Communications
in
Commun.
Math.
Phys.
128,
197-212
(1990)
Mathematical
Physics
©
Springer-Verlag
1990
From
Geometric
Quantization
to
Conformal
Field
Theory
A.
Alekseev
and S.
Shatashvili
Leningrad
Steklov
Mathematical
Institute,
Fontanka27,
SU-191011
Leningrad,
USSR
Abstract. Investigation
of 2d
conformal
field
theory
in
terms
of
geometric
quantization
is
given.
We
quantize
the
so-called model space
of the
compact
Lie
group, Virasoro group
and
Kac-Moody group.
In
particular,
we
give
a
geometrical interpretation
of the
Virasoro discrete series
and
explain that this
type
of
geometric quantization reproduces
the
chiral
part
of CFT
(minimal
models,
2d-gravity,
WZNW theory).
In the
appendix
we
discuss
the
relation
between classical (constant)
r-matrices
and
this geometrical
approach.
1.
Introduction
In
this
paper
we
continue
an
investigation
of
2d
conformal
field
theories
in
terms
of
geometric quantization (see
[1-3]).
As
demonstrated
in our
previous
papers,
the
standard geometric quantization method
[4] can be
reformulated
in
terms
of the
path integral approach.
In [2] the
correspondence between
the
coadjoint orbit
and
the
irreducible representation
of
compact
Lie
groups
was
explicitly realized
by
means
of the
functional
integral. More precisely,
we
constructed
in [2] a
quantum
mechanical system, such that
the
path integral with boundary conditions gives
matrix
coefficients
of the
corresponding irreducible representation.
The
action
functional
of
this system
is
defined
by the
canonical symplectic structure
Ω on the
given coadjoint orbit
and a
Hamiltonian
H(X\
which
is a
function
on the
orbit:
S
=
$d~
l
Ω
—
$H(X)dtι
this action
is a
functional
of
trajectories
on the
orbit. Later
in
[3]
using
the
same rules,
we
described quantum
field
theory
on the
coadjoint
orbit
of
infinite
dimensional
Lie
groups (Virasoro, Kac-Moody)
and the
properties
of
the
corresponding action
functional
investigated.
In
particular,
we
have shown
that
for the
Virasoro
group
the
geometrical action, written
in
terms
of
group
variables
F(x)€diϊϊS
1
differs
from
the
action
in 2d
gravity
[5] by the
extra term
J
b
0
FF'dxdt,
where
the
number
b
0
parametrizes generic coadjoint orbits.
(A
similar
statement
is
true also
for
Kac-Moody group
and
WZNW model.)
In the
language
of
geometric quantization
the
appearance
of
SL(2,
R)
current
algebra
in 2d
gravity
is
the
consequence
of
symplectic geometry,
and as it was
shown
in [3]
Virasoro
198 A.
Alekseev
and S.
Shatashvili
geometrical
action
can be
obtained
from
the
SL(2,R)
Kac-Moody
one by the
Lagrangian version
of the
Drinfeld-Sokolov Hamiltonian reduction (see also
[6]).
Here
we
will give
a
slightly
different
type
of
geometric construction
in
which
all
representations
of the
group
are
considered simultaneously
and on the
same
footing.
More precisely, using
the
path integral
approach,
we
will
quantize
the so-
called model space, i.e. such space that
its
quantization yields
all
representations
of
the
group with multiplicity one. This space
is
larger than
the
coadjoint
orbit
(roughly
speaking,
it
contains
an
extra variable which parametrizes
the
orbits
and
the
conjugate moment).
The
corresponding Hubert space splits into
the
direct
sum
of
all
irreducible representations. Model spaces
for
compact semisimple
Lie
groups have been studied earlier
in
[7].
(So
far,
we
have been unable
to
compare
our
construction with those
in
[7].)
We
believe that
the
study
of
model space
provides
a
more natural language
for
geometric quantization, especially
in the
infinite
dimensional case.
It
seems
likely
that
for
Virasoro
and
Kac-Moody groups
the
quantization
of
individual orbits gives rise only
to
Verma modules.
By
contrast, quantization
of the
model space gives rise (see Sects.
3 and 4
below)
to
their irreducible quotients. Moreover, this type
of
geometric quantization,
as
will
be
explained
in
Sects.
3 and 4
reproduces
the
chiral
part
in
conformal
field
theories
and
therefore
it is
also
a
natural language
for
them. Physically, this construction
means that
path
integrals
are
averaged over
the set of all
orbits. More exactly,
the
parameter which lables
the
Virasoro coadjoint orbit becomes
a
quantum-
mechanical dynamical variable.
The
geometrical action,
defined
in our
previous
papers (more accurately
its
trivial generalization) plays
in
this construction
a
crucial
role.
It
defines
the
symplectic structure
on the
model space. Using
the
technique developed
in
[1,2,3]
it is
possible
to
introduce
the
"Darboux"
variables
for
this symplectic
form
both
in the finite
dimensional case
and in the
Virasoro case
(in
the
Kac-Moody group case
-
only
for the
quantum-mechanical part).
As a
result
the
path
integral reduces
to a sum
over blocks, where each
of
them
is
also
path
integral,
but
over
a
special
set of
orbits.
For the
case
of the
Kac-Moody group
this
sum is finite and
each block corresponds
to an
integrable representation
of the
group (this construction
gives
precisely
a
chiral
part
in
WZNW model).
For the
case
of the
Virasoro group with
C
q
<\
we
also
get a finite
sum, with each block
corresponding
to an
irreducible representation
of the
group (for exact
an
statement,
see
below
in
Sect.
3) and the
central charge also quantized
C
q
= 1
—
6———
the sum is
over Virasoro discrete series. These properties
are
similar
to
those
for the finite
dimensional case.
This
paper
is
organized
as
follows.
In
Sect.
2 we
consider
the
quantization
of
the
cotangent bundle
T*G and
model space
for the
compact
Lie
groups.
In
Sects.
3
and 4
using
the
path
integral approach
for the
quantization
of
model space
for
Virasoro
and
Kac-Moody groups correspondingly
we
reproduce
the finite
sums
in
RCFT.
In the
appendix
we
discuss
an
interesting question
on the
relation
between
the
geometrical actions
and
classical
r-matrices
(without
the
spectral
parameter).
Our
main observation
is
that
the
geometrical action
defines
a
nondegenerate symplectic structure
on the
group
(in the
infinite-dimensional
case)
where
corresponding
Poisson
brackets
are of the
r-matrix nature.
It
means that
there
is a
relation between
RCFT
and
quantum groups just
on the
classical level.
From Geometric Quantization
to
Conformal Field Theory
199
We
think that this observation together with
the
geometrical point
of
view
on the
conformal
field
theory
can
explain
in the
future
the
appearance
of
quantum
groups
in
RCFT.
We
will return
to
this subject
in
future
publications.
2.
Quantization
of the
J*G
and the
Model
Space
The
most natural symplectic manifold, related
to the
group
is
T*G
- the
cotangent
bundle
of the
group
G (in
this section
we
will consider
a
compact
Lie
group
G). By
using Hamiltonian reduction
we
replace
it
with
a
smaller space, which will
be
called
the
model
space.
The
quantization
of
this space yields
all
irreducible unitary
representations
of the
group with multiplicity one.
The
quantization
of the
model
space
is
performed
via the
path integral method.
For
concreteness
we
shall
consider below only
the
case
G =
SU(n).
A
generalization
to
arbitrary simple
groups
is
straightforward (cf.
[2]).
The
canonical symplectic
2-form
on T*G is
defined
by
Ω=%trdXdg.g-
l
+tιX(dg.g-
l
)
2
)
9
(1)
where
g
e
G, X
e
<&*
is the
right-invariant moment,
^*
is the
space dual
to the
corresponding
Lie
algebra
^.
As in
[1-3]
we
define
the
geometric action
on
T*G
as
a
functional
of
trajectories
on the T*G
-^Jα,
(2)
here
doc
= Ω.
This geometrical action
possesses
two
symmetries:
1)
X(t)->X(t)
9
g(t)-+g(t)h
R
,
2)
Here
h
R
and
h
L
are
constant matrices
from
G.
Using
the
parametrization
X-fXoΓ
1
,
feG
9
where
X
0
is a
diagonal matrix
we get
and in the
notation
g=f~
1
g
the
action acquires
the
form
(4)
/ is
defined
modulo transformation
/-»/Λ,
with
heH;
H is the
stationary
subgroup
of
X
0
,
i.e.
h is a
diagonal matrix
from
the
Cartan subgroup
of G.
In
new
variables
the
global symmetries
(3) are
expressed
by
2)
g(tHg(ί),
It
means that
g(/)
possesses
only right
(left)
symmetry.
There
is
also
an
additional local symmetry
in the
action
Kί)-»goWg(t),
(6)
200 A.
Alekseev
and S.
Shatashvili
where
g
0
(ί)
belongs
to the
stabilizer
of the
Cartan subalgebra,
i.e.
g
0
is
such
an
element
of the
group that
X
0
=
g
0
^ogo
1
is
again diagonal.
The
eigenvalues
of the
matrices
X
0
and
X
0
coincide
and
both
are
diagonal,
it
means that
the
subgroup
of
G
formed
by
these transformations contains
a
component, which acts
on the
eigenvalues
of
X
0
as a
permutation,
and
therefore this action
can be
identified
with
the
corresponding
Weyl
group.
Let
e\
i
=
1,
. .
.,
n be an
orthonormal
basis
in the
n-dimensional
complex vector
n
space,
in
which
X
0
is a
diagonal with eigenvalues
2m?(ί
ί
),
. .
.,2m°
f
;
Σ
m|
0)
= 0.
Using
i
the
symmetry with respect
to the
permutations, discussed above,
we may
impose
the
restriction
m°
^
w°
^
. . .
§;
w°,
which
is a
fundamental domain
of the
Weyl
group,
the
Weyl chamber. Then
it can be
shown
(see
[2])
that each term
in (4)
acquires
the
form
i
tr
X
0
dgg
^
=
\i
mf(t)
{_(da
b
a
t
)
-
(a
t
da^
, (7)
^
i
where
a\t)
=
g(t)e
l
.
The new
vectors,
a
1
,
are
also orthonormal.
Let us
parametrize
the
last component
of all
these vectors
by the
angles
φf:
£
=
<&%.,
(8)
where
aj,
- are
real:
Imαj,
= 0.
Then
we get
itrA
odgΓ
*
=£™W+!iX[(da,
a
;
)-(a,
da,.)]
. (9)
1
ί
1
Using
the
construction, developed
in
[2],
for the
second term
in (9) we
obtain
the
expression
,
(10)
i,k
where
φf,
φ
(
V
are
angles,
0
^
φf
}
£Ξ
2π and the
variables
m[
k)
form
a
Polyhedron
Π
Γ
Λ
.
n
2
— n
dimG
—
rankG
of
the
dimension
2
2
«2-Π
Π
:mf-
1)
^m[
fc)
^mf
+
-ι
1)
,
(H)
which
is the
classical analog
of the
Gelfand-Zetlin basis.
The
same construction
can be
applied
to the
second term
in (4) and we finally
Set
/»
\
S
=
$
I
Σ
»ii(dφ?
+
dψf)
+ Σ
n^dφV
+ Σ
"ΪW?
)
,
\1
ί,k
i,k J
where
nf
}
form
the
polyhedron
77
Π
,
defined
by the
"highest"
weight
(
—
w£,
. .
.,
— m?)
and
φf
}
are the
corresponding angle variables.
Now
we
must remember that
/ was
defined
module transformations
f-+fh:heH
and
therefore
we can
choose
v?|
0)
=
0,
i =
i,
...,π
— 1.
Conditions
detg=
1 and
det/=
1 in
terms
of m,
φ
variables
can be
written
as
From
Geometric
Quantization
to
Conformal
Field
Theory
201
This
finally
yields
the
following
expression
for the
action:
S
= ί
i
Δ
,
<M°>
+ Σ
Δ^dφM
+ Σ
Wdψΐ>
,
(13)
\1
i
t
k
i,k J
where
A
,
=
m|
0)
-
m<
0)
,
Δf
>
=
m|
k)
-
mi,
0
),
and
JJ*>
=
nf
}
+
m<
0)
;
Zl
1
^zl
2
^...^zl
n
_
1
^0;zle77
and the
corresponding regions
for
J[
n)
and
zlf
}
we
denote
by
Π
Δ
,
U-
Δ
.
Thus
we
constructed
the
"Darboux"
variables,
and so we can
easily reduce this
system
to a
smaller one.
The first is the
reduction over
the
constraints
\pf
}
=
const.
These constraints kill
the
field
/ The
resulting system
is the
so-called model
space.
The
corresponding Hubert
space
will
be
J^
= 0
J^
b
where
J
^
are all
irreducible
i
representations
of the
group.
If we
impose
an
extra condition
mj
0)
=
M
t
=
const
we
will
come
back
to the
orbit, corresponding
to the
point
X
0
=
diag(M
i
).
Let us now
consider
the
path integral quantization
of the
system under
consideration. Following
[2] the
Hamiltonian
is
chosen
to be a
linear combination
of
Cartan elements
H
t
:H
=
α^/ί^m)
+
βfl^n).
In the
SU(ri)
case
Cartan elements
H
t
are
Gelfand-Zetlin
parametrization
and are
given
by the sum
H
t
=
£
mf
}
over
the
rows
i of the
classical Gelfand-Zetlin table. Thus
l
(14)
ί,k
i,k
/
As in the
case
of the
coadjoint orbit (see
[2])
the
path integral
G(φ°",
φ",
ip"
I
φ°',
φ\
ιp'}
=
J
dA
.dφ^dή^dΔ^dφ^dip^^
(1
5)
with
the
boundary conditions:
φ|
0)
(0)
=
φ|
0)
,
φf>(0)-φ|
k)
,
ψ
(
f\0)
=
ψf
γ
,
and
can be
easily calculated
and we get
G(φ°",
φ",
ψ"
I
φ°',
φ',
ψ')
= Σ
expf
i
Σ^(ψ
m
"
~
<P
(
°
Y
)
ΔeΠz
i
(16)
,k
/
exp
i
"
'
where
71^
z
is
integer valued Gelfand-Zetlin table. From this
we
obtain that
the
Hubert
space
is
direct
sum
J^
=
0
J^
L
(χ)^,
where
^
and
^
are
irreducible
representations
of the
group where
left
and
right translations
are
acting. This
decomposition
of
L
2
(G)
is
well known.
[Character
can be
obtained
from
(15)
if we
put
φ'
=
φ",
\p'
=
\p"
and
integrate over
φ,
φ.]
Remark that this situation
is
quite
similar
to
that
in
CFT. Maybe
the
study
of the
same construction
for the
infinite-
dimensional
Lie
groups
(Virasoro, Kac-Moody)
and
their cosets,
or
Drinfeld-
Sokolov reductions will help
us to
understand
the
geometrical nature
of
CFT.