82 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 show that the fuzzification of a
projective toric variety amounts to a quantization of its toric base.Comment: 1+25 pages, extended version, to appear in JHE
On the Mini-Superambitwistor Space and N=8 Super Yang-Mills Theory
We construct a new supertwistor space suited for establishing a Penrose-Ward
transform between certain bundles over this space and solutions to the N=8
super Yang-Mills equations in three dimensions. This mini-superambitwistor
space is obtained by dimensional reduction of the superambitwistor space, the
standard superextension of the ambitwistor space. We discuss in detail the
construction of this space and its geometry before presenting the Penrose-Ward
transform. We also comment on a further such transform for purely bosonic
Yang-Mills-Higgs theory in three dimensions by considering third order formal
"sub-neighborhoods" of a mini-ambitwistor space.Comment: 24 pages, revised and published versio
The Mini-Superambitwistor Space
We present the construction of the mini-superambitwistor space, which is
suited for establishing a Penrose-Ward transform between certain bundles over
this space and solutions to the N=6 super Yang-Mills equations in three
dimensions.Comment: 8 pages, talk given at the International Workshop "Supersymmetries
and Quantum Symmetries" (SQS'05), Dubna, July 27-31 2005; to appear in the
proceeding
Towards an M5-Brane Model I: A 6d Superconformal Field Theory
We present an action for a six-dimensional superconformal field theory
containing a non-abelian tensor multiplet. All of the ingredients of this
action have been available in the literature. We bring these pieces together by
choosing the string Lie 2-algebra as a gauge structure, which we motivated in
previous work. The kinematical data contains a connection on a categorified
principal bundle, which is the appropriate mathematical description of the
parallel transport of self-dual strings. Our action can be written down for
each of the simply laced Dynkin diagrams, and each case reduces to a
four-dimensional supersymmetric Yang--Mills theory with corresponding gauge Lie
algebra. Our action also reduces nicely to an M2-brane model which is a
deformation of the ABJM model. While this action is certainly not the desired
M5-brane model, we regard it as a key stepping stone towards a potential
construction of the (2,0)-theory.Comment: 1+39 pages, v3: minor improvements, published versio
-Algebra Models and Higher Chern-Simons Theories
We continue our study of zero-dimensional field theories in which the fields
take values in a strong homotopy Lie algebra. In a first part, we review in
detail how higher Chern-Simons theories arise in the AKSZ-formalism. These
theories form a universal starting point for the construction of
-algebra models. We then show how to describe superconformal field
theories and how to perform dimensional reductions in this context. In a second
part, we demonstrate that Nambu-Poisson and multisymplectic manifolds are
closely related via their Heisenberg algebras. As a byproduct of our
discussion, we find central Lie -algebra extensions of .
Finally, we study a number of -algebra models which are physically
interesting and which exhibit quantized multisymplectic manifolds as vacuum
solutions.Comment: 44 pages, minor corrections, published versio
The Non-Abelian Self-Dual String and the (2,0)-Theory
We argue that the relevant higher gauge group for the non-abelian
generalization of the self-dual string equation is the string 2-group. We then
derive the corresponding equations of motion and discuss their properties. The
underlying geometric picture is a string structure, i.e. a categorified
principal bundle with connection whose structure 2-group is the string 2-group.
We readily write down the explicit elementary solution to our equations, which
is the categorified analogue of the 't Hooft-Polyakov monopole. Our solution
passes all the relevant consistency checks; in particular, it is globally
defined on and approaches the abelian self-dual string of charge
one at infinity. We note that our equations also arise as the BPS equations in
a recently proposed six-dimensional superconformal field theory and we show
that with our choice of higher gauge structure, the action of this theory can
be reduced to four-dimensional supersymmetric Yang-Mills theory.Comment: v3: 1+42 pages, presentation improved, typos fixed, published versio
Lie 2-algebra models
In this paper, we begin the study of zero-dimensional field theories with
fields taking values in a semistrict Lie 2-algebra. These theories contain the
IKKT matrix model and various M-brane related models as special cases. They
feature solutions that can be interpreted as quantized 2-plectic manifolds. In
particular, we find solutions corresponding to quantizations of R^3, S^3 and a
five-dimensional Hpp-wave. Moreover, by expanding a certain class of Lie
2-algebra models around the solution corresponding to quantized R^3, we obtain
higher BF-theory on this quantized space.Comment: 47 pages, presentation improved, version published in JHE
A Multitrace Matrix Model from Fuzzy Scalar Field Theory
We present the analytical approach to scalar field theory on the fuzzy sphere
which has been developed in arXiv:0706.2493 [hep-th]. This approach is based on
considering a perturbative expansion of the kinetic term in the partition
function. After truncating this expansion at second order, one arrives at a
multitrace matrix model, which allows for an application of the saddle-point
method. The results are in agreement with the numerical findings in the
literature.Comment: 8 pages, talk given by CS at the International Workshop
"Supersymmetries and Quantum Symmetries" (SQS'07), Dubna, July 30 - August 4
2007; to appear in the proceeding
The Phase Diagram of Scalar Field Theory on the Fuzzy Disc
Using a recently developed bootstrapping method, we compute the phase diagram
of scalar field theory on the fuzzy disc with quartic even potential. We find
three distinct phases with second and third order phase transitions between
them. In particular, we find that the second order phase transition happens
approximately at a fixed ratio of the two coupling constants defining the
potential. We compute this ratio analytically in the limit of large coupling
constants. Our results qualitatively agree with previously obtained numerical
results.Comment: 1+17 pages, v2: typos fixed, published versio
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