L∞L_\infty-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism

We review in detail the Batalin-Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an $L_\infty$-algebra and how quasi-isomorphisms between $L_\infty$-algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern-Simons theories and give some useful shortcuts in usually rather involved computations.Comment: v3: 131 pages, minor improvements, published versio

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

Synaptic Sampling of Neural Networks

Probabilistic artificial neural networks offer intriguing prospects for enabling the uncertainty of artificial intelligence methods to be described explicitly in their function; however, the development of techniques that quantify uncertainty by well-understood methods such as Monte Carlo sampling has been limited by the high costs of stochastic sampling on deterministic computing hardware. Emerging computing systems that are amenable to hardware-level probabilistic computing, such as those that leverage stochastic devices, may make probabilistic neural networks more feasible in the not-too-distant future. This paper describes the scANN technique -- \textit{sampling (by coinflips) artificial neural networks} -- which enables neural networks to be sampled directly by treating the weights as Bernoulli coin flips. This method is natively well suited for probabilistic computing techniques that focus on tunable stochastic devices, nearly matches fully deterministic performance while also describing the uncertainty of correct and incorrect neural network outputs.Comment: 9 pages, accepted to 2023 IEEE International Conference on Rebooting Computin

Induced Distributions from Generalized Unfair Dice

In this paper we analyze the probability distributions associated with rolling (possibly unfair) dice infinitely often. Specifically, given a $q$-sided die, if $x_i\in\{0,\ldots,q-1\}$ denotes the outcome of the $i^{\text{th}}$ toss, then the distribution function is $F(x)=\mathbb{P}[X\leq x]$, where $X = \sum_{i=1}^\infty x_i q^{-i}$. We show that $F$ is singular and establish a piecewise linear, iterative construction for it. We investigate two ways of comparing $F$ to the fair distribution -- one using supremum norms and another using arclength. In the case of coin flips, we also address the case where each independent flip could come from a different distribution. In part, this work aims to address outstanding claims in the literature on Bernoulli schemes. The results herein are motivated by emerging needs, desires, and opportunities in computation to leverage physical stochasticity in microelectronic devices for random number generation.Comment: 18 pages, 1 figur

Neurogenesis Deep Learning

Neural machine learning methods, such as deep neural networks (DNN), have achieved remarkable success in a number of complex data processing tasks. These methods have arguably had their strongest impact on tasks such as image and audio processing - data processing domains in which humans have long held clear advantages over conventional algorithms. In contrast to biological neural systems, which are capable of learning continuously, deep artificial networks have a limited ability for incorporating new information in an already trained network. As a result, methods for continuous learning are potentially highly impactful in enabling the application of deep networks to dynamic data sets. Here, inspired by the process of adult neurogenesis in the hippocampus, we explore the potential for adding new neurons to deep layers of artificial neural networks in order to facilitate their acquisition of novel information while preserving previously trained data representations. Our results on the MNIST handwritten digit dataset and the NIST SD 19 dataset, which includes lower and upper case letters and digits, demonstrate that neurogenesis is well suited for addressing the stability-plasticity dilemma that has long challenged adaptive machine learning algorithms.Comment: 8 pages, 8 figures, Accepted to 2017 International Joint Conference on Neural Networks (IJCNN 2017

Modular classes of skew algebroid relations

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E* which is invariant with respect to all Hamiltonian vector fields if and only if E is modular, i.e. mod(E)=0. Further, relative modular class of a subalgebroid is introduced and studied together with its application to holonomy, as well as modular class of a skew algebroid relation. These notions provide, in particular, a unified approach to the concepts of a modular class of a Lie algebroid morphism and that of a Poisson map.Comment: 20 page

Topological Field Theories and Geometry of Batalin-Vilkovisky Algebras

The algebraic and geometric structures of deformations are analyzed concerning topological field theories of Schwarz type by means of the Batalin-Vilkovisky formalism. Deformations of the Chern-Simons-BF theory in three dimensions induces the Courant algebroid structure on the target space as a sigma model. Deformations of BF theories in $n$ dimensions are also analyzed. Two dimensional deformed BF theory induces the Poisson structure and three dimensional deformed BF theory induces the Courant algebroid structure on the target space as a sigma model. The deformations of BF theories in $n$ dimensions induce the structures of Batalin-Vilkovisky algebras on the target space.Comment: 25 page

Classical BV theories on manifolds with boundary

In this paper we extend the classical BV framework to gauge theories on spacetime manifolds with boundary. In particular, we connect the BV construction in the bulk with the BFV construction on the boundary and we develop its extension to strata of higher codimension in the case of manifolds with corners. We present several examples including electrodynamics, Yang-Mills theory and topological field theories coming from the AKSZ construction, in particular, the Chern-Simons theory, the $BF$ theory, and the Poisson sigma model. This paper is the first step towards developing the perturbative quantization of such theories on manifolds with boundary in a way consistent with gluing.Comment: The second version has many typos corrected, references added. Some typos are probably still there, in particular, signs in examples. In the third version more typoes are corrected and the exposition is slightly change