Gauge-Invariant Coordinates on Gauge-Theory Orbit Space

A gauge-invariant field is found which describes physical configurations, i.e. gauge orbits, of non-Abelian gauge theories. This is accomplished with non-Abelian generalizations of the Poincare'-Hodge formula for one-forms. In a particular sense, the new field is dual to the gauge field. Using this field as a coordinate, the metric and intrinsic curvature are discussed for Yang-Mills orbit space for the (2+1)- and (3+1)-dimensional cases. The sectional, Ricci and scalar curvatures are all formally non-negative. An expression for the new field in terms of the Yang-Mills connection is found in 2+1 dimensions. The measure on Schroedinger wave functionals is found in both 2+1 and 3+1 dimensions; in the former case, it resembles Karabali, Kim and Nair's measure. We briefly discuss the form of the Hamiltonian in terms of the dual field and comment on how this is relevant to the mass gap for both the (2+1)- and (3+1)-dimensional cases.Comment: Typos corrected, more about the non-Abelian decomposition and inner products, more discussion of the mass gap in 3+1 dimensions. Now 23 page

Beyond the String Genus

In an earlier work we used a path integral analysis to propose a higher genus generalization of the elliptic genus. We found a cobordism invariant parametrized by Teichmuller space. Here we simplify the formula and study the behavior of our invariant under the action of the mapping class group of the Riemann surface. We find that our invariant is a modular function with multiplier just as in genus one.Comment: 40 pages, 1 figur

The supersymmetric sigma model and the geometry of the Weyl-Kac character formula

Field theoretic and geometric ideas are used to construct a chiral supersymmetric field theory whose ground state is a specified irreducible representation of a centrally extended loop group. The character index of the associated supercharge (an appropriate Dirac operator on $LG/T$) is the Weyl-K\v{a}c character formula which we compute explicitly by the steepest descent approximation.Comment: 40 page

The fundamental gap of simplices

The fundamental gap conjecture was recently proven by Andrews and Clutterbuck: for any convex domain in $\R^n$ normalized to have unit diameter, the difference between the first two Dirichlet eigenvalues of the Laplacian is bounded below by that of the interval. In this work, we focus on the moduli spaces of simplices in all dimensions, and later specialize to the moduli space of Euclidean triangles. Our first theorem is a compactness result for the gap function on the moduli space of simplices in any dimension. Our second main result verifies a recent conjecture of Antunes-Freitas: for any Euclidean triangle normalized to have unit diameter, the fundamental gap is uniquely minimized by the equilateral triangle.Comment: Final version, Journal ref adde

A study of the zero modes of the Faddeev-Popov operator in Euclidean Yang-Mills theories in the Landau gauge in d=2,3,4 dimensions

Examples of normalizable zero modes of the Faddeev-Popov operator in SU(2) Euclidean Yang-Mills theories in the Landau gauge are constructed in d=2,3,4 dimensions.Comment: 18 pages. Text modifications. References added. Version accepted for publication in the EPJ

Gauge equivalence in QCD: the Weyl and Coulomb gauges

The Weyl-gauge ($A_0^a=0)$ QCD Hamiltonian is unitarily transformed to a representation in which it is expressed entirely in terms of gauge-invariant quark and gluon fields. In a subspace of gauge-invariant states we have constructed that implement the non-Abelian Gauss's law, this unitarily transformed Weyl-gauge Hamiltonian can be further transformed and, under appropriate circumstances, can be identified with the QCD Hamiltonian in the Coulomb gauge. We demonstrate an isomorphism that materially facilitates the application of this Hamiltonian to a variety of physical processes, including the evaluation of $S$-matrix elements. This isomorphism relates the gauge-invariant representation of the Hamiltonian and the required set of gauge-invariant states to a Hamiltonian of the same functional form but dependent on ordinary unconstrained Weyl-gauge fields operating within a space of ``standard'' perturbative states. The fact that the gauge-invariant chromoelectric field is not hermitian has important implications for the functional form of the Hamiltonian finally obtained. When this nonhermiticity is taken into account, the ``extra'' vertices in Christ and Lee's Coulomb-gauge Hamiltonian are natural outgrowths of the formalism. When this nonhermiticity is neglected, the Hamiltonian used in the earlier work of Gribov and others results.Comment: 25 page

Gauge Orbit Types for Theories with Classical Compact Gauge Group

We determine the orbit types of the action of the group of local gauge transformations on the space of connections in a principal bundle with structure group O(n), SO(n) or $Sp(n)$ over a closed, simply connected manifold of dimension 4. Complemented with earlier results on U(n) and SU(n) this completes the classification of the orbit types for all classical compact gauge groups over such space-time manifolds. On the way we derive the classification of principal bundles with structure group SO(n) over these manifolds and the Howe subgroups of SO(n).Comment: 57 page

Analytic and Reidemeister torsion for representations in finite type Hilbert modules

For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-M\"uller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, the L2 analytic and Reidemeister torsions are equal.Comment: 78 pages, AMSTe

Towards Solving QCD - The Transverse Zero Modes in Light-Cone Quantization

We formulate QCD in (d+1) dimensions using Dirac's front form with periodic boundary conditions, that is, within Discretized Light-Cone Quantization. The formalism is worked out in detail for SU(2) pure glue theory in (2+1) dimensions which is approximated by restriction to the lowest {\it transverse} momentum gluons. The dimensionally-reduced theory turns out to be SU(2) gauge theory coupled to adjoint scalar matter in (1+1) dimensions. The scalar field is the remnant of the transverse gluon. This field has modes of both non-zero and zero {\it longitudinal} momentum. We categorize the types of zero modes that occur into three classes, dynamical, topological, and constrained, each well known in separate contexts. The equation for the constrained mode is explicitly worked out. The Gauss law is rather simply resolved to extract physical, namely color singlet states. The topological gauge mode is treated according to two alternative scenarios related to the In the one, a spectrum is found consistent with pure SU(2) gluons in (1+1) dimensions. In the other, the gauge mode excitations are estimated and their role in the spectrum with genuine Fock excitations is explored. A color singlet state is given which satisfies Gauss' law. Its invariant mass is estimated and discussed in the physical limit.Comment: LaTex document, 26 pages, one figure (obtainable by contacting authors). To appear in Physical. Review

Accessing directly the properties of fundamental scalars in the confinement and Higgs phase

The properties of elementary particles are encoded in their respective propagators and interaction vertices. For a SU(2) gauge theory coupled to a doublet of fundamental complex scalars these propagators are determined in both the Higgs phase and the confinement phase and compared to the Yang-Mills case, using lattice gauge theory. Since the propagators are gauge-dependent, this is done in the Landau limit of 't Hooft gauge, permitting to also determine the ghost propagator. It is found that neither the gauge boson nor the scalar differ qualitatively in the different cases. In particular, the gauge boson acquires a screening mass, and the scalar's screening mass is larger than the renormalized mass. Only the ghost propagator shows a significant change. Furthermore, indications are found that the consequences of the residual non-perturbative gauge freedom due to Gribov copies could be different in the confinement and the Higgs phase.Comment: 11 pages, 6 figures, 1 table; v2: one minor error corrected; v3: one appendix on systematic uncertainties added and some minor changes, version to appear in EPJ