Recursive representation of the torus 1-point conformal block

arXiv:0911.2353v2 [hep-th] 22 Nov 2009

Recursive representation of the torus 1-point conformal block

Leszek Hadasz†1 , Zbigniew Jask´olski‡2 and Paulina Suchanek‡3 †

M. Smoluchowski Institute of Physics, Jagiellonian University Reymonta 4, 30-059 Krak´ow, Poland, ‡

Institute of Theoretical Physics, University of Wroclaw pl. M. Borna, 50-204 Wroclaw, Poland.

Abstract The recursive relation for the 1-point conformal block on a torus is derived and used to prove the identities between conformal blocks recently conjectured by Poghossian in [1]. As an illustration of the efficiency of the recurrence method the modular invariance of the 1-point Liouville correlation function is numerically analyzed.

PACS: 11.25.Hf, 11.30.Pb 1

e-mail : [email protected] e-mail : [email protected] 3 e-mail : [email protected] 2

1

1

Introduction

An interesting duality between certain class of N = 2 superconformal field theories in four dimensions and the two-dimensional Liouville field theory has been recently discovered by Alday, Gaiotto and Tachikawa [2]. An essential part of this relation actively studied in a number of papers [3–11] is an exact correspondence between instanton parts of Nekrasov partition functions in N=2 SCFT and conformal blocks of the 2-dimensional CFT. This in particular concerns two basic objects of CFT: the 4-point conformal block on a sphere and the 1-point conformal block on a torus. These special cases has been recently analyzed by Poghossian [1] who applied the recursive relation for the 4-point blocks on a sphere discovered long time ago by Alexei Zamolodchikov in CFT [12–14] to the instanton part of the Nekrasov partition function with four fundamentals. He also proposed and verified to certain order a recursive relation for the Nekrasov function with one adjoint hypermultiplet. It was observed that by the AGT duality this yields previously unknown recursive relation for conformal blocks in 2-dimensional CFT. The aim of the present paper is to provide a complete derivation of the recursive relation for 1-point conformal blocks on a torus conjectured by Poghossian [1]. Our method is based on the algebraic properties of the 3-point conformal blocks and the analytic structure of the inverse of the Gram matrices in Verma modules [15]. It parallels to large extend Zamolodchikov’s original derivation for the 4-point blocks [12–14]. We set our notation in Section 2 and present the derivation in Section 3. As an application of the recurrence relations we prove two important identities between conformal blocks conjectured by Poghossian [1]. The first one relates the 1-point block on a torus with certain 4-point elliptic blocks on a sphere. The second relates 4-point elliptic blocks for different values of central charges, external weights and the elliptic variable q. It had been motivated by the intriguing relation between the Liouville 1-point correlation function on the torus and the Liouville 4-point correlation function on the sphere recently proposed by Fateev, Litvinov, Neveu and Onofri [16]. The derivation of both relations is presented in Section 4. They in particular imply4 : 1 λ  √
2b′ 2 2 λ ′ ′ (q) , q = H Hc,∆ 1 c ,∆α′ 1 α 2b′

2b′

b′ =

√b 2

, α′ =

√

2α ,

(1)

√ 2α .

(2)

and

λ q Hc,∆ α 4


2

= Hc′ ,∆′α′

"

b′ 2 b′ 2

λ √ 2 b′ 2

#

See the next section for our notation conventions.

2

(q) ,

b′ =

√

2b , α′ =

These identities, which can be seen as chiral versions of the identity for the Liouville correlation functions proposed in [16], constitute the most interesting results of the present work. They provide for instance a new tool for analyzing virtually untouched problem of modular invariance of nontrivial torus 1-point functions in CFT. We hope to report on some of they consequences in the forthcoming paper. Beside theoretical applications the recursive relation for 1-point block on a torus gives a very efficient numerical method of analyzing the modular bootstrap in CFT. As an illustration of this method we present in Section 5 some numerical checks of modular bootstrap in the Liouville theory.

2

1-point functions

Consider a primary field φλ,λ¯ with the conformal weights (∆λ , ∆λ¯ ) for which the following parametrization is assumed: ∆λ =


1 Q 2 − λ2 , 4

c = 1 + 6Q2 , Q = b + b−1 .

In terms of the CFT on the complex plane the 1-point correlation function of φλ,λ¯ on a torus takes the form:     c ˆ ˆ ¯ hφλ,λ¯ i = Tr e−(Imτ )H+i(Reτ )P φλ (1, 1) = (q q¯)− 24 Tr q L0 q¯L0 φλ (1, 1) , where τ is the torus modular parameter and ˆ = 2π(L0 + L ¯ 0 ) − πc , H 6

¯ 0 ), Pˆ = 2π(L0 − L

q = e2πiτ .

The trace can be calculated in the standard bases of Verma modules: ν∆,M = L−M ν∆ ≡ L−mj . . . L−m1 ν∆ , where M = {m1 , m2 , . . . , mj } ⊂ N stands for an arbitrary ordered set of indices mj ≤ . . . ≤ m2 ≤ m1 , and ν∆ ∈ V∆ is the highest weight state. This yields: c

hφλ,λ¯ i = (q q¯)− 24 ×

X

∞ X X

¯

q ∆+n q¯∆+n

(3)

¯ n=0 (∆,∆)

n=|M |=|N | ¯ ¯ n=|M|=| N|

iM N h h iM¯ N¯

n ¯n ¯ φ ¯ (1, 1)|ν∆,N ⊗ ν¯ ¯ ¯ i B Bc,∆ ν ⊗ ν ¯ ¯ ¯ ∆,M ∆, M ∆,N λ,λ c,∆

3

¯ from the spectrum of the theory is assumed, |M | = where the sum over all weights (∆, ∆) 

n m1 + . . . + mj and Bc,∆

at the level n.



MN

n Bc,∆



is the inverse of the Gram matrix MN

= hν∆,N |ν∆,M i ,

|M | = |N | = n,

The 3-point correlation function in (3) can be expressed as a product of the 3-point (“left” and “right”) conformal blocks and a structure constant:

¯ λ,λ ν∆,M ⊗ ν¯∆, ¯∆, ¯ M ¯ ) C∆,∆ ¯ , ν∆, ¯ N ¯ , νλ ¯ M ¯ φλ,λ ¯ N ¯ i = ρ(ν∆,N , νλ , ν∆,M ) ρ(ν∆, ¯ (1, 1)|ν∆,N ⊗ ν ¯, ¯

λ,λ C∆, ¯ i. ¯ | φλ,λ ¯ (1, 1)|ν∆ ⊗ ν∆ ¯ = hν∆ ⊗ ν∆ ∆

Introducing the 1-point conformal block: c

λ Fc,∆ (q) = q ∆− 24 λ,n Fc,∆

=

X

∞ X

λ,n , q n Fc,∆

n=0

ρ(ν∆,N , νλ , ν∆,M )

n=|M |=|N |



M N n Bc,∆

(4) ,

one can write the 1-point correlation function on the torus in the following form: hφλ,λ¯ i =

3

X

¯ (∆,∆)

¯

λ,λ λ Fc,∆ (q) Fc,λ∆ q ) C∆, ¯ (¯ ¯ . ∆

Recursive relations

For arbitrary vectors ξi ∈ V∆i the 3-point block ρ(ξ3 , ξ2 , ξ1 ) is a polynomial function of the weights ∆i , completely determined by the conformal Ward identities [15] and the normalλ,n ization condition ρ(ν3 , ν2 , ν1 ) = 1. It follows from (4) that the block coefficients Fc,∆ are

polynomials in the external weight ∆λ and rational functions of the central charge c and the intermediate weight ∆ with the locations of poles determined by the zeroes of the determinant i h n . of the Gram matrix Bc,∆ M,N

In the generic case the Verma module V∆rs +rs is not reducible and the only singularities iM,N h n as a function of ∆ are simple poles at of Bc,∆ ∆rs (c) =

s 2 Q2 1  rb + − 4 4 b

where r, s ∈ Z, r ≥ 1, s ≥ 1, 1 ≤ rs ≤ n.

4

As a function of c the inverse Gram matrix at the level n has simple poles at the locations 2  1 , crs (∆) = 1 + 6 brs (∆) + brs (∆)   q 1 2 2 2 brs (∆) = rs − 1 + 2 ∆ + (r − s) + 4 (r s − 1) ∆ + 4 ∆ . 1 − r2 where r, s ∈ Z, r ≥ 2, s ≥ 1, 1 ≤ rs ≤ n.

λ,n The block’s coefficient Fc,∆ can be expressed either as a sum over the poles in the inter-

mediate weight: X

λ,n = hλ,n Fc,∆ c +

1≤rs≤n

Rλ,n c, rs , ∆ − ∆rs (c)

(5)

or in the central charge: λ,n Fc,∆

=

λ,n f∆

X

+

1 rs. Working in this basis one gets [15]: λ,n Rλ,n c, rs = lim (∆ − ∆rs (c)) Fc,∆ = Ars (c) ∆→∆rs

X

iM N h n−rs ρ(L−N χrs , νλ , L−M χrs ) Bc,∆ , +rs rs

n−rs=|M |=|N |

where Ars (c) =

 !−1 ∆ χ∆ rs |χrs . ∆ − ∆rs (c)

lim

∆→∆rs

5

(7)

It is convenient to normalize the singular vector χrs such that the coefficient in front of (L−1 )rs equals 1. For this normalization the exact form of the coefficient Ars (c) was first proposed by Al. Zamolodchikov in [12] and then justified in [17]. It reads: Ars (c) =

r Y

1 2

p=1−r

s  Y q −1 . pb + b q=1−s

(p,q)6=(0,0),(r,s)

Using the factorization formula: ρ(L−N χrs , νλ , L−M χrs ) = ρ(L−N ν∆rs +rs , νλ , L−M ν∆rs +rs ) ρ(χrs , νλ , χrs ) one can show that the residue is proportional to the lower order block coefficient: λ,n−rs Rλ,n c, rs = Ars (c) ρ(χrs , νλ , χrs ) Fc,∆rs +rs .

For the normalized singular vector χrs one gets [15]: ρ(χrs , νλ , χrs ) = ρ(χrs , νλ , ν∆rs +rs ) ρ(ν∆rs , νλ , χrs ) = Pcrs

h

∆λ ∆rs +rs

i

Pcrs

h

∆λ ∆rs

i

where the fusion polynomials are defined by: h

2 Pcrs ∆ ∆1

i

r−1 Y

=

s−1 Y

p=1−r

q=1−s



λ2 + λ1 + pb + qb−1 2



λ2 − λ1 + pb + qb−1 2



p+r=1 mod 2 q+s=1 mod 2

and ∆i = h

1 4


Q2 − λ2i . In the case under consideration:

λ Pcrs ∆rs∆+rs

i

h

λ Pcrs ∆ ∆rs

i

2r−1 Y k=1

2s−1 Y

k=1 mod 2

l=1 mod 2

=

×





l=1

λ − kb − lb−1 2



λ + kb + lb−1 2

λ + kb − lb−1 2





λ − kb + lb−1 2

(8)



.

The last step in our derivation is to find the non-singular terms in the equations (5), (6). Let λ,n us start with the expansion (6). Since f∆ does not depend on the central charge it can be

calculated from the c → ∞ limit: ∞ X

n=0

c

λ,n λ (q). = lim q −∆+ 24 Fc,∆ q n f∆ c→∞

iM,N h n . Note that the block’s coefficients (4) depend on c only via the inverse Gram matrix Bc,∆ Analyzing the polynomial dependence of the Gram matrix minors on c one can show [15] that 6

the only element of the inverse Gram matrix which does not vanish in the limit c → ∞ is the diagonal one corresponding to the state Ln−1 |ν∆ i: i1I 1I h 1 1 n , = = lim Bc,∆ n n c→∞ hν∆ | L1 L−1 |ν∆ i n!(2∆)n

where (a)n =

Γ(a+n) Γ(a)

is the Pochhammer symbol. The 3-point block for this state is given by:

ρ(Ln−1 ν∆ , νλ , Ln−1 ν∆ )

=

n X k=0

1 (n − k)!



n! k!

2

Γ(2∆ + n) Γ(∆λ + k) . Γ(2∆ + k) Γ(∆λ − k)

Thus the c → ∞ limit of the 1-point block reads: c

λ lim q −∆+ 24 Fc,∆ (q) =

c→∞

which yields:

∞ X

λ,n q n f∆

n=0

∞ X

n=0

qn

n X k=0

Γ(2∆) Γ(∆λ + k) n! , 2 (n − k)!(k!) Γ(2∆ + k) Γ(∆λ − k)

  1 q = . 2 F1 ∆λ , 1 − ∆λ ; 2∆; 1−q q−1

The ∆ → ∞ asymptotics of the block is even easier to obtain and leads to a more convenient recursion. With the help of the Ward identities one easily shows that for ∆ → ∞ :
ρ(ν∆,N , νλ , ν∆,M ) = ρ(ν∆,N , νQ , ν∆,M ) 1 + O(∆−1 )

where νQ is the highest weight state in the vacuum Verma module (∆Q = 0). Since (for |M | = |N | = n):

 n  ρ(ν∆,N , νQ , ν∆,M ) = Bc,∆ NM

we get:

λ,n hλ,n = lim Fc,∆ = c ∆→∞

X

|M |=|N |=n



n Bc,∆



NM



n Bc,∆

M N

=

X

N δN = p(n)

|N |=n

where p(n) denotes the number of ways n can be written as a sum of positive integers. Thus lim

∆→∞

∞  c  Y λ (1 − q n )−1 . q 24 −∆ Fc,∆ (q) = n=1

This suggest the following definition of the elliptic 1-point block on a torus: λ Hc,∆ (q) = q

c−1 −∆ 24

λ η(q) Fc,∆ (q) =

∞ X

λ,n q n Hc,∆ ,

n=0

λ,n where η(q) is the Dedekind eta function. One easily checks that the coefficients Hc,∆ have λ,n essentially the same pole structure as the coefficients Fc,∆ and the following recursive formula

holds: λ,n Hc,∆ = δ0n +

X Ars (c) Pcrs

1≤rs≤n

h

∆λ ∆rs +rs

i

Pcrs

∆ − ∆rs (c)

7

h

∆λ ∆rs

i

λ,n−rs Hc,∆ . rs +rs

(9)

Let us note that the form of the regular terms in (9) can be also deduced from the limiting case ∆ → 0 (λ → Q) where the fusion polynomials, and therefore all the residua, vanish and the 1-point block become the Virasoro character.

4

Poghossian identities

Using (8) one can check the identity: h

λ Pcrs ∆rs∆+rs

i

h

λ Pcrs ∆ ∆rs

i

r−1 Y

=

s−1 Y

p=1−r

q=1−s



λ Q + pb + qb−1 + 2 2



p+r=1 mod 2 q+s=1 mod 2

×



Q λ + pb + qb−1 − 2 2



b 1 λ + pb + qb−1 + − 2 2 2b



b 1 λ + pb + qb−1 − + 2 2 2b

The r.h.s. can be identified as a product of fusion polynomials: h i h i h i i h rs ∆3 rs rs ∆2 λ λ P = (16) P Pcrs ∆ Pcrs ∆rs∆+rs c ∆4 c ∆1 ∆rs



.

(10)

provided that one of the equalities:

(λ1 , λ2 , λ3 , λ4 ) = (λ1 , λ2 , λ3 , λ4 ) = (λ1 , λ2 , λ3 , λ4 ) =







Q λ λ b 1 , , , − 2 2 2 2 2b



,

1 λ b λ b 1 , + , − , 2b 2 2 2 2 2b b λ 1 λ 1 b , + , − , 2 2 2b 2 2b 2





(11)

, ,

hold. Let us now recall that in the case of the 4-point elliptic block on the sphere Hc,∆

h

i

∆3 ∆2 ∆4 ∆1

(q) = 1 +

∞ X

n (16q)n Hc,∆

n=1

h

∆3 ∆2 ∆4 ∆1

i

the recursive relation takes the form [13, 14]: n Hc,∆

h

∆3 ∆2 ∆4 ∆1

i

= δ0n +

X Ars (c) Pcrs

1≤rs≤n

h

∆2 ∆1

i

Pcrs

∆ − ∆rs (c)

h

∆3 ∆4

i

n−rs Hc,∆ rs +rs

h

∆3 ∆2 ∆4 ∆1

i

.

(12)

Since the solutions to the recursive formulae (9) and (12) are unique we get by comparing (9), (10) and (12) that λ,n n Hc,∆ = 16n Hc,∆

and therefore λ Hc,∆ (q)

= Hc,∆



1 λ 2

1 λ 2

1 (b− 1b ) 12 (b+ 1b ) 2



(q) = Hc,∆

1 2

8

h

(λ−b) 1 2b

∆3 ∆2 ∆4 ∆1

i

1 (λ+b) 2 1 2b



(q) = Hc,∆

1

2

(λ− 1b ) 1 b 2

1 (λ+ 1b ) 2 1 b 2

 (q).

(13)

We have thus obtained a simple proof of the first relation proposed in [1]. In the present notation the second relation conjectured in [1] reads:  1   1 √ µ √1 µ µ 2 2 2b ′ ′ Hc,∆ 1 1 (q) = Hc ,∆ 1 (b′ − 1 ) 1 (b′ + 1 ) (q 2 ) 2b 2b 2 2 b′ b′ 1  1 1 ′ √ µ− b √ µ+ 1 b′ 2 2 2 2 = Hc′ ,∆′ (q 2 ) 1 1 2b′ 2b′  1 √ µ− 1 ′ √1 µ+ 1 ′ 2b 2b 2 2 (q 2 ). = Hc′ ,∆′ 1 ′ 1 ′ b b 2

(14)

2

where: ′

c

′

b

 (b′ + b1′ )2 (λ′ )2 1 2 ∆′λ′ = − , = 1+6 b + ′ , b 4 4 √ λ 2b , λ′ = √ . = 2 

′

There is also another relation of a similar origin with b′ = Hc,∆

b

2 b 2

µ b 2



(q) = Hc′ ,∆′ = Hc′ ,∆′ = H

c′ ,∆′







(15)

√b : 2

 (q 2 )  √1 µ+ 1 b′ 2 2 (q 2 ) 1 2b′  √1 µ+ 1 ′ 2b 2 (q 2 ). 1 ′ b

(16)

i

(17)

√1 µ √1 µ 2 2 1 ′ 1 ′ 1 ) (b − (b + b1′ ) ′ 2 2 b √1 µ− 1 b′ 2 2 1 2b′ √1 µ− 1 ′ 2b 2 1 ′ b 2

2

We shall show that relations of the form: Hc,∆ are to large extent unique.

η µ  ηη

(q) = Hc′ ,∆′

h

λ′3 λ′2 λ′4 λ′1

(q 2 )

Let us first observe that the residua of the coefficients of

Hc,∆ [ηη µη ] (q) contain the fusion polynomial  

Pcrs ηη

=

r−1 Y

s−1 Y

p=1−r

q=1−s



2η + pb + qb−1 2



pb + qb−1 2



p+r=1 mod 2 q+s=1 mod 2

which always vanishes if both r and s are odd. Moreover if η = 1 [η µ ] = 0, it follows that for η = all s.5 Since Hc,∆ ηη

1 2b

1 2b

it vanishes for odd r and

all the odd coefficients of Hc,∆ [ηη µη ] (q)

vanish and the even ones satisfy the recursive relation: 2m Hc,∆

5

The case η =

b 2

η µ  ηη

= δ0n +

X

k,s∈N 1≤ks≤m

Rc2k,s [ηη µη ] 2m−2ks η µ  . H ∆ − ∆2k,s c,∆2k,s +2ks η η

leading to the relation (16) can be analyzed in a similar way.

9

This is to be compared with the recursive relation: h i k,s λ′3 λ′2 h ′ ′i h ′ ′i R X ′ ′ ′ c λ4 λ1 λ3 λ2 λ λ m−ks H Hbm′ ,∆′ λ3′ λ2′ = δ0n + . ′ ′ +ks λ′4 λ′1 b ,∆ ′ ′ 4 1 k,s ∆ − ∆rs k,s∈N

1≤ks≤m

The numbers of terms in both relations coincide. Moreover for the identification of parameters (15):

 s 2 λ2 1 √ = 2(∆′ − ∆′k,s), (∆ − ∆2k,s ) = 2 ( 2kb + √ ) − 4 8 2b

(18)

∆2ks + 2ks = ∆′ks + ks.

(19)



and:

The fusion polynomials can be expressed in the form: Pc2k,s

1 2b 1 2b

Pc2k,s

  µ

1 2b

2k−1 Y

−ks

= 4

s−1 Y

q=1−s

p=1−2k



pb + (q + 1)b−1 2



pb + qb−1 2



p+2k=1 mod 2 q+s=1 mod 2

×

s−1 Y

k−1 Y

q=1−s

p=1−kr



1 q b′ µ √ + ′ − + pb′ + ′ 2 b 2 2b



µ 1 q b′ √ − ′ − + pb′ + ′ 2 b 2 2b



p+k=1 mod 2 q+s=1 mod 2



1 b′ q µ √ + ′ + + pb′ + ′ 2 b 2 2b



µ 1 b′ q √ − ′ + + pb′ + ′ 2 b 2 2b

! .

As in the case of our previous derivation the last two lines can be identified as the fusion polynomials: Pc2k,s

1 2b 1 2b

Pc2k,s

  µ

1 2b

−ks

= 4

2k−1 Y

s−1 Y

q=1−s

p=1−2k



pb + (q + 1)b−1 2



pb + qb−1 2



p+2k=1 mod 2 q+s=1 mod 2 ks

× (16)

h i

λ′2 Pcks ′ λ′1

h i

(20)

λ′3 Pcks ′ λ′4

where one of the following choices is assumed:   µ b′ 1 µ b′ 1 ′ ′ ′ ′ √ , + ′, √ , − ′ , (λ1 , λ2 , λ3 , λ4 ) = 2b 2b 2 2 2 2  
b′ 1 µ b′ 1 µ ′ ′ ′ ′ √ + , ′, √ − , ′ , λ1 , λ2 , λ3 , λ4 = 2 2b 2 2b 2 2  
µ 1 b′ µ 1 b′ ′ ′ ′ ′ √ + ′, , √ − ′, λ1 , λ2 , λ3 , λ4 = . 2 2b 2 2 2b 2 10

(21)

Finally calculating the coefficients Ars one gets: A2k,s (c) = 21−2ks Aks (c′ )

2k−1 Y

s−1 Y

p=1−2k

pb + (q + 1)b−1

q=1−s


−1

pb + qb−1


−1

,

(22)

p+2k=1 mod 2 q+s=1 mod 2

where: 1 Aks (c ) = 2 ′

s Y

k Y

(pb′ +

p=1−k q=1−s

q −1 ) . b′

(p,q)6=(0,0),(k,s)

Taking into account formulae (18), (19), (20) and (22) one obtains: 1  h i 2k,s 2b µ k,s λ′3 λ′2 Rc 1 1 R ′ ′ ′ c λ λ 2b 2b = 16−ks ′ 4 ′ 1 . ∆ − ∆rs (∆ − ∆rs ) h ′ ′i λ3 λ2 2m [η µ ] and (16)−m H m with the weights (21) satisfy the Hence the coefficients Hc,∆ ′ ′ ηη b ,∆ λ′ λ′ 4

1

same recursive relations. This completes our proof of the formula (14). Formula (16) can be derived along the same lines starting with η = 2b .

5

Modular bootstrap in Liouville theory

The modular invariance of the 1-point function on the torus: hφλ,λ¯ i− 1 = (−1)∆λ −∆λ¯ τ ∆λ τ¯∆λ¯ hφλ,λ¯ iτ

(23)

τ

along with the crossing invariance of the 4-point function on the sphere form the basic consistency conditions for any CFT on closed surfaces [18]. In the case of the Liouville theory the 1-point function can be expressed in terms of the elliptic blocks as follows: 2 Z dα ∆α − Q2 −1 λ Cλ , 4 q η(q) H (q) hφλ iτ = c,∆ α ∆α 2i iR+

where q = e2πiτ . Condition (23) then takes the form: Z

iR+

2 2 Z dα − α2 λ dα − α2 λ 2∆ +1 λ λ λ q˜ 4 Hc,∆α (˜ q 4 Hc,∆α (q) C∆ q ) C∆α = |τ | α 2i 2i iR

1

where and q˜ = e−2πi τ . The Liouville structure constant reads [19]: λ C∆ α

h

2

= πµγ(b )b

2−2b2

i− Q+λ 2b

×

Υ2



Q 2

Υ0 Υ(Q + λ)Υ(α)Υ(−α)     λ + + + α Υ Q Υ Q 2 2 2 +

11

λ 2

λ 2

 . −α

(24)

Separating the α-dependent part i− Q+λ Υ Υ(Q + λ) h 2b 0 λ 2 2−2b2   r(α) × C∆ = 4 πµγ(b )b α λ Υ2 Q + 2 2  2 
(b −1)t ! tλ Z ∞ − cosh 4 2 2 cosh 2 2 2b dt 1 + b − b (λ + 2) −t α e + cosh(αt) r(α) = − exp 2 bt t 4 t 2b sinh( 2 ) sinh( 2b ) 0 one can write (24) as: Z

iR

2 2 Z dα − α2 λ dα − α2 λ 2∆ +1 λ q˜ 4 Hc,∆α (˜ q 4 Hc,∆α (q) r(α) . q ) r(α) = |τ | 2i 2i

(25)

iR

This relation can be numerically analyzed with the help of the recursion relations derived in Section 3. Due to the rapidly oscillating integrand in (25) the numerical calculation of function r(t) has to be carefully done. We present a sample of the calculations for c = 2, λ = i and for the modular parameter τ along the imaginary axis in the range [0.2i, 5i]. The results for the elliptic block expanded up to the term q n , n = 4, 5, 6 are presented on Fig.1. where the relative difference of the left and the right side of (25) is plotted. 0.003

n=4 0.002 0.001

n=6 2

3

4

5

n=5 -0.001

-0.002

-0.003

Fig.1 Numerical check of the modular bootstrap.

Acknowledgements This work was supported by the Polish State Research Committee (KBN) grant no. N N202 0859 33. The work of L.H. was also supported by MNII grant 189/6.PRUE/2007/7.

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