Shadow formalism for supersymmetric conformal blocks

Abstract

A bstract
Shadow formalism is a technique in two-dimensional CFT allowing straightforward computation of conformal blocks in the limit of infinitely large central charge. We generalize the construction of shadow operator for superconformal field theories. We demonstrate that shadow formalism yields known expressions for the large-c limit of the four-point superconformal block on a plane and of the one-point superconformal block on a torus. We also explicitly find the two-point global torus superconformal block in the necklace channel and check it against the Casimir differential equation.

Full text

JHEP11(2024)048
Published for SISSA by Springer
Received: August 18, 2024
Accepted: October 8, 2024
Published: November 7, 2024
Shadow formalism for supersymmetric conformal blocks
V. Belavin , J. Ramos Cabezas and B. Runov
Physics Department, Ariel University,
Ariel 40700, territories administered by Israel
E-mail: vladimirbe@ariel.ac.il,juanjose.ramoscab@msmail.ariel.ac.il,
borisru@ariel.ac.il
Abstract: Shadow formalism is a technique in two-dimensional CFT allowing straightforward
computation of conformal blocks in the limit of infinitely large central charge. We generalize
the construction of shadow operator for superconformal field theories. We demonstrate
that shadow formalism yields known expressions for the large-c limit of the four-point
superconformal block on a plane and of the one-point superconformal block on a torus. We
also explicitly find the two-point global torus superconformal block in the necklace channel
and check it against the Casimir differential equation.
Keywords: AdS-CFT Correspondence, Conformal and W Symmetry, Field Theories in
Lower Dimensions, Supersymmetry and Duality
ArXiv ePrint: 2408.07684
Open Access,©The Authors.
Article funded by SCOAP3.https://doi.org/10.1007/JHEP11(2024)048

JHEP11(2024)048
Contents
1 Introduction 1
2sl(2) shadow formalism 2
2.1 Shadow formalism on a sphere 2
2.2 Shadow formalism on a torus 4
3 Superconformal field theory 5
4 Shadow formalism for supersymmetric case 8
4.1 Supersymmetric shadow operator 8
4.2 Identity decomposition 9
5 Supersymmetric conformal blocks 10
6 Torus superconformal blocks 12
6.1 One-point superconformal blocks 13
6.2 Two-point superconformal blocks 14
7 Torus superconformal blocks via shadow formalism 17
7.1 osp(1|2) torus shadow formalism 17
7.2 One-point torus superconformal blocks 17
7.3 Two-point torus superconformal blocks 18
8 Casimir operator for superconformal blocks 21
9 Conclusion and outlook 24
A Calculation of the four-point superconformal block 25
B Integral representation of sl(2) two-point torus conformal block 26
1 Introduction
A conformal field theory (CFT) in two dimensions is greatly simplified by the presence of
extended symmetry w.r.t. local conformal transformations [
1
]. This symmetry allows to
express any correlation function in terms of three-point structure constants and a set of
functions called conformal blocks. The latter depend on conformally invariant cross-ratios of
coordinates, the central charge of the theory and conformal dimensions of the fields involved,
but do not depend on the three-point constants. Knowledge of four-point blocks leads to
a complete solution of the theory via bootstrap equations [
2
]. However, while the series
expansion of conformal block can be computed term by term, the rapid growth of complexity
of the computation makes it impractical beyond the first few terms. Several alternative
approaches, such as recursion representation [
3
,
4
], have been investigated. The full four-point
conformal block is not known for a general CFT with an arbitrary choice of dimensions.
Higher-point conformal blocks, as well as conformal blocks on torus [
5
,
6
] and Riemann
surfaces of higher genus, are also of significant interest.
In the large central charge limit, the Virasoro conformal blocks reduce to so-called global
conformal blocks [
7
]. They are analogous to the conformal blocks in higher dimensions, as only
– 1 –

JHEP11(2024)048
descendants of intermediate field generated by global conformal subalgebra contribute to the
block in this limit. Global conformal blocks have been extensively studied both on a sphere
and on a torus [
8
–
16
]. It turns out that they are relevant in the holographic context [
17
–
20
].
In particular, as it was shown in [
21
–
23
], they compute geodesic Witten diagrams in
AdS3
.
The shadow formalism was originally proposed in [
24
–
26
] to compute conformal blocks of
scalar fields in a CFT in dimension greater than two and was subsequently generalized to
fields with spin in [
27
]. For a 2D CFT it was demonstrated that the shadow formalism can
be used to compute global conformal blocks [
28
,
29
]. It was also successfully applied to CFTs
based on W-algebras [
30
,
31
] and Galilean CFT [
32
].
The key element of the shadow formalism is a shadow operator, which is, in Virasoro
case, a quasi-primary composite field of dimension 1
−h
, which can be constructed for any
conformal primary of dimension
h
. The key property of the shadow operator is that its
two-point function with the corresponding primary field is a two-dimensional delta function.
This property allows (in the large central charge limit) to construct explicitly the projector
from the Hilbert space of the theory onto the highest weight module over subalgebra of global
conformal transformations (
sl
(2) for Virasoro case) in terms of the shadow operator.
Supersymmetric conformal field theories (SCFTs) are a key element of superstring
theory [
33
,
34
], they arise also in the context of AdS/CFT duality [
35
–
42
]. The goal of the
present paper is to generalize the shadow formalism to two-dimensional
N
= 1 superconformal
field theory in the Neveu-Schwarz sector.
The paper is organized as follows. In section 2, we review the shadow formalism for
two-dimensional conformal field theories in the Virasoro case. Section 3contains key facts
about
N
= 1 superconformal field theory in the Neveu-Schwarz sector. In section 4, we
introduce the supersymmetric shadow operator and construct the projector onto Verma
supermodules corresponding to primary superfields. In section 5, we compute the four-point
conformal block on a sphere via shadow formalism. In section 6, we recall the definition
of the torus superconformal blocks. In section 7, using the shadow formalism, we compute
one- and two-point torus superconformal blocks. Our results for the four-point spherical
superconformal block and one-point torus superconformal block are in agreement with known
results [
12
,
43
,
44
] obtained by other methods. The obtained representation for the two-point
torus superconformal block is new. In section 8, we verify that it satisfies the required
differential equations, which follow from the consideration based on the
osp
(1
|
2) Casimir
operator. In section 9, we present our conclusions and comments on further research directions.
2sl(2) shadow formalism
2.1 Shadow formalism on a sphere
Let us consider a conformal field theory on a Riemann sphere.
1
For a primary field
Oh,¯
h
of
dimension (
h, ¯
h
)the corresponding shadow dual field
˜
Oh,¯
h
is defined as [
29
]
˜
Oh,¯
h(z, ¯z) = Zd2wOh,¯
h(w, ¯w)
(z−w)2−2h(¯z−¯w)2−2¯
h.(2.1)
1
Let us recall that the terms “CFT on a plane” and “CFT on a sphere” are often used interchangeably, as
local conformal transformations allow us to map one theory to the other without altering key properties. Here,
we use the term “sphere” to emphasize the distinction between a CFT on a genus-zero surface and a CFT on
a genus-one surface (i.e., torus), which will be discussed below.
– 2 –

JHEP11(2024)048
It is a quasi-primary non-local field of dimension (
h∗,¯
h∗
), related to (
h, ¯
h
)as follows:
h∗= 1 −h , ¯
h∗= 1 −¯
h . (2.2)
It can be demonstrated that, upon appropriate regularization, the two-point function of the
shadow field and the corresponding primary field is equal to a two-dimensional delta function,
while the two-point function with any other primary obviously vanishes.
⟨Oh,¯
h(z, ¯z)˜
Oh,¯
h(w, ¯w)⟩=δ2(z−w).(2.3)
Therefore, the following operator
Πh,¯
h=Zd2wOh,¯
h(w, ¯w)|0⟩⟨0|˜
Oh,¯
h(w, ¯w)(2.4)
is invariant under global conformal transformations and acts as a projector onto irreducible
sl
(2) modules, satisfying
Πh1,¯
h1Πh2,¯
h2=δh1,h2δ¯
h1¯
h2Πh1,¯
h1.(2.5)
Throughout the rest of this section, we will assume that all considered primary fields are
diagonal, i.e., their holomorphic and antiholomorphic conformal dimensions coincide and
omit the antiholomorphic dimensions where it wouldn’t lead to confusion.
A multi-point correlation function of the primary fields can be represented as a sum
of conformal partial waves
⟨ϕh1(z1,¯z1). . . ϕhn(zn,¯zn)⟩=X
∆1,∆2,...,∆n−3
Ψh1,...,hn
∆1,...,∆n−3(z1, . . . , zn|¯z1,...,¯zn).(2.6)
In the limit of large central charge, one can decompose the identity operator as a sum of the
projectors Π
h
corresponding to the primary fields of the theory. Using this decomposition, it
is straightforward to obtain an integral representation for a conformal partial wave [
28
]:
Ψh1,...,hn
∆1,...,∆n−3(z1,...,zn|¯z1,...,¯zn) = Zn−3
Y
i=1
d2wiVh1,h2,∆1(z1,¯z1, z2,¯z2, w1,¯w1)
×
n−4
Y
i=1
V∆∗
i,hi+2,∆i+1 (wi,¯wi, zi+2 ,¯zi+2, wi+1 ,¯wi+1)
×V∆∗
n−3,hn−1,hn(wn−3,¯wn−3,zn−1,¯zn−1, zn,¯zn).
(2.7)
The symbol
Vh1,h2,h3
above stands for the three-point function
Vh1,h2,h3(z1,¯z1, z2,¯z2, z3,¯z3) = Ch1h2h3|vh1,h2,h3(z1, z2, z3)|2
=⟨ϕh1(z1,¯z1)ϕh2(z2,¯z2)ϕh3(z3,¯z3)⟩,(2.8)
and
vh1,h2,h3
above denotes the holomorphic dependence of
Vh1,h2,h3
, namely
vh1,h2,h3(z1, z2, z3) = 1
zh1+h2−h3
12 zh1+h3−h2
13 zh2+h3−h1
23
.(2.9)
– 3 –

JHEP11(2024)048
The conformal partial wave can be factorized into a product of “leg factor” ensuring
invariance w.r.t. global conformal transformations, three-point constants, and the model-
independent conformal block. For example, for the four-point conformal partial wave, we
have (assuming all primary fields involved are diagonal)
Ψh1,...,h4
∆(z1, . . . , z4|¯z1,...,¯z4) = Lh1,...,h4(z1, . . . , z4|¯z1,...,¯z4)Ch1h2∆C∆h3h4|Fh1,...,h4
∆(x)|2
(2.10)
where
L
is the “leg factor”,
F
stands for the four-point conformal block, and
x
is the
conformally invariant cross-ratio.
2.2 Shadow formalism on a torus
A correlation function on the torus is defined as follows
⟨ϕh1(z1,¯z1). . . ϕhn(zn,¯zn)⟩τ=TrHhqL0¯q¯
L0ϕh1(z1,¯z1). . . ϕhn(zn,¯zn)i,(2.11)
where the trace is computed over the Hilbert space of the theory, and the number
q
is related
to the modular parameter of the torus
τ
as follows:
q=e2πiτ .(2.12)
Similar to the correlation functions on the sphere considered above, the correlation functions
on the torus can be decomposed into a sum of torus conformal partial waves:
⟨ϕh1(z1,¯z1). . . ϕhn(zn,¯zn)⟩τ=X
∆1,...,∆n
Yh1,...,hn
∆1,...,∆n(q, ¯q , z1,¯z1, . . . , zn,¯zn),(2.13)
which in turn factorize into products of structure constants and torus conformal blocks, e.g.,
Yh1,h2
∆1,∆2(q, ¯q , z1,¯z1, z2,¯z2) = C∆1h1∆2C∆2h2∆1|Fh1,h2
∆1,∆2(q, z1, z2)|2.(2.14)
As demonstrated in [
29
,
31
], integral representations of global torus conformal blocks can
be obtained by inserting projectors Π
h
, defined by (2.4), between primary fields within the
trace expression of the
n
-point conformal blocks. The insertions of operators Π
h
are analogous
to the insertions of the resolution of identity
PhPsl(2)
h
, where
Psl(2)
h
is the projector onto the
sl
(2) module with the highest weight
h
, within the trace when one is considering global
sl
(2)
n
-point torus conformal blocks in the so-called necklace channel. The main difference is that
for projectors (2.4), the first projector is inserted between the trace and the first primary
field, whereas for the insertions of the resolution of identity, these occur only between the
primary fields. The procedure of inserting projectors (2.4) results in expressing the torus
conformal blocks in terms of torus conformal partial waves. For the one- and two-point torus
conformal partial waves, we have the expressions
2
Yh1
∆1(q, ¯q, z1,¯z1) = q∆1¯q∆1Zd2wv1−∆1,h1,∆1(w, z1, qw)2,(2.15)
Yh1,h2
∆1,∆2(q, ¯q, z1,¯z1, z2,¯z2) =
q∆1¯q∆1Zd2w1d2w2v1−∆1,h1,∆2(w1,z1, w2)2v1−∆2,h2,∆1(w2,z2,qw1)2.(2.16)
2
Since we are interested in the description of global conformal blocks by using (2.15), (2.16), we omit the
structure constants in those equations.
– 4 –

JHEP11(2024)048
Notice that in the above equations, the subscripts 1
−
∆
i
represent the conformal dimensions
of the corresponding shadow field as denoted in (2.2). To extract the holomorphic global
sl
(2) one- and two-point conformal blocks, one does not need to take the full two-dimensional
integrals in (2.15), (2.16). Instead, it is sufficient to work only with the holomorphic part
of the integrands and then take the integral over
wi
over an appropriate domain. For the
global
sl
(2) one-point torus conformal block, the integral representation reads
Fh1
∆1(q) = 1
c1
q∆1Zz1
0
dw1v1−∆1,h1,∆1(w1, z1, qw1),(2.17)
where
c1
is a normalization constant given by
c1(h1,∆1) = Γ (2∆1−h1) Γ (h1) (−1)2∆1(−z1)h1
Γ (2∆1).(2.18)
From the holomorphic part of (2.16), the integral representation for
sl
(2) two-point torus
conformal block reads
Fh1,h2
∆1,∆2(q1, z1, z2) =
=q∆1
c2(h1, h2,∆1,∆2)ZC1
dw1ZC2
dw2v1−∆1,h2,∆2(w1, z1, w2)v1−∆2,h2,∆1(w2, z2, w1q)
= zh2
1zh1
2(1 −q)h1+h2
zh1+h2
12 (z2−qz1)h1+h2!ρ∆1
1ρ∆2
2F4h∆1+∆2−h1,∆2+∆1−h2
2∆1,2∆2|ρ1, ρ2i,(2.19)
where
c2
is also a normalization constant given by (B.9), the integration domains C
1
and
C
2
are defined accordingly
C2:w2∈[w1q, z2],
C1:w1∈[z2, z1].(2.20)
The variables
ρ1, ρ2
are given by
ρ1=q(z12)2
z1z2(1 −q)2, ρ2=(z2−qz1)2
z1z2(1 −q)2,(2.21)
and
F4
is the Appell function defined as
F4[a1,a2
c1,c2|x1, x2] =
∞
X
m1,m2=0
(a1)m1+m2(a2)m1+m2
(c1)m1(c2)m2
xm1
1
m1!
xm2
2
m2!,(2.22)
where, (
ai
)
m
stands for the Pochhammer symbol. For the
osp
(1
|
2) discussion, the inte-
gral (2.19) will pay a key role. Therefore, we will review it in detail in appendix B.
3 Superconformal field theory
In this section we list key facts about
N
= 1 two-dimensional superconformal field theory in
Neveu-Schwartz sector, following review [
45
]. The
N
= 1 super-Virasoro algebra in NS sector
– 5 –

JHEP11(2024)048
is comprised of generators
Lk
and
Gk+1
2
, obeying the following commutation relations [
46
]
[Lm, Ln]=(m−n)Lm+n+δm+n,0
ˆc
8m3−m,(3.1)
[Lm, Gr] = m
2−rGm+r,(3.2)
{Gr, Gs}= 2Lr+s+ˆc
2r2−1
4.(3.3)
It is a central extension of the algebra of generators of local superconformal transformations of
C1|1
superspace. The latter can be parametrized by two real and two Grassmann numbers. It
is natural to introduce holomorphic and antiholomorphic supercoordinates on the superspace
Z= (z, θ),¯
Z= (¯z, ¯
θ)(3.4)
and superderivatives
D=∂θ+θ∂z,¯
D=∂¯
θ+¯
θ∂¯z(3.5)
obeying
D¯
Z=¯
DZ = 0 .(3.6)
A function
f
(
Z, ¯
Z
)on superspace is called superanalytic if it satisfies
¯
Df (Z, ¯
Z)=0.(3.7)
Superanalytic functions admit Taylor-like series expansion
f(Z1) =
∞
X
k=0
Zk
12
k!∂k
2(1 + (θ1−θ2)D2)f(Z2),(3.8)
where the quantity
Z12
(which is the supersymmetric generalization of the difference of
coordinates) depends on
Z1, Z2
as
Z12 =z12 −θ12 (3.9)
and
z12 =z1−z2, θ12 =θ1θ2.(3.10)
Superconformal transformations
Z7→ ˜
Z= (˜z(Z),˜
θ(Z)) (3.11)
are defined as transformations preserving the superderivative:
D=D˜
θ˜
D . (3.12)
One can also define a superdifferential
dZ
transforming as
d˜
Z=D˜
θdZ (3.13)
under superconformal transformations (3.11). The subgroup of global superconformal trans-
formations is isomorphic to
OSp
(1
|
2) and is comprised of linear fractional transformations
of the form
˜z=az +b+αθ
cz +dβθ ,˜
θ=¯αz +¯
β+¯
Aθ
cz +d+βθ ,(3.14)
– 6 –

JHEP11(2024)048
with
¯α=aβ −cα
√ad −bc ,¯
β=bβ −dα
√ad −bc ,¯
A=pad −bc −3αβ . (3.15)
It has five independent parameters, with corresponding generators given by
L±1, L0, G±1
2
.
The fields of the superconformal field theory are operator-valued functions on the superspace.
Since all functions of Grassmann variables are linear, any superfield can be decomposed
into a linear combination of ordinary fields as follows
3
Φh(Z, ¯
Z) = ϕh(z, ¯z) + θψh(z, ¯z) + ¯
θ¯
ψh(z, ¯z) + θ¯
θ˜
ϕh(z, ¯z).(3.16)
A superprimary field of dimensions (
h, ¯
h
)is defined by the requirement that the differential
Φh(Z, ¯
Z)dZ2hd¯
Z2¯
h(3.17)
is invariant under superconformal transformations. Super Virasoro algebra contains ordinary
Virasoro algebra with central charge
c=3ˆc
2(3.18)
as a subalgebra. The components (3.16) of a superprimary field are Virasoro primaries
with respective conformal dimensions (
h, ¯
h
),(
h
+
1
2,¯
h
),(
h, ¯
h
+
1
2
),(
h
+
1
2,¯
h
+
1
2
). As in
the non-supersymmetric case, global superconformal symmetry fixes two- and three-point
functions up to several constants:
⟨Φh1(Z1,¯
Z1)Φh2(Z2,¯
Z2)⟩=δh1,h2δ¯
h1,¯
h2
Z2h1
12 ¯
Z2¯
h1
12
=
=δh1,h2δ¯
h1,¯
h2(z12 + 2h1θ1θ2)(¯z12 + 2¯
h1¯
θ1¯
θ2)
(z12)2h+1 ,(3.19)
Vh1,h2,h3(Z1,¯
Z1;Z2,¯
Z2;Z3,¯
Z3) = ⟨Φh1(Z1,¯
Z1)Φh2(Z2,¯
Z2)Φh3(Z3,¯
Z3)⟩=
=Ch1h2h3+η123 ¯η123 ˜
Ch1h2h3
Zγ123
12 ¯
Z¯γ123
12 Zγ312
13 ¯
Z¯γ312
13 Zγ231
23 ¯
Z¯γ231 ,(3.20)
where we have used shorthand notations (3.9), the numbers
γijk
are defined as
γijk =hi+hj−hk,(3.21)
the symbol
η123
denotes an odd conformally invariant cross-ratio
η123 =θ1Z23 +θ2Z31 +θ3Z12 +θ1θ2θ3
(Z12Z13 Z32)1
2
(3.22)
and
Ch1h2h3
,
˜
Ch1h2h3
are two independent three-point structure constants.
A Verma supermodule associated with a superfield Φ
∆i
will be denoted by
H∆i=V∆i⊗V∆i,(3.23)
3
In the notation of primary superfields (3.16), for simplicity of writing, we omit the dependence on the
antiholomorphic conformal dimension ¯
h.
– 7 –

JHEP11(2024)048
and is spanned by the basis of descendant states
|M, ¯
M , ∆i⟩=Li1
−n1···Lil
−nlGj1
−r1···Gjk
−rkׯ
L¯
i1
−¯n1··· ¯
L¯
il
−¯nl
¯
G¯
j1
−¯r1··· ¯
G¯
jk
−¯rk|∆i⟩,
|M|=n1i1+· ·· +nlil+r1j1+· ·· +rkjk,
|¯
M|= ¯n1¯
i1+· ·· + ¯nl¯
il+ ¯r1¯
j1+· ·· + ¯rk¯
jk,|M|,|¯
M| ∈ N+
2,
(3.24)
where
|∆i⟩= Φ∆i(0,0)|0⟩(3.25)
is the highest weight state, which satisfies
L0|
∆
i⟩
= ∆
i|
∆
i⟩
, and it is annihilated by the
generators
Ln,¯
Ln
for
n >
0.
N+
stands for non negative integers. If one focuses only on
the holomorphic sector
V∆i
of (3.23), then the descendant states of the supermodule
V∆i
can be written as
|M, ∆i⟩=Li1
−n1···Lil
−nlGj1
−r1···Gjk
−rk|∆i⟩.(3.26)
To compute the global conformal blocks it is sufficient to consider only the
osp
(1
|
2) subsector
of the
N
= 1 supersymmetric theory. For this, it will be convenient to write the basis of
states of the
osp
(1
|
2) supermodule
Vosp
∆i
just as
|M, ∆i⟩=Lm
−1Gk
−1/2|∆i⟩, M = (m, k), m ∈N+, k = 0,1.(3.27)
Clearly, the
osp
(1
|
2) supermodule factorizes into two
sl
(2) modules, as follows
Vosp
∆i=Vsl(2)
∆i⊗Vsl(2)
∆i+1
2
,(3.28)
where the
Vsl(2)
∆i
module is spanned by states
|M,
∆
i⟩
with
k
= 0 and the
Vsl(2)
∆i+1
2
with
k
= 1,
according to the notation (3.27). These two sectors correspond to the even and odd parts
of the
osp
(1
|
2) supermodule, respectively.
4 Shadow formalism for supersymmetric case
4.1 Supersymmetric shadow operator
In the spirit of the original shadow formalism, we seek the shadow operator in the form of
an integral over coordinate space of the corresponding primary multiplied by some function
of coordinates, and the projector as
Πh,¯
h=Zd2wZd2ξOh,¯
h(W, ξ)|0⟩⟨0|˜
Oh,¯
h(W, ξ)(4.1)
In order for the nonlocal operator of the form
˜
Oh,¯
h(z, ¯z, θ, ¯
θ) = Zd2wZd2ξf (z, ¯z, θ, ¯
θ;w, ¯w, ξ, ¯
ξ)Oh,¯
h(w, ¯w, ξ, ¯
ξ)(4.2)
to be a quasi-primary it must be equal to
˜
Oh,¯
h(z, ξ, ¯z, ¯
ξ) = NhZd2wZd2θOh,¯
h(w, θ, ¯w, ¯
θ)
(z−w−θξ)1−2h(¯z−¯w−¯
θ¯
ξ)1−2¯
h(4.3)
– 8 –

JHEP11(2024)048
for some value of the normalization constant
Nh
. The superconformal dimension (
h∗,¯
h∗
)of
the shadow operator (4.3) is related to the dimension of the superprimary field
O
as follows
h∗=1
2−h , ¯
h∗=1
2−¯
h . (4.4)
Then the operator (4.1) has superconformal dimension (0
,
0). In the remainder of the paper,
for the sake of clarity, we will restrict ourselves to the spinless fields, so that
h=¯
h(4.5)
and completely omit the dependence on
¯
h
. Nevertheless, with minimal effort our results
can be generalized to fields with non-zero spin.
The definition (4.3) implies the following relation between the structure constants
˜
C∗
hh1h2= 24h−2(2h−1)2I0(h1+h−h2, h2+h−h1)NhChh1h2,(4.6)
C∗
hh1h2= 24hI0h1+h−h2+1
2, h2+h−h1+1
2Nh˜
Chh1h2,(4.7)
where
C∗
hh1h2
,
˜
C∗
hh1h2
are three-point constants corresponding to the correlation function
involving the shadow operator
˜
Oh
and two superprimary fields Φ
h1
and Φ
h2
. The function
I0
(
h1, h2
)in the last equation is defined as follows
I0(h1, h2)=4−h1−h2+1πΓ(1 −h1)Γ(1 −h2)Γ(h1+h2−1)
Γ(h1)Γ(h2)Γ(2 −h1−h2),(4.8)
and admits the following integral representation
I0(h1, h2) = Zd2w|w−1|−2h1|w+ 1|−2h2(4.9)
within the domain of convergence of the r.h.s. of the eq. (4.9).
4.2 Identity decomposition
One can prove that the operator (4.1) is a projector onto irreducible highest weight
osp
(1
|
2)
module by demonstrating that (2.3) can be generalized to supersymmetric case as follows:
⟨˜
Oh(Z1,¯
Z1)Oh(Z2,¯
Z2)⟩=δ2(z1−z2)δ2(θ1−θ2).(4.10)
Indeed, consider the integral representation
⟨˜
Oh(Z1,¯
Z1)Oh(Z2,¯
Z2)⟩=NhZd2ξd2w|z1−w−θ1ξ|−2+4h|w−z2−ξθ2|−4h.(4.11)
As it is clearly divergent, we regularize this expression by replacing
h
with
hϵ=h−ϵ
2(4.12)
in the first factor of the integrand in the r.h.s. of (4.11):
⟨˜
Oh(Z1,¯
Z1)Oh(Z2,¯
Z2)⟩ϵ=NhZd2ξd2w|z1−w−θ1ξ|−2+4h−2ϵ|w−z2−ξθ2|−4h.(4.13)
– 9 –

JHEP11(2024)048
The integral above can be computed explicitly:
⟨˜
Oh(Z1,¯
Z1)Oh(Z2,¯
Z2)⟩ϵ=|z12|
2−2+2ϵ
×h−4h2θ2¯
θ2I0(2h∗
ϵ,2h+ 1) −2hh∗
ϵ(θ1¯
θ2+θ2¯
θ1)I0(2h∗
ϵ,2h+ 1)
−2hh∗
ϵ(θ1¯
θ2+θ2¯
θ1)I0(1 + 2h∗
ϵ,2h)−4(h∗
ϵ)2θ1¯
θ1I0(1 + 2h∗
ϵ,2h)
+ 8hh∗
ϵ(θ1¯
θ2+θ2¯
θ1)I0(1 + 2h∗
ϵ,1+2h)i,(4.14)
where we’ve used shortcut notation
h∗
ϵ=1
2−hϵ.(4.15)
In the limit
ϵ→
0the function
I0
is proportional to two-dimensional delta function [
47
],
while delta function of Grassmann variables is simply a linear function:
δ(θ1−θ2) = θ1−θ2,Zdθ1f(θ1)δ(θ1−θ2) = f(θ2).(4.16)
Therefore, if the normalization factor
Nh
is chosen as
Nh=−1
4π2,(4.17)
then, in this limit, the two-point function becomes a product of delta functions:
⟨˜
Oh(z1, θ1)Oh(z2, θ2)⟩=δ2(z1−z2)δ2(θ1−θ2).(4.18)
Thus, the identity operator admits the decomposition into a sum of projectors (4.1)
I=X
h
Πh.(4.19)
5 Supersymmetric conformal blocks
The supersymmetric four-point correlation function of spinless primary superfields can be
expressed as
⟨
4
Y
i=1
Φhi(Zi,¯
Zi)⟩=|Lh1,h2,h3,h4(Z12, Z34 , Z24, Z13)|2
×X
h
Gh(h1, h2, h3, h4|X, ¯
X, η, ¯η , η′,¯η′),
(5.1)
where the supercoordinates read in components
Zi= (zi, θi),¯
Zi= (¯zi,¯
θi),(5.2)
the variables
X
,
η
and
η′
are
OSp
(1
|
2) invariant cross-ratios
X=Z34Z21
Z31Z24
,(5.3)
η=η124 , η′= (1 −X)1
2η123 ,(5.4)
and the symbol
Lh1,h2,h3,h4
Lh1,h2,h3,h4=Z−h1−h2
12 Z−h3−h4
34 Zh1−h2
24 Z−h3+h4
13 Z−h1+h2+h3−h4
14 (5.5)
– 10 –

JHEP11(2024)048
stands for the “leg factor” ensuring correct transformation properties w.r.t. global supercon-
formal transformations, and
Gh
is a superconformal block. Restoring the antiholomorphic
dependence and taking into account that the whole correlation function must be even, one
gets the following general form of the conformal block:
Gh(X, ¯
X, η, ¯η , η′,¯η′) = g(0,0)
h(X, ¯
X) + g(1,0)
h(X, ¯
X)η¯η+f(1,1)
h(X, ¯
X)ηη′+f(−1,−1)
h(X, ¯
X)¯η¯η′
+f(1,−1)
h(X, ¯
X)η¯η′+f(−1,1)
h(X, ¯
X)¯ηη′
+g(0,1)
h(X, ¯
X)η′¯η′+g(1,1)
h(X, ¯
X)ηη′¯η¯η′.
(5.6)
Inserting identity decomposition (4.19) between Φ
h2
and Φ
h3
we represent the four-point
function as a sum of superconformal partial waves
⟨
4
Y
i=1
Φhi(Zi,¯
Zi)⟩=X
h
Ψh1,...,h4
h(Z1, . . . , Z4;¯
Z1,..., ¯
Z4),(5.7)
Ψh1,...,h4
h(Z1, . . . , Z4;¯
Z1,..., ¯
Z4) = Zd2z0Zd2θ0Vh1,h2,h(Z1,¯
Z1;Z2,¯
Z2;Z0,¯
Z0)
× Vh∗,h3,h4(Z0,¯
Z0;Z3,¯
Z3;Z4,¯
Z4).(5.8)
To compute the components
g(i,j)
h
(
X, ¯
X
)appearing in the expansion (5.6) we can set some
of the Grassmann coordinates
θi
and
¯
θi
to zero, reducing functions of
X, ¯
X
to functions
of
x, ¯x
, where
x=z43z21
z31z24
(5.9)
and use the superanalyticity (3.8) of the conformal block to restore dependence on Grassmann
variables. In particular, the function
g(0,0)
h
(
X, ¯
X
)can be computed by setting all four
holomorphic Grassmann variables, along with their antiholomorphic counterparts, to zero.
At this point of the superspace the invariant cross-ratios evaluate to complex numbers:
X=x , η = 0 , η′= 0 .(5.10)
The conformal partial wave then admits the following integral representation
Ψh1,...,h4
h(z1, . . . , z4; ¯z1,...,¯z4) = Zd2z0
˜
C12hC∗
h34|z12 |2h−2h1−2h2+1|z34 |1−2h−2h3−2h4
|z10|1+2p1|z20 |1+2p2|z03|2p3|z04 |2p4
+Zd2z0
C12h˜
C∗
h34|z12 |2h−2h1−2h2)|z34|2−2h−2h3−2h4
|z10|2p1|z20 |2p2|z03|1+2p3|z04 |1+2p4
(5.11)
with the powers in the denominators given by
p1=h+h12 , p2=h−h12 , p3=1
2−h+h34 , p4=1
2−h−h34 ,(5.12)
and the shortcut notation
hij
is used for the difference of conformal dimensions:
hij =hi−hj.(5.13)
– 11 –

JHEP11(2024)048
The conformal partial wave is known to contain contributions from the global conformal
block
Gh
and the so-called shadow global conformal block
Gh∗
[
48
], and this holds true
for the superconformal partial waves as well, as will be demonstrated by the calculations
below. Performing a linear fractional transformation mapping the points
z1, z2, z3, z4
to
∞,
1
, x,
0respectively and stripping off the leg factor we obtain the following expression
for the function
g(0,0)
h
g(0,0)
h(X, ¯
X) + g(0,0)
h∗(X, ¯
X) = ˜
Chh1h2C∗
hh3h4|X|F4pt h+1
2, h12, h34 X, ¯
X
+Chh1h2˜
C∗
hh3h4|X|F4pt h, h12, h34 |X, ¯
X,
(5.14)
where
F4pt(h, h12 , h34|X, ¯
X) = Y(1 −h, h12)X1
2−h2F1(1 −h+h34,1−h−h12 ,2−2h|X)2
+Y(h, h34)Xh−1
22F1(h+h34, h −h12,2h|X)2.
(5.15)
Here,
2F1
stands for the hypergeometric function, and the coefficients
Y
are given by
Y(h, h′) = πΓ(1 −2h)Γ(h+h′)Γ(h−h′)
Γ(2h)Γ(1 −h−h′)Γ(1 −h+h′).(5.16)
Taking into account relations (4.6), (4.7) between the structure constants one can verify
that the function
F4pt
satisfies the following identity
C∗
hh1h2˜
Chh3h4F4pt 1
2−h, h12, h34 X, ¯
X=˜
Chh1h2C∗
hh3h4F4pt h+1
2, h12, h34 X, ¯
X.
(5.17)
Applying this identity to the r.h.s. of eq. (5.14) we see that the conformal partial wave
is invariant under interchange of conformal dimension of the intermediate primary field
h
and its shadow dual
h∗
. Therefore, we are justified in interpreting this result as a sum of
contributions from the global conformal block and the shadow global conformal block. The
former can be extracted from the whole conformal partial wave by selecting terms with correct
asymptotic behaviour. Explicitly, the function
g(0,0)
h
can be expressed as follows:
g(0,0)
h(X, ¯
X) = ˜
C012 ˜
C034 G(o)
0,0(h, h12, h34 |X)2+C012C034 G(e)
0,0(h, h12, h34 |X)2,(5.18)
where
G(e)
0,0(h, h12, h34 |X) = Xh2F1(h34 +h, −h12 +h, 2h|X),(5.19)
G(o)
0,0(h, h12, h34 |X) = 1
2hG(e)
0,0h+1
2, h12, h34 X.(5.20)
The expressions (5.19), (5.20) for the components of the four-point superconformal block are
(up to a choice of order of points
z1, z2, z3, z4
) in agreement with earlier results [
43
,
44
,
49
].
6 Torus superconformal blocks
In the following sections, we study superconformal field theory on a two-dimensional torus.
We will focus on the one and two-point torus superconformal blocks. Our goal is to describe
the torus superconformal blocks using shadow formalism.
– 12 –

JHEP11(2024)048
Similarly to the spherical case, the correlation functions on torus can be decomposed into a
sum over intermediate primary superfields, but even the one-point function on a torus is already
nontrivial (i.e., involves a sum over all superprimary fields in the theory). The one-point
and two-point torus correlation functions of primary superfields Φ
h1
(
Z1,¯
Z1
)
,
Φ
h2
(
Z2,¯
Z2
)
can be written as
⟨Φh1(Z1,¯
Z1)⟩τ=X
∆1
strH∆1hqL0¯q¯
L0Φh1(Z1,¯
Z1)i,(6.1)
⟨Φh1(Z1,¯
Z1)Φh2(Z2,¯
Z2)⟩τ=X
∆1
strH∆1hqL0¯q¯
L0Φh1(Z1,¯
Z1)Φh2(Z2,¯
Z2)i,(6.2)
where the
strH∆1
stands for the supertrace taken over the supermodule
H∆1
. Consider-
ing (3.16), one can decompose the torus correlation functions (6.1), (6.2) into different
components obtained from the superfields. By parity arguments in Grassmann variables,
only terms with an even number of Grassmann variables contribute.
4
Thus, the one-point
function (6.1) can be written as
⟨Φh1(Z1,¯
Z1)⟩τ=⟨ϕh1(z1,¯z1)⟩τ+θ1¯
θ1⟨˜
ϕh1(z1,¯z1)⟩τ.(6.3)
For the two-point function (6.2), the decomposition reads
⟨Φh1(Z1,¯
Z1)Φh2(Z2,¯
Z2)⟩τ=
=⟨ϕh1(z1,¯z1)ϕh2(z2,¯z2)⟩τ+θ1θ2⟨ψh1(z1,¯z1)ψh2(z2,¯z2)⟩τ+θ1¯
θ1⟨˜
ϕh1(z1,¯z1)ϕh2(z2,¯z2)⟩τ
+¯
θ1¯
θ2⟨¯
ψh1(z1,¯z1)¯
ψh2(z2,¯z2)⟩τ+θ2¯
θ2⟨ϕh1(z1,¯z1)˜
ϕh2(z2,¯z2)⟩τ+θ1¯
θ2⟨ψh1(z1,¯z1)¯
ψh2(z2,¯z2)⟩τ
+¯
θ1θ2⟨¯
ψh1(z1,¯z1)ψh2(z2,¯z2)⟩τ+θ1¯
θ1θ2¯
θ2⟨˜
ϕh1(z1,¯z1)˜
ϕh2(z2,¯z2)⟩.
(6.4)
In turn, each of the terms on the r.h.s. of (6.3), (6.4) can be decomposed into torus super-
conformal blocks, as we will discuss.
6.1 One-point superconformal blocks
For the one-point function (6.3), the decomposition into superconformal blocks reads
⟨ϕh1(z1,¯z1)⟩τ=X
∆1
C∆1h1∆1|B0(h1,∆1|q)|2,
⟨˜
ϕ∆1(z1,¯z1)⟩τ=X
∆1
˜
C∆1h1∆1|B1(h1,∆1|q)|2,(6.5)
where
B0
and
B1
are the holomorphic one-point lower and upper superconformal blocks
B0(h1,∆1|q) = 1
⟨∆1|ϕh1|∆1⟩str∆1hqL0ϕh1i,
B1(h1,∆1|q) = 1
⟨∆1|ψh1|∆1⟩str∆1hqL0ψh1i.
(6.6)
4
This argument can be verified by noting that the parity, w.r.t. the parity operator (
−
1)
F⊗
(
−
1)
¯
F
, of the
matrix elements representing the structure constants
C∆ihi∆i,˜
C∆ihi∆i
is always even. Notice also that the
same behaviour occurs in the sphere two-point function (3.19) and four-point function (5.6), where only even
terms in Grassmann variables contribute.
– 13 –

JHEP11(2024)048
Here the graded supertrace
str∆1
is evaluated over the supermodule
V∆1
, and we have used
the fact that one can work with
ψh1
to describe purely holomorphic contribution from
˜
ϕh1
.
In the
osp
(1
|
2) sector, one can compute closed-form expressions for the trace (see [
12
]). Thus,
one can write explicitly (6.6) in this sector as
B0(h1,∆1|q) = 1
⟨∆1|ϕh1|∆1⟩X
n=N+
2X
|N|=|M|=n
(−1)2nBN|M
∆1⟨∆1, N |qL0ϕh1|M, ∆1⟩,
B1(h1,∆1|q) = 1
⟨∆1|ψh1|∆1⟩X
n=N+
2X
|N|=|M|=nBN|M
∆1⟨∆1, N |qL0ψh1|M, ∆1⟩,
(6.7)
where the sum is over the states (3.27), and
BN|M
∆i
is matrix element (
N, M
) of the inverse of
the Gram matrix. Closed-form expressions for (6.7) are given in terms in linear combinations
of
sl
(2) one-point torus conformal blocks, namely
B0(h1,∆1|q) = Fh1
∆1(q)−(2∆1−h1)
2∆1Fh1
∆1+1
2
(q),
B1(h1,∆1|q) = Fh1+1
2
∆1(q)−2∆1+h1−1
2
2∆1Fh1+1
2
∆1+1
2
(q),
(6.8)
where
Fh1
∆1
(
q
)is the
sl
(2) one-point torus conformal block
Fh1
∆1(q) = q∆1
(1 −q)1−h12F1(h1, h1+ 2∆1−1,2∆1|q).(6.9)
We notice that the linear combinations (6.8) can be obtained by splitting the sum over
n
in (6.7) into the even and odd parts according to (3.28). Each term obtained after
the splitting can be written in terms of (6.9), and using the relations between matrix
elements (6.19), (6.20), (6.21), one can obtain precisely (6.8). The same idea can be applied
to higher-point superconformal blocks.
6.2 Two-point superconformal blocks
Higher-point correlation functions can be decomposed into superconformal blocks in different
channels. In this work, we are interested in the necklace channel decompositions. For this,
one inserts the following resolution of identity between the primary fields
I=X
∆2
∞
X
n,m∈N+
2X
|M|=|N|=nX
|¯
M|=|¯
N|=m|N, ¯
N , ∆2⟩BN|M
∆2
¯
B¯
N|¯
M
∆2⟨∆2, M, ¯
M|,(6.10)
into the terms of r.h.s. of (6.4). Let us explain this construction for the first and second terms
of (6.4). The discussion for the other terms follows the same idea. For the purely bosonic
contribution, the decomposition into conformal blocks can be written as
⟨ϕh1(z1,¯z1)ϕh2(z2,¯z2)⟩τ
=X
∆1
strH∆1hqL0¯q¯
L0ϕh1(z1,¯z1)Iϕh2(z2,¯z2)i
=X
∆1,∆2
C∆1h1∆2C∆2h2∆1|B(1)
00 (q1, z1, z2)|2+˜
C∆1h1∆2˜
C∆2h2∆1|B(2)
00 (q1, z1, z2)|2,
(6.11)
– 14 –

JHEP11(2024)048
where
B(1)
00
, and
B(2)
00
are the holomorphic two-point superconformal blocks, which provide
contributions from odd and even parts of the trace
strH∆1
and the operator (6.10). These
two conformal blocks are given by
B(1)
00 (q, z1,z2) =
=z∆1−∆2−h1
1z−∆1+∆2−h2
2
⟨∆1|ϕh1(z1)|∆2⟩⟨∆2|ϕh2(z2)|∆1⟩
×X
n,m=0
n+m=NX
|M|=|N|=n
|S|=|T|=m
(−1)2nBM|N
∆1⟨∆1,M |qL0ϕh1(z1)|S,∆2⟩BS|T
∆2⟨∆2, T |ϕh2(z2)|N, ∆1⟩,
(6.12)
B(2)
00 (q, z1,z2) =
=z∆1−∆2−h1−1
2
1z−∆1+∆2−h2+1
2
2
⟨∆1|ϕh1(z1)|∆2+1
2⟩⟨∆2+1
2|ϕh2(z2)|∆1⟩
×X
n,m=0
2(n+m)=odd X
|M|=|N|=n
|S|=|T|=m
(−1)2nBM|N
∆1⟨∆1,M |qL0ϕh1(z1)|S,∆2⟩BS|T
∆2⟨∆2, T |ϕh2(z2)|N, ∆1⟩,
(6.13)
In (6.2), (6.13) and below, given that we mostly emphasize the holomorphic dependence, we
omit the antiholomorphic coordinate of the field, and we also use the notion that the
holomorphic coordinate dependence of the matrix element
⟨
∆
1|ϕh1
(
zi
)
|
∆
2⟩
is given by
⟨
∆
1|ϕh1
(
zi
)
|
∆
2⟩
=
⟨
∆
1|ϕh1
(1)
|
∆
2⟩z∆1−h1−∆2
i
. Notice that in (6.2), the condition
n
+
m
=
N
indicates that sum over
n
and
m
is performed such that 2
n
and 2
m
have the same parity, while
in (6.13) the condition 2(
n
+
m
) =
odd
indicates that the parity of 2
m
and 2
n
is different.
For the second term of (6.4), we have
⟨ψh1(z1,¯z1)ψh2(z2,¯z2)⟩τ=X
∆1
strH∆1hqL0¯q¯
L0ψh1(z1,¯z1)Iψh2(z2,¯z2)i=
=X
∆1,∆2
˜
C∆1h1∆2˜
C∆2h2∆1B(2)
θ1θ2(q, z1, z2)¯
B(2)
00 (¯q, ¯z1,¯z2)
+C∆1h1∆2C∆2h2∆1B(1)
θ1θ2(q, z1, z2)¯
B(1)
00 (¯q, ¯z1,¯z2),
(6.14)
where the two holomorphic superconformal blocks
B(2)
θ1θ2
(
q, z1, z2
)
, B(1)
θ1θ2
(
q, z1, z2
)similarly
to (6.2), (6.2), are given by
B(1)
θ1θ2(q, z1,z2) =
=z∆1−∆2−h1−1
1z−∆1+∆2−h2
2
⟨∆1|ψh1(z1)|∆2+1
2⟩⟨∆2+1
2|ψh2(z2)|∆1⟩
×X
n,m=0
2(n+m)=odd X
|M|=|N|=n
|S|=|T|=m
(−1)2nBM|N
∆1⟨∆1,M |qL0ψh1(z1)|S,∆2⟩BS|T
∆2⟨∆2, T |ψh2(z2)|N, ∆1⟩,
(6.15)
B(2)
θ1θ2(q, z1,z2) =
=z∆1−∆2−h1−1
2
1z−∆1+∆2−h2−1
2
2
⟨∆1|ψh1(z1)|∆2⟩⟨∆2|ψh2(z2)|∆1⟩
×X
n,m=0
n+m=NX
|M|=|N|=n
|S|=|T|=m
(−1)2nBM|N
∆1⟨∆1,M |qL0ψh1(z1)|S,∆2⟩BS|T
∆2⟨∆2, T |ψh2(z2)|N, ∆1⟩.
(6.16)
– 15 –

JHEP11(2024)048
By concentrating on the
osp
(1
|
2) submodule, we can utilize various methods to derive closed-
form expressions for ((6.2), (6.13), (6.14), (6.15)). In the following section, we will employ
the shadow formalism to compute them. This technique proved efficient for calculating
global higher-point torus conformal blocks [
29
]. Since no known expressions exist, we will
ensure that the expressions derived using the shadow formalism precisely correspond to
the conformal blocks under consideration. On the one hand, we can derive differential
equations for superconformal blocks from the
osp
(1
|
2) Casimir operator [
6
,
12
,
20
], which
these superconformal blocks must satisfy. We will analyze this in section 8.
On the other hand, one can obtain closed-form expressions for the global
osp
(1
|
2) two-
point superconformal blocks by splitting the sum over the descendant states in ((6.2), (6.13),
(6.14), (6.15)) into
sl
(2) modules, and then express the obtained terms using
sl
(2) two-point
torus conformal blocks. Let us explain this simple rationale for the purely bosonic part (6.2).
The sums of (6.2) can be regrouped in even and odd parts as follows
B(1)
00 (q, z1, z2) = z∆1−∆2−h1
1z−∆1+∆2−h2
2
⟨∆1|ϕh1(z1)|∆2⟩⟨∆2|ϕh2(z2)|∆1⟩
×X
n,m=N+X
|M|=|N|=n
|S|=|T|=m



BM|N
∆1⟨∆1,M |qL0ϕh1(z1)|S, ∆2⟩BS|T
∆2⟨∆2, T |ϕh2(z2)|N, ∆1⟩
| {z }
(1)
−BM|N
∆1+1
2⟨∆1+1
2,M |qL0ϕh1(z1)|S, ∆2+1
2⟩BS|T
∆2+1
2⟨∆2+1
2, T |ϕh2(z2)|N, ∆1+1
2⟩
| {z }
(2)




,
(6.17)
where in the second term, we used the notation
|N, ∆i+1
2⟩= (L−1)|N|G−1/2|∆i⟩.(6.18)
One can check that first term of (6.17) is proportional to the
sl
(2) two-point torus con-
formal block
Fh1,h2
∆1,∆2
(
q, z1, z2
)[
11
] given by (2.19) while the second term is proportional to
Fh1,h2
∆1+1
2,∆2+1
2
(
q, z1, z2
). By using the following relations
⟨∆i+1
2|∆i+1
2⟩= 2∆i⟨∆i|∆i⟩= 2∆i,(6.19)
⟨∆1+1
2|ϕh1(1)|∆2+1
2⟩= (∆1+ ∆2−h1)⟨∆1|ϕh1(1)|∆2⟩,(6.20)
one gets that (6.17) is given by the linear combination (7.9). For the other superconformal
blocks, e.g., (6.15), (6.16), one can repeat the same rationale and obtain similar expressions,
which will be detailed below. For (6.15), (6.16) one requires the relations
⟨∆1|ψh1(1)|∆2+1
2⟩= (∆1−h1−∆2)⟨∆1|ϕh1(1)|∆2⟩,
⟨∆1+1
2|ψh1(1)|∆2⟩= (∆1+h1−∆2)⟨∆1|ϕh1(1)|∆2⟩,
⟨∆1+1
2|ψh1(1)|∆2+1
2⟩=−∆1+h1+ ∆2−1
2⟨∆1|ψh1(1)|∆2⟩.
(6.21)
– 16 –

JHEP11(2024)048
7 Torus superconformal blocks via shadow formalism
In this section, we apply shadow formalism to compute global one- and two-point torus
superconformal blocks. The generalization of the shadow formalism to the
osp
(1
|
2) case
follows a similar approach to that of the
sl
(2) case.
7.1 osp(1|2) torus shadow formalism
A straightforward generalization of (2.15), (2.16) to the
osp
(1
|
2) case involves replacing the
three-point function
v
with the supersymmetric three-point function (3.20). Thus, for the
supersymmetric case, in analogy with (2.15), (2.16), one can define the one- and two-point
torus superconformal partial waves as follows
Wh1
∆1q, ¯q, Z1,¯
Z1=q∆1¯q∆1Zd2w1d2ξ1V∆∗
1,h1,∆1W1,¯
W1;Z1,¯
Z1;q·W1,¯q·¯
W1,(7.1)
Wh1,h2
∆1,∆2q, ¯q, Z1,¯
Z1,Z2,¯
Z2=
q∆1¯q∆1Zd2w1d2w2d2ξ1d2ξ2V∆∗
1,h1,∆2W1,¯
W1;Z1,¯
Z1;W2,¯
W2
×V∆∗
2,h2,∆1W2,¯
W2;Z2,¯
Z2;q·W1,¯q·¯
W1.(7.2)
For supersymmetric shadow formalism, we find that the conformal dimension ∆
∗
i
of the
shadow field is given by the relation (4.4), and the product
q·Wi
is defined as follows
q·Wi= (qwi,√qξi).(7.3)
We will see in the discussion below that the definition (7.3)
5
is relevant for computing the
superconformal blocks. Since we are interested in the holomorphic superconformal blocks, we
will focus only on the holomorphic parts of (7.1), (7.2) and apply the same logic discussed
for the global
sl
(2) conformal blocks.
7.2 One-point torus superconformal blocks
For the one-point torus superconformal block, we first expand the integrand of (7.1) in
Grassmann variables and then take the integral over the two variables
ξ1
and
¯
ξ1
. This
results in expressing (7.1) as
Wh1
∆1q, ¯q , Z1,¯
Z1=q∆1¯q∆1Zd2w1˜
C∗
∆1h1∆1|b0|2−θ1¯
θ2C∗
∆1h1∆1|b1|2,(7.4)
where
b1, b2
are given by
b0=v1−∆1,h1,∆1(w1, z1, qw1) + √qv1
2−∆1,h1,∆1+1
2(w1, z1, qw1),
b1=(−2∆1+h1+ 1/2)v1−∆1,h1+1
2,∆1(w1, z1, qw1)+
+ (2∆1+h1−1/2)√qv 1
2−∆1,h1+1
2,∆1+1
2(w1, z1, qw1).
(7.5)
5A similar relation to (7.3) was found in [31] in the discussion of the shadow formalism for W3CFT.
– 17 –

JHEP11(2024)048
Integrating (7.5) in the same way as discussed for the
sl
(2) case, we obtain that the one-point
lower and upper superconformal blocks (6.7) are computed by
B0(h1,∆1|q) = 1
c1(h1,∆1)q∆1Zz1
0
b0dw1,
B1(h1,∆1|q) = 1
c1h1+1
2,∆1(−2∆1+h1+ 1/2)q∆1Zz1
0
b1dw1.(7.6)
7.3 Two-point torus superconformal blocks
We repeat the procedure applied to the one-point superconformal blocks to find the two-point
torus superconformal blocks. We first expand the integrand of (7.2) in the Grassmann
variables and then take the integral over
ξ1,¯
ξ1
and
ξ2,¯
ξ2
. This results in expressing (7.2)
in terms of eight independent terms
Wh1,h2
∆1,∆2(q, ¯q, Z1,¯
Z1,Z2,¯
Z2) = q∆1¯q∆1Zd2w1d2w2f1+θ1θ2f2+θ1¯
θ1f3+θ2¯
θ2f4+
+θ1¯
θ2f5+¯
θ1θ2f6+¯
θ1¯
θ2f7+θ1¯
θ1θ2¯
θ2f8.
(7.7)
Each term
fi
can be used to compute superconformal blocks corresponding to different
parts of (6.4). Here, we will analyze in detail the terms
f1
and
f2
. For other terms, we
will provide the final result since the analysis follows the same rationale. For the purely
bosonic term, i.e., the term
f1
, we obtain
f1=˜
C∗
∆1h1∆2˜
C∗
∆2h2∆1v1−∆1,h1,∆2(w1, z1, w2)v1−∆2,h2,∆1(w2, z2, qw1)−
−√qv1/2−∆1,h1,∆2+1/2(w1, z1, w2)v1/2−∆2,h2,∆1+1/2(w2, z2, qw1)2
+C∗
∆1h1∆2C∗
∆2h2∆1
×(1/2−∆1+ ∆2−h1)v1−∆1,h1,∆2+1/2(w1, z1, w2)v1/2−∆2,h2,∆1(w2, z2, qw1) +
−(1/2+∆1−∆2−h2)√qv1/2−∆1,h1,∆2(w1, z1, w2)v1−∆2,h2,∆1+1/2(w2, z2, qw1)2.
(7.8)
Since the structure constants
˜
C∗
∆ihj∆k, C∗
∆ihj∆k
are independent,
f1
consists of two indepen-
dent terms, each providing different conformal blocks. Focusing on the holomorphic part of
f1
and taking the integral as discussed for the
sl
(2) case, we obtain the integral representation
for
B(1)
00
from the term proportional to
˜
C∗
∆1h1∆2
˜
C∗
∆2h2∆1
, as follows
B(1)
00 (q, z1, z2)
=q∆1
c2(h1, h2,∆1,∆2)ZC1
dw1ZC2
dw2v1−∆1,h1,∆2(w1, z1, w2)v1−∆2,h2,∆1(w2, z2, qw1)
−√qv1/2−∆1,h1,∆2+1/2(w1, z1, w2)v1/2−∆2,h2,∆1+1/2(w2, z2, qw1)
=Fh1,h2
∆1,∆2(q, z1, z2)
−(∆1+ ∆2−h1)(∆1+ ∆2−h2)
4∆1∆2Fh1,h2
∆1+1/2,∆2+1/2(q, z1, z2).
(7.9)
– 18 –

JHEP11(2024)048
Similarly, from the term proportional to
C∗
∆1h1∆2C∗
∆2h2∆1
, we obtain the integral represen-
tation for
B(2)
00
:
B(2)
00 (q, z1, z2) = q∆1
c2(h1,h2,∆1,∆2+1/2)
×ZC1
dw1ZC2
dw2v1−∆1,h1,∆2+1/2(w1,z1, w2)v1/2−∆2,h2,∆1(w2, z2, qw1)
−(1/2+∆1−∆2−h2)
(1/2−∆1+∆2−h1)√qv1/2−∆1,h1,∆2(w1, z1, w2)v1−∆2,h2,∆1+1/2(w2,z2, qw1)
=Fh1,h2
∆1,∆2+1/2(q, z1, z2)−∆2
∆1Fh1,h2
∆1+1/2,∆2(q, z1, z2).
(7.10)
Notice that in the limit
h2→
0
,
∆
2→
∆
1
, the expression (7.9) reduces to
B0
from (6.8), which
is the desired relation for the purely bosonic two-point torus superconformal block. For (7.10),
such a limit cannot be imposed since ∆
1,
∆
2
differ by 1
/
2in both terms on r.h.s. of (7.10).
Contribution
θ1θ2
.Now we proceed with the term
f2
from (7.7). This term is given by
f2¯q→0
=˜
C∗
∆1h1∆2˜
C∗
∆2h2∆1˜a(1)
6v1−∆1,h1+1/2,∆2+1/2(w1, z1, w2)v1/2−∆2,h2+1/2,∆1(w2, z2, qw1)
−˜a(2)
6√qv1/2−∆1,h1+1/2,∆2(w1, z1, w2)v1−∆2,h2+1/2,∆1+1/2(w2, z2, qw1)
×¯v1−∆1,h1,∆2( ¯w1,¯z1,¯w2) ¯v1−∆2,h2,∆1( ¯w2,¯z2,0)
+C∗
∆1h1∆2C∗
∆2h2∆1a(1)
6v1−∆1,h1+1/2,∆2(w1, z1, w2)v1−∆2,h2+1/2,∆1(w2, z2, qw1)
+a(2)
6√qv1/2−∆1,h1+1/2,∆2+1/2(w1, z1, w2)v1/2−∆2,h2+1/2,∆1+1/2(w2, z2, qw1)
×¯v1−∆1,h1,∆2+1/2( ¯w1,¯z1,¯w2) ¯v1/2−∆2,h2,∆1( ¯w2,¯z2,0),
(7.11)
where we used
˜a(1)
6= 2 (−∆1+ ∆2+h1), a(1)
6=−−∆1−∆2+h1+1
2−∆1−∆2+h2+1
2,
˜a(2)
6= 2 (∆1−∆2+h2), a(2)
6=∆1+ ∆2+h1−1
2∆1+ ∆2+h2−1
2.
(7.12)
From the holomorphic part of (7.11) we obtain the integral expressions for (6.15), (6.16).
Thus, from the term proportional to
˜
C∗
∆1h1∆2
˜
C∗
∆2h2∆1
, we obtain
B(1)
θ1θ2=q∆1
c2(h1+1/2, h2+1/2,∆1,∆2+1/2)
×ZC1
dw1ZC2
dw2v1−∆1,h1+1/2,∆2+1/2(w1,z1, w2)v1/2−∆2,h2+1/2,∆1(w2,z2, qw1)
−˜a(2)
6
˜a(1)
6
√qv1/2−∆1,h1+1/2,∆2(w1, z1,w2)v1−∆2,h2+1/2,∆1+1/2(w2, z2,qw1)
=Fh1+1/2,h2+1/2
∆1,∆2+1/2(q, z1,z2)−∆2(∆1−∆2+h1) (∆2−∆1−h2)
∆1(∆1−∆2−h1)(∆2−∆1+h2)Fh1+1/2,h2+1/2
∆1+1/2,∆2(q, z1,z2),
(7.13)
– 19 –

JHEP11(2024)048
and similarly, from the term proportional to
C∗
∆1h1∆2C∗
∆2h2∆1
, we obtain the integral rep-
resentation for
B(2)
θ2θ1
B(2)
θ1θ2=q∆1
c2(h1+1/2, h2+1/2,∆1,∆2)
×ZC1
dw1ZC2
dw2v1−∆1,h1+1/2,∆2(w1,z1, w2)v1−∆2,h2+1/2,∆1(w2,z2, qw1)
+a(2)
6
a(1)
6
√qv1/2−∆1,h1+1/2,∆2+1/2(w1, z1,w2)v1/2−∆2,h2+1/2,∆1+1/2(w2, z2,qw1)
=Fh1+1/2,h2+1/2
∆1,∆2(q, z1,z2)−α3Fh1+1/2,h2+1/2
∆1+1/2,∆2+1/2(q, z1,z2),
(7.14)
where
α3=(2∆1+ 2∆2+ 2h1−1) (2∆1+ 2∆2+ 2h2−1)
16∆1∆2
.(7.15)
In the next section, it will be necessary to work with the redefined
B(1),(2)
θ1θ2
obtained by
multiplying them by the following constants
˜
B(1)
θ1θ2=α4B(1)
θ1θ2,
e
B(2)
θ1θ2=α5B(2)
θ1θ2,
(7.16)
where
α4=(∆1−∆2−h1) (−∆1+ ∆2+h2)
2∆2
,
α5=−2∆2.
(7.17)
The constants (7.17) arise when considering equation (6.19) and choosing the same nor-
malization for equations (6.15) and (6.16) (the denominators of those equations) as for
equations (6.2) and (6.13), respectively.
We can repeat the same procedure for the other terms
fi
of (7.7). The results we obtained
for the other holomorphic superconformal blocks are listed as follows:
Contribution
θ1¯
θ1
.From the term
f3
we obtain the superconformal blocks
B(1)
θ1¯
θ1=Fh1+1/2,h2
∆1,∆2(q, z1, z2)
−(−1 + 2∆1+ 2∆2+ 2h1) (∆1+ ∆2−h2)
8∆1∆2Fh1+1/2,h2
∆1+1/2,∆2+1/2(q, z1, z2),(7.18)
B(2)
θ1¯
θ1=Fh1+1/2,h2
∆1,∆2+1/2(q, z1, z2)
−∆2(∆1−∆2+h1)
∆1(∆1−∆2−h1)Fh1+1/2,h2
∆1+1/2,∆2(q, z1, z2).(7.19)
In the limit
h2→
0
,
∆
2→
∆
1
, the expression (7.18) reduces to
B1
from (6.8), which is
the desired relation.
– 20 –

JHEP11(2024)048
Contribution
θ2¯
θ2
.From the term
f4
we obtain the superconformal blocks
B(1)
θ2¯
θ2=Fh1,h2+1
2
∆1,∆2(q, z1, z2)
−(∆1+ ∆2−h1) (2∆1+ 2∆2+ 2h2−1)
8∆1∆2Fh1,h2+1
2
∆1+1
2,∆2+1
2
(q, z1, z2),(7.20)
B(2)
θ2¯
θ2=Fh1,h2+1
2
∆1,∆2+1
2
(q, z1, z2)−∆2(∆1−∆2+h2)
∆1(∆1−∆2−h2)Fh1,h2+1
2
∆1+1
2,∆2(q, z1, z2).(7.21)
Contribution
θ1¯
θ2
.From the term
f5
we obtain the superconformal blocks
B(1)
θ1¯
θ2=B(1)
θ1¯
θ1,(7.22)
B(2)
θ1¯
θ2=B(2)
θ1¯
θ1.(7.23)
Contribution
¯
θ1θ2
.From the term
f6
we obtain the superconformal blocks
B(1)
¯
θ1θ2=B(1)
θ2¯
θ2,(7.24)
B(2)
¯
θ1θ2=B(2)
θ2¯
θ2.(7.25)
Contribution
¯
θ1¯
θ2
.From the term
f7
we obtain the superconformal blocks
B(1)
¯
θ1¯
θ2=B(1)
00 ,(7.26)
B(2)
¯
θ1¯
θ2=B(2)
00 .(7.27)
Contribution
θ1¯
θ1θ2¯
θ2
.From the term
f8
we obtain the superconformal blocks
B(1)
θ1¯
θ1θ2¯
θ2=B(1)
θ1θ2,(7.28)
B(2)
θ1¯
θ1θ2¯
θ2=B(2)
θ1θ2.(7.29)
8 Casimir operator for superconformal blocks
In this section, we will check that the superconformal blocks ((7.9), (7.10) (7.16)) satisfy
differential equations derived from the Casimir operator. First, let us derive the differential
equations. The
osp
(1
|
2) Casimir operator is given by
S2=−L2
0+1
2(L1L−1+L−1L1) + 1
4G−1/2G1/2−G1/2G−1/2.(8.1)
One can insert the Casimir operator in the following two ways
str∆1hS2qL0ϕh1P∆2ϕh2i=−∆1∆1−1
2str∆1hqL0ϕh1P∆2ϕh2i,(8.2)
str∆1hqL0ϕh1S2P∆2ϕh2i=−∆2∆2−1
2str∆1hqL0ϕh1P∆2ϕh2i.(8.3)
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JHEP11(2024)048
For each particular insertion, we obtain different eigenvalues as described by the r.h.s. of
the above equations. Here,
P∆2
stands for the projector onto
V∆2
. Next, we write the l.h.s.
of (8.2), (8.3) as differential operators. We do this using the standard procedure described
in [
6
,
12
,
20
]. For this, we require the following commutation and anticommutation relations
[Lm, ϕhi(zi)] = L(i)
mϕhi(zi),L(i)
m=zm
i(zi∂zi+ (m+ 1)hi),(8.4)
[Gr, ϕhi(zi)] = zr+1/2
iψhi(zi),(8.5)
[Lm, ψhi(zi)] = zm
izi∂zi+ (m+ 1) hi+1
2ψhi(zi),(8.6)
{Gr, ψhi(zi)}=G(i)
rϕhi(zi),G(i)
r=zr−1
2
i(zi∂zi+ (2r+ 1)hi).(8.7)
The simplest term of the Casimir operator to treat is
L2
0
, for which we have
str∆1hL2
0qL0ϕh1ϕh2i= (q∂q)2str∆1hqL0ϕh1ϕh2i.(8.8)
The insertions of other operators, e.g.,
L2
0, L1L−1
, can be computed by moving these operators
to the most-right side of the trace (this is done by using the relations (8.4)) and then using
the graded cyclic property of the supertrace
str∆1
. One can show that these insertions
result in the equations
str∆1hqL0ϕh1L2
0ϕh2i=L(2)
02+ 2L(2)
0q∂q+ (q∂q)2str∆1hqL0ϕh1ϕh2i,(8.9)
and
str∆1hL1L−1qL0ϕh1ϕh2i=2
1−qq∂q−q
(1−q)2ˆ
A1str∆1hqL0ϕh1ϕh2i,(8.10)
str∆1hqL0ϕh1L1L−1ϕh2i=2
1−qq∂q+2
1−qL(2)
0−1
(1−q)2ˆ
A2str∆1hqL0ϕh1ϕh2i,(8.11)
where we used
ˆ
A1=L(1)
−1+L(2)
−1L(1)
1+L(2)
1,
ˆ
A2=L(1)
−1+qL(2)
−1L(2)
1+qL(1)
1.
(8.12)
The insertion of the operators
G−1/2G1/2
in (8.2), (8.3) generates also terms proportional to
str∆1hqL0ψh1ψh2i
. By using the same idea used for generators
Li
, one obtains the following
relations
str∆1hG−rGrqL0ϕh1ϕh2i=
2q∂q
(1 −q−r)+qr
(1 −qr)2z−r+1
2
1G(1)
r+z−r+1
2
2G(2)
r!str∆1hqL0ϕh1ϕh2i
+qr
(1 −qr)2zr+1
2
1z−r+1
2
2−z−r+1
2
1zr+1
2
2str∆1hqL0ψh1ψh2i,
(8.13)
– 22 –

JHEP11(2024)048
and
str∆1hqL0ϕh1G1
2G−1
2ϕh2i=


2(L(2)
0+q∂q)
1−q1
2+q1
2
1−q1
22(z2∂z2+z1∂z1)

str∆1hqL0ϕh1ϕh2i
+1
1−q1
22(z2−qz1) str∆1hqL0ψh1ψh2i.
(8.14)
From Ward’s identity, we also have
(z1∂z1+z2∂z2) str∆1hqL0ϕh1ϕh2i=−(h1+h2) str∆1hqL0ϕh1ϕh2i.(8.15)
By substituting ((8.8), (8.9), (8.10), (8.11), (8.13), (8.14)) into (8.2), (8.3), using (8.15) and
writing the resulting equations in components, we obtain from (8.2) the first differential
equation for the superconformal blocks
−q∂q−q2∂2
q+q(1 −q1/2)
2(1 + q1/2)∂q−q
(1 −q)2ˆ
A1+q1/2
2(1 −q1/2)2(h1+h2)
+ ∆1∆1−1
2B(i)
00 +q1/2
2(1 −q1/2)2(z1−z2)˜
B(i)
θ1θ2= 0,
(8.16)
and from (8.3), we obtain the second differential equation
−q2∂2
q+2q2
1−q∂q−1
(1 −q)2ˆ
A2+(1 + q)
1−qL(2)
0−(L(2)
0)2+ 2qL(2)
0∂q
+ ∆2(∆2−1/2) + 1
2(L(2)
0+q∂q)−1
1−q1/2(L(2)
0+q∂q)
+q1
2
2(1 −q1/2)2(h1+h2)!B(i)
00 −1
2(1 −q1/2)2(z2−qz1)˜
B(i)
θ1θ2= 0.
(8.17)
where
i
= 1
,
2. It is straightforward to verify these differential equations perturbatively
by expanding the superconformal blocks in
z2/z1
and
q
. For this type of expansion, it is
convenient to use the representation (B.10) for
F
. One can also prove equations (8.16), (8.17)
exactly. For the general proof, it is convenient to use the representation (2.19). The proof
can be summarized in three key steps, outlined below:
Step 1.
We rewrite (8.16), (8.17) in terms of
∂q, ∂ρ1, ∂ρ2
. Using the representation (2.19),
we substitute ((7.9), (7.10) (7.16)) into (8.16), (8.17). After this substitution,
we get rid of the overall factor in front of Appell function
F4
in (2.19), obtaining
that (8.16), (8.17) become differential equations for functions
F4
, each of the obtained
differential equation involves four different functions
F4
(this is so, because,
B(i)
00
and
˜
B(i)
θ1θ2
are given by combinations of two Appell functions
F4
with different
arguments). This step can be performed straightforwardly.
Step 2.
We rewrite the differential equations obtained in the previous step entirely in terms
of the variables
ρ1, ρ2
. This is achieved by splitting each differential equation into
– 23 –

JHEP11(2024)048
two terms: one containing only integer powers of
q
and the other containing only
half-integer powers of
q
. Each of these terms can then be written as a differential
equation involving only
ρ1, ρ2
. Due to the non simple relations (2.21) between
z1, z2, q
, and
ρ1, ρ2
, this task turns out to be intricate. Even though the expressions
obtained do not simplify in a simple way, the change of variables can be performed.
Step 3.
In the final step, we verify that the differential equations obtained in the previous
step are all satisfied. We do this by converting the differential equations into
recurrence relations for the series coefficients of the functions F4.
9 Conclusion and outlook
In this work, we have generalized the shadow formalism to
N
= 1 two-dimensional super-
conformal field theory in the Neveu-Schwarz sector and used it to compute global
osp
(1
|
2)
superconformal blocks, which arise in the large central charge limit of the superconformal
theory. An essential ingredient of the shadow formalism is the so-called shadow operator
that we have explicitly constructed in (4.3) for the scalar superfields. We demonstrated
that the two-point function of a superfield with its shadow factorizes into a product of
delta functions (4.10) of spatial and Grassmann coordinates, and therefore the projector-like
operator (4.1) can be constructed. The shadow operator allows us to construct an identity-like
operator (4.19), which can be used to decompose correlation functions of superfields into
superconformal partial waves. In sections 5and 7, we have applied this formalism to compute
the four-point superconformal block on a plane, and one- and two-point blocks on a torus.
For four-point spherical superconformal blocks and one-point torus superconformal blocks,
our results agree with earlier results obtained by other methods. For two-point torus super-
conformal blocks, we have verified that the expressions obtained via shadow formalism satisfy
required nontrivial relations for torus superconformal blocks. In particular, in section 8, we
showed that the two-point torus superconformal blocks (involving bosonic
ϕhi
and fermionic
ψhi
components) satisfy the differential equations which follow from the
osp
(1
|
2) Casimir
operator. These results show that the constructed supersymmetric shadow formalism provides
correct integral representations of superconformal blocks both on a plane and on a torus.
There are several related problems that we plan to explore. To have the complete
picture about
N
= 1 supersymmetry theory we need to consider the shadow formalism in
the Ramond sector. From the holography perspective, it would be interesting to see the
explicit interpretation of the global higher-point superconformal blocks as dual geodesic
diagrams. There are questions about shadow formalism relevant beyond the supersymmetric
case. For instance, whether the shadow formalism can be generalized to the full Virasoro
algebra, enabling us to go beyond the semiclassical limit.
Acknowledgments
We thank Mikhail Pavlov for fruitful discussions.
– 24 –

JHEP11(2024)048
A Calculation of the four-point superconformal block
In the section 5, we have computed the component
g(0,0)
h
of the four-point superconformal
block. Below we demonstrate how the rest can be computed using the superanalyticity
of the correlation function. The component
g(1,0)
h
(
X, ¯
X
)of the superconformal block (5.1)
can be computed by setting
θ1=θ2=θ3= 0 ,¯
θ1=¯
θ2=¯
θ3= 0 ,(A.1)
which implies
X=x , η =z
1
2
12z−1
2
14 z−1
2
24 θ4, η′= 0 .(A.2)
The conformal partial wave, in this case, evaluates to
Ψh1,...,h4
h(z1,...,Z4; ¯z1,..., ¯
Z4) =
= Ψh1,...,h4
h(z1,...,z4; ¯z1,...,¯z4)
+θ4¯
θ4(h∗−h34)2Zd2z0
Ch1h2hC∗
hh3h4|z12|2h−2h1−2h2|z34 |1−2h−2h3−2h4
|z10|2p1|z20 |2p2|z03|2p3|z04 |2p4+2
+θ4¯
θ4Zd2z0
˜
Ch1h2h˜
C∗
hh3h4|z12|2h−2h1−2h2+1 |z34|−2h−2h3−2h4
|z10|1+2p1|z20 |1+2p2|z03|2p3−1|z04 |1+2p4.
(A.3)
Evaluating the integrals, we get
Ψh1,...,h4
h(z1,...,Z4; ¯z1,..., ¯
Z4) = Ψh1,...,h4
h(z1,...,z4; ¯z1,...,¯z4)
+η¯ηLh1,h2,h3,h4(h∗−h34 )2Ch1h2hC∗
hh3h4F4pt h,h12,h34 −1
2X, ¯
X
+η¯ηLh1,h2,h3,h4˜
Ch1h2h˜
C∗
hh3h4F4pt h+1
2,h12,h34 −1
2X, ¯
X.
(A.4)
Only the last two terms contribute to
g(1,0)
h
. Moreover, there are no other contributions, as
the factor
θ4¯
θ4
couldn’t have arisen from the expansion of the leg factor, as the holomorphic
Grassmann variables in the expansion of the latter always come in pairs, i.e.
θiθj
, but all
such pairs vanish at the chosen point (A.1) of the superspace. Separating the contributions
of the global conformal block and that of the shadow global conformal block, we obtain
the following expression for the function
g(1,0)
h
:
g(1,0)
h(X, ¯
X) = ˜
Chh1h2Chh3h4|G(o)
1,0(h, h12, h34 |X)|2+Chh1h2˜
Chh3h4|G(e)
1,0(h, h12, h34 |X)|2,
(A.5)
where even and odd parts of holomorphic conformal block read respectively
G(e)
1,0(h, h12, h34 |X) = Xh−1
22F1h34 −1
2+h, −h12 +h, 2hX,(A.6)
G(o)
1,0(h, h12, h34 |X) = h−h34
2hG(e)
1,0h+1
2, h12, h34 X.(A.7)
– 25 –

JHEP11(2024)048
To compute the component
g(0,1)
h
of the superblock we set
θ1=θ2=θ4= 0 ,¯
θ1=¯
θ2=¯
θ4= 0 .(A.8)
Then superconformal partial wave evaluates to
Ψh1,...,h4
h(z1, . . . , Z3, z4; ¯z1,..., ¯
Z3,¯z4) =
= Ψh1,...,h4
h(z1, . . . , z4; ¯z1,...,¯z4)
+η′¯η′Lh1,h2,h3,h4(h∗+h34)2Chh1h2C∗
hh3h4F4pt h, h12, h34 +1
2X, ¯
X
+η′¯η′Lh1,h2,h3,h4˜
Chh1h2˜
C∗
hh3h4F4pt h+1
2, h12, h34 +1
2X, ¯
X,
(A.9)
which leads to the following expression for the function
g(0,1)
h
:
g(0,1)
h(X, ¯
X) = ˜
Chh1h2Chh3h4|G(o)
0,1(h, h12, h34 |X)|2+Chh1h2˜
Chh3h4|G(e)
0,1(h, h12, h34 |X)|2,
(A.10)
with even and odd components of the conformal block given by
G(e)
0,1(h, h12, h34 |X) = Xh2F1h34 −1
2+h, −h12 +h, 2hX,(A.11)
G(o)
0,1(h, h12, h34 |X) = h+h34
2hG(e)
0,1h+1
2, h12, h34 X.(A.12)
B Integral representation of sl(2) two-point torus conformal block
In this section, we briefly describe some integrals that arise in shadow formalism, which yield
global
sl
(2) two-point conformal blocks. We define the integral
Ψh1,h2,h3,h4
∆(z1,z2, z3,z4) = Zz3
z4
dwvh1,h2,∆(z1, z2, w)v1−∆,h3,h4(w,z3, z4)
=Zz3
z4
dw z−h1−h2+∆
12 z−h3−h4+1−∆
34
(w−z1)∆+h12 (w−z2)∆−h12 (w−z3)1−∆+h34 (w−z4)1−∆−h34 .
(B.1)
By performing the change of variables (see, [
28
]),
w→z1zz4w+z4z12
z24w+z12
,(B.2)
the integral (B.1) becomes
Ψh1,h2,h3,h4
∆(z1, z2, z3, z4) = 1
zh1+h2
12 zh3+h4
34 z24
z14 h12 z14
z13 h34
×Zχ1
0
dw χ1−∆
1
w1−∆−h34 (1 −w)∆−h12 (χ1−w)1−∆+h34 ,
(B.3)
– 26 –

JHEP11(2024)048
where
χ1
= (
z12z34
)
/
(
z13z24
), and it is assumed that
χ1<
1, hence
z1> z2> z3> z4
.
By taking (B.3) one finds
Ψh1,h2,h3,h4
∆(z1, z2, z3, z4)
=α0(∆, h34)
zh1+h2
12 zh3+h4
34 z24
z14 h12 z14
z13 h34
χ∆
12F1(∆ −h12,∆ + h34 ,2∆, χ1),(B.4)
where
α0(∆, h) = πcsc (π(∆ + h)) Γ (∆ −h)
Γ(2∆)Γ (−∆−h+ 1) .(B.5)
The expression obtained from shadow formalism [
29
] that reproduces
sl
(2) two-point
torus conformal blocks (up to an overall factor) is given by
Fh1,h2
∆1,∆2(q, z1, z2) = q∆1ZC1
dw1ZC2
dw2v1−∆1,h1,∆2(w1, z1, w2)v1−∆2,h2,∆1(w2, z2, w1q),
(B.6)
where the integration domains
C2,
C
2
are given by (2.20). The integration over
w2
clearly
has the form (B.1), and hence we can write
Fh1,h2
∆1,∆2(q, z1, z2) = q∆1ZC1
dw1Ψ1−∆1,h1,h2,∆1
∆2(w1, z1, z2, w1q).(B.7)
The integration over
w1
is similar and also takes the form of (B.1). To see this, one needs
to expand in series the hypergeometric function present in the expression for
Ψ
1−∆1,h1,h2,∆1
∆2
(
w1, z1, z2, w1q
). After integrating over
w1
, one obtains that
Fh1,h2
∆1,∆2(q, z1, z2) = c2(h1, h2,∆1,∆2)Fh1,h2
∆1,∆2(q, z1, z2),(B.8)
where the coefficient
c2
is
c2(h1, h2,∆1,∆2) = α0(∆2, h2−∆1)α0(∆1, h1−∆2)
=π2csc (π(∆1−∆2+h1)) csc (π(−∆1+ ∆2+h2))Γ (−h1+ ∆1+ ∆2)
Γ(2∆1)Γ(2∆2)Γ(−h2+ ∆1−∆2+ 1)Γ(−h1−∆1+ ∆2+ 1)
×Γ(−h2+ ∆1+ ∆2),
(B.9)
and
Fh1,h2
∆1,∆2
(
q, z1, z2
)is given by (2.19).
Finally, let us recall that there exists another representation for the
sl
(2) two-point torus
conformal blocks in the necklace channel [
11
], namely
Fh1,h2
∆1,∆2(q, z1, z2) = q∆1z∆1−∆2−h1
1z−∆1+∆2−h2
2
×
∞
X
n=0
∞
X
m=0
qnz2
z1m−nτm,n(∆2, h2,∆1)τn,m (∆1, h1,∆2)
m!n!(2∆2)m(2∆1)n
,
(B.10)
where
τn,m (a, b, c) =
min(m,n)
X
p=0
n!(m)(p)(2c+m−1)(p)(−a+b+c)m−p(a+b−c−m+p)n−p
p!(n−p)! ,
(a)(m)=
m−1
Y
i=0
(a−i).(B.11)
This representation is useful when one is interested in expanding the conformal blocks in
variables
z2/z1
and
q
.
– 27 –

JHEP11(2024)048
Open Access. This article is distributed under the terms of the Creative Commons Attri-
bution License (CC-BY4.0), which permits any use, distribution and reproduction in any
medium, provided the original author(s) and source are credited.
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