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Finite projective geometries, especially the Fano plane, have been observed to arise in the context of certain quantum gate operators. We use Clifford algebras to explain why these geometries, both planar and higher dimensional, appear in the context of multi-qubit composite systems., Comment: 7 pages, 3 figures

A framework of connections, gauge transformations, and more, is developed over cell complexes. The relationship between this and the continuum theory is explained. A 2-form over the space of flat connections over certain types of cell complexes is defined and its significant properties proved; this 2-form is induced by the Atiyah–Bott symplectic structure on the space of connections over closed, oriented surfaces.

In this paper we present the central ideas and results of a rigorous theory of the Chern–Simons functional integral. In particular, we show that it is possible to define the Wilson loop observables (WLOs) for pure Chern–Simons models with base manifold M = ℝ3 rigorously as infinite dimensional oscillatory integrals by exploiting an "axial gauge fixing" and applying certain regularization techniques like "loop-smearing" and "framing". The (values of the) WLOs can be computed explicitly. If the structure group G of the model is Abelian one obtains well-known linking number expressions for the WLOs. If G is Non-Abelian one obtains expressions which are similar but not identical to the state model representations for the Homfly and Kauffman polynomials given in [19, 21, 31].

For a closed surface Σ obtained by pasting together two surfaces Σ1 and Σ2 along their boundary components, it is proven that the Yang–Mills measure for a bundle of specified topology over the closed surface Σ results from a "convolution" of the corresponding measures for the surfaces Σ1 and Σ2.

Let {α(n); n≥0} be a sequence of positive numbers satisfying certain conditions. A Gel'fand triple [Formula: see text] associated with the sequence {α (n); n ≥ 0} has been introduced on a white noise space (ℰ′,μ) by Cochran, Kuo and Sengupta. In this paper we obtain additional conditions on the sequence {α(n); n≥0} in order to carry out white noise distribution theory on the space (ℰ′,μ). Moreover, we show that the Bell numbers satisfy these additional conditions.

The Segal–Bargmann transform of certain Wilson loop variables for quantum Yang–Mills theory on the plane is determined explicitly.

We consider the Yang–Mills action functional S YM on the infinite-dimensional space of connections on a bundle over a compact surface. We find a lower bound for S YM (ω) in terms of holonomies of the connection ω, the topology of the surface, and the topology of the bundle. An intermediate 'energy inequality' becomes an equality if and only if the connection is a critical point of the Yang–Mills action. Yang–Mills minima can also be understood using these inequalities.

The S-transform is studied as a mapping from a space of tensors to a space of functions over a complex space. The range of this transform is characterized in terms of analyticity and growth. These results are applied to a broad class of generalized functions in white noise analysis. These correspond to completions of the Gaussian L2-space which preserve orthogonality of Hermite polynomials. The S-transform is defined for the new generalized functions, and the range of this S-transform is identified in terms of analyticity and growth. Examples of the new spaces of generalized functions are given; these include distributions considered by Kondratiev and Streit, as well as new classes of distributions whose S-transforms have growth bounded by iterated exponentials.

We construct rigorously an infinite dimensional distribution which corresponds to the Chern-Simons (CS) functional integral associated with a principal fiber bundle over R 3 with structure group a compact connected Lie group. We determine the ‘moments’ of the CS distribution and show that these coincide with those used in informal studies of the CS integral. A locality property of the CS distribution is proven. The complexified theory of Frohlich and King is also discussed within our framework.

The quantum field measure for gauge fields over a compact surface with boundary, with holonomy around the boundary components specified, is constructed. Loop expectation values for general loop configurations are computed. For a compact oriented surface with one boundary component, let \(\) be the moduli space of flat connections with boundary holonomy lying in a conjugacy class \(\) in the gauge group G. We prove that a certain natural closed 2-form on \(\), introduced in an earlier work by C. King and the author, is a symplectic structure on the generic stratum of \(\) for generic \(\). We then prove that the quantum Yang-Mills measure, with the boundary holonomy constrained to lie in \(\), converges in a natural sense to the corresponding symplectic volume measure in the classical limit. We conclude with a detailed treatment of the case \(\), and determine the symplectic volume of this moduli space.

Yang–Mills connections over closed oriented surfaces of genus ≥1, for compact connected gauge groups, are constructed explicitly. The resulting formulas for Yang–Mills connections are used to carry out a Marsden–Weinstein type procedure. An explicit formula is obtained for the resulting 2-form on the moduli space. It is shown that this 2-form provides a symplectic structure on appropriate subsets of the moduli space.

For compact surfaces with one boundary component, and semisimple gauge groups, we construct a closed gauge invariant 2-form on the space of flat connections whose boundary holonomy lies in a fixed conjugacy class. This form descends to the moduli space under the action of the full gauge group, and provides an explicit description of a symplectic structure for this moduli space.

A natural gauge-invariant 2-form is introduced on the space of connections over a compact oriented surface with boundary. It is shown that this 2-form descends to the moduli space of flat connections, under a group of based gauge transformations. An explicit expression (in terms of holonomy variations) is given for the resulting 2-form, and it is related to the symplectic structure of the extended moduli space introduced by L. C. Jeffrey.

We prove that for $SU(2)$ and $SO(3)$ quantum gauge theory on a torus, holonomy expectation values with respect to the Yang-Mills measure $d\mu_T(\o) =N_T^{-1}e^{-S_{YM}(\o)/T}[{\cal D}\o]$ converge, as $T\downarrow 0$, to integrals with respect to a symplectic volume measure $\mu_0$ on the moduli space of flat connections on the bundle. These moduli spaces and the symplectic structures are described explicitly., Comment: 18 pages

The moduli space of flat connections on a principal G‐bundle over a compact oriented surface of genus g≥1 is considered herein. Using the holonomies around noncontractible loops, the moduli space is described as a quotient of a submanifold of G2g. An explicit expression is obtained for the symplectic form on the smooth part of moduli space, and several properties of this form are established.

We develop a differential geometric framework for parallel transport over path spaces and a corresponding discrete theory, an integrated version of the continuum theory, using a category-theoretic framework., 27pp 3fig pdflatex only; v2: rewritten with several clarifications; v3: minor changes, added references. Version to be published, 30pp

There has been growing recent interest in probabilistic interpretations of kernel-based methods as well as learning in Banach spaces. The absence of a useful Lebesgue measure on an infinite-dimensional reproducing kernel Hilbert space is a serious obstacle for such stochastic models. We propose an estimation model for the ridge regression problem within the framework of abstract Wiener spaces and show how the support vector machine solution to such problems can be interpreted in terms of the Gaussian Radon transform.

We relate the semiclassical limit of the quantum Yang-Mills partition function on a compact oriented surface to the symplectic volume of the moduli space of flat connections, by using an explicit expression for the symplectic form. This gives an independent proof of some recent results of Witten and Forman., 10 pages