Connections on a trivial bundle MxG can be identified with their holonomy maps, i.e. with homomorphisms of a groupoid of paths in M into the gauge group G.
For a connected compact G, various algebras depending on the set of the smooth connections through their holonomy maps have been introduced in the literature, called cylindrical and holonomy algebras. We discuss the relations between these algebras and the consistence of their spectra.
We introduce the holonomy-diffeomorphism algebra, a C*-algebra generated by flows of vectorfields and the compactly supported smooth functions on a manifold. We show that the separable representations of the holonomy-diffeomorphism algebra are given by measurable connections, and that the unitary equivalence of the representations corresponds to measured gauge equivalence of the measurable connections. We compare the setup to Loop Quantum Gravity and show that the generalized connections found there are not contained in the spectrum of the holonomy-diffeomorphism algebra in dimensions higher t…
We introduce the Quantum Holonomy-Diffeomorphism *-algebra, which is generated by holonomy-diffeomorphisms on a 3-dimensional manifold and translations on a space of SU(2)-connections. We show that this algebra encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Furthermore, we show that semi-classical states exist on the holonomy-diffeomorphism part of the algebra but that these states cannot be extended to the full algebra. Via a Dirac type operator we derive a certain class of unbounded operators that act in the GNS constructio…
In this paper we continue the development of Quantum Holonomy Theory, which is a candidate for a fundamental theory, by constructing separable strongly continuous representations of its algebraic foundation, the quantum holonomy-diffeomorphism algebra. Since the quantum holonomy-diffeomorphism algebra encodes the canonical commutation relations of a gauge theory these representations provide a possible framework for the kinematical sector of a quantum gauge theory. Furthermore, we device a method of constructing physically interesting operators such as the Yang-Mills Hamilton operator. This es…
The SW(3/2,3/2,2) superconformal algebra is a W algebra with two free parameters. It consists of 3 superconformal currents of spins 3/2, 3/2 and 2. The algebra is proved to be the symmetry algebra of the coset (su(2)+su(2)+su(2))/su(2). At the central charge c=21/2 the algebra coincides with the superconformal algebra associated to manifolds of G_2 holonomy. The unitary minimal models of the SW(3/2,3/2,2) algebra and their fusion structure are found. The spectrum of unitary representations of the G_2 holonomy algebra is obtained.
We present a new formulation of quantum holonomy theory, which is a candidate for a non-perturbative and background independent theory of quantum gravity coupled to matter and gauge degrees of freedom. The new formulation is based on a Hilbert space representation of the QHD(M) algebra, which is generated by holonomy-diffeomorphisms on a 3-dimensional manifold and by canonical translation operators on the underlying configuration space over which the holonomy-diffeomorphisms form a non-commutative C*-algebra. A proof that the state that generates the representation exist is left for later publ…
In this article a new C*-algebra derived from the basic quantum variables: holonomies along paths and group-valued quantum flux operators in the framework of Loop Quantum Gravity is constructed. This development is based on the theory of cross-products and C*-dynamical systems. The author has presented a set of actions of the flux group associated to a surface set on the analytic holonomy C*-algebra, which define C*-dynamical systems. These objects are used to define the holonomy-flux cross-product C*-algebra associated to a surface set. Furthermore surface-preserving path- and graph-diffeomor…
In this article the holonomy-flux *-algebra, which has been introduced by Lewandowski, Okolow, Sahlmann and Thiemann, is modificated. The new *-algebra is called the holonomy-flux cross-product *-algebra. This algebra is an abstract cross-product *-algebra. It is given by the universal algebra of the algebra of continuous and differentiable functions on the configuration space of generalised connections and the universal enveloping flux algebra associated to a surface set, and some canonical commutator relations. There is a uniqueness result for a certain path- and graph-diffeomorphism invaria…
In the project AQV the issue of quantum constraints, KMS-states and algebras of quantum configuration and momentum variables in Loop Quantum Gravity has been argued. There a physical algebra has been required to contain complete observables and the quantum constraints, or at least the quantum constraints are affilliated with this algebra. In this context a first conjecture for a physical algebra is presented in this article. A new *-algebra for LGQ, which is called the localised holonomy-flux cross-product *-algebra, is studied. A suggestion for a physical *-algebra, which contains the localis…
A new approach to a unified theory of quantum gravity based on noncommutative geometry and canonical quantum gravity is presented. The approach is built around a *-algebra generated by local holonomy-diffeomorphisms on a 3-manifold and a quantized Dirac type operator; the two capturing the kinematics of quantum gravity formulated in terms of Ashtekar variables. We prove that the separable part of the spectrum of the algebra is contained in the space of measurable connections modulo gauge transformations and we give limitations to the non-separable part. The construction of the Dirac type opera…
The theorem by Lewandowski et al. stating uniqueness of a diffeomorphism invariant state on an algebra of quantum observables for background independent theories of connections is based on some technical assumptions imposed on the algebra and the diffeomorphisms. In this paper we present a class of diffeomorphism invariant states on an algebra of this sort, which exist when the algebra and the diffeomorphisms satisfy alternative assumptions.
We present quantum holonomy theory, which is a non-perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a C*-algebra that involves holonomy-diffeomorphisms on a 3-dimensional manifold and which encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Employing a Dirac type operator on the configuration space of Ashtekar connections we obtain a semi-classical state and a kinematical Hilbert space via its GNS construction. We use the Dirac type operator, which provides a metric structure …
Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms. While this result is particularly imp…
The holonomy algebra $\g$ of an indecomposable Lorentzian (n+2)-dimensional manifold $M$ is a weakly-irreducible subalgebra of the Lorentzian algebra $\so_{1,n+1}$. L. Berard Bergery and A. Ikemakhen divided weakly-irreducible not irreducible subalgebras into 4 types and associated with each such subalgebra $\g$ a subalgebra $\h\subset \so_n$ of the orthogonal Lie algebra. We give a description of the spaces $\R(\g)$ of the curvature tensors for algebras of each type in terms of the space $¶(\h)$ of $\h$-valued 1-forms on $\Real^n$ that satisfy the Bianchi identity and reduce the classificatio…
Quantum holonomy theory is a candidate for a non-perturbative theory of quantum gravity coupled to fermions. The theory is based on the QHD(M) algebra, which essentially encodes how matter degrees of freedom are moved on a three-dimensional manifold. In this paper we commence the development of a lattice-independent formulation. We first introduce a flow-dependent version of the QHD(M) algebra and formulate necessary conditions for a state to exist hereon. We then use the GNS construction to build a kinematical Hilbert space. Finally we find that operators, that correspond to the Dirac and gra…
We review the motivation, construction and physical interpretation of a semi-finite spectral triple obtained through a rearrangement of central elements of loop quantum gravity. The triple is based on a countable set of oriented graphs and the algebra consists of generalized holonomy loops in this set. The Dirac type operator resembles a global functional derivation operator and the interaction between the algebra of holonomy loops and the Dirac type operator reproduces the structure of a quantized Poisson bracket of general relativity. Finally we give a heuristic argument as to how a natural …
We study the chiral de Rham complex (CDR) over a manifold $M$ with holonomy $\rm G_2$. We prove that the vertex algebra of global sections of the CDR associated to $M$ contains two commuting copies of the Shatashvili-Vafa $\rm G_2$ superconformal algebra. Our proof is a tour de force, based on explicit computations.
Interpreting the chiral de Rham complex (CDR) as a formal Hamiltonian quantization of the supersymmetric non-linear sigma model, we suggest a setup for the study of CDR on manifolds with special holonomy. We show how to systematically construct global sections of CDR from differential forms, and investigate the algebra of the sections corresponding to the covariantly constant forms associated with the special holonomy. As a concrete example, we construct two commuting copies of the Odake algebra (an extension of the N=2 superconformal algebra) on the space of global sections of CDR of a Calabi…
We investigate lattice and continuous quantum gauge theories on the Euclidean plane with a structure group that is replaced by a $H$-algebra; non-commutative analogues of groups and contain the class of Voiculescu's dual groups. We are interested in non-commutative analogues of random gauge fields, which we describe through the random Holonomy that they induce. We propose a general definition of a Quantum Holonomy Hield over a $H$-algebra and construct such fields starting from a quantum Lévy process on a $H$-algebra. As an application, we define higher-dimensional generalizations of the so-ca…
There are two different notions of holonomy in supergeometry, the supergroup introduced by Galaev and our functorial approach motivated by super Wilson loops. Either theory comes with its own version of invariance of vectors and subspaces under holonomy. By our first main result, the Twofold Theorem, these definitions are equivalent. Our proof is based on the Comparison Theorem, our second main result, which characterises Galaev's holonomy algebra as an algebra of coefficients, building on previous results. As an application, we generalise some of Galaev's results to S-points, utilising the ho…
It is shown that the description of certain class of representations of the holonomy Lie algebra associated to hyperplane arrangement $Δ$ is essentially equivalent to the classification of $\vee$-systems associated to $Δ.$ The flat sections of the corresponding $\vee$-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any $\vee$-system is free in Saito's sense and show this for all known $\vee$-systems and for a special class of $\vee$-systems called harmonic, which includes all Coxeter systems. In the irred…
The algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and quantizes the character variety of the Riemann surface $Σ_{g,n}\!\setminus\! D$ ($D$ is an open disk). In this article we define a holonomy map in that quantized setting, which associates a tensor with components in $\mathcal{L}_{g,n}(H)$ to tangles in $(Σ_{g,n}\!\setminus\!D) \times [0,1]$, generalizing previous works of Buffenoir-Roche and Bullock-Frohman-Kania-Bartoszynska. We show that holonomy behaves well for the stack product and the action of the mapping class group; then we specializ…
The Wess-Zumino term in two-dimensional conformal field theory is best understood as a surface holonomy of a bundle gerbe. We define additional structure for a bundle gerbe that allows to extend the notion of surface holonomy to unoriented surfaces. This provides a candidate for the Wess-Zumino term for WZW models on unoriented surfaces. Our ansatz reproduces some results known from the algebraic approach to WZW models.
Holonomy groups and holonomy algebras for connections on locally free sheaves over supermanifolds are introduced. A one-to-one correspondence between parallel sections and holonomy-invariant vectors, and a one-to-one correspondence between parallel locally direct subsheaves and holonomy-invariant vector supersubspaces are obtained. As the special case, the holonomy of linear connections on supermanifolds is studied. Examples of parallel geometric structures on supermanifolds and the corresponding holonomies are given. For Riemannian supermanifolds an analog of the Wu theorem is proved. Berger …
A crossed module constitutes a strict $2$-groupoid $\mathcal{G}$ and a $\mathcal{G}$-valued cocycle on a manifold defines a $2$-bundle. A $2$-connection on this $2$-bundle is given by a Lie algebra $\mathfrak g$ valued $1$-form $A $ and a Lie algebra $\mathfrak h$ valued $2$-form $B $ over each coordinate chart together with $2$-gauge transformations between them, which satisfy the compatibility condition. Locally, the path-ordered integral of $A $ gives us the local $1$-holonomy, and the surface-ordered integral of $(A ,B )$ gives us the local $2$-holonomy. The transformation of local $2$-hol…
Narasimhan and Ramadas showed that the restricted holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group when one considers the product principal bundle $S^3 \times SU(2)\to S^3$. Instead of a base manifold S^3, we consider here a base manifold of dimension $n\ge 2$ with a boundary and use Dirichlet boundary conditions on connections as defined by Marini. A key step in the method of Narasimhan and Ramadas consisted in showing that the linear space spanned by the curvature form at one specially chosen connection is dense in the holonomy L…
We construct a cycle in higher Hochschild homology associated to the 2-dimensional torus which represents 2-holonomy of a non-abelian gerbe in the same way the ordinary holonomy of a principal G-bundle gives rise to a cycle in ordinary Hochschild homology. This is done using the connection 1-form of Baez-Schreiber. A crucial ingredient in our work is the possibility to arrange that in the structure crossed module mu: g -> h of the principal 2-bundle, the Lie algebra h is abelian, up to equivalence of crossed modules.
