Charging up the functional bootstrap

We revisit the problem of bootstrapping CFT correlators of charged fields. After discussing in detail how bounds for uncharged fields can be recycled to the charged case, we introduce two sets of analytic functional bases for correlators on the line. The first, which we call “simple”, is essentially a direct sum of analytic functionals for the uncharged case. We use it to establish very general bounds on the OPE density appearing in charged correlators. The second basis is dual to generalized free fields and we explain how it is related to a charged version of the Polyakov bootstrap. We apply these functionals to map out the space of correlators and obtain new improved bounds on the 3d Ising twist defect.

On holomorphic reflexivity conditions for complex Lie groups

We consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components, that (1) the topological Hopf algebra of holomorphic functions on $G$ is holomorphically reflexive if and only if $G$ is linear; (2) the dual cocommutative topological Hopf algebra of exponential analytic functional on $G$ is holomorphically reflexive. We give a counterexample, which shows that the first criterion cannot be extended to the case of infinitely many components. Nevertheless, we conjecture that, in general, the question can be solved in terms of the Banach-algebra linearity of $G$ .

Spectrum and Analytic Functional Calculus for Clifford Operators via Stem Functions

The main purpose of this work is the construction of an analytic functional calculus for Clifford operators, which are operators acting on certain modules over Clifford algebras. Unlike in some preceding works by other authors, we use a spectrum defined in the complex plane, and also certain stem functions, analytic in neighborhoods of such a spectrum. The replacement of the slice regular functions, having values in a Clifford algebra, by analytic stem functions becomes possible because of an isomorphism induced by a Cauchy type transform, whose existence is proved in the first part of this work.

Topographic gradients of intrinsic dynamics across neocortex

The intrinsic dynamics of neuronal populations are shaped by both microscale attributes and macroscale connectome architecture. Here we comprehensively characterize the rich temporal patterns of neural activity throughout the human brain. Applying massive temporal feature extraction to regional haemodynamic activity, we systematically estimate over 6000 statistical properties of individual brain regions’ time-series across the neocortex. We identify two robust spatial gradients of intrinsic dynamics, one spanning a ventromedial-dorsolateral axis and dominated by measures of signal autocorrelation, and the other spanning a unimodal-transmodal axis and dominated by measures of dynamic range. These gradients reflect spatial patterns of gene expression, intracortical myelin and cortical thickness, as well as structural and functional network embedding. Importantly, these gradients are correlated with patterns of meta-analytic functional activation, differentiating cognitive versus affective processing and sensory versus higher-order cognitive processing. Altogether, these findings demonstrate a link between microscale and macroscale architecture, intrinsic dynamics, and cognition.

How to succeed at Witten diagram recursions without really trying

Witten diagrams are basic objects for studying dynamics in AdS space, and also play key roles in the analytic functional bootstrap. However, these diagrams are notoriously hard to evaluate, making it extremely difficult to search for recursion relations among them. In this note, we present simple methods to obtain recursion relations for exchange Witten diagrams from conformal block recursion relations. We discover a variety of new relations, including the dimensional reduction formulae for exchange Witten diagrams. In particular, we find a five-term recursion relation relating exchange Witten diagrams in d and d − 2 dimensions. This gives the holographic analogue of a similar formula for conformal blocks due to Parisi-Sourlas supersymmetry. We also extend the analysis to two-point functions in CFTs with conformal boundaries, and obtain similar results.