Phase and Scaling Properties of Determinants Arising in Topological Field Theories

In topological field theories determinants of maps with negative as well as positive eigenvalues arise. We give a generalisation of the zeta-regularisation technique to derive expressions for the phase and scaling-dependence of these determinants. For theories on odd-dimensional manifolds a simple formula for the scaling dependence is obtained in terms of the dimensions of certain cohomology spaces. This enables a non-perturbative feature of Chern-Simons gauge theory to be reproduced by path-integral methods.Comment: 12 pages, Latex. To appear in Physics Letters

The Index Theorem and Universality Properties of the Low-lying Eigenvalues of Improved Staggered Quarks

We study various improved staggered quark Dirac operators on quenched gluon backgrounds in lattice QCD generated using a Symanzik-improved gluon action. We find a clear separation of the spectrum into would-be zero modes and others. The number of would-be zero modes depends on the topological charge as expected from the Index Theorem, and their chirality expectation value is large (approximately 0.7). The remaining modes have low chirality and show clear signs of clustering into quartets and approaching the random matrix theory predictions for all topological charge sectors. We conclude that improvement of the fermionic and gauge actions moves the staggered quarks closer to the continuum limit where they respond correctly to QCD topology.Comment: 4 pages, 3 figure

Eta invariants for flat manifolds

Using H. Donnelly result from the article "Eta Invariants for G-Spaces" we calculate the eta invariants of the signature operator for almost all 7-dimensional flat manifolds with cyclic holonomy group. In all cases this eta invariants are an integer numbers. The article was motivated by D. D. Long and A. Reid article "On the geometric boundaries of hyperbolic 4-manifolds, Geom. Topology 4, 2000, 171-178Comment: 18 pages, a new version with referees comment

Instanton-Meron Hybrid in the Background of Gravitational Instantons

When it comes to the topological aspects, gravity may have profound effects even at the level of particle physics despite its negligibly small relative strength well below the Planck scale. In spite of this intriguing possibility, relatively little attempt has been made toward the exhibition of this phenomenon in relevant physical systems. In the present work, perhaps the simplest and the most straightforward new algorithm for generating solutions to (anti) self-dual Yang-Mills (YM) equation in the typical gravitational instanton backgrounds is proposed and then applied to find the solutions practically in all the gravitational instantons known. Solutions thus obtained turn out to be some kind of instanton-meron hybrids possessing mixed features of both. Namely, they are rather exotic type of configurations obeying first order (anti) self-dual YM equation which are everywhere non-singular and have finite Euclidean YM actions on one hand while exhibiting meron-like large distance behavior and carrying generally fractional topological charge values on the other. Close inspection, however, reveals that the solutions are more like instantons rather than merons in their generic natures.Comment: 33pages, Revtex, typos correcte

Distribution of fermionic and topological observables on the lattice

We analyze the topological and fermionic vacuum structure of four-dimensional QCD on the lattice by means of correlators of fermionic observables and topological densities. We show the existence of strong local correlations between the topological charge and monopole density on the one side and the quark condensate, charge and chiral density on the other side. Visualization of individual gauge configurations demonstrates that instantons (antiinstantons) carry positive (negative) chirality, whereas the quark charge density fluctuates in sign within instantons.Comment: 10 pages, 5 eps figures, to appear in Phys. Lett.

A G_2 Unification of the Deformed and Resolved Conifolds

We find general first-order equations for G_2 metrics of cohomogeneity one with S^3\times S^3 principal orbits. These reduce in two special cases to previously-known systems of first-order equations that describe regular asymptotically locally conical (ALC) metrics \bB_7 and \bD_7, which have weak-coupling limits that are S^1 times the deformed conifold and the resolved conifold respectively. Our more general first-order equations provide a supersymmetric unification of the two Calabi-Yau manifolds, since the metrics \bB_7 and \bD_7 arise as solutions of the {\it same} system of first-order equations, with different values of certain integration constants. Additionally, we find a new class of ALC G_2 solutions to these first-order equations, which we denote by \wtd\bC_7, whose topology is an \R^2 bundle over T^{1,1}. There are two non-trivial parameters characterising the homogeneous squashing of the T^{1,1} bolt. Like the previous examples of the \bB_7 and \bD_7 ALC metrics, here too there is a U(1) isometry for which the circle has everywhere finite and non-zero length. The weak-coupling limit of the \wtd\bC_7 metrics gives S^1 times a family of Calabi-Yau metrics on a complex line bundle over S^2\times S^2, with an adjustable parameter characterising the relative sizes of the two S^2 factors.Comment: Latex, 14 pages, Major simplification of first-order equations; references amende

Non-Trivial SU(N) Instantons and Exotic Skyrmions

The classical Yang-Mills equations are solved for arbitrary semi-simple gauge groups in the Schwinger-Fock gauge. A new class of SU(N) instantons is presented which are not embeddings of SU(N-1) instantons but have non-trivial SU(N) color structure and carry winding number $n=N(N^{2}-1)/6$. Explicit configurations are given for SU(3) and SU(4) gauge groups. By means of the Atiyah Manton procedure Skyrmion fields are constructed from the SU(N) instantons. These Skyrmions represent exotic baryon states.Comment: 10 LaTex pages (1 figure available on request), UNITUE-THEP-10-199

Localization and Diagonalization: A review of functional integral techniques for low-dimensional gauge theories and topological field theories

We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaat-Heckman theorem, and the Weyl integral formula respectively. In each case, we first introduce the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups), and describe the finite dimensional integration formulae. We then discuss some applications to path integrals and give an overview of the relevant literature. The applications we deal with include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two-dimensional Yang-Mills theory.Comment: 72 pages (60 A4 pages), LaTeX (to appear in the Journal of Mathematical Physics Special Issue on Functional Integration (May 1995)

Possible Origin of Fermion Chirality and Gut Structure From Extra Dimensions

The fundamental chiral nature of the observed quarks and leptons and the emergence of the gauge group itself are most puzzling aspects of the standard model. Starting from general considerations of topological properties of gauge field configurations in higher space-time dimensions, it is shown that the existence of non-trivial structures in ten dimensions would determine a class of models corresponding to a grand unified GUT structure with complex fermion representations with respect to $SU(3)_C \otimes SU(2)_L \otimes U(1)_Y$. The discussion is carried out within the framework of string theories with characteristic energy scales below the Planck mass. Avoidance of topological obstructions upon continuous deformation of field configurations leads to global chiral symmetry breaking of the underlying fundamental theory, imposes rigorous restrictions on the structure of the vacuum and space-time itself and determines uniquely the gauge structure and matter content.Comment: final version to appear in Phys. Rev.