Trace Algebra Research Articles

From Hagedorn to Lee-Yang: partition functions of mathcal{N} = 4 SYM theory at finite N

We study the thermodynamics of the maximally supersymmetric Yang-Mills theory with gauge group U(N ) on ℝ × S3, dual to type IIB superstring theory on AdS5× S5. While both theories are well-known to exhibit Hagedorn behavior at infinite N , we find evidence that this is replaced by Lee-Yang behavior at large but finite N : the zeros of the partition function condense into two arcs in the complex temperature plane that pinch the real axis at the temperature of the confinement-deconfinement transition. Concretely, we demonstrate this for the free theory via exact calculations of the (unrefined and refined) partition functions at N ≤ 7 for the mathfrak{su} (2) sector containing two complex scalars, as well as at N ≤ 5 for the mathfrak{su} (2|3) sector containing 3 complex scalars and 2 fermions. In order to obtain these explicit results, we use a Molien-Weyl formula for arbitrary field content, utilizing the equivalence of the partition function with what is known to mathematicians as the Poincaré series of trace algebras of generic matrices. Via this Molien-Weyl formula, we also generate exact results for larger sectors.

The elliptic Hall algebra and the deformed Khovanov Heisenberg category

We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined by Licata and Savage. We also show that as an algebra, it is isomorphic to "half" of a central extension of the elliptic Hall algebra of Burban and Schiffmann, specialized at $\sigma = \bar\sigma^{-1} = q$. A key step in the proof may be of independent interest: we show that the sum (over $n$) of the Hochschild homologies of the positive affine Hecke algebras $\mathrm{AH}_n^+$ is again an algebra, and that this algebra injects into both the elliptic Hall algebra and the trace of the $q$-Heisenberg category. Finally, we show that a natural action of the trace algebra on the space of symmetric functions agrees with the specialization of an action constructed by Schiffmann and Vasserot using Hilbert schemes.

Unifying theories of time with generalised reactive processes

Hoare and He's theory of reactive processes provides a unifying foundation for the formal semantics of concurrent and reactive languages. Though highly applicable, their theory is limited to models that can express event histories as discrete sequences. In this paper, we show how their theory can be generalised by using an abstract trace algebra. We show how the algebra, notably, allows us to consider continuous-time traces and thereby facilitate models of hybrid systems. We then use this algebra to reconstruct the theory of reactive processes in our generic setting, and prove characteristic laws for sequential and parallel processes, all of which have been mechanically verified in the Isabelle/HOL proof assistant.

Quantization of noncompact coverings and its physical applications

A rigorous algebraic definition of noncommutative coverings is developed. In the case of commutative algebras this definition is equivalent to the classical definition of topological coverings of locally compact spaces. The theory has following nontrivial applications:• Coverings of continuous trace algebras,• Coverings of noncommutative tori,• Coverings of the quantum SU(2) group,• Coverings of foliations,• Coverings of isospectral deformations of Spin – manifolds.The theory supplies the rigorous definition of noncommutative Wilson lines.

Research on formalization of efficient query application problems with compound condition in software development

This paper provides a formalized definition of the application problem of compound condition query (CCQ) and a formal method of applying requirements elicitation based on trace information space derived from trace algebra. With the formalized process of solving the application problem of CCQ, formal requirements specification of application of CCQ is given, a formalized and automatic mapping of the results of requirements elicitation to the formal requirements specification is performed, the software system model and the application code are developed. Through a sample application of comprehensive query on housing information, the feasibility of formalized and automatic software development for the application problem of CCQ is proved. The result has important implications for the other problems regarding formalization and automatic software development.

Crossed Module Actions on Continuous Trace C*-Algebras

We lift an action of a torus $\mathbb{T}^n$ on the spectrum of a continuous trace algebra to an action of a certain crossed module of Lie groups that is an extension of $\mathbb{R}^n$. We compute equivariant Brauer and Picard groups for this crossed module and describe the obstruction to the existence of an action of $\mathbb{R}^n$ in our framework.

On higher real and stable ranks for $CCR$ $C^*-$algebras

We calculate the real rank and stable rank of CCR algebras which either have only finite dimensional irreducible representations or have finite topological dimension. We show that either rank of A is determined in a good way by the ranks of an ideal I and the quotient A/I in four cases: When A is CCR; when I has only finite dimensional irreducible representations; when I is separable, of generalized continuous trace and finite topological dimension, and all irreducible representations of I are infinite dimensional; or when I is separable, stable, has an approximate identity consisting of projections, and has the corona factorization property. We also present a counterexample on higher ranks of M(A), A subhomogeneous, and a theorem of P. Green on generalized continuous trace algebras.

Algebraic Geometry of Matrix Product States

We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state's amplitudes which hold if and only if the state is a translation invariant matrix product state or a limit of such states. For systems with few qubits, we give these equations explicitly, considering both periodic and open boundary conditions. Using the classical theory of trace varieties and trace algebras, we explain the relationship between MPS and hidden Markov models and exploit this relationship to derive useful parameterizations of MPS. We make four conjectures on the identifiability of MPS parameters.

A presentation of the trace algebra of three [formula omitted] matrices

The trace algebra Cnd is generated by all traces of products of d generic n×n matrices. Minimal generating sets of Cnd and their defining relations are known for n<3 and n=3, d=2. This paper states a minimal generating set and their defining relations for n=d=3. Furthermore the computations yield a description of C33 as a free module over the ring generated by a homogeneous system of parameters.

Invertibility threshold for $H^{∞}$ trace algebras, and effective matrix inversions

Invertibility threshold for $H^{∞}$ trace algebras, and effective matrix inversions

DEFINING RELATIONS OF LOW DEGREE OF INVARIANTS OF TWO 4 × 4 MATRICES

The trace algebra Cnd over a field of characteristic 0 is generated by all traces of products of d generic n × n matrices, n, d ≥ 2. Minimal sets of generators of Cnd are known for n = 2 and 3 for any d and for n = 4 and 5 and d = 2. The explicit defining relations between the generators are found for n = 2 and any d and for n = 3, d = 2 only. Defining relations of minimal degree for n = 3 and any d are also known. The minimal degree of the defining relations of any homogeneous minimal generating set of C42 is equal to 12. Starting with the generating set given recently by Drensky and Sadikova, we have determined all relations of degree ≤ 14. For this purpose we have developed further algorithms based on representation theory of the general linear group and easy computer calculations with standard functions of Maple.

Defining relations of minimal degree of the trace algebra of [formula omitted] matrices

The trace algebra C n d over a field of characteristic 0 is generated by all traces of products of d generic n × n matrices, n , d ⩾ 2 . Minimal sets of generators of C n d are known for n = 2 and n = 3 for any d as well as for n = 4 and n = 5 and d = 2 . The defining relations between the generators are found for n = 2 and any d and for n = 3 , d = 2 only. Starting with the generating set of C 3 d given by Abeasis and Pittaluga in 1989, we have shown that the minimal degree of the set of defining relations of C 3 d is equal to 7 for any d ⩾ 3 . We have determined all relations of minimal degree. For d = 3 we have also found the defining relations of degree 8. The proofs are based on methods of representation theory of the general linear group and easy computer calculations with standard functions of Maple.

KK-theoretic duality for proper twisted actions

Let \({\mathcal{A}}\) be a smooth continuous trace algebra, with a Riemannian manifold spectrum X, equipped with a smooth action by a discrete group G such that G acts on X properly and isometrically. Then \({\mathcal{A}}^{-1}\rtimes G \) is KK-theoretically Poincare dual to \(\big(\mathcal A {\hat {\otimes}_{C_0(X)}} C_\tau (X)\big) \rtimes G\) , where \({\mathcal{A}}^{-1}\) is the inverse of \({\mathcal{A}}\) in the Brauer group of Morita equivalence classes of continuous trace algebras equipped with a group action. We deduce this from a strengthening of Kasparov’s duality theorem. As applications we obtain a version of the above Poincare duality with X replaced by a compact G-manifold M and Poincare dualities for twisted group algebras if the group satisfies some additional properties related to the Dirac dual-Dirac method for the Baum- Connes conjecture.

Curves of fixed points of trace maps

We study curves of fixed points for certain diffeomorphisms of ${\mathbb{R}}^3$ that are induced by automorphisms of a trace algebra. We classify these curves. There is a function $E$ which is invariant under all such trace maps and the level surfaces $E_t: E=t$ are invariant; a point of $E_t$ will be said to have level $t$. The surface $E_1$ is significant. Then most fixed points on $E_1$ are actually on a curve $\gamma$ of fixed points interior to $E_1$. We describe the possibilities for the other end of $\gamma$ on $E_1$.

Entire cyclic homology of stable continuous trace algebras

A central result in this paper is the computation of the entire cyclic homology of canonical smooth subalgebras of stable continuous trace C*-algebras having smooth manifolds M as their spectrum. More precisely, the entire cyclic homology is shown to be canonically isomorphic to the continuous periodic cyclic homology for these algebras. By an earlier result of the authors, one concludes that the entire cyclic homology of the algebra is canonically isomorphic to the twisted de Rham cohomology of M.

An action of subgroups of mapping class groups on polynomial algebras

We study a certain subgroup of the mapping class group of a surface with boundary by obtaining an action of this subgroup on a polynomial algebra. This action comes from a lift of the action of the subgroup on a trace algebra, Magnus having done a similar thing for the braid groups. We then investigate the invariant theory for this action and various representations that this action affords. In particular, we obtain finite permutation representations and infinite linear representations of these subgroups that are non-trivial on subgroups of the Torelli group.

Poincaré series of some pure and mixed trace algebras of two generic matrices

We work over a field K of characteristic zero. The Poincaré series for the algebra C n , 2 of GL n -invariants and the algebra T n , 2 of GL n -concomitants of two generic n × n matrices x and y are computed for n ⩽ 6 . Both simply graded and bigraded cases are included. The cases n ⩽ 4 were known previously. For C 4 , 2 and C 5 , 2 we construct a minimal set of generators, and give an application to Specht's theorem on unitary similarity of matrices. By identifying the space M n 2 of pairs of n × n matrices with M n ⊗ K 2 , we extend the action of GL n to GL n × GL 2 . For n ⩽ 5 , we compute the Poincaré series for the polynomial invariants of this action when restricted to the subgroups GL n × SL 2 and GL n × Δ 1 , where Δ 1 is the maximal torus of SL 2 consisting of diagonal matrices. Five conjectures are proposed concerning the numerators and denominators of various Poincaré series mentioned above. Some heuristic formulas and open problems are stated.

Defining relations of the noncommutative trace algebra of two [formula omitted] matrices

The noncommutative (or mixed) trace algebra T n d is generated by d generic n × n matrices and by the algebra C n d generated by all traces of products of generic matrices, n , d ⩾ 2 . It is known that over a field of characteristic 0 this algebra is a finitely generated free module over a polynomial subalgebra S of the center C n d . For n = 3 and d = 2 we have found explicitly such a subalgebra S and a set of free generators of the S-module T 32 . We give also a set of defining relations of T 32 as an algebra and a Gröbner basis of the corresponding ideal. The proofs are based on easy computer calculations with standard functions of Maple, the explicit presentation of C 32 in terms of generators and relations, and methods of representation theory of the general linear group.

On a generalised Connes–Hochschild–Kostant–Rosenberg theorem

The central result of this paper is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalising some fundamental results of Connes and Hochschild–Kostant–Rosenberg. The Connes–Chern character is also identified here with the twisted Chern character.

Trace identities from identities for determinants

We present new identities for determinants of matrices (Ai,j) with entries (Ai,j) equal to (ai,j) or ai,0a0,j−ai,j, where the ai,j’s are indeterminates. We show that these identities are behind trace identities for SL(2,C) matrices found earlier by Magnus in his study of trace algebras.