Two-Dimensional Conformal Field Theory - Math · Two-dimensional conformal ﬁeld theories The major problems solved Open problems Two-Dimensional Conformal Field Theory Yi-Zhi Huang

Two-dimensional conformal field theories The major problems solved Open problems

Two-Dimensional Conformal Field Theory

Yi-Zhi Huang

Department of MathematicsRutgers University

Piscataway, NJ 08854, USAand

Beijing International Center for Mathematical ResearchPeking University

Beijing, China

June 3, 2015

Institute of Mathematics, Chinese Academy of Sciences


Outline

1 Two-dimensional conformal field theoriesQuantum field theories in mathematicsA definition and early conjecturesProblems and a mathematical program

2 The major problems solvedThe geometry of vertex operator algebrasIntertwining operators and vertex tensor categoriesModular invariance and Verlinde formulaRigidity and modularityFull and open-closed conformal field theoriesCohomology and deformation theory

3 Open problemsHigher-genus theories and locally convex completionsModuli space of conformal field theoriesNonrational conformal field theories


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Part 1

Two-dimensionalconformal field theories


Quantum field theories in mathematics

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Early formulations

The early mathematical study of quantum field theoriesstarted in 1950’s. It tried to put the “operator-valued fields”on a rigorous mathematical foundation using functionalanalysis. The Wightman axioms, theOsterwalder-Schrader theorem and the Haag-Kastleraxiomatic system all appeared during this period.Important works on the construction of theories satisfyingsome of these axioms were done by I. Segal, Jaffe, Glimmand others in 1970’s.Unfortunately, interesting quantum field theories such asfour-dimensional Yang-Mills theoreies still cannot betreated using these methods.



Early formulations




Early formulations




Early formulations




Geomrtric formulations

Starting from 1980’s, motivated by the path integralapproach in physics, Kontsevich, G. Segal and Atiyahproposed that quantum field theories are functors fromgeometric categories to categories formed by Hilbertspaces.Such a definition and the subsequent construction andstudy are very successful in the case of topologicalquantum field theories. The main reason for this success isthat the Hilbert space for a topological quantum field theoryis typically finite dimensional.For nontrivial nontopological quantum field theories, theHilbert space must be infinite dimensional. Theconstruction and study of these theories are much moredifficult.















Mathematical problems

One of the Millennium Prize Problems: Establish rigorouslythe existence of the quantum Yang-Mills theory and provethat there is a mass gap in this theory.Many mathematical conjectures arising from string theoryare in fact arising from two dimensional superconformalfield theories or higher-dimensional supersymmetricYang-Mills theories.To understand these mathematical conjectures completely,we need to construct the correspoding quantum fieldtheories.The simplest nontopological quantum field theories aretwo-dimensional conformal field theories. Manyconjectures in mathematics have been derived based onthe stronger conjectures that the correspondingtwo-dimensional conformal field theories exist.


















A definition and early conjectures

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A definition of two-dimensional conformal field theory

1987, Kontsevich and G. Segal: Definition oftwo-dimensional conformal field theory.1988, G. Segal: Definitions of modular functor and weaklyconformal field theory.A two-dimensional conformal field theory in the sense ofKontsevich-Segal is

a locally convex topological vector space H,a nondegenerate hermitian form,a projective functor from the category whose morphismsare Riemann surfaces with parametrized boundaries to thecategory of tensor powers of H and traceclass maps,

satisfying additional but natural conditions.From now on, for simplicity, we omit the words“two-dimensional” so that conformal field theories meantwo-domensional conformal field theories.



















































Conjectures of Verlinde and Moore-Seiberg

1987, E. Verlinde:Verlinde conjecture: For a rational conformal field theory,the modular transformaton S on the space of vacuumcharacters associated to τ 7→ −1/τ diagonaizes thematrices formed by fusion rules.Verlinde formula for fusion rules: Express fusion rules interms of matrix elements of the modular transformaton S.

1988, Moore and Seiberg:Obtained Moore-Seiberg polynomial equations usingconjectures on operator product expansion and modularinvariance for intertwining operators.Derived Verlinde conjecture and Verlinde fromula fromthese polynomial equations.Conjectured the existence of modular tensor category.






































Conjectures of Witten

1989, Witten:Obtained knot and three-manifold invariants fromconjectures for rational conformal field theories.Conjectured that in the case of Wess-Zumino-Wittenmodels, these invariants should be the same as those fromthe corresponding Chern-Simons theories.

1991, Witten:Holomorphic factorzation of Wess-Zumino-Witten models,assuming the nondegeneracy of hermitian forms on thespaces of conformal blocks.




























Conjectures on nonlinear σ-models associated toCalabi-Yau manifolds

1985, Friedan, Candelas, Horowitz, Strominger, Witten,Alvarez-Gaumé, Coleman, Ginsparg, Nemeschansky, Senand so on: Nonlinear σ-models with Calabi-Yau manifoldsas targets are N = 2 superconformal field theories.1987, Gepner: A suitable orbifold and tensor productconstruction from an N = 2 superconformal minimal modelgives the nonlinear σ-model for the quantic Calabi-Yaumanifold.1989, Green and Plesser: Gepner conjecture, orbifoldconstruction and deformations of Calabi-Yau manifolds andN = 2 superconformal field theories give mirror symmetry.














Problems and a mathematical program

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The mathematical problems

Problem 1: Formulate precisely and prove the conjecturesof Verlinde, Moore-Seiberg and Witten.Problem 2: Give a construction of conformal field theoriessatisfying the axioms of Kontsevich and Segal, or at leastprove the existence of such conformal field theories. Inparticular, give a construction of the Wess-Zumino-Wittenmodels and the minimal models, or at least prove theexistence of these theories.Problem 3: Study the moduli space of conformal fieldtheories.Problem 4: Construct the N = 2 superconformal fieldtheories associated to Calabi-Yau manifolds, proveGepner’s conjecture, turn the ideas of Green-Plesser into amathematical construction of mirror symmetry.



















A long term program

If there exists a conformal field theory satisfying thedefinition of Kontsevich and G. Segal, then the space ofmeromorphic fields form an algebraic structure calledvertex operator algebra in mathematics and chiral algebrain physics.The first part of the program: Construct and studyconformal field theories using the representation theory ofvertex operator algebras.The second part of the program: Study the moduli space ofconformal field theories using the cohomology theory anddeformation theory of vertex operator algebras.



A long term program




A long term program




A long term program




A long term program

Rational conformal field theories have been constructedfrom modules and intertwining operators for vertexoperator algebras, except that there are still someconjectures involving higher-genus Riemann surfaces tobe proved. We believe that several classes of non-rationalconformal field theories can also be constructed using therepresentation theory of vertex operator algebras.A cohomology theory and a deformation theory for vertexoperator algebras and conformal field theories are beingdeveloped. We believe that the moduli space of conformalfield theories can be studied using these theories.In this talk, I will survey the main results and openproblems in this long term program.



A long term program




A long term program




A long term program



Part 2

The major problems solved


The geometry of vertex operator algebras

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Vertex operator algebras

1984, Belavin, Polyakov and Zamolodchikov: Operatorproduct expansion of conformal fields. Vertex operatoralgebras were implicitly there.1986, Borcherds: Vertex algebras. Examples from evenlattices. Insight on the moonshine module.1988, Frenkel, Lepowsky and Meurman: Vertex operatoralgebras. Examples, including the moonshine modulevertex operator algebra. Construction of modules andtwisted modules.Since then, the representation theory of vertex operatoralgebras has been rapidly developed by manymathemticians and physicists.




















Vertex operator algebras have a purely algebraic definition.They are analogues of commutative associative algebrasand Lie algebras, but involve additional variables. Here Iomit this definition.Mathematicians have developed systematic methods toconstruct vertex operator algebras.Examples of vertex operator algebras were constructed inphysics and mathematics from representations ofHeisenberg algebras, Clifford algebras, affine Lie algebras,Virasoro algebra, superconformal algebras andW-algebras. They were also constructed from lattices(corresponding to certain types of tori). One remarkableexample is the moonshine module whose automorphismgroup is the largest sporadic finite simple group, theMonster.















The early work of I. Frenkel and Tsukada

1986, I. Frenkel and Tsukada: Started a program toconstruct conformal field theories mathematically usingpath integrals. Obtained a geometric interpretation ofmeromorphic vertex operators and their basic properties.1988, Tsukada: Constructed vertex operator algebrasassociated to positive-definite even lattices using pathintegrals.However, they did not solve the problem of giving ageometric formulation of the conformal element, theVirasoro algebra or, especially, the central charge for avertex operator algebra. This geometric formulation isnecessary if we want to construct conformal field theoriesin the sense of Kontsevich and G. Segal.















The first major problem in the program

1988, Segal: The central charge of a conformal field theoryshould be interpreted as twice the power of thedeterminant line bundle over the moduli space of Riemannsurfaces with parametrized boundaries.The first major problem to be solved in this program: Fromthe works above, one can easily make a conjecture onwhat the geometric formulation of a vertex operatoralgebra (including the conformal element, the Virasoroalgebra and the central charge) should be. Prove that thepurely algebraic formulation of a vertex operator algebra isequivalent to this infinite-dimensional analytic andgeometric formulation. This turned out to be a very difficultproblem.











The solution

1991, H.: Solved this problem completely.Main hard part: Prove that certain formal series obtainedfrom vertex operators and the Virasoro operators areexpansions of certain analytic functions coming fromgenus-zero Riemann surfaces and the determinant linebundle. This was done by using a theorem of Fischer andGrauert in the deformation theory of complex manifoldsand the holomorphicity of the sewing isomorphisms for thedeterminant lines.Geometric definition: A vertex operator algebra of centralcharge c is roughly speaking a meromorphicrepresentation of the c/2-th power of the determinant linebundle over the moduli space of Riemann sphere withpunctures and local coordinates vanishing at thepunctures, equipped with the natural sewing operation.



The solution




The solution




The solution



Intertwining operators and vertex tensor categories

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Intertwining operators

The graded dimension (vacuum character) of a vertexoperator algebra in general is not modular invariant andthus is not enough to construct a genus-one conformalfield theory. For affine Lie algebras and the Virasoroalgebra, one needs to use all modules to obtain a modularinvariant vector space. The modular invariancerequirement forces us to consider modules for the vertexoperator algebra, not just the algebra itself.Consequently, we have to study “vertex operators” amongdifferent modules. These “vertex operators” were calledchiral vertex operators by Moore and Seiberg andintertwining operators by Frenkel, Lepowsky and myself.











The operator product expansion conjecture of Mooreand Seiberg

1988, Moore and Seiberg: “Consider the operator productexpansion:

Φα,a(z1)Φβ,b(z2) =∑

k

∑c∈V l

kr ;d∈V kij

Fpk

[i jl r

]cd

ab

×∑

K∈Hk

ξα,βp,K ,d

[i jl r

](z1, z2, z3)ΦK ,c(z3) (3.9)

.... This expansion is an asymptotic expansion which isbelieved to have a finite radius of convergence. It is valid forz1 ∼ z2 ∼ z3.”



The operator product expansion conjecture of Mooreand Seiberg

1988, Moore and Seiberg: “Consider the operator productexpansion:

Φα,a(z1)Φβ,b(z2) =∑

k

∑c∈V l

kr ;d∈V kij

Fpk

[i jl r

]cd

ab

×∑

K∈Hk

ξα,βp,K ,d

[i jl r

](z1, z2, z3)ΦK ,c(z3) (3.9)

.... This expansion is an asymptotic expansion which isbelieved to have a finite radius of convergence. It is valid forz1 ∼ z2 ∼ z3.”



The second major problem

There was no explanation or even discussion as to whythis operator product expansion must hold. It was used byMoore and Seiberg as an additional hypothesis, not aresult. Mathematically, it was clearly a conjecture.This operator product expansion is in fact equivalent to theassociativity for intertwining operators:

Y1(w1, z1)Y2(w2, z2) = Y3(Y4(w1, z1 − z2)w2, z2)

in the region |z1| > |z2| > |z1 − z2| > 0.This was the second major problem to be solved. Sinceintertwining operators are in general multivalued and formvector spaces, the usual purely algebraic method used tostudy vertex operator algebras and modules does notwork. It was necessary to develop a new method.





Y1(w1, z1)Y2(w2, z2) = Y3(Y4(w1, z1 − z2)w2, z2)






Y1(w1, z1)Y2(w2, z2) = Y3(Y4(w1, z1 − z2)w2, z2)






Y1(w1, z1)Y2(w2, z2) = Y3(Y4(w1, z1 − z2)w2, z2)




The “intermediate modules” and the tensor productmodules

To prove the associativity for intertwining operators, we firsthad to construct the “intermediate module.” Theintermediate modue can in fact be taken to be the tensorproduct, if it exists, of two of the modules involved.1991, H. and Lepowsky: The tensor product modules wasconstructed for a vertex operator algebra satisfying certainfinite reductivity conditions.











The early results

1995, H.: Proved the associativity for intertwiningoperators using a characterization of intertwining operatorsobtained in the construction of tensor product modules andassuming certain “convergence and extension properties”in addition to certain finite reductivity conditions.1995, H.: Proved the associativity for intertwining operatorsin the case of minimal models. A new construction of themoonshine module was obtained using this result.1997, H. and Lepowsky: Proved the associativity forintertwining operators in the case of theWess-Zumino-Witten models.1999 and 2000, H. and Milas: Proved the associativity forintertwining operators in the case of N = 1 and N = 2superconformal minimal models.



The early results




The early results




The early results




The early results




The solution

2002, H.: Solved completely this second major problem byproving that the convergence and extension propertieshold when the vertex operator algebra or its modulessatisfy a certain purely algebraic cofiniteness condition inaddition to certain other natural and purely algebraicconditions.Main idea: Derive differential equations with regularsingular points and then use these differential equations toprove the convergence and extension properties.



The solution




The solution




Braided tensor categories and vertex tensorcategories

The proof of the associativity for intertwining operators alsogave immediately natural associativity isomorphisms forthe tensor product bifunctors constructed by Lepowsky andme. The coherence for the associativity isomorphismsfollows easily from a characterization of the associativityisomorphisms.The braiding isomorphism can be obtained easily from theskew-symmetry of intertwining operators. In particular, thecategory of modules has a natural structure of a braidedtensor category.In fact, since the tensor product bifunctor constructed byLepowsky and me depends on a sphere with puncturesand local coordinates vanishing at the punctures, what weobtain is what we call a “vertex tensor category.”














Modular invariance and Verlinde formula

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The modular invariance conjecture of Moore andSeiberg

1988, Moore and Seiberg: “The final equation is obtained fromthe two-point function on the torus. The conformal blocks forthe two-point function of β1 ∈ Hj1 , β2 ∈ Hj2 are given by

Tri

[qL0− c

24

(i

j1p

)z1

(β1 ⊗ ·)(

pj2i

)z2

(β2 ⊗ ·)

]·

·(dz1)∆β1 (dz2)

∆β2 . (4.13)”



The modular invariance conjecture of Moore andSeiberg

1988, Moore and Seiberg: “The final equation is obtained fromthe two-point function on the torus. The conformal blocks forthe two-point function of β1 ∈ Hj1 , β2 ∈ Hj2 are given by

Tri

[qL0− c

24

(i

j1p

)z1

(β1 ⊗ ·)(

pj2i

)z2

(β2 ⊗ ·)

]·

·(dz1)∆β1 (dz2)

∆β2 . (4.13)”



The third major problem

By stating that the conformal blocks for the two-pointfunction are given by the traces above, Moore and Seibergin fact assumed the modular invaraince of the spacespanned by these traces. This modular invariance wasused as an additional hypothesis, not a result.Mathematically, it was clearly a powerful conjecture. Manyof the deep results in this program depend on the solutionto this conjecture. This was the third major problem to besolved.Note that even for Wess-Zumino-Witten and minimalmodels, this was a conjecture, not a theorem.















Zhu’s theorem

In 1990, in his important Ph.D. thesis work, Zhu proved apartial result on the modular invariance conjecture ofMoore and Seiberg.This partial result stated that when the vertex operatoralgebra satisfies a positive energy condition, a completereducibility condition and a condition now calledC2-cofiniteness condition, the q-traces of products of nsuitable modified vertex operators for irreducible modulescan be analytically extended to meromorphicdoubly-periodic functions on the plane with periods 1 andτ = (log q)/2πi and span a vector space that is invariantunder the action of the full modular group SL(2,Z).



Zhu’s theorem




Zhu’s theorem




The failure of Zhu’s method

2000, Miyamoto: Generalized Zhu’s partial result to thepartial result for one intertwining operator and n vertexoperators for modules, using Zhu’s method.Unfortunately, the method developed by Zhu cannot beused or adapted to prove the (full) modular invarianceconjecture of Moore and Seiberg mentioned above.The reason that Zhu’s method cannot be used or adaptedis the following: Zhu’s method uses the commutatorformula for vertex operators (acting on modules) to reducethe construction of genus-one n-point functions to theconstruction of genus-one one-point functions. But there isno commutator formula for general intertwining operators.















The solution

2003, H.: Proved the full modular invariance conjecture ofMoore and Seiberg under conditions that are slightlyweaker than those in Zhu’s partial result.In particular, the full modular invariance conjectures forWess-Zumino-Witten and minimal models are theorems asof 2003.The proof depended on the associativity for intertwiningoperators and used a new method that is completelydifferent from the one by Zhu.



The solution




The solution




The solution




The new method

Prove that the q-traces of products of intertwiningoperators satisfy certain systems of modular invariantdifferential equations with regular singular points.Prove that the q-traces of products of intertwiningoperators are absolutely convergent and have genus-oneassociativity for intertwining operators (or genus-oneoperator product expansions ) using the systems ofdifferential equations and the associativity for intertwiningoperators.Reduce the proof of the modular invariance for genus-onen-point functions to the modular invariance for genus-oneone-point functions (already proved by Zhu and Miyamoto)using the genus-one associativity for intertwiningoperators.



The new method




The new method




The new method




The fourth major problem

Moore and Seiberg showed that the Verlinde conjectureand Verlinde formula could indeed be derived from somebasic conjectures—for example, the operator productexpansion for chiral vertex operators and the full modularinvariance conjecture on rational conformal field theories.But since Moore and Seiberg did not prove theseconjectures, the Verlinde conjecture and Verlinde formulawere still not proved in their paper.It turned out that the Verlinde conjecture and Verlindeformula were an important step in the program to bfconstruct rational conformal field theories fromrepresentations of vertex operator algebras. Thus, theyshould be viewed as the fourth major problem to be solvedin this program.











The soultion

2004, H.: Proved the Verlinde conjecture and Verlindeformula for a vertex operator algebra satisfying the sameconditions as in the full modular invariance theoremtogether with another condition that the contragredientmodule of the vetrex operator algebra viewed as a moduleis equivalent to the vertex operator algebra itself.This work in fact proved that the Moore-Seiberg polynomialequations hold for all such vertex operator algebras. Thusmuch stronger results were obtained. These strongerresults played an important role in the proof of the rigidityand modularity conjecture which I will discuss next.



The soultion




The soultion



Rigidity and modularity

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The fifth major problem

The rigidity of braided tensor category structure on thecategory of modules for a vertex operator algebra was anopen problem for many years.Another closely related hard open problem (let’s call itmodularity) was the nondegenracy property and theidentification of the S-matrix obtained from the ribbontensor category structure with the action of the modulartransformation associated to τ 7→ −1/τ on the spacespanned by the graded dimension.These were the fifth major problem to be solved.















The solution

2005, H.: Proved the rigidity and modularity of the braidedtensor category of modules for a vertex operator algebrasatisfying the three conditions in the modular invariancetheorem. In particular, combining this result with the workof Turaev, we obtain knot and three manifold invariantsproposed first by Witten.The proof used a strong version of the Verlinde formulaand thus logically used the modular invariance theorem.This was a surprise.For many years, there were widely circulated claims thatthe rigidity and modularity for the Wess-Zumino-Wittenmodels had been proved, and that for general rationalconformal field theories they could be proved in the sameway. Such claims have been shown to be wrong andacknowledged as such in recent years.



The solution




The solution




The solution




The case of Wess-Zumino-Witten models

In the case of Wess-Zumino-Witten models, Finkelberg’stensor-category-equivalence theorem together withKazhdan-Lusztig’s rigidity theorem for negative levels hadbeen thought to prove the rigidity for almost all (but not all)cases. But it turned out that Finkelberg’s paper had a gapand it required either the Verlinde formula proved byFaltings, Teleman and me or, alternatively, the rigidityproved by me, to fill the gap and prove the equivalencetheorem (again, for almost all, but not all, cases).Note that, as is mentioned above, my proof of the Verlindeformula or my proof of the rigidity needs the full modularinvariance theorem.Finkelberg’s tensor-category-equivalence theorem, aftercorrection, still does not cover a few exceptional cases,including in particular, the E8 level 2 case.














Full and open-closed conformal field theories

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The sixth major problem

The results discussed above are for chiral conformal fieldtheories. Chiral conformal field theories are not enough.We need to put chiral and antichiral conformal field theoriestogether in a suitable way to construct full conformal fieldtheories. Here antichiral conformal field theories are justsome chiral conformal field theories that will become theantichiral parts of full conformal field theories.Since the conformal fields in full conformal field theoriesmust be single-valued but intertwining operators are ingeneral multivalued, the construction of full field theoriesfrom chiral and antichiral conformal field theories are highlynontrivial.This was the sixth major problem to be solved.















The solution

2005 and 2006, Kong and H.: Solved this major problemfor genus-zero and genus-one theories, respectively, in theso-called diagonal case, that is, the case that the statespace of a full conformal field theory is the direct sum ofthe tensor products of irreducible modules for a vertexoperator algebra and its contragredient modules.The main work is to construct a nondegenerate bilinearform on the space of intertwining operators satisfyingnatural properties. The difficult part is the proof of thenondegeneracy of the bilinear form. Recall that in the workof Witten on holomorphc factorization ofWess-Zumino-Witten models, the nondegeracy of thehermitian form on the space of conformal blocks was anassumption, not a result.



The solution




The solution




A surprise

It was very surprising to us that the proof of thenodegeneracy of the bilinear form, a property forgenus-zero chiral conformal field theories, needs the (full)modular invariance theorem, a genus-one property.The nondegeneracy of the bilinear form is in fact equivalentto the rigidity of the corresponding braided tensor category.This is another indication why the proof of the rigidityneeded the (full) modular invariance theorem and was sodifficult, even in the case of Wess-Zumino-Witten models.



A surprise




A surprise




The seventh major problem

The problems and solutions discussed above are all forclosed conformal field theories. We also need to constructopen-closed conformal field theories.One needs to construct open-string vertex operatoragebras. Modules for open-string vertex algebras in factcorrespond exactly to the important D-branes introducedand studied by string theorists.The connection between the open part and the closed partof an open-closed conformal field theory is given by what iscalled Cardy condition, studied first by Cardy.The seventh major problem was to construct open-stringvertex operator algebras and open-closed conformal fieldtheories.



















The solution

2004, Kong and H.: Constructed open-string vertexoperator algebras from modules and intertwining operatorsfor suitable vertex operator algebras.2006, Kong: Gave tensor categorical and geometricformulations of open-closed conformal field theories.Studied the Cardy condition in terms of the underlyingmodular tensor categorical structure discussed above onthe category of modules for the underlying vertex operatoralgebra.



The solution




The solution



Cohomology and deformation theory

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The eighth major problem

To study the moduli space of conformal field theories, oneneeds a deformation theory of conformal field theories. Inparticular, one needs a deformation theory of vertexoperator algebras.Just as for associative algebras, Lie algebras and otheralgebras, one needs a correct cohomology theory forvertex operator algebras to develop a deformation theoryand to study the structure and representation theory ofvertex operator algebras. There had been proposals forsuch a theory before 2010, but unfortunately theseproposals are not the correct cohomology theory.Finding a correct cohomology theory for vertex operatoralgebras is the eighth major problem in the program.















Cohomology theory

2010, H.: Introduced a cohomology theory and proved thatthis cohomology theory has the properties a cohomologytheory must have. Thus this is the correct cohomologytheory.The crucial idea in this cohomology theory is that insteadof linear maps from the tensor powers of the algebra to abimodule for the algebra, we use the linear maps from thetensor powers of the algebra to the rational functionsvalued in the algebraic completion of the bimodule.



Cohomology theory




Cohomology theory




Deformation theory

2010, H.: Proved that infinitesimal deformations of a vertexoperator algebra correspond to elements of the secondcohomology of the algebra with coefficient in the algebra.Quantizations of free bosons, WZNW models and minimalmodels correspond to deformations of vertex operatoralgebras. In particular, these deformations are determinedby elements of the second cohomologies of thecorresponding vertex operator algebras.2011, H. (not circulated): The third cohomology and acertain convergence is the obstruction for an infinitesimaldeformation to be the first order deformation of a formaldeformation.



Deformation theory




Deformation theory




Deformation theory



Part 3

Open problems


Higher-genus theories and locally convex completions

Outline






Higher-genus theories

The major problem to be solved in the higher-genus caseis a convergence problem similar to the convergenceproblem for products and iterates of intertwining operatorsin the genus-zero case and the convergence problem fortraces of products and iterates of intertwining operators.To prove this convergence, one needs to prove someconjectures on certain types of functions on theinfinite-dimensional Teichmüller spaces and moduli spacesof Riemann surfaces with parametrized boundaries.Recently Radnell, Schipper and Staubach have beenmaking good progress in the study of these Teichmüllerspaces and moduli spaces. I hope that they will soon beable to establish those conjectures as theorems onfunctions on these spaces.















Locally convex completions

1998 and 2000, H.: Constructed topological completions ofvertex operator algebras and modules. This constructioncan be generalized easily to construct topologicalcompletions of the state spaces of the genus-zero chiraland full conformal field theories discussed above. But toobtain the full topological completions, we need firstconstruct higher-genus theories.On the other hand, in the case that the state space has anatural inner product, there is another completion given bythe corresponding norm.Conjecture (H.): The topological completion obtained fromhigher-genus theory using the method giving topologicalcompletions of vertex operator algebras is the same as thetopological completion obtained from the norm given bythe inner product.














Moduli space of conformal field theories

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Analytic deformations, topology of the moduli spaceand rigidity of the moonshine module

Finding conditions under which formal deformations ofvertex operator algebras are convergent so that we obtainanalytic deformations.Douglas: Give a "correct" topology to the moduli space ofconformal field theories.Rigidity conjecture for the moonshine module (H.): Asimple vertex operator algebra with only one irreduciblemodule, of central charge 24 and without weight 1elements is rigid. This conjecture is a consequence of theuniqueness conjecture of Frenkel, Lepowsky and Meurmanfor the moonshine module vertex operator algebra.To prove this coonjecture, one needs only prove that thesecond cohomology of such a vertex operator algebravanishes.


















Nonrational conformal field theories

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Logarithmic conformal field theories

For non-rational conformal field theories, the main resultswe have now are on logarithmic conformal field theories.io2015, Fiordalisi: Proved modular invariance for logarithmicconformal field theories with nonzero central charge, usingq-pseudo-traces of Miyamoto instead of q-traces.2003, H., Lepowsky and Zhang: Proved the logarithmicoperator product expansion and constructed the vertextensor category structure.Conjecture (H.): Analytic extensions of suitably generalizedq-pseudo-traces of products of logarithmic intertwiningoperators span a modular invariant vector space. Fiordalisihas made substantial progress. Conjecture (H.): Rigidityholds in the C2-cofinite logarithmic case.



















Calabi-Yau superconformal field theories

One of the most important unsolved problem is certainlythe construction or at least the proof of the existence of theN = 2 superconformal field theories associated toCalabi-Yau manifolds. In this case, even the correct vertexoperator superalgebras are not constructed.The Gepner conjecture on the superconformal field theoryfor the quintic needs to be proved.The Green-Plesser construction should be givenmathematically.For K 3 surfaces, there is a Mathieu moonshine conjecture.Only after the vertex operator superalgebras areconstructed, we might be able to start to understand theMathieu moonshine and the other related moonshinephenomena.



















Thanks!

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Two-Dimensional Conformal Field Theory - Math · Two-dimensional conformal ﬁeld theories The major problems solved Open problems Two-Dimensional Conformal Field Theory Yi-Zhi Huang