Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 1/20
Four-Point Functions in LCFT
Crossing Symmetry and SL(2,
C
) covariance
Michael Flohr
Physics Institute
University of Bonn
Marco Krohn
Institute for Theoretical Physics
University of Hannover
Global Conformal Invariance ●Ward Identities ●Recurrences of Inhomogeneities ●Correlators Four-Point Functions Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 2/20
Global Conformal Invariance ●Ward Identities ●Recurrences of Inhomogeneities ●Correlators Four-Point Functions Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 3/20
Ward Identities
n
Correlation functions have to satisfy the
global conformal
Ward identities
, i.e. for
m
=
−
1
,
0
,
1
we must have
0
=
Lm
h
Ψ
1
(
z
1
)
. . .
Ψ
n
(
zn
)
i
=
n
X
i
=1
z
i
m
h
z
i
∂
i
+ (
m
+ 1)(
h
i
+
δ
ˆ
h
i
)
i
h
Ψ
1
(
z
1
)
. . .
Ψ
n
(
z
n
)
i
.
n
In case of rank
r >
1
Jordan cells of indecomposable
representations with respect to
Vir
, we have
ˆ
δ
h
i
Ψ
(h
j
;k
j
)
=
(
δi,j
Ψ
(h
j
;k
j
−
1)
if
1
≤
kj
≤
r
−
1
,
0
if
k
j
= 0
.
n
Equivalently,
L
0
|
h;
k
i
=
h
|
h;
k
i
+ (1
−
δk,
0
)
|
h;
k
−
1
i
.
Global Conformal Invariance ●Ward Identities ●Recurrences of Inhomogeneities ●Correlators Four-Point Functions Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 3/20
Ward Identities
n
Correlation functions have to satisfy the
global conformal
Ward identities
, i.e. for
m
=
−
1,
0,
1
we must have
0
=
Lm
h
Ψ
1
(z
1
)
. . .
Ψn
(zn
)
i
=
n
X
i=1
z
i
m
h
z
i
∂
i
+ (m
+ 1)(h
i
+ ˆ
δ
h
i
)
i
h
Ψ
1
(z
1
)
. . .
Ψ
n
(z
n
)
i
.
n
In case of rank
r >
1
Jordan cells of indecomposable
representations with respect to
Vir
, we have
ˆ
δ
h
i
Ψ
(
h
j
;
k
j
)
=
(
δi,j
Ψ
(
h
j
;
k
j
−1)
if
1
≤
kj
≤
r
−
1
,
0
if
k
j
= 0
.
n
Equivalently,
L
0
|
h;
k
i
=
h
|
h;
k
i
+ (1
−
δk,
0
)
|
h;
k
−
1
i
.
Global Conformal Invariance ●Ward Identities ●Recurrences of Inhomogeneities ●Correlators Four-Point Functions Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 3/20
Ward Identities
n
Correlation functions have to satisfy the
global conformal
Ward identities
, i.e. for
m
=
−
1,
0,
1
we must have
0
=
Lm
h
Ψ
1
(z
1
)
. . .
Ψn
(zn
)
i
=
n
X
i=1
z
i
m
h
z
i
∂
i
+ (m
+ 1)(h
i
+ ˆ
δ
h
i
)
i
h
Ψ
1
(z
1
)
. . .
Ψ
n
(z
n
)
i
.
n
In case of rank
r >
1
Jordan cells of indecomposable
representations with respect to
Vir
, we have
ˆ
δ
h
i
Ψ
(h
j
;k
j
)
=
(
δi,j
Ψ
(h
j
;k
j
−
1)
if
1
≤
kj
≤
r
−
1
,
0
if
k
j
= 0
.
n
Equivalently,
L
0
|
h
;
k
i
=
h
|
h
;
k
i
+ (1
−
δk,
0
)
|
h
;
k
−
1
i
.
Global Conformal Invariance ●Ward Identities ●Recurrences of Inhomogeneities ●Correlators Four-Point Functions Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 4/20
Recurrences of Inhomogeneities
n
Ward identities become
inhomogeneous
in LCFT. The
inhomogeneities are given by correlation functions with
total
Jordan-level
K
=
P
n
i
=1
ki
decreased by one,
Ψ
(
h
1
;
k
1
)
(
z
1
)
. . .
Ψ
(
h
n
;
k
n
)
(
zn
)
≡ h
k
1
k
2
. . . kn
i
,
1
(
m
+ 1)
L
0
m
h
k
1
k
2
. . . k
n
i
=
−
z
m
1
h
k
1
−
1
, k
2
. . . k
n
i
−
z
2
m
h
k
1
,
k
2
−
1
, k
3
. . . kn
i
−
. . .
−
z
n
m
h
k
1
. . . kn
−1
,
k
n
−
1
i
.
n
We obtain a hierarchical scheme of solutions, starting with
Global Conformal Invariance ●Ward Identities ●Recurrences of Inhomogeneities ●Correlators Four-Point Functions Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 4/20
Recurrences of Inhomogeneities
n
Ward identities become
inhomogeneous
in LCFT. The
inhomogeneities are given by correlation functions with
total
Jordan-level
K
=
P
n
i=1
ki
decreased by one,
Ψ
(h
1
;k
1
)
(z
1
)
. . .
Ψ
(h
n
;k
n
)
(zn
)
≡ h
k
1
k
2
. . . kn
i
,
1
(m
+ 1)
L
0
m
h
k
1
k
2
. . . k
n
i
=
−
z
m
1
h
k
1
−
1, k
2
. . . k
n
i
−
z
2
m
h
k
1
, k
2
−
1, k
3
. . . kn
i
−
. . .
−
z
n
m
h
k
1
. . . kn
−
1
, kn
−
1
i
.
n
We obtain a hierarchical scheme of solutions, starting with
Global Conformal Invariance ●Ward Identities ●Recurrences of Inhomogeneities ●Correlators Four-Point Functions Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 5/20
Correlators
n
Generic form of 1, 2 and 3pt functions for fields forming
Jordan cells, pre-logarithmic fields and fermionic fields in
arbitrary rank
r
LCFT uniquely fixed:
h
Ψ
(
h
;
k
)
i
=
δh,
0
δk,
r
−
1
,
h
Ψ
(
h
;
k
)
(
z
)Ψ
(
h
0
;
k
0
)
(0)
i
=
δ
hh
0
k
+
k
0
X
j
=r
−
1
D
(
h
;
j
)
X
0
≤
i
≤
k,
0
≤
i
0 ≤
k
0
i
+
i
0
=
k
+
k
0 −j
(
∂
h
)
i
i
!
(
∂
h
0
)
i
0
i
0
!
z
−
h
−
h
0
,
h
Ψ
(
h
1
;
k
1
)
(
z
1
)Ψ
(
h
2
;
k
2
)
(
z
2
)Ψ
(
h
3
;
k
3
)
(
z
3
)
i
=
k1
+
k2
+
k3
X
j
=r
−
1
C
(
h
1
h
2
h
3
;
j
)
×
X
0
≤
il
≤
kl,l
=1
,
2
,
3
i
1+
i
2 +
i
3 =
k
1 +
k
2 +
k
3
−j
(
∂
h
1
)
i
1
i
1
!
(
∂
h
2
)
i
2
i
2
!
(
∂
h
3
)
i
3
i
3
!
Y
σ
∈
S
3
σ
(1)
<σ
(2)
(
z
σ
(1)
σ
(2)
)
h
σ
(3)
−
h
σ
(1)
−
h
σ
(2)
.
Global Conformal Invariance ●Ward Identities ●Recurrences of Inhomogeneities ●Correlators Four-Point Functions Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 5/20
Correlators
n
Generic form of 1, 2 and 3pt functions for fields forming
Jordan cells, pre-logarithmic fields and fermionic fields in
arbitrary rank
r
LCFT uniquely fixed:
h
Ψ
(
h
;
k
)
i
=
δh,
0
δk,
r
−
1
,
h
Ψ
(
h
;
k
)
(
z
)Ψ
(
h
0
;
k
0
)
(0)
i
=
δ
hh
0
k
+
k
0
X
j
=r
−
1
D
(
h
;
j
)
X
0
≤
i
≤
k,
0
≤
i
0 ≤
k
0
i
+
i
0
=
k
+
k
0 −j
(
∂
h
)
i
i
!
(
∂
h
0
)
i
0
i
0
!
z
−
h
−
h
0
,
h
Ψ
(
h
1
;
k
1
)
(
z
1
)Ψ
(
h
2
;
k
2
)
(
z
2
)Ψ
(
h
3
;
k
3
)
(
z
3
)
i
=
k1
+
k2
+
k3
X
j
=r
−
1
C
(
h
1
h
2
h
3
;
j
)
×
X
0
≤
il
≤
kl,l
=1
,
2
,
3
i
1+
i
2 +
i
3 =
k
1 +
k
2 +
k
3
−j
(
∂
h
1
)
i
1
i
1
!
(
∂
h
2
)
i
2
i
2
!
(
∂
h
3
)
i
3
i
3
!
Y
σ
∈
S
3
σ
(1)
<σ
(2)
(
z
σ
(1)
σ
(2)
)
h
σ
(3)
−
h
σ
(1)
−
h
σ
(2)
.
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 6/20
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 7/20
Beyond 3pt Functions
n
To find a useful algorithm to fix the generic form of 4pt
functions, visualize a logarithmic field
Ψ
(
h
;
k
)
by a vertex with
k
outgoing lines.
n
Contractions of logarithmic fields give rise to logarithms in
the correlators. The possible powers with which
log(z
ij
)
may
occur, can be determined by graph combinatorics.
h;k
(
)
Ψ
h’;k’
(
)
Ψ
k-i
k’-i’
i
i’
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 7/20
Beyond 3pt Functions
n
To find a useful algorithm to fix the generic form of 4pt
functions, visualize a logarithmic field
Ψ
(
h
;
k
)
by a vertex with
k
outgoing lines.
n
Contractions of logarithmic fields give rise to logarithms in
the correlators. The possible powers with which
log(
z
ij
)
may
occur, can be determined by graph combinatorics.
h;k
(
)
Ψ
h’;k’
(
)
Ψ
k-i
k’-i’
i
i’
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 7/20
Beyond 3pt Functions
n
To find a useful algorithm to fix the generic form of 4pt
functions, visualize a logarithmic field
Ψ
(
h
;
k
)
by a vertex with
k
outgoing lines.
n
Contractions of logarithmic fields give rise to logarithms in
the correlators. The possible powers with which
log(z
ij
)
may
occur, can be determined by graph combinatorics.
h;k
(
)
Ψ
h’;k’
(
)
Ψ
k-i
k’-i’
i
i’
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 8/20
Graphs
n
Terms of generic form of
n
-pt function given by sum over all
admissible graphs subject to the rules:
u
Each
k
out
-vertex may receive
k
in
0
≤
(r
−
1)
lines.
u
Vertices with
k
out
= 0
(proper primary fiels) do not receive
any legs.
u
Vertex
i
can receive legs from vertex
j
only for
j
6
=
i.
u
Precisely
r
−
1
lines in correlator remain open.
n
Example:
4pt function for
r
= 2
and all fields logarithmic
yields, upto permutations, the graphs
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 8/20
Graphs
n
Terms of generic form of
n-pt function given by sum over all
admissible graphs subject to the rules:
u
Each
k
out
-vertex may receive
k
in
0
≤
(
r
−
1)
lines.
u
Vertices with
k
out
= 0
(proper primary fiels) do not receive
any legs.
u
Vertex
i
can receive legs from vertex
j
only for
j
6
=
i.
u
Precisely
r
−
1
lines in correlator remain open.
n
Example:
4pt function for
r
= 2
and all fields logarithmic
yields, upto permutations, the graphs
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 8/20
Graphs
n
Terms of generic form of
n-pt function given by sum over all
admissible graphs subject to the rules:
u
Each
k
out
-vertex may receive
k
in
0
≤
(
r
−
1)
lines.
u
Vertices with
k
out
= 0
(proper primary fiels) do not receive
any legs.
u
Vertex
i
can receive legs from vertex
j
only for
j
6
=
i.
u
Precisely
r
−
1
lines in correlator remain open.
n
Example:
4pt function for
r
= 2
and all fields logarithmic
yields, upto permutations, the graphs
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 8/20
Graphs
n
Terms of generic form of
n-pt function given by sum over all
admissible graphs subject to the rules:
u
Each
k
out
-vertex may receive
k
in
0
≤
(
r
−
1)
lines.
u
Vertices with
k
out
= 0
(proper primary fiels) do not receive
any legs.
u
Vertex
i
can receive legs from vertex
j
only for
j
6
=
i
.
u
Precisely
r
−
1
lines in correlator remain open.
n
Example:
4pt function for
r
= 2
and all fields logarithmic
yields, upto permutations, the graphs
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 8/20
Graphs
n
Terms of generic form of
n-pt function given by sum over all
admissible graphs subject to the rules:
u
Each
k
out
-vertex may receive
k
in
0
≤
(
r
−
1)
lines.
u
Vertices with
k
out
= 0
(proper primary fiels) do not receive
any legs.
u
Vertex
i
can receive legs from vertex
j
only for
j
6
=
i
.
u
Precisely
r
−
1
lines in correlator remain open.
n
Example:
4pt function for
r
= 2
and all fields logarithmic
yields, upto permutations, the graphs
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 8/20
Graphs
n
Terms of generic form of
n-pt function given by sum over all
admissible graphs subject to the rules:
u
Each
k
out
-vertex may receive
k
in
0
≤
(r
−
1)
lines.
u
Vertices with
k
out
= 0
(proper primary fiels) do not receive
any legs.
u
Vertex
i
can receive legs from vertex
j
only for
j
6
=
i.
u
Precisely
r
−
1
lines in correlator remain open.
n
Example:
4pt function for
r
= 2
and all fields logarithmic
yields, upto permutations, the graphs
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 9/20
The Algorithm
n
Linking numbers
A
ij
(
g
)
of given graph
g
yield
upper bounds
for power with which logarithms occur.
n
Recursive procedure:
start with all ways
f
i
to choose
r
−
1
free legs, find at each level
K
0
and for each configuration
fi
all graphs, which connect the remaining
K
−
K
0
−
(r
−
1)
legs to vertices.
n
Write down corresponding monomial in
log(zij
), multiplied
with an as yet undetermined constant
C
(g)
for each graph
g.
n
Determine some constants by imposing global conformal
invariance.
n
Fix further constants by imposing admissible permutation
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 9/20
The Algorithm
n
Linking numbers
A
ij
(
g
)
of given graph
g
yield
upper bounds
for power with which logarithms occur.
n
Recursive procedure:
start with all ways
f
i
to choose
r
−
1
free legs, find at each level
K
0
and for each configuration
f
i
all graphs, which connect the remaining
K
−
K
0
−
(
r
−
1)
legs to vertices.
n
Write down corresponding monomial in
log(zij
), multiplied
with an as yet undetermined constant
C
(g)
for each graph
g.
n
Determine some constants by imposing global conformal
invariance.
n
Fix further constants by imposing admissible permutation
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 9/20
The Algorithm
n
Linking numbers
A
ij
(
g
)
of given graph
g
yield
upper bounds
for power with which logarithms occur.
n
Recursive procedure:
start with all ways
f
i
to choose
r
−
1
free legs, find at each level
K
0
and for each configuration
f
i
all graphs, which connect the remaining
K
−
K
0
−
(
r
−
1)
legs to vertices.
n
Write down corresponding monomial in
log(
zij
)
, multiplied
with an as yet undetermined constant
C
(
g
)
for each graph
g
.
n
Determine some constants by imposing global conformal
invariance.
n
Fix further constants by imposing admissible permutation
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 9/20
The Algorithm
n
Linking numbers
A
ij
(
g
)
of given graph
g
yield
upper bounds
for power with which logarithms occur.
n
Recursive procedure:
start with all ways
f
i
to choose
r
−
1
free legs, find at each level
K
0
and for each configuration
f
i
all graphs, which connect the remaining
K
−
K
0
−
(
r
−
1)
legs to vertices.
n
Write down corresponding monomial in
log(
zij
)
, multiplied
with an as yet undetermined constant
C
(
g
)
for each graph
g
.
n
Determine some constants by imposing global conformal
invariance.
n
Fix further constants by imposing admissible permutation
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 9/20
The Algorithm
n
Linking numbers
A
ij
(
g
)
of given graph
g
yield
upper bounds
for power with which logarithms occur.
n
Recursive procedure:
start with all ways
f
i
to choose
r
−
1
free legs, find at each level
K
0
and for each configuration
f
i
all graphs, which connect the remaining
K
−
K
0
−
(
r
−
1)
legs to vertices.
n
Write down corresponding monomial in
log(
zij
)
, multiplied
with an as yet undetermined constant
C
(
g
)
for each graph
g
.
n
Determine some constants by imposing global conformal
invariance.
n
Fix further constants by imposing admissible permutation
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 10/20
Generic Form
n
Generic form of the LCFT 4pt functions is
h
k
1
k
2
k
3
k
4
i ≡ h
Ψ
(
h
1
;
k
1
)
(
z
1
)
. . .
Ψ
(
h
4
;
k
4
)
(
z
4
)
i
=
Y
i<j
(
zij
)
µ
ij
X
(
k
1
0
,k
0
2
,k
3
0
,k
4
0
)
X
g
∈G
K
−
K
0
C
(
g
)
Y
i<j
log
A
ij
(
g
)
(
zij
)
F
k
0
1
k
0
2
k
0
3
k
4
0
(
x
)
,
where
u
GK
−
K
0
is set of graphs for
(k
1
−
k
1
0
, . . . , k
4
−
k
4
0
),
u
A
ij
(g)
is linking number of vertices
i, j
of graph
g,
u
x
is the crossing ratio
x
=
z
12
z
34
z
14
z
23
,
u
µij
is typically
µij
=
1
3
(
P
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 10/20
Generic Form
n
Generic form of the LCFT 4pt functions is
h
k
1
k
2
k
3
k
4
i ≡ h
Ψ
(
h
1
;
k
1
)
(
z
1
)
. . .
Ψ
(
h
4
;
k
4
)
(
z
4
)
i
=
Y
i<j
(
zij
)
µ
ij
X
(
k
1
0
,k
0
2
,k
3
0
,k
4
0
)
X
g
∈G
K
−
K
0
C
(
g
)
Y
i<j
log
A
ij
(
g
)
(
zij
)
F
k
0
1
k
0
2
k
0
3
k
4
0
(
x
)
,
where
u
GK
−
K
0
is set of graphs for
(
k
1
−
k
1
0
, . . . , k
4
−
k
4
0
)
,
u
A
ij
(g)
is linking number of vertices
i, j
of graph
g,
u
x
is the crossing ratio
x
=
z
12
z
34
z
14
z
23
,
u
µij
is typically
µij
=
1
3
(
P
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 10/20
Generic Form
n
Generic form of the LCFT 4pt functions is
h
k
1
k
2
k
3
k
4
i ≡ h
Ψ
(
h
1
;
k
1
)
(
z
1
)
. . .
Ψ
(
h
4
;
k
4
)
(
z
4
)
i
=
Y
i<j
(
zij
)
µ
ij
X
(
k
1
0
,k
0
2
,k
3
0
,k
4
0
)
X
g
∈G
K
−
K
0
C
(
g
)
Y
i<j
log
A
ij
(
g
)
(
zij
)
F
k
0
1
k
0
2
k
0
3
k
4
0
(
x
)
,
where
u
GK
−
K
0
is set of graphs for
(
k
1
−
k
1
0
, . . . , k
4
−
k
4
0
)
,
u
A
ij
(
g
)
is linking number of vertices
i, j
of graph
g
,
u
x
is the crossing ratio
x
=
z
12
z
34
z
14
z
23
,
u
µij
is typically
µij
=
1
3
(
P
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 10/20
Generic Form
n
Generic form of the LCFT 4pt functions is
h
k
1
k
2
k
3
k
4
i ≡ h
Ψ
(
h
1
;
k
1
)
(
z
1
)
. . .
Ψ
(
h
4
;
k
4
)
(
z
4
)
i
=
Y
i<j
(
zij
)
µ
ij
X
(
k
1
0
,k
0
2
,k
3
0
,k
4
0
)
X
g
∈G
K
−
K
0
C
(
g
)
Y
i<j
log
A
ij
(
g
)
(
zij
)
F
k
0
1
k
0
2
k
0
3
k
4
0
(
x
)
,
where
u
GK
−
K
0
is set of graphs for
(
k
1
−
k
1
0
, . . . , k
4
−
k
4
0
)
,
u
A
ij
(
g
)
is linking number of vertices
i, j
of graph
g
,
u
x
is the crossing ratio
x
=
z
12
z
34
z
14
z
23
,
u
µij
is typically
µij
=
1
3
(
P
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 10/20
Generic Form
n
Generic form of the LCFT 4pt functions is
h
k
1
k
2
k
3
k
4
i ≡ h
Ψ
(
h
1
;
k
1
)
(
z
1
)
. . .
Ψ
(
h
4
;
k
4
)
(
z
4
)
i
=
Y
i<j
(
zij
)
µ
ij
X
(
k
1
0
,k
0
2
,k
3
0
,k
4
0
)
X
g
∈G
K
−
K
0
C
(
g
)
Y
i<j
log
A
ij
(
g
)
(
zij
)
F
k
0
1
k
0
2
k
0
3
k
4
0
(
x
)
,
where
u
GK
−
K
0
is set of graphs for
(
k
1
−
k
1
0
, . . . , k
4
−
k
4
0
)
,
u
A
ij
(
g
)
is linking number of vertices
i, j
of graph
g
,
u
x
is the crossing ratio
x
=
z
12
z
34
z
14
z
23
,
u
µ
ij
is typically
µ
ij
=
1
3
(
P
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 11/20
Simplification and Recursion
n
Generic structure
=
⇒
set
h
i
= 0
.
n
L
−1
is translation invariance
=
⇒
z
ij
.
n
Initial conditions:
h
k
1
k
2
k
3
k
4
i
=
F
0
(
x
)
for
P
i
k
i
=
r
−
1
n
and
h
k
1
k
2
k
3
k
4
i
= 0
for
P
i
k
i
<
r
−
1
.
O
0
h
. . .
i
≡
P
i
z
i
∂
i
h
. . .
i
=
−
P
i
δ
ˆ
h
i
h
. . .
i
O
1
h
. . .
i
≡
P
i
z
i
2
∂i
h
. . .
i
=
−
2
P
i
zi
δ
ˆ
h
i
h
. . .
i
h
0110
i
=
h
1100
i
=
h
1010
i
=
h
2000
i
=
F
0
(l=0)
@
@
R
?
@
@
R
@
R
@
?
h
1110
i
h
2100
i ∼ h
2010
i
(l=1)
@
@
R ?
h
2110
i
(l=2)
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 11/20
Simplification and Recursion
n
Generic structure
=
⇒
set
h
i
= 0
.
n
L
−1
is translation invariance
=
⇒
z
ij
.
n
Initial conditions:
h
k
1
k
2
k
3
k
4
i
=
F
0
(
x
)
for
P
i
k
i
=
r
−
1
n
and
h
k
1
k
2
k
3
k
4
i
= 0
for
P
i
k
i
<
r
−
1
.
O
0
h
. . .
i
≡
P
i
z
i
∂
i
h
. . .
i
=
−
P
i
δ
ˆ
h
i
h
. . .
i
O
1
h
. . .
i
≡
P
i
z
i
2
∂i
h
. . .
i
=
−
2
P
i
zi
δ
ˆ
h
i
h
. . .
i
h
0110
i
=
h
1100
i
=
h
1010
i
=
h
2000
i
=
F
0
(l=0)
@
@
R
?
@
@
R
@
R
@
?
h
1110
i
h
2100
i ∼ h
2010
i
(l=1)
@
@
R ?
h
2110
i
(l=2)
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 12/20
Rank
r
= 2
h
1000
i
=
F
0
h
1100
i
=
P
S
2
n
1
2
F
1100
−
r
r
r
r
F
0
o
h
1110
i
=
P
S
3
n
1
6
F
1110
+ (
1
2
P
(13)
−
1)
r
r
r
r
F
0110
+
r
r
r
r
−
1
2
r
r
r
r
F
0
o
h
1111
i
=
P
S
4
n
1
24
F
1111
+ (
1
6
P
(13)
−
1
3
)
r
r
r
r
F
0111
+
1
2
(P
(24)
−
1)
r
r
r
r
+ (1
−
1
2
P
(14))
r
r
r
r
−
1
4
r
r
r
r
F
0011
+
1
2
r
r
r
r
+
1
3
r
r
r
r
−
r
r
r
r
F
0
o
.
Global Conformal Invariance Four-Point Functions ●Beyond 3pt Functions ●Graphs ●The Algorithm ●Generic Form
●Simplification and Recursion
●Rankr= 2
●Redundancy
Symmetries & Graphs
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 13/20
Redundancy
Interestingly, there remain
free constants
, when all four fields
are logarithmic!
h
1111
i
=
F
1111
+
P
(1234)
˘ˆ
(
−
`
12
−
`
34
+
`
23
+
`
14
)
C
1
+ (
`
13
+
`
24
−
`
12
−
`
34
)
C
2
−
`
14
+
`
34
−
`
13
)
˜
F
0111
¯
+
P
(12)(34)
˘ˆ
(
`
2
13
+
`
2
24
−
`
2
14
−
`
2
23
+ 2(
−
`
34
`
24
−
`
12
`
24
+
`
34
`
14
+
`
13
`
24
−
`
13
`
34
+
`
23
`
34
+
`
12
`
23
−
`
12
`
13
−
`
23
`
14
+
`
12
`
14
))
C
3
+ (
−
(
`
23
+
`
14
)
2
+
`
23
`
34
+
`
12
`
14
−
`
13
`
34
+
`
34
`
14
+
`
13
`
14
−
`
34
`
24
−
`
12
`
13
−
`
12
`
24
+
`
23
`
24
+
`
23
`
13
+
`
12
`
23
+
`
24
`
14
))
C
4
−
`
2
34
−
`
2
23
−
`
14
2
+ 2
`
23
`
34
+ 2
`
34
`
14
−
2
`
12
`
34
−
`
23
`
14
+
`
23
`
24
−
`
12
`
13
+
`
12
`
14
+
`
12
`
23
−
`
12
`
24
+
`
13
`
14
+
`
13
`
24
)
˜
F
1100
¯
+
ˆ
2(
`
12
`
24
`
14
−
`
23
`
13
`
14
+
`
23
`
34
`
24
−
`
24
`
13
`
34
−
`
23
`
34
`
14
−
`
12
`
23
`
34
−
`
12
`
34
`
24
−
`
23
`
13
`
24
+
`
12
`
23
`
13
+
`
13
`
34
`
14
−
`
13
`
14
`
24
−
`
23
`
24
`
14
−
`
12
`
13
`
24
−
`
12
`
23
`
14
−
`
12
`
13
`
34
−
`
12
`
34
`
14
)
+ 2(
`
2
13
`
24
+
`
2
12
`
34
+
`
2
14
`
23
+
`
2
23
`
14
+
`
2
34
`
12
+
`
2
24
`
13
)
˜
F
0
Global Conformal Invariance Four-Point Functions
Symmetries & Graphs
●Additional constants ●Graphical Solution ●Rankr= 3 ●<2222> :: part I ●<2222> :: part VI ●Towardsc= 0
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 14/20
Global Conformal Invariance Four-Point Functions Symmetries & Graphs
●Additional constants ●Graphical Solution ●Rankr= 3 ●<2222> :: part I ●<2222> :: part VI ●Towardsc= 0
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 15/20
Additional Constants
O
0
h
k
1
k
2
k
3
k
4
i
=
−
P
i
δ
ˆ
h
i
h
k
1
k
2
k
3
k
4
i
,
O
1
h
k
1
k
2
k
3
k
4
i
=
−
2
P
i
zi
δ
ˆ
h
i
h
k
1
k
2
k
3
k
4
i
.
Are there any
f
∈ F
log
with
O
0
f
= 0
and
O
1
f
= 0
?
ker
F
log
,g
O
=
(
g
X
i
=0
ai
K
1
i
K
2
g
−
i
:
ak
∈
R
)
K
1
≡
l
12
+
l
34
−
l
13
−
l
24
= log
|
x
| −
log
|
1
−
x
|
K
2
≡
l
12
+
l
34
−
l
14
−
l
23
= log
|
x
|
.
This means that all four fields have to be of logarithmic type.
Symmetry considerations and combinatorical restrictions in
fact constrain the number of additional constants further.
Ker
h1111i
=
P
S
4
n
Global Conformal Invariance Four-Point Functions Symmetries & Graphs
●Additional constants ●Graphical Solution ●Rankr= 3 ●<2222> :: part I ●<2222> :: part VI ●Towardsc= 0
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 15/20
Additional Constants
O
0
h
k
1
k
2
k
3
k
4
i
=
−
P
i
δ
ˆ
h
i
h
k
1
k
2
k
3
k
4
i
,
O
1
h
k
1
k
2
k
3
k
4
i
=
−
2
P
i
zi
δ
ˆ
h
i
h
k
1
k
2
k
3
k
4
i
.
Are there any
f
∈ F
log
with
O
0
f
= 0
and
O
1
f
= 0
?
ker
F
log
,g
O
=
(
g
X
i
=0
ai
K
1
i
K
2
g
−
i
:
ak
∈
R
)
K
1
≡
l
12
+
l
34
−
l
13
−
l
24
= log
|
x
| −
log
|
1
−
x
|
K
2
≡
l
12
+
l
34
−
l
14
−
l
23
= log
|
x
|
.
This means that all four fields have to be of logarithmic type.
Symmetry considerations and combinatorical restrictions in
fact constrain the number of additional constants further.
Ker
h1111i
=
P
S
4
n
Global Conformal Invariance Four-Point Functions Symmetries & Graphs
●Additional constants ●Graphical Solution ●Rankr= 3 ●<2222> :: part I ●<2222> :: part VI ●Towardsc= 0
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 15/20
Additional Constants
O
0
h
k
1
k
2
k
3
k
4
i
=
−
P
i
δ
ˆ
h
i
h
k
1
k
2
k
3
k
4
i
,
O
1
h
k
1
k
2
k
3
k
4
i
=
−
2
P
i
zi
δ
ˆ
h
i
h
k
1
k
2
k
3
k
4
i
.
Are there any
f
∈ F
log
with
O
0
f
= 0
and
O
1
f
= 0
?
ker
F
log
,g
O
=
(
g
X
i
=0
ai
K
1
i
K
2
g
−
i
:
ak
∈
R
)
K
1
≡
l
12
+
l
34
−
l
13
−
l
24
= log
|
x
| −
log
|
1
−
x
|
K
2
≡
l
12
+
l
34
−
l
14
−
l
23
= log
|
x
|
.
This means that all four fields have to be of logarithmic type.
Symmetry considerations and combinatorical restrictions in
fact constrain the number of additional constants further.
Ker
h1111i
=
P
S
4
n
Global Conformal Invariance Four-Point Functions Symmetries & Graphs
●Additional constants ●Graphical Solution ●Rankr= 3 ●<2222> :: part I ●<2222> :: part VI ●Towardsc= 0
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 16/20
Graphical Solution
n
Now, the final results simply read
h
1110
i
=
F
1110
−
P
S3
(
`
12
)
F
0011
+
P
S3
(2
`
12
`
23
−
`
2
12
)
F
0
=
F
1110
−
P
S
3
(
r
r
r
r
)
F
0011
+
P
S
3
(2
r
r
r
r
−
r
r
r
r
)
F
0
,
h
1111
i
=
F
1111
−
1
6
P
S4
(
r
r
r
r
)
F
0111
+
1
4
P
S4
(
r
r
r
r
+
K
S
(2)
4
)
F
0011
+
P
S
4
(
1
2
r
r
r
r
+
1
3
r
r
r
r
−
r
r
r
r
)
F
0
.
n
Note the appearance of a term
K
∈
ker
L
offdiag
m
in the kernel of
the nilpotent part of the Virasoro generators:
ker(L
m
−
L
0
m
) =
h
K
1
≡
log(x),
K
2
≡ −
log(1
−
1/x)
i
=
h
`
12
+
`
34
−
`
14
−
`
23
, `
12
+
`
34
−
`
13
−
`
24
i
;
Global Conformal Invariance Four-Point Functions Symmetries & Graphs
●Additional constants ●Graphical Solution ●Rankr= 3 ●<2222> :: part I ●<2222> :: part VI ●Towardsc= 0
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 16/20
Graphical Solution
n
Now, the final results simply read
h
1110
i
=
F
1110
− P
S
3
(`
12
)F
0011
+
P
S
3
(2`
12
`
23
−
`
2
12
)F
0
=
F
1110
− P
S
3
(
r
r
r
r
)F
0011
+
P
S
3
(2
r
r
r
r
−
r
r
r
r
)F
0
,
h
1111
i
=
F
1111
−
1
6
P
S
4
(
r
r
r
r
)F
0111
+
1
4
P
S
4
(
r
r
r
r
+
K
(2)
S
4
)F
0011
+
P
S
4
(
1
2
r
r
r
r
+
1
3
r
r
r
r
−
r
r
r
r
)F
0
.
n
Note the appearance of a term
K
∈
ker
L
offdiag
m
in the kernel of
the nilpotent part of the Virasoro generators:
ker(
L
m
−
L
0
m
) =
h
K
1
≡
log(
x
)
,
K
2
≡ −
log(1
−
1
/x
)
i
=
h
`
12
+
`
34
−
`
14
−
`
23
, `
12
+
`
34
−
`
13
−
`
24
i
;
Global Conformal Invariance Four-Point Functions Symmetries & Graphs
●Additional constants ●Graphical Solution ●Rankr= 3 ●<2222> :: part I ●<2222> :: part VI ●Towardsc= 0
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 17/20
Rank
r
= 3
Again, if all
ki
>
0
, free constants remain:
h
1111
i
=
F
1111
+
P
(1234)
˘ˆ
(
`
13
−
`
12
+
`
24
−
`
34
)
C
1
+ (
`
23
+
`
14
−
`
34
−
`
12
)
C
2
−
`
14
+
`
24
−
`
12
˜
F
0111
¯
+
ˆ`
`
2
12
+
`
2
24
+
`
2
34
+
`
2
13
+ 2(
`
12
`
13
+
`
13
`
24
−
`
13
`
34
−
`
34
`
24
−
`
12
`
24
+
`
12
`
34
)
´
C
3
+
`
−
2
`
13
`
14
+
`
2
24
+ 2
`
23
`
14
−
2
`
23
`
24
+
`
2
23
+ 2
`
13
`
24
+
`
2
13
−
2
`
23
`
13
+
`
2
14
−
2
`
24
`
14
´
C
4
+
`
(
`
24
+
`
13
)
2
+
`
12
`
14
−
`
23
`
24
−
`
12
`
24
−
`
24
`
14
−
`
23
`
13
+
`
34
`
14
−
`
13
`
34
−
`
13
`
14
+
`
23
`
34
−
`
34
`
24
+
`
12
`
23
−
`
12
`
13
´
C
5
+ 2(
`
13
`
24
+
`
23
`
14
+
`
12
`
34
)
˜
F
0
.
h
1111
i
=
F
1111
−
1
6
P
S
4
(
`
12
)
F
0111
+
n
P
S
4
h
−
1
4
`
2
34
+
1
2
`
34
`
24
i
+
K
S
(2)
4
o
F
0
.
Global Conformal Invariance Four-Point Functions Symmetries & Graphs
●Additional constants ●Graphical Solution ●Rankr= 3 ●<2222> :: part I ●<2222> :: part VI ●Towardsc= 0
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 17/20
Rank
r
= 3
Again, if all
ki
>
0
, free constants remain:
h
1111
i
=
F
1111
+
P
(1234)
˘ˆ
(
`
13
−
`
12
+
`
24
−
`
34
)
C
1
+ (
`
23
+
`
14
−
`
34
−
`
12
)
C
2
−
`
14
+
`
24
−
`
12
˜
F
0111
¯
+
ˆ`
`
2
12
+
`
2
24
+
`
2
34
+
`
2
13
+ 2(
`
12
`
13
+
`
13
`
24
−
`
13
`
34
−
`
34
`
24
−
`
12
`
24
+
`
12
`
34
)
´
C
3
+
`
−
2
`
13
`
14
+
`
2
24
+ 2
`
23
`
14
−
2
`
23
`
24
+
`
2
23
+ 2
`
13
`
24
+
`
2
13
−
2
`
23
`
13
+
`
2
14
−
2
`
24
`
14
´
C
4
+
`
(
`
24
+
`
13
)
2
+
`
12
`
14
−
`
23
`
24
−
`
12
`
24
−
`
24
`
14
−
`
23
`
13
+
`
34
`
14
−
`
13
`
34
−
`
13
`
14
+
`
23
`
34
−
`
34
`
24
+
`
12
`
23
−
`
12
`
13
´
C
5
+ 2(
`
13
`
24
+
`
23
`
14
+
`
12
`
34
)
˜
F
0
.
h
1111
i
=
F
1111
−
1
6
P
S
4
(
`
12
)
F
0111
+
n
P
S
4
h
−
1
4
`
2
34
+
1
2
`
34
`
24
i
+
K
S
(2)
4
o
F
0
.
Global Conformal Invariance Four-Point Functions Symmetries & Graphs
●Additional constants ●Graphical Solution ●Rankr= 3 ●<2222> :: part I ●<2222> :: part VI ●Towardsc= 0
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 18/20
<2222> :: part I
h
2222
i
=
P
S
4
n
1
24
F
2222
+ (
1
6
P
(13)
−
1
3
)
r
r
r
r
F
1222
+
(
1
3
P
(12)
+
1
6
P
(14))
r
r
r
r
−
1
3
r
r
r
r
−
12
1
P
(13)
r
r
r
r
F
0222
+
1
2
P
(24)
r
r
r
r
+ (
1
2
P
(23)
−
P
(24))
r
r
r
r
+
1
4
P
(13)(24)
r
r
r
r
F
1122
+
(3P
(34)
+ 3 +
P
(14))
r
r
r
r
−
5
r
r
r
r
−
(
13
6
+
5
2
P
(34))
r
r
r
r
+
(5 + 3P
(24)
)
r
r
r
r
−
r
r
r
r
−
(3 +
3
2
P
(14))
r
r
r
r
)
F
1112
+
1
2
(P
(124)
−
11
−
9P
(12)
−
7P
(123)
−
P
(132)
−
3P
(142)
−
7P
(14)
−
7P
(13)(24)+
P
(13)
−
8P
(14)(23))
r
r
r
r
+ (5 +
3
2
P
(24)
+ 3P
(14)
+
9
2
P
(13)(24))
r
r
r
r
+
(6 + 7P
(23)
+ 5P
(12)
)
r
r
r
r
+ (
3
2
+
P
(13
)
+
5
4
P
(13)(24))
r
r
r
r
+
. . .
Global Conformal Invariance Four-Point Functions Symmetries & Graphs
●Additional constants ●Graphical Solution ●Rankr= 3 ●<2222> :: part I ●<2222> :: part VI ●Towardsc= 0
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 19/20
<2222> :: part VI
. . .
+
1
2
(P
(243)
−
P
(24)
−
P
(13)
+
P
(14)
)
r
r
r
r
+ (P
(34)
−
P
(13
)
+
P
(13)(24))
r
r
r
r
+
(2P
(34)
+
P
(234)
−
P
(243)
+
P
(13)
−
P
(14))
r
r
r
r
−
(1 +
3
4
P
(13)(24))
r
r
r
r
+
(3 +
1
4
P
(234)
−
P
(243)
+
1
2
P
(13))
r
r
r
r
−
(P
(24)
+
P
(1324))
r
r
r
r
+
1
2
P
(1324)
r
r
r
r
−
3
8
r
r
r
r
−
P
(13)
r
r
r
r
F
0012
+
r
r
r
r
+
r
r
r
r
−
r
r
r
r
+
1
4
r
r
r
r
−
1
4
r
r
r
r
−
1
6
r
r
r
r
+
r
r
r
r
+
1
2
r
r
r
r
−
1
2
r
r
r
r
−
3
2
r
r
r
r
+
1
4
r
r
r
r
−
r
r
r
r
+
1
4
r
r
r
r
−
5
r
r
r
r
−
3
4
r
r
r
r
−
r
r
r
r
−
1
2
r
r
r
r
−
r
r
r
r
+
1
2
r
r
r
r
+ 3
r
r
r
r
+ 2
r
r
r
r
+
5
2
r
r
r
r
+
1
2
r
r
r
r
F
0
o
Global Conformal Invariance Four-Point Functions Symmetries & Graphs
●Additional constants ●Graphical Solution ●Rankr= 3 ●<2222> :: part I ●<2222> :: part VI ●Towardsc= 0
Michael Flohr :: Santiago :: EUCLID :: 16 Sep 2005 LCFT :: 4pt-Functions :: p. 20/20
Towards
c
= 0
n
That is all very elegant, very short, and very simple
;-)
n
Remember initial conditions:
u
h
k
1
k
2
k
3
k
4
i
= 0
∀
k
1
+
k
2
+
k
3
+
k
4
< r
−
1,
u
h
k
1
k
2
k
3
k
4
i
=
h
k
0
1
k
0
2
k
3
0
k
4
0
i
∀
k<