15 research outputs found
The trouble with twisting (2,0) theory
We consider a twisted version of the abelian theory placed upon a
Lorenzian six-manifold with a product structure, . This is
done by an investigation of the free tensor multiplet on the level of equations
of motion, where the problem of its formulation in Euclidean signature is
circumvented by letting the time-like direction lie in the two-manifold and
performing a topological twist along alone. A compactification on is
shown to be necessary to enable the possibility of finding a topological field
theory. The hypothetical twist along a Euclidean is argued to amount to the
correct choice of linear combination of the two supercharges scalar on .
This procedure is expected and conjectured to result in a topological field
theory, but we arrive at the surprising conclusion that this twisted theory
contains no -exact and covariantly conserved stress tensor unless has
vanishing curvature. This is to our knowledge a phenomenon which has not been
observed before in topological field theories. In the literature, the setup of
the twisting used here has been suggested as the origin of the conjectured
AGT-correspondence, and our hope is that this work may somehow contribute to
the understanding of it.Comment: 25 pages, v2: Some further clarifications including an extended
discussion on the relation to other topological twistings. References adde
Off-shell structure of twisted (2,0) theory
A -exact off-shell action is constructed for twisted abelian (2,0) theory
on a Lorentzian six-manifold of the form , where is
a flat two-manifold and is a general Euclidean four-manifold. The
properties of this formulation, which is obtained by introducing two auxiliary
fields, can be summarised by a commutative diagram where the Lagrangian and its
stress-tensor arise from the -variation of two fermionic quantities and
. This completes and extends the analysis in
[arXiv:1311.3300].Comment: 15 pages, 2 figure
Geometric deep learning and equivariant neural networks
We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds M using principal bundles with structure group K and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces M= G/ K , which are instead equivariant with respect to the global symmetry G on M . Group equivariant layers can be interpreted as intertwiners between induced representations of G, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case M= S2= SO (3) / SO (2) . Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch–Gordan coefficients for G= SO (3) , illustrating the power of representation theory for deep learning
HEAL-SWIN: A Vision Transformer On The Sphere
High-resolution wide-angle fisheye images are becoming more and more
important for robotics applications such as autonomous driving. However, using
ordinary convolutional neural networks or vision transformers on this data is
problematic due to projection and distortion losses introduced when projecting
to a rectangular grid on the plane. We introduce the HEAL-SWIN transformer,
which combines the highly uniform Hierarchical Equal Area iso-Latitude
Pixelation (HEALPix) grid used in astrophysics and cosmology with the
Hierarchical Shifted-Window (SWIN) transformer to yield an efficient and
flexible model capable of training on high-resolution, distortion-free
spherical data. In HEAL-SWIN, the nested structure of the HEALPix grid is used
to perform the patching and windowing operations of the SWIN transformer,
resulting in a one-dimensional representation of the spherical data with
minimal computational overhead. We demonstrate the superior performance of our
model for semantic segmentation and depth regression tasks on both synthetic
and real automotive datasets. Our code is available at
https://github.com/JanEGerken/HEAL-SWIN.Comment: Main body: 10 pages, 7 figures. Appendices: 4 pages, 2 figure
(2,0) theory on circle fibrations
We consider (2,0) theory on a manifold M_6 that is a fibration of a spatial
S^1 over some five-dimensional base manifold M_5. Initially, we study the free
(2,0) tensor multiplet which can be described in terms of classical equations
of motion in six dimensions. Given a metric on M_6 the low energy effective
theory obtained through dimensional reduction on the circle is a Maxwell theory
on M_5. The parameters describing the local geometry of the fibration are
interpreted respectively as the metric on M_5, a non-dynamical U(1) gauge field
and the coupling strength of the resulting low energy Maxwell theory. We derive
the general form of the action of the Maxwell theory by integrating the reduced
equations of motion, and consider the symmetries of this theory originating
from the superconformal symmetry in six dimensions. Subsequently, we consider a
non-abelian generalization of the Maxwell theory on M_5. Completing the theory
with Yukawa and phi^4 terms, and suitably modifying the supersymmetry
transformations, we obtain a supersymmetric Yang-Mills theory which includes
terms related to the geometry of the fibration.Comment: 24 pages, v2 References added, typos correcte
Fast convolutional neural networks on FPGAs with hls4ml
We introduce an automated tool for deploying ultra low-latency, low-power
deep neural networks with convolutional layers on FPGAs. By extending the
hls4ml library, we demonstrate an inference latency of s using
convolutional architectures, targeting microsecond latency applications like
those at the CERN Large Hadron Collider. Considering benchmark models trained
on the Street View House Numbers Dataset, we demonstrate various methods for
model compression in order to fit the computational constraints of a typical
FPGA device used in trigger and data acquisition systems of particle detectors.
In particular, we discuss pruning and quantization-aware training, and
demonstrate how resource utilization can be significantly reduced with little
to no loss in model accuracy. We show that the FPGA critical resource
consumption can be reduced by 97% with zero loss in model accuracy, and by 99%
when tolerating a 6% accuracy degradation.Comment: 18 pages, 18 figures, 4 table