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Andrew R. Linshaw

Publications and source records attributed to Andrew R. Linshaw.

54 records · Page 3Linked to original sources
arXiv2015

Arc spaces and the vertex algebra commutant problem

Given a vertex algebra $\mathcal{V}$ and a subalgebra $\mathcal{A}\subset \mathcal{V}$, the commutant $\text{Com}(\mathcal{A},\mathcal{V})$ is the subalgebra of $\mathcal{V}$ which commutes with all elements of $\mathcal{A}$. This construction is analogous to the ordinary commutant in the theory of associative algebras, and is important in physics in the construction of coset conformal field theories. When $\mathcal{A}$ is an affine vertex algebra, $\text{Com}(\mathcal{A},\mathcal{V})$ is closely related to rings of invariant functions on arc spaces. We find strong finite generating sets for a family of examples where $\mathcal{A}$ is affine and $\mathcal{V}$ is a $βγ$-system, $bc$-system, or $bcβγ$-system.

math.RT↗
arXiv2015

Orbifolds of symplectic fermion algebras

We present a systematic study of the orbifolds of the rank $n$ symplectic fermion algebra $\mathcal{A}(n)$, which has full automorphism group $Sp(2n)$. First, we show that $\mathcal{A}(n)^{Sp(2n)}$ and $\mathcal{A}(n)^{GL(n)}$ are $\mathcal{W}$-algebras of type $\mathcal{W}(2,4,\dots, 2n)$ and $\mathcal{W}(2,3,\dots, 2n+1)$, respectively. Using these results, we find minimal strong finite generating sets for $\mathcal{A}(mn)^{Sp(2n)}$ and $\mathcal{A}(mn)^{GL(n)}$ for all $m,n\geq 1$. We compute the characters of the irreducible representations of $\mathcal{A}(mn)^{Sp(2n)\times SO(m)}$ and $\mathcal{A}(mn)^{GL(n)\times GL(m)}$ appearing inside $\mathcal{A}(mn)$, and we express these characters using partial theta functions. Finally, we give a complete solution to the Hilbert problem for $\mathcal{A}(n)$; we show that for any reductive group $G$ of automorphisms, $\mathcal{A}(n)^G$ is strongly finitely generated.

math.RT↗
arXiv2013

A commutant realization of Odake's algebra

The bcβγ-system W of rank 3 has an action of the affine vertex algebra V_0(sl_2), and the commutant vertex algebra C =Com(V_0(sl_2), W) contains copies of V_{-3/2}(sl_2) and Odake's algebra O. Odake's algebra is an extension of the N=2 superconformal algebra with c=9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V_{-3/2}(sl_2) and O form a Howe pair (i.e., a pair of mutual commutants) inside C. More generally, any finite-dimensional representation of a Lie algebra g gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of sl_2.

math.QA↗
arXiv2013

The super W_{1+\infty} algebra with integral central charge

The Lie superalgebra SD of regular differential operators on the super circle has a universal central extension \hat{SD}. For each c\in C, the vacuum module M_c(\hat{SD}) of central charge c admits a vertex superalgebra structure, and M_c(\hat{SD}) \cong M_{-c}(\hat{SD}). The irreducible quotient V_c(\hat{SD}) of the vacuum module is known as the super W_{1+\infty} algebra. We show that for each integer n>0, V_n(\hat{SD}) has a minimal strong generating set consisting of 4n fields, and we identify it with a W-algebra associated to the purely odd simple root system of gl(n|n). Finally, we realize V_n(\hat{SD}) as the limit of a family of commutant vertex algebras that generically have the same graded character and possess a minimal strong generating set of the same cardinality.

math.QA↗
arXiv2012

Invariant subalgebras of affine vertex algebras

Given a finite-dimensional complex Lie algebra g equipped with a nondegenerate, symmetric, invariant bilinear form B, let V_k(g,B) denote the universal affine vertex algebra associated to g and B at level k. For any reductive group G of automorphisms of V_k(g,B), we show that the invariant subalgebra V_k(g,B)^G is strongly finitely generated for generic values of k. This implies the existence of a new family of deformable W-algebras W(g,B,G)_k which exist for all but finitely many values of k.

math.QA↗
arXiv2012

A commutant realization of W^(2)_n at critical level

For n\geq 2, there is a free field realization of the affine vertex superalgebra A associated to psl(n|n) at critical level inside the bcβγsystem W of rank n^2. We show that the commutant C=Com(A,W) is purely bosonic and is freely generated by n+1 fields. We identify the Zhu algebra of C with the ring of invariant differential operators on the space of n\times n matrices under SL_n \times SL_n, and we classify the irreducible, admissible C-modules with finite dimensional graded pieces. For n\leq 4, C is isomorphic to the W_n^{(2)}-algebra at critical level, and we conjecture that this holds for all n.

math.QA↗
arXiv2011

Fermionic Coset, Critical Level W^(2)_4-Algebra and Higher Spins

The fermionic coset is a limit of the pure spinor formulation of the AdS5xS5 sigma model as well as a limit of a nonlinear topological A-model, introduced by Berkovits. We study the latter, especially its symmetries, and map them to higher spin algebras. We show the following. The linear A-model possesses affine $\AKMSA{pgl}{4}{4}_0$ symmetry at critical level and its $\AKMSA{psl}{4}{4}_0$ current-current perturbation is the nonlinear model. We find that the perturbation preserves $\mathcal{W}^{(2)}_4$-algebra symmetry at critical level. There is a topological algebra associated to $\AKMSA{pgl}{4}{4}_0$ with the properties that the perturbation is BRST-exact. Further, the BRST-cohomology contains world-sheet supersymmetric symplectic fermions and the non-trivial generators of the $\mathcal{W}^{(2)}_4$-algebra. The Zhu functor maps the linear model to a higher spin theory. We analyze its $\SLSA{psl}{4}{4}$ action and find finite dimensional short multiplets.

hep-th↗
arXiv2011

Invariant theory and the Heisenberg vertex algebra

The invariant subalgebra H^+ of the Heisenberg vertex algebra H under its automorphism group Z/2Z was shown by Dong-Nagatomo to be a W-algebra of type W(2,4). Similarly, the rank n Heisenberg vertex algebra H(n) has the orthogonal group O(n) as its automorphism group, and we conjecture that H(n)^{O(n)} is a W-algebra of type W(2,4,6,...,n^2+3n). We prove our conjecture for n=2 and n=3, and we show that this conjecture implies that H(n)^G is strongly finitely generated for any reductive group G\subset O(n).

math.RT↗
arXiv2011

Invariant theory and the W_{1+\infty} algebra with negative integral central charge

The vertex algebra W_{1+\infty,c} with central charge c may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer n\geq 1, it was conjectured in the physics literature that W_{1+\infty,-n} should have a minimal strong generating set consisting of n^2+2n elements. Using a free field realization of W_{1+\infty,-n} due to Kac-Radul, together with a deformed version of Weyl's first and second fundamental theorems of invariant theory for the standard representation of GL_n, we prove this conjecture. A consequence is that the irreducible, highest-weight representations of W_{1+\infty,-n} are parametrized by a closed subvariety of C^{n^2+2n}.

math.RT↗
arXiv2011

Chiral equivariant cohomology of a point: a first look

The chiral equivariant cohomology contains and generalizes the classical equivariant cohomology of a manifold M with an action of a compact Lie group G. For any simple G, there exist compact manifolds with the same classical equivariant cohomology, which can be distinguished by this invariant. When M is a point, this cohomology is an interesting conformal vertex algebra whose structure is still mysterious. In this paper, we scratch the surface of this object in the case G=SU(2).

math.QA↗
arXiv2010

A Hilbert theorem for vertex algebras

Given a simple vertex algebra A and a reductive group G of automorphisms of A, the invariant subalgebra A^G is strongly finitely generated in most examples where its structure is known. This phenomenon is subtle, and is generally not true of the classical limit of A^G, which often requires infinitely many generators and infinitely many relations to describe. Using tools from classical invariant theory, together with recent results on the structure of the W_{1+\infty} algebra, we establish the strong finite generation of a large family of invariant subalgebras of βγ-systems, bc-systems, and bcβγ-systems.

math.RT↗
arXiv2010

Chiral Equivariant Cohomology III

This is the third of a series of papers on a new equivariant cohomology that takes values in a vertex algebra, and contains and generalizes the classical equivariant cohomology of a manifold with a Lie group action a la H. Cartan. In this paper, we compute this cohomology for spheres and show that for any simple connected group G, there is a sphere with infinitely many actions of G which have distinct chiral equivariant cohomology, but identical classical equivariant cohomology. Unlike the classical case, the description of the chiral equivariant cohomology of spheres requires a substantial amount of new structural theory, which we fully develop in this paper. This includes a quasi-conformal structure, equivariant homotopy invariance, and the values of this cohomology on homogeneous spaces. These results rely on crucial features of the underlying vertex algebra valued complex that have no classical analogues.

math.DG↗
arXiv2008

Invariant chiral differential operators and the W_3 algebra

Attached to a vector space V is a vertex algebra S(V) known as the beta-gamma system or algebra of chiral differential operators on V. It is analogous to the Weyl algebra D(V), and is related to D(V) via the Zhu functor. If G is a connected Lie group with Lie algebra g, and V is a linear G-representation, there is an action of the corresponding affine algebra on S(V). The invariant space S(V)^{g[t]} is a commutant subalgebra of S(V), and plays the role of the classical invariant ring D(V)^G. When G is an abelian Lie group acting diagonally on V, we find a finite set of generators for S(V)^{g[t]}, and show that S(V)^{g[t]} is a simple vertex algebra and a member of a Howe pair. The Zamolodchikov W_3 algebra with c=-2 plays a fundamental role in the structure of S(V)^{g[t]}.

math.RT↗
arXiv2008

Chiral Equivariant Cohomology II

This is the second in a series of papers on a new equivariant cohomology that takes values in a vertex algebra. In an earlier paper, the first two authors gave a construction of the cohomology functor on the category of O(sg) algebras. The new cohomology theory can be viewed as a kind of "chiralization'' of the classical equivariant cohomology, the latter being defined on the category of G^* algebras a la H. Cartan. In this paper, we further develop the chiral theory by first extending it to allow a much larger class of algebras which we call sg[t] algebras. In the geometrical setting, our principal example of an O(sg) algebra is the chiral de Rham complex Q(M) of a G manifold M. There is an interesting subalgebra of Q(M) which does not admit a full O(sg) algebra structure but retains the structure of an sg[t] algebra, enough for us to define its chiral equivariant cohomology. The latter then turns out to have many surprising features that allows us to delineate a number of interesting geometric aspects of the G manifold M, sometimes in ways that are quite different from the classical theory.

math.DG↗
arXiv2007

Howe Pairs in the Theory of Vertex Algebras

For any vertex algebra V and any subalgebra A of V, there is a new subalgebra of V known as the commutant of A in V. This construction was introduced by Frenkel-Zhu, and is a generalization of an earlier construction due to Kac-Peterson and Goddard-Kent-Olive known as the coset construction. In this paper, we interpret the commutant as a vertex algebra notion of invariant theory. We present an approach to describing commutant algebras in an appropriate category of vertex algebras by reducing the problem to a question in commutative algebra. We give an interesting example of a Howe pair (ie, a pair of mutual commutants) in the vertex algebra setting.

math.RT↗
arXiv2006

Chiral Equivariant Cohomology I

We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, with the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai-Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern-Weil map. We give interesting cohomology classes in the new theory that have no classical analogues.

math.DG↗
arXiv2006

The Cohomology Algebra of the Semi-infinite Weil Complex

In 1993, Lian-Zuckerman constructed two cohomology operations on the BRST complex of a conformal vertex algebra with central charge 26. They gave explicit generators and relations for the cohomology algebra equipped with these operations in the case of the c=1 model. In this paper, we describe another such example, namely, the semi-infinite Weil complex of the Virasoro algebra. The semi-infinite Weil complex of a tame Z-graded Lie algebra was defined in 1991 by Feigin-Frenkel, and they computed the linear structure of its cohomology in the case of the Virasoro algebra. We build on this result by giving an explicit generator for each non-zero cohomology class, and describing all algebraic relations in the sense of Lian-Zuckerman, among these generators.

math.QA↗