rather complete picture for the minimal models we want to investigate. In particular, the irreducible highest-weight modules (including the relaxed highest-weight modules, which result in a continuous spectrum) are classified, their characters and Grothendieck fusion rules are computed. The genuine fusion products and the projective covers of the irreducibles are conjectured. The thesis concludes with a vision of how this method can be used for the study of other affine superalgebras. This provides a promising approach to solving superconformal field theories that are currently little known in the literature.The thesis presents the study of the N=2 and osp(1|2) minimal models at admissible levels using the method of coset constructions. These sophisticated minimal models are rich in mathematical structure and come with various interesting features for us to investigate. First, some general principles of conformal field theory are reviewed, notations used throughout the thesis are established. The ideas are then illustrated with three examples of bosonic conformal field theories, namely, the free boson, the Virasoro minimal models, and the admissible-level Wess-Zumino-Witten models of affine sl(2). The concept of supersymmetry is then introduced, and examples of fermionic conformal field theories are discussed. Of the two minimal models of interest, the N=2 minimal model, tensored with a free boson, can be extended into an sl(2) minimal model tensored with a pair of fermionic ghosts, whereas an osp(1|2) minimal model is an extension of the tensor product of certain Virasoro and sl(2) minimal models. We can therefore induce the known structures of the representations of the coset components and get a rather complete picture for the minimal models we want to investigate. In particular, the irreducible highest-weight modules (including the relaxed highest-weight modules, which result in a continuous spectrum) are classified, their characters and Grothendieck fusion rules are computed. The genuine fusion products and the projective covers of the irreducibles are conjectured. The thesis concludes with a vision of how this method can be used for the study of other affine superalgebras. This provides a promising approach to solving superconformal field theories that are currently little known in the literature.
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ItemCritical dense polymers on the strip and cylinder( 2013)This thesis is concerned with the study of solvable critical dense polymers using Yang-Baxter techniques in two distinct topologies, namely the strip and the cylinder. Critical dense polymers is the first member ${\cal LM}(1,2)$ of the Yang-Baxter integrable series of logarithmic minimal models ${\cal LM}(p,p')$. The associated logarithmic conformal field theory admits an infinite family of Kac representations labelled by the Kac labels $r,s=1,2,\ldots$. We begin by explicitly constructing the conjugate boundary conditions on the strip. The boundary operators are labelled by the Kac fusion \mbox{labels} $(r,s)=(r,1)\otimes (1,s)$ 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 in the infinitely extended Kac table $\Delta_{r,s}=\frac{(2r-s)^2-1}{8}$ for $r,s=1,2,3,\ldots$. 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 $\vec J$. The number of defects $\ell$ thus separates the theory into Ramond ($\ell/2$ even), Neveu-Schwarz ($\ell/2$ odd) and $\mathbb{Z}_4$ ($\ell$ odd) sectors. For the case of loop fugacity $\alpha=2$, the inversion identity is solved exactly sector by sector 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. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler-Maclaurin formulas. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge $c=-2$ and conformal weights $\Delta,\bar\Delta=\Delta_t=(t^2-1)/8$. Here $t=\ell/2$ and $t=2r-s\in\mathbb{N}$ in the $\ell$ even sectors with Kac labels $r=1,2,3,\ldots; s=1,2$ while $t\in\mathbb{Z}-\half$ in the $\ell$ odd sectors. Strikingly, the $\ell/2$ odd sectors exhibit a ${\cal W}$-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.