# Dr Simon Wood

Lecturer

*Email:*- woodsi@cardiff.ac.uk
*Telephone:*- +44 (0)29 2087 5312
*Fax:*- 02920874199

### Research Interests

My research focuses on symmetries in the laws of physics. Such symmetries are fundamental to our understanding of the laws nature (they have arguably been the driving force behind almost all advances in theoretical physics for well over 100 years) and are also a source of beautiful mathematics. Specifically, I mainly study the mathematics of 2 dimensional conformal quantum field theories and all their myriad connections to Lie theory, vertex operator algebras, representation theory, modular forms and modular tensor categories to name but a few. Recently I have also begun working on integrability and its connections to the much celebrated Yang-Baxter equation.

### Research Group

Geometry, Algebra, Mathematical Physics & Topology Research Group.

### 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:

2015, Discovery Project, Australian Research Council

2013, Discovery Early Career Researcher Award, Australian Research Council

2011, JSPS Postdoctoral Fellowship for Foreign Researchers, Japan Society of the Promotion of Science

2010, SNSF Fellowship for Prospective Researchers; Swiss National Science Foundation

### PhD students

- 2014 - present, Steve Siu (University of Melbourne) joint with David Ridout
- 2015 - present, Tianshu Liu (University of Melbourne) joint with David Ridout

### Masters and project students

- 2017 - present, 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

- Foundations of Mathematics I, 2017/18
- Groups, 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

### My Research

Most of my work focuses on the rigorous algebraic underpinnings of two dimensional conformal field theory and more recently also integrability and its connections to the Yang-Baxter equation.

#### 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 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 module categories 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. Two big endeavours in this context are module classification and analysing the additional structures that these modules admit (characters, fusion products, Verlinde formulae, etc).

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 publication in this line of research include:

- 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

Modules over rational vertex algebras satisfy the much celebrated Verlinde formula, which 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:

- 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

#### The Yang-Baxter equation

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 G. Lechner, U. Pennig and I classified all such representations which in addition satisfy that they are unitary representations of the infinite symmetric group.

### Conference organisation

- 07/2018 RIMS Gasshuku-style Seminar "Vertex Operator Algebras and Conformal Field Theory", Sapporo, 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

**2018**

- TBA, Geometric and categorical aspects of CFTs, Oaxaca, Mexico
- TBA, RIMS Workshop: Vertex Operator Algebras and Symmetries, RIMS Kyoto University, Japan
- TBA, International conference on Vertex Operator Algebras, Number Theory and Related Topics, Sacramento, USA
- "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

**2017**

- "Vertex algebra module theory made easy-ish", University of Glasgow
- "Representation theory in conformal field theories", Durham University
- "Module classification in conformal field theory through symmetric polynomials", King's College London
- "Yang-Baxter equations and the symmetric groups", University of Melbourne

**2016**

- "Universal vertex algebras and free field realisations", University of Alberta, Edmonton
- "Universal vertex algebras and free field realisations", Kavli IMPU, Tokyo
- "The rationality of the N=1 minimal model vertex algebra and its connection to symmetric functions", Rutgers University
- "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
- "Symmetric polynomials and their relation to conformal field theory", Australian National University
- "The Verlinde formula in logarithmic conformal field theory", Kavli IPMU

**2014**

- "Conformal Symmetry in Physics", CSU Chico
- "Symmetric polynomials in free field theories", Laval University
- "Symmetric polynomials in free field VOAs", University of Montreal
- "A working Verlinde Formula for logarithmic CFT", TU Wien
- "Conformal Symmetry in Physics", University of Queensland
- "Jack symmetric polynomials and their connection to the Lie algebra of infinitesimal conformal transformations", University of Queensland

**2013 and earlier**

- 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