Free Field Construction for the ABF Models in Regime II

The Wakimoto construction for the quantum affine algebra U $_q$ ( $(\widehat{\mathfrak{s}\mathfrak{l}_2 })$ ) admits a reduction to the q-deformed parafermion algebras. We interpret the latter theory as a free field realization of the Andrews–Baxter–Forrester models in regime II. We give multi-particle form factors of some local operators on the lattice and compute their scaling limit, where the models are described by a massive field theory with $\mathbb{Z}$ $_k$ symmetric minimal scattering matrices.     

Singular $$ \mathcal{R}$$ -Matrices and Drinfeld's Comultiplication

We compute the $\mathcal{R}$ -matrix which intertwines two dimensional evaluation representations with Drinfeld comultiplication for ${\text{U}}_q \left( {\widehat{{\text{sl}}}_{\text{2}} } \right)$ . This $\mathcal{R}$ -matrix contains terms proportional to the δ-function. We construct the algebra $A\left( \mathcal{R} \right)$ generated by the elements of the matrices L^±(z) with relations determined by $\mathcal{R}$ . In the category of highest-weight representations, there is a Hopf algebra isomorphism between $A\left( \mathcal{R} \right)$ and an extension $\overline {\text{U}} _q \left( {\widehat{{\text{sl}}}_{\text{2}} } \right)$ of Drinfeld's algebra.     

Back to the Amitsur–Levitzki Theorem: a Super Version for the Orthosymplectic Lie Superalgebra $${\mathfrak{o}}{\mathfrak{s}}{\mathfrak{p}}\left( {1,2n} \right)$$

We prove an Amitsur–Levitzki type theorem for the Lie superalgebras $\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {1,2n} \right)$ ) inspired by Kostant's cohomological interpretation of the classical theorem. We show that the Lie superalgebras $\mathfrak{g}\mathfrak{l}\left( {p,q} \right)$ cannot satisfy an Amitsur–Levitzki type super identity if pq≠0 and conjecture that neither can any other classical simple Lie superalgebra with the exception of $\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {1,2n} \right)$ .     

Fomenko–Mischenko Theory, Hessenberg Varieties, and Polarizations

The symmetric algebra ${S(\mathfrak{g})}$ over a Lie algebra ${\mathfrak{g}}$ has the structure of a Poisson algebra. Assume ${\mathfrak{g}}$ is complex semisimple. Then results of Fomenko–Mischenko (translation of invariants) and Tarasov construct a polynomial subalgebra ${{\mathcal {H}} = {\mathbb C}[q_1,\ldots,q_b]}$ of ${S(\mathfrak{g})}$ which is maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of ${\mathfrak{g}}$ . Let G be the adjoint group of ${\mathfrak{g}}$ and let ? = rank ${\mathfrak{g}}$ . Using the Killing form, identify ${\mathfrak{g}}$ with its dual so that any G-orbit O in ${\mathfrak{g}}$ has the structure (KKS) of a symplectic manifold and ${S(\mathfrak{g})}$ can be identified with the affine algebra of ${\mathfrak{g}}$ . An element ${x\in \mathfrak{g}}$ will be called strongly regular if ${\{({\rm d}q_i)_x\},\,i=1,\ldots,b}$ , are linearly independent. Then the set ${\mathfrak{g}^{\rm{sreg}}}$ of all strongly regular elements is Zariski open and dense in ${\mathfrak{g}}$ and also ${\mathfrak{g}^{\rm{sreg}}\subset \mathfrak{g}^{\rm{ reg}}}$ where ${\mathfrak{g}^{\rm{reg}}}$ is the set of all regular elements in ${\mathfrak{g}}$ . A Hessenberg variety is the b-dimensional affine plane in ${\mathfrak{g}}$ , obtained by translating a Borel subalgebra by a suitable principal nilpotent element. Such a variety was introduced in Kostant (Am J Math 85:327–404, 1963). Defining Hess to be a particular Hessenberg variety, Tarasov has shown that ${{\rm{Hess}}\subset \mathfrak{g}^{\rm{sreg}}}$ . Let R be the set of all regular G-orbits in ${\mathfrak{g}}$ . Thus if ${O\in R}$ , then O is a symplectic manifold of dimension 2n where n = b ? ?. For any ${O\in R}$ let ${O^{\rm{sreg}} = \mathfrak{g}^{\rm{sreg}} \cap O}$ . One shows that O ^sreg is Zariski open and dense in O so that O ^sreg is again a symplectic manifold of dimension 2n. For any ${O\in R}$ let ${{\rm{Hess}}(O) = {\rm{Hess}}\cap O}$ . One proves that Hess(O) is a Lagrangian submanifold of O ^sreg and that $${\rm{Hess}} = \sqcup_{O\in R}{\rm{Hess}}(O).$$ The main result of this paper is to show that there exists simultaneously over all ${O\in R}$ , an explicit polarization (i.e., a “fibration” by Lagrangian submanifolds) of O ^sreg which makes O ^sreg simulate, in some sense, the cotangent bundle of Hess(O).     

Indecomposable representations with invariant inner product

Consequences of the existence of an invariant (necessarily indefinite) non-degenerate inner product for an indecomposable representation π of a groupG on a space \(\mathfrak{H}\) are studied. If π has an irreducible subrepresentation π₁ on a subspace \(\mathfrak{H}_1 \) , it is shown that there exists an invariant subspace \(\mathfrak{H}_2 \) of \(\mathfrak{H}\) containing \(\mathfrak{H}_1 \) and satisfying the following conditions: (1) the representation π ₁ ^# =π mod \(\mathfrak{H}_2 \) on \(\mathfrak{H}\) mod \(\mathfrak{H}_2 \) is conjugate to the representation (π₁, \(\mathfrak{H}_1 \) ), (2) \(\mathfrak{H}_1 \) is a null space for the inner product, and (3) the induced inner product on \(\mathfrak{H}_2 \) mod \(\mathfrak{H}_1 \) is non-degenerate and invariant for the representation $$\pi _2 = (\pi _2 |_{\mathfrak{H}_2 } )\bmod \mathfrak{H}_1 ,$$ a special example being the Gupta-Bleuler triplet for the one-particle space of the free classical electromagnetic field with \(\mathfrak{H}_1 \) =space of longitudinal photons and \(\mathfrak{H}_2 \) =the space defined by the subsidiary condition.     

From nonassociativity to solutions of the KP hierarchy

A recently observed relation between ‘weakly nonassociative’ algebras $\mathbb{A}$ (for which the associator ( $\mathbb{A},\mathbb{A}^2 ,\mathbb{A}$ ) vanishes) and the KP hierarchy (with dependent variable in the middle nucleus $\mathbb{A}$ ′ of { $\mathbb{A}$ ) is recalled. For any such algebra there is a nonassociative hierarchy of ODEs, the solutions of which determine solutions of the KP hierarchy. In a special case, and with matrix algebra $\mathbb{A}$ ′, this becomes a matrix Riccati hierarchy which is easily solved. The matrix solution then leads to solutions of the scalar KP hierarchy. We discuss some classes of solutions obtained in this way.     

Unfolding the singularities in superspace

A method is described for unfolding the singularities in superspace, \(\mathcal{G} = \mathfrak{M}/\mathfrak{D}\) , the space of Riemannian geometries of a manifoldM. This unfolded superspace is described by the projection $$\mathcal{G}_{F\left( M \right)} = \frac{{\mathfrak{M} \times F\left( M \right)}}{\mathfrak{D}} \to \frac{\mathfrak{M}}{\mathfrak{D}} = \mathcal{G}$$ whereF(M) is the frame bundle ofM. The unfolded space \(\mathcal{G}_{F\left( M \right)}\) is infinite-dimensional manifold without singularities. Moreover, as expected, the unfolding of \(\mathcal{G}_{F\left( M \right)}\) at each geometry [g _o] ∈ \(\mathcal{G}\) is parameterized by the isometry groupIg _o (M) of g₀. Our construction is natural, is generally covariant with respect to all coordinate transformations, and gives the necessary information at each geometry to make \(\mathcal{G}\) a manifold. This construction is a canonical and geometric model of a nonrelativistic construction that unfolds superspace by restricting to those coordinate transformations that fix a frame at a point. These particular unfoldings are tied together by an infinite-dimensional fiber bundleE overM, associated with the frame bundleF(M), with standard fiber \(\mathcal{G}_{F\left( M \right)}\) , and with fiber at a point inM being the particular noncanonical unfolding of \(\mathcal{G}\) based at that point. ThusE is the totality of all the particular unfoldings, and so is a grand unfolding of \(\mathcal{G}\) .     

Strange and non-strange (Anti-) baryon production at 200 A GeV/c

Rapidity distributions of net hyperons $\left( {\Lambda - \bar \Lambda } \right)$ are compared to distributions of participant protons $\left( {p - \bar p} \right)$ . Strangeness production (mean multiplicities of produced Λ/Σ⁰ hyperons and $\left\langle {K + \bar K} \right\rangle $ in central nucleusnucleus collisions is shown for different collision systems at different energies. An enhanced production of $\bar \Lambda $ compared to $\bar p$ is observed at 200 GeV per nucleon.     

The Bloch–Okounkov Correlation Functions, a Classical Half-integral Case

Bloch and Okounkov’s correlation function on the infinite wedge space has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, and certain character functions of ${\widehat{ \mathfrak{gl} }_\infty}$ -modules of level one. Recent works have calculated these character functions for higher levels for ${\widehat{ \mathfrak{gl} }_\infty}$ and its Lie subalgebras of classical type. Here we obtain these functions for the subalgebra of type D of half-integral levels and as a byproduct, obtain q-dimension formulas for integral modules of type D at half-integral level.     

On q-Deformed {\mathfrak{gl}_{\ell+1}}-Whittaker Function II

A representation of a specialization of a q-deformed class one lattice ${\mathfrak{gl}_{\ell+1}}$ -Whittaker function in terms of cohomology groups of line bundles on the space ${\mathcal{QM}_d(\mathbb{P}^{\ell})}$ of quasi-maps ${\mathbb{P}^1 \to \mathbb{P}^{\ell}}$ of degree d is proposed. For ? = 1, this provides an interpretation of the non-specialized q-deformed ${\mathfrak{gl}_{2}}$ -Whittaker function in terms of ${\mathcal{QM}_d(\mathbb{P}^1)}$ . In particular the (q-version of the) Mellin-Barnes representation of the ${\mathfrak{gl}_2}$ -Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of Γ-function as a topological genus in semi-infinite geometry. A relation with the Givental-Lee universal solution (J-function) of q-deformed ${\mathfrak{gl}_2}$ -Toda chain is also discussed.     

Equivalent Forms of the Bessis–Moussa–Villani Conjecture

The BMV conjecture for traces, which states that ${\text{Tr}}\;{\text{exp}}\left( {A - \lambda B} \right)$ is the Laplace transform of a positive measure, is shown to be equivalent to two other statements: (i) The polynomial $\lambda \mapsto {\text{Tr}}\;\left( {A + \lambda B} \right)^p$ has only non-negative coefficients for all $A,B \geqslant 0,p \in \mathbb{N}$ and (ii) $\lambda \mapsto {\text{Tr}}\;\left( {A + \lambda B} \right)^{ - p}$ is the Laplace transform of a positive measure for $A,B \geqslant 0,p > 0$ .     

The Percolation Transition in the Zero-Temperature Domany Model

We analyze a deterministic cellular automaton σ ^?=(σ ⁿ :n≥0) corresponding to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice $\mathbb{N}$ . The state space $\mathcal{S}_\mathbb{H} = \left\{ { - 1, + 1} \right\}^\mathbb{H}$ consists of assignments of ?1 or +1 to each site of $\mathbb{H}$ and the initial state $\sigma ^0 = \left\{ {\sigma _{^x }^0 } \right\}_{x \in \mathbb{H}}$ is chosen randomly with P(σ ⁰ _x=+1)=p∈[0,1]. The sites of $\mathbb{H}$ are partitioned in two sets $\mathcal{A}$ and $\mathcal{B}$ so that all the neighbors of a site x in $\mathcal{A}$ belong to $\mathcal{B}$ and vice versa, and the discrete time dynamics is such that the σ ^? _x 's with ${x \in \mathcal{A}}$ (respectively, $\mathcal{B}$ ) are updated simultaneously at odd (resp., even) times, making σ ^? _x agree with the majority of its three neighbors. In ref. 1 it was proved that there is a percolation transition at p=1/2 in the percolation models defined by σ ⁿ , for all times n∈[1,∞]. In this paper, we study the nature of that transition and prove that the critical exponents β, ν, and η of the dependent percolation models defined by σ ⁿ , n∈[1,∞], have the same values as for standard two-dimensional independent site percolation (on the triangular lattice).     

Finite W-Superalgebras and Truncated Super Yangians

Let ${Y_{m|n}^{\ell}}$ be the super Yangian of general linear Lie superalgebra for ${\mathfrak{gl}_{m|n}}$ . Let ${e \in \mathfrak{gl}_{m\ell|n\ell}}$ be a “rectangular” nilpotent element and ${\mathcal{W}_e}$ be the finite W-superalgebra associated to e. We show that ${Y_{m|n}^{\ell}}$ is isomorphic to ${\mathcal{W}_e}$ .     

Dirac multimode ket-bra operators’ \mathfrak{Q}-ordered and \mathfrak{P}-ordered integration theory and general squeezing operator

We develop quantum mechanical Dirac ket-bra operator’s integration theory in $\mathfrak{Q}$ -ordering or $\mathfrak{P}$ -ordering to multimode case, where $\mathfrak{Q}$ -ordering means all Qs are to the left of all Ps and $\mathfrak{P}$ -ordering means all Ps are to the left of all Qs. As their applications, we derive $\mathfrak{Q}$ -ordered and $\mathfrak{P}$ -ordered expansion formulas of multimode exponential operator $e^{ - iP_l \Lambda _{lk} Q_k } $ . Application of the new formula in finding new general squeezing operators is demonstrated. The general exponential operator for coordinate representation transformation $\left| {\left. {\left( {_{q_2 }^{q_1 } } \right)} \right\rangle \to } \right|\left. {\left( {_{CD}^{AB} } \right)\left( {_{q_2 }^{q_1 } } \right)} \right\rangle $ is also derived. In this way, much more correpondence relations between classical coordinate transformations and their quantum mechanical images can be revealed.     

On AB Bond Percolation on the Square Lattice and AB Site Percolation on Its Line Graph

We prove that AB site percolation occurs on the line graph of the square lattice when $p \in (1 - \sqrt {1 - p_c } ,\sqrt {1 - p_c } )$ , where p _c is the critical probability for site percolation in $\mathbb{Z}^2$ . Also, we prove that AB bond percolation does not occur on $\mathbb{Z}^2$ for p = $\frac{1}{2}$ .     

Reichenbachian Common Cause Systems

A partition C_{i i∈ I} of a Boolean algebra $\mathcal{S}$ in a probability measure space $(\mathcal{S},p)$ is called a Reichenbachian common cause system for the correlated pair A,B of events in $\mathcal{S}$ if any two elements in the partition behave like a Reichenbachian common cause and its complement, the cardinality of the index set I is called the size of the common cause system. It is shown that given any correlation in $(\mathcal{S},p)$ , and given any finite size n>2, the probability space $(\mathcal{S},p)$ can be embedded into a larger probability space in such a manner that the larger space contains a Reichenbachian common cause system of size n for the correlation. It also is shown that every totally ordered subset in the partially ordered set of all partitions of $\mathcal{S}$ contains only one Reichenbachian common cause system. Some open problems concerning Reichenbachian common cause systems are formulated.     

Normal and locally normal states

It is shown that if \(\mathfrak{A}\) is an irreducibleC* algebra on a Hilbert space ? andN is the set of normal states of \(\mathfrak{A}\) then the weak and uniform topologies onN coincide and are identical to the weak*- \(\mathfrak{A}\) topology for each \(\mathfrak{A} \supset \mathfrak{L}\mathfrak{C}\) (?). It is further shown that all weak* topologies coincide with the uniform topology on the set of normal states with finite energy or with finite conditional entropy. A number of continuity properties of the spectra of density matrices, the mean energy, and the conditional entropy are also derived. The extension of these results to locally normal states is indicated and it is established that locally normal factor states are characterized by a doubly uniform clustering property.     

Renormalization of parameters of a soft breakdown of supersymmetry in the regime of strong yukawa coupling within a nonminimal supersymmetric standard model

Within a nonminimal supersymmetric (SuSy) model, the renormalization of trilinear coupling constants A _i(t) for scalar fields and of specific combinations $\mathfrak{M}_i^2 (t)$ of the scalar-particle masses is investigated in the regime of strong Yukawa coupling. The dependence of these parameters on their initial values at the Grand Unification scale disappears as solutions to the renormalization-group equations approach infrared quasifixed points with increasing Y _i(0). In the vicinities of quasifixed points for $\tilde \alpha _{GUT} \ll Y_i (0) \ll 1$ , all solutions A _i(t) and $\mathfrak{M}_i^2 (t)$ are concentrated near some straight lines or planes in the space of parameters of a soft breakdown of supersymmetry. This behavior of the solutions in question is explained by a sufficiently slow disappearance of the A _i(0) and $\mathfrak{M}_i^2 (t)$ dependence of the trilinear coupling constants and combinations of the scalar-particle masses. A method is proposed for deriving equations describing the aforementioned straight lines and planes, and the process of their formation is discussed by considering the example of exact and approximate solutions to the renormalization-group equations within a nonminimal supersymmetric standard model.