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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

(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. 2/20

(3)

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

.

(4)

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

.

(5)

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

.

(6)

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

(7)

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

(8)

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

(

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

(

h

2

;

k

2

)

(

z

2

(

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)

(2)

(

z

σ

(1)

σ

(2)

)

h

σ

(3)

h

σ

(1)

h

σ

(2)

.

(9)

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

(

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

(

h

2

;

k

2

)

(

z

2

(

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)

(2)

(

z

σ

(1)

σ

(2)

)

h

σ

(3)

h

σ

(1)

h

σ

(2)

.

(10)

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

(11)

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’

(12)

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’

(13)

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’

(14)

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

(15)

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

(16)

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

(17)

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

(18)

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

(19)

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

(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. 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

(21)

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

(22)

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

(23)

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

(24)

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

(25)

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

(26)

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

(27)

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

(28)

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

(29)

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

(30)

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)

(31)

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)

(32)

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

.

(33)

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

(34)

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

(35)

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

(36)

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

(37)

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

(38)

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

;

(39)

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

;

(40)

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

.

(41)

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

.

(42)

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

+

. . .

(43)

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

(44)

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<

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