arXiv subjects

Simon Wood

Publications and source records attributed to Simon Wood.

At least 19 recordsLinked to original sources

arXiv2024

Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras

We prove that the categories of weight modules over the simple $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator algebras at fractional admissible levels and central charges are rigid (and hence the categories of weight modules are braided ribbon categories) and that the decomposition formulae of fusion products of simple projective modules conjectured by Thomas Creutzig, David Ridout and collaborators hold (including when the decomposition involves summands that are indecomposable yet not simple). In addition to solving this old open problem, we develop new techniques for the construction of intertwining operators by means of integrating screening currents over certain cycles, which are expected to be of independent interest, due to their applicability to many other algebras. In the example of $\mathfrak{sl}(2)$ these new techniques allow us to give explicit formulae for a logarithmic intertwining operator from a pair of simple projective modules to the projective cover of the tensor unit, namely, the vertex operator algebra as a module over itself.

math.QA↗

arXiv2024

The Visual Monitoring Camera (VMC) on Mars Express: a new science instrument made from an old webcam orbiting Mars

The Visual Monitoring Camera (VMC) is a small imaging instrument onboard Mars Express with a field of view of ~40x30 degrees. The camera was initially intended to provide visual confirmation of the separation of the Beagle 2 lander and has similar technical specifications to a typical webcam of the 2000s. In 2007, a few years after the end of its original mission, VMC was turned on again to obtain full-disk images of Mars to be used for outreach purposes. As VMC obtained more images, the scientific potential of the camera became evident, and in 2018 the camera was given an upgraded status of a new scientific instrument, with science goals in the field of Martian atmosphere meteorology. The wide Field of View of the camera combined with the orbit of Mars Express enable the acquisition of full-disk images of the planet showing different local times, which for a long time has been rare among orbital missions around Mars. The small data volume of images also allows videos that show the atmospheric dynamics of dust and cloud systems to be obtained. This paper is intended to be the new reference paper for VMC as a scientific instrument, and thus provides an overview of the updated procedures to plan, command and execute science observations of the Martian atmosphere. These observations produce valuable science data that is calibrated and distributed to the community for scientific use.

astro-ph.IM↗

arXiv2024

Grothendieck-Verdier module categories, Frobenius algebras and relative Serre functors

We develop the theory of module categories over a Grothendieck-Verdier category, i.e. a monoidal category with a dualizing object and hence a duality structure more general than rigidity. Such a category C comes with two monoidal structures which are related by non-invertible morphisms and which we treat on an equal footing. Quite generally, non-invertible structure morphisms play a dominant role in this theory. In any Grothendieck-Verdier module category M we find two important subcategories M' and M''. The internal End of an object in M' that is a C-generator is an algebra such that its category of modules is equivalent to M as a module category. We also introduce a partially defined relative Serre functor S which furnishes an equivalence between M' and M''. Any isomorphism between an object m of M' and S(m) in M'' endows the internal End of m with the structure of a Grothendieck-Verdier Frobenius algebra.

math.CT↗

arXiv2024

The modular properties of $\mathfrak{sl}(2)$ torus $1$-point functions

Conformal field theory and its axiomatisation in terms of vertex operator algebras or chiral algebras are most commonly considered on the Riemann sphere. However, an important constraint in physics and an interesting source of mathematics is the fact that conformal field theories are expected to be well defined on any Riemann surface. To this end, a thorough understanding of chiral torus 1-point functions, ideally including explicit formulae, is a prerequisite for a detailed understanding of higher genera. These are distinguished from characters or vacuum torus 1-point functions because the insertion point is explicitly allowed to be labelled by any module over the vertex operator algebra rather than just the vertex operator algebra itself. Compellingly, chiral torus 1-point functions exhibit interesting modular properties, which we explore here in the example of the simple affine $\mathfrak{sl}(2)$ vertex operator algebras at non-negative integral levels. We determine the dimension of the space spanned by such functions, choose a natural basis to construct vector-valued modular forms and describe the congruence properties of these forms. In particular, we find explicit generators for the spaces of all vector-valued modular forms of dimension at most three, when the insertion comes from a simple module other than the vertex operator algebra. Finally, we use the fact that categories of modules over rational vertex operator algebras are modular tensor categories to give explicit formulae for the action of the modular group on chiral torus 1-point functions entirely in terms of categorical data. The usual modular $\mathsf{S}$ and $\mathsf{T}$ matrices of characters are known not to be complete invariants of modular tensor categories, so these generalised modular data are good candidates for more fine-grained invariants.

math.QA↗

arXiv2023

Grothendieck-Verdier duality in categories of bimodules and weak module functors

Various monoidal categories, including suitable representation categories of vertex operator algebras, admit natural Grothendieck-Verdier duality structures. We recall that such a Grothendieck-Verdier category comes with two tensor products which should be related by distributors obeying pentagon identities. We discuss in which circumstances these distributors are isomorphisms. This is achieved by taking the perspective of module categories over monoidal categories, using in particular the natural weak module functor structure of internal Homs and internal coHoms. As an illustration, we exhibit these concepts concretely in the case of categories of bimodules over associative algebras.

math.CT↗

arXiv2023

Algebraic structures in two-dimensional conformal field theory

This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics. We review the following algebraic structures which appear in two-dimensional conformal field theory (CFT): The symmetries of two-dimensional conformal field theories (CFTs) can be formalised as chiral algebras, vertex operator algebras or nets of observable algebras. Their representation categories are abelian categories having additional structures, which are induced by properties of conformal blocks, i.e. of vector bundles over the moduli space of curves with marked points, which can be constructed from the symmetry structure. These mathematical notions pertain to the description of chiral CFTs. In a full local CFT one deals in addition with correlators, which are specific elements in the spaces of conformal blocks. In fact, a full CFT is the same as a consistent system of correlators for arbitrary conformal surfaces with any number and type of field insertions in the bulk as well as on boundaries and on topological defect lines. We present algebraic structures that allow one to construct such systems of correlators.

math.QA↗

arXiv2021

A note on the modeling of the effects of experimental time in psycholinguistic experiments

Thul et al. (2020) called attention to problems that arise when chronometric experiments implementing specific factorial designs are analysed with the generalized additive mixed model (GAMM), using factor smooths to capture trial-to-trial dependencies. From a series of simulations incorporating such dependencies, they conclude that GAMMs are inappropriate for between-subject designs. They argue that in addition GAMMs come with too many modeling possibilities, and advise using the linear mixed model (LMM) instead. We address the questions raised by Thul et al. (2020), who clearly demonstrated that problems can indeed arise when using factor smooths in combination with factorial designs. We show that the problem does not arise when using by-smooths. Furthermore, we have traced a bug in the implementation of factor smooths in the mgcv package, which will have been removed from version 1.8-36 onwards. To illustrate that GAMMs now produce correct estimates, we report simulation studies implementing different by-subject longitudinal effects. The maximal LMM emerges as slightly conservative compared to GAMMs, and GAMMs provide estimated coefficients that can be less variable across simulation runs. We also discuss two datasets where time-varying effects interact with numerical predictors in a theoretically informative way. Furthermore, we argue that the wide range of tools that GAMMs make available to researcher across all domains of scientific inquiry do not come with uncontrolled researcher degrees of freedom once confronted with a specific psycholinguistic datasets. We also introduce a distinction between replicable and non-replicable non-linear effects. We conclude that GAMMs are an excellent and reliable tool for understanding experimental data, including chronometric data with time-varying effects.

stat.ME↗

arXiv2021

Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson

Huang, Lepowsky and Zhang have developed a module theory for vertex operator algebras that endows suitably chosen module categories with the structure of braided monoidal categories. Included in the theory is a functor which assigns to discretely strongly graded modules a contragredient module, obtained as a gradewise dual. In this paper, we show that this gradewise dual endows the module category with the structure of a ribbon Grothendieck-Verdier category. This duality structure is more general than that of a rigid monoidal category; in contrast to rigidity, it naturally accommodates the fact that a vertex operator algebra and its gradewise dual need not be isomorphic as modules and that the tensor product of modules over vertex operator algebras need not be exact. We develop criteria which allow the detection of ribbon Grothendieck-Verdier equivalences and use them to explore ribbon Grothendieck-Verdier structures in the example of the rank $n$ Heisenberg vertex operator algebra or chiral free boson on a not necessarily full rank even lattice with arbitrary choice of conformal vector. We show that these categories are equivalent, as ribbon Grothendieck-Verdier categories, to certain categories of graded vector spaces and categories of modules over a certain Hopf algebra.

math.QA↗

arXiv2021

Admissible-level $\mathfrak{sl}_3$ minimal models

The first part of this work uses the algorithm recently detailed in arXiv:1906.02935 to classify the irreducible weight modules of the minimal model vertex operator algebra $L_k(\mathfrak{sl}_3)$, when the level $k$ is admissible. These are naturally described in terms of families parametrised by up to two complex numbers. We also determine the action of the relevant group of automorphisms of $\hat{\mathfrak{sl}}_3$ on their isomorphism classes and compute explicitly the decomposition into irreducibles when a given family's parameters are permitted to take certain limiting values. Along with certain character formulae, previously established in arXiv:2003.10148, these results form the input data required by the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level $\mathfrak{sl}_3$ minimal models. The second part of this work applies the standard module formalism to compute these explicitly when $k=-\frac32$. We expect that the methodology developed here will apply in much greater generality.

math.QA↗

arXiv2021

Photodynamically Active Electrospun Fibres for Antibiotic-Free Infection Control

Antimicrobial biomaterials are critical to aid in the regeneration of oral soft tissue and prevent or treat localised bacterial infections. With the rising trend in antibiotic resistance, there is a pressing clinical need for new antimicrobial chemistries and biomaterial design approaches enabling on-demand activation of antibiotic-free antimicrobial functionality following an infection that are environment-friendly, flexible and commercially-viable. This study explores the feasibility of integrating a bioresorbable electrospun polymer scaffold with localised antimicrobial photodynamic therapy (aPDT) capability. To enable aPDT, we encapsulated a photosensitiser (PS) in polyester fibres in the PS inert state, so that the antibacterial function would be activated on-demand via a visible light source. Fibrous scaffolds were successfully electrospun from FDA-approved polyesters, either poly(epsilon-caprolactone (PCL) or poly[(rac-lactide)-co-glycolide] (PLGA) with encapsulated PS (either methylene blue (MB) or erythrosin B (ER)). The electrospun fibres achieved an ~100 wt.% loading efficiency of PS, which significantly increased their tensile modulus and reduced their average fibre diameter and pore size with respect to PS-free controls. In vitro, PS release varied between a burst release profile to limited release within 100 hours depending on the selected scaffold formulation. Exposure of PS-encapsulated PCL fibres to visible light successfully led to at least a 1 log reduction in E. coli viability after 60 minutes of light exposure whereas PS-free electrospun controls did not inactive microbes. This study successfully demonstrates the significant potential of PS-encapsulated electrospun fibres as photodynamically active biomaterial for antibiotic-free infection control.

q-bio.TO↗

arXiv2021

Hydrolytic Degradability, Cell Tolerance and On-Demand Antibacterial Effect of Electrospun Photodynamically Active Fibres

Photodynamically active fibres (PAFs) are a novel class of stimulus-sensitive systems capable of triggering antibiotic-free antibacterial effect on-demand when exposed to light. Despite their relevance in infection control, however, the broad clinical applicability of PAFs has not yet been fully realised due to the limited control in fibrous microstructure, cell tolerance and antibacterial activity in the physiologic environment. We addressed this challenge by creating semicrystalline electrospun fibres with varying content of poly[(L-lactide)-co-(glycolide)] (PLGA), poly(epsilon-caprolactone) (PCL) and methylene blue (MB), whereby the effect of polymer morphology, fibre composition and photosensitiser (PS) uptake on wet state fibre behaviour and functions was studied. The presence of crystalline domains and PS-polymer secondary interactions proved key to accomplishing long-lasting fibrous microstructure, controlled mass loss and controlled MB release profiles (37 degC, pH 7.4, 8 weeks). PAFs with equivalent PLGA:PCL weight ratio successfully promoted attachment and proliferation of L929 cells over a 7-day culture with and without light activation, while triggering up to 2.5 and 4 log reduction in E. coli and S. mutans viability, respectively. These results support the therapeutic applicability of PAFs for frequently encountered bacterial infections, opening up new opportunities in photodynamic fibrous systems with integrated wound healing and infection control capabilities.

q-bio.TO↗

arXiv2020

Bosonic ghostbusting -- The bosonic ghost vertex algebra admits a logarithmic module category with rigid fusion

The rank 1 bosonic ghost vertex algebra, also known as the $\beta \gamma$ ghosts, symplectic bosons or Weyl vertex algebra, is a simple example of a conformal field theory which is neither rational, nor $C_2$-cofinite. We identify a module category, denoted category $\mathscr{F}$, which satisfies three necessary conditions coming from conformal field theory considerations: closure under restricted duals, closure under fusion and closure under the action of the modular group on characters. We prove the second of these conditions, with the other two already being known. Further, we show that category $\mathscr{F}$ has sufficiently many projective and injective modules, give a classification of all indecomposable modules, show that fusion is rigid and compute all fusion products. The fusion product formulae turn out to perfectly match a previously proposed Verlinde formula, which was computed using a conjectured generalisation of the usual rational Verlinde formula, called the standard module formalism. The bosonic ghosts therefore exhibit essentially all of the rich structure of rational theories despite satisfying none of the standard rationality assumptions such as $C_2$-cofiniteness, the vertex algebra being isomorphic to its restricted dual or having a one-dimensional conformal weight 0 space. In particular, to the best of the authors' knowledge this is the first example of a proof of rigidity for a logarithmic non-$C_2$-cofinite vertex algebra.

math.QA↗

arXiv2019

Unitary and non-unitary $N=2$ minimal models

The unitary $N = 2$ superconformal minimal models have a long history in string theory and mathematical physics, while their non-unitary (and logarithmic) cousins have recently attracted interest from mathematicians. Here, we give an efficient and uniform analysis of all these models as an application of a type of Schur-Weyl duality, as it pertains to the well-known Kazama-Suzuki coset construction. The results include straightforward classifications of the irreducible modules, branching rules, (super)characters and (Grothendieck) fusion rules.

math-ph↗

arXiv2018

Admissible level $\mathfrak{osp}(1|2)$ minimal models and their relaxed highest weight modules

The minimal model $\mathfrak{osp}(1|2)$ vertex operator superalgebras are the simple quotients of affine vertex operator superalgebras constructed from the affine Lie super algebra $\widehat{\mathfrak{osp}}(1|2)$ at certain rational values of the level $k$. We classify all isomorphism classes of $\mathbb{Z}_2$-graded simple relaxed highest weight modules over the minimal model $\mathfrak{osp}(1|2)$ vertex operator superalgebras in both the Neveu-Schwarz and Ramond sectors. To this end, we combine free field realisations, screening operators and the theory of symmetric functions in the Jack basis to compute explicit presentations for the Zhu algebras in both the Neveu-Schwarz and Ramond sectors. Two different free field realisations are used depending on the level. For $k<-1$, the free field realisation resembles the Wakimoto free field realisation of affine $\mathfrak{sl}(2)$ and is originally due to Bershadsky and Ooguri. It involves 1 free boson (or rank 1 Heisenberg vertex algebra), one $\beta\gamma$ bosonic ghost system and one $bc$ fermionic ghost system. For $k>-1$, the argument presented here requires the bosonisation of the $\beta\gamma$ system by embedding it into an indefinite rank 2 lattice vertex algebra.

math.QA↗

arXiv2018

Singular vectors for the $W_N$ algebras

In this paper, we use free field realisations of the A-type principal, or Casimir, $W_N$ algebras to derive explicit formulae for singular vectors in Fock modules. These singular vectors are constructed by applying screening operators to Fock module highest weight vectors. The action of the screening operators is then explicitly evaluated in terms of Jack symmetric functions and their skew analogues. The resulting formulae depend on sequences of pairs of integers that completely determine the Fock module as well as the Jack symmetric functions.

math-ph↗

arXiv2017

Yang-Baxter representations of the infinite symmetric group

Every unitary involutive solution of the quantum Yang-Baxter equation ("R-matrix") defines an extremal character and a representation of the infinite symmetric group $S_\infty$. We give a complete classification of all such Yang-Baxter characters and determine which extremal characters of $S_\infty$ are of Yang-Baxter form. Calling two involutive R-matrices equivalent if they have the same character and the same dimension, we show that equivalence classes are classified by pairs of Young diagrams, and construct an explicit normal form R-matrix for each class. Using operator-algebraic techniques (subfactors), we prove that two R-matrices are equivalent if and only if they have similar partial traces. Furthermore, we describe the algebraic structure of the equivalence classes of all involutive R-matrices, and discuss several classes of examples. These include unitary Yang-Baxter representations of the Temperley-Lieb algebra at loop parameter $\delta=2$, which can be completely classified in terms of their rank and dimension.

math.QA↗

arXiv2017

An admissible level $\widehat{\mathfrak{osp}} \left( 1 \middle\vert 2 \right)$-model: modular transformations and the Verlinde formula

The modular properties of the simple vertex operator superalgebra associated to the affine Kac-Moody superalgebra $\widehat{\mathfrak{osp}} \left( 1 \middle\vert 2 \right)$ at level $-\frac{5}{4}$ are investigated. After classifying the relaxed highest-weight modules over this vertex operator superalgebra, the characters and supercharacters of the simple weight modules are computed and their modular transforms are determined. This leads to a complete list of the Grothendieck fusion rules by way of a continuous superalgebraic analogue of the Verlinde formula. All Grothendieck fusion coefficients are observed to be non-negative integers. These results indicate that the extension to general admissible levels will follow using the same methodology once the classification of relaxed highest-weight modules is completed.

hep-th↗

arXiv2016

Superconformal minimal models and admissible Jack polynomials

We give new proofs of the rationality of the N=1 superconformal minimal model vertex operator superalgebras and of the classification of their modules in both the Neveu-Schwarz and Ramond sectors. For this, we combine the standard free field realisation with the theory of Jack symmetric functions. A key role is played by Jack symmetric polynomials with a certain negative parameter that are labelled by admissible partitions. These polynomials are shown to describe free fermion correlators, suitably dressed by a symmetrising factor. The classification proofs concentrate on explicitly identifying Zhu's algebra and its twisted analogue. Interestingly, these identifications do not use an explicit expression for the non-trivial vacuum singular vector. While the latter is known to be expressible in terms of an Uglov symmetric polynomial or a linear combination of Jack superpolynomials, it turns out that standard Jack polynomials (and functions) suffice to prove the classification.

hep-th↗