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We obtain results describing the behavior of the action of rotation generators on polynomials over a commutative ring. We also explore harmonic polynomials in a purely algebraic setting., 53 pages

A gauge transformation of categorical principal bundles arises from a functorial isomorphism between such bundles. We determine the geometric nature of such gauge transformations. For a twisted-product categorical principal bundle whose structure group is given by a pair of Lie groups G and H we show that a pair consisting of a traditional gauge transformation θ , given by a G -valued function, and an L ( H ) -valued 1-form Λ H determine a categorical gauge transformation. More general gauge transformations are also studied.

For a principal bundle $P\to M$ equipped with a connection ${\bar A}$, we study an infinite dimensional bundle ${\mathcal P}^{\rm dec}_{\bar A}P$ over the space of paths on $M$, with the points of ${\mathcal P}^{\rm dec}_{\bar A}P$ being horizontal paths on $P$ decorated with elements of a second structure group. We construct parallel transport processes on such bundles and study holonomy bundles in this setting. We explain the relationship with categorical geometry and explore the notion of categorical connections on categorical principal bundles in a concrete differential geometric way., Comment: 53pp, 3 figs

The Gaussian Radon transform is an analog of the traditional Radon transform in an infinite dimensional Gaussian setting. We show that this infinite dimensional transform can be obtained as a limit of Radon transforms on finite dimensional spheres.

We show that a natural class of orthogonal polynomials on large spheres in N dimensions tend to Hermite polynomials in the large-N limit. We determine the behavior of the spherical Laplacian as well as zonal harmonic polynomials in the large-N limit.

We show that the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finite-codimension affine subspace in infinite dimensions.

We show that for a suitable class of functions of finitely-many variables, the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finite-codimension affine subspace in infinite dimensions.

Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical bundles. We study categorical principal bundles that are product bundles in the categorical sense, developing the relationship between functorial sections of such bundles and trivializations. We construct functorial cocycles with values in categorical groups using a suitable family of locally defined functions on the object space of the base category. Categorical product bundles being too rigid to give a widely applicable model for local triviality, we introduce the notion of a twisted-product categorical bundle. We relate such bundles to decorated categorical bundles that contain more information, specifically parallel transport data., Comment: v2:36 pages, 4 figures added to clarify statements about functorial cocycles and twisted product bundles

This is the written version of a set of four lectures given at the CIB in Lausanne in April 2015. The aim of these lectures was to present some of the mathematical-physical ideas underlying low-dimensional gauge theories, and some of the mathematical results which were obtained in the last twenty years on 2-dimensional Yang–Mills and 3-dimensional Chern–Simons theories.

We study a type of connection forms, given by Chen integrals, over pathspaces by placing such forms within a category-theoretic framework of principal bundles and connections. We introduce a notion of 'decorated' principal bundles, develop parallel transport on such bundles, and explore specific examples in the context of pathspaces., Comment: 50 pages. Updated and revised on 16th August 2012

We provide a recipe for “fattening” a category that leads to the construction of a double category. Motivated by an example where the underlying category has vector spaces as objects, we show how a monoidal category leads to a law of composition, satisfying certain coherence properties, on the object set of the fattened category.

We present a scenic but practical guide through nuclear spaces and their dual spaces, examining useful, unexpected, and often unfamiliar results both for nuclear spaces and their strong and weak duals.

We prove that in infinite dimensions, if a bounded continuous function has zero Gaussian integral over all hyperplanes outside a closed bounded convex set then the function is zero outside this set. This is an infinite-dimensional form of the well-known Helgason support theorem for Radon transforms in finite dimensions.

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

We prove an infinite-dimensional integral identity equating the integral of a function on a subspace of a linear space to the integral of its Segal-Bargmann transform over the orthogonal complement.

An account of the Lagrangian and Hamiltonian dynamics of the pure Yang–Mills system is presented. This framework is applied to the case of (1+1)-dimensional cylindrical spacetime. Hamiltonian dynamics on the space of connections over a circle is often identified with dynamics on the cotangent bundle of the gauge group by means of the holonomy. In support of this procedure we show that the symplectic structure for Hamiltonian dynamics for connections on a circle is identifiable with the natural symplectic structure on the cotangent bundle of the gauge group.

We obtain Dirac’s classic monopole charge quantization from the point of view of geometric quantization and demonstrate how this leads to the conclusion that the electromagnetic field is a U(1)-gauge field.

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].

We prove that integration over the moduli space of flat connections can be obtained as a limit of integration with respect to the Yang–Mills measure defined in terms of the heat-kernel for the gauge group. In doing this we also give a rigorous proof of Witten’s formula for the symplectic volume of the moduli space of flat connections. Our proof uses an elementary identity connecting determinants of matrices along with a careful accounting of certain dense subsets of full measure in the moduli space.

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.

Background: Genetic instability of chromosome 11 is a frequent event in many solid tumours, including head and neck squamous cell carcinoma (HNSCC). Aims: To perform allelic imbalance analysis of cytogenetically mapped altered regions of human chromosome 11 in patients with HNSCC from eastern India. Methods: Genomic alterations were investigated using highly polymorphic microsatellite markers in both HNSCC and leukoplakia tissues. Results: Microsatellite markers D11S1758 from 11p13‐15 and D11S925 from 11q23.3‐24 had the highest frequency (38% and 32%, respectively) of loss of heterozygosity among all the markers analysed. Allelic loss at the marker D11S925 was seen in both leukoplakia and in all stages of HNSCC tumour tissues suggesting that it is an early event in HNSCC tumorigenesis. Microsatellite size alteration was also found to be high (> 20%) in several markers. In leukoplakia samples microsatellite instability was seen at a higher frequency than loss of the allele, indicating such alterations might initiate the process of tumorigenesis in HNSCC. Conclusions: The high rate of chromosomal alterations at 11q21‐24 in HNSCC suggests the presence of a putative tumour suppressor gene in this region.

A 2-form is constructed on the space of connections on a principal bundle over an oriented surface with boundary. This induces a symplectic structure for the moduli space of flat connections with boundary holonomies lying in prescribed conjugacy classes. The Yang–Mills quantum field measure is described for this situation. This measure converges to the normalized symplectic volume measure in the “classical” limit.

We study the Gaussian Radon transform in the classical Wiener space of Brownian motion. We determine explicit formulas for transforms of Brownian functionals specified by stochastic integrals. A Fock space decom- position is also established for Gaussian measure conditioned to closed affine subspaces in Hilbert spaces.

From a differential geometric approach parallel transport on path spaces has been addressed. Integral version establishes the category theoretic frame work. We have glossed over the topic of representations of categorical groups and its relation with categorical framework of path space parallel transport.

A formula is derived which expresses the symplectic volume measure on the moduli space of flat connections over a closed Riemann surface Σ in terms of the corresponding volume measures for the surfaces Σ 1 and Σ 2 , when the latter have one boundary component each.

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.

All the connected components of the moduli space of flat connections on SU (2) and SO (3) (trivial and non-trivial) bundles over closed oriented surfaces are determined. The symplectic structure and volumes of the non-maximal strata of the moduli space are also determined.

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.

LetKbe a connected Lie group of compact type and letW(K) denote the set of continuous paths inK, starting at the identity and with time-interval [0, 1]. ThenW(K) forms an infinite-dimensional group under the operation of pointwise multiplication. Letρdenote the Wiener measure onW(K). We construct an analog of the Segal–Bargmann transform forW(K). LetKCbe the complexification ofK,W(KC) the set of continuous paths inKCstarting at the identity, andμthe Wiener measure onW(KC). Our transform is a unitary map ofL2(W(K),ρ) onto the “holomorphic” subspace ofL2(W(KC),μ). By analogy with the classical transform, our transform is given by convolution with the Wiener measure, followed by analytic continuation. We prove that the transform forW(K) is nicely related by means of the Itô map to the classical Segal–Bargmann transform for the path-space in the Lie algebra ofK.

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

We develop a Radon transform on Banach spaces using Gaussian measure and prove that if a bounded continuous function on a separable Banach space has zero Gaussian integral over all hyperplanes outside a closed bounded convex set in the Hilbert space corresponding to the Gaussian measure then the function is zero outside this set.

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.

It is shown that, for certain gauge groups, central functions of the holonomy variables do not determine connections up to gauge equivalence. It is also shown that, for a large class of compact groups, such functions do determine connections up to gauge equivalence. It is then shown that, for the latter type of gauge groups, the Euclidean quantum gauge field measure is determined by the expectation values of the Wilson loop variables (products of characters evaluated on holonomies).