Dr Simon Wood

Publication

• Date
• Type

Research

My Research

Most of my work focuses on the rigorous algebraic underpinnings of two dimensional conformal field theory in the form of vertex operator algebras and tensor categories.

Conformal symmetry

The algebraic axiomatisation of the symmetries underlying a two dimensional conformal field theory is called a vertex (operator) algebra. Vertex algebras can be thought of as a kind of generalisation of associative commutative algebras but different from associative non-commutative algebras. As with associative algebras, much can be learnt from studying modules and many questions in the study of conformal field theory boil down question in vertex algebra module theory.

The most studied vertex algebras are the so called rational vertex algebras. These are distinguished by the fact that their categories of modules are semisimple with only a finite number of isomorphism classes of simple modules. I focus on vertex algebras for which neither the semi-simplicity nor the finite number of simple modules assumption need hold. Vertex algebras for which the semi-simplicity assumption fails are called logarithmic vertex algebras and the conformal field theories associated to them are called logarithmic conformal field theories. Some big endeavours in this context include module classification, analysing the additional structures that these modules admit (characters, fusion products, Verlinde formulae, etc) and finding the right abstract categorical tools which will enable a general structure theory.

My work on vertex algebra module classification makes use of certain associative algebras, called Zhu algebras, which encode a lot of information about vertex algebra module theory. Zhu algebras are notoriously hard to work with in practice and I have developed methods which recast hard Zhu algebra questions into comparatively easier quetions in terms of the combinatorics of symmetric functions. Some representative publications in this line of research include:

• Admissible level osp(1|2) minimal models and their relaxed highest weight modules
• Superconformal minimal models and admissible Jack polynomials
• Relaxed singular vectors, Jack symmetric functions and fractional level sl2 models
• From Jack polynomials to minimal model spectra
• On the extended W-algebra of type sl2 at positive rational level

Categories of modules over rational vertex algebras are so called modular tensor categories. Among many other things, this implies that the much celebrated Verlinde formula holds. This formula relates the fusion product of modules (a kind of tensor product) to an action of the modular group, SL(2,Z), on module characters. My work aims to generalise this Verlinde formula to logarithmic vertex algebras. Some representative publication in this line of research include:

• Admissible-level sl3 minimal models
• An admissible level osp(1|2)-model: modular transformations and the Verlinde formula
• On Regularised Quantum Dimensions of the Singlet Vertex Operator Algebra and False Theta Functions
• The Verlinde formula in logarithmic CFT
• Bosonic Ghosts at c=2 as a Logarithmic CFT
• Modular Transformations and Verlinde Formulae for Logarithmic (p+,p−)-Models 

Vertex algebras and their modules are infinite dimensional vector spaces (with much additional structure). This means that it is very easy to get lost in technical details. Category theory is the perfect antidote to this as it ignores the internal structure of objects being studied and tries to understand them solely via the maps between objects. So while vertex algebra modules are infinite dimensional the spaces of maps between them, the dimension of spaces of maps between them is usually finite, so this is a large reduction of complexit, if one can find the correct categorical tools. The following papers show that well certain logarithmic vertex algebras with well chosen categories of modules admit structures that are as rich as those of rational vertex algebras.

• Bosonic ghostbusting -- The bosonic ghost vertex algebra admits a logarithmic module category with rigid fusion
• The tensor structure on the representation category of the $\mathcal{W}_p$ triplet algebra

While categories of modules over vertex algebras will not be modular tensor categories, if the vertex algebra is not rational, they still admit many rich structures that generalise those of modular tensor categories. Recent work of mine shows that categories of modules over vertex algebras admit a duality structure called Grothendieck-Verdier duality. Exploring the implications of this duality structure is one of the main aims of a Humboldt Fellowship project that I am currently undertaking at Hamburg University.

• Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson
• Grothendieck-Verdier duality in categories of bimodules and weak module functors

Connections to integrability

As mentioned above symmetric functions have proved immensely helpful in module classification problems for vertex algebras. However, they are also ubiquitous in integrable models and are believed to be the source of a large family of correspondences between integrable models and vertex algebras. I have a joint EPSRC funded project with Prof. Christian Korff of the University of Glasgow that aims to fully understand and systematise these correspondences.

Symmetric functions were also key to joint with Prof. Gandalf Lechner and Dr. Ulrich Pennig on Yang-Baxter equations. The Yang-Baxter equation is remarkably ubiquitous throughout mathematical physics and some areas of pure mathematics. In its simplest parameter independent form it is equivalent braiding of the braid group. Solutions to the Yang-Baxter equation therefore give rise to representations of the (infinite) braid group. There is still much that is unkown about braid group representations and so the Yang-Baxter equation has the potential to be a great source of interesting representations. In  Yang-Baxter representations of the infinite symmetric group all such representations which in addition satisfy that they are unitary representations of the infinite symmetric group were classified.

Conference organisation

• 12/2019 Beyond Rationality ∞: exploring the many roads to postrational conformal field theory, Utrecht, The Netherlands
• 05/2019 Beyond Rationality 2: Indecomposability and Post-Rational Conformal Field Theory, Utrecht, The Netherlands
• 07/2018 RIMS Gasshuku-style Seminar "Vertex Operator Algebras and Conformal Field Theory", Kyoto, Japan
• 03/2018 22nd UK Meeting on Integrable Models, Conformal Field Theory and Related Topics, Cardiff, UK
• 12/2017 LMS South West & South Wales Regional Meeting and Workshop: Algebraic Structures and Quantum Physics, Cardiff, UK
• 12/2016 Yang-Baxter Equations: A Tangle of Physics and Mathematics, Cardiff, UK
• 07/2015 The mathematics of conformal field theory, ANU, Canberra, Australia

Conference talks

2023

• "Duality structures on tensor categories coming from vertex operator algebras", Representation Theory XVIII, Dubrovnik, Croatia
• "Duality structures on tensor categories coming from vertex operator algebras", Groups, Rings and the Yang-Baxter equation 2023, Blankenberge, Belgium

2022

• "The classification of simple (semi)relaxed admissible affine sl(3) modules and their modular properties at level − 3/2" , Representation Theory XVII, Dubrovnik, Croatia
• "Modular properties for well chosen non-rational vertex operator algebras", Modular Forms and Representation Theory, Manigod, France

2021

• "Grothendieck-Verdier duality in categories of VOA modules with examples", Quantum Field Theories and Quantum Topology Beyond Semisimplicity, Banff International Research Station, Canada

2019

• "I ain't afraid of no ghost", The Mathematical Foundations of Conformal Field Theory and Related Topics - A conference in honor of Yi-Zhi Huang, Chern Institute, Nankai University.

2018

• "Presentations of Zhu algebras from free field realisations", Workshop on vertex algebras and infinite-dimensional Lie algebras, University of Split
• "Logarithmic Conformal Field Theory and the Verlinde Formula", 11th Seminar on Conformal Field Theory, Friedrich-Alexander-Universität, Erlangen-Nürnberg
• "The standard module formalism and affine sl3 at level −3/2, Vertex Operator Algebras and Symmetries", RIMS Workshop: Vertex Operator Algebras and Symmetries, RIMS Kyoto University, Japan
• "Conference, N = 2 minimal models at unitary and beyond", International conference on Vertex Operator Algebras, Number Theory and Related Topics, Sacramento, USA
• "Admissible level osp(1|2) minimal models and their relaxed highest weightmodules", Vertex algebras and related topics, University of Zagreb, Croatia

• "Module classification through free fields and symmetric functions", Conformal field theories and categorical structures beyond rationality, Woudschoten, Netherlands

2017

• "Classifying positive energy modules in conformal field theory", Shanks Workshop: Subfactors and Applications, Vanderbilt University
• "Affine vertex operator superalgebras at admissible levels", Representation Theory XIV, Dubrovnik
• "What to expect from logarithmic conformal field theory", Operator algebras: subfactors and their applications, Isaac Newton Institute, Cambridge
• "What to expect from logarithmic conformal field theory", Quantum Field Theory: Concepts, Constructions & Curved Spacetimes, York
• "Fusion by hand: The NGK algorithm", Tensor Categories and Field Theory, Melbourne

2016

• "Symmetric functions and their relation to free field vertex algebras", AMS Sectional Meeting, Stony Brook
• "The rationality of N=1 minimal models through symmetric polynomials", BIRS Workshop: Vertex Algebras and Quantum Groups, Banff

2015

• “Symmetric polynomials and modules over affine sl2 at admissible levels”, Conference on Lie Algebras, Vertex Operator Algebras, and Related Topics, University of Notre Dame
• “Two-dimensional conformal field theory with affine Lie algebra symmetry”, Symmetries and Spinors Interactions between Geometry and Physics, University of Adelaide
• "Minimal models from free fields", ANZAMP Meeting 2015, University of Newcastle.

2014

• “From free field theory to symmetric polynomials”, Australia New Zealand Mathematics Convention 2014, Melbourne
• “From free field theory to symmetric polynomials”, String-Math 2014, University of Alberta, Edmonton
• “Rational logarithmic extensions of the minimal models and their simple modules”, Modern Trends in TQFT, Erwin Schrödinger Institute, Vienna

2013 and earlier

• 2013, “On the extended W-algebra of type sl2 at positive rational level”, String theory, Integrable systems and representation theory, RIMS Symposium, The University of Kyoto
• 2012,  Understanding logarithmic CFT, String-Math 2012, Hausdorff center for Mathematics, Bonn
• 2011, Vertex operator algebras for logarithmic CFT, Vertex Operator Algebras, Finite Groups and Related Topics, Academia Sinica, Taipei

Seminar and colloquium talks

2023

• "Symmetric function combinatorics and conformal field theory", Pure Mathematics Colloquium, Hamburg University
  

2022

• "Tensor structures in the wild", Hamburg research seminar in algebra and mathematical physics, Hamburg University

2021

• "There is always more than can be learnt fro the free boson", Rocky Mountain Representation Theory Seminar, University of Colorado Boulder.
• "From vertex operator algebras to tensor products", Algebra and Mathematical Physics Seminar, Hamburg University.

2020

• "Vertex algebras with nice structure despite failing all conventional niceness criteria", Algebra Seminar, University of Aberdeen.

2019

• "Logarithmic vs rational conformal field theory – Who really wants to be rational anyway?", School of Physics and Astronomy, Queen Mary University.

2018

• βγ ghost algebras and the Verlinde formula, Algebra and Mathematical Physics Seminar, Hamburg University.
• Non-rational conformal field theory with sl3 symmetry, Mathematical Physics Seminar, Glasgow University.
• Classifying simple positive energy modules over vertex operator superalgebras, Mathematics Seminar, University of Melbourne.
• Yang-Baxter equations and symmetric groups, Mathematics - String Theory seminar, IPMU University of Tokyo.
• Conformal field theory from affine Lie algebras at fractional levels, Quantum Field Theory Seminar, University of Oxford.

2017 and earlier

• 2017, "Vertex algebra module theory made easy-ish", University of Glasgow
• 2017, "Representation theory in conformal field theories", Durham University
• 2017, "Module classification in conformal field theory through symmetric polynomials", King's College London
• 2017, "Yang-Baxter equations and the symmetric groups", University of Melbourne
• 2016, "Universal vertex algebras and free field realisations", University of Alberta, Edmonton
• 2016, "Universal vertex algebras and free field realisations",  Kavli IMPU, Tokyo
• 2016, "The rationality of the N=1 minimal model vertex algebra and its connection to symmetric functions", Rutgers University
• 2016, "Universal vertex algebras and free field realisations", University of Notre Dame, Notre Dame.
• 2015, "Classifying simple modules at minimal model central charges through symmetric polynomials",  University of Queensland
• 2015, "Symmetric polynomials and their relation to conformal field theory", Australian National University
• 2015, "The Verlinde formula in logarithmic conformal field theory", Kavli IPMU
• 2014, "Conformal Symmetry in Physics", CSU Chico
• 2014, "Symmetric polynomials in free field theories", Laval University
• 2014, "Symmetric polynomials in free field VOAs", University of Montreal
• 2014, "A working Verlinde Formula for logarithmic CFT", TU Wien
• 2014, "Conformal Symmetry in Physics", University of Queensland
• 2014, "Jack symmetric polynomials and their connection to the Lie algebra of infinitesimal conformal transformations", University of Queensland
• 2013, "On the extended W-algebra of type sl2 at positive rational level", University of Alberta
• 2013, "On the extended W-algebra of type sl2 at positive rational level", SUNY at Albany
• 2013, "On the extended W-algebra of type sl2 at positive rational level", The University of Tokyo
• 2013, "On the extended W-algebra of type sl2 at positive rational level", SCGP Stony Brook
• 2012, "M(p+ ,p−) the extended W-algebra of sl2 type at rational leve"l, Conformal Field Theory and Moonshine Trimester, Hausdorff Research Institute for Mathematics
• 2012, "Logarithmic versus non-logarithmic CFT", Australian National University

Teaching

PhD projects

I currently have no open PhD postion. However, interested students are still welcome to contact me with any queries. Please note that I get many such queries, so you are much more likely to get a reply, if you include a brief description of your research interests, the courses you have taken and why you are interested in doing a PhD with me specifically. I will then be able to inform you, if a position unexpectedly becomes available or I might be able redirect you to another suitable supervisor with an open position.

PhD students

• 2020 - present, Jamal Shafiq
• 2018 - 2022, Robert Allen, now at University of Bristol
• 2015 - 2019, Tianshu Liu (University of Melbourne) joint with David Ridout

Masters and project students

• 2023 - ongoing, Jack Brown (MMath project, Cardiff)
  Project: "Symmetric Polynomials"
• 2021 - 2022, Daniel Townley-Keogh (MMath project, Cardiff)
  Project: "Galois Theory and its Applications to Classifying Modular Invariants"
• 2020 - 2021, Ieuan Fishlock (MMath project, Cardiff)
  Project: "Representation Theory of Finite Groups"
• 2019 - 2020, Owen Tanner (MMath project, Cardiff)
  Project: "Knizhnik-Zamolodchikov Equations"
• 2018 - 2019, Tudur Lewis (MMath project, Cardiff)
  Project: "Reflection Groups"
• 2017 - 2018, Anna Clancy (MMath project, Cardiff)
  Project: "Symmetric Polynomials"
• 2015 - 2016, John Snadden (Masters student, ANU) joint with David Ridout
• 2015, Matthew Geleta (Honours student, ANU) joint with David Ridout
  Project: "The Coulomb gas formalism"

Courses

• Algebra II: Rings 2023/24
• Algebra II: Rings 2022/23
• Algebra I: Groups 2021/22
• Algebra II: Rings 2021/22
• Algebra I: Groups 2020/21
• Algebra II: Rings 2020/21
• Groups 2019/20
• Rings and Fields 2019/20
• Groups, 2018/19
• Foundations of Mathematics I, 2018/19
• Groups, 2017/18
• Foundations of Mathematics I, 2017/18
• Foundations of Mathematics I, 2016/17
• Honours course on Lie algebras and representation theory 2015/16
• Honours course on conformal field theory 2015
• Honours course on Lie algebras and representation theory 2014

Biography

Qualifications:

02/2011 Doctor of Science, ETH, Zurich
03/2008 Masters in Physics, ETH, Zurich

Previous Positions:

2014 - 2016, PostDoc, The Australian National University, Canberra, Australia
2011 - 2013, PostDoc, Kavli IPMU, University of Tokyo, Japan

Grants and awards:

2021, "Integrable models and deformations of vertex algebras via symmetric functions", EPSRC Standard Grant, EP/V053787/1
2020, "Exploring quantum group structures in logarithmic conformal field theory", Humboldt Fellowship for Experienced Researchers
2015, "Towards higher rank logarithmic conformal field theories", Discovery Project, Australian Research Council
2013, "The Algebraic Structure of Logarithmic Conformal Field Theory", Discovery Early Career Researcher Award, Australian Research Council
2011, JSPS Postdoctoral Fellowship for Foreign Researchers, Japan Society of the Promotion of Science
2010, "Conformal field theory, vertex operators algebras and quantum groups", SNSF Fellowship for Prospective Researchers; Swiss National Science Foundation