Symplectic Dirac-K\"ahler Fields

For the description of space-time fermions, Dirac-K\"ahler fields (inhomogeneous differential forms) provide an interesting alternative to the Dirac spinor fields. In this paper we develop a similar concept within the symplectic geometry of phase-spaces. Rather than on space-time, symplectic Dirac-K\"ahler fields can be defined on the classical phase-space of any Hamiltonian system. They are equivalent to an infinite family of metaplectic spinor fields, i.e. spinors of Sp(2N), in the same way an ordinary Dirac-K\"ahler field is equivalent to a (finite) mulitplet of Dirac spinors. The results are interpreted in the framework of the gauge theory formulation of quantum mechanics which was proposed recently. An intriguing analogy is found between the lattice fermion problem (species doubling) and the problem of quantization in general.Comment: 86 pages, late

Twisted Superspace for N=D=2 Super BF and Yang-Mills with Dirac-K\"ahler Fermion Mechanism

We propose a twisted D=N=2 superspace formalism. The relation between the twisted super charges including the BRST charge, vector and pseudo scalar super charges and the N=2 spinor super charges is established. We claim that this relation is essentially related with the Dirac-K\"ahler fermion mechanism. We show that a fermionic bilinear form of twisted N=2 chiral and anti-chiral superfields is equivalent to the quantized version of BF theory with the Landau type gauge fixing while a bosonic bilinear form leads to the N=2 Wess-Zumino action. We then construct a Yang-Mills action described by the twisted N=2 chiral and vector superfields, and show that the action is equivalent to the twisted version of the D=N=2 super Yang-Mills action, previously obtained from the quantized generalized topological Yang-Mills action with instanton gauge fixing.Comment: 36 page

On finitely ambiguous B\"uchi automata

Unambiguous B\"uchi automata, i.e. B\"uchi automata allowing only one accepting run per word, are a useful restriction of B\"uchi automata that is well-suited for probabilistic model-checking. In this paper we propose a more permissive variant, namely finitely ambiguous B\"uchi automata, a generalisation where each word has at most $k$ accepting runs, for some fixed $k$. We adapt existing notions and results concerning finite and bounded ambiguity of finite automata to the setting of $\omega$-languages and present a translation from arbitrary nondeterministic B\"uchi automata with $n$ states to finitely ambiguous automata with at most $3^n$ states and at most $n$ accepting runs per word

Positivity and conservation of superenergy tensors

Two essential properties of energy-momentum tensors T_{\mu\nu} are their positivity and conservation. This is mathematically formalized by, respectively, an energy condition, as the dominant energy condition, and the vanishing of their divergence \nabla^\mu T_{\mu\nu}=0. The classical Bel and Bel-Robinson superenergy tensors, generated from the Riemann and Weyl tensors, respectively, are rank-4 tensors. But they share these two properties with energy momentum tensors: the Dominant Property (DP) and the divergence-free property in the absence of sources (vacuum). Senovilla defined a universal algebraic construction which generates a basic superenergy tensor T{A} from any arbitrary tensor A. In this construction the seed tensor A is structured as an r-fold multivector, which can always be done. The most important feature of the basic superenergy tensors is that they satisfy automatically the DP, independently of the generating tensor A. In a previous paper we presented a more compact definition of T{A} using the r-fold Clifford algebra. This form for the superenergy tensors allowed to obtain an easy proof of the DP valid for any dimension. In this paper we include this proof. We explain which new elements appear when we consider the tensor T{A} generated by a non-degree-defined r-fold multivector A and how orthogonal Lorentz transformations and bilinear observables of spinor fields are included as particular cases of superenergy tensors. We find some sufficient conditions for the seed tensor A, which guarantee that the generated tensor T{A} is divergence-free. These sufficient conditions are satisfied by some physical fields, which are presented as examples.Comment: 19 pages, no figures. Language and minor changes. Published versio

On local boundary CFT and non-local CFT on the boundary

The holographic relation between local boundary conformal quantum field theories (BCFT) and their non-local boundary restrictions is reviewed, and non-vacuum BCFT's, whose existence was conjectured previously, are constructed.Comment: 16 pages. Contribution to "Rigorous Quantum Field Theory", Symposium in honour of J. Bros, Paris, July 2004. Based on joint work math-ph/0405067 with R. Long

Dirac-Hestenes spinor fields in Riemann-Cartan spacetime

In this paper we study Dirac-Hestenes spinor fields (DHSF) on a four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields must be defined as certain equivalence classes of even sections of the Clifford bundle (over the RCST), thereby being certain particular sections of a new bundle named Spin-Clifford bundle (SCB). The conditions for the existence of the SCB are studied and are shown to be equivalent to the famous Geroch's theorem concerning to the existence of spinor structures in a Lorentzian spacetime. We introduce also the covariant and algebraic Dirac spinor fields and compare these with DHSF, showing that all the three kinds of spinor fields contain the same mathematical and physical information. We clarify also the notion of (Crumeyrolle's) amorphous spinors (Dirac-K\"ahler spinor fields are of this type), showing that they cannot be used to describe fermionic fields. We develop a rigorous theory for the covariant derivatives of Clifford fields (sections of the Clifford bundle (CB)) and of Dirac-Hestenes spinor fields. We show how to generalize the original Dirac-Hestenes equation in Minkowski spacetime for the case of a RCST. Our results are obtained from a variational principle formulated through the multiform derivative approach to Lagrangian field theory in the Clifford bundle.Comment: 45 pages, special macros kapproc.sty and makro822.te

New Results in Sasaki-Einstein Geometry

This article is a summary of some of the author's work on Sasaki-Einstein geometry. A rather general conjecture in string theory known as the AdS/CFT correspondence relates Sasaki-Einstein geometry, in low dimensions, to superconformal field theory; properties of the latter are therefore reflected in the former, and vice versa. Despite this physical motivation, many recent results are of independent geometrical interest, and are described here in purely mathematical terms: explicit constructions of infinite families of both quasi-regular and irregular Sasaki-Einstein metrics; toric Sasakian geometry; an extremal problem that determines the Reeb vector field for, and hence also the volume of, a Sasaki-Einstein manifold; and finally, obstructions to the existence of Sasaki-Einstein metrics. Some of these results also provide new insights into Kahler geometry, and in particular new obstructions to the existence of Kahler-Einstein metrics on Fano orbifolds.Comment: 31 pages, no figures. Invited contribution to the proceedings of the conference "Riemannian Topology: Geometric Structures on Manifolds"; minor typos corrected, reference added; published version; Riemannian Topology and Geometric Structures on Manifolds (Progress in Mathematics), Birkhauser (Nov 2008

Polygenic overlap between schizophrenia risk and antipsychotic response: a genomic medicine approach

Therapeutic treatments for schizophrenia do not alleviate symptoms for all patients and efficacy is limited by common, often severe, side-effects. Genetic studies of disease can identify novel drug targets, and drugs for which the mechanism has direct genetic support have increased likelihood of clinical success. Large-scale genetic studies of schizophrenia have increased the number of genes and gene sets associated with risk. We aimed to examine the overlap between schizophrenia risk loci and gene targets of a comprehensive set of medications to potentially inform and improve treatment of schizophrenia

N=2 Supersymmetric Model with Dirac-Kahler Fermions from Generalized Gauge Theory in Two Dimensions

We investigate the generalized gauge theory which has been proposed previously and show that in two dimensions the instanton gauge fixing of the generalized topological Yang-Mills action leads to a twisted N=2 supersymmetric action. We have found that the R-symmetry of N=2 supersymmetry can be identified with the flavour symmetry of Dirac-Kahler fermion formulation. Thus the procedure of twist allows topological ghost fields to be interpreted as the Dirac-Kahler matter fermions.Comment: 22 pages, LaTe