# Geometric and Categorical Aspects of CFTs (18w5131)

Arriving in Oaxaca, Mexico Sunday, September 23 and departing Friday September 28, 2018

## Organizers

Jorgen Ellegaard Andersen (Aarhus University)

Ana Ros Camacho (Mathematisch Instituut, Universiteit Utrecht)

Gaetan Borot (Max Planck Institute for Mathematics)

David Ridout (University of Melbourne)

## Objectives

The two main aims that motivate our workshop proposal are:

1) To discuss the interrelations between the algebraic approaches, specifically vertex operator algebras and their representation theory, and the geometric approaches to two-dimensional conformal field theories.

2) To relate certain constructions relevant to low-dimensional topological quantum field theories, specifically modular functors, cohomological field theories}, topological recursion and factorisation homology, in which axioms involving the degeneration or glueing of surfaces play a prominent role, in order to disseminate mathematical techniques between communities.

Description:

--------------------------------------------

A. Two-dimensional conformal field theories (CFTs) are a special class of quantum field theories in which the usual Lorentz symmetry is enhanced to conformal symmetry. There has been substantial efforts over the last thirty years to study this symmetry --- rare in nature --- and the rich mathematical structures that it relates to. From an algebraic point of view, CFT has strongly motivated research on certain types of infinite--dimensional Lie algebras and is intimately connected to the theory of vertex operator algebras (VOAs), themselves introduced as a key tool to prove the monstrous moonshine conjectures of Conway and Norton.

The representation theory of VOAs has turned out to be unexpectedly rich with applications to enumerative combinatorics and q-series, modular forms and their generalisations, quantum groups and knot theory, to name a few. More recently, categorical aspects of VOA modules have been studied intensively, particularly in the case of rational VOAs (where the module category is semisimple). Much of the excitement here centres on the fact (first proved by Huang) that the modules of a rational VOA form a modular tensor category (MTC), a beautiful but subtle structure which manifests itself, e.g., via the celebrated Verlinde formula. Interestingly, MTCs also arise naturally in the study of subfactors and quantum groups.

Conformal field theories also have a strong geometric flavour deriving from their applications to (perturbative) string theory. This was originally developed for rational CFTs in the pioneering work of Knizhnik-Zamolodchikov, Beilinson-Drinfeld, Segal, Friedan-Shenker, Sonoda, Moore-Seiberg, Cardy-Lewellen, and many others. In their setup, the algebraic side (e.g., the VOA) encapsulates the local data of the CFT, while the global picture is codified through sewing relations. More precisely, the local structures are associated to punctured spheres which are then sewed/glued together to describe CFTs on general two-dimensional surfaces. This procedure, or rather its inverse factorisation forms a prototype for an idea that has led to remarkable progress in many mathematical disciplines.

B. The geometric approach to CFT has also given impetus to the study of low-dimensional topological quantum field theories (TQFTs). For instance, the results of Moore-Seiberg in part led Reshetikhin and Turaev to invent the notion of a MTC and turned CFTs into an effective machinery to design interesting 3-manifold invariants and 3d TQFTs. The topological content of a rational CFT is encoded in the notion of a modular functor (MF), a symmetric monoidal functor from a certain category of surfaces to the category of vector spaces that enjoys factorisation properties when the surfaces are pinched. MFs provide representations of towers of mapping class groups of surfaces whose properties are still being actively investigated.

The most famous example of a TQFT is 3d Chern-Simons theory for a compact gauge group, whose restriction to a 2d boundary gives the (rational) Wess-Zumino-Witten CFTs. For instance, proving the equivalence of several different constructions (via geometric quantisation of moduli spaces, via VOAs, via quantum groups) of MFs for these rather simple CFTs is nevertheless a very rich mathematical problem, which is still generating significant activity in algebraic geometry (see e.g. Andersen-Ueno). Exploring and exploiting this activity is one of the themes of the workshop.

C. The factorisation of rational CFTs can also be formulated as a property of the vector bundle of conformal blocks over the moduli space of curves when approaching its Deligne-Mumford boundary. The same factorisation axioms at the level of Chern characters are captured in the notion of cohomological field theory (CohFT). CohFTs were in fact invented by Kontsevich and Manin as an axiomatic approach to Gromov-Witten theory for symplectic manifolds. This represents a shadow of deeper relations, expected from the early stages of topological string theory, between CFTs and enumerative problems in algebraic geometry.

The theory of CohFTs has greatly advanced in recent years, with the classification of semisimple CohFTs, due to works of Givental and of Teleman. This fundamental result was instrumental in deriving new relations in the tautological subring of the cohomology of the moduli space of curves. It was also exploited by Dunin-Barkowski-Orantin-Shadrin-Spitz. to establish a dictionary between semisimple CohFTs and the theory of topological recursion (TR) initiated by Eynard and Orantin. In recent years, the same TR structure has been uncovered in an increasing number of problems in enumerative geometry, low-dimensional TQFTs and Gromov-Witten theory. Moreover, precise relationships between CFT and TR are now being actively investigated.

D. The theory of factorisation algebras and factorisation homology, recently developed by Costello-Gwilliam and Lurie, initially from the perspective of homotopy theory and higher categories, has led to an ambitious dream to unify parts of homological algebra and topology. These theories have already been successfully applied to the construction of 3d and 4d TQFTs (Ben-Zvi-Brochier-Jordan), the definition of new categorical invariants of knots, the construction of VOAs (Costello-Gwilliam), and in the quantisation of moduli spaces (character varieties) and algebraic structures (Lie bialgebras). We want to discuss these exciting techniques and specifically their applications and relations to CFTs in the workshop.

E. We would also like to discuss other geometric realisations of CohFTs and certain relations between them, with at least two directions supported by recent work. First, on the B-side of mirror symmetry, Polishchuk and Vaintrob have produced CohFTs out of orbifold Landau-Ginzburg (LG) models. These are induced by functors from categories of matrix factorisations (attached to the LG model) to the derived category of coherent sheaves on the moduli space of curves.

Second, Marian-Oprea-Pandharipande-Pixton-Zvonkinee (for Wess-Zumino-Witten CFTs) and Andersen-Borot-Orantin (for any MF) construct a semisimple CohFT, obtained as the Chern character of the bundle of conformal blocks. The connections between those two types of constructions should be clarified by the Landau-Ginzburg/CFT correspondence, which is the conjectural equivalence of monoidal categories of matrix factorisations (on the LG side, often easy to compute with) and categories of representations of superconformal algebras (on the CFT side, usually harder to control), see e.g. Ros Camacho. An important question in this field is whether one can find a LG model description for the CohFT attached to a given MF. This should be closely related to the properties of the kernel of the mapping class group representations produced by the MF, itself an important object of study.

F. Despite all these mathematical advances, CFTs and VOAs are only reasonably well understood in the rational (semi-simple) case. Physicists realised long ago that non-rational examples, known as logarithmic CFTs, arise far more frequently in applications. A key question that is currently attracting a lot of attention is how the Verlinde formula and the notions of MTCs and MFs generalise to the logarithmic setting (Fuchs-Hwang-Semikhatov-Tipunin, Creutzig-Ridout, Creutzig-Gannon, Gainutdinov-Runkel). We will propose a 2h introductory lecture on the topic of logarithmic CFTs and generalisations of the Verlinde formula and MTCs.

We would also like to discuss how the CohFT -> MF construction should be amended for logarithmic CFTs; the result should involve non-semisimple CohFTs. Although the notion of non-semisimple CohFTs makes perfect sense, they are considerably less well understood despite many important examples coming from Gromov-Witten theory and mirror symmetry.

Similar questions also arise for TQFTs with an infinite-dimensional space of states, like 3d Chern-Simons theory with a complex gauge group. An example of hint in this direction is the Verlinde-type formula found in Andersen-Gukov-Pei for the dimension of the space of states obtained by the geometric quantisation of the moduli stack of Higgs bundle.

All the problems mentioned above have in common the prominence of factorisation properties. The workshop will be a unique opportunity to disseminate and discuss important recent advances among several communities of researchers working in areas related to geometric and categorical aspects of CFT, MTCs, VOAs and representation theory, quantum topology, higher categories and mathematical physics. Given the rapid development of these fields, there is indeed a constant need for unification and comparison in the techniques being developed and used to understand these mathematical structures.

To help foster these interactions, we plan to invite several experts to present introductory level expositions, on

1) logarithmic CFTs and generalisations of the Verlinde formula and MTCs

2) CohFTs from matrix factorizations of LG models

3) Factorization homology and applications in QFTs

and aimed specifically at younger researchers in different, but related, areas. These lectures will then complement the high level research talks and discussion sessions that will form the core of the workshop.

1) To discuss the interrelations between the algebraic approaches, specifically vertex operator algebras and their representation theory, and the geometric approaches to two-dimensional conformal field theories.

2) To relate certain constructions relevant to low-dimensional topological quantum field theories, specifically modular functors, cohomological field theories}, topological recursion and factorisation homology, in which axioms involving the degeneration or glueing of surfaces play a prominent role, in order to disseminate mathematical techniques between communities.

Description:

--------------------------------------------

A. Two-dimensional conformal field theories (CFTs) are a special class of quantum field theories in which the usual Lorentz symmetry is enhanced to conformal symmetry. There has been substantial efforts over the last thirty years to study this symmetry --- rare in nature --- and the rich mathematical structures that it relates to. From an algebraic point of view, CFT has strongly motivated research on certain types of infinite--dimensional Lie algebras and is intimately connected to the theory of vertex operator algebras (VOAs), themselves introduced as a key tool to prove the monstrous moonshine conjectures of Conway and Norton.

The representation theory of VOAs has turned out to be unexpectedly rich with applications to enumerative combinatorics and q-series, modular forms and their generalisations, quantum groups and knot theory, to name a few. More recently, categorical aspects of VOA modules have been studied intensively, particularly in the case of rational VOAs (where the module category is semisimple). Much of the excitement here centres on the fact (first proved by Huang) that the modules of a rational VOA form a modular tensor category (MTC), a beautiful but subtle structure which manifests itself, e.g., via the celebrated Verlinde formula. Interestingly, MTCs also arise naturally in the study of subfactors and quantum groups.

Conformal field theories also have a strong geometric flavour deriving from their applications to (perturbative) string theory. This was originally developed for rational CFTs in the pioneering work of Knizhnik-Zamolodchikov, Beilinson-Drinfeld, Segal, Friedan-Shenker, Sonoda, Moore-Seiberg, Cardy-Lewellen, and many others. In their setup, the algebraic side (e.g., the VOA) encapsulates the local data of the CFT, while the global picture is codified through sewing relations. More precisely, the local structures are associated to punctured spheres which are then sewed/glued together to describe CFTs on general two-dimensional surfaces. This procedure, or rather its inverse factorisation forms a prototype for an idea that has led to remarkable progress in many mathematical disciplines.

B. The geometric approach to CFT has also given impetus to the study of low-dimensional topological quantum field theories (TQFTs). For instance, the results of Moore-Seiberg in part led Reshetikhin and Turaev to invent the notion of a MTC and turned CFTs into an effective machinery to design interesting 3-manifold invariants and 3d TQFTs. The topological content of a rational CFT is encoded in the notion of a modular functor (MF), a symmetric monoidal functor from a certain category of surfaces to the category of vector spaces that enjoys factorisation properties when the surfaces are pinched. MFs provide representations of towers of mapping class groups of surfaces whose properties are still being actively investigated.

The most famous example of a TQFT is 3d Chern-Simons theory for a compact gauge group, whose restriction to a 2d boundary gives the (rational) Wess-Zumino-Witten CFTs. For instance, proving the equivalence of several different constructions (via geometric quantisation of moduli spaces, via VOAs, via quantum groups) of MFs for these rather simple CFTs is nevertheless a very rich mathematical problem, which is still generating significant activity in algebraic geometry (see e.g. Andersen-Ueno). Exploring and exploiting this activity is one of the themes of the workshop.

C. The factorisation of rational CFTs can also be formulated as a property of the vector bundle of conformal blocks over the moduli space of curves when approaching its Deligne-Mumford boundary. The same factorisation axioms at the level of Chern characters are captured in the notion of cohomological field theory (CohFT). CohFTs were in fact invented by Kontsevich and Manin as an axiomatic approach to Gromov-Witten theory for symplectic manifolds. This represents a shadow of deeper relations, expected from the early stages of topological string theory, between CFTs and enumerative problems in algebraic geometry.

The theory of CohFTs has greatly advanced in recent years, with the classification of semisimple CohFTs, due to works of Givental and of Teleman. This fundamental result was instrumental in deriving new relations in the tautological subring of the cohomology of the moduli space of curves. It was also exploited by Dunin-Barkowski-Orantin-Shadrin-Spitz. to establish a dictionary between semisimple CohFTs and the theory of topological recursion (TR) initiated by Eynard and Orantin. In recent years, the same TR structure has been uncovered in an increasing number of problems in enumerative geometry, low-dimensional TQFTs and Gromov-Witten theory. Moreover, precise relationships between CFT and TR are now being actively investigated.

D. The theory of factorisation algebras and factorisation homology, recently developed by Costello-Gwilliam and Lurie, initially from the perspective of homotopy theory and higher categories, has led to an ambitious dream to unify parts of homological algebra and topology. These theories have already been successfully applied to the construction of 3d and 4d TQFTs (Ben-Zvi-Brochier-Jordan), the definition of new categorical invariants of knots, the construction of VOAs (Costello-Gwilliam), and in the quantisation of moduli spaces (character varieties) and algebraic structures (Lie bialgebras). We want to discuss these exciting techniques and specifically their applications and relations to CFTs in the workshop.

E. We would also like to discuss other geometric realisations of CohFTs and certain relations between them, with at least two directions supported by recent work. First, on the B-side of mirror symmetry, Polishchuk and Vaintrob have produced CohFTs out of orbifold Landau-Ginzburg (LG) models. These are induced by functors from categories of matrix factorisations (attached to the LG model) to the derived category of coherent sheaves on the moduli space of curves.

Second, Marian-Oprea-Pandharipande-Pixton-Zvonkinee (for Wess-Zumino-Witten CFTs) and Andersen-Borot-Orantin (for any MF) construct a semisimple CohFT, obtained as the Chern character of the bundle of conformal blocks. The connections between those two types of constructions should be clarified by the Landau-Ginzburg/CFT correspondence, which is the conjectural equivalence of monoidal categories of matrix factorisations (on the LG side, often easy to compute with) and categories of representations of superconformal algebras (on the CFT side, usually harder to control), see e.g. Ros Camacho. An important question in this field is whether one can find a LG model description for the CohFT attached to a given MF. This should be closely related to the properties of the kernel of the mapping class group representations produced by the MF, itself an important object of study.

F. Despite all these mathematical advances, CFTs and VOAs are only reasonably well understood in the rational (semi-simple) case. Physicists realised long ago that non-rational examples, known as logarithmic CFTs, arise far more frequently in applications. A key question that is currently attracting a lot of attention is how the Verlinde formula and the notions of MTCs and MFs generalise to the logarithmic setting (Fuchs-Hwang-Semikhatov-Tipunin, Creutzig-Ridout, Creutzig-Gannon, Gainutdinov-Runkel). We will propose a 2h introductory lecture on the topic of logarithmic CFTs and generalisations of the Verlinde formula and MTCs.

We would also like to discuss how the CohFT -> MF construction should be amended for logarithmic CFTs; the result should involve non-semisimple CohFTs. Although the notion of non-semisimple CohFTs makes perfect sense, they are considerably less well understood despite many important examples coming from Gromov-Witten theory and mirror symmetry.

Similar questions also arise for TQFTs with an infinite-dimensional space of states, like 3d Chern-Simons theory with a complex gauge group. An example of hint in this direction is the Verlinde-type formula found in Andersen-Gukov-Pei for the dimension of the space of states obtained by the geometric quantisation of the moduli stack of Higgs bundle.

All the problems mentioned above have in common the prominence of factorisation properties. The workshop will be a unique opportunity to disseminate and discuss important recent advances among several communities of researchers working in areas related to geometric and categorical aspects of CFT, MTCs, VOAs and representation theory, quantum topology, higher categories and mathematical physics. Given the rapid development of these fields, there is indeed a constant need for unification and comparison in the techniques being developed and used to understand these mathematical structures.

To help foster these interactions, we plan to invite several experts to present introductory level expositions, on

1) logarithmic CFTs and generalisations of the Verlinde formula and MTCs

2) CohFTs from matrix factorizations of LG models

3) Factorization homology and applications in QFTs

and aimed specifically at younger researchers in different, but related, areas. These lectures will then complement the high level research talks and discussion sessions that will form the core of the workshop.