Conformal Field Theory
and Operator Algebras
Yasuyuki Kawahigashi
Department of Mathematical Sciences
University of Tokyo, Komaba, Tokyo, 153-8914, Japan
e-mail: yasuyuki@ms.u-tokyo.ac.jp
Abstract
We review recent progress in operator algebraic approach to con-
formal quantum field theory. Our emphasis is on use of representation
theory in classification theory. This is based on a series of joint works
with R. Longo.
1 Introduction
A mathematically rigorous approach to quantum field theory based on op-
erator algebras is called an algebraic quantum field theory. It has a long
history since pioneering works of Araki, Haag, Kastker. (See [22] for a gen-
eral treatment of algebraic quantum field theory.) This theory works on
Minkowski spaces on any spacetime dimension, and there have been some
recent results on curved spacetimes or even noncommutative spacetimes. In
the case of 1+1-dimensional Minkowski space with higher spacetime symme-
try, conformal symmetry, we have conformal field theory and there we have
seen many new developments in the recent years, so we survey such results
here. Our emphasis is on representation theoretic aspects of the theory and
we make various comparison with another mathematically rigorous and more
recent approach to conformal field theory, that is, theory of vertex operator
algebras.
∗Supported in part by JSPS.
http://arxiv.org/abs/0704.0097v1
Roughly speaking, a mathematical study of quantum field theory is a
study of Wightman fields, which are certain type of operator-valued distri-
butions on a spacetime with covariance with respect to a given spacetime
symmetry group. We have mathematically rigorous axioms for such Wight-
man fields, but they involve distributions and unbounded operators, so these
cause various kinds of technical difficulty. In contrast, in the algebraic quan-
tum field theory, our fundamental object is a net of von Neumann algebras
of bounded linear operators on a Hilbert space. (See [46] for general the-
ory of von Neumann algebras.) Technical problems on definition domains of
unbounded operators do not arise in this approach.
A basic idea is as follows. Suppose we have a Wightman field Φ on a
spacetime. Fix a bounded region O in the space time and consider a test
function ϕ with support contained in O. Then the pairing 〈Φ, ϕ〉 produces
an (unbounded) operator. We have many Φ and ϕ for a fixed O and obtain
many unbounded operators from such pairing. Then we consider a von Neu-
mann algebra of bounded linear operators on this Hilbert space generated
by these unbounded operators. (For example, if we have a self-adjoint un-
bounded operators, we consider its spectral projections which are obviously
all bounded. In this way, we deal with only bounded operators.) This is re-
garded as a von Neumann algebra generated by observables in the spacetime
region O. A von Neumann algebra is an algebra of bounded linear operators
which is closed under the adjoint operation and the strong operator topol-
ogy. In this way, we have a family {A(O)} of von Neumann algebras on the
same Hilbert space parameterized by spacetime regions. Since the spacetime
regions make a net with respect to the inclusion order, we call such a family a
net of von Neumann algebras. Now we forget Wightman fields and consider
only a net of von Neumann algebras. We have some expected properties for
such nets of von Neumann algebras from a physical consideration, and now
we use these properties as axioms. So our mathematical object is a net of
von Neumann algebras subject to certain set of axioms. Our mathematical
aim is to study such nets of von Neumann algebras.
2 Conformal Quantum Field Theory
We first explain formulation of full conformal quantum field theory on the
1 + 1-dimensional Minkowski space in algebraic quantum field theory. As a
spacetime region O above, it is enough to consider only open rectangles O
with edges parallel to t = ±x in (1 + 1)-dim Minkowski space. In this way,
we get a family {A(O)} of operator algebras parameterized by spacetime
regions O (rectangles). In order to realize conformal symmetry, we have to
make a partial compactification of the 1+1-dimensional Minkowski space. If
two rectangles are spacelike separated, then we have no interactions between
them even at the speed of light, so our axiom requires that the corresponding
two von Neumann algebras commute with each other. This is the locality
axiom. Since this is not our main object in this paper, we omit details of the
other axioms. See [29] for full details.
Next we briefly explain that boundary conformal field theory can be han-
dled within the same framework. Now we consider the half-space {(x, t) |
x > 0} in the 1+1-dimensional Minkowski space and only rectangles O con-
tained in this half-space. In this way, we have a similar net of von Neumann
algebras {A(O)} parameterized with rectangles in the half-space. See [38]
for full details of the axioms.
If we have a net of von Neumann algebras over the 1 + 1-dimensional
Minkowski space, we can restrict the net of von Neumann algebras to two
chiral conformal field theories on the light cones {x = ±t}. In this way, we
have two nets of von Neumann algebras on the compactified S1 as description
of two chiral conformal field theories. Since this net is our main mathematical
object in this article, we give a full set of axioms. (See [29] for details of this
“restriction” procedure.)
Now our “spacetime” is S1 and a “spacetime region” is an interval I,
which means a non-empty, non-dense open connected subset of S1. We have
a family {A(I)} of von Neumann algebras on a fixed Hilbert space H . These
von Neumann algebras are simple and such von Neumann algebras are called
factors, so the family {A(I)} satisfying the axioms below is called a net of
factors (or an irreducible local conformal net of factors, strictly speaking).
Actually, the set of intervals on S1 is not directed with respect to inclusions,
so the terminology net is not mathematically appropriate, but is widely used.
1. (isotony) For intervals I1 ⊂ I2, we have A(I1) ⊂ A(I2).
2. (locality) For intervals I1, I2 with I1∩I2 = ∅, we have [A(I1),A(I2)] = 0
3. (Möbius covariance) There exists a strongly continuous unitary repre-
sentation U of PSL(2,R) on H satisfying U(g)A(I)U(g)∗ = A(gI) for
any g ∈ PSL(2,R) and any interval I.
4. (positivity of energy) The generator of the one-parameter rotation sub-
group of U , called the conformal Hamiltonian, is positive.
5. (existence of the vacuum) There exists a unit U -invariant vector Ω in
H , called the vacuum vector, and the von Neumann algebra
I∈S1 A(I)
generated by all A(I)’s is B(H).
6. (conformal covariance) There exists a projective unitary representation
U of Diff(S1) on H extending the unitary representation of PSL(2,R)
such that for all intervals I, we have
U(g)A(I)U(g)∗ = A(gI), g ∈ Diff(S1),
U(g)AU(g)∗ = A, A ∈ A(I), g ∈ Diff(I ′),
where Diff(S1) is the group of orientation-preserving diffeomorphisms
of S1 and Diff(I ′) is the group of diffeomorphisms g of S1 with g(t) = t
for all t ∈ I.
The isotony axiom is natural because we have more test functions (or more
observables) for a larger interval. The locality axiom takes this simple form on
S1. The choice of the spacetime symmetry is not unique, and we can use the
Poincaré symmetry on the Minkowski space or the Möbius covariance on S1,
for example, but in the conformal field theory, we use conformal symmetry,
which means diffeomorphism covariance as above. This set of axioms imply
various nice conditions such as the Reeh-Schlieder property, the Bisognano-
Wichmann property and the Haag duality. See [28] and references there for
details.
In the usual situation, all the von Neumann algebras A(I) are isomorphic
to the so-called Araki-Woods type III1 factor for all nets A and all intervals
I. So each von Neumann algebra does not contain any information about
the conformal field theory, but it is the relative position of the von Neumann
algebras in the family that encodes the physical information of the theory.
(It is similar to subfactor theory of Jones where we study a relative position
of one factor in another.)
At the end of this section, we compare our formulation of conformal
quantum field theory with another mathematically rigorous approach, the-
ory of vertex operator algebras. A vertex operator algebra is an algebraic
axiomatization of Wightman fields on S1, called vertex operators. If we
have an operator valued distribution on S1, its Fourier expansion should give
countably many (possibly unbounded) operators as the Fourier coefficients.
Under the so-called state-field correspondence, any vector in the space of
“states” should give an operator-valued distribution, a quantum “field”, and
its Fourier expansion gives countably many operators. In this way, one vector
should give countably many operators on the space of these vectors. In other
words, for two vectors v, w we have countably many binary operations v(n)w,
n ∈ Z, the action of the n-th operator given by v on w. An axiomatization
of this idea gives a notion of vertex operator algebra. (See [16] for a precise
definition. There is a slightly weaker notion of a vertex algebra. See [27]
for its precise definition and related results.) In theory of vertex operator
algebra, one considers a vector space of states without an inner product and
even when we have a positive definite inner product, one considers this vec-
tor space without completion. Here in comparison to nets of factors, we are
interested in the case where we have positive definite inner products on the
spaces of states. We say that such a vertex operator algebra is unitary.
Both of one (unitary) vertex operator algebra and one net of factors
should describe one chiral conformal field theory. So unitary vertex operator
algebras and nets of factors should be in a bijective correspondence, at least
under some “nice” additional conditions, but no general theorems have been
known for such a correspondence, though there is a recent progress due to S.
Carpi and M. Weiner. However, if we have one construction or an idea on one
side, we can often “translate” it to the other side, though it can be highly non-
trivial from a technical viewpoint. Fundamental sources of constructions for
vertex operator algebras are affine Kac-Moody algebras and integral lattices.
The corresponding constructions for nets of factors have been done by A.
Wassermann [47] and his students, and Dong-Xu [12], respectively, after the
initial construction of Buchholz-Mack-Todorov [5]. If we have examples with
some nice properties, we canoften construct new examples from them, and
as such methods of constructions of vertex operator algebras, we have simple
current extensions, the coset construction, and the orbifold construction. The
simple current extensions for nets of factors are simply crossed products by
DHR-automorphisms and easy to realize. (See the next section for a notion
of DHR-endomorphisms.) The coset and orbifold constructions for nets of
factors have been studied in detail by F. Xu [50, 51, 52].
For nets of factors, we have introduced a new construction of examples
in [28] based on Longo’s notion of Q-systems [36]. Further examples have
been constructed by Xu [55] with this method. This can be translated to the
setting of vertex operator algebras, as we will see in this article later.
3 Representation Theory
An important tool to study nets of factors is a representation theory. For a
net of factors {A(I)}, all the algebras A(I) act on the initial Hilbert space
H from the beginning, but we also consider their representations on another
Hilbert space, that is, a family {πI} of representations πI : A(I) → B(K),
where K is another Hilbert space, common for all I. For I1 ⊂ I2, we must
have that the restriction of πI2 on A(I1) is equal to πI1 . The representation on
the initial Hilbert space is called the vacuum representation and plays a role of
a trivial representation. We also have to take care of the spacetime symmetry
group when we consider a representation, but this part is often automatic
(see [20]), so we now ignore it for simplicity. See [20] for a more detailed
treatment. Note that a representation of a net of factors is a counterpart of
a module over a vertex operator algebra.
Notions of irreducibility and a direct sum for such representations are
easy to formulate. Non-trivial notions are dimensions and tensor products.
Each representation {πI} is in a bijective correspondence to a certain endo-
morphism λ of an infinite dimensional operator algebra, called a Doplicher-
Haag-Roberts (DHR) endomorphism [13, 15], and we can restrict λ to a single
factor A(I) for an arbitrarily but fixed interval I. Then λ(A(I)) ⊂ A(I) is
a subfactor and we have its Jones index [26]. (See [14, 41, 43] for general
theory of subfactors.) The square root of this Jones index plays the role of
the dimension of the representation [35]. In algebraic quantum field theory,
such a dimension was called a statistical dimension, and it is analogous to
a quantum dimension in the theory of quantum groups. It is a positive real
numbers in the interval [1,∞]. We can also compose endomorphisms and
this composition gives the correct notion of tensor products. We then get a
braided tensor category as in [15].
In representation theory of a vertex operator algebra (and also a quantum
group), it sometimes happens that we have only finitely many irreducible rep-
resentations. Such finiteness is often called rationality, possibly with some
extra assumptions on some finite dimensionality. This also plays an impor-
tant role in theory of quantum invariants in low dimensional topology. In [32],
we have introduced an operator algebraic condition for such rationality for
nets of factors as follows and we called it complete rationality. We split the
circle into four intervals I1, I2, I3, I4 in this order, say, counterclockwise. Then
complete rationality is given by the finiteness of the Jones index for a subfac-
tor A(I1)∨A(I3) ⊂ (A(I2)∨A(I4))
′ where ′ means the commutant, together
with the split property. The split property is known to hold if the vacuum
character,
n=0(dimHn)q
n, is convergent for |q| < 1 by [9], so it usually
holds and is easy to verify. (Here H =
n=0Hn is the eigenspace decompo-
sition of the original Hilbert space for the positive generator of the rotation
group. So this convergence property can be verified simply by looking at
the Hilbert space, not the von Neumann algebras.) In the original definition
of complete rationality in [32], we required another condition called strong
additivity, but it was proved to be redundant by Longo-Xu [39]. We have
proved in [32] that this complete rationality implies that we have a modular
tensor category as a representation category of {A(I)}. A modular tensor
category produces a 3-dimensional topological quantum field theory. (See [45]
for general theory of topological quantum field theory.) The SU(N)k-net of
Wassermann has been shown to be completely rational by [49].
We now introduce an important notion of α-induction. For an inclusion
of nets of factors, A(I) ⊂ B(I), we have an induction procedure analogous
to the group representation. So from a representation of the smaller net A,
we would like to construct a representation of the larger net B, but what
we actually obtain is not a genuine representation of the larger net B in
general, and is something weaker called solitonic. This induction procedure
is called the α-induction and depends a choice of braiding, so we write α+
and α−. This was first defined in Longo-Rehren [37] and studied in detail
in Xu [48]. Then Böckenhauer-Evans [1] made a further study, and [2, 3]
unified this study with Ocneanu’s graphical method [42]. The intersection
of the irreducible endomorphisms appearing in the images of α+-induction
and α−-induction gives the true representation category of {B(I)} if A is
completely rational by [2, 32].
This α-induction opens an important and new connection with theory
of modular invariants. A modular tensor category produces a unitary rep-
resentation π of SL(2,Z) through its braiding as in [44], and its dimension
is the number of irreducible objects. So a completely rational net of fac-
tors produces such a unitary representation. (Note that our representation
of SL(2,Z) comes from the braiding structure, not from the action of this
group on the characters through change of variables τ 7→
aτ + b
cτ + d
, though in
all the “nice” known examples, these two representations coincide. See [30]
for a discussion on this matter.)
It has been proved in [2] that the matrix (Zλ,µ) defined by
Zλ,µ = dimHom(α
λ , α
is in the commutant of the representation π, using Ocneanu’s graphical cal-
culus [42]. Such a matrix Z is called a modular invariant, and we have only
finitely many such Z for a given π. For any completely rational net {A(I)},
any extension {B(I) ⊃ A(I)} produces such Z. Matrices Z are certainly
much easier to classify than extensions and this is a source of classification
theory in the next section.
4 Classification Theory
For a net of factors, we can naturally define a central charge and it is well-
known to take discrete values 1− 6/m(m+ 1), m = 3, 4, 5, . . . , below 1 and
all values in [1,∞) by [17, 18]. We have the Virasoro net {Virc(I)} for each
such c and it is the operator algebraic counterpart of the Virasoro vertex
operator algebra with the same c. Any net of factors {A(I)} with central
charge c is an extension of the Virasoro net with the same central charge and
it is automatically completely rational if c < 1, as shown in [28]. So we can
apply the above theory and we get the following complete classification list
for the case c < 1 as in [28].
1. The Virasoro nets {Virc(I)} with c < 1.
2. The simple current extensions of the Virasoro nets with index 2.
3. Four exceptionals at c = 21/22, 25/26, 144/145, 154/155.
The unitary representations of SL(2,Z) for the Virasoro nets are the well-
known ones, and all the modular invariants for these have been classified by
[6]. Our result shows that each of the so-called type I modular invariants in
the classification list of [6] corresponds to a net of factors uniquely. They
are labeled with pairs of A-D2n-E6,8 Dynkin diagrams with Coxeter numbers
differing by 1. Three in (3) of the above list have been identified with coset
models, but the remaining one does not seem to be related to any other
known constructions. This is constructed with “extension by Q-system”.
Xu [55] recently applied this construction to many other coset models and
obtained infinitely many new examples based on [54], called mirror exten-
sions. Classification for the case c = 1 has been also done under some extra
assumption [7, 53].
This classification theorem also implies a classification of certain types of
vertex operator algebras as follows.
Let V be a (rational) vertex operator algebra and Wi be its irreducible
modules. We would like to classify all vertex operator algebras arising from
putting a vertex operator algebra structure on
i niWi and using the same
Virasoro element as V , where ni is multiplicity and W0 = V , n0 = 1. From
a viewpoint of tensor category, this classification problem of extensions of
a vertex operator algebras is the “same” as the classification problem of
extensions of a completely rational net of factors, as shown in [24].
So the above classification theorem of local conformal nets implies a clas-
sification theorem of extensions of the Virasoro vertex operator algebras with
c < 1 as above, and we obtain the same classification list. That is, besides
the Virasoro vertex operator algebras themselves, we have their simple cur-
rent extensions, and four exceptionals at c = 21/22, 25/26, 144/145, 154/155.
With the usual notation of L(c, h) for a module with central charge c and
conformal weight h of the Virasoro vertex operator algebras with c < 1, the
four exceptionals are listed as follows.
1. L(21/22, 0)⊕L(21/22, 8). It has 15 irreducible representations and has
two coset realizations, from SU(9)2 ⊂ (E8)2 and (E8)3 ⊂ (E8)2⊗(E8)1.
2. L(25/26, 0) ⊕ L(25/26, 10). It has 18 irreducible representations and
has a coset realization from SU(2)11 ⊂ SO(5)1 ⊗ SU(2)1.
3. L(144/145, 0)⊕L(144/145, 24)⊕L(144/145, 78)⊕ L(144/145, 189). It
has 28 irreducible representations and no coset realization has been
known.
4. L(154/155, 0)⊕L(154/155, 26)⊕L(154/155, 84)⊕ L(154/155, 203). It
has 30 irreducible representations and has a coset realization from
SU(2)29 ⊂ (G2)1 ⊗ SU(2)1.
Note that it is not obvious that the representation category of the Virasoro
net Virc and the representation category of the Virasoro vertex operator
algebra L(c, 0) are isomorphic, but as long as the two are braided tensor
category and have the same S- and T -matrices, the arguments in [28] work,
so we obtain the above classification result for vertex operator algebras.
Using the above results and more techniques, we can also completely
classify full conformal field theories within the framework algebraic quantum
field theory for the case c < 1. Full conformal field theories are given as
certain nets of factors on 1 + 1-dimensional Minkowski space. Under natu-
ral symmetry and maximality conditions, those with c < 1 are completely
labeled with the pairs of A-D-E Dynkin diagrams with the difference of
their Coxeter numbers equal to 1, as shown in [29]. We now naturally have
D2n+1, E7 as labels, unlike in the chiral case. The main difficulty in this
work lies in proving uniqueness of the structure for each modular invariant
in the Cappelli-Itzykson-Zuber list [6]. This is done through 2-cohomology
vanishing for certain tensor categories. in the spirit of [25].
Furthermore, using the above results and more techniques we can also
completely classify boundary conformal field theories for the case c < 1.
Boundary conformal field theories are given as certain nets of factors on a 1+
1-dimensional Minkowski half-space. Under a natural maximality condition,
these with c < 1 are now completely labeled with the pairs of A-D-E Dynkin
diagrams with distinguished vertices having the difference of their Coxeter
numbers equal to 1, as shown in [33] based on a general theory in [38]. The
“chiral fields” in a boundary conformal field theory should produce a net
of factors on the boundary (which is compactified to S1) as in the operator
algebraic approach. Then a general boundary conformal field theory restricts
to this boundary to produce a non-local extension of this chiral conformal
field theory on the boundary.
5 Moonshine Conjecture
The Moonshine conjecture, formulated by Conway-Norton [8], is about mys-
terious relations between finite simple groups and modular functions, since
an observation due to McKay.
Today the classification of all finite simple groups is complete and the
classification list contains 26 sporadic groups in addition to several infinite
series. The largest group among the 26 sporadic groups is called the Monster
group and its order is about 8× 1053
One the one hand, the non-trivial irreducible representation of the Mon-
ster having the smallest dimension is 196883 dimensional. On the other
hand, the following function, called j-function, has been classically studied
in algebra.
j(τ) = q−1 + 744 + 196884q +
21493760q2 + 864299970q3 + · · ·
For q = exp(2πiτ), Im τ > 0, we have modular invariance property, j(τ) =
aτ + b
cτ + d
∈ SL(2,Z), and this is the only function, up to
the constant term, satisfying this property and starting with q−1,
McKay noticed 196884 = 196883 + 1, and similar simple relations for
other coefficients of the j-function and dimensions of irreducible represen-
tations of the Monster group turned out to be true. Then Conway-Norton
[8] formulated the Moonshine conjecture roughly as follows, which has been
now proved by Borcherds [4] in 1992.
1. We have a “natural” infinite dimensional graded vector space V =
Vn with some algebraic structure having a Monster action pre-
serving the grading and each Vn is finite dimensional.
2. For any element g in the Monster, the power series
n=0(Tr g|Vn)q
is a special function called a Hauptmodul for some discrete subgroup
of SL(2,R). When g is the identity element, the series is the j-function
minus constant term 744.
For the part (1) of this conjecture, Frenkel-Lepowsky-Meurman [16] gave
a precise definition of “some algebraic structure” as a vertex operator algebra
and constructed a particular example V , which is now called the Moonshine
vertex operator algebras and denoted by V ♮.
The construction roughly goes as follows. In dimension 24, we have an
exceptional lattice Λ called the Leech lattice. Then there is a general con-
struction of a vertex operator algebra from a certain lattice, and the one for
the Leech lattice gives something very close to our final object V ♮. Then we
take a fixed point algebra under a natural action of Z/2Z arising from the
lattice symmetry, and then make a simple current extension of order 2. The
resulting vertex operator algebra is the Moonshine vertex operator algebra
V ♮. (The final step is called a twisted orbifold construction). The series
n=0(dimV
n−1 is indeed the j-function minus constant term 744.
Miyamoto [40] has a new realization of V ♮ as an extension of a tensor
power of the Virasoro vertex operator algebra with c = 1/2, L(1/2, 0)⊗48
(based on Dong-Mason-Zhu [11]). This kind of extension of a Virasoro tensor
power is called a framed vertex operator algebra as in [10].
We have given an operator algebraic counterpart of such a construction
in [31].
We realize a Leech lattice net of factors on S1 as an extension of Vir1/2
using certain Z4-code. Then we can perform the twisted orbifold construction
in the operator algebraic sense to obtain a net of factors, the Moonshine net
A♮. Theory of α-induction is used for obtaining various decompositions. We
then get a Miyamoto-type description of this construction, as an operator
algebraic counterpart of the framed vertex operator algebras. We then have
the following properties.
1. c = 24.
2. The representation theory is trivial.
3. The automorphism group is the Monster.
4. The Hauptmodul property (as above).
Outline of the proof of these four properties is as follows.
It is immediate to get c = 24. We can show complete rationality passes
to an extension (and an orbifold) in general with control over the size of the
representation category, using the Jones index. With this, we obtain (2) very
easily. Such a net is called holomorphic. Property (3) is the most difficult
part. For the Virasoro VOA L(1/2, 0), the vertex operator is indeed a well-
behaved Wightman field and smeared fields produce the Virasoro net Vir1/2.
Using this property and the fact that
g g(L(1/2, 0)
⊗48) for all g ∈ Aut(V ♮)
generate the entire Moonshine VOA V ♮, we can prove that the automorphism
group as a vertex operator algebra and the automorphism group as a net
of factors are indeed the same. Then (4) is now a trivial corollary of the
Borcherds theorem [4].
We note that the Baby Monster, the second largest among the 26 sporadic
finite simple groups, can be treated similarly with Höhn’s construction of the
shorter Moonshine super vertex operator algebra.
Still, these examples are treated with various tricks case by case. We
expect a bijective correspondence between vertex operator algebras and nets
of factors on S1 under some nice conditions. On the side of vertex operator
algebras, the most natural candidate for such a “nice” condition is the C2-
finiteness condition of Zhu [56] (with unitarity). On the operator algebraic
side, our complete rationality in [32] seems to be such a “nice” condition, but
the actual relations between the two notions are not clear at this moment.
The essential condition for complete rationality is the finiteness of the Jones
index arising from four intervals on the circle, and this finiteness somehow
has formal similarity to the finiteness appearing in the definition of the C2-
finiteness.
At the end, we list some open problems. The operator algebraic approach
has an advantage in control of representation theory, but is behind of theory
of vertex operator algebras in the theory of characters.
For a net of factors, we can naturally define a notion of a character for
each representation. But even convergence of these characters have not been
proved in general, and the modular invariance property, the counterpart of
Zhu’s result [56], is unknown, though we certainly expect it to be true. We
also expect the Verlinde identity holds, which has been proved in the context
of vertex operator algebras recently by Huang [23]. We would need an S-
matrix version of the spin-statistics theorem [21] for nets of factors.
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	Introduction
	Conformal Quantum Field Theory
	Representation Theory
	Classification Theory
	Moonshine Conjecture

  We review recent progress in operator algebraic approach to conformal quantum
field theory. Our emphasis is on use of representation theory in classification
theory. This is based on a series of joint works with R. Longo.

Introduction
A mathematically rigorous approach to quantum field theory based on op-
erator algebras is called an algebraic quantum field theory. It has a long
history since pioneering works of Araki, Haag, Kastker. (See [22] for a gen-
eral treatment of algebraic quantum field theory.) This theory works on
Minkowski spaces on any spacetime dimension, and there have been some
recent results on curved spacetimes or even noncommutative spacetimes. In
the case of 1+1-dimensional Minkowski space with higher spacetime symme-
try, conformal symmetry, we have conformal field theory and there we have
seen many new developments in the recent years, so we survey such results
here. Our emphasis is on representation theoretic aspects of the theory and
we make various comparison with another mathematically rigorous and more
recent approach to conformal field theory, that is, theory of vertex operator
algebras.
∗Supported in part by JSPS.
http://arxiv.org/abs/0704.0097v1
Roughly speaking, a mathematical study of quantum field theory is a
study of Wightman fields, which are certain type of operator-valued distri-
butions on a spacetime with covariance with respect to a given spacetime
symmetry group. We have mathematically rigorous axioms for such Wight-
man fields, but they involve distributions and unbounded operators, so these
cause various kinds of technical difficulty. In contrast, in the algebraic quan-
tum field theory, our fundamental object is a net of von Neumann algebras
of bounded linear operators on a Hilbert space. (See [46] for general the-
ory of von Neumann algebras.) Technical problems on definition domains of
unbounded operators do not arise in this approach.
A basic idea is as follows. Suppose we have a Wightman field Φ on a
spacetime. Fix a bounded region O in the space time and consider a test
function ϕ with support contained in O. Then the pairing 〈Φ, ϕ〉 produces
an (unbounded) operator. We have many Φ and ϕ for a fixed O and obtain
many unbounded operators from such pairing. Then we consider a von Neu-
mann algebra of bounded linear operators on this Hilbert space generated
by these unbounded operators. (For example, if we have a self-adjoint un-
bounded operators, we consider its spectral projections which are obviously
all bounded. In this way, we deal with only bounded operators.) This is re-
garded as a von Neumann algebra generated by observables in the spacetime
region O. A von Neumann algebra is an algebra of bounded linear operators
which is closed under the adjoint operation and the strong operator topol-
ogy. In this way, we have a family {A(O)} of von Neumann algebras on the
same Hilbert space parameterized by spacetime regions. Since the spacetime
regions make a net with respect to the inclusion order, we call such a family a
net of von Neumann algebras. Now we forget Wightman fields and consider
only a net of von Neumann algebras. We have some expected properties for
such nets of von Neumann algebras from a physical consideration, and now
we use these properties as axioms. So our mathematical object is a net of
von Neumann algebras subject to certain set of axioms. Our mathematical
aim is to study such nets of von Neumann algebras.
2 Conformal Quantum Field Theory
We first explain formulation of full conformal quantum field theory on the
1 + 1-dimensional Minkowski space in algebraic quantum field theory. As a
spacetime region O above, it is enough to consider only open rectangles O
with edges parallel to t = ±x in (1 + 1)-dim Minkowski space. In this way,
we get a family {A(O)} of operator algebras parameterized by spacetime
regions O (rectangles). In order to realize conformal symmetry, we have to
make a partial compactification of the 1+1-dimensional Minkowski space. If
two rectangles are spacelike separated, then we have no interactions between
them even at the speed of light, so our axiom requires that the corresponding
two von Neumann algebras commute with each other. This is the locality
axiom. Since this is not our main object in this paper, we omit details of the
other axioms. See [29] for full details.
Next we briefly explain that boundary conformal field theory can be han-
dled within the same framework. Now we consider the half-space {(x, t) |
x > 0} in the 1+1-dimensional Minkowski space and only rectangles O con-
tained in this half-space. In this way, we have a similar net of von Neumann
algebras {A(O)} parameterized with rectangles in the half-space. See [38]
for full details of the axioms.
If we have a net of von Neumann algebras over the 1 + 1-dimensional
Minkowski space, we can restrict the net of von Neumann algebras to two
chiral conformal field theories on the light cones {x = ±t}. In this way, we
have two nets of von Neumann algebras on the compactified S1 as description
of two chiral conformal field theories. Since this net is our main mathematical
object in this article, we give a full set of axioms. (See [29] for details of this
“restriction” procedure.)
Now our “spacetime” is S1 and a “spacetime region” is an interval I,
which means a non-empty, non-dense open connected subset of S1. We have
a family {A(I)} of von Neumann algebras on a fixed Hilbert space H . These
von Neumann algebras are simple and such von Neumann algebras are called
factors, so the family {A(I)} satisfying the axioms below is called a net of
factors (or an irreducible local conformal net of factors, strictly speaking).
Actually, the set of intervals on S1 is not directed with respect to inclusions,
so the terminology net is not mathematically appropriate, but is widely used.
1. (isotony) For intervals I1 ⊂ I2, we have A(I1) ⊂ A(I2).
2. (locality) For intervals I1, I2 with I1∩I2 = ∅, we have [A(I1),A(I2)] = 0
3. (Möbius covariance) There exists a strongly continuous unitary repre-
sentation U of PSL(2,R) on H satisfying U(g)A(I)U(g)∗ = A(gI) for
any g ∈ PSL(2,R) and any interval I.
4. (positivity of energy) The generator of the one-parameter rotation sub-
group of U , called the conformal Hamiltonian, is positive.
5. (existence of the vacuum) There exists a unit U -invariant vector Ω in
H , called the vacuum vector, and the von Neumann algebra
I∈S1 A(I)
generated by all A(I)’s is B(H).
6. (conformal covariance) There exists a projective unitary representation
U of Diff(S1) on H extending the unitary representation of PSL(2,R)
such that for all intervals I, we have
U(g)A(I)U(g)∗ = A(gI), g ∈ Diff(S1),
U(g)AU(g)∗ = A, A ∈ A(I), g ∈ Diff(I ′),
where Diff(S1) is the group of orientation-preserving diffeomorphisms
of S1 and Diff(I ′) is the group of diffeomorphisms g of S1 with g(t) = t
for all t ∈ I.
The isotony axiom is natural because we have more test functions (or more
observables) for a larger interval. The locality axiom takes this simple form on
S1. The choice of the spacetime symmetry is not unique, and we can use the
Poincaré symmetry on the Minkowski space or the Möbius covariance on S1,
for example, but in the conformal field theory, we use conformal symmetry,
which means diffeomorphism covariance as above. This set of axioms imply
various nice conditions such as the Reeh-Schlieder property, the Bisognano-
Wichmann property and the Haag duality. See [28] and references there for
details.
In the usual situation, all the von Neumann algebras A(I) are isomorphic
to the so-called Araki-Woods type III1 factor for all nets A and all intervals
I. So each von Neumann algebra does not contain any information about
the conformal field theory, but it is the relative position of the von Neumann
algebras in the family that encodes the physical information of the theory.
(It is similar to subfactor theory of Jones where we study a relative position
of one factor in another.)
At the end of this section, we compare our formulation of conformal
quantum field theory with another mathematically rigorous approach, the-
ory of vertex operator algebras. A vertex operator algebra is an algebraic
axiomatization of Wightman fields on S1, called vertex operators. If we
have an operator valued distribution on S1, its Fourier expansion should give
countably many (possibly unbounded) operators as the Fourier coefficients.
Under the so-called state-field correspondence, any vector in the space of
“states” should give an operator-valued distribution, a quantum “field”, and
its Fourier expansion gives countably many operators. In this way, one vector
should give countably many operators on the space of these vectors. In other
words, for two vectors v, w we have countably many binary operations v(n)w,
n ∈ Z, the action of the n-th operator given by v on w. An axiomatization
of this idea gives a notion of vertex operator algebra. (See [16] for a precise
definition. There is a slightly weaker notion of a vertex algebra. See [27]
for its precise definition and related results.) In theory of vertex operator
algebra, one considers a vector space of states without an inner product and
even when we have a positive definite inner product, one considers this vec-
tor space without completion. Here in comparison to nets of factors, we are
interested in the case where we have positive definite inner products on the
spaces of states. We say that such a vertex operator algebra is unitary.
Both of one (unitary) vertex operator algebra and one net of factors
should describe one chiral conformal field theory. So unitary vertex operator
algebras and nets of factors should be in a bijective correspondence, at least
under some “nice” additional conditions, but no general theorems have been
known for such a correspondence, though there is a recent progress due to S.
Carpi and M. Weiner. However, if we have one construction or an idea on one
side, we can often “translate” it to the other side, though it can be highly non-
trivial from a technical viewpoint. Fundamental sources of constructions for
vertex operator algebras are affine Kac-Moody algebras and integral lattices.
The corresponding constructions for nets of factors have been done by A.
Wassermann [47] and his students, and Dong-Xu [12], respectively, after the
initial construction of Buchholz-Mack-Todorov [5]. If we have examples with
some nice properties, we canoften construct new examples from them, and
as such methods of constructions of vertex operator algebras, we have simple
current extensions, the coset construction, and the orbifold construction. The
simple current extensions for nets of factors are simply crossed products by
DHR-automorphisms and easy to realize. (See the next section for a notion
of DHR-endomorphisms.) The coset and orbifold constructions for nets of
factors have been studied in detail by F. Xu [50, 51, 52].
For nets of factors, we have introduced a new construction of examples
in [28] based on Longo’s notion of Q-systems [36]. Further examples have
been constructed by Xu [55] with this method. This can be translated to the
setting of vertex operator algebras, as we will see in this article later.
3 Representation Theory
An important tool to study nets of factors is a representation theory. For a
net of factors {A(I)}, all the algebras A(I) act on the initial Hilbert space
H from the beginning, but we also consider their representations on another
Hilbert space, that is, a family {πI} of representations πI : A(I) → B(K),
where K is another Hilbert space, common for all I. For I1 ⊂ I2, we must
have that the restriction of πI2 on A(I1) is equal to πI1 . The representation on
the initial Hilbert space is called the vacuum representation and plays a role of
a trivial representation. We also have to take care of the spacetime symmetry
group when we consider a representation, but this part is often automatic
(see [20]), so we now ignore it for simplicity. See [20] for a more detailed
treatment. Note that a representation of a net of factors is a counterpart of
a module over a vertex operator algebra.
Notions of irreducibility and a direct sum for such representations are
easy to formulate. Non-trivial notions are dimensions and tensor products.
Each representation {πI} is in a bijective correspondence to a certain endo-
morphism λ of an infinite dimensional operator algebra, called a Doplicher-
Haag-Roberts (DHR) endomorphism [13, 15], and we can restrict λ to a single
factor A(I) for an arbitrarily but fixed interval I. Then λ(A(I)) ⊂ A(I) is
a subfactor and we have its Jones index [26]. (See [14, 41, 43] for general
theory of subfactors.) The square root of this Jones index plays the role of
the dimension of the representation [35]. In algebraic quantum field theory,
such a dimension was called a statistical dimension, and it is analogous to
a quantum dimension in the theory of quantum groups. It is a positive real
numbers in the interval [1,∞]. We can also compose endomorphisms and
this composition gives the correct notion of tensor products. We then get a
braided tensor category as in [15].
In representation theory of a vertex operator algebra (and also a quantum
group), it sometimes happens that we have only finitely many irreducible rep-
resentations. Such finiteness is often called rationality, possibly with some
extra assumptions on some finite dimensionality. This also plays an impor-
tant role in theory of quantum invariants in low dimensional topology. In [32],
we have introduced an operator algebraic condition for such rationality for
nets of factors as follows and we called it complete rationality. We split the
circle into four intervals I1, I2, I3, I4 in this order, say, counterclockwise. Then
complete rationality is given by the finiteness of the Jones index for a subfac-
tor A(I1)∨A(I3) ⊂ (A(I2)∨A(I4))
′ where ′ means the commutant, together
with the split property. The split property is known to hold if the vacuum
character,
n=0(dimHn)q
n, is convergent for |q| < 1 by [9], so it usually
holds and is easy to verify. (Here H =
n=0Hn is the eigenspace decompo-
sition of the original Hilbert space for the positive generator of the rotation
group. So this convergence property can be verified simply by looking at
the Hilbert space, not the von Neumann algebras.) In the original definition
of complete rationality in [32], we required another condition called strong
additivity, but it was proved to be redundant by Longo-Xu [39]. We have
proved in [32] that this complete rationality implies that we have a modular
tensor category as a representation category of {A(I)}. A modular tensor
category produces a 3-dimensional topological quantum field theory. (See [45]
for general theory of topological quantum field theory.) The SU(N)k-net of
Wassermann has been shown to be completely rational by [49].
We now introduce an important notion of α-induction. For an inclusion
of nets of factors, A(I) ⊂ B(I), we have an induction procedure analogous
to the group representation. So from a representation of the smaller net A,
we would like to construct a representation of the larger net B, but what
we actually obtain is not a genuine representation of the larger net B in
general, and is something weaker called solitonic. This induction procedure
is called the α-induction and depends a choice of braiding, so we write α+
and α−. This was first defined in Longo-Rehren [37] and studied in detail
in Xu [48]. Then Böckenhauer-Evans [1] made a further study, and [2, 3]
unified this study with Ocneanu’s graphical method [42]. The intersection
of the irreducible endomorphisms appearing in the images of α+-induction
and α−-induction gives the true representation category of {B(I)} if A is
completely rational by [2, 32].
This α-induction opens an important and new connection with theory
of modular invariants. A modular tensor category produces a unitary rep-
resentation π of SL(2,Z) through its braiding as in [44], and its dimension
is the number of irreducible objects. So a completely rational net of fac-
tors produces such a unitary representation. (Note that our representation
of SL(2,Z) comes from the braiding structure, not from the action of this
group on the characters through change of variables τ 7→
aτ + b
cτ + d
, though in
all the “nice” known examples, these two representations coincide. See [30]
for a discussion on this matter.)
It has been proved in [2] that the matrix (Zλ,µ) defined by
Zλ,µ = dimHom(α
λ , α
is in the commutant of the representation π, using Ocneanu’s graphical cal-
culus [42]. Such a matrix Z is called a modular invariant, and we have only
finitely many such Z for a given π. For any completely rational net {A(I)},
any extension {B(I) ⊃ A(I)} produces such Z. Matrices Z are certainly
much easier to classify than extensions and this is a source of classification
theory in the next section.
4 Classification Theory
For a net of factors, we can naturally define a central charge and it is well-
known to take discrete values 1− 6/m(m+ 1), m = 3, 4, 5, . . . , below 1 and
all values in [1,∞) by [17, 18]. We have the Virasoro net {Virc(I)} for each
such c and it is the operator algebraic counterpart of the Virasoro vertex
operator algebra with the same c. Any net of factors {A(I)} with central
charge c is an extension of the Virasoro net with the same central charge and
it is automatically completely rational if c < 1, as shown in [28]. So we can
apply the above theory and we get the following complete classification list
for the case c < 1 as in [28].
1. The Virasoro nets {Virc(I)} with c < 1.
2. The simple current extensions of the Virasoro nets with index 2.
3. Four exceptionals at c = 21/22, 25/26, 144/145, 154/155.
The unitary representations of SL(2,Z) for the Virasoro nets are the well-
known ones, and all the modular invariants for these have been classified by
[6]. Our result shows that each of the so-called type I modular invariants in
the classification list of [6] corresponds to a net of factors uniquely. They
are labeled with pairs of A-D2n-E6,8 Dynkin diagrams with Coxeter numbers
differing by 1. Three in (3) of the above list have been identified with coset
models, but the remaining one does not seem to be related to any other
known constructions. This is constructed with “extension by Q-system”.
Xu [55] recently applied this construction to many other coset models and
obtained infinitely many new examples based on [54], called mirror exten-
sions. Classification for the case c = 1 has been also done under some extra
assumption [7, 53].
This classification theorem also implies a classification of certain types of
vertex operator algebras as follows.
Let V be a (rational) vertex operator algebra and Wi be its irreducible
modules. We would like to classify all vertex operator algebras arising from
putting a vertex operator algebra structure on
i niWi and using the same
Virasoro element as V , where ni is multiplicity and W0 = V , n0 = 1. From
a viewpoint of tensor category, this classification problem of extensions of
a vertex operator algebras is the “same” as the classification problem of
extensions of a completely rational net of factors, as shown in [24].
So the above classification theorem of local conformal nets implies a clas-
sification theorem of extensions of the Virasoro vertex operator algebras with
c < 1 as above, and we obtain the same classification list. That is, besides
the Virasoro vertex operator algebras themselves, we have their simple cur-
rent extensions, and four exceptionals at c = 21/22, 25/26, 144/145, 154/155.
With the usual notation of L(c, h) for a module with central charge c and
conformal weight h of the Virasoro vertex operator algebras with c < 1, the
four exceptionals are listed as follows.
1. L(21/22, 0)⊕L(21/22, 8). It has 15 irreducible representations and has
two coset realizations, from SU(9)2 ⊂ (E8)2 and (E8)3 ⊂ (E8)2⊗(E8)1.
2. L(25/26, 0) ⊕ L(25/26, 10). It has 18 irreducible representations and
has a coset realization from SU(2)11 ⊂ SO(5)1 ⊗ SU(2)1.
3. L(144/145, 0)⊕L(144/145, 24)⊕L(144/145, 78)⊕ L(144/145, 189). It
has 28 irreducible representations and no coset realization has been
known.
4. L(154/155, 0)⊕L(154/155, 26)⊕L(154/155, 84)⊕ L(154/155, 203). It
has 30 irreducible representations and has a coset realization from
SU(2)29 ⊂ (G2)1 ⊗ SU(2)1.
Note that it is not obvious that the representation category of the Virasoro
net Virc and the representation category of the Virasoro vertex operator
algebra L(c, 0) are isomorphic, but as long as the two are braided tensor
category and have the same S- and T -matrices, the arguments in [28] work,
so we obtain the above classification result for vertex operator algebras.
Using the above results and more techniques, we can also completely
classify full conformal field theories within the framework algebraic quantum
field theory for the case c < 1. Full conformal field theories are given as
certain nets of factors on 1 + 1-dimensional Minkowski space. Under natu-
ral symmetry and maximality conditions, those with c < 1 are completely
labeled with the pairs of A-D-E Dynkin diagrams with the difference of
their Coxeter numbers equal to 1, as shown in [29]. We now naturally have
D2n+1, E7 as labels, unlike in the chiral case. The main difficulty in this
work lies in proving uniqueness of the structure for each modular invariant
in the Cappelli-Itzykson-Zuber list [6]. This is done through 2-cohomology
vanishing for certain tensor categories. in the spirit of [25].
Furthermore, using the above results and more techniques we can also
completely classify boundary conformal field theories for the case c < 1.
Boundary conformal field theories are given as certain nets of factors on a 1+
1-dimensional Minkowski half-space. Under a natural maximality condition,
these with c < 1 are now completely labeled with the pairs of A-D-E Dynkin
diagrams with distinguished vertices having the difference of their Coxeter
numbers equal to 1, as shown in [33] based on a general theory in [38]. The
“chiral fields” in a boundary conformal field theory should produce a net
of factors on the boundary (which is compactified to S1) as in the operator
algebraic approach. Then a general boundary conformal field theory restricts
to this boundary to produce a non-local extension of this chiral conformal
field theory on the boundary.
5 Moonshine Conjecture
The Moonshine conjecture, formulated by Conway-Norton [8], is about mys-
terious relations between finite simple groups and modular functions, since
an observation due to McKay.
Today the classification of all finite simple groups is complete and the
classification list contains 26 sporadic groups in addition to several infinite
series. The largest group among the 26 sporadic groups is called the Monster
group and its order is about 8× 1053
One the one hand, the non-trivial irreducible representation of the Mon-
ster having the smallest dimension is 196883 dimensional. On the other
hand, the following function, called j-function, has been classically studied
in algebra.
j(τ) = q−1 + 744 + 196884q +
21493760q2 + 864299970q3 + · · ·
For q = exp(2πiτ), Im τ > 0, we have modular invariance property, j(τ) =
aτ + b
cτ + d
∈ SL(2,Z), and this is the only function, up to
the constant term, satisfying this property and starting with q−1,
McKay noticed 196884 = 196883 + 1, and similar simple relations for
other coefficients of the j-function and dimensions of irreducible represen-
tations of the Monster group turned out to be true. Then Conway-Norton
[8] formulated the Moonshine conjecture roughly as follows, which has been
now proved by Borcherds [4] in 1992.
1. We have a “natural” infinite dimensional graded vector space V =
Vn with some algebraic structure having a Monster action pre-
serving the grading and each Vn is finite dimensional.
2. For any element g in the Monster, the power series
n=0(Tr g|Vn)q
is a special function called a Hauptmodul for some discrete subgroup
of SL(2,R). When g is the identity element, the series is the j-function
minus constant term 744.
For the part (1) of this conjecture, Frenkel-Lepowsky-Meurman [16] gave
a precise definition of “some algebraic structure” as a vertex operator algebra
and constructed a particular example V , which is now called the Moonshine
vertex operator algebras and denoted by V ♮.
The construction roughly goes as follows. In dimension 24, we have an
exceptional lattice Λ called the Leech lattice. Then there is a general con-
struction of a vertex operator algebra from a certain lattice, and the one for
the Leech lattice gives something very close to our final object V ♮. Then we
take a fixed point algebra under a natural action of Z/2Z arising from the
lattice symmetry, and then make a simple current extension of order 2. The
resulting vertex operator algebra is the Moonshine vertex operator algebra
V ♮. (The final step is called a twisted orbifold construction). The series
n=0(dimV
n−1 is indeed the j-function minus constant term 744.
Miyamoto [40] has a new realization of V ♮ as an extension of a tensor
power of the Virasoro vertex operator algebra with c = 1/2, L(1/2, 0)⊗48
(based on Dong-Mason-Zhu [11]). This kind of extension of a Virasoro tensor
power is called a framed vertex operator algebra as in [10].
We have given an operator algebraic counterpart of such a construction
in [31].
We realize a Leech lattice net of factors on S1 as an extension of Vir1/2
using certain Z4-code. Then we can perform the twisted orbifold construction
in the operator algebraic sense to obtain a net of factors, the Moonshine net
A♮. Theory of α-induction is used for obtaining various decompositions. We
then get a Miyamoto-type description of this construction, as an operator
algebraic counterpart of the framed vertex operator algebras. We then have
the following properties.
1. c = 24.
2. The representation theory is trivial.
3. The automorphism group is the Monster.
4. The Hauptmodul property (as above).
Outline of the proof of these four properties is as follows.
It is immediate to get c = 24. We can show complete rationality passes
to an extension (and an orbifold) in general with control over the size of the
representation category, using the Jones index. With this, we obtain (2) very
easily. Such a net is called holomorphic. Property (3) is the most difficult
part. For the Virasoro VOA L(1/2, 0), the vertex operator is indeed a well-
behaved Wightman field and smeared fields produce the Virasoro net Vir1/2.
Using this property and the fact that
g g(L(1/2, 0)
⊗48) for all g ∈ Aut(V ♮)
generate the entire Moonshine VOA V ♮, we can prove that the automorphism
group as a vertex operator algebra and the automorphism group as a net
of factors are indeed the same. Then (4) is now a trivial corollary of the
Borcherds theorem [4].
We note that the Baby Monster, the second largest among the 26 sporadic
finite simple groups, can be treated similarly with Höhn’s construction of the
shorter Moonshine super vertex operator algebra.
Still, these examples are treated with various tricks case by case. We
expect a bijective correspondence between vertex operator algebras and nets
of factors on S1 under some nice conditions. On the side of vertex operator
algebras, the most natural candidate for such a “nice” condition is the C2-
finiteness condition of Zhu [56] (with unitarity). On the operator algebraic
side, our complete rationality in [32] seems to be such a “nice” condition, but
the actual relations between the two notions are not clear at this moment.
The essential condition for complete rationality is the finiteness of the Jones
index arising from four intervals on the circle, and this finiteness somehow
has formal similarity to the finiteness appearing in the definition of the C2-
finiteness.
At the end, we list some open problems. The operator algebraic approach
has an advantage in control of representation theory, but is behind of theory
of vertex operator algebras in the theory of characters.
For a net of factors, we can naturally define a notion of a character for
each representation. But even convergence of these characters have not been
proved in general, and the modular invariance property, the counterpart of
Zhu’s result [56], is unknown, though we certainly expect it to be true. We
also expect the Verlinde identity holds, which has been proved in the context
of vertex operator algebras recently by Huang [23]. We would need an S-
matrix version of the spin-statistics theorem [21] for nets of factors.
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	Introduction
	Conformal Quantum Field Theory
	Representation Theory
	Classification Theory
	Moonshine Conjecture
