Solitons and nonsmooth diffeomorphisms in conformal nets

We show that any solitonic representation of a conformal (diffeomorphism covariant) net on S^1 has positive energy and construct an uncountable family of mutually inequivalent solitonic representations of any conformal net, using nonsmooth diffeomorphisms. On the loop group nets, we show that these representations induce representations of the subgroup of loops compactly supported in S^1 \ {-1} which do not extend to the whole loop group. In the case of the U(1)-current net, we extend the diffeomorphism covariance to the Sobolev diffeomorphisms D^s(S^1), s > 2, and show that the positive-energy vacuum representations of Diff_+(S^1) with integer central charges extend to D^s(S^1). The solitonic representations constructed above for the U(1)-current net and for Virasoro nets with integral central charge are continuously covariant with respect to the stabilizer subgroup of Diff_+(S^1) of -1 of the circle.Comment: 33 pages, 3 TikZ figure

Positive energy representations of Sobolev diffeomorphism groups of the circle

We show that any positive energy projective representation of Diff(S^1) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms D^s(S^1) with s>3, and in particular to C^k-diffeomorphisms Diff^k(S^1) with k >= 4. A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on S^1 is covariant with respect to D^s(S^1), s > 3. Moreover every direct sum of irreducible representations of a conformal net is also D^s(S^1)-covariant.Comment: 30 pages, 1 TikZ figur

On truncated tt-free Fock spaces: spectrum of position operators and shift-invariant states

The ergodic properties of the shift on both full and $m$-truncated $t$-free $C^*$-algebras are analyzed. In particular, the shift is shown to be uniquely ergodic with respect to the fixed-point algebra. In addition, for every $m\geq 1$, the invariant states of the shift acting on the $m$-truncated $t$-free $C^*$-algebra are shown to yield a $m+1$-dimensional Choquet simplex, which collapses to a segment in the full case. Finally, the spectrum of the position operators on the $m$-truncated $t$-free Fock space is also determined.Comment: 15 page

Low energy spectrum of the XXZ model coupled to a magnetic field

It is shown that, for a class of Hamiltonians of XXZ chains in an external magnetic field that are small perturbations of an Ising Hamiltonian, the spectral gap above the ground-state energy remains strictly positive when the perturbation is turned on, uniformly in the length of the chain. The result is proven for both the ferromagnetic and the antiferromagnetic Ising Hamiltonian; in the latter case the external magnetic field is required to be small, and for an even number of sites the two-fold degenerate ground-state energy of the unperturbed Hamiltonian may split into two energy levels whose difference is small. This result is proven by using a new, quite subtle refinement of a method developed in earlier work and used to iteratively block-diagonalize Hamiltonians of ever larger subsystems with the help of local unitary conjugations. One novel ingredient of the method presented in this paper consists of the use of Lieb-Robinson bounds.Comment: 6 figure

Freedman's theorem for unitarily invariant states on the CCR algebra

The set of states on ${\rm CCR}(\ch)$, the CCR algebra of a separable Hilbert space $\ch$, is here looked at as a natural object to obtain a non-commutative version of Freedman's theorem for unitarily invariant stochastic processes. In this regard, we provide a complete description of the compact convex set of states of ${\rm CCR}(\ch)$ that are invariant under the action of all automorphisms induced in second quantization by unitaries of $\ch$. We prove that this set is a Bauer simplex, whose extreme states are either the canonical trace of the CCR algebra or Gaussian states with variance at least $1$.Comment: 22 page

Infinite index extensions of local nets and defects

Subfactor theory provides a tool to analyze and construct extensions of Quantum Field Theories, once the latter are formulated as local nets of von Neumann algebras. We generalize some of the results of [LR95] to the case of extensions with infinite Jones index. This case naturally arises in physics, the canonical examples are given by global gauge theories with respect to a compact (non-finite) group of internal symmetries. Building on the works of Izumi, Longo, Popa [ILP98] and Fidaleo, Isola [FI99], we consider generalized Q-systems (of intertwiners) for a semidiscrete inclusion of properly infinite von Neumann algebras, which generalize ordinary Q-systems introduced by Longo [Lon94] to the infinite index case. We characterize inclusions which admit generalized Q-systems of intertwiners and define a braided product among the latter, hence we construct examples of QFTs with defects (phase boundaries) of infinite index, extending the family of boundaries in the grasp of [BKLR16].Comment: 50 page

Comparison Theorems for Stochastic Chemical Reaction Networks

Continuous-time Markov chains are frequently used as stochastic models for chemical reaction networks, especially in the growing field of systems biology. A fundamental problem for these Stochastic Chemical Reaction Networks (SCRNs) is to understand the dependence of the stochastic behavior of these systems on the chemical reaction rate parameters. Towards solving this problem, in this paper we develop theoretical tools called comparison theorems that provide stochastic ordering results for SCRNs. These theorems give sufficient conditions for monotonic dependence on parameters in these network models, which allow us to obtain, under suitable conditions, information about transient and steady state behavior. These theorems exploit structural properties of SCRNs, beyond those of general continuous-time Markov chains. Furthermore, we derive two theorems to compare stationary distributions and mean first passage times for SCRNs with different parameter values, or with the same parameters and different initial conditions. These tools are developed for SCRNs taking values in a generic (finite or countably infinite) state space and can also be applied for non-mass-action kinetics models. When propensity functions are bounded, our method of proof gives an explicit method for coupling two comparable SCRNs, which can be used to simultaneously simulate their sample paths in a comparable manner. We illustrate our results with applications to models of enzymatic kinetics and epigenetic regulation by chromatin modifications.Comment: Compared to the first version, the Supplementary Information (SI) file has been adde

The web of laughter: frontal and limbic projections of the anterior cingulate cortex revealed by cortico-cortical evoked potential from sites eliciting laughter

According to an evolutionist approach, laughter is a multifaceted behaviour affecting social, emotional, motor and speech functions. Albeit previous studies have suggested that high-frequency electrical stimulation (HF-ES) of the pregenual anterior cingulate cortex ( pACC) may induce bursts of laughter—suggesting a crucial contribution of this region to the cortical con- trol of this behaviour—the complex nature of laughter implies that outward connections from the pACC may reach and affect a complex network of fron- tal and limbic regions. Here, we studied the effective connectivity of the pACC by analysing the cortico-cortical evoked potentials elicited by single-pulse electrical stimulation of pACC sites whose HF-ES elicited laugh- ter in 12 patients. Once these regions were identified, we studied their clinical response to HF-ES, to reveal the specific functional target of pACC representation of laughter. Results reveal that the neural representation of laughter in the pACC interacts with several frontal and limbic regions, including cingulate, orbitofrontal, medial prefrontal and anterior insular regions—involved in interoception, emotion, social reward and motor be- haviour. These results offer neuroscientific support to the evolutionist approach to laughter, providing a possible mechanistic explanation of the interplay between this behaviour and emotion regulation, speech production and social interactions. This article is part of the theme issue ‘Cracking the laugh code: laughter through the lens of biology, psychology, and neuroscience’