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 \({{\mathbb{T}^{+}}}\).     

A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

For a Hopf algebra B, we endow the Heisenberg double \({\mathcal{H}(B^*)}\) with the structure of a module algebra over the Drinfeld double \({\mathcal{D}(B)}\). Based on this property, we propose that \({\mathcal{H}(B^*)}\) is to be the counterpart of the algebra of fields on the quantum-group side of the Kazhdan–Lusztig duality between logarithmic conformal field theories and quantum groups. As an example, we work out the case where B is the Taft Hopf algebra related to the \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\) quantum group that is Kazhdan–Lusztig-dual to (p,1) logarithmic conformal models. The corresponding pair \({(\mathcal{D}(B),\mathcal{H}(B^*))}\) is “truncated” to \({(\overline{\mathcal{U}}_{\mathfrak{q}} s\ell2,\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2))}\), where \({\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)}\) is a \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\) module algebra that turns out to have the form \({\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)=\mathbb{C}_{\mathfrak{q}}[z,\partial]\otimes\mathbb{C}[\lambda]/(\lambda^{2p}-1)}\), where \({\mathbb{C}_{\mathfrak{q}}[z,\partial]}\) is the \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\)-module algebra with the relations z ^p  = 0, ? ^p  = 0, and \({\partial z = \mathfrak{q}-\mathfrak{q}^{-1} + \mathfrak{q}^{-2} z\partial}\).     

Fluctuations for the Ginzburg-Landau $${\nabla \phi}$$ Interface Model on a Bounded Domain

We study the massless field on \({D_n = D \cap \tfrac{1}{n} \mathbf{Z}^2}\), where \({D \subseteq \mathbf{R}^2}\) is a bounded domain with smooth boundary, with Hamiltonian \({\mathcal {H}(h) = \sum_{x \sim y} \mathcal {V}(h(x) - h(y))}\). The interaction \({\mathcal {V}}\) is assumed to be symmetric and uniformly convex. This is a general model for a (2 + 1)-dimensional effective interface where h represents the height. We take our boundary conditions to be a continuous perturbation of a macroscopic tilt: h(x) = n x · u + f(x) for \({x \in \partial D_n,\,u \in \mathbf{R}^2}\), and f : R ² → R continuous. We prove that the fluctuations of linear functionals of h(x) about the tilt converge in the limit to a Gaussian free field on D, the standard Gaussian with respect to the weighted Dirichlet inner product \({(f,g)_\nabla^\beta = \int_D \sum_i \beta_i \partial_i f_i \partial_i g_i}\) for some explicit β = β(u). In a subsequent article, we will employ the tools developed here to resolve a conjecture of Sheffield that the zero contour lines of h are asymptotically described by SLE(4), a conformally invariant random curve.     

Asymptotic Behavior of the Selberg Zeta Functions for Degenerating Families of Hyperbolic Manifolds

Let {M _k } be a degenerating sequence of finite volume, hyperbolic manifolds of dimension d, with d = 2 or d = 3, with finite volume limit M _∞. Let \({Z_{M_{k}} (s)}\) be the associated sequence of Selberg zeta functions, and let \({{\mathcal{Z}}_{k} (s)}\) be the product of local factors in the Euler product expansion of \({Z_{M_{k}} (s)}\) corresponding to the pinching geodesics on M _k . The main result in this article is to prove that \({Z_{M_{k}} (s)/{\mathcal{Z}}_{k} (s)}\) converges to \({Z_{M_{\infty}} (s)}\) for all \({s \in \mathbf{C}}\)with Re(s) > (d ? 1)/2. The significant feature of our analysis is that the convergence of \({Z_{M_{k}} (s)/{\mathcal{Z}}_{k} (s)}\) to \({Z_{M_{\infty}} (s)}\) is obtained up to the critical line, including the right half of the critical strip, a region where the Euler product definition of the Selberg zeta function does not converge. In the case d = 2, our result reproves by different means the main theorem in Schulze (J Funct Anal 236:120–160, 2006).     

Quantum Fisher Information for <Emphasis Type="Italic">s</Emphasis><Emphasis Type="Italic">u</Emphasis>(2) Atomic Coherent States and <Emphasis Type="Italic">s</Emphasis><Emphasis Type="Italic">u</Emphasis>(1, 1) Coherent States

We investigate quantum Fisher information (QFI) for s u(2) atomic coherent states and s u(1, 1) coherent states. In this work, we find that for s u(2) atomic coherent states, the QFI with respect to \(\vartheta ~(\mathcal {F}_{\vartheta })\) is independent of φ, the QFI with respect to \(\varphi (\mathcal {F}_{\varphi })\) is governed by ??. Analogously, for s u(1,1) coherent states, \(\mathcal {F}_{\tau }\) is independent of φ, and \(\mathcal {F}_{\varphi }\) is determined by τ. Particularly, our results show that \(\mathcal {F}_{\varphi }\) is symmetric with respect to ?? = π/2 for s u(2) atomic coherent states. And for s u(1,1) coherent states, \(\mathcal {F}_{\varphi }\) also possesses symmetry with respect to τ = 0.     

Associated Production of the Charged and Neutral Higgs Bosons at the ILC within the Higgs Triplet Model

The Higgs Triplet Model (HTM) predicts the existences of the extra neutral scalars H_i(H_i = H, A) and the charged Higgs bosons (H^± and H^±±). In this work, we make a systematic investigation for the associated production of the singly-charged and neutral Higgs bosons via the processes: \(e^{+}e^{-}\rightarrow H^{+}W^{-}H\) and \(e^{+}e^{-}\rightarrow H^{+}W^{-}A\). From the numerical evaluations for the production cross sections and relevant phenomenological analysis we find that (i) the production rates of these processes can reach the level of several fb with reasonable parameter values; (ii) due to the large production rates and small backgrounds, the signals of these scalars might be detected via these processes at the future ILC experiments; and (iii) for the case of \(m_{H_{i}}> m_{H^{\pm }}> m_{H^{\pm \pm }}\), the cascade decay modes \(H_{i}\to H^{\pm }W^{\mp \ast }\) with \(H^{\pm }\to H^{\pm \pm }W^{\mp \ast }\) would lead to production of H⁺⁺H^?? accompanied by several virtual W bosons. Such characteristic feature can help us to distinguish the HTM from the Two-Higgs-Doublet Model (2HDM) and the Minimal Supersymmetric Model (MSSM).     

Domain and Range of the Modified Wave Operator for Schrödinger Equations with a Critical Nonlinearity

We study the final problem for the nonlinear Schrödinger equation

$i{\partial }_{t}u+\frac{1}{2}\Delta u=\lambda|u|^{\frac{2}{n}}u,\quad (t,x)\in {\mathbf{R}}\times \mathbf{R}^{n},$

where\(\lambda \in{\bf R},n=1,2,3\). If the final data\(u_{+}\in {\bf H}^{0,\alpha }=\left\{ \phi \in {\bf L}^{2}:\left( 1+\left\vert x\right\vert \right) ^{\alpha }\phi \in {\bf L}^{2}\right\} \) with\(\frac{ n}{2} < \alpha < \min \left( n,2,1+\frac{2}{n}\right) \) and the norm\(\Vert \widehat{u_{+}}\Vert _{{\bf L}^{\infty }}\) is sufficiently small, then we prove the existence of the wave operator in L ². We also construct the modified scattering operator from H ^0,α to H ^0,δ with\(\frac{n}{2} < \delta < \alpha\).

     

Kinetically modified non-minimal inflation with exponential frame function

We consider supersymmetric (SUSY) and non-SUSY models of chaotic inflation based on the \(\phi ^n\) potential with \(n=2\) or 4. We show that the coexistence of an exponential non-minimal coupling to gravity \(f_\mathcal{R}=\mathrm{e}^{c_\mathcal{R}\phi ^{p}}\) with a kinetic mixing of the form \(f_{\mathrm{K}}=c_{\mathrm{K}}f_\mathcal{R}^m\) can accommodate inflationary observables favored by the Planck and Bicep2/Keck Array results for \(p=1\) and 2, \(1\le m\le 15\) and \(2.6\times 10^{-3}\le r_{\mathcal {R}\mathrm{K}}=c_\mathcal{R}/c_{\mathrm{K}}^{p/2}\le 1,\) where the upper limit is not imposed for \(p=1\). Inflation is of hilltop type and it can be attained for subplanckian inflaton values with the corresponding effective theories retaining the perturbative unitarity up to the Planck scale. The supergravity embedding of these models is achieved employing two chiral gauge singlet supefields, a monomial superpotential and several (semi)logarithmic or semi-polynomial Kähler potentials.     

Likelihood analysis of the sub-GUT MSSM in light of LHC 13-TeV data

We describe a likelihood analysis using MasterCode of variants of the MSSM in which the soft supersymmetry-breaking parameters are assumed to have universal values at some scale \(M_\mathrm{in}\) below the supersymmetric grand unification scale \(M_\mathrm{GUT}\), as can occur in mirage mediation and other models. In addition to \(M_\mathrm{in}\), such ‘sub-GUT’ models have the 4 parameters of the CMSSM, namely a common gaugino mass \(m_{1/2}\), a common soft supersymmetry-breaking scalar mass \(m_0\), a common trilinear mixing parameter A and the ratio of MSSM Higgs vevs \(\tan \beta \), assuming that the Higgs mixing parameter \(\mu > 0\). We take into account constraints on strongly- and electroweakly-interacting sparticles from \(\sim 36\)/fb of LHC data at 13 TeV and the LUX and 2017 PICO, XENON1T and PandaX-II searches for dark matter scattering, in addition to the previous LHC and dark matter constraints as well as full sets of flavour and electroweak constraints. We find a preference for \(M_\mathrm{in}\sim 10^5\) to \(10^9 \,\, \mathrm {GeV}\), with \(M_\mathrm{in}\sim M_\mathrm{GUT}\) disfavoured by \(\Delta \chi ^2 \sim 3\) due to the \(\mathrm{BR}(B_{s, d} \rightarrow \mu ^+\mu ^-)\) constraint. The lower limits on strongly-interacting sparticles are largely determined by LHC searches, and similar to those in the CMSSM. We find a preference for the LSP to be a Bino or Higgsino with \(m_{\tilde{\chi }^0_{1}} \sim 1 \,\, \mathrm {TeV}\), with annihilation via heavy Higgs bosons H / A and stop coannihilation, or chargino coannihilation, bringing the cold dark matter density into the cosmological range. We find that spin-independent dark matter scattering is likely to be within reach of the planned LUX-Zeplin and XENONnT experiments. We probe the impact of the \((g-2)_\mu \) constraint, finding similar results whether or not it is included.     

Synthesis,characterization and molecular docking studies of thiouracil derivatives as potent thymidylate synthase inhibitors and potential anticancer agents

Thymidylate synthase (TS), one of folate-dependent enzymes, is a key and well-recognized target for anticancer agents. In this study, a series of 6-aryl-5-cyano thiouracil derivatives were designed and synthesized in accordance with essential pharmacophoric features of known TS inhibitors. Nineteen compounds were screened in vitro for their anti-proliferative activities toward HePG-2, MCF-7, HCT-116, and PC-3 cell lines. Compounds \(\mathbf{21}_{\mathbf{c}}\), \(\mathbf{21}_{\mathbf{d}}\), and 24 exhibited high anti-proliferative activity, comparable to that of 5-fluorouracil. Additionally, ten compounds with potent anti-proliferative activities were further evaluated for their ability to inhibit TS enzyme. Six compounds (\(\mathbf{21}_{\mathbf{b}}\), \(\mathbf{21}_{\mathbf{c}}\), \(\mathbf{21}_{\mathbf{d}}\), 22, 23 and 24) demonstrated potent dose-related TS inhibition with \(\hbox {IC}_{50}\) values ranging from 1.57 to \(3.89\,\upmu \hbox {M}\). The in vitro TS activity results were consistent with those of the cytotoxicity assay where the most potent anti-proliferative compounds of the series showed good TS inhibitory activity comparable to that of 5-fluorouracil. Furthermore, molecular docking studies were carried out to investigate the binding pattern of the designed compounds with the prospective target, TS (PDB-code: 1JU6).     

Flat Bi-Hamiltonian Structures and Invariant Densities

A bi-Hamiltonian structure is a pair of Poisson structures \({{\mathcal P}}\), \({{\mathcal Q}}\) which are compatible, meaning that any linear combination \({\alpha {\mathcal P} + \beta {\mathcal Q}}\) is again a Poisson structure. A bi-Hamiltonian structure \({({\mathcal P}, {\mathcal Q})}\) is called flat if \({{\mathcal P}}\) and \({{\mathcal Q}}\) can be simultaneously brought to a constant form in a neighborhood of a generic point. We prove that a generic bi-Hamiltonian structure \({({\mathcal P}, {\mathcal Q})}\) on an odd-dimensional manifold is flat if and only if there exists a local density which is preserved by all vector fields Hamiltonian with respect to \({{\mathcal P}}\), as well as by all vector fields Hamiltonian with respect to \({{\mathcal Q}}\).     

Generic Representation of Y(so(3)) Based on the Lie Algebraic Basis of so(3)

We focus on constructing a generic representation of Y(so(3)) based on the Lie algebraic basis of so(3) basis, and further developing transition of Yangian operator \(\hat {\mathbf {Y}}\). As an application of Y(so(3)), we calculate all the matrix elements of unit vector operators \(\hat {\mathbf {n}}\) in angular momentum basis. It is also discovered that the Yangian operator \(\hat {\mathbf {Y}}\) may work in quantum vector space. In addition, some shift operators \(\hat {O}^{(\pm )}_{\mu }\) are naturally built on the basis of the representation of Y(so(3)). As an another application of Y(so(3)), we can derive the CG cofficients of two coupled angular momenta from the down-shift operator \(\hat {O}^{(-)}_{-1}\) acting on a so(3)-coupled tensor basis. This not only explores that Yangian algebras can work in quantum tensor space, but also provides a novel approach to solve CG coefficients for two coupled angular momenta.     

Exact holography of the mass-deformed M2-brane theory

We test the holographic relation between the vacuum expectation values of gauge invariant operators in \({\mathcal {N}} = 6\) U\(_k(N)\times \mathrm{U}_{-k}(N)\) mass-deformed ABJM theory and the LLM geometries with \({\mathbb {Z}}_k\) orbifold in 11-dimensional supergravity. To do so, we apply the Kaluza–Klein reduction to construct a 4-dimensional gravity theory and implement the holographic renormalization procedure. We obtain an exact holographic relation for the vacuum expectation values of the chiral primary operator with conformal dimension \(\Delta = 1\), which is given by \(\langle {\mathcal {O}}^{(\Delta =1)}\rangle = N^{\frac{3}{2}} \, f_{(\Delta =1)}\), for large N and \(k=1\). Here the factor \(f_{(\Delta )}\) is independent of N. Our results involve an infinite number of exact dual relations for all possible supersymmetric Higgs vacua and so provide a non-trivial test of gauge/gravity duality away from the conformal fixed point. We extend our results to the case of \(k\ne 1\) for LLM geometries represented by rectangular-shaped Young diagrams. We also discuss the exact mapping of the gauge/gravity at finite N for classical supersymmetric vacuum solutions in field theory side and corresponding classical solutions in gravity side.     

Searching for cosmological preferred axis using cosmographic approach

Recent released Planck data and other astronomical observations show that the universe may be anisotropic on large scales. This hints a cosmological privileged axis in our anisotropic expanding universe. This paper proceeds a modified redshift in anisotropic cosmological model as \( 1+\tilde{z}(t,\hat{\mathbf{p }})=\frac{a(t_{0)}}{a(t)}(1-A(\hat{\mathbf{n }}.\hat{\mathbf{p }}))\) (where A is the magnitude of anisotropy, \(\hat{\mathbf{n }}\) is the direction of privileged axis, and \(\hat{\mathbf{p }}\) is the direction of each SNe Ia sample to galactic coordinates) along with anisotropic parameter \(\delta =\frac{A(\hat{\mathbf{n }}.\hat{\mathbf{p }})}{1+A(\hat{\mathbf{n }}.\hat{\mathbf{p }})}\). The luminosity distance is expanded with model-independent cosmographic parameters as a function of modified redshift \(\tilde{z}\). As the transformation matrix \(M(n\times n)\) is obtained to convert the Taylor series coefficients of isotropic luminosity distance to corresponding anisotropic parameters. These results culminate the magnitude of anisotropy about \(\mid A\mid \simeq 10^{-3}\) and the direction of preferred axis as \((l,b)=\left( 297^{-34}_{+34},3^{-28}_{+28}\right) \), which are consistent with other studies in \(1-\sigma \) confidence level.     

On <Emphasis Type="Italic">τ</Emphasis>-Compactness of Products of <Emphasis Type="Italic">τ</Emphasis>-Measurable Operators

Let \(\mathcal {M}\) be a von Neumann algebra of operators on a Hilbert space \(\mathcal {H}\), τ be a faithful normal semifinite trace on \(\mathcal {M}\). We obtain some new inequalities for rearrangements of τ-measurable operators products. We also establish some sufficient τ-compactness conditions for products of selfadjoint τ-measurable operators. Next we obtain a τ-compactness criterion for product of a nonnegative τ-measurable operator with an arbitrary τ-measurable operator. We construct an example that shows importance of nonnegativity for one of the factors. The similar results are obtained also for elementary operators from \(\mathcal {M}\). We apply our results to symmetric spaces on \((\mathcal {M}, \tau )\). The results are new even for the *-algebra \(\mathcal {B}(\mathcal {H})\) of all linear bounded operators on \(\mathcal {H}\) endowed with the canonical trace τ = tr.     

Pseudo-Hermitian Systems with $\mathcal {P}\mathcal {T}$-Symmetry: Degeneracy and Krein Space

We show in the present paper that pseudo-Hermitian Hamiltonian systems with even \(\mathcal {P}\mathcal {T}\)-symmetry \((\mathcal {P}^{2}=1,\mathcal {T}^{2}=1)\) admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd \(\mathcal {P}\mathcal {T}\)-symmetric systems \((\mathcal {P}^{2}=1,\mathcal {T}^{2}=-1)\) which is appropriate to the fermions (Scolarici and Solombrino, Phys. Lett. A 303, 239 2002; Jones-Smith and Mathur, Phys. Rev. A 82, 042101 2010). We establish that the pseudo-Hermitian Hamiltonians with even \(\mathcal {P}\mathcal {T}\)-symmetry admit a degeneracy structure if the operator \(\mathcal {PT}\) anticommutes with the metric operator η _σ which is necessarily indefinite. We also show that the Krein space formulation of the Hilbert space is the convenient framework for the implementation of unbroken \(\mathcal {P}\mathcal {T}\)-symmetry. These general results are illustrated with great details for four-level pseudo-Hermitian Hamiltonian with even \(\mathcal {P}\mathcal {T}\) -symmetry.     

Pentagon Equation Arising from State Equations of a C*-Bialgebra

The direct sum \({{\mathcal O}_{*}}\) of all Cuntz algebras has a non-cocommutative comultiplication \({\Delta_{\varphi}}\) such that there exists no antipode of any dense subbialgebra of the C*-bialgebra \({({\mathcal O}_{*},\Delta_{\varphi})}\). From states equations of \({{\mathcal O}_{*}}\) with respect to the tensor product, we construct an operator W for \({({\mathcal O}_{*},\Delta_{\varphi})}\) such that W* is an isometry, \({W(x\otimes I)W^{*}=\Delta_{\varphi}(x)}\) for each \({x\in {\mathcal O}_{*}}\) and W satisfies the pentagon equation.     

Consequences of R-parity violating interactions for anomalies in $${\bar{B}}\rightarrow D^{(*)} \tau {\bar{\nu }}$$ and $$b\rightarrow s \mu ^+\mu ^-$$b→sμ+μ-

We investigate the possibility of explaining the enhancement in semileptonic decays of \({\bar{B}} \rightarrow D^{(*)} \tau {\bar{\nu }}\), the anomalies induced by \(b\rightarrow s\mu ^+\mu ^-\) in \({\bar{B}}\rightarrow (K, K^*, \phi )\mu ^+\mu ^-\) and violation of lepton universality in \(R_K = \mathrm{Br}({\bar{B}}\rightarrow K \mu ^+\mu ^-)/\mathrm{Br}({\bar{B}}\rightarrow K e^+e^-)\) within the framework of R-parity violating MSSM. The exchange of down type right-handed squark coupled to quarks and leptons yields interactions which are similar to leptoquark induced interactions that have been proposed to explain the \({\bar{B}} \rightarrow D^{(*)} \tau {\bar{\nu }}\) by tree level interactions and \(b\rightarrow s \mu ^+\mu ^-\) anomalies by loop induced interactions, simultaneously. However, the Yukawa couplings in such theories have severe constraints from other rare processes in B and D decays. Although this interaction can provide a viable solution to the \(R(D^{(*)})\) anomaly, we show that with the severe constraint from \({\bar{B}} \rightarrow K \nu {\bar{\nu }}\), it is impossible to solve the anomalies in the \(b\rightarrow s \mu ^+\mu ^-\) process simultaneously.