More differential equations for correlators

The results of the previous sections can be extended to compute more general correlators. In this section, we relax the condition for the fields in the correlator to be bare twist-fields. We first generalise the previous analysis to the case where only one of the fields is the bare twist field. We then discuss how the computation can be adapted if the null vector occurs at level 2. Finally, we study multi-cycle and higher-point correlators of bare twist fields.

One twist field

As we already noticed in Section 9.2, we only made use of the null vector L−1/w3σg₃ =0. It follows that eq. (9.15) can be easily generalised to

α₃∂_u+ (h₁γ₁+h₂γ₂+h⁰₃γ₃−h₄γ₄)
×

× O_g1(0)O_g2(1)σg₃(u)O_g4(_∞)=0 . (9.80) where O_gj for j = 1, 2, 4 is a primary field of conformal weight h_j in the w_j-twisted sector.

The functions f_k(z)can be chosen as in eqs. (9.16) and (9.17). We find the differential equation

"

∂_x+h₁∂_xa₁

a₁ +h₂∂_xa₂ a₂ + ^c

24

w²₃−1 w₃

∂_xa₃ a₃

−h₄∂_xa₄ a₄ − ^c

12

 w₃²−1 w₃

 b₃ a₃

#

×

×O_g1(0)O_g2(1)σg₃(u)O_g4(_∞)=0 . (9.81)

Making use of identity (F.3), this becomes

"

∂_x+^_h1− ^c

24(_w1−1)^∂xa1

a₁ +^_h2− ^c

24(_w2−1)^∂xa2 a₂ + ^c

24

 w3−1 w₃

∂xa3

a₃ −^h₄− ^c

24(w₄−1)^∂xa4 a₄ + ^c

12

∑

i

∂xC_i C_i

#

× O_g1(₀)O_g2(₁)σ_g3(u)O_g4(_∞)=_{0 , (9.82)} and taking into account holomorphic and antiholomorphic contributions we obtain

O_g1(0)O_g2(1)σ_g3(u)O_g4(_∞)=const.|a₁|^{−2h1+12c(w1−1)}

× |a₂|^{−2h2+12c(w2−1)}|a₃|^{−12c w3}

−1

w3 |a₄|^{2h4−12c(w4−1)}

∏

i

|C_i|

!−^c₆

, (9.83)

where the integration constant is left undetermined. The normalisation of (9.83) depends now also on the specific seed theory we are taking the orbifold of.

A twisted BPZ equation

In the derivation of eq. (9.83), as well as in the one of (9.23), our method relied on the existence of the null vector L_−1/wσ_gw. One can wonder if it is possible to derive differential equations for correlators by exploiting the existence of higher level null vectors. We find that this is actually possible.

To illustrate this, we consider the case where the seed theoryM has only Virasoro symmetry. We will show how to derive and solve a differential equation for the correlator

O_g1(0)O_g2(1)µ_g3(u)O_g4(_∞) , (9.84) where µ_g3 has a null vector at level 2. We shall adopt a convenient way to parametrise the conformal weight as follows,

h_j =h⁰_j + ^∆j

w_j . (9.85)

This is motivated by the fact that ∆j can be interpreted as the conformal weight in the covering space in the covering space method (see (4.72)). More-over, as is customary for a Virasoro symmetry, we parametrise the central charge as c=1+6(b+b⁻¹)². The different fields are assumed to be defined in the twisted sectors w_j and we leave h_j (or equivalently∆j) for j6=3 generic.

The ‘covering space conformal weight’ of µ_g3(z)is fixed by the existence of the null vector at level 2 and reads

∆3= −^{ 1} 2+³

4b⁻²



. (9.86)

For this value of the conformal weight, the field



is null and hence decouples from correlation functions

 We are going to show how a differential equation for the correlator (9.84) can be deduced from this null-vector constraint.

Let us start by noticing that repeating the same arguments made in Section 9.2 to deduce eq. (9.8), we find

I where the term proportional to β₃ does not vanish in this case. Similarly, from

where the term proportional to δ₃ is due to the contribution of

Putting together eqs. (9.90) and (9.92) we find

β²₃ In order to express in terms of differential operators the term containing L₋2

w3

in eq. (9.88), we use a relation similar to (9.89), but with different functions f_k, which we denote by ˜f_k(z). They obey the following properties:

1. Close to the insertion points z_j, for all k ∈g_j

˜f_k(e^2πiz+z_j) = ˜f_gj(k)(z+z_j), j=1, . . . , 4 . (9.95) This ensures single-valuedness of the integrand.

2. ˜f_k(z)has no poles except at infinity, where the behaviour is specified through item 3.

where the w_r values of k correspond to the different branches.

Thus, the only difference compared to f_k is the existence of the coefficient ˜η₃, Combining (9.88), (9.90), (9.94) and (9.98) we find

"

Constructing the functions ˜f_k

The functions f_k(z) can once more be chosen according to eqs. (9.16) and (9.17). Regarding the functions ˜f_k(z), item 1 is satisfied by the ansatz

˜f_k = ˜h◦_Γ⁻_k¹ , (9.100) where ˜h is a meromorphic function on the covering space. One can show that items 2 and 3 can be satisfied by setting

˜h(t) = ^t(t−₁)

t−x ∂_tΓ(t). (9.101)

The coefficients entering (9.99) can be expressed in terms of the covering map

For the coefficients entering the expansion of f_k(z), in addition to eq. (9.18), we have

In order to simplify eq. (9.99), it is convenient to make the ansatz O_g1(0)O_g2(1)µ_g3(u)O_g4(_∞)= |a₁|^{−2h1+12c(w1−1)}|a₂|^{−2h2+12c(w2−1)} and solve for M(x, ¯x). Here, we stripped off the conformal factors that are expected from the analysis of [113, 211] and M(x, ¯x)is the correlation function in the covering space. Compare this to eq. (9.83). Making use of the identities (F.3), (F.9) and of eqs. (9.18), (9.102), (9.103), the differential equation (9.99) collapses to the BPZ [76] equation for M(x, ¯x),

As described in Chapter 4, this differential equation can be solved in terms of hypergeometric functions. See also e.g. [201] for a detailed discussion. This result hence makes equation (9.2) precise, since the final correlator indeed has a universal prefactor given by covering map data, which we factored out in (9.104), times the correlator evaluated on the covering space.

Higher-point and multi-cycle correlation functions

In this section we will show that the method described in Section 9.2 to compute four-point functions of twist operators can be directly extended to higher-point functions. We find that the answer has exactly the same structure as in the four-point function case. In particular, as already alluded to in Section 4.3, the coincidence limit of higher-point functions also yields correlators of multi-cycle twist fields. This is because e.g. for z₁→z₂,

σ_{(1···w1)(w1+1···w1+w2)}(z₂) ∼σ_(1···w1)(z₁)σ_{(w1+1···w1+w2)}(z₂), (9.106) see eq. (4.55). As a consequence the answer for the sphere contribution to the twist correlators is completely universal and is valid for arbitrary higher-point functions and for multi-cycle twist fields.

For simplicity, let us illustrate our method by deriving differential equations for the five-point function

G(u₁, u2) =σg₁(0)σg₂(1)σg₃(u₁)σg₄(u2)σg₅(_∞) . (9.107) The derivation for higher-point functions will be analogous. Note that the correlation function (9.107) depends now on two cross-ratios, u₁ and u₂. The arguments of Section 9.2 can be repeated and an essentially identical derivation implies in the present context

α₃^a∂_u1+α^a₄∂_u2+h⁰₁γ₁^a+h⁰₂γ^a₂+h⁰₃γ^a₃+h⁰₄γ₄^a−h⁰₅γ^a₅
G(u₁, u2) =0 , (9.108) where α^a₃, α₄^a and γ₁^a, . . . , γ₅^a enter the expansion of a function f_k^a(z)satisfying the following properties:

1. Close to the insertion points z_j, for k=1, . . . , n f_k^a(e^2πiz+z_j) = f_g^a

j(k)(z+z_j), j=1, . . . , 5 . (9.109) 2. f_k^a(z)has no poles except at infinity, where the behaviour is specified

through item 3.

3. Close to the insertion points z_j, for k=1, . . . , n, the functions f_k^a have because there are now two independent functions obeying the requirements in items 1–3, namely

f_k¹ =∂_x1Γ◦_Γ⁻_k¹ , f_k² =∂_x2Γ◦_Γ⁻_k¹, (9.111) where (4.91) is now replaced by

Γ(t) ∼0+a₁t^w1+b₁t^w1+1+ O(t^w1+2) t ∼0 , (9.112a) 2. By rewriting the parameters entering eq. (9.110) in terms of covering map data and making use of identity (F.6), the differential equations in (9.108) become for a=1, 2,

contribution from left and right movers, we find

The result is analogous for an m-point function of bare twist operators, σ_g1(0, 0)σ_g2(1, 1)σ_g3(u₁, ¯u₁). . . σ_gm−1(u_m−3, ¯u_m−3)σ_gm(_{∞, ∞})

The constant Cw₁,...,wm can again be fixed by requiring factorisation and one can show that

In this section, we discuss how our method can be applied if the covering surface has higher genus.

Parameter counting

Let us begin by analysing more structurally the constraints on the functions f_k, which played a crucial role in evaluating the null-vector constraint in Section 9.2. We discuss for simplicity the four-point function case. The analysis for higher-point functions is very similar.

Deriving differential equations for correlators started from writing⁸ I where the contour C encircles all insertion points. The fieldsO_gi are primary twisted fields, twisted by the group element g_i. We chose the functions f_k(z) such that the resulting correlator becomes single-valued. This is achieved by setting

f_k =h◦_Γ⁻_k¹ , (9.118)

where k is the k-th branch of the inverse of the covering map. h is a mero-morphic function on the covering space. We imposed that h can have at most double poles at the poles ofΓ(t), since this leads to an asymptotic growth

8We keep the insertion points general, since the following discussion is somewhat easier if no field is at∞. The result is the same in either case.

In document Research Collection. Different perspectives on AdS3/CFT2. Doctoral Thesis. ETH Library. Author(s): Dei, Andrea. Publication Date: 2020 (Page 190-198)