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).     

Semi-Classical Twists for {\mathfrak{sl}}_{3} and {\mathfrak{sl}}_{4} Boundary r-Matrices of Cremmer–Gervais Type

We obtain explicit formulas for the semi-classical twists deforming the coalgebraic structure of $U({\mathfrak{sl}}_{3})$ and $U({\mathfrak{sl}}_{4})$ . In rank 2 and 3 the corresponding universal R-matrices quantize the boundary r-matrices of Cremmer–Gervais type defining Lie Frobenius structures on the maximal parabolic subalgebras in ${\mathfrak{sl}}_{n}$ .     

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.     

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 ${\mathcal{W}_e}$ .     

The Weil Algebra of a Hopf Algebra I: A Noncommutative Framework

We generalize the notion, introduced by Henri Cartan, of an operation of a Lie algebra ${\mathfrak{g}}$ in a graded differential algebra Ω. We define the notion of an operation of a Hopf algebra ${\mathcal{H}}$ in a graded differential algebra Ω which is referred to as a ${\mathcal{H}}$ -operation. We then generalize for such an operation the notion of algebraic connection. Finally we discuss the corresponding noncommutative version of the Weil algebra: The Weil algebra ${W(\mathcal{H})}$ of the Hopf algebra ${\mathcal{H}}$ is the universal initial object of the category of ${\mathcal{H}}$ -operations with connections.     

Self-Avoiding Walk is Sub-Ballistic

We prove that self-avoiding walk on ${\mathbb{Z}^d}$ is sub-ballistic in any dimension d ≥ 2. That is, writing ${\| u \|}$ for the Euclidean norm of ${u \in \mathbb{Z}^d}$ , and ${\mathsf{P_{SAW}}_n}$ for the uniform measure on self-avoiding walks ${\gamma : \{0, \ldots, n\} \to \mathbb{Z}^d}$ for which γ ₀ = 0, we show that, for each v > 0, there exists ${\varepsilon > 0}$ such that, for each ${n \in \mathbb{N}, \mathsf{P_{SAW}}_n \big( {\rm max}\big\{\| \gamma_k \| : 0 \leq k \leq n\big\} \geq vn \big) \leq e^{-\varepsilon n}}$ .     

Quantum Sphere {\mathbb{S}^4} as a Non-Levi Conjugacy Class

We construct a ${U_\hbar(\mathfrak{sp}(4))}$ -equivariant quantization of the four-dimensional complex sphere ${\mathbb{S}^4}$ regarded as a conjugacy class, Sp(4)/Sp(2) ×?Sp(2), of a simple complex group with non-Levi isotropy subgroup, through an operator realization of the quantum polynomial algebra ${\mathbb{C}_\hbar[\mathbb{S}^4]}$ on a highest weight module of ${U_\hbar(\mathfrak{sp}(4))}$ .     

Lagrangian Subalgebras and Quasi-trigonometric r-Matrices

It was proved by Montaner and Zelmanov that up to classical twisting Lie bialgebra structures on ${\mathfrak{g}[u]}$ fall into four classes. Here ${\mathfrak{g}}$ is a simple complex finite-dimensional Lie algebra. It turns out that classical twists within one of these four classes are in a one-to-one correspondence with the so-called quasi-trigonometric solutions of the classical Yang–Baxter equation. In this paper we give a complete list of the quasi-trigonometric solutions in terms of sub-diagrams of the certain Dynkin diagrams related to ${\mathfrak{g}}$ . We also explain how to quantize the corresponding Lie bialgebra structures.     

Multiplicity-free Quantum 6j-Symbols for {U_q(\mathfrak{sl}_N)}

We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6j-symbols for ${U_q(\mathfrak{sl}_N)}$ . The expression is a natural generalization of the quantum 6j-symbols for ${U_q(\mathfrak{sl}_2)}$ obtained by Kirillov and Reshetikhin. Our conjectured form enables computation of colored HOMFLY polynomials for various knots and links carrying arbitrary symmetric representations.     

On Dirichlet-to-Neumann Maps, Nonlocal Interactions, and Some Applications to Fredholm Determinants

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrödinger operators describing nonlocal interactions in ${L^2(\Omega; d^n x)}$ , where ${\Omega \subset \mathbb{R}^n}$ , ${n\in\mathbb{N}}$ , ${n\geq 2}$ , are open sets with a compact, nonempty boundary ${\partial\Omega}$ satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of Fredholm perturbation determinants associated with operators in ${L^2(\Omega; d^{n} x)}$ to Fredholm perturbation determinants associated with operators in ${L^2(\partial\Omega; d^{n-1} \sigma)}$ , ${n\in\mathbb{N}}$ , ${n\geq 2}$ . This leads to an extension of a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with a Schrödinger operator on the half-line ${(0,\infty)}$ , in the case of local interactions, to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation.     

On Representations of Quantum Conjugacy Classes of GL(n)

Let O be a closed Poisson conjugacy class of the complex algebraic Poisson group GL(n) relative to the Drinfeld-Jimbo factorizable classical r-matrix. Denote by T the maximal torus of diagonal matrices in GL(n). With every ${a \in O \cap T}$ we associate a highest weight module M _a over the quantum group ${U_q \bigl(\mathfrak{g} \mathfrak{l}(n)\bigr)}$ and an equivariant quantization ${\mathbb{C}_{\hbar,a}[O]}$ of the polynomial ring ${\mathbb{C}[O]}$ realized by operators on M _a . All quantizations ${\mathbb{C}_{\hbar,a}[O]}$ are isomorphic and can be regarded as different exact representations of the same algebra, ${\mathbb{C}_{\hbar}[O]}$ . Similar results are obtained for semisimple adjoint orbits in ${\mathfrak{g} \mathfrak{l}(n)}$ equipped with the canonical GL(n)-invariant Poisson structure.     

Joint System Quantum Descriptions Arising from Local Quantumness

We study the entropy flux in the stationary state of a finite one-dimensional sample ${\mathcal{S}}$ connected at its left and right ends to two infinitely extended reservoirs ${\mathcal{R}_{l/r}}$ at distinct (inverse) temperatures ${\beta_{l/r}}$ and chemical potentials ${\mu_{l/r}}$ . The sample is a free lattice Fermi gas confined to a box [0, L] with energy operator ${h_{\mathcal{S}, L}= - \Delta + v}$ . The Landauer-Büttiker formula expresses the steady state entropy flux in the coupled system ${\mathcal{R}_l + \mathcal{S} + \mathcal{R}_r}$ in terms of scattering data. We study the behaviour of this steady state entropy flux in the limit ${L \to \infty}$ and relate persistence of transport to norm bounds on the transfer matrices of the limiting half-line Schrödinger operator ${h_\mathcal{S}}$ .     

On Endomorphisms of Quantum Tensor Space

We give a presentation of the endomorphism algebra ${\rm End}_{\mathcal {U}_{q}(\mathfrak {sl}_{2})}(V^{\otimes r})$ , where V is the three-dimensional irreducible module for quantum ${\mathfrak {sl}_2}$ over the function field ${\mathbb {C}(q^{\frac{1}{2}})}$ . This will be as a quotient of the Birman–Wenzl–Murakami algebra BMW _r (q) : =  BMW _r (q ^?4, q ² ? q ^?2) by an ideal generated by a single idempotent Φ _q . Our presentation is in analogy with the case where V is replaced by the two-dimensional irreducible ${\mathcal {U}_q(\mathfrak {sl}_{2})}$ -module, the BMW algebra is replaced by the Hecke algebra H _r (q) of type A _r-1, Φ _q is replaced by the quantum alternator in H ₃(q), and the endomorphism algebra is the classical realisation of the Temperley–Lieb algebra on tensor space. In particular, we show that all relations among the endomorphisms defined by the R-matrices on ${V^{\otimes r}}$ are consequences of relations among the three R-matrices acting on ${V^{\otimes 4}}$ . The proof makes extensive use of the theory of cellular algebras. Potential applications include the decomposition of tensor powers when q is a root of unity.     

Coset Constructions of Logarithmic (1, p) Models

One of the best understood families of logarithmic onformal field theories consists of the (1, p) models (p =  2, 3, . . .) of central charge c _{1, p} =1 ? 6(p ? 1)²/p. This family includes the theories corresponding to the singlet algebras ${\mathcal{M}(p)}$ and the triplet algebras ${\mathcal{W}(p)}$ , as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realised through a coset construction. The ${W^{(2)}_n}$ algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of ${\widehat{\mathfrak{sl}}(n)_k}$ , generalising the Bershadsky–Polyakov algebra ${W^{(2)}_3}$ . Inspired by work of Adamovi? for p = 3, vertex algebras ${\mathcal{B}_p}$ are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p≤5, the algebra ${\mathcal{B}_p}$ is a quotient of ${W^{(2)}_{p-1}}$ at level ?(p ? 1)²/p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p> 5 as well. The triplet algebra ${\mathcal{W}(p)}$ is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra ${\mathcal{M}(p)}$ is similarly realised inside ${\mathcal{B}_p}$ . As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p =  2 and 3.     

A Strong Central Limit Theorem for a Class of Random Surfaces

This paper is concerned with d = 2 dimensional lattice field models with action ${V(\nabla\phi(\cdot))}$ , where ${V : \mathbf{R}^d \rightarrow \mathbf{R}}$ is a uniformly convex function. The fluctuations of the variable ${\phi(0) - \phi(x)}$ are studied for large |x| via the generating function given by ${g(x, \mu) = \ln \langle e^{\mu(\phi(0) - \phi(x))}\rangle_{A}}$ . In two dimensions ${g'' (x, \mu) = \partial^2g(x, \mu)/\partial\mu^2}$ is proportional to ${\ln\vert x\vert}$ . The main result of this paper is a bound on ${g''' (x, \mu) = \partial^3 g(x, \mu)/\partial \mu^3}$ which is uniform in ${\vert x \vert}$ for a class of convex V. The proof uses integration by parts following Helffer–Sjöstrand and Witten, and relies on estimates of singular integral operators on weighted Hilbert spaces.     

Haag Duality and the Distal Split Property for Cones in the Toric Code

We prove that Haag duality holds for cones in the toric code model. That is, for a cone ??, the algebra ${\mathcal{R}_{\Lambda}}$ of observables localized in ?? and the algebra ${\mathcal{R}_{\Lambda^c}}$ of observables localized in the complement ?? ^c generate each other??s commutant as von Neumann algebras. Moreover, we show that the distal split property holds: if ${\Lambda_1 \subset \Lambda_2}$ are two cones whose boundaries are well separated, there is a Type I factor ${\mathcal{N}}$ such that ${\mathcal{R}_{\Lambda_1} \subset \mathcal{N} \subset \mathcal{R}_{\Lambda_2}}$ . We demonstrate this by explicitly constructing ${\mathcal{N}}$ .     

Canonical Basis for Quantum {\mathfrak{osp}(1|2)}

We introduce a modified quantum enveloping algebra as well as a (modified) covering quantum algebra for the ortho-symplectic Lie superalgebra ${\mathfrak{osp}(1|2)}$ . Then we formulate and compute the corresponding canonical bases, and relate them to the counterpart for ${\mathfrak{sl}(2)}$ . This provides a first example of canonical basis for quantum superalgebras.     

Courant–Dorfman Algebras and their Cohomology

We introduce a new type of algebra, the Courant–Dorfman algebra. These are to Courant algebroids what Lie–Rinehart algebras are to Lie algebroids, or Poisson algebras to Poisson manifolds. We work with arbitrary rings and modules, without any regularity, finiteness or non-degeneracy assumptions. To each Courant–Dorfman algebra ${(\mathcal{R}, \mathcal{E})}$ we associate a differential graded algebra ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ in a functorial way by means of explicit formulas. We describe two canonical filtrations on ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ , and derive an analogue of the Cartan relations for derivations of ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ ; we classify central extensions of ${\mathcal{E}}$ in terms of ${H^2(\mathcal{E}, \mathcal{R})}$ and study the canonical cocycle ${\Theta \in \mathcal{C}^3(\mathcal{E}, \mathcal{R})}$ whose class ${[\Theta]}$ obstructs re-scalings of the Courant–Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ ; for Courant–Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ is isomorphic to the one constructed in Roytenberg (On the structure of graded symplectic supermanifolds and Courant algebroids. American Mathematical Society, Providence, 2002) using graded manifolds.     

Equivariant GW Theory of Stacky Curves

We extend Okounkov and Pandharipande’s work on the equivariant Gromov–Witten theory of ${\mathbb{P}^1}$ to a class of stacky curves ${\mathcal{X}}$ . Our main result uses virtual localization and the orbifold ELSV formula to express the tau function ${\tau_\mathcal{X}}$ as a vacuum expectation on a Fock space. As corollaries, we prove the decomposition conjecture for these ${\mathcal{X}}$ , and prove that ${\tau_\mathcal{X}}$ satisfies a version of the 2-Toda hierarchy. Coupled with degeneration techniques, the result should lead to treatment of general orbifold curves.