One-dimensional backreacting holographic <Emphasis Type="Italic">p</Emphasis>-wave superconductors

We analytically and numerically study the properties of one-dimensional holographic p-wave superconductors in the presence of backreaction. We employ the Sturm–Liouville eigenvalue problem for the analytical calculation and the shooting method for the numerical investigations. We apply the \(\hbox {AdS}_{{3}}\)/\(\hbox {CFT}_{{2}}\) correspondence and determine the relation between the critical temperature \(T_{c}\) and the chemical potential \(\mu \) for different values of the mass m of a charged spin-1 field \(\rho _{\mu }\) and backreacting parameters. We observe that the data of both analytical and numerical studies are in good agreement. We find that increasing the backreaction and the mass parameter causes the greater values for \({T_{c}}/{\mu }\). Thus, it makes the condensation harder to form. In addition, the analytical and numerical approaches show that the value of the critical exponent \( \beta \) is 1 / 2, which is the same as in the mean field theory. Moreover, both methods confirm the existence of a second order phase transition.     

On the Convexity of the KdV Hamiltonian

Motivated by perturbation theory, we prove that the nonlinear part \({H^{*}}\) of the KdV Hamiltonian \({H^{kdv}}\), when expressed in action variables \({I = (I_{n})_{n \geqslant 1}}\), extends to a real analytic function on the positive quadrant \({\ell^{2}_{+}(\mathbb{N})}\) of \({\ell^{2}(\mathbb{N})}\) and is strictly concave near \({0}\). As a consequence, the differential of \({H^{*}}\) defines a local diffeomorphism near 0 of \({\ell_{\mathbb{C}}^{2}(\mathbb{N})}\). Furthermore, we prove that the Fourier-Lebesgue spaces \({\mathcal{F}\mathcal{L}^{s,p}}\) with \({-1/2 \leqslant s \leqslant 0}\) and \({2 \leqslant p < \infty}\), admit global KdV-Birkhoff coordinates. In particular, it means that \({\ell^{2}_+(\mathbb{N})}\) is the space of action variables of the underlying phase space \({\mathcal{F}\mathcal{L}^{-1/2,4}}\) and that the KdV equation is globally in time \({C^{0}}\)-well-posed on \({\mathcal{F}\mathcal{L}^{-1/2,4}}\).     

On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture

In their recent paper, Alday et al. (Lett Math Phys 91:167–197, 2010) proposed a relation between \({\mathcal{N}=2}\) four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired by the exact form of the instanton part of the Nekrasov partition function. In this paper we study the origin of such an expansion from a CFT point of view. We consider the algebra \({\mathcal{A}={\sf Vir} \otimes\mathcal{H}}\) which is the tensor product of mutually commuting Virasoro and Heisenberg algebras and discover the special orthogonal basis of states in the highest weight representations of \({\mathcal{A}}\). The matrix elements of primary fields in this basis have a very simple factorized form and coincide with the function called \({Z_{{\sf bif}}}\) appearing in the instanton counting literature. Having such a simple basis, the problem of computation of the conformal blocks simplifies drastically and can be shown to lead to the expansion proposed in Alday et al. (2010). We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin–Ono integrable hierarchy.     

A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index { < 4}

It is well known that the quantum double \({D(N\subset M)}\) of a finite depth subfactor \({N\subset M}\), or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. It is big open conjecture that all (unitary) modular tensor categories arise from conformal field theory. We show that for every subfactor \({N\subset M}\) with index \({[M:N] < 4}\) the quantum double \({D(N\subset M)}\) is realized as the representation category of a completely rational conformal net. In particular, the quantum double of \({E_6}\) can be realized as a \({\mathbb{Z}_2}\)-simple current extension of \({{{\rm SU}(2)}_{10}\times {{\rm Spin}(11)}_1}\) and thus is not exotic in any sense. As a byproduct, we obtain a vertex operator algebra for every such subfactor. We obtain the result by showing that if a subfactor \({N\subset M }\) arises from \({\alpha}\)-induction of completely rational nets \({\mathcal{A}\subset \mathcal{B}}\) and there is a net \({\tilde{\mathcal{A}}}\) with the opposite braiding, then the quantum \({D(N\subset M)}\) is realized by completely rational net. We construct completely rational nets with the opposite braiding of \({{{\rm SU}(2)}_k}\) and use the well-known fact that all subfactors with index \({[M:N] < 4}\) arise by \({\alpha}\)-induction from \({{{\rm SU}(2)}_k}\).     

Holographic screening length in a hot plasma of two sphere

We study the screening length \(L_{\mathrm{max}}\) of a moving quark–antiquark pair in a hot plasma, which lives in a two sphere, \(S^2\), using the AdS/CFT correspondence in which the corresponding background metric is the four-dimensional Schwarzschild–AdS black hole. The geodesic of both ends of the string at the boundary, interpreted as the quark–antiquark pair, is given by a stationary motion in the equatorial plane by which the separation length L of both ends of the string is parallel to the angular velocity \(\omega \). The screening length and total energy H of the quark–antiquark pair are computed numerically and show that the plots are bounded from below by some functions related to the momentum transfer \(P_c\) of the drag force configuration. We compare the result by computing the screening length in the reference frame of the moving quark–antiquark pair, in which the background metrics are “Boost-AdS” and Kerr–AdS black holes. Comparing both black holes, we argue that the mass parameters \(M_{\mathrm{Sch}}\) of the Schwarzschild–AdS black hole and \(M_{\mathrm{Kerr}}\) of the Kerr–AdS black hole are related at high temperature by \(M_{\mathrm{Kerr}}=M_{\mathrm{Sch}}(1-a^2l^2)^{3/2}\), where a is the angular momentum parameter and l is the AdS curvature.     

Entanglement Rates and the Stability of the Area Law for the Entanglement Entropy

We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed \({\mathfrak{gl}(1|1)}\) spin-chain and its continuum limit—the \({c=-2}\) symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245–288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289–329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones–Temperley–Lieb algebra JTL _N , goes over in the continuum limit to a bigger algebra than \({\boldsymbol{\mathcal{V}}}\), the product of the left and right Virasoro algebras. This algebra, \({\mathcal{S}}\)—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field \({S(z,\bar{z})\equiv S_{\alpha\beta} \psi^\alpha(z)\bar{\psi}^\beta(\bar{z})}\), with a symmetric form \({S_{\alpha\beta}}\) and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL _N in the \({\mathfrak{gl}(1|1)}\) spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of \({\mathfrak{sp}_{N-2}}\). The semi-simple part of JTL _N is represented by \({U \mathfrak{sp}_{N-2}}\), providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL _N image represented in the spin-chain. On the continuum side, simple modules over \({\mathcal{S}}\) are identified with “fundamental” representations of \({\mathfrak{sp}_\infty}\).     

T-Duality Simplifies Bulk-Boundary Correspondence

We extend and apply a rigorous renormalisation group method to study critical correlation functions, on the 4-dimensional lattice \({{{\mathbb{Z}}}^{4}}\), for the weakly coupled n-component \({|\varphi|^{4}}\) spin model for all \({n \ge 1}\), and for the continuous-time weakly self-avoiding walk. For the \({|\varphi|^{4}}\) model, we prove that the critical two-point function has |x|^?2 (Gaussian) decay asymptotically, for \({n \ge 1}\). We also determine the asymptotic decay of the critical correlations of the squares of components of \({\varphi}\), including the logarithmic corrections to Gaussian scaling, for \({n \ge 1}\). The above extends previously known results for n = 1 to all \({n \ge 1}\), and also observes new phenomena for n > 1, all with a new method of proof. For the continuous-time weakly self-avoiding walk, we determine the decay of the critical generating function for the “watermelon” network consisting of p weakly mutually- and self-avoiding walks, for all \({p \ge 1}\), including the logarithmic corrections. This extends a previously known result for p = 1, for which there is no logarithmic correction, to a much more general setting. In addition, for both models, we study the approach to the critical point and prove the existence of logarithmic corrections to scaling for certain correlation functions. Our method gives a rigorous analysis of the weakly self-avoiding walk as the n = 0 case of the \({|\varphi|^{4}}\) model, and provides a unified treatment of both models, and of all the above results.     

Dorey’s Rule and the q-Characters of Simply-Laced Quantum Affine Algebras

Let \({U_q(\widehat{\mathfrak g})}\) be the quantum affine algebra associated to a simply-laced simple Lie algebra \({\mathfrak{g}}\) . We examine the relationship between Dorey’s rule, which is a geometrical statement about Coxeter orbits of \({\mathfrak{g}}\) -weights, and the structure of q-characters of fundamental representations V _i,a of \({U_q(\widehat{\mathfrak g})}\) . In particular, we prove, without recourse to the ADE classification, that the rule provides a necessary and sufficient condition for the monomial 1 to appear in the q-character of a three-fold tensor product \({V_{i,a}\otimes V_{j,b}\otimes V_{k,c}}\) .     

Non-Negative Integral Level Affine Lie Algebra Tensor Categories and Their Associativity Isomorphisms

For a finite-dimensional simple Lie algebra \({\mathfrak{g}}\), we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra \({{\widehat{\mathfrak{g}}}}\) at a fixed level \({\ell\in\mathbb{N}}\) with a certain tensor category of finite-dimensional \({\mathfrak{g}}\)-modules. More precisely, the category of level ? standard \({{\widehat{\mathfrak{g}}}}\)-modules is the module category for the simple vertex operator algebra \({L_{\widehat{\mathfrak{g}}}(\ell, 0)}\), and as is well known, this category is equivalent as an abelian category to \({\mathbf{D}(\mathfrak{g},\ell)}\), the category of finite-dimensional modules for the Zhu’s algebra \({A{(L_{\widehat{\mathfrak{g}}}(\ell, 0))}}\), which is a quotient of \({U(\mathfrak{g})}\). Our main result is a direct construction using Knizhnik–Zamolodchikov equations of the associativity isomorphisms in \({\mathbf{D}(\mathfrak{g},\ell)}\) induced from the associativity isomorphisms constructed by Huang and Lepowsky in \({{L_{\widehat{\mathfrak{g}}}(\ell, 0) - \mathbf{mod}}}\). This construction shows that \({\mathbf{D}(\mathfrak{g},\ell)}\) is closely related to the Drinfeld category of \({U(\mathfrak{g})}\)[[h]]-modules used by Kazhdan and Lusztig to identify categories of \({{\widehat{\mathfrak{g}}}}\)-modules at irrational and most negative rational levels with categories of quantum group modules.     

On a Lagrangian formulation of gravitoelectromagnetism

We examine a Lagrangian formulation of gravity based on an approach analogous to electromagnetism, called Gravitoelectromagnetism (GEM). The gravitational analogue of the electromagnetic field tensor is a three-index tensor, \({\mathcal {F}_{\mu\nu\lambda}}\), defined in terms of a two-index gravitoelectromagnetic potential, \({\mathcal {A}_{\mu\nu}}\). The energy-momentum tensor is derived and is symmetric. We construct a Lagrangian which allows us to describe interactions between fermions, photons and gravitons. We calculate transition amplitudes of various processes involving gravitons: gravitational Møller scattering, gravitational Compton scattering, and the graviton photoproduction.     

Hölder Regularity of the Solutions of the Cohomological Equation for Roth Type Interval Exchange Maps

There is a general method for constructing a soliton hierarchy from a splitting \({{L_{\pm}}}\) of a loop group as positive and negative sub-groups together with a commuting linearly independent sequence in the positive Lie algebra \({\mathcal{L}_{+}}\). Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each f in the negative subgroup \({L_-}\) a solution \({u_{f}}\) of the hierarchy. When there is a 2 co-cycle of the Lie algebra that vanishes on both sub-algebras, Wilson constructed a tau function \({\tau_{f}}\) for each element \({f \in L_-}\). In this paper, we give integral formulas for variations of \({\ln\tau_{f}}\) and second partials of \({\ln\tau_{f}}\), discuss whether we can recover solutions \({u_{f}}\) from \({\tau_{f}}\), and give a general construction of actions of the positive half of the Virasoro algebra on tau functions. We write down formulas relating tau functions and formal inverse scattering solutions and the Virasoro vector fields for the \({GL(n,\mathbb{C})}\)-hierarchy.     

Disappearance of Mott Oscillations in Sub-barrier Elastic Scattering of Identical Nuclei and Atomic Ions

The scattering of identical nuclei at low energies exhibits conspicuous Mott oscillations which can be used to investigate the presence of components in the predominantly Coulomb interaction arising from several physical effects. It is found that at a certain critical value of the Sommerfeld parameter the Mott oscillations disappear and the cross section becomes quite flat. We call this effect Transverse Isotropy (TI). The critical value of the Sommerfeld parameter at which TI sets in is found to be \({\eta_{c} = \sqrt{3s + 2}}\), where s is the spin of the nuclei participating in the scattering. No TI is found in the Mott scattering of identical Fermionic nuclei. The critical center of mass energy corresponding to \({\eta_c}\) is found to be \({E_c = 0.40}\) MeV for \({\alpha + \alpha}\) (s = 0), 1.2 MeV for \({^{6}}\)Li + \({^{6}}\)LI (s = 1) and 7.1 MeV for \({^{10}}\)B + \({^{10}}\)B (s = 3). We further found that the inclusion of the nuclear interaction induces a significant modification in the TI. We suggest measurements at these sub-barrier energies for the purpose of extracting useful information about the nuclear interaction between light heavy ions. We also suggest extending the study of the TI to the scattering of identical atomic ions.     

The Structure of the Kac–Wang–Yan Algebra

The Lie algebra \({\mathcal{D}}\) of regular differential operators on the circle has a universal central extension \({\hat{\mathcal{D}}}\). The invariant subalgebra \({\hat{\mathcal{D}}^+}\) under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum \({\hat{\mathcal{D}}^+}\)-module with central charge \({c \in \mathbb{C}}\), and its irreducible quotient \({\mathcal{V}_c}\), possess vertex algebra structures, and \({\mathcal{V}_c}\) has a nontrivial structure if and only if \({c \in \frac{1}{2}\mathbb{Z}}\). We show that for each integer \({n > 0}\), \({\mathcal{V}_{n/2}}\) and \({\mathcal{V}_{-n}}\) are \({\mathcal{W}}\)-algebras of types \({\mathcal{W}(2, 4,\dots,2n)}\) and \({\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}\), respectively. These results are formal consequences of Weyl’s first and second fundamental theorems of invariant theory for the orthogonal group \({{\rm O}(n)}\) and the symplectic group \({{\rm Sp}(2n)}\), respectively. Based on Sergeev’s theorems on the invariant theory of \({{\rm Osp}(1, 2n)}\) we conjecture that \({\mathcal{V}_{-n+1/2}}\) is of type \({\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}\), and we prove this for \({n = 1}\). As an application, we show that invariant subalgebras of \({\beta\gamma}\)-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.     

A Categorification of the Boson–Fermion Correspondence via Representation Theory of sl(∞)

In recent years different aspects of categorification of the boson–fermion correspondence have been studied. In this paper we propose a categorification of the boson–fermion correspondence based on the category of tensor modules of the Lie algebra sl(∞) of finitary infinite matrices. By \({\mathbb{T}^{+}}\) we denote the category of “polynomial” tensor sl(∞)-modules. There is a natural “creation” functor \({{\mathcal{T}_{N}} : {\mathbb{T}^{+}} \to {\mathbb{T}^{+}}}\), \({M \mapsto N \otimes M, \quad M,N \in \mathbb{T}^{+}}\). The key idea of the paper is to employ the entire category \({\mathbb{T}}\) of tensor sl(∞)-modules in order to define the “annihilation” functor \({{\mathcal{D}_{N}} : {\mathbb{T}^{+}} \to {\mathbb{T}^{+}}}\) corresponding to \({{\mathcal{T}_{N}}}\). We show that the relations allowing one to express fermions via bosons arise from relations in the cohomology of complexes of linear endofunctors on \({{\mathbb{T}^{+}}}\).     

Positive Casimir and Central Characters of Split Real Quantum Groups

We consider the one parameter family \({\alpha \mapsto T_{\alpha}}\) (\({\alpha \in [0,1)}\)) of Pomeau-Manneville type interval maps \({T_{\alpha}(x) = x(1+2^{\alpha} x^{\alpha})}\) for \({x \in [0,1/2)}\) and \({T_{\alpha}(x)=2x-1}\) for \({x \in [1/2, 1]}\), with the associated absolutely continuous invariant probability measure \({\mu_{\alpha}}\). For \({\alpha \in (0,1)}\), Sarig and Gouëzel proved that the system mixes only polynomially with rate \({n^{1-1/{\alpha}}}\) (in particular, there is no spectral gap). We show that for any \({\psi \in L^{q}}\), the map \({\alpha \to \int_0^{1} \psi\, d \mu_{\alpha}}\) is differentiable on \({[0,1-1/q)}\), and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula for the SRB measure is obtained in the setting of slowly mixing dynamics. Our argument shows how cone techniques can be used in this context. For \({\alpha \ge 1/2}\) we need the \({n^{-1/{\alpha}}}\) decorrelation obtained by Gouëzel under additional conditions.     

On {J^{PC}=0^{--}} exotic glueball

The mass spectrum of the gluonium with \({J^{PC}=0^{--}}\) is examined in three bottom-up AdS/QCD models. The results are used to identify several production and decay modes useful for searching this state. Moreover, the properties of such glueball in a hot and dense quark medium are discussed.     

Limits of Gaudin Algebras,Quantization of Bending Flows,Jucys–Murphy Elements and Gelfand–Tsetlin Bases

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra \({U(\mathfrak {g})}\) of a semisimple Lie algebra \({\mathfrak {g}}\). This family is parameterized by collections of pairwise distinct complex numbers z ₁, . . . , z _n . We obtain some new commutative subalgebras in \({U(\mathfrak {g})^{\otimes n}}\) as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the Hamiltonians of bending flows and to the Gelfand–Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.     

Generic Rigidity for Circle Diffeomorphisms with Breaks

We prove that \({C^r}\)-smooth (\({r > 2}\)) circle diffeomorphisms with a break, i.e., circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity, are generically, i.e., for almost all irrational rotation numbers, not \({C^{1+\varepsilon}}\)-rigid, for any \({\varepsilon > 0}\). This result complements our recent proof, joint with Khanin (Geom Funct Anal 24:2002–2028, 2014), that such maps are generically \({C^1}\)-rigid. It stands in remarkable contrast to the result of Yoccoz (Ann Sci Ec Norm Sup 17:333–361, 1984) that \({C^r}\)-smooth circle diffeomorphisms are generically \({C^{r-1-\varkappa}}\)-rigid, for any \({\varkappa > 0}\).     

Braidings of Tensor Spaces

Let V be a braided vector space, i.e., a vector space together with a solution \({\hat{R}\in {{End}}(V\otimes V)}\) of the Yang–Baxter equation. Denote \({T(V):=\bigoplus_k V^{\otimes k}}\) . We associate to \({\hat{R}}\) a one-parameter family of solutions \({T(\hat{R})\in {\rm End}(T(V)\otimes T(V))}\) of the Yang–Baxter equation on the tensor space T (V). Main ingredients of the solution are braid analogues of the binomial coefficients and of the Pochhammer symbols. The association \({\hat{R}\rightsquigarrow T(\hat{R})}\) is functorial with respect to V.     

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment

A quantum system (with Hilbert space \({\mathcal {H}_{1}}\)) entangled with its environment (with Hilbert space \({\mathcal {H}_{2}}\)) is usually not attributed to a wave function but only to a reduced density matrix \({\rho_{1}}\). Nevertheless, there is a precise way of attributing to it a random wave function \({\psi_{1}}\), called its conditional wave function, whose probability distribution \({\mu_{1}}\) depends on the entangled wave function \({\psi \in \mathcal {H}_{1} \otimes \mathcal {H}_{2}}\) in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of \({\mathcal {H}_{2}}\) but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about \({\mu_{1}}\), e.g., that if the environment is sufficiently large then for every orthonormal basis of \({\mathcal {H}_{2}}\), most entangled states \({\psi}\) with given reduced density matrix \({\rho_{1}}\) are such that \({\mu_{1}}\) is close to one of the so-called GAP (Gaussian adjusted projected) measures, \({GAP(\rho_{1})}\). We also show that, for most entangled states \({\psi}\) from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval \({[E, E+ \delta E]}\)) and most orthonormal bases of \({\mathcal {H}_{2}}\), \({\mu_{1}}\) is close to \({GAP(\rm {tr}_{2} \rho_{mc})}\) with \({\rho_{mc}}\) the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then \({\mu_{1}}\) is close to \({GAP(\rho_\beta)}\) with \({\rho_\beta}\) the canonical density matrix on \({\mathcal {H}_{1}}\) at inverse temperature \({\beta=\beta(E)}\). This provides the mathematical justification of our claim in Goldstein et al. (J Stat Phys 125: 1193–1221, 2006) that GAP measures describe the thermal equilibrium distribution of the wave function.