3D N = 1 SYM Chern-Simons theory on the Lattice

We present a method to implement 3-dimensional N = 1 SUSY Yang-Mills theory (a theory with two real supercharges containing gauge fields and an adjoint Majorana fermion) on the lattice, including a way to implement the Chern-Simons term present in this theory. At nonzero Chern-Simons number our implementation suffers from a sign problem which will make the numerical effort grow exponentially with volume. We also show that the theory with vanishing Chern-Simons number is anomalous; its partition function identically vanishes.Comment: v2, minor changes: expanded discussion in section III c, typos corrected, 17 pages, 9 figure

Low Energy Dynamics of N=2 Supersymmetric Monopoles

It is argued that the low-energy dynamics of $k$ monopoles in N=2 supersymmetric Yang-Mills theory are determined by an N=4 supersymmetric quantum mechanics based on the moduli space of $k$ static monople solutions. This generalises Manton's ``geodesic approximation" for studying the low-energy dynamics of (bosonic) BPS monopoles. We discuss some aspects of the quantisation and in particular argue that dolbeault cohomology classes of the moduli space are related to bound states of the full quantum field theory.Comment: 20 pages, EFI-93-0

A Newman-Penrose Calculator for Instanton Metrics

We present a Maple11+GRTensorII based symbolic calculator for instanton metrics using Newman-Penrose formalism. Gravitational instantons are exact solutions of Einstein's vacuum field equations with Euclidean signature. The Newman-Penrose formalism, which supplies a toolbox for studying the exact solutions of Einstein's field equations, was adopted to the instanton case and our code translates it for the computational use.Comment: 13 pages. Matches the published version. The web page of the codes is changed as https://github.com/tbirkandan/NPInstanto

Elliptic operators in odd subspaces

An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with boundary, or as spectral subspaces for self-adjoint elliptic differential operators of odd order. Index formulas are obtained for operators in odd subspaces on closed manifolds and for general boundary value problems. We prove that the eta-invariant of operators of odd order on even-dimesional manifolds is a dyadic rational number.Comment: 27 page

Quantum cohomology of flag manifolds and Toda lattices

We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice.Comment: 35 page

Theory of Anomalous Quantum Hall Effects in Graphene

Recent successes in manufacturing of atomically thin graphite samples (graphene) have stimulated intense experimental and theoretical activity. The key feature of graphene is the massless Dirac type of low-energy electron excitations. This gives rise to a number of unusual physical properties of this system distinguishing it from conventional two-dimensional metals. One of the most remarkable properties of graphene is the anomalous quantum Hall effect. It is extremely sensitive to the structure of the system; in particular, it clearly distinguishes single- and double-layer samples. In spite of the impressive experimental progress, the theory of quantum Hall effect in graphene has not been established. This theory is a subject of the present paper. We demonstrate that the Landau level structure by itself is not sufficient to determine the form of the quantum Hall effect. The Hall quantization is due to Anderson localization which, in graphene, is very peculiar and depends strongly on the character of disorder. It is only a special symmetry of disorder that may give rise to anomalous quantum Hall effects in graphene. We analyze the symmetries of disordered single- and double-layer graphene in magnetic field and identify the conditions for anomalous Hall quantization.Comment: 13 pages (article + supplementary material), 5 figure

Equivariant geometric K-homology for compact Lie group actions

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K^G_*(X), using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the "official" equivariant K-homology groups) and show that these are isomorphism.Comment: 25 pages. v2: some mistakes corrected, more detail added, Michael Walter as author added. To appear in Abhandlungen aus dem Mathematischen Seminar der Universit\"at Hambur

Spectral curves and the mass of hyperbolic monopoles

The moduli spaces of hyperbolic monopoles are naturally fibred by the monopole mass, and this leads to a nontrivial mass dependence of the holomorphic data (spectral curves, rational maps, holomorphic spheres) associated to hyperbolic multi-monopoles. In this paper, we obtain an explicit description of this dependence for general hyperbolic monopoles of magnetic charge two. In addition, we show how to compute the monopole mass of higher charge spectral curves with tetrahedral and octahedral symmetries. Spectral curves of euclidean monopoles are recovered from our results via an infinite-mass limit.Comment: 43 pages, LaTeX, 3 figure

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.