Dr Simon Wood
Senior Lecturer
 woodsi@cardiff.ac.uk
 +44 (0)29 2087 5312
 Fax:
 02920874199
 3.60, Abacws, Senghennydd Road, Cathays, Cardiff, CF24 4AG
Overview
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 YangBaxter equation.
Research Group
Geometry, Algebra, Mathematical Physics & Topology Research Group.
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, EPSRC Standard Grant, "Integrable models and deformations of vertex algebras via symmetric functions", EP/V053787/1
2020, Humboldt Fellowship for Experienced Researchers, "Exploring quantum group structures in logarithmic conformal field theory"
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
Publications
2020
 Wood, S. 2020. Admissible level osp(12) minimal models and their relaxed highest weight modules. Transformation Groups 25, pp. 887943. (10.1007/s00031020095673)
2019
 Lechner, G., Pennig, U. and Wood, S. 2019. YangBaxter representations of the infinite symmetric group. Advances in Mathematics 355, article number: 106769. (10.1016/j.aim.2019.106769)
 Creutzig, T.et al. 2019. Unitary and nonunitary N = 2 minimal models. Journal of High Energy Physics 2019(6), article number: 24. (10.1007/JHEP06(2019)024)
2018
 Snadden, J., Ridout, D. and Wood, S. 2018. An admissible level osp (12)model: modular transformations and the Verlinde formula. Letters in Mathematical Physics 108(11), pp. 23632423. (10.1007/s1100501810975)
 Ridout, D., Siu, S. and Wood, S. 2018. Singular vectors for the WN algebras. Journal of Mathematical Physics 59(3), article number: 31701. (10.1063/1.5019278)
2017
 BlondeauFournier, O.et al. 2017. Superconformal minimal models and admissible Jack polynomials. Advances in Mathematics 314, pp. 71123. (10.1016/j.aim.2017.04.026)
 Creutzig, T., Milas, A. and Wood, S. 2017. On regularised quantum dimensions of the singlet vertex operator algebra and false theta functions. International Mathematics Research Notices 2017(5), pp. 13901432. (10.1093/imrn/rnw037)
2016
 BlondeauFournier, O.et al. 2016. The superVirasoro singular vectors and Jack superpolynomials relationship revisited. Nuclear Physics B 913, pp. 3463. (10.1016/j.nuclphysb.2016.09.003)
2015
 Tsuchiya, A. and Wood, S. 2015. On the extended Walgebra of type sl2 at positive rational level. International Mathematics Research Notices 2015(14), pp. 53575435. (10.1093/imrn/rnu090)
 Ridout, D. and Wood, S. 2015. Relaxed singular vectors, Jack symmetric functions and fractional level sl(2) models. Nuclear Physics B 894, pp. 621664. (10.1016/j.nuclphysb.2015.03.023)
 Ridout, D. and Wood, S. 2015. Bosonic ghosts at c = 2 as a logarithmic CFT. Letters in Mathematical Physics 105, pp. 279307. (10.1007/s110050140740z)
 Ridout, D. and Wood, S. 2015. The Verlinde formula in logarithmic CFT. Journal of Physics: Conference Series 597, article number: 12065. (10.1088/17426596/597/1/012065)
 Ridout, D. and Wood, S. 2015. From Jack polynomials to minimal model spectra. Journal of Physics A: Mathematical and Theoretical 48(4), article number: 45201. (10.1088/17518113/48/4/045201)
2014
 Creutzig, T., Ridout, D. and Wood, S. 2014. Coset constructions of logarithmic (1, p) models. Letters in Mathematical Physics 104(5), pp. 553583. (10.1007/s1100501406807)
 Ridout, D. and Wood, S. 2014. Modular transformations and Verlinde formulae for logarithmic (p+,p_)models. Nuclear Physics B 880, pp. 175202. (10.1016/j.nuclphysb.2014.01.010)
 Runkel, I., Gaberdiel, M. R. and Wood, S. 2014. Logarithmic Bulk and Boundary Conformal Field Theory and the Full Centre Construction. Presented at: Conformal field theories and tensor categories workshop, Beijing, China, 1418 June 2011 Presented at Bai, C. et al. eds.Conformal Field Theories and Tensor Categories: Proceedings of a Workshop Held at Beijing International Center for Mathematical Research. Mathematical Lectures from Peking University Berlin: Springer pp. 93168., (10.1007/9783642393839_4)
2013
 Tsuchiya, A. and Wood, S. 2013. The tensor structure on the representation category of the $\mathcal {W}_p$ triplet algebra. Journal of Physics A: Mathematical and Theoretical 46(44), article number: 445203. (10.1088/17518113/46/44/445203)
2011
 Gaberdiel, M. R., Runkel, I. and Wood, S. 2011. A modular invariant bulk theory for the \boldsymbol{c=0} triplet model. Journal of Physics A: Mathematical and Theoretical 44(1), article number: 15204. (10.1088/17518113/44/1/015204)
2010
 Wood, S. 2010. Fusion rules of the {\cal W}_{p,q} triplet models. Journal of Physics A: Mathematical and Theoretical 43(4), article number: 45212. (10.1088/17518113/43/4/045212)
2009
 Baumgartl, M. and Wood, S. 2009. Moduli webs and superpotentials for fivebranes. Journal of High Energy Physics 2009(6), article number: 52. (10.1088/11266708/2009/06/052)
 Gaberdiel, M. R., Runkel, I. and Wood, S. 2009. Fusion rules and boundary conditions in thec= 0 triplet model. Journal of Physics A: Mathematical and Theoretical 42(32), article number: 325403. (10.1088/17518113/42/32/325403)
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  present, Robert Allen
 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
 2021  present, Daniel TownleyKeogh (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: "KnizhnikZamolodchikov 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 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
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 but different from associative noncommutative 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 semisimplicity nor the finite number of simple modules assumption need hold. Vertex algebras for which the semisimplicity 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 Walgebra 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(12)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 YangBaxter equation
The YangBaxter 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 YangBaxter 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 YangBaxter equation has the potential to be a great source of interesting representations. In YangBaxter 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 Gasshukustyle 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 YangBaxter Equations: A Tangle of Physics and Mathematics, Cardiff, UK
 07/2015 The mathematics of conformal field theory, ANU, Canberra, Australia
Conference talks
2019
 "I ain't afraid of no ghost", The Mathematical Foundations of Conformal Field Theory and Related Topics  A conference in honor of YiZhi Huang, Chern Institute, Nankai University.
2018
 "Presentations of Zhu algebras from free field realisations", Workshop on vertex algebras and infinitedimensional Lie algebras, University of Split
 "Logarithmic Conformal Field Theory and the Verlinde Formula", 11th Seminar on Conformal Field Theory, FriedrichAlexanderUniversität, ErlangenNürnberg
 "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(12) 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
 “Twodimensional 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”, StringMath 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 Walgebra of type sl2 at positive rational level”, String theory, Integrable systems and representation theory, RIMS Symposium, The University of Kyoto
 2012, Understanding logarithmic CFT, StringMath 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
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.
 Nonrational 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.
 YangBaxter 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
 "Vertex algebra module theory made easyish", University of Glasgow
 "Representation theory in conformal field theories", Durham University
 "Module classification in conformal field theory through symmetric polynomials", King's College London
 "YangBaxter 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 Walgebra of type sl2 at positive rational level", University of Alberta
 2013, "On the extended Walgebra of type sl2 at positive rational level", SUNY at Albany
 2013, "On the extended Walgebra of type sl2 at positive rational level", The University of Tokyo
 2013, "On the extended Walgebra of type sl2 at positive rational level", SCGP Stony Brook
 2012, "M(p+ ,p−) the extended Walgebra of sl2 type at rational leve"l, Conformal Field Theory and Moonshine Trimester, Hausdorff Research Institute for Mathematics
 2012, "Logarithmic versus nonlogarithmic CFT", Australian National University