Publications

Selected publications

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

Published in Advances in Mathematics, 2024
Joint with Hiromu Nakano, Florencia Orosz Hunziker, Ana Ros Camacho
Journal Article | Preprint

Settles the long-standing rigidity and fusion conjectures for weight modules of admissible-level affine \(\mathfrak{sl}(2)\) and the \(N=2\) superconformal algebra, using a screening-current method expected to apply much more widely.

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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.

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

Published in Selecta Mathematica, 2021
Joint with R. Allen, S. Lentner and C. Schweigert
Journal Article | Preprint

Identifies Grothendieck–Verdier duality as the natural duality on categories of vertex operator algebra modules when rigidity fails, laying foundations for a better tensor-categorical understanding of logarithmic and non-rational conformal field theories.

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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.

On the extended \(W\)-algebra of type \(\mathfrak{sl}_2\) at positive rational level

Published in International Mathematical Research Notices, 2013
Joint with A. Tsuchiya
Journal Article | Preprint

Proves that the triplet algebras, logarithmic \(\mathcal{W}\)-algebras extending the Virasoro minimal models, are \(C_2\)-cofinite and classifies all simple modules. Establishes module classification methods involving the combinatorics of Jack symmetric functions that were subsequently applied to many other algebras.

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The extended W-algebra of type sl_2 at positive rational level, denoted by M_{p_+,p_-}, is a vertex operator algebra that was originally proposed in [1]. This vertex operator algebra is an extension of the minimal model vertex operator algebra and plays the role of symmetry algebra for certain logarithmic conformal field theories. We give a construction of M_{p_+,p_-} in terms of screening operators and use this construction to prove that M_{p_+,p_-} satisfies Zhu’s c_2-cofiniteness condition, calculate the structure of the zero mode algebra (also known as Zhu’s algebra) and classify all simple M_{p_+,p_-}-modules.

Book Chapters

Algebraic structures in two-dimensional conformal field theory

Published in Encyclopedia of Mathematical Physics (2nd Ed.), 2023
Joint with J. Fuchs, C. Schweigert and Y. Yang
Journal Article | Preprint

A review for the Encyclopedia of Mathematical Physics of the algebraic structures underlying two-dimensional conformal field theory. It covers vertex operator algebras and nets of observables, the categorical structure of their representations, and the tools used to build consistent correlators for full conformal field theories with boundaries and defects.

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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.

Conference Proceedings

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

Published in Quantum Symmetries, Contemporary Mathematics 813, 2023
Journal Article | Preprint

Studies Grothendieck–Verdier duality, a generalisation of rigidity arising in vertex operator algebra representation theory, which comes with two tensor products linked by distributor maps. Treating internal Homs and coHoms as weak module functors, it finds conditions under which the distributors are isomorphisms, with explicit formulae for bimodules over finite-dimensional algebras.

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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.

The Verlinde formula in logarithmic CFT

Published in Proceedings for the 30th ICGTMP (Ghent, 2014), 2014
Joint with D. Ridout
Journal Article | Preprint

A review that refines, through simple examples, the standard module formalism for modular transformations and Verlinde formulae in logarithmic conformal field theory. It also explains how fusion rules behave under simple current extensions, reaching logarithmic theories not covered directly by the standard module formalism.

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In rational conformal field theory, the Verlinde formula computes the fusion coefficients from the modular S-transformations of the characters of the chiral algebra’s representations. Generalising this formula to logarithmic models has proven rather difficult for a variety of reasons. Here, a recently proposed formalism (arXiv:1303.0847 [hep-th]) for the modular properties of certain classes of logarithmic theories is reviewed, and refined, using simple examples. A formalism addressing fusion rules in simple current extensions is also reviewed as a means to tackle logarithmic theories to which the proposed modular formalism does not directly apply.

Logarithmic bulk and boundary conformal field theory and the full centre construction

Published in Conformal Field Theories and Tensor Categories, 2012
Joint with M.R. Gaberdiel and I. Runkel
Journal Article | Preprint

Reviews bulk and boundary conformal field theory in a form that includes logarithmic theories, defining the largest bulk theory compatible with a given boundary theory via the full centre of an algebra in a braided tensor category. This lets complicated bulk theories be built from simpler boundary ones, as illustrated by the central charge zero triplet model.

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We review the definition of bulk and boundary conformal field theory in a way suited for logarithmic conformal field theory. The notion of a maximal bulk theory which can be non-degenerately joined to a boundary theory is defined. The purpose of this construction is to obtain the more complicated bulk theories from simpler boundary theories. We then describe the algebraic counterpart of the maximal bulk theory, namely the so-called full centre of an algebra in an abelian braided monoidal category. Finally, we illustrate the previous discussion in the example of the W(2,3)-model with central charge 0.

Journal Articles

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

Published in Advances in Mathematics, 2024
Joint with Hiromu Nakano, Florencia Orosz Hunziker, Ana Ros Camacho
Journal Article | Preprint

Proves that the categories of weight modules over the admissible-level affine \(\mathfrak{sl}(2)\) and \(\mathcal{N}=2\) superconformal vertex operator superalgebras are rigid, and hence braided ribbon categories, confirming the long-conjectured fusion rules of Creutzig, Ridout and collaborators. The proof constructs intertwining operators by integrating screening currents, a technique expected to apply more widely.

Show abstract

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.

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

Published in Advances in Mathematics, 2024
Joint with J. Fuchs, G. Schaumann and C. Schweigert
Journal Article | Preprint

Develops a theory of module categories over Grothendieck–Verdier categories, showing that suitable module categories can be realised as modules over an internal End algebra. A partially defined relative Serre functor then gives such algebras a Grothendieck–Verdier Frobenius structure.

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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.

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

Published in Transactions of the American Mathematical Society, 2024
Journal Article | Preprint

Studies chiral torus 1-point functions of the rational affine \(\mathfrak{sl}(2)\) vertex operator algebras with insertions from any module, determining the dimension of their span and building vector-valued modular forms from them. Their modular transformations are also expressed purely in terms of modular tensor category data, giving candidate invariants finer than the usual S and T matrices.

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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.

Admissible-level \(\mathfrak{sl}(3)\) minimal models

Published in Letters in Mathematical Physics, 2021
Joint with K. Kawasetsu and D. Ridout
Journal Article | Preprint

Classifies the irreducible weight modules of the admissible-level \(\mathfrak{sl}_3\) minimal model vertex operator algebras, providing the input needed for the standard module formalism. At level \(-\tfrac{3}{2}\) this gives modular transformations and conjectural Grothendieck fusion rules, the first application of the formalism to a genuinely rank-two example.

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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.

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

Published in Selecta Mathematica, 2021
Joint with R. Allen, S. Lentner and C. Schweigert
Journal Article | Preprint

Shows that taking contragredient duals gives categories of vertex operator algebra modules the structure of ribbon Grothendieck–Verdier categories, a notion generalising rigidity that allows non-exact tensor products. Develops criteria for equivalences of such categories and uses them to describe free boson (Heisenberg) categories with arbitrary conformal vector in terms of graded vector spaces and Hopf algebra modules.

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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.

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

Published in Communications in Mathematical Physics, 2020
Joint with R. Allen
Journal Article | Preprint

Shows that the bosonic ghost (\(\beta\gamma\)) vertex algebra, although neither rational nor \(C_2\)-cofinite, has a module category that is closed under fusion and in which fusion is rigid. Additionaly all indecomposable modules are classified and all fusion products are computed. This is the first proof of rigidity for a logarithmic, non-\(C_2\)-cofinite vertex algebra.

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The rank 1 bosonic ghost vertex algebra, also known as the \(βγ\) 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.

Unitary and non-unitary \(N=2\) minimal models

Published in Journal of High Energy Physics, 2019
Joint with T. Creutzig, T. Liu and D. Ridout
Journal Article | Preprint

Treats the unitary and non-unitary \(N=2\) superconformal minimal models in one uniform framework by applying a Schur–Weyl-type duality for Heisenberg cosets to the Kazama–Suzuki coset construction. Obtains classifications of irreducible modules, branching rules, (super)characters and Grothendieck fusion rules, including those of logarithmic models.

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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.

Admissible level \(\mathfrak{osp}(1\vert2)\) minimal models and their relaxed highest weight modules

Published in Transformation Groups, 2018
Journal Article | Preprint

Gives the first classification of the simple relaxed highest-weight modules over the \(\mathfrak{osp}(1\vert 2)\) minimal model vertex operator superalgebras at all admissible levels, in both the Neveu–Schwarz and Ramond sectors. Zhu’s algebras are computed explicitly using free field realisations, screening operators and Jack symmetric functions.

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The minimal model \(\mathfrak{osp}(1\vert 2)\) vertex operator superalgebras are the simple quotients of affine vertex operator superalgebras constructed from the affine Lie super algebra \(\widehat{\mathfrak{osp}}(1\vert 2)\) 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 \(\mathfrak{osp}(1\vert 2)\) 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 \(βγ\) bosonic ghost system and one \(bc\) fermionic ghost system. For \(k>-1\), the argument presented here requires the bosonisation of the \(βγ\) system by embedding it into an indefinite rank 2 lattice vertex algebra.

Singular vectors for the \(W_N\) algebras

Published in Journal of Mathematical Physics, 2017
Joint with D. Ridout and S. Siu
Journal Article | Preprint

Derives explicit formulae for singular vectors in Fock modules of the \(W_N\) algebras, the higher-rank generalisations of the Virasoro algebra, as combinations of Jack symmetric functions and their skew versions obtained from screening operators. This lays the groundwork for classifying modules of higher-rank vertex operator algebras.

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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.

Yang-Baxter representations of the infinite symmetric group

Published in Advances in Mathematics, 2017
Joint with G. Lechner and U. Pennig
Journal Article | Preprint

Classifies all unitary involutive solutions of the Yang–Baxter equation up to a natural representation-theoretic equivalence, with classes labelled by pairs of Young diagrams and an explicit normal form for each. The proof uses characters of the infinite symmetric group and subfactor techniques.

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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 \(δ=2\), which can be completely classified in terms of their rank and dimension.

An admissible level \(\widehat{\mathfrak{osp}}(1\vert2)\)-model: modular transformations and the Verlinde formula

Published in Letters in Mathematical Physics, 2017
Joint with D. Ridout and J. Snadden
Journal Article | Preprint

A detailed case study of the affine \(\mathfrak{osp}(1\vert 2)\) vertex operator superalgebra at admissible level \(-\tfrac{5}{4}\), testing the standard module formalism for logarithmic conformal field theories with fermions. Classifies the relaxed highest-weight modules, computes their modular transformations and obtains Grothendieck fusion rules with non-negative integer coefficients from a Verlinde-type formula.

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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.

Superconformal minimal models and admissible Jack polynomials

Published in Advances in Mathematics, 2016
Joint with O. Blondeau-Fournier, P. Mathieu and D. Ridout
Journal Article | Preprint

Gives new proofs of the rationality of the \(\mathcal{N}=1\) superconformal minimal model vertex operator superalgebras and classifies their simple modules in both the Neveu–Schwarz and Ramond sectors, without coset constructions. Zhu’s algebras are identified by combining free field realisations with Jack symmetric polynomials, without needing an explicit formula for the vacuum singular vector.

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We give new proofs of the rationality of the \(\mathcal{N}=1\) 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.

The super-Virasoro singular vectors and Jack superpolynomials relationship revisited

Published in Nuclear Physics B, 2016
Joint with O. Blondeau-Fournier, P. Mathieu and D. Ridout
Journal Article | Preprint

Extends a free field derivation of Virasoro singular vectors to the \(\mathcal{N}=1\) superconformal algebra, expressing the singular vectors in both the Neveu–Schwarz and Ramond sectors as a simple differential operator acting on a single Jack superpolynomial. This proves formulae that are more compact than those previously conjectured.

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A recent novel derivation of the representation of Virasoro singular vectors in terms of Jack polynomials is extended to the supersymmetric case. The resulting expression of a generic super-Virasoro singular vector is given in terms of a simple differential operator (whose form is characteristic of the sector, Neveu-Schwarz or Ramond) acting on a Jack superpolynomial. The latter is indexed by a superpartition depending upon the two integers r,s that specify the reducible module under consideration. The corresponding singular vector (at grade rs/2), when expanded as a linear combination of Jack superpolynomials, results in an expression that (in addition to being proved) turns out to be more compact than those that have been previously conjectured. As an aside, in relation with the differential operator alluded to above, a remarkable property of the Jack superpolynomials at alpha=-3 is pointed out.

Relaxed singular vectors, Jack symmetric functions and fractional level \(\widehat{\mathfrak{sl}}(2)\) models

Published in Nuclear Physics B, 2015
Journal Article | Preprint

Combines Wakimoto’s free field realisation with Jack symmetric functions to study fractional level affine \(\mathfrak{sl}(2)\) models, giving explicit singular vectors in relaxed modules and an explicit presentation of Zhu’s algebra. This yields a new, simpler proof of the classification of simple relaxed highest-weight modules.

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The fractional level models are (logarithmic) conformal field theories associated with affine Kac-Moody (super)algebras at certain levels \(k \in \mathbb{Q}\). They are particularly noteworthy because of several longstanding difficulties that have only recently been resolved. Here, Wakimoto’s free field realisation is combined with the theory of Jack symmetric functions to analyse the fractional level \(\widehat{\mathfrak{sl}}(2)\) models. The first main results are explicit formulae for the singular vectors of minimal grade in relaxed Wakimoto modules. These are closely related to the minimal grade singular vectors in relaxed (parabolic) Verma modules. Further results include an explicit presentation of Zhu’s algebra and an elegant new proof of the classification of simple relaxed highest weight modules over the corresponding vertex operator algebra. These results suggest that generalisations to higher rank fractional level models are now within reach.

On Regularised Quantum Dimensions of the Singlet Vertex Operator Algebra and False Theta Functions

Published in International Mathematical Research Notices, 2014
Joint with T. Creutzig and A. Milas
Journal Article | Preprint

Links singlet vertex operator algebras, quantum modular forms and modular tensor categories by computing regularised quantum dimensions from characters expressed as false theta functions. Depending on the regularisation, these reproduce either the Verlinde fusion ring or the fusion ring of a rational vertex operator algebra, which can be read as a semisimplification.

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We study a family of non-C2-cofinite vertex operator algebras, called the singlet vertex operator algebras, and connect several important concepts in the theory of vertex operator algebras, quantum modular forms, and modular tensor categories. More precisely, starting from explicit formulae for characters of modules over the singlet vertex operator algebra, which can be expressed in terms of false theta functions and their derivatives, we first deform these characters by using a complex parameter ε. We then apply modular trans- formation properties of regularised partial theta functions to study asymptotic behaviour of regularised characters of irreducible modules and compute their regularised quantum dimensions. We also give a purely geometric description of the regularisation parameter as a uniformisation parameter of the fusion variety coming from atypical blocks. It turns out that the quantum dimensions behave very differently depending on the sign of the real part of ε. The map from the space of characters equipped with the Verlinde product to the space of regularised quantum dimensions turns out to be a genuine ring isomorphism for positive real part of ε while for sufficiently negative real part of ε its surjective image gives the fusion ring of a rational vertex operator algebra. The category of modules of this rational vertex operator algebra should be viewed as obtained through the process of a semi-simplification procedure widely used in the theory of quantum groups. Interestingly, the modular tensor category structure constants of this vertex operator algebra, can be also detected from vector valued quantum modular forms formed by distinguished atypical characters.

From Jack polynomials to minimal model spectra

Published in Journal of Physics A, 2014
Joint with D. Ridout
Journal Article | Preprint

An introductory account of how free field realisations combine with Jack symmetric polynomials to give much shorter proofs of known results: explicit Virasoro singular vector formulae and the classification of irreducible modules of the Virasoro minimal model vertex operator algebras. The same methods extend to logarithmic and affine settings.

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In this note, a deep connection between free field realisations of conformal field theories and symmetric polynomials is presented. We give a brief introduction into the necessary prerequisites of both free field realisations and symmetric polynomials, in particular Jack symmetric polynomials. Then we combine these two fields to classify the irreducible representations of the minimal model vertex operator algebras as an illuminating example of the power of these methods. While these results on the representation theory of the minimal models are all known, this note exploits the full power of Jack polynomials to present significant simplifications of the original proofs in the literature.

Bosonic Ghosts at \(c=2\) as a Logarithmic CFT

Published in Letters in Mathematical Physics, 2014
Joint with D. Ridout
Journal Article | Preprint

Analyses the \(c=2\) bosonic ghost system, a building block of many free field realisations, as a logarithmic conformal field theory using the standard module formalism. Derives the modular transformations of characters, non-negative integer Verlinde coefficients and infinitely many modular invariant partition functions, with explicit fusion computations confirming that the theory is logarithmic.

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Motivated by Wakimoto free field realisations, the bosonic ghost system of central charge \(c=2\) is studied using a recently proposed formalism for logarithmic conformal field theories. This formalism addresses the modular properties of the theory with the aim being to determine the (Grothendieck) fusion coefficients from a variant of the Verlinde formula. The key insight, in the case of bosonic ghosts, is to introduce a family of parabolic Verma modules which dominate the spectrum of the theory. The results include S-transformation formulae for characters, non-negative integer Verlinde coefficients, and a family of modular invariant partition functions. The logarithmic nature of the corresponding ghost theories is explicitly verified using the Nahm-Gaberdiel-Kausch fusion algorithm.

Modular Transformations and Verlinde Formulae for Logarithmic \((p_+,p_−)\)-Models

Published in Nuclear Physics B, 2013
Joint with D. Ridout
Journal Article | Preprint

Computes the modular transformations of the characters of the uncountably many simple modules of the logarithmic \((p_+,p_-)\) singlet algebras and derives their Grothendieck fusion rules from a continuous Verlinde formula. Lifting to the triplet algebras via simple current extensions recovers the previously proposed triplet fusion rules.

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The \((p_+,p_-)\) singlet algebra is a vertex operator algebra that is strongly generated by a Virasoro field of central charge \(1-6(p_+-p_-)^2/p_+p_-\) and a single Virasoro primary field of conformal weight \((2p_+-1)(2p_--1)\). Here, the modular properties of the characters of the uncountably many simple modules of each singlet algebra are investigated and the results used as the input to a continuous analogue of the Verlinde formula to obtain the “fusion rules” of the singlet modules. The effect of the failure of fusion to be exact in general is studied at the level of Verlinde products and the rules derived are lifted to the \((p_+,p_-)\) triplet algebras by regarding these algebras as simple current extensions of their singlet cousins. The result is a relatively effortless derivation of the triplet “fusion rules” that agrees with those previously proposed in the literature.

Coset Constructions of Logarithmic \((1,p)\)-Models

Published in Letters in Mathematical Physics, 2013
Joint with T. Creutzig and D. Ridout
Journal Article | Preprint

Realises the singlet and triplet algebras of the logarithmic \((1,p)\)-models as cosets inside lattice vertex algebras, along the way constructing vertex algebras that, for small \(p\), are quotients of Feigin–Semikhatov W-algebras. Explicit character decompositions are worked out for \(p=2\) and \(p=3\).

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One of the best understood families of logarithmic conformal field theories is that consisting of the (1,p) models (p = 2, 3, …) of central charge c_{1,p} = 1 - 6 (p-1)^2 / p. This family includes the theories corresponding to the singlet algebras M(p) and the triplet algebras W(p), as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realized through a coset construction. The W^(2)n algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of affine sl(n)_k, generalising the Bershadsky-Polyakov algebra W^(2)_3. Inspired by work of Adamovic for p=3, vertex algebras B_p are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p <= 5, the algebra B_p is a homomorphic image of W^(2){p-1} at level -(p-1)^2 / p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p>5 as well. The triplet algebra W(p) is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra M(p) is similarly realised inside B_p. As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p=2 and 3.

On the extended \(W\)-algebra of type \(\mathfrak{sl}_2\) at positive rational level

Published in International Mathematical Research Notices, 2013
Joint with A. Tsuchiya
Journal Article | Preprint

Constructs, using screening operators on lattice vertex operator algebras, a family of vertex operator algebras that extend the Virasoro minimal models and serve as symmetries of logarithmic conformal field theories. Proves they are \(C_2\)-cofinite, determines their Zhu algebras and classifies their simple modules.

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The extended W-algebra of type sl_2 at positive rational level, denoted by M_{p_+,p_-}, is a vertex operator algebra that was originally proposed in [1]. This vertex operator algebra is an extension of the minimal model vertex operator algebra and plays the role of symmetry algebra for certain logarithmic conformal field theories. We give a construction of M_{p_+,p_-} in terms of screening operators and use this construction to prove that M_{p_+,p_-} satisfies Zhu’s c_2-cofiniteness condition, calculate the structure of the zero mode algebra (also known as Zhu’s algebra) and classify all simple M_{p_+,p_-}-modules.

The tensor structure on the representation category of the \(W_p\) triplet algebra

Published in Journal of Physics A, 2012
Joint with A. Tsuchiya
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Makes the Nahm–Gaberdiel–Kausch algorithm for computing fusion rigorous, giving a systematic method that works when representations are not completely reducible. Uses it to prove that the representation category of the \(\mathcal{W}_p\) triplet algebras is rigid and to prove the conjectured fusion formulae for all simple and projective modules.

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We study the braided monoidal structure that the fusion product induces on the abelian category \(\mathcal{W}_p\)-mod, the category of representations of the triplet \(W\)-algebra \(\mathcal{W}_p\). The \(\mathcal{W}_p\)-algebras are a family of vertex operator algebras that form the simplest known examples of symmetry algebras of logarithmic conformal field theories. We formalise the methods for computing fusion products, developed by Nahm, Gaberdiel and Kausch, that are widely used in the physics literature and illustrate a systematic approach to calculating fusion products in non-semi-simple representation categories. We apply these methods to the braided monoidal structure of \(\mathcal{W}_p\)-mod, previously constructed by Huang, Lepowsky and Zhang, to prove that this braided monoidal structure is rigid. The rigidity of \(\mathcal{W}_p\)-mod allows us to prove explicit formulae for the fusion product on the set of all simple and all projective \(\mathcal{W}_p\)-modules, which were first conjectured by Fuchs, Hwang, Semikhatov and Tipunin; and Gaberdiel and Runkel.

A modular invariant bulk theory for the \(c=0\) triplet model

Published in Journal of Physics A, 2010
Joint with M.R. Gaberdiel and I. Runkel
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Proposes the bulk state space of the logarithmic triplet model at central charge zero, built from a known consistent boundary theory and with a modular invariant partition function.

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A proposal for the bulk space of the logarithmic W(2,3)-triplet model at central charge zero is made. The construction is based on the idea that one may reconstruct the bulk theory of a rational conformal field theory from its boundary theory. The resulting bulk space is a quotient of the direct sum of projective representations, which is isomorphic, as a vector space, to the direct sum of tensor products of the irreducible representations with their projective covers. As a consistency check of our analysis we show that the partition function of the bulk theory is modular invariant, and that the boundary state analysis is compatible with the proposed annulus partition functions of this model.

Fusion Rules of the \(\mathcal{W}_{p,q}\) Triplet Models

Published in Journal of Physics A, 2009
Journal Article | Preprint

Extends the analysis of the central charge zero triplet model to the whole family of logarithmic \(\mathcal{W}_{p,q}\) triplet models, proposing associative fusion rules and identifying the projective covers. It also determines where fusion gives a consistent product on the Grothendieck group and suggests a candidate modular invariant partition function.

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In this paper we determine the fusion rules of the logarithmic \(\mathcal{W}_{p,q}\) triplet theory and construct the Grothendieck group with subgroups for which consistent product structures can be defined. The fusion rules are then used to determine projective covers. This allows us also to write down a candidate for a modular invariant partition function. Our results demonstrate that recent work on the \(\mathcal{W}_{2,3}\) model generalises naturally to arbitrary (p,q).

Fusion rules and boundary conditions in the \(c=0\) triplet model

Published in Journal of Physics A, 2008
Joint with M.R. Gaberdiel and I. Runkel
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Determines from first principles the fusion rules of the logarithmic triplet model at central charge zero, finding a finite set of representations that closes under fusion. Applied to boundary conditions, it shows that, unlike in rational theories, only some representations give consistent boundary conditions.

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The logarithmic triplet model W_2,3 at c=0 is studied. In particular, we determine the fusion rules of the irreducible representations from first principles, and show that there exists a finite set of representations, including all irreducible representations, that closes under fusion. With the help of these results we then investigate the possible boundary conditions of the W_2,3 theory. Unlike the familiar Cardy case where there is a consistent boundary condition for every representation of the chiral algebra, we find that for W_2,3 only a subset of representations gives rise to consistent boundary conditions. These then have boundary spectra with non-degenerate two-point correlators.

Moduli Webs and Superpotentials for Five-Branes

Published in Journal of High Energy Physics, 2008
Joint with M. Baumgartl
Journal Article | Preprint

Uses matrix factorisations of Landau–Ginzburg models to study D5-branes wrapping lines in one-parameter Calabi–Yau manifolds such as the quintic, whose moduli spaces form webs of curves meeting at permutation branes. This yields open–closed superpotentials to all orders and shows which branes survive bulk deformations.

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We investigate the one-parameter Calabi-Yau models and identify families of D5-branes which are associated to lines embedded in these manifolds. The moduli spaces are given by sets of Riemann curves, which form a web whose intersection points are described by permutation branes. We arrive at a geometric interpretation for bulk-boundary correlators as holomorphic differentials on the moduli space and use this to compute effective open-closed superpotentials to all orders in the open string couplings. The fixed points of D5-brane moduli under bulk deformations are determined.