Conformal Field Theory

This appendix demonstrates an explicit mapping from the E₈ root system to the SU 3) gauge subgroup-illustrating how your E₈-based substrate naturally contains Standard Model symmetries. We provide step-by-step algebraic decompositions and explanatory notes. E₈ has rank 8 and 240 roots in an 8-dimensional Euclidean space. Standard construction: These 240 vectors satisfy. Explanatory Note: The root system defines the geometric structure underlying the E₈ Lie algebra. Each root corresponds to a generator in the algebra. The E₈ Dynkin diagram contains an A₂ subdiagram, corresponding to SU 3. One standard embedding removes six nodes to isolate two connected nodes: o-o-o-o-o-o-o | o Retain an A₂ chain: two adjacent nodes in the long branch. Explanatory Note: A₂ is the Dynkin type for SU 3. By selecting an A₂ subdiagram within E₈ʼs diagram, we identify the SU 3) subalgebra.

We consider the O(n) theory in the n → 0 limit. We show that the theory is described by logarithmic conformal field theory, and that the correlation functions have logarithmic singularities. The explicit forms of the two-, three-and four-point correlation functions of the scaling fields and the corresponding logarithmic partners are derived.

We consider the O(n) theory in the n → 0 limit. We show that the theory is described by logarithmic conformal field theory, and that the correlation functions have logarithmic singularities. The explicit forms of the two-, three-and four-point correlation functions of the scaling fields and the corresponding logarithmic partners are derived.

Using a description of defects in solids in terms of three-dimensional gravity, we study the propagation of electrons in the background of disclinations and screw dislocations. We study the situations where there are bound states that are effectively localized on the defect and hence can be described in terms of an effective 1 + 1 dimensional field theory for the low energy excitations. In the case of screw dislocations, we find that these excitations are chiral and can be described by an effective field theory of chiral fermions. Fermions of both chirality occur even for a given direction of the magnetic field. The "net" chirality of the system however is not always the same for a given direction of the magnetic field, but changes from one sign of the chirality through zero to the other sign as the Fermi momentum or the magnitude of the magnetic flux

This supplementary document is intended to accompany the submitted work Yang–Mills Millennium Prob-
lem: Potential Solution. Its role is to provide clarification, expansion, and targeted commentary on aspects
of the original paper that warrant additional detail in view of the standards expected for peer review.
The emphasis throughout is on transparency, completeness, and methodological rigor. Where the original
manuscript presents the core argument in a compact form, the supplement develops underlying proofs,
articulates operator-theoretic assumptions, and situates the construction against related approaches. By
making explicit the points most likely to arise in referee inquiry, it ensures that the presentation meets the
benchmarks of constructive quantum field theory as required by the Clay Millennium Problem criteria.
The supplement should therefore be read not as a revision, but as a companion intended to facilitate
scrutiny, reproducibility, and verification by the broader mathematical physics community.

A homological selection theorem for C-spaces, as well as a finite-dimensional homological selection theorem is established. We apply the finite-dimensional homological selection theorem to obtain fixed-point theorems for usco homologically U V n set-valued maps.

Solitons emerge as non-perturbative solutions of non-linear wave equations in classical and quantun theories. These are non-dispersive and localised packets of energy-remarkable properties for solutions of non-linear differential equations. In presence of such objects, the solutions of Dirac equation lead to the curious phenomenon of 'fractional fermion number', a number which under normal conditions takes strictly integral values. In this article, we describe this accidental discovery and its manifestation in polyacetylene chains, which has led to the development of organic conductors.

The generalized massive Thirring model (GMT) with Nf[=number of positive roots of su(n)] fermion species is bosonized in the context of the functional integral and operator formulations and shown to be equivalent to a generalized sine-Gordon model (GSG) with Nf interacting soliton species. The generalized Mandelstam-Halpern soliton operators are constructed and the fermion-boson mapping is established through a set of generalized bosonization rules in a quotient positive definite Hilbert space of states. Each fermion species is mapped to its corresponding soliton in the spirit of particle/soliton duality of Abelian bosonization. In the semi-classical limit one recovers the so-called SU(n) affine Toda model coupled to matter fields (ATM) from which the classical GSG and GMT models were recently derived in the literature. The intermediate ATM like effective action possesses some spinors resembling the higher grading fields of the ATM theory which have non-zero chirality. These fields ...

The generalized massive Thirring model (GMT) with three fermion species is bosonized in the context of the functional integral and operator formulations and shown to be equivalent to a generalized sine-Gordon model (GSG) with three interacting soliton species. The generalized Mandelstam soliton operators are constructed and the fermion-boson mapping is established through a set of generalized bosonization rules in a quotient positive definite Hilbert space of states. Each fermion species is mapped to its corresponding soliton in the spirit of particle/soliton duality of Abelian bosonization. In the semi-classical limit one recovers the so-called SU (3) affine Toda model coupled to matter fields (ATM) from which the classical GSG and GMT models were recently derived in the literature. The intermediate ATM like effective action possesses some spinors resembling the higher grading fields of the ATM theory which have nonzero chirality. These fields are shown to disappear from the physical spectrum, thus providing a bag model like confinement mechanism and leading to the appearance of the massive fermions (solitons). The ordinary MT/SG duality turns out to be related to each SU (2) sub-group. The higher rank Lie algebra extension is also discussed.

We consider the Lagrangian description of the soliton sector of the so-called affine ŝl(3) Toda model coupled to matter (Dirac) fields (ATM). The theory is treated as a constrained system in the contexts of the Faddeev-Jackiw, the symplectic, as well as the master Lagrangian approaches. We exhibit the master Lagrangian nature of the model from which generalizations of the sine-Gordon (GSG) or the massive Thirring (GMT) models are derivable. The GMT model describes N f = 3 [number of positive roots of su(3)] massive Dirac fermion species with current-current interactions amongst all the U (1) species currents; on the other hand, the GSG theory corresponds to N b = 2 [rank of the su(3) Lie algebra] independent Toda fields (bosons) with a potential given by the sum of three SG cosine terms. The dual description of the model is further emphasized by providing some on shell relationships between bilinears of the GMT spinors and the relevant expressions of the GSG fields. In this way, in the first part of the chapter, we exhibit the strong/weak coupling phases and the (generalized) soliton/particle correspondences of the model at the classical level. In the second part of the chapter we give a full Lie algebraic formulation of the duality at the level of the equations of motion written in matrix form. The effective off-critical ŝl(3) ATM action is written in terms of the Wess-Zumino-Novikov-Witten (WZNW) action plus some kinetic terms for the spinors and scalar-spinor interaction terms. Moreover, this theory still presents a remarkable equivalence between the Noether and topological currents, describes the soliton sector of the original model and turns out to be the master Lagrangian describing the GMT and GSG models.

We propose that there exist generalized Seiberg-Witten equations in the Liouville conformal field theory, which allow the computation of correlation functions from the resolution of certain Ward identities. These identities involve a multivalued spin one chiral field, which is built from the energy-momentum tensor. We solve the Ward identities perturbatively in an expansion around the heavy asymptotic limit, and check that the first two terms of the Liouville three-point function agree with the known result of Dorn, Otto, Zamolodchikov and Zamolodchikov. We argue that such calculations can be interpreted in terms of the geometry of non-commutative spectral curves.

In this article which is the first of a series of three, we consider W(sl d )-symmetric conformal field theory in topological regimes for a generic value of the background charge, where W(sl d ) is the W-algebra associated to the affine Lie algebra sl d , whose vertex operator algebra is included to that of the affine Lie algebra g 1 at level 1. In such regimes, the theory admits a free field representation. We show that the generalized Ward identities assumed to be satisfied by chiral conformal blocks with current insertions can be solved perturbatively in topological regimes. This resolution uses a generalization of the topological recursion to non-commutative, or quantum, spectral curves. In turn, special geometry arguments yields a conjecture for the perturbative reconstruction of a particular chiral block. 1 Lightning review of conformal field theory 2 2 W-algebras and associated conformal field theories 10 2.1 From Virasoro to W-algebras .

2D quantum gravity is the idea that a set of discretized surfaces (called map, a graph on a surface), equipped with a graph measure, converges in the large size limit (large number of faces) to a conformal field theory (CFT), and in the simplest case to the simplest CFT known as pure gravity, also known as the gravity dressed (3,2) minimal model. Here we consider the set of planar Strebel graphs (planar trivalent metric graphs) with fixed perimeter faces, with the measure product of Lebesgue measure of all edge lengths, submitted to the perimeter constraints. We prove that expectation values of a large class of observables indeed converge towards the CFT amplitudes of the (3,2) minimal model. Contents 1 Introduction 2 2 Strebel graphs with uniform perimeters 5 2.1 General definitions -Strebel graphs, moduli space of Riemann surfaces

In this article which is the first of a series of three, we consider W(sl d )-symmetric conformal field theory in topological regimes for a generic value of the background charge, where W(sl d ) is the W-algebra associated to the affine Lie algebra sl d , whose vertex operator algebra is included to that of the affine Lie algebra g 1 at level 1. In such regimes, the theory admits a free field representation. We show that the generalized Ward identities assumed to be satisfied by chiral conformal blocks with current insertions can be solved perturbatively in topological regimes. This resolution uses a generalization of the topological recursion to non-commutative, or quantum, spectral curves. In turn, special geometry arguments yields a conjecture for the perturbative reconstruction of a particular chiral block. 1 Lightning review of conformal field theory 2 2 W-algebras and associated conformal field theories 10 2.1 From Virasoro to W-algebras .

In this article which is the first of a series of two, we consider $\mathcal W({\mathfrak{sl}_d})$-symmetric conformal field theory in topological regimes for a generic value of the background charge, where $\mathcal W({\mathfrak{sl}_d})$ is the W-algebra associated to the affine Lie algebra $\widehat{\mathfrak{sl}_d}$, whose vertex operator algebra is included to that of the affine Lie algebra $\widehat{\mathfrak{g}_1}$ at level 1. In such regimes, the theory admits a free field representation. We show that the generalized Ward identities assumed to be satisfied by chiral conformal blocks with current insertions can be solved perturbatively in topological regimes. This resolution uses a generalization of the topological recursion to non-commutative, or quantum, spectral curves. In turn, special geometry arguments yields a conjecture for the perturbative reconstruction of a particular chiral block.

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We propose that there exist generalized Seiberg-Witten equations in the Liouville conformal field theory, which allow the computation of correlation functions from the resolution of certain Ward identities. These identities involve a multivalued spin one chiral field, which is built from the energy-momentum tensor. We solve the Ward identities perturbatively in an expansion around the heavy asymptotic limit, and check that the first two terms of the Liouville three-point function agree with the known result of Dorn, Otto, Zamolodchikov, and Zamolodchikov. We argue that such calculations can be interpreted in terms of the geometry of non-commutative spectral curves.

The Wavy Universe Consciousness Cycle (WUCC) and Binary Paradox Theory propose a unified conceptual framework for understanding the structure and dynamics of the cosmos through the dual nature of consciousness and the binary foundations of universal reality. The WUCC describes existence as an oscillatory cycle in which consciousness, matter, and antimatter evolve within a wave-like continuum, governed by phases of expansion, contraction, and transformation. The Binary Paradox introduces a fundamental code—expressed as “1 0 1 0”—where duality (matter/antimatter, existence/non-existence, observer/observed) is resolved through paradoxical overlap. This overlap accounts for asymmetries in matter–antimatter distribution, the persistence of a 5% matter dominance, and the cosmic microwave background as a residual echo of annihilation. Together, the theories suggest that reality is not linear but recursive, encoded through paradoxical binaries that sustain universal cycles. By linking metaphysics with cosmology, the WUCC and Binary Paradox Theory aim to offer a novel perspective on the origin, structure, and evolution of the universe while positioning consciousness as a central, active principle in cosmic dynamics.

For 2-2 scattering in quantum field theories, the usual fixed t dispersion relation exhibits only two-channel symmetry. This paper considers a crossing symmetric dispersion relation, reviving certain old ideas in the 1970s. Rather than the fixed t dispersion relation, this needs a dispersion relation in a different variable z, which is related to the Mandelstam invariants s,t,u via a parametric cubic relation making the crossing symmetry in the complex z plane a geometric rotation. The resulting dispersion is manifestly three-channel crossing symmetric. We give simple derivations of certain known positivity conditions for effective field theories, including the null constraints, which lead to two sided bounds and derive a general set of new non-perturbative inequalities. We show how these inequalities enable us to locate the first massive string state from a low energy expansion of the four dilaton amplitude in type II string theory. We also show how a generalized (numerical) Froiss...

We investigate constraints imposed by entanglement on gravity in the context of holography. First, by demanding that relative entropy is positive and using the Ryu-Takayanagi entropy functional, we find certain constraints at a nonlinear level for the dual gravity. Second, by considering Gauss-Bonnet gravity, we show that for a class of small perturbations around the vacuum state, the positivity of the two point function of the field theory stress tensor guarantees the positivity of the relative entropy. Further, if we impose that the entangling surface closes off smoothly in the bulk interior, we find restrictions on the coupling constant in Gauss-Bonnet gravity. We also give an example of an anisotropic excited state in an unstable phase with broken conformal invariance which leads to a negative relative entropy.

Quasi-topological gravity is a new gravitational theory including curvature-cubed interactions and for which exact black hole solutions were constructed. In a holographic framework, classical quasi-topological gravity can be thought to be dual to the large N c limit of some non-supersymmetric but conformal gauge theory. We establish various elements of the AdS/CFT dictionary for this duality. This allows us to infer physical constraints on the couplings in the gravitational theory. Further we use holography to investigate hydrodynamic aspects of the dual gauge theory. In particular, we find that the minimum value of the shearviscosity-to-entropy-density ratio for this model is η/s ≃ 0.4140/(4π).

We consider a SU (N )×SU (M ) generalization of the multichannel single-impurity Kondo model which we solve analytically in the limit N → ∞, M → ∞, with γ = M/N fixed. Non-Fermi liquid behavior of the single electron Green function and of the local spin and flavor susceptibilities occurs in both regimes, N ≤ M and N > M , with leading critical exponents identical to those found in the conformal field theory solution for all N and M (with M ≥ 2). We explain this remarkable agreement and connect it to "spin-flavor separation", the essential feature of the non-Fermi-liquid fixed point of the multichannel Kondo problem.

We extend the observer-relative formulation of black hole complementarity by integrating TomitaTakesaki modular theory into the geometric structure of causal diamonds. By treating modular time as a connection over diamond-space, we show that trajectory-dependent holonomies encode a form of "memory" of the observers path, with curvature given by the commutator of modular generators. This provides a rigorous algebraic framework for the relational encoding of information accessibility across horizons, while maintaining compatibility with algebraic quantum field theory (AQFT) and conformal symmetry. Initial computations in a 1 + 1 CFT setup reveal how bilocal entanglement kernels dynamically deform under evolving observer horizons, suggesting new avenues for exploring quantum information flow in curved spacetimes.

We study logarithmic operators in Coulomb gas models, and show that they occur when the "puncture" operator of the Liouville theory is included in the model. We also consider WZNW models for SL(2, R), and for SU (2) at level 0, in which we find logarithmic operators which form Jordan blocks for the Kac-Moody as well as the Virasoro algebra.

For M := Riem(M ) the space of Riemannian metrics over a compact 3-manifold without boundary M , we study topological properties of the dense open subspace M ′ of metrics which possess no Killing vectors. Given the stratification of M, we work under the condition that, in a sense defined in the text, the connected components of each stratum do not accumulate. Given this condition we find that one of the most fundamental results regarding the topology of M, namely that it has trivial homotopy groups, would still be true for M ′ . This would make the topology of M ′ completely understood. Coupled with the fact that for M ′ , Diff(M ) ֒→ M ′ π → M ′ /Diff(M ) is a principal fiber bundle, which makes M ′ /Diff(M ) a proper manifold (as opposed to M/Diff(M )), we would have that π n (M ′ /D(M )) = π n-1 (D(M )), which reflects the topology of M . These results would render the space of metrics with no symmetries subject to the above condition, M ′ , as an ideal setting for geometrodynamical analysis.

In the second half of 2007 a major upgrade has been implemented on the Frascati DA NE collider in order to test the novel idea of Crab-Waist collisions. New vacuum chambers and permanent quadrupole magnets have been designed, built and installed to realize the new configuration. At the same time the performances of relevant hardware components, such as fast injection kickers and shielded bellows have been improved relying on new design concepts. The collider has been successfully commissioned in this new configuration. The paper describes several experimental results about linear and non-linear optics setup and optimization, damping of beam-beam instabilities and discusses the obtained luminosity performances.

After a long preparatory phase, including a wide hardware consolidation program, the Italian lepton collider DAFNE, is now systematically delivering data to the KLOE-2 experiment. In approximately 200 days of operation 1 fb-1 has been given to the detector limiting the background to a level compatible with an efficient data acquisition. Instantaneous and maximum daily integrated luminosity measured, so far, are considerably higher with respect to the previous KLOE runs, and are: L(inst) ~ 2.0 1032 cm-2s-1, and L(day) ~ 12.5 pb-1 respectively. A general review concerning refurbishing activities, machine optimization efforts and data taking performances is presented and discussed.

We propose and develop a framework in which the electromagnetic properties of the vacuum are not fundamental constants, but emergent functions of quantum entanglement entropy. Using the renormalized smallball limit of entanglement entropy as a bona fide local scalar, we introduce a constitutive function χ(Sent) multiplying the Maxwell kinetic term. Replica trick and modular Hamiltonian arguments fix the susceptibility of Sent to coupling variations, while local renormalization group analysis and QED anomaly coefficients determine the flow equation for χ. To leading order, this yields an exponential law χ(S) = χ0 exp[-κ(S-S0)] with slope κ proportional to the QED beta function. Variations in the finestructure constant then follow the linear relation ∆α/α ≃ κ ∆Sent. We derive the modified field equations, stress-energy tensor, and boundary terms explicitly, showing that gauge invariance, current conservation, and Weyl invariance are preserved. Worked examples in thermal states and near-horizon Rindler regimes demonstrate how entanglement gradients induce controlled variations in α. Phenomenological translation yields direct bounds on ∆Sent from atomic clocks, cavity QED, quasar absorption spectra, and Oklo data, while allowing substantial local gradients in extreme environments consistent with global homogeneity. This construction embeds electromagnetism into the broader conformal-entropic paradigm previously applied to gravity, suggesting a unifying principle: that both geometry and gauge response are determined by quantum entanglement, with anomaly coefficients providing the bridge between microphysics and effective constants of nature.

The reconciliation of quantum mechanics with general relativity remains one of the most profound challenges in modern theoretical physics. Among the various proposals, the holographic principle has emerged as a revolutionary paradigm suggesting that the fundamental description of spacetime and gravity is encoded on lower-dimensional boundaries. This review provides a comprehensive and detailed discussion of the holographic principle, its foundations in black hole thermodynamics, and its realization in the AdS/CFT correspondence. We further analyze its implications for quantum gravity, black hole information paradox, and the role of quantum entanglement. The paper also highlights open problems and future directions in the study of holography as a guiding framework toward a consistent quantum theory of gravity.

UMCP v1.0.1 consolidates the v1.0 kernel and tightens auditability without altering core results. The release fixes the generator convention to column-vector evolution with column-sum-zero (mass conservation), distinguishes the canonical normal integrity face (p=3) from the exact finite-wall face with ε>0, formalizes return-delay as a hazard/in-law rate, codifies normalization and the contract test under unit changes, certifies boundary behavior via either pre-clip differencing (preferred) or post-clip differencing with a wall-guard, normalizes multi-field integrity aggregation by a geometric mean, states κ-transport as a field identity that requires declared closures for prediction, and extends the audit manifest for exact reruns. All weld guarantees and qualitative conclusions from v1.0 remain intact.

We study coupled geometric flows involving the metric, dilaton, and flux fields arising from worldsheet β-functions in string theory. Extending the Ricci flow formalism, we derive parabolic evolution equations governing these fields and prove short-time existence and uniqueness for SU(3)-structure compactifications. We establish monotonicity properties of flow functionals analogous to Perelman's entropy and identify conditions for moduli stabilization in type II backgrounds. Our results unify Ricci-type flow techniques with flux compactifications and suggest new mathematical tools for analyzing dynamical string backgrounds and quantum gravity.

This article establishes a parallel between Kosmos Theory and contemporary physics, in particular topological physics and the AdS/CFT holographic duality, proposing that life and consciousness be understood as topological phases of spacetime. These phases are sustained by Cosmological Consciousness (CC)the ontological foundation associated with the geometric volume of the cosmosand manifested in the nonorientable interface between mirror universes (matter and antimatter), modeled by the topology of the Klein bottle. The analysis takes as its starting point Wolfgang Pauli's famous phrase-"God made the volume; the surface was invented by the devil"reinterpreting it in cosmological, physical, and phenomenological terms. The volume (bulk) corresponds to the continuous domain of gravitational geometry, where Cosmological Consciousness resides as the stability of Being. The surface (boundary) corresponds to the holographic and informational domain, the paradoxical and creative locus where life, information, and embodied consciousness emerge. On this surface, the Unitary Phenomenal Self (UPS) manifests as a protected edge state, robust against local perturbations, analogous to Majorana modes in topological systems. The article integrates concepts from general relativity (gravity as geometric curvature), condensed matter physics (topological phases and edge states), black hole thermodynamics (horizons as informational surfaces), and AdS/CFT holography (bulkboundary). These physical references are articulated with contributions from Husserl and Merleau-Ponty's phenomenology (consciousness as experience of a boundary), Lupasco's logic of the included third (paradoxical interface of opposites), and the philosophy of Heidegger and Jung (ek-sistence and Unus Mundus). It concludes that consciousness should not be understood as a neurobiological epiphenomenon, but as a structural and topological dimension of the cosmos. Gravity curves the volume, life stabilizes the topological surface, and consciousness emerges as the holographic correlation between volume and surface. The UPS, as a topological Dasein, is the intersection where geometry, information, and sentience intertwine, making the cosmos capable of reflecting upon itself.

In the current study, our aim is to define new generalized extended beta and hypergeometric types of functions. Next, we methodically determine several integral representations, Mellin transforms, summation formulas, and recurrence relations. Moreover, we provide log-convexity, Turán type inequality for the generalized extended beta function and differentiation formulas, transformation formulas, differential and difference relations for the generalized extended hypergeometric type functions. Also, we additionally suggest a generating function. Further, we provide the generalized extended beta distribution by making use of the generalized extended beta function as an application to statistics and obtaining variance, coefficient of variation, moment generating function, characteristic function, cumulative distribution function, and cumulative distribution function’s complement.

We present a strengthened HLV (Helix-Light-Vortex) formulation that unifies spiral time with a discrete quasicrystalline embedding. New results: (i) an explicit, unitary-safe spiral correction of the free propagator; (ii) a one-loop renormalization-group (RG) flow modified by a stationary U 3 bandwidth mode; (iii) detector-centric double-slit signatures (visibility sinc law and harmonic sidebands). We also give a discrete G-lattice embedding that acts as a natural regulator while preserving falsifiable predictions. All limits reduce continuously to standard quantum theory.

The higher fusion level logarithmic minimal models LM(P, P ′ ; n) have recently been constructed as the diagonal GKO cosets (A 1 ) k+n where n ≥ 1 is an integer fusion level and k = nP P ′ -P -2 is a fractional level. For n = 1, these are the well-studied logarithmic minimal models LM(P, P ′ ) ≡ LM(P, P ′ ; 1). For n ≥ 2, we argue that these critical theories are realized on the lattice by n × n fusion of the n = 1 models. We study the critical fused lattice models LM(p, p ′ ) n×n within a lattice approach and focus our study on the n = 2 models. We call these logarithmic superconformal minimal models LSM(p, p ′ ) ≡ LM(P, P ′ ; 2) where P = |2p -p ′ |, P ′ = p ′ and p, p ′ are coprime. These models share the central charges c = c P,P ′ ;2 = 3 2 1 -2(P ′ -P ) 2 P P ′ of the rational superconformal minimal models SM(P, P ′ ). Lattice realizations of these theories are constructed by fusing 2 × 2 blocks of the elementary face operators of the n = 1 logarithmic minimal models LM(p, p ′ ). Algebraically, this entails the fused planar Temperley-Lieb algebra which is a spin-1 Birman-Murakami-Wenzl tangle algebra with loop fugacity β 2 = [x] 3 = x 2 + 1 + x -2 and twist ω = x 4 where x = e iλ and λ = (p ′ -p)π p ′ . The first two members of this n = 2 series are superconformal dense polymers LSM(2, 3) with c = -5 2 , β 2 = 0 and superconformal percolation LSM(3, 4) with c = 0, β 2 = 1. We calculate the bulk and boundary free energies analytically. By numerically studying finite-size conformal spectra on the strip with appropriate boundary conditions 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 of the form q -c P,P ′ ;2 24

Working in the dense loop representation, we use the planar Temperley-Lieb algebra to build integrable lattice models called logarithmic minimal models LM(p, p ′ ). Specifically, we construct Yang-Baxter integrable Temperley-Lieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the Temperley-Lieb algebra are inherently non-local and not (timereversal) symmetric. We argue that, in the continuum scaling limit, they yield logarithmic conformal field theories with central charges c = 1 -6(p-p ′ ) 2 pp ′ where p, p ′ = 1, 2, . . . are coprime. The first few members of the principal series LM(m, m + 1) are critical dense polymers (m = 1, c = -2), critical percolation (m = 2, c = 0) and logarithmic Ising model (m = 3, c = 1/2). For the principal series, we find an infinite family of integrable and conformal boundary conditions organized in an extended Kac table with conformal weights ∆ r,s = ((m+1)r-ms) 2 -1 4m(m+1) , r, s = 1, 2, . . .. The associated conformal partition functions are given in terms of Virasoro characters of highest-weight representations. Individually, these characters decompose into a finite number of characters of irreducible representations. We show with examples how indecomposable representations arise from fusion.

Functional equations, in the form of fusion hierarchies, are studied for the transfer matrices of the fused restricted A (1) n−1 lattice models of Jimbo, Miwa and Okado. Specifically, these equations are solved analytically for the finite-size scaling spectra, central charges and some conformal weights. The results are obtained in terms of Rogers dilogarithm and correspond to coset conformal field theories based on the affine Lie algebra A (1) n−1 hep-th/9502067 1

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 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.

The higher fusion level logarithmic minimal models LM(P, P ′ ; n) have recently been constructed as the diagonal GKO cosets (A 1 ) k+n where n ≥ 1 is an integer fusion level and k = nP P ′ -P -2 is a fractional level. For n = 1, these are the well-studied logarithmic minimal models LM(P, P ′ ) ≡ LM(P, P ′ ; 1). For n ≥ 2, we argue that these critical theories are realized on the lattice by n × n fusion of the n = 1 models. We study the critical fused lattice models LM(p, p ′ ) n×n within a lattice approach and focus our study on the n = 2 models. We call these logarithmic superconformal minimal models LSM(p, p ′ ) ≡ LM(P, P ′ ; 2) where P = |2p -p ′ |, P ′ = p ′ and p, p ′ are coprime. These models share the central charges c = c P,P ′ ;2 = 3 2 1 -2(P ′ -P ) 2 P P ′ of the rational superconformal minimal models SM(P, P ′ ). Lattice realizations of these theories are constructed by fusing 2 × 2 blocks of the elementary face operators of the n = 1 logarithmic minimal models LM(p, p ′ ). Algebraically, this entails the fused planar Temperley-Lieb algebra which is a spin-1 Birman-Murakami-Wenzl tangle algebra with loop fugacity β 2 = [x] 3 = x 2 + 1 + x -2 and twist ω = x 4 where x = e iλ and λ = (p ′ -p)π p ′ . The first two members of this n = 2 series are superconformal dense polymers LSM(2, 3) with c = -5 2 , β 2 = 0 and superconformal percolation LSM(3, 4) with c = 0, β 2 = 1. We calculate the bulk and boundary free energies analytically. By numerically studying finite-size conformal spectra on the strip with appropriate boundary conditions 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 of the form q -c P,P ′ ;2 24

Working in the dense loop representation, we use the planar Temperley-Lieb algebra to build integrable lattice models called logarithmic minimal models LM(p, p ′ ). Specifically, we construct Yang-Baxter integrable Temperley-Lieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the Temperley-Lieb algebra are inherently non-local and not (timereversal) symmetric. We argue that, in the continuum scaling limit, they yield logarithmic conformal field theories with central charges c = 1 -6(p-p ′ ) 2 pp ′ where p, p ′ = 1, 2, . . . are coprime. The first few members of the principal series LM(m, m + 1) are critical dense polymers (m = 1, c = -2), critical percolation (m = 2, c = 0) and logarithmic Ising model (m = 3, c = 1/2). For the principal series, we find an infinite family of integrable and conformal boundary conditions organized in an extended Kac table with conformal weights ∆ r,s = ((m+1)r-ms) 2 -1 4m(m+1) , r, s = 1, 2, . . .. The associated conformal partition functions are given in terms of Virasoro characters of highest-weight representations. Individually, these characters decompose into a finite number of characters of irreducible representations. We show with examples how indecomposable representations arise from fusion.

The correspondence between theories in anti-de Sitter space and conformal field theories in physical space-time leads to an analytic, semiclassical model for strongly-coupled QCD which has scale invariance at short distances and color confinement at large distances. Light-front holography is a remarkable feature of AdS/CFT: it allows hadronic amplitudes in the AdS fifth dimension to be mapped to frame-independent light-front wavefunctions of hadrons in physical space-time, thus providing a relativistic description of hadrons at the amplitude level. Some novel features of QCD are discussed, including the consequences of confinement for quark and gluon condensates and the behavior of the QCD coupling in the infrared. We suggest that the spatial support of QCD condensates is restricted to the interior of hadrons, since they arise due to the interactions of confined quarks and gluons. Chiral symmetry is thus broken in a limited domain of size 1/m π , in analogy to the limited physical extent of superconductor phases. A new method for computing the hadronization of quark and gluon jets at the amplitude level, an event amplitude generator, is outlined.

Ultrarelativistic heavy-ion collisions at the Relativistic Heavy-Ion Collider (RHIC) are thought to have produced a state of matter called the Quark-Gluon-Plasma (QGP). The QGP forms when nuclear matter governed by Quantum Chromodynamics (QCD) reaches a temperature and baryochemical potential necessary to achieve the transition of hadrons (bound states of quarks and gluons) to deconfined quarks and gluons. Such conditions have been achieved at RHIC, and the resulting QGP created exhibits properties of a near perfect fluid. In particular, strong evidence shows that the QGP exhibits a very small shear viscosity to entropy density ratio η/s, near the lower bound predicted for that quantity by Anti-deSitter space/Conformal Field Theory (AdS/CFT) methods of η/s = 4πk B , where is Planck's constant and k B is Boltzmann's constant. As the produced matter expands and cools, it evolves through a phase described by a hadron gas with rapidly increasing η/s. This thesis presents robust calculations of η/s for hadronic and partonic media as a function of temperature using the Green-Kubo formalism. An analysis is performed for the behavior of η/s to mimic situations of the hadronic media at RHIC evolving out of chemical equilibrium, and systematic uncertainties are assessed for our method. In addition, preliminary results are presented for the bulk viscosity to entropy density ratio ζ/s, whose behavior is not well-known in a relativistic heavy ion collisions. The diffusion coefficient for baryon number is investigated, and an algorithm is presented to improve upon the previous work of investigation of heavy quark diffusion in a iv thermal QGP. By combining the results of my investigations for η/s from our microscopic transport models with what is currently known from the experimental results on elliptic flow from RHIC, I find that the trajectory of η/s in a heavy ion collision has a rich structure, especially near the deconfinement transition temperature T c . I have helped quantify the viscous hadronic effects to enable investigators to constrain the value of η/s for the QGP created at RHIC. v To Soliman M. Demir, who shall always be remembered in our lives. vi

In this paper, we consider and extend some fixed point results in F-complete F-metric spaces by relaxing the symmetry of complete metric spaces. We generalize α,β-admissible mappings in the setting of F-metric spaces. The derived results are supplemented with suitable examples, and the obtained results are applied to find the existence of the solution to the integral equation. The analytical results are compared through numerical simulation. We pose certain open problems for extending and applying our results in the future.

Using the Bekenstein upper bound for the ratio of the entropy $S$ of any bounded system, with energy $E = Mc^2$ and effective size $R$, to its energy $E$ i.e. $S/E < 2\pi k R/\hbar c$, we combine it with the holographic principle (HP) bound ('t Hooft and Susskind) which is $S \le \pi k c^3R^2/\hbar G$. We find that, if both bounds are identical, such bounded system is a black hole (BH). For a system that is not a BH the two upper bounds are different. The entropy of the system must obey the lowest bound. If the bounds are proportional, the result is the proportionality between the mass M of the system and its effective size $R$. When the constant of proportionality is $2G/c^2$ the system in question is a BH, and the two bounds are identical. We analyze the case for a universe. Then the universe is a BH in the sense that its mass $M$ and its Hubble size $R \approx ct$, t the age of the universe, follow the Schwarzschild relation $2GM/c^2 = R$. Finally, for a BH, the Hawking an...