Vertex (operator) algebras
are a fundamental class of algebraic
structures that arose in mathematics and physics in the
1980s.
These
algebras and their representations are deeply related to many
directions in mathematics and physics, in particular, the
representation theory of the
Fischer–Griess Monster
simple finite group
and the connection with the phenomena of
"Monstrous Moonshine"
(cf. also
Moonshine conjectures),
the representation theory of the
Virasoro algebra
and
affine Kac–Moody Lie algebras
(cf. also
Kac–Moody algebra),
modular functions (cf. also
Modular function),
the theory of Riemann surfaces (cf. also
Riemann surface),
knot invariants and invariants of three-manifolds (cf. also
Knot theory;
Three-dimensional manifold),
quantum groups,
monodromy associated with differential equations, and
conformal
and
topological field theory
and string theory in
physics. In fact, the theory of vertex operator algebras and their
representations can be thought of as an algebraic foundation of a
great number of constructions in these theories. Various equivalent
definitions of the notion of vertex algebra and of the variant notion
of vertex operator algebra are given below.
The notion of vertex algebra was defined by
R. Borcherds
[a1]
and is a mathematically precise algebraic counterpart of the concept
of
"chiral algebra"
in two-dimensional conformal
quantum field theory
(a physical theory foundational in
string theory
[a9]
and in
two-dimensional statistical mechanics), as formalized by
A. Belavin,
A.M. Polyakov
and
A. Zamolodchikov
[a3].
This fundamental notion
reflects deep features of the traditional notions of commutative
associative algebra and at the same time of Lie algebra. The
theory of vertex operator algebras has a distinctly non-classical
flavour, which can be thought of as analogous to the non-classical
flavour of string theory. The elements of a vertex operator algebra
correspond to (abstract)
"vertex operators" ,
which in special cases include many of the
"vertex operators"
introduced by physicists in
the early days of string theory to describe hypothesized interactions
of certain elementary particles at a
"vertex"
(cf. also
Vertex operator).
One of the main original motivations for the introduction of the
notion of vertex (operator) algebra arose from the problem of
realizing the Monster
as a symmetry group of a certain
infinite-dimensional graded vector space with natural
additional
structure. The additional structure can be expressed in terms of the
axioms defining these new algebraic objects (which are not algebras,
not even non-associative algebras, in the usual sense). This problem
arose in the study of the remarkable subject of Monstrous Moonshine.
In
1978–1979,
J. McKay,
J.G. Thompson,
J.H. Conway
and
S.P. Norton
[a4]
conjectured the existence of a natural infinite-dimensional
-graded
representation (call it
)
of the then-conjectured Monster
group such that the formal series
would be the
modular function
and such that the action of every element of the Monster on
would give rise to a certain
modular function with special properties. After
R. Griess
[a11]
proved the existence of
by constructing it as an
automorphism group of a remarkable new algebra of dimension
,
I. Frenkel,
J. Lepowsky
and
A. Meurman
[a7]
gave a construction, incorporating a vertex operator realization of
the
Griess algebra, of what they called a
"moonshine module"
for
having the desired relation with
,
and they showed that the action on
of certain elements of
gave rise to modular functions. This structure
was interpreted by physicists as a
"toy model"
physical
theory of a
-dimensional string
compactified on a
-dimensional
"orbifold"
associated with the
Leech lattice,
so that
turned out to be the symmetry group of an idealized physical theory.
Then Borcherds introduced the axiomatic notion of vertex algebra
[a1]
and perceived that
could be endowed with an
-invariant
vertex algebra structure. An
-invariant
vertex operator algebra structure on
was indeed constructed in
[a8].
The Monster is in
fact the full symmetry group of this special vertex operator algebra
,
just as the Mathieu finite group
(cf. also
Mathieu group)
is the symmetry group of a special error-correcting code, the
Golay code,
and the
Conway finite group
is the symmetry group of a special positive definite even lattice, the
Leech lattice.
All three of these
special objects possess and can be characterized by the following
properties (the uniqueness being conjectural in the case of
):
a)
"self-dual" ,
b)
"rank 24" ,
c)
"having no small elements" .
These
properties have appropriate definitions for
each of the three types of mathematical structures. In fact, these
structures and analogies enter into the construction of
([a7],
[a8]).
Using the vertex operator algebra structure on
and other ideas and results, Borcherds completed in
[a2]
the proof of the
Conway–Norton Monstrous Moonshine conjecture
(cf. also
Moonshine conjectures)
concerning the modular functions associated to elements of
,
acting on
.
The notion of vertex operator algebra
([a8],
[a6])
is a modification of the notion of vertex algebra. There are several
equivalent formulations of these notions, including formulations in
terms of
"minimal"
and
"maximal"
axioms. The canonical
"maximal"
axiom is a formal-variable identity called the
"Jacobi identity"
or the
"Jacobi–Cauchy identity"
([a8],
[a6]),
on which the
fundamental principles are heavily based. This identity contains the
"full"
necessary information in compact form; it is analogous to the
classical Jacobi identity in the definition of the notion of
Lie algebra;
and it is invariant in a natural sense under the symmetric
group on three letters. The Jacobi identity and the underlying
formal-variable calculus are discussed in detail in
[a8]
and
[a6].
The definition of vertex algebra in
[a1]
involved certain special cases of the Jacobi identity (see below).
There are also
"minimal"
axioms, stemming from the fact that the
(suitably formulated)
"commutativity"
of vertex operators implies
"associativity"
(again suitably formulated) and hence the Jacobi
identity, as explained in
[a8],
[a6]
(cf.
[a3],
[a10]).
The simplest
"minimal"
axiom in the definition of the
notion of vertex algebra (see v) below) was found in
[a5].
A general and systematic approach and solution to the problem of
efficiently constructing examples of vertex operator algebras and
their modules, and related problems, was first carried out in
[a15].
A program to construct (geometric) conformal field theory using vertex
operator algebras and the
"sewing"
of Riemann spheres with punctures
was initiated by Frenkel. A precise geometric formulation of the
notion of vertex operator algebra in terms of partial operads of
complex powers of the determinant line bundle over the moduli space of
Riemann spheres with punctures and local coordinates was given in
[a12],
[a14],
[a13].
Complete definitions of the notions of vertex algebra
and vertex operator algebra, using commuting formal variables
,
and
are given below. The definition of vertex algebra with either
"minimal"
or
"maximal"
axioms is equivalent to Borcherds' definition in
[a1];
see below.
Vertex algebras.
A
vertex algebra
is a
vector space
(over
,
say) equipped with a linear mapping
(the vector space of formal Laurent series in the formal variable
with coefficients in
),
written as
(the
vertex operator
corresponding to the element
),
and a distinguished vector
,
satisfying the following conditions for
:
i)
the formal series
involves only finitely many negative powers of
;
ii)
(the identity operator on
);
iii)
involves only non-negative powers of
and its constant term is
;
iv)
,
where
is the mapping such that
is the constant term of
for
;
v)
there exists a non-negative integer
(depending on
and
)
such that
In this definition, conditions i–v) are
often called the
truncation condition,
the
vacuum property,
the
creation property,
the
-bracket-derivative formula,
and
weak commutativity,
respectively. The last condition was formulated and exploited in
[a5].
The
"main property"
of a vertex algebra, the
Jacobi identity,
states:
vi)
For
,
where
is the formal Laurent series
and, more precisely,
and similarly for the other two
-function
expressions. (All the expressions are well
defined.)
It can be proved that the axioms ii), iv) and v) can be
replaced by the Jacobi identity in the definition. This definition in
terms of the Jacobi identity is the definition with
"maximal"
axioms.
The Jacobi identity is actually the generating function of an infinite
family of identities. The use of the three formal variables, rather
than complex variables (which can also be used, but with changes in
the formulas), allows the full symmetry of the Jacobi identity to
reveal itself, as explained in
[a8]
and
[a6].
Vertex algebras also satisfy a
"commutativity"
condition, which asserts, roughly speaking, that for
,
where
"~"
denotes equality up to a suitable kind of generalized
analytic continuation, and also an
"associativity"
condition,
where the right-hand side and the generalized analytic
continuation
have to be understood suitably, as explained in
[a8]
and
[a6]
(cf.
[a3],
[a10]).
The associativity condition corresponds to the
"operator product expansion"
for holomorphic fields in conformal field theory, together with its
"associativity" .
As mentioned above, these two conditions, and even the commutativity
condition alone, essentially imply the Jacobi identity. Commutativity
and associativity are intimately related to the geometric
interpretation of the notion of vertex operator algebra.
Vertex operator algebras.
A
vertex operator algebra
is a
vertex algebra
equipped with a
-grading
and a distinguished vector
satisfying the following additional conditions:
vii)
for
and
for
sufficiently small;
viii)
for
,
where
,
,
are the operators on
defined by
and where
;
ix)
for
and
;
x)
.
Conditions vii–x) are often called the
grading-restriction conditions,
the
Virasoro algebra relations,
the
-grading property
and the
-derivative property,
respectively.
In
[a1],
Borcherds gave the following definition: A
vertex algebra
is a vector space
(actually, this definition works over
or any other commutative ring) equipped with an element
,
linear operators
on
for
and bilinear operations
from
to
,
for
,
satisfying the following relations a)–e) for
and
:
a)
for
sufficiently large (depending on
and
);
b)
if
,
if
;
c)
;
d)
;
e)
.
It can be proved that the other definitions above of the notion of
vertex algebra are equivalent to this one, over a field of
characteristic
.
The relation between the vertex operator mapping and
the bilinear operations is given by
for
.
Equating the coefficient of the formal variable
on the two sides of the Jacobi identity above recovers
e), in generating-function form.
Among the sources of examples of vertex (operator) algebras are
conformal field theory and string theory and related mathematical
structures: suitable representations of the
Virasoro algebra,
of
Heisenberg Lie algebras
(cf. also
Commutation and anti-commutation relationships, representation of)
and of Kac–Moody algebras, including affine
Lie algebras, and analogues, generalizations and modifications of such
structures. Vertex algebras are typically
infinite-dimensional,
although any commutative associative algebra with derivation carries
the structure of a vertex algebra
[a1].