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This is the second in a series of papers on natural modification of the normal tractor connection in a parabolic geometry, which naturally prolongs an underlying overdetermined system of invariant differential equations. We give a short…
We use the embedding formalism to construct conformal fields in $D$ dimensions, by restricting Lorentz-invariant ensembles of homogeneous neural networks in $(D+2)$ dimensions to the projective null cone. Conformal correlators may be…
We describe applications of (perturbed) conformal field theories to two-dimensional disordered systems. We present various methods of study~: (i) {\it A direct method} in which we compute the explicit disorder dependence of the correlation…
We show that every 2nd order ODE defines a 4-parameter family of projective connections on its 2-dimensional solution space. In a special case of ODEs, for which a certain point transformation invariant vanishes, we find that this family of…
Conformal fields are a recently discovered class of representations of the algebra of vector fields in $N$ dimensions. Invariant first-order differential operators (exterior derivatives) for conformal fields are constructed.
For conformal field theories in arbitrary dimensions, we introduce a method to derive the conformal blocks corresponding to the exchange of a traceless symmetric tensor appearing in four point functions of operators with spin. Using the…
In this note, we consider how the bundle geometry of field space interplays with the covariant phase space methods so as to allow to write results of some generality on the presymplectic structure of invariant gauge theories coupled to…
Developing on the ideas of R. Stora and coworkers, a formulation of two dimensional field theory endowed with extended conformal symmetry is given, which is based on deformation theory of holomorphic and Hermitian spaces. The geometric…
Many examples of rank two bundles on ${\bf P}^4$ are constructed in positive characteristic. Construction depends on constructing certain special bundles on ${\bf P}^3$ which is shown to be equivalent to constructing bundles on ${\bf P}^4$…
When trained on language data, do transformers learn some arbitrary computation that utilizes the full capacity of the architecture or do they learn a simpler, tree-like computation, hypothesized to underlie compositional meaning systems…
There are two methods to study families of conformal theories in the operator formalism. In the first method we begin with a theory and a family of deformed theories is defined in the state space of the original theory. In the other there…
The decomposition of correlation functions into conformal blocks is an indispensable tool in conformal field theory. For spinning correlators, non-trivial tensor structures are needed to mediate between the conformal blocks, which are…
In this paper, projective representations of generalized chain geometries are investigated, using the concepts and results of part I. In particular, we study under which conditions such a projective representation maps the chains of a…
This review paper is concerned with the generalizations to field theory of the tangent and cotangent structures and bundles that play fundamental roles in the Lagrangian and Hamiltonian formulations of classical mechanics. The paper…
Transformer networks have seen great success in natural language processing and machine vision, where task objectives such as next word prediction and image classification benefit from nuanced context sensitivity across high-dimensional…
We study the conformal classes of 2-dimensional Lorentzian tori with (non zero) Killing fields. We define a map that associate to such a class a vector field on the circle (up to a scalar factor). This map is not injective but has finite…
We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this…
We provide a brief but self-contained review of two-dimensional conformal field theory, from the basic principles to some of the simplest models. From the representations of the Virasoro algebra on the one hand, and the state-field…
In work by Ausoni, Dundas and Rognes a half magnetic monopole is discovered and describes an obstruction to creating a determinant K(ku) \to ku*. In fact it is an obstruction to creating a determinant gerbe map from K(ku) to K(Z,3). We…
We associate bicomplexes with several integrable models in such a way that conserved currents are obtained by a simple iterative construction. Gauge transformations and dressings are discussed in this framework and several examples are…