75 research outputs found
Fuzzy toric geometries
We describe a construction of fuzzy spaces which approximate projective toric varieties. The construction uses the canonical embedding of such varieties into a complex projective space: The algebra of fuzzy functions on a toric variety is obtained by a restriction of the fuzzy algebra of functions on the complex projective space appearing in the embedding. We give several explicit examples for this construction; in particular, we present fuzzy weighted projective spaces as well as fuzzy Hirzebruch and del Pezzo surfaces. As our construction is actually suited for arbitrary subvarieties of complex projective spaces, one can easily obtain large classes of fuzzy Calabi-Yau manifolds and we comment on fuzzy K3 surfaces and fuzzy quintic three-folds. Besides enlarging the number of available fuzzy spaces significantly, we find evidence for the conjecture that the fuzzification of a projective toric variety amounts to a quantization of its toric base
Lorentz meets Lipschitz
We show that maximal causal curves for a Lipschitz continuous Lorentzian
metric admit a -parametrization and that they solve the
geodesic equation in the sense of Filippov in this parametrization. Our proof
shows that maximal causal curves are either everywhere lightlike or everywhere
timelike. Furthermore, the proof demonstrates that maximal causal curves for an
-H\"older continuous Lorentzian metric admit a
-parametrization.Comment: 25 pages; v2: minor improvements of the presentatio
Six-Dimensional (1,0) Superconformal Models and Higher Gauge Theory
We analyze the gauge structure of a recently proposed superconformal field
theory in six dimensions. We find that this structure amounts to a weak
Courant-Dorfman algebra, which, in turn, can be interpreted as a strong
homotopy Lie algebra. This suggests that the superconformal field theory is
closely related to higher gauge theory, describing the parallel transport of
extended objects. Indeed we find that, under certain restrictions, the field
content and gauge transformations reduce to those of higher gauge theory. We
also present a number of interesting examples of admissible gauge structures
such as the structure Lie 2-algebra of an abelian gerbe, differential crossed
modules, the 3-algebras of M2-brane models and string Lie 2-algebras.Comment: 31+1 pages, presentation slightly improved, version published in JM
Quantized Nambu-Poisson Manifolds and n-Lie Algebras
We investigate the geometric interpretation of quantized Nambu-Poisson
structures in terms of noncommutative geometries. We describe an extension of
the usual axioms of quantization in which classical Nambu-Poisson structures
are translated to n-Lie algebras at quantum level. We demonstrate that this
generalized procedure matches an extension of Berezin-Toeplitz quantization
yielding quantized spheres, hyperboloids, and superspheres. The extended
Berezin quantization of spheres is closely related to a deformation
quantization of n-Lie algebras, as well as the approach based on harmonic
analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms
of foliations of R^n by fuzzy spheres, fuzzy hyperboloids, and noncommutative
hyperplanes. Some applications to the quantum geometry of branes in M-theory
are also briefly discussed.Comment: 43 pages, minor corrections, presentation improved, references adde
Non-Abelian Tensor Multiplet Equations from Twistor Space
We establish a Penrose-Ward transform yielding a bijection between
holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual
tensor fields on six-dimensional flat space-time. Extending the twistor space
to supertwistor space, we derive sets of manifestly N=(1,0) and N=(2,0)
supersymmetric non-Abelian constraint equations containing the tensor
multiplet. We also demonstrate how this construction leads to constraint
equations for non-Abelian supersymmetric self-dual strings.Comment: v3: 23 pages, revised version published in Commun. Math. Phy
Multiple M2-branes and Generalized 3-Lie algebras
We propose a generalization of the Bagger-Lambert-Gustavsson action as a
candidate for the description of an arbitrary number of M2-branes. The action
is formulated in terms of N=2 superfields in three dimensions and corresponds
to an extension of the usual superfield formulation of Chern-Simons matter
theories. Demanding gauge invariance of the resulting theory does not imply the
total antisymmetry of the underlying 3-Lie algebra structure constants. We
relax this condition and propose a class of examples for these generalized
3-Lie algebras. We also discuss how to associate various ordinary Lie algebras.Comment: 1+19 pages, version published in Phys. Rev.
Fuzzy Scalar Field Theory as a Multitrace Matrix Model
We develop an analytical approach to scalar field theory on the fuzzy sphere
based on considering a perturbative expansion of the kinetic term. This
expansion allows us to integrate out the angular degrees of freedom in the
hermitian matrices encoding the scalar field. The remaining model depends only
on the eigenvalues of the matrices and corresponds to a multitrace hermitian
matrix model. Such a model can be solved by standard techniques as e.g. the
saddle-point approximation. We evaluate the perturbative expansion up to second
order and present the one-cut solution of the saddle-point approximation in the
large N limit. We apply our approach to a model which has been proposed as an
appropriate regularization of scalar field theory on the plane within the
framework of fuzzy geometry.Comment: 1+25 pages, replaced with published version, minor improvement
Drinfeld-Twisted Supersymmetry and Non-Anticommutative Superspace
We extend the analysis of hep-th/0408069 on a Lorentz invariant
interpretation of noncommutative spacetime to field theories on
non-anticommutative superspace with half the supersymmetries broken. By
defining a Drinfeld-twisted Hopf superalgebra, it is shown that one can restore
twisted supersymmetry and therefore obtain a twisted version of the chiral
rings along with certain Ward-Takahashi identities. Moreover, we argue that the
representation content of theories on the deformed superspace is identical to
that of their undeformed cousins and comment on the consequences of our
analysis concerning non-renormalization theorems.Comment: 1+17 pages; typos fixed, minor correction
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