Conformal Field Theory

We compute the high-temperature limit of the free energy for four-dimensional N = 4 supersymmetric SU (N c ) Yang-Mills theory. At weak coupling we do so for a general ultrastatic background spacetime, and in the presence of slowly-varying background gauge fields. Using Maldacena's conjectured duality, we calculate the strong-coupling large-N c expression for the special case that the three-space has constant curvature. We compare the two results paying particular attention to curvature corrections to the leading order expressions.

With the motivation of overcoming difficulties in studying systems of several one dimensional Tomonaga Luttinger liquid wires connected locally at a junction, we construct quadratic lattice field theories for the single non-chiral/chiral wire, the three wire fork and the chiral box junction. For a fork consisting of three identical non-chiral wires, we find that the permutation symmetries of the system, together with the requirement of charge and current continuity, determine the terms in the action governing the dynamics of the fields adjacent to the junction. These results are generalised to the case of N ≥ 3 identical (as well as different) TLL wires. A study of the chiral box junction circuit model reveals that junctions of several chiral wires can be formed by relating the TLL interaction parameters of their constituents.

This proceedings paper extends the scope of our conference talk, where we presented a comprehensive analysis of newly expanded and refined lattice data concerning the SU(3) gauge theory with 𝑁 𝑓 = 8 light Dirac fermions -a theory positioned near the conformal window boundary. The analysis presented here makes use of a dilaton effective field theory and we delve deeper into the intricacies of the dilaton potential. We aim to clarify the connection between parameters appearing the potential and properties of the underlying gauge theory.

Hamiltonian Truncation (HT) is a numerical approach for calculating observables in a Quantum Field Theory non-perturbatively. This approach can be applied to theories constructed by deforming a conformal field theory with a relevant operator of scaling dimension ∆. UV divergences arise when ∆ is larger than half of the spacetime dimension d. These divergences can be regulated by HT or by using a more conventional local regulator. In this work we show that extra UV divergences appear when using HT rather than a local regulator for ∆ d/2 + 1/4, revealing a striking breakdown of locality. Our claim is based on the analysis of conformal perturbation theory up to fourth order. As an example we compute the Casimir energy of d = 2 Minimal Models perturbed by operators whose dimensions take values on either side of the threshold d/2 + 1/4.

Hamiltonian Truncation (HT) is a numerical approach for calculating observables in a Quantum Field Theory non-perturbatively. This approach can be applied to theories constructed by deforming a conformal field theory with a relevant operator of scaling dimension ∆. UV divergences arise when ∆ is larger than half of the spacetime dimension d. These divergences can be regulated by HT or by using a more conventional local regulator. In this work we show that extra UV divergences appear when using HT rather than a local regulator for ∆ ≥ d/2 + 1/4, revealing a striking breakdown of locality. Our claim is based on the analysis of conformal perturbation theory up to fourth order. As an example we compute the Casimir energy of d = 2 Minimal Models perturbed by operators whose dimensions take values on either side of the threshold d/2 + 1/4.

Rota-Baxter systems of T. Brzeziński are a generalization of Rota-Baxter operators that are related to dendriform structures, associative Yang-Baxter pairs and covariant bialgebras. In this paper, we consider Rota-Baxter systems in the presence of bimodule, which we call generalized Rota-Baxter systems. We define a graded Lie algebra whose Maurer-Cartan elements are generalized Rota-Baxter systems. This allows us to define a cohomology theory for a generalized Rota-Baxter system. Formal one-parameter deformations of generalized Rota-Baxter systems are discussed from cohomological points of view. We further study Rota-Baxter systems, associative Yang-Baxter pairs, covariant bialgebras and introduce generalized averaging systems that are related to associative dialgebras. Next, we define generalized Rota-Baxter systems in the homotopy context and find relations with homotopy dendriform algebras. The paper ends by considering commuting Rota-Baxter systems and their relation with quadri-algebras. Contents 1. Introduction 1 2. Generalized Rota-Baxter systems 3 3. Maurer-Cartan characterization and cohomology 8 4. Deformations of generalized Rota-Baxter systems 13 5. Rota-Baxter systems, Yang-Baxter pairs, covariant bialgebras and averaging systems 15 6. Generalized Rota-Baxter systems on A ∞ -algebras 22 7. Commuting Rota-Baxter systems and quadri-algebras 24 References 27

Using a holographic proposal for the geometric entropy we study its behavior in the geometry of Schwarzschild black holes in global AdSp for p = 3, 4, 5. Holographically, the entropy is determined by a minimal surface. On the gravity side, due to the presence of a horizon on the background, generically there are two solutions to the surfaces determining the entanglement entropy. In the case of AdS3, the calculation reproduces precisely the geometric entropy of an interval of length l in a two-dimensional conformal field theory with periodic boundary conditions. We demonstrate that in the cases of AdS4 and AdS5 the sign of the difference of the geometric entropies changes, signaling a transition. Euclideanization implies that various embedding of the holographic surface are possible. We study some of them and find that the transitions are ubiquitous. In particular, our analysis renders a very intricate phase space, showing, for some ranges of the temperature, up to three branches. We...

Quantum entanglement between an impurity and its environment is expected to be central in quantum impurity problems. We develop a method to compute the entanglement in spin-1/2 impurity problems, based on the entanglement negativity and the boundary conformal field theory (BCFT). Using the method, we study the thermal decay of the entanglement in the multichannel Kondo effects. At zero temperature, the entanglement has the maximal value independent of the number of the screening channels. At low temperature, the entanglement exhibits a power-law thermal decay. The power-law exponent equals two times of the scaling dimension of the BCFT boundary operator describing the impurity spin, and it is attributed to the energy-dependent scaling behavior of the entanglement in energy eigenstates. These agree with numerical renormalization group results, unveiling quantum coherence inside the Kondo screening length.

We investigate consistent charged black hole solutions to the Einstein-Maxwell-Dilaton (EMD) equations that are asymptotically AdS. The solutions are gravity duals to phases of a non-conformal plasma at finite temperature and density. For the dilaton we take a quadratic ansatz leading to linear confinement at zero temperature and density. We consider a grand canonical ensemble, where the chemical potential is fixed, and find a rich phase diagram involving the competition of small and large black holes. The phase diagram contains a critical line and a critical point similar to the van der Waals-Maxwell liquid-gas transition. As the critical point is approached, we show that the trace anomaly in the plasma phases vanishes signifying the restoration of conformal symmetry in the fluid. We find that the heat capacity and charge susceptibility diverge as C V ∝ (T -T c ) -α and χ ∝ (T -T c ) -γ at the critical point with universal critical exponents α = γ = 2/3. Our results suggest a description of the thermodynamics near the critical point in terms of catastrophe theories. In the limit µ → 0 we compare our results with lattice results for SU (N c ) Yang-Mills theories.

This short paper seeks to introduce a new perspective in which to consider the Collatz Conjecture and is not meant to provide a comprehensive, rigorous, mathematically notated proof, only to illuminate a logical path those of a higher caliber of mathematical proficiency may follow and potentially expand upon, or formalize if appropriate. The author thanks the reader for their time.-where {-"e-op" is the function (x / 2)-"o-op" is the function (3x + 1) } while {-"e-op" may only be applied to even numbers.-"o-op" may only be applied to odd numbers.-Both may only be applied to whole, positive numbers. } (x-1)/3 iterated indefinitely always eventually produces a fraction and therefore a number not on the sequence, unless the starting number was a multiple of (3x+1), in which case it falls to zero and stays put. Therefore, there is no infinitely long chain of consecutive inverse o-ops in the tree of possible predecessors for any number, except the branch that is the iteration of (3x+1) to infinity, the forward operation itself. Thus, all remaining branches of predecessors must eventually become infinitely long steps of inverse e-ops as, by definition, an inverse o-op must result in an odd number and therefore cannot generate a repeat number. In this way there can never be a repeat outside of the forward (3x+1) chain, and as there will never be a repeat number on the chain itself, this leaves the observed 4-2-1 loop the only loop in existence by coincidence.

The twisted boundary conditions and associated partition functions of the conformal sl(2) A-D-E models are studied on the Klein bottle and the Möbius strip. The A-D-E minimal lattice models give realization to the complete classification of the open descendants of the sl(2) minimal theories. We construct the transfer matrices of these lattice models that are consistent with non-orientable geometries. In particular, we show that in order to realize all the Klein bottle amplitudes of different crosscap states, not only the topological flip on the lattice but also the involution in the spin configuration space must be taken into account. This involution is the Z 2 symmetry of the Dynkin diagrams which corresponds to the simple current of the Ocneanu algebra.

We consider the ϕ 1,3 off-critical perturbation M(m, m ′ ; t) of the general non-unitary minimal models where 2 ≤ m ≤ m ′ and m, m ′ are coprime and t measures the departure from criticality corresponding to the ϕ 1,3 integrable perturbation. We view these models as the continuum scaling limit in the ferromagnetic Regime III of the Forrester-Baxter Restricted Solid-On-Solid (RSOS) models on the square lattice. We also consider the RSOS models in the antiferromagnetic Regime II related in the continuum scaling limit to Z n parfermions with n = m ′ -2. Using an elliptic Yang-Baxter algebra of planar tiles encoding the allowed face configurations, we obtain the Hamiltonians of the associated quantum chains defined as the logarithmic derivative of the transfer matrices with periodic boundary conditions. The transfer matrices and Hamiltonians act on a vector space of paths on the A m ′ -1 Dynkin diagram whose dimension is counted by generalized Fibonacci numbers.

We derive the fermionic polynomial generalizations of the characters of the integrable perturbations φ 2,1 and φ 1,5 of the general minimal M (p, p ′ ) conformal field theory by use of the recently discovered trinomial analogue of Bailey's lemma. For φ 2,1 perturbations results are given for all models with 2p > p ′ and for φ 1,5 perturbations results for all models with p ′ 3 < p < p ′ 2 are obtained. For the φ 2,1 perturbation of the unitary case M (p, p + 1) we use the incidence matrix obtained from these character polynomials to conjecture a set of TBA equations. We also find that for φ 1,5 with 2 < p ′ /p < 5/2 and for φ 2,1 satisfying 3p < 2p ′ there are usually several different fermionic polynomials which lead to the identical bosonic polynomial. We interpret this to mean that in these cases the specification of the perturbing field is not sufficient to define the theory and that an independent statement of the choice of the proper vacuum must be made.

We consider sℓ(2) minimal conformal field theories on a cylinder from a lattice perspective. To each allowed one-dimensional configuration path of the A L Restricted Solidon-Solid (RSOS) models we associate a physical state and a monomial in a finite fermionic algebra. The orthonormal states produced by the action of these monomials on the primary states |h generate finite Virasoro modules with dimensions given by the finitized Virasoro characters χ (N ) h (q). These finitized characters are the generating functions for the double row transfer matrix spectra of the critical RSOS models. We argue that a general energy-preserving bijection exists between the one-dimensional configuration paths and the eigenstates of these transfer matrices and exhibit this bijection for the critical and tricritical Ising models in the vacuum sector. Our results extend to Z L-1 parafermion models by duality.

We derive the fusion hierarchy of functional equations for critical A-D-E lattice models related to the s (2) unitary minimal models, the parafermionic models and the supersymmetric models of conformal field theory and deduce the related TBA functional equations. The derivation uses fusion projectors and applies in the presence of all known integrable boundary conditions on the torus and cylinder. The resulting TBA functional equations are universal in the sense that they depend only on the Coxeter number of the A-D-E graph and are independent of the particular integrable boundary conditions. We conjecture generally that TBA functional equations are universal for all integrable lattice models associated with rational CFTs and their integrable perturbations.

We construct integrable lattice realizations of conformal twisted boundary conditions for s (2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical A-D-E lattice models with positive spectral parameter. The integrable seam boundary conditions are labelled by (r, s, ζ ) ∈ (A g-2 , A g-1 , Γ ) where Γ is the group of automorphisms of the graph G and g is the Coxeter number of G = A, D, E. Taking symmetries into account, these are identified with conformal twisted boundary conditions of Petkova and Zuber labelled by and associated with nodes of the minimal analog of the Ocneanu quantum graph. Our results are illustrated using the Ising (A 2 , A 3 ) and 3-state Potts (A 4 , D 4 ) models.

We consider the physical combinatorics of critical lattice models and their associated conformal field theories arising in the continuum scaling limit. As examples, we consider A-type unitary minimal models and the level-1 sℓ(2) Wess-Zumino-Witten (WZW) model. The Hamiltonian of the WZW model is the U q (sℓ(2)) invariant XXX quantum spin chain. For simplicity, we consider these theories only in their vacuum sectors on the strip. Combinatorially, fermionic particles are introduced as certain features of RSOS paths. They are composites of dual-particles and exhibit the properties of quasiparticles. The particles and dual-particles are identified, through a conjectured energy preserving bijection, with patterns of zeros of the eigenvalues of the fused transfer matrices in their analyticity strips. The associated (m, n) systems arise as geometric packing constraints on the particles. The analyticity encoded in the patterns of zeros is the key to the analytic calculation of the excitation energies through the Thermodynamic Bethe Ansatz (TBA). As a by-product of our study, in the case of the WZW or XXX model, we find a relation between the location of the Bethe root strings and the location of the transfer matrix 2-strings.

The twisted boundary conditions and associated partition functions of the conformal sl(2) A-D-E models are studied on the Klein bottle and the Möbius strip. The A-D-E minimal lattice models give realization to the complete classification of the open descendants of the sl(2) minimal theories. We construct the transfer matrices of these lattice models that are consistent with non-orientable geometries. In particular, we show that in order to realize all the Klein bottle amplitudes of different crosscap states, not only the topological flip on the lattice but also the involution in the spin configuration space must be taken into account. This involution is the Z 2 symmetry of the Dynkin diagrams which corresponds to the simple current of the Ocneanu algebra.

We discuss the errors introduced by level truncation in the study of boundary renormalization group (RG) flows using the truncated conformal space approach (TCSA). We show that the TCSA results can have the qualitative form of a sequence of RG flows between different conformal boundary conditions. In the case of a perturbation by the field φ (13) , we propose a RG equation for the coupling constant which predicts a fixed point at a finite value of the TCSA coupling constant and we compare the predictions with data obtained using thermodynamic Bethe ansatz equations.

We study integrable realizations of conformal twisted boundary conditions for sℓ(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical G = A, D, E lattice models with positive spectral parameter u > 0 and Coxeter number g. Integrable seams are constructed by fusing blocks of elementary local face weights. The usual A-type fusions are labelled by the Kac labels (r, s) and are associated with the Verlinde fusion algebra. We introduce a new type of fusion in the two braid limits u → ±i∞ associated with the graph fusion algebra, and labelled by nodes a, b ∈ G respectively. When combined with automorphisms, they lead to general integrable seams labelled by x = (r, a, b, κ) ∈ (A g-2 , H, H, Z 2 ) where H is the graph G itself for Type I theories and its parent for Type II theories. Identifying our construction labels with the conformal labels of Petkova and Zuber, we find that the integrable seams are in one-to-one correspondence with the conformal seams. The distinct seams are thus associated with the nodes of the Ocneanu quantum graph. The quantum symmetries and twisted partition functions are checked numerically for |G| ≤ 6. We also show, in the case of D 2ℓ , that the noncommutativity of the Ocneanu algebra of seams arises because the automorphisms do not commute with the fusions.

We study the conformal spectra of the critical square lattice Ising model on the Klein bottle and Möbius strip using Yang–Baxter techniques and the solution of functional equations. In particular, we obtain expressions for the finitized conformal partition functions in terms of finitized Virasoro characters. This demonstrates that Yang–Baxter techniques and functional equations can be used to study the conformal

By considering the continuum scaling limit of the A4 RSOS lattice model of Andrews-Baxter-Forrester with integrable boundaries, we derive excited state TBA equations describing the boundary flows of the tricritical Ising model. Fixing the bulk weights to their critical values, the integrable boundary weights admit two parameters ξ and η which play the role of the perturbing boundary fields φ1,3 and φ1,2 and induce the renormalization group flow between boundary fixed points. The boundary TBA equations determining the RG flows are derived in the [Formula: see text] example. The induced map between distinct Virasoro characters of the theory are specified in terms of distribution of zeros of the double row transfer matrix.

We discuss a one-dimensional model of a fluctuating interface with a dynamic exponent z = 1. The events that occur are adsorption, which is local, and desorption which is nonlocal and may take place over regions of the order of the system size. In the thermodynamic limit, the time dependence of the system is given by characters of the c = 0 logarithmic conformal field theory of percolation. This implies in a rigorous way a connection between logarithmic conformal field theory and stochastic processes. The finite-size scaling behavior of the average height, interface width and other observables are obtained. The avalanches produced during desorption are analyzed and we show that the probability distribution of the avalanche sizes obeys finite-size scaling with new critical exponents.

We study the scaling limits of the L-state Restricted Solid-on-Solid (RSOS) lattice models and their fusion hierarchies in the off-critical regimes. Starting with the elliptic functional equations of Klümper and Pearce, we derive the Thermodynamic Bethe Ansatz (TBA) equations of Zamolodchikov. Although this systematic approach, in principle, allows TBA equations to be derived for all the excited states we restrict our attention here to the largest eigenvalue or groundstate in Regimes III and IV. In Regime III the TBA equations are massive while in Regime IV there is massless scattering describing the renormalization group flow between distinct A (1) 1 coset conformal field theories. Regimes I and II, pertaining to Z L-1 parafermions, will be treated in a subsequent paper.

An analytic method of calculating conformal partition functions of solvable lattice models is presented. Our technique involves the solution of transfer matrix functional equations in the scaling limit, and unifies the work of Albertini et al. on the classification of transfer matrix eigenvalues with that of Kl~imper and Pearce on finite-size corrections. As an illustration, conlormal partition functions of the tricritical hard square model with fixed boundaries are derived. The resulting Virasoro characters arise in their fermionic representation and agree with the Coulomb gas calculations of Saleur and Bauer. @ 1997 Elsevier Science B.V.

We study integrable and conformal boundary conditions for ŝℓ(2) Z k parafermions on a cylinder. These conformal field theories are realized as the continuum scaling limit of critical A-D-E lattice models with negative spectral parameter. The conformal boundary conditions labelled by (a, m) ∈ (G, Z 2k ) are identified with associated integrable lattice boundary conditions labelled by (r, a) ∈ (A g-2 , G) where g is the Coxeter number of the A-D-E graph G. We obtain analytically the boundary free energies, present general expressions for the parafermion cylinder partition functions and confirm these results by numerical calculations.

We compute the partition functions of N = 1 gauge theories on S^2 ×R^2_ε using supersymmetric localization. The path integral reduces to a sum over vortices at the poles of S^2 and at the origin of R^2_ε. The exact partition functions allow us to test Seiberg duality beyond the supersymmetric index. We propose the N = 1 partition functions on the Ω-background, and show that the Nekrasov partition functions can be recovered from these building blocks.

This paper presents a complete and rigorous proof of the Poincaré Conjecture, confirming that any simply connected, closed 3-manifold is homeomorphic to the 3-sphere S 3 . Building upon Ricci flow techniques introduced by Hamilton and the breakthrough of Perelman, we refine the arguments to ensure the full classification of 3-manifolds, closing any remaining gaps in the proof and providing a comprehensive understanding of geometric flows in three dimensions.

Appreciation of Stochastic Loewner evolution (SLEκ), as a powerful tool to check for conformal invariant properties of geometrical features of critical systems has been rising. In this paper we use this method to check conformal invariance in sandpile models. Avalanche frontiers in Abelian sandpile model (ASM) are numerically shown to be conformally invariant and can be described by SLE with diffusivity κ = 2. This value is the same as value obtained for loop erased random walks (LERW). The fractal dimension and Schramm's formula for left passage probability also suggest the same result. We also check the same properties for Zhang's sandpile model.

Using the recently developed notion of a fractional Virasoro algebra, we explore the implied operator product expansions in nonlocal conformal field theories and their geometric meaning. We probe the interplay between classical nonlocality in the functionalanalytic sense and quantization in a two-dimensional setting and find that nonlocal quantum dynamics realize this fractional Virasoro algebra exclusively with a state dependent central charge. Notably, we prove that the widely studied free Gaussian fixed points with a fractional Laplacian kinetic term does not fit this criterion but that the RG flow associated non-Gaussian fixed points do.

We introduce the impact parameter representation for conformal field theory correlators of the form This representation is appropriate in the eikonal kinematical regime, and approximates the conformal partial wave decomposition in the limit of large spin and dimension of the exchanged primary. Using recent results on the two-point function O 1 O 1 shock in the presence of a shock wave in Anti-de Sitter, and its relation to the discontinuity of the four-point amplitude A across a kinematical branch cut, we find the high spin and dimension conformal partial wave decomposition of all tree-level Anti-de Sitter Witten diagrams. We show that, as in flat space, the eikonal kinematical regime is dominated by the T -channel exchange of the massless particle with highest spin (graviton dominance). We also compute the anomalous dimensions of the high spin O 1 O 2 composites. Finally, we conjecture a formula re-summing crossed-ladder Witten diagrams to all orders in the gravitational coupling.

Global properties of vacuum static, spherically symmetric configurations are studied in a general class of scalar-tensor theories (STTs) of gravity in various dimensions. The conformal mapping between the Jordan and Einstein frames is used as a tool. Necessary and sufficient conditions are found for the existence of solutions admitting a conformal continuation (CC). The latter means that a singularity in the Einstein-frame manifold maps to a regular surface Strans in the Jordan frame, and the solution is then continued beyond this surface. Strans can be an ordinary regular sphere or a horizon. In the second case, Strans connect two epochs of a Kantowski-Sachs type cosmology. It is shown that the list of possible types of global causal structure of vacuum space–times in any STT, with any potential function U(φ), is the same as in general relativity with a cosmological constant. This is even true for conformally continued solutions. A traversable wormhole is shown to be one of the gen...

The classification of rational conformal field theories is reconsidered from the standpoint of boundary conditions. Solving Cardy's equation expressing the consistency condition on a cylinder is equivalent to finding integer valued representations of the fusion algebra. A complete solution not only yields the admissible boundary conditions but also gives valuable information on the bulk properties.

In this paper we show that there exists a new class of topological field theories, whose correlators are intersection numbers of cohomology classes in a constrained moduli space. Our specific example is a formulation of 2D topological gravity. The constrained modulispace is the Poincaré dual of the top Chern-class of the bundle E hol -→ M g , whose sections are the holomorphic differentials. Its complex dimension is 2g -3, rather then 3g -3. We derive our model by performing the A-topological twist of N=2 supergravity, that we identify with N=2 Liouville theory, whose rheonomic construction is also presented. The peculiar field theoretical mechanism, rooted in BRST cohomology, that is responsible for the constraint on moduli space is discussed, the key point being the fact that the graviphoton becomes a Lagrange multiplier after twist. The relation with conformal field theories is also explored. Our formulation of N=2 Liouville theory leads to a representation of the N=2 superconformal algebra with c = 6, instead of the value c = 9 that is obtained by untwisting the Verlinde and Verlinde formulation of topological gravity. The reduced central charge is the shadow, in conformal field theory, of the constraint on moduli space. Our representation of the N=2 algebra can be split into the direct sum of a minimal model with c = 3/2 and a "maximal" model with c = 9/2. Considerations on the matter coupling of constrained topological gravity are also presented. Their study requires the analysis of both the A-twist and the B-twist of N=2 matter coupled supergravity, that we postpone to future work.

We formulate conformal field theories on the infinite-dimensional grassmannian manifold. Besides recovering the known results for the central charge and correlation functions of the b-c system this formalism immediately lends itself to further generalization. The grassmannian manifold is in fact an ad hoc model for the geometrical interpretation of the irreducible representations of an infinite-dimensional Kac-Moody algebra which, in turn, admit an intertwining action of a Virasoro algebra. We further give a proof of bosonization from a purely grassmannian point of view.

Lie conformal algebras are useful tools for studying vertex operator algebras and their representations. In this paper, we establish close relations between Poisson conformal algebras and representations of Lie conformal algebras. We also calculate explicitly Poisson conformal brackets on the associated graded conformal algebras of universal associative conformal envelopes of Virasoro conformal algebra and Neveu-Schwartz conformal superalgebra.

We reformulate the Ω-deformation of four-dimensional gauge theory in a way that is valid away from fixed points of the associated group action. We use this reformulation together with the theory of coisotropic A-branes to explain recent results linking the Ω-deformation to integrable Hamiltonian systems in one direction and Liouville theory of two-dimensional conformal field theory in another direction.

We show that spin generalization of elliptic Calogero-Moser system, elliptic extension of Gaudin model and their cousins can be treated as a degenerations of Hitchin systems. Applications to the constructions of integrals of motion, angle-action variables and quantum systems are discussed. The constructions are motivated by the Conformal Field Theory, and their quantum counterpart can be treated as a degeneration of the critical level Knizhnik-Zamolodchikov-Bernard equations.

Atomic physics and hadronic physics are both governed by the Yang Mills gauge theory Lagrangian; in fact, Abelian quantum electrodynamics can be regarded as the zero-color limit of quantum chromodynamics. I review a number of areas where the techniques of atomic physics can provide important insight into hadronic eigenstates in QCD. For example, the Dirac-Coulomb equation, which predicts the spectroscopy and structure of hydrogenic atoms, has an analog in hadron physics in the form of frame-independent light-front relativistic equations of motion consistent with light-front holography which give a remarkable first approximation to the spectroscopy, dynamics, and structure of light hadrons. The production of antihydrogen in flight can provide important insight into the dynamics of hadron production in QCD at the amplitude level. The renormalization scale for the running coupling is unambiguously set in QED; an analogous procedure sets the renormalization scales in QCD, leading to scheme-independent scale-fixed predictions. Conversely, many techniques which have been developed for hadron physics, such as scaling laws, evolution equations, the quark-interchange process and light-front quantization have important applicants for atomic physics and photon science, especially in the relativistic domain.

We discuss a one-dimensional model of a fluctuating interface with a dynamic exponent z = 1. The events that occur are adsorption, which is local, and desorption which is nonlocal and may take place over regions of the order of the system size. In the thermodynamic limit, the time dependence of the system is given by characters of the c = 0 logarithmic conformal field theory of percolation. This implies in a rigorous way a connection between logarithmic conformal field theory and stochastic processes. The finite-size scaling behavior of the average height, interface width and other observables are obtained. The avalanches produced during desorption are analyzed and we show that the probability distribution of the avalanche sizes obeys finite-size scaling with new critical exponents.

We study the scaling limits of the L-state Restricted Solid-on-Solid (RSOS) lattice models and their fusion hierarchies in the off-critical regimes. Starting with the elliptic functional equations of Klümper and Pearce, we derive the Thermodynamic Bethe Ansatz (TBA) equations of Zamolodchikov. Although this systematic approach, in principle, allows TBA equations to be derived for all the excited states we restrict our attention here to the largest eigenvalue or groundstate in Regimes III and IV. In Regime III the TBA equations are massive while in Regime IV there is massless scattering describing the renormalization group flow between distinct A (1) 1 coset conformal field theories. Regimes I and II, pertaining to Z L-1 parafermions, will be treated in a subsequent paper.

In this note we study the possible connection between functions appearing in diagrammatic expansion and the conformal correlator expansion. To study the connection we propose a generating function which can be expanded to construct a basis. This basis can be utilized to expand, I) the four point function of scalars near the Wilson-Fisher fixed point in d = 4 -ǫ as in [1] and II) integrals for loop diagrams for massless φ 4 theory in position space in four dimensions. This suggests that a linear combination of one expansion can be recast in terms of a linear combination of the other. As a by-product, we also derive the Mellin space representation for the twist-2 higher spin conformal blocks. We also discuss the higher derivative contact terms in the present scenario.

The periodic system of chemical elements is represented within the framework of the weight diagram of the Lie algebra of the fourth rank of the rotation group of an eightdimensional pseudo-Euclidean space. The hydrogen realization of the Cartan subalgebra and Weyl generators of the group algebra is studied. The root structure of the subalgebras of the group algebra of a conformal group in the framework of a twofold covering is analyzed. Based on the analysis, the Cartan-Weyl basis of the group algebra is determined. The root and weight diagrams are constructed. A mass formula associated with each node of the weight diagram is introduced. Spin is interpreted as the fourth generator of the Cartan subalgebra, whose two eigenvalues correspond to two three-dimensional projections of the weight diagram containing elements of the periodic system from hydrogen to moscovium (the first projection) and from helium to oganesson (the second projection). One of the main advantages of the proposed group-theoretic construction of the periodic system is the natural inclusion of antimatter in the general scheme.

Sufficient conditions for existence of common fixed point on complex partial b-metric spaces are obtained. Our results generalize and extend several well-known results. In the end we explore applications of our key results to solve a system of Urysohn type integral equations.

This paper studies the concept of fuzzy generalized topologies, which are generalizations of smooth topologies and Chang's fuzzy topologies. A basis of fuzzy generalized topological space will be defined as functions from the family of all fuzzy subsets of a non-empty set X to [0, 1] and some basic properties of their structure will be obtained. Some characterizations of the basis of fuzzy generalized topology, fuzzy generalized cotopology and the product of fuzzy generalized topological spaces will also be shown.

In the framework of six-dimensional conformal field theories with $$ \mathcal{N}=\left(1,\ 0\right) $$ N = 1 , 0 supersymmetry we develop the map between the holographic description, the field theoretical description and the associated Hanany-Witten set-ups. General expressions that calculate various observables are presented. The study of string solitons singles out a special background of Massive IIA on which we show (by explicitly finding a Lax pair) that the Neveu-Schwarz part of the string sigma model is classically integrable. We study the particular dual conformal field theory and compute some of its observables.