arXiv subjects

Alexander Alldridge

Publications and source records attributed to Alexander Alldridge.

At least 19 recordsLinked to original sources

arXiv2019

Bulk-boundary correspondence for disordered free-fermion topological phases

Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C$^\ast$-algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator $K$-theory (or $KR$-theory): a bulk class, using Van Daele's picture, along with a boundary class, using Kasparov's Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these $KR$-theory classes to each other. This implies bulk-boundary correspondence, in the presence of disorder, for both the "strong" and the "weak" invariants.

math-ph↗

arXiv2018

Schur $Q$-functions and the Capelli eigenvalue problem for the Lie superalgebra $\mathfrak q(n)$

Let $\mathfrak l:= \mathfrak q(n)\times\mathfrak q(n)$, where $\mathfrak q(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of type $Q(n)$ has a left and right action of $\mathfrak q(n)$, and hence is equipped with a canonical $\mathfrak l$-module structure. We consider a distinguished basis $\{D_λ\}$ of the algebra of $\mathfrak l$-invariant super-polynomial differential operators on $V$, which is indexed by strict partitions of length at most $n$. We show that the spectrum of the operator $D_λ$, when it acts on the algebra $\mathscr P(V)$ of super-polynomials on $V$, is given by the factorial Schur $Q$-function of Okounkov and Ivanov. This constitutes a refinement and a new proof of a result of Nazarov, who computed the top-degree homogeneous part of the Harish-Chandra image of $D_λ$. As a further application, we show that the radial projections of the spherical super-polynomials corresponding to the diagonal symmetric pair $(\mathfrak l,\mathfrak m)$, where $\mathfrak m:=\mathfrak q(n)$, of irreducible $\mathfrak l$-submodules of $\mathscr P(V)$ are the classical Schur $Q$-functions.

math.RT↗

arXiv2016

Non-Euclidean Fourier inversion on super-hyperbolic space

For the super-hyperbolic space in any dimension, we introduce the non-Euclidean Helgason--Fourier transform. We prove an inversion formula exhibiting residue contributions at the poles of the Harish-Chandra c-function, signalling discrete parts in the spectrum. The proof is based on a detailed study of the spherical superfunctions, using recursion relations and localization techniques to normalize them precisely, careful estimates of their derivatives, and a rigorous analysis of the boundary terms appearing in the polar coordinate expression of the invariant integral

math.RT↗

arXiv2016

Superorbits

We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy (or stabiliser) supergroups and orbits through general points. In this setting, we show that the coadjoint orbits always admit a (relative) supersymplectic structure of Kirillov-Kostant-Souriau type. Applying a family version of Kirillov's orbit method, we decompose the regular representation of an odd Abelian supergroup into an odd direct integral of characters and construct universal families of representations, parametrised by a supermanifold, for two different super variants of the Heisenberg group.

math.DG↗

arXiv2016

Fréchet globalisations of Harish-Chandra supermodules

For any Lie supergroup whose underlying Lie group is reductive, we prove an extension of the Casselman-Wallach globalisation theorem: There is an equivalence between the category of Harish-Chandra modules and the category of SF-representations (smooth Fréchet representations of moderate growth) whose module of finite vectors is Harish-Chandra. As an application, we extend to Lie supergroups a general general form of the Gel'fand-Kazhdan criterion due to Sun-Zhu.

math.RT↗

arXiv2014

Asymptotics of spherical superfunctions on rank one Riemannian symmetric superspaces

We compute the Harish-Chandra $c$-function for a generic class of rank-one purely non-compact Riemannian symmetric superspaces $X=G/K$ in terms of Euler $Γ$ functions, proving that it is meromorphic. Compared to the even case, the poles of the $c$-function are shifted into the right half-space. We derive the full asymptotic Harish-Chandra series expansion of the spherical superfunctions on $X$. In the case where the multiplicity of the simple root is an even negative number, they have a closed expression as Jacobi polynomials for an unusual choice of parameters.

math.RT↗

arXiv2014

Spherical representations of Lie supergroups

The classical Cartan-Helgason theorem characterises finite-dimensional spherical representations of reductive Lie groups in terms of their highest weights. We generalise the theorem to the case of a reductive symmetric supergroup pair $(G,K)$ of even type. Along the way, we compute the Harish-Chandra $c$-function of the symmetric superspace $G/K$. By way of an application, we show that all spherical representations are self-dual in type AIII|AIII.

math.RT↗

arXiv2014

Singular superspaces

We introduce a wide category of superspaces, called locally finitely generated, which properly includes supermanifolds but enjoys much stronger permanence properties, as are prompted by applications. Namely, it is closed under taking finite fibre products (i.e. is finitely complete) and thickenings by spectra of Weil superalgebras. Nevertheless, in this category, morphisms with values in a supermanifold are still given in terms of coordinates. This framework gives a natural notion of relative supermanifolds over a locally finitely generated base. Moreover, the existence of inner homs, whose source is the spectrum of a Weil superalgebra, is established; they are generalisations of the Weil functors defined for smooth manifolds.

math.DG↗

arXiv2014

Superbosonisation via Riesz superdistributions

The superbosonisation identity of Littelmann-Sommers-Zirnbauer is a new tool to study universality of random matrix ensembles via supersymmetry, which is applicable to non-Gaussian invariant distributions. In this note, we identify the right-hand side with a super-generalisation of the Riesz distribution. Using the Laplace transformation and tools from harmonic superanalysis, we give a short and conceptual new proof of the formula.

math.RT↗

arXiv2013

Superbosonisation, Riesz superdistributions, and highest weight modules (extended version)

This is the extended version of a survey prepared for publication in the Springer INdAM series. Superbosonisation, introduced by Littelmann-Sommers-Zirnbauer, is a generalisation of bosonisation, with applications in Random Matrix Theory and Condensed Matter Physics. We link the superbosonisation identity to Representation Theory and Harmonic Analysis and explain two new proofs, one via the Laplace transform and one based on a multiplicity freeness statement.

math-ph↗

arXiv2011

A convenient category of supermanifolds

With a view towards applications in the theory of infinite-dimensional representations of finite-dimensional Lie supergroups, we introduce a new category of supermanifolds. In this category, supermanifolds of `maps' and `fields' (fibre bundle sections) exist. In particular, loop supergroups can be realised globally in this framework. It also provides a convenient setting for induced representations of supergroups, allowing for a version of Frobenius reciprocity. Finally, convolution algebras of finite-dimensional Lie supergroups are introduced and applied to a prove a supergroup Dixmier-Malliavin Theorem: The space of smooth vectors of a continuous representation of a supergroup pair equals the Garding space given by the convolution with compactly supported smooth supergroup densities.

math.DG↗

arXiv2011

Integration on Non-Compact Supermanifolds

We investigate the Berezin integral of non-compactly supported quantities. In the framework of supermanifolds with corners, we give a general, explicit and coordinate-free repesentation of the boundary terms introduced by an arbitrary change of variables. As a corollary, a general Stokes's theorem is derived - here, the boundary integral contains transversal derivatives of arbitrarily high order.

math.DG↗

arXiv2011

The Harish-Chandra isomorphism for reductive symmetric superpairs

We consider symmetric pairs of Lie superalgebras which are strongly reductive and of even type, and introduce a graded Harish-Chandra homomorphism. We prove that its image is a certain explicit filtered subalgebra of the Weyl invariants on a Cartan subspace whose associated graded is the image of Chevalley's restriction map on symmetric invariants. This generalises results of Harish-Chandra and V. Kac, M. Gorelik.

math.RT↗

arXiv2011

Harmonic analysis on Heisenberg--Clifford Lie supergroups

We define a Fourier transform and a convolution product for functions and distributions on Heisenberg--Clifford Lie supergroups. The Fourier transform exchanges the convolution and a pointwise product, and is an intertwining operator for the left regular representation. We generalize various classical theorems, including the Paley--Wiener--Schwartz theorem, and define a convolution Banach algebra.

math.RT↗

arXiv2011

Convex polytopes and the index of Wiener-Hopf operators

We study the C$^*$-algebra of Wiener-Hopf operators $A_Ω$ on a cone $Ω$ with polyhedral base $P$. As is known, a sequence of symbol maps may be defined, and their kernels give a filtration by ideals of $A_Ω$, with liminary subquotients. One may define $K$-group valued 'index maps' between the subquotients. These form the $E^1$ term of the Atiyah-Hirzebruch type spectral sequence induced by the filtration. We show that this $E^1$ term may, as a complex, be identified with the cellular complex of $P$, considered as CW complex by taking convex faces as cells. It follows that $A_Ω$ is $KK$-contractible, and that $A_Ω/\mathbb K$ and $S$ are $KK$-equivalent. Moreover, the isomorphism class of $A_Ω$ is a complete invariant for the combinatorial type of $P$.

math.OA↗

arXiv2010

Infinite-dimensional supermanifolds over arbitrary base fields

In his recent investigation of a super Teichmüller space, Sachse (2007), based on work of Molotkov (1984), has proposed a theory of Banach supermanifolds using the `functor of points' approach of Bernstein and Schwarz. We prove that the the category of Berezin-Kostant-Leites supermanifolds is equivalent to the category of finite-dimensional Molotkov-Sachse supermanifolds. Simultaneously, using the differential calculus of Bertram-Glöckner-Neeb (2004), we extend Molotkov-Sachse's approach to supermanifolds modeled on Hausdorff topological super-vector spaces over an arbitrary non-discrete Hausdorff topological base field of characteristic zero. We also extend to locally k-omega base fields the `DeWitt' supermanifolds considered by Tuynman in his monograph (2004), and prove that this leads to a category which is isomorphic to the full subcategory of Molokov-Sachse supermanifolds modeled on locally k-omega spaces.

math.DG↗

arXiv2009

Invariant Berezin integration on homogeneous supermanifolds

Let G be a Lie supergroup and H a closed subsupergroup. We study the unimodularity of the homogeneous supermanifold G/H, i.e. the existence of G-invariant sections of its Berezinian line bundle. To that end, we express this line bundle as a G-equivariant associated bundle of the principal H-bundle G over G/H. We also study the fibre integration of Berezinians on oriented fibre bundles. As an application, we prove a formula of `Fubini' type: the invariant integral over G can be expressed (up to sign) by a succesive invariant integration over H and G/H. Moreover, we derive analogues of integral formulae for the transformation under local isomorphisms of unimodular homogeneous superspaces G/H and S/T, and under the products of subsupergroups of Lie supergroups. The classical counterparts of these formulae have numerous applications in harmonic analysis.

math.DG↗

arXiv2009

An index theorem for Wiener--Hopf operators

We study multivariate generalisations of the classical Wiener--Hopf algebra, which is the C$^*$-algebra generated by the Wiener--Hopf operators, given by the convolutions restricted to convex cones. By the work of Muhly and Renault, this C$^*$-algebra is known to be isomorphic to the reduced C$^*$-algebra of a certain restricted action groupoid. In a previous paper, we have determined a composition series of this C$^*$-algebra, and compute the $K$-theory homomorphisms induced by the `symbol' maps given by the subquotients of the composition series in terms of the analytical index of a continuous family of Fredholm operators. In this paper, we obtain a topological expression for these index maps in terms of geometric-topological data naturally associated to the underlying convex cone. The resulting index formula is expressed in the framework of Kasparov's bivariant $KK$-theory. Our proof relies heavily on groupoid methods.

math.OA↗