# Crossing antisymmetric Polyakov blocks + Dispersion relation

###### Abstract

Many CFT problems, e.g. ones with global symmetries, have correlation
functions with a crossing antisymmetric sector. We show that such a crossing
antisymmetric function can be expanded in terms of manifestly crossing
antisymmetric objects, which we call the ‘$+$ type Polyakov blocks’. These
blocks are built from AdSd+1 Witten diagrams. In 1d they encode the ‘$+$ type’
analytic functionals which act on crossing antisymmetric functions. In general
$d$ we establish this Witten diagram basis from a crossing antisymmetric
dispersion relation in Mellin space. Analogous to the crossing symmetric case,
the dispersion relation imposes a set of independent ‘locality constraints’ in
addition to the usual CFT sum rules given by the ‘Polyakov conditions’. We use
the Polyakov blocks to simplify more general analytic functionals in $d>1$ and
global symmetry functionals.

Apratim Kaviraj

DESY Hamburg, Theory Group,
Notkestraße 85, D-22607 Hamburg, Germany

<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 1d crossing antisymmetric functionals
    1. 2.1 Basic kinematics
    2. 2.2 1d analytic functionals and Polyakov Bootstrap
    3. 2.3 The ‘+ type’ Polyakov blocks
      1. 2.3.1 Fermionic case
      2. 2.3.2  Bosonic case
  3. 3 General $d$ Polyakov blocks and dispersion relation
    1. 3.1 Contact diagrams
    2. 3.2 Dispersion relation
    3. 3.3 Polyakov Conditions
  4. 4 Applications to functionals
    1. 4.1 Product Functionals in even $d$
    2. 4.2 Global symmetry - Simple functionals
      1. 4.2.1 $d=1$
      2. 4.2.2 $d=2$ with product functionals
      3. 4.2.3 General $d$
  5. 5 Conclusion
  6. A Details of Witten diagrams
  7. B Derivation of the dispersion relation
    1. B.1 General case - review
    2. B.2 Imposing crossing (anti)symmetry
  8. C Checks with the dispersion relation
    1. C.1 Derivation of Witten diagrams for $\ell=5$
    2. C.2 Numerical checks
  9. D Product functionals in $d=4$ and $d=6$
    1. D.1 $d=4$
    2. D.2 $d=6$
    3. D.3 Bounds with product functionals

##  1 Introduction

One of the most important tools available to theoretical physicists in CFT
problems is crossing symmetry or the Conformal Bootstrap. For numerical
analysis one typically approaches the 4-point crossing problem by introducing
a set of functionals, which are simply a set of derivatives w.r.t. the cross
ratios. From the recent works of [1, 2, 3] (see also [4] and futher
generalizations [5, 6, 7, 8, 9]) we have learnt that there exists a more
efficient set of functionals, the analytic functionals. These are built to
optimize OPE coefficients, and are ‘dual’ to GFF or Generalized Free Fields
(i.e. a basis expansion in terms of double trace conformal blocks).

The analytic functionals are perhaps most well understood in 1d, where they
are related to another formulation of crossing symmetry: the Polyakov
bootstrap [10]. The latter is an idea that a crossing symmetric correlator can
be expanded in terms of a manifestly crossing symmetric sum of Witten diagrams
(a Polyakov block) which must be equal to the usual conformal block expansion
[11, 12, 13]. The validity of this has been rigorously proven by the 1d
analytic functionals. In 1d, Polyakov blocks also give a simple way to
determine action of the functionals, which are otherwise hard to compute. This
connection has been recently used in [14] for general global symmetries to
obtain a number of interesting numerical results.

Recently in [15, 16] a crossing symmetric dispersion relation has been
proposed which establishes the Polyakov Bootstrap in general $d$. The main
idea here, which is based on [17], is to work in Mellin space and map the
Mellin variables to new variables that are manifestly crossing symmetric but
with nonlocal singularities. A dispersion relation is then obtained in these
variables, which reduces to a crossing symmetric sum of exchange and contact
Witten diagrams when one sets the nonlocal singularities to zero (locality
constraints). The dispersion relation is also used in the QFT context
(replacing Mellin variables with Mandelstam variables) where these locality
constraints have been used to derive interesting bounds on Effective Field
Theories (EFTs).

Let us now address a situation where crossing symmetry of a 4-point function
involves more than just crossing symmetric functions. E.g. consider a CFT with
global symmetry with a correlator $\mathcal{G}_{ijkl}(z,\bar{z})$ of charged
fields $\phi_{i}$ transforming in some irrep $\mathfrak{r}$. If
$\mathfrak{r}\otimes\mathfrak{r}$ contains the irreps labelled $\mathfrak{a}$
then the correlator decomposes as

$\mathcal{G}_{ijkl}(z,\bar{z})=\sum_{\mathfrak{a}}T^{\mathfrak{a}}_{ij,kl}\mathcal{G}^{\mathfrak{a}}(z,\bar{z})$
(1.1)

where $T^{\mathfrak{a}}$ denotes the associated tensor structure of
$\mathfrak{a}$. We use the notations of usual CFT kinematics, reviewed in
section 2. The crossing equation for $\mathcal{G}^{\mathfrak{a}}$ can be
written as:

$\mathcal{G}^{\mathfrak{a}}(z,\bar{z})=C^{\mathfrak{a}\mathfrak{b}}\mathcal{G}^{\mathfrak{b}}(1-z,1-\bar{z})\,.$
(1.2)

Here $C^{\mathfrak{a}\mathfrak{b}}$ is a crossing matrix (see e.g. [14]) that
has eigenvalues $\pm 1$ corresponding to eigenvectors
$e^{\mathfrak{a}}_{\pm,s}$ where $s$ is a label. Now, while the functions
$\mathcal{G}_{-}(z,\bar{z})=e^{\mathfrak{a}}_{+,s}\mathcal{G}^{\mathfrak{a}}(z,\bar{z})$,
for any $s$, obey usual crossing symmetry
$\mathcal{G}_{-}(z,\bar{z})=\mathcal{G}_{-}(1-z,1-\bar{z})$, the combinations
$\mathcal{G}_{+}(z,\bar{z})=e^{\mathfrak{a}}_{-,s}\mathcal{G}^{\mathfrak{a}}(z,\bar{z})$
are functions that have crossing antisymmetry:

$\mathcal{G}_{+}(z,\bar{z})=-\mathcal{G}_{+}(1-z,1-\bar{z})\,.$ (1.3)

An obvious question to ask is: does $\mathcal{G}_{+}$ also allow a Witten
diagram like expansion? In 1d there exists a set of analytic functionals,
called $+$ type functionals, which can bootstrap such antisymmetric functions
although it is not understood if they are related to a Polyakov block.111In
[14] this was bypassed by considering analytic functionals dual to GFF for a
certain global symmetry which are related to crossing symmetric Polyakov
blocks of the same symmetry [18, 19]. It would be nice to understand this more
generally in $d\geq 1$.

In this paper we show that, analogous to the crossing symmetric case, crossing
antisymmetric functions can be expanded in terms of manifestly crossing
antisymmetric objects as follows:

$\mathcal{G}_{+}(z,\bar{z})=\sum_{\Delta,\ell}a_{\Delta,\ell}\mathcal{P}_{+,\Delta,\ell}(z,\bar{z})\,,$
(1.4)

where $\mathcal{P}_{+,\Delta,\ell}$, the ‘$+$ type Polyakov block’, is a
linear combination of AdSd+1 Witten diagrams. In 1d these blocks involve a
finite number of diagrams similar to its crossing symmetric counterpart. They
are related to the $+$ type analytic functionals. We construct two sets of
blocks corresponding to the bosonic and fermionic functional bases.

The idea of a $+$ type Polyakov block is then extended to general dimensions,
which requires an infinite number of exchange Witten diagrams and crossing
antisymmetric contact diagrams. For this we show that a crossing antisymmetric
function can be obtained from a dispersion relation in Mellin space. Analogous
to the crossing symmetric case, it is characterized by an infinite number of
nonlocal singularities that lead to a new set of locality constraints. The
dispersion relation can be reformulated as the Witten diagram expansion once
the locality conditions are satisfied. Finally we give a set of crossing
antisymmetric sum rules.

We have numerically demonstrated how locality conditions work, and how a known
crossing antisymmetric solution satisfies the new sum rules. We show how the
$+$ type Polyakov blocks simplify computation of ‘product functionals’ which
exist in even dimensions and are built by combining 1d functionals [20]. We
also show one can obtain a set of ‘simple functionals’ for global symmetry
problems in general dimensions from the Polyakov blocks extending the
arguments of [14].

The paper is organized as follows: in section 2 we review the $\pm$ type 1d
functionals and introduce the $+$ type bosonic and fermionic Polyakov blocks.
In section 3 we generalize the $+$ type Polyakov blocks to arbitrary
dimensions and introduce a crossing antisymmetric dispersion relation. In
section 4 we show how the previous findings simplify various $d\geq 1$
functionals. We conclude in section 5. There are four appendices elaborating
on the many technical details and numerical checks.

Notations: Throughout this paper we will use the subscripts $-$/$+$ to
indicate crossing symmetric/antisymmetric equations or quantities. This,
rather counter-intuitive, notation is for consistency with the analytic
functional literature.

##  2 1d crossing antisymmetric functionals

###  2.1 Basic kinematics

Let us define some basic kinematics and notations. Consider a 4-point function
$\mathcal{G}(x_{1},x_{2},x_{3},x_{4})$ of scalars (say $\phi(x_{i})$) with
identical dimension $\Delta_{\phi}$ in a $d$-dimensional CFT which is given by

$\mathcal{G}(x_{1},x_{2},x_{3},x_{4})=\frac{\mathcal{G}(z,\bar{z})}{x_{13}^{2\Delta_{\phi}}x_{24}^{2\Delta_{\phi}}}\,,\
\ z\bar{z}=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}\,,\
(1-z)(1-\bar{z})=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}\,.$ (2.1)

We will sometimes use the notations $u=z\bar{z}$ and $v=(1-z)(1-\bar{z})$. The
quantity $\mathcal{G}(z,\bar{z})$ can be decomposed into conformal blocks as
follows:

$\mathcal{G}(z,\bar{z})=\sum_{\Delta,\ell}a_{\Delta,\ell}G_{\Delta,\ell}^{(d)}(z,\bar{z}|\Delta_{\phi})\,.$
(2.2)

Here $G_{\Delta,\ell}^{(d)}(z,\bar{z}|\Delta_{\phi})$ is the $d$-dimensional
conformal block (defined with a factor of $(z\bar{z})^{-\Delta_{\phi}}$) and
$a_{\Delta,\ell}$ denotes square of OPE coefficients. Typically the function
$\mathcal{G}(z,\bar{z})=\mathcal{G}_{-}(z,\bar{z})$ is crossing symmetric i.e.
$\mathcal{G}_{-}(z,\bar{z})=\mathcal{G}_{-}(1-z,1-\bar{z})$ (symmetry under
$x_{1}\leftrightarrow x_{3}$ exchange) and we may write

$\sum_{\Delta,\ell}a_{\Delta,\ell}F_{-,\Delta,\ell}(z,\bar{z}|\Delta_{\phi})=0$
(2.3)

where we defined the crossing symmetric bootstrap vector

$F_{-,\Delta,\ell}(z,\bar{z}|\Delta_{\phi})=G_{\Delta,\ell}(z,\bar{z}|\Delta_{\phi})-G_{\Delta,\ell}(1-z,1-\bar{z}|\Delta_{\phi})\,.$
(2.4)

As shown in (1.3) it is often necessary to consider
$\mathcal{G}(z,\bar{z})=\mathcal{G}_{+}(z,\bar{z})$ which is antisymmetric
under $x_{1}\leftrightarrow x_{3}$ exchange i.e.
$\mathcal{G}_{+}(z,\bar{z})=-\mathcal{G}_{+}(1-z,1-\bar{z})$. If it decomposes
into conformal blocks like (2.2) we have the crossing antisymmetry equation

$\sum_{\Delta,\ell}a_{\Delta,\ell}F_{+,\Delta,\ell}(z,\bar{z}|\Delta_{\phi})=0$
(2.5)

where we have the crossing antisymmetric bootstrap vector

$F_{+,\Delta,\ell}(z,\bar{z}|\Delta_{\phi})=G_{\Delta,\ell}(z,\bar{z}|\Delta_{\phi})+G_{\Delta,\ell}(1-z,1-\bar{z}|\Delta_{\phi})\,.$
(2.6)

We will sometimes loosely refer to the crossing antisymmetric equation as
‘anticrossing equation’. It becomes important in many interesting problems
like bootstrapping CFTs with global symmetries or multiple correlators.

For the rest of this section we will focus on 1d CFTs. Here we have a single
cross-ratio, so we set $z=\bar{z}$. The (anti)crossing equation is written as

$\sum_{\Delta}a_{\Delta}F_{\pm,\Delta}(z|\Delta_{\phi})=0$ (2.7)

where we have

$F_{\pm}(z|\Delta_{\phi})=G_{\Delta}(z|\Delta_{\phi})\pm
G_{\Delta}(1-z|\Delta_{\phi})$ (2.8)

and
$G_{\Delta}(z|\Delta_{\phi})=z^{\Delta-2\Delta_{\phi}}{}_{2}F_{1}(\Delta,\Delta,2\Delta,z)$
is the $SL(2,\mathbb{R})$ conformal block .

###  2.2 1d analytic functionals and Polyakov Bootstrap

The usual approach to constrain CFT data from the (anti)crossing equation is
by introducing a set of linear functionals as follows
($\omega(\Delta|\Delta_{\phi})\equiv\omega[F_{\Delta}(z|\Delta_{\phi})]$)

$\omega_{\pm}\Big{[}\sum_{\Delta}a_{\Delta}F_{\pm,\Delta}(z|\Delta_{\phi})\Big{]}=\sum_{\Delta}a_{\Delta}\omega_{\pm}(\Delta|\Delta_{\phi})=0$
(2.9)

In the standard conformal bootstrap approach one chooses a basis of
derivatives $\omega_{\pm}=\\{\partial_{z},\partial_{z}^{2},\cdots\\}$ at
$z=\frac{1}{2}$ . In [2, 3] a set of functionals that gives optimal bounds on
OPE coefficients were proposed. These are the analytic functionals which
correspond to a basis expansion of bootstrap vectors in terms of double trace
operator blocks as follows:

$F_{\pm,\Delta}(z|\Delta_{\phi})=\sum_{n}\Big{[}\alpha^{B,F}_{\pm,n}(\Delta)\,F_{\pm,\Delta_{n}^{B,F}}(z|\Delta_{\phi})+\beta^{B,F}_{\pm,n}\,\partial_{\Delta}F_{\pm,\Delta_{n}^{B,F}}(z|\Delta_{\phi})\Big{]}\,.$
(2.10)

Here we have shown two set of bases: Bosonic ($B$) and Fermionic ($F$) which
correspond to $\Delta_{n}^{B}=2\Delta_{\phi}+2n$ and
$\Delta_{n}^{F}=2\Delta_{\phi}+2n+1$. Each analytic functional
$\omega_{\pm}=\alpha_{\pm},\beta_{\pm}$ is dual to a basis element above. It
is defined as follows:

$\omega_{\pm}(\Delta|\Delta_{\phi})=\int_{\frac{1}{2}}^{\frac{1}{2}+i\infty}dz\,f_{\pm}(z)F_{\pm,\Delta}(z|\Delta_{\phi})+\int_{\frac{1}{2}}^{1}dz\,g_{\pm}(z)F_{\pm,\Delta}(z|\Delta_{\phi})\,.$
(2.11)

The functions $f$ and $g$ satisfy the conditions

$f_{\pm}(z)=\mp f_{\pm}(1-z)\,,\ \mathcal{R}_{z}f_{\pm}(z)=-g_{\pm}(z)\pm
g_{\pm}(1-z)\,,\
g_{\pm}(z)=\epsilon(1-z)^{2\Delta_{\phi}-2}f_{\pm}(\frac{z}{z-1})\,.$ (2.12)

Here $\mathcal{R}_{z}$ denotes the real part, and $\epsilon=+1(-1)$ for
bosonic (fermionic) case. It is then possible to choose the kernel $f$ so that
the functionals satisfy the orthogonality properties:

$\displaystyle\alpha_{\pm,n}^{B}(\Delta_{m}^{B})$
$\displaystyle=\delta_{mn}\,,\hskip
28.45274pt\partial_{\Delta}\alpha_{\pm,n}^{B}(\Delta_{m}^{B})=-c^{B}_{\pm,n}(\Delta_{\phi})\delta_{m0}\,,$
$\displaystyle\beta_{\pm,n}^{B}(\Delta_{m}^{B})$ $\displaystyle=0\,,\hskip
39.83368pt\partial_{\Delta}\beta_{\pm,n}^{B}(\Delta_{m}^{B})=\delta_{mn}-d_{\pm,n}^{B}(\Delta_{\phi})\delta_{m,0}\,.$
(2.13)

Here we show the bosonic case with $\beta^{B}_{\pm,0}=0$ and
$c^{B}_{\pm,n},d^{B}_{\pm,n}$ are known. The fermionic functionals satisfy
similar conditions replacing $B\to F$ and with $\beta^{F}_{\pm,0}\neq 0$,
$c^{F}_{\pm,n}=d^{F}_{\pm,n}=0$ . Explicit forms of all kernels can be found
in Appendix A of [20].

A feature of the kernels that will be important in our discussion is the Regge
limit $z\to i\infty$. These are as follows:

$\displaystyle\omega_{-}:$ $\displaystyle\ \ f(z)\stackrel{{\scriptstyle z\to
i\infty}}{{\sim}}O(z^{-2})$ $\displaystyle\omega_{+}:$ $\displaystyle\ \
f(z)\stackrel{{\scriptstyle z\to i\infty}}{{\sim}}O(z^{-3})\,.$ (2.14)

This feature implies that $\omega_{-}$ ($\omega_{+}$) functionals can
bootstrap (anti)crossing solutions with an $O(z^{0})$ ($O(z)$) large $z$
behavior.

It has been established that ‘$-$ type’ analytic functionals are related to 1d
Polyakov Bootstrap in an interesting way. The latter is the idea that a
crossing symmetric correlator can be expanded in terms of manifestly crossing
symmetric functions, called Polyakov blocks, in the following way

$\mathcal{G}_{-}(z)=\sum_{\Delta}a_{\Delta}\mathcal{P}^{B,F}_{-,\Delta}(z)=\sum_{\Delta}a_{\Delta}G_{\Delta}(z)\,.$
(2.15)

The equality on the right is the usual OPE. The Polyakov block
$\mathcal{P}_{-,\Delta}(z)$ is given by a crossing symmetric sum of 4-point
Witten diagrams

$\mathcal{P}^{B,F}_{-,\Delta}(z)=W_{\Delta,0}^{(s),B,F}(z)+W_{\Delta,0}^{(t),B,F}(z)+W_{\Delta,0}^{(u),B,F}(z)+\kappa_{-}^{B,F}\,\mathcal{C}_{-}(z)\,.$
(2.16)

Here $W_{\Delta,0}^{(i),B,F}(z)$ is a Witten exchange diagram (spin $0$ and
dimension $\Delta$ exchange in $i=s,t,u$ channels) with bosonic/fermionic legs
and drawn with a suitable choice of vertex. The $\mathcal{C}_{-}(z)$ is a
crossing symmetric 4-point contact diagram. We require the Polyakov block to
be Regge bounded i.e. $\mathcal{P}_{-,\Delta}(z)<\infty$ as $z\to i\infty$.
This allows a single contact diagram with bosonic external legs with
$\Phi^{4}$ vertex. We define the Witten diagrams in Appendix A.

The Witten diagrams above can be decomposed into conformal blocks of
dimensions $\Delta_{n}^{B,F}$. We can choose Polyakov blocks such that their
block decomposition computes functional actions (dropping $\Delta_{\phi}$
dependence for convenience) as follows:

$\mathcal{P}^{B,F}_{-,\Delta}(z)=G_{\Delta}(z)-\sum_{n}\Big{[}\alpha^{B,F}_{-,n}(\Delta)G_{\Delta^{B,F}_{n}}(z)+\beta^{B,F}_{-,n}(\Delta)\partial_{\Delta}G_{\Delta_{n}^{B,F}}(z)\Big{]}\,.$
(2.17)

So the statement of Polyakov Bootstrap (2.15) is identical to the functional
bootstrap equations. While for the fermionic case this is automatic
($\kappa_{-}^{F}=0$), for the bosonic case one can choose $\kappa_{-}^{B}$
such that $\beta^{B}_{-,0}=0$ and have the above correspondence. Notice that
the basis (2.10) is nothing but the crossing symmetry equation for
$\mathcal{P}_{-,\Delta}(z)$ . For more details of Polyakov bootstrap in 1d see
[3, 14].

###  2.3 The ‘+ type’ Polyakov blocks

An obvious question that one can now ask is if there exists an analogue of
Polyakov blocks that computes ‘+ type’ functional actions. Indeed there is
such a function which we call a ‘$+$ type Polyakov block’. In this section we
show how to build it from familiar Witten diagrams.

We introduce the following notation for convenience:

$\displaystyle\Delta_{n,\ell}=2\Delta_{\phi}+2n+\ell\,.$ (2.18)

This is for consistency with the following sections. Note that we have
$\Delta_{n,0}=\Delta^{B}_{n}$ and $\Delta_{n,1}=\Delta^{F}_{n}$ .

####  2.3.1 Fermionic case

Let us first discuss the case of fermionic functionals. Since $+$ type
functionals act on crossing antisymmetric vectors, the $+$ type Polyakov
blocks must be crossing antisymmetric. Consider spin 1 exchange Witten
diagrams in AdS2 with _bosonic_ external legs. They have the conformal block
decomposition (for this subsection we write $W_{\Delta,1}^{(i),B}\to
W_{\Delta,1}^{(i)}$):

$W_{\Delta,1}^{(s)}=G_{\Delta}(z)+\sum_{n}\big{(}a_{n,1}^{(s)}G_{\Delta_{n,1}}(z)+b_{n,1}^{(s)}\partial_{\Delta}G_{\Delta_{n,1}}(z)\big{)}\,.$
(2.19)

In the crossed channels we have

$\displaystyle W_{\Delta,1}^{(-)}(z)$
$\displaystyle=\sum_{n}\big{(}a_{n,1}^{(t)}G_{\Delta_{n,0}}(z)+b_{n,1}^{(t)}\partial_{\Delta}G_{\Delta_{n,0}}(z)\big{)}\,,$
$\displaystyle W_{\Delta,1}^{(+)}(z)$
$\displaystyle=\sum_{n}\big{(}\bar{a}_{n,1}^{(t)}G_{\Delta_{n,1}}(z)+\bar{b}_{n,1}^{(t)}\partial_{\Delta}G_{\Delta_{n,1}}(z)\big{)}\,.$
(2.20)

Here we defined $W_{\Delta,1}^{(\pm)}=\frac{1}{2}\big{(}W_{\Delta,1}^{(t)}\pm
W_{\Delta,1}^{(u)}\big{)}$ . The expressions for $W^{(s)}_{\Delta,1}$ and
other useful details are given Appendix A (see [14, 21, 22] for more on the
conformal block decomposition). The crossed channel diagrams are related to
the $s$-channel via:

$W_{\Delta,1}^{(t)}(z)=W_{\Delta,1}^{(s)}(1-z)\,,\hskip
28.45274ptW_{\Delta,1}^{(u)}(z)=(1-z)^{-2\Delta_{\phi}}\text{Re}W_{\Delta,1}^{(s)}\big{(}\frac{1}{1-z}\big{)}\,.$
(2.21)

We also have

$W_{\Delta,1}^{(u)}(1-z)=-W_{\Delta,1}^{(u)}(z)\,.$ (2.22)

A natural crossing antisymmetric Polyakov block is then

$\displaystyle\mathcal{P}_{+,\Delta}^{F}(z)$
$\displaystyle=W_{\Delta,1}^{(s)}(z)-W_{\Delta,1}^{(t)}(z)-W_{\Delta,1}^{(u)}(z)\,.$
$\displaystyle=W_{\Delta,1}^{(s)}(z)-2W_{\Delta,1}^{(+)}(z)\,.$ (2.23)

Notice that this object is totally crossing antisymmetric i.e.
$\mathcal{P}_{+}(z)\to-\mathcal{P}_{+}(z)$ under $x_{1}\leftrightarrow x_{3}$
or $x_{1}\leftrightarrow x_{4}$. It is also antisymmetric in
$x_{1}\leftrightarrow x_{2}$ which is not obvious from (2.3.1) although it is
clear from their Mellin amplitudes (see Appendix A and the next
section).222All $x_{i}\leftrightarrow x_{j}$ are equivalent to simple
transformations of Mellin variables, e.g.: $x_{1}\leftrightarrow x_{3}$ is
equivalent to $s\leftrightarrow t+\Delta_{\phi}$. The anticrossing property
implies the following:

$\displaystyle\mathcal{P}_{+\Delta}(z)=G_{\Delta}(z)-\sum_{n}\Big{[}\alpha^{F}_{+,n}(\Delta)G_{\Delta_{n,1}}(z)+\beta^{F}_{+,n}(\Delta)\partial_{\Delta}G_{\Delta_{n,1}}(z)\Big{]}$
$\displaystyle\implies$ $\displaystyle
F_{+,\Delta}(z)=\sum_{n}\Big{[}\alpha^{F}_{+,n}(\Delta|\Delta_{\phi})F_{+,\Delta_{n,1}}(z)+\beta^{F}_{+,n}(\Delta|\Delta_{\phi})\partial_{\Delta}F_{+,\Delta_{n,1}}(z)\Big{]}\,,$
(2.24)

where the coefficients are given by

$\displaystyle\alpha^{F}_{+,n}(\Delta|\Delta_{\phi})$
$\displaystyle=-a_{n,1}^{(s)}+2\bar{a}_{n,1}^{(t)}\,,$
$\displaystyle\beta^{F}_{+,n}(\Delta|\Delta_{\phi})$
$\displaystyle=-b_{n,1}^{(s)}+2\bar{b}_{n,1}^{(t)}\,.$ (2.25)

The coefficients $\alpha^{F}_{+,n}(\Delta|\Delta_{\phi})$ and
$\beta^{F}_{+,n}(\Delta|\Delta_{\phi})$ are named as such since they are
equivalent to the + type fermionic functional actions. It can be verified by
explicitly evaluating the actions from (2.11) that they match the expressions
(2.3.1).

We now point out an interesting feature. For the $-$ type bosonic functionals
the missing $\beta_{-,0}^{B}$ functional is attributed to the ambiguity of
addition by a $\Phi^{4}$ contact Witten diagram to any crossing solution that
preserves Regge boundedness [3]. Similarly for the fermionic case there is no
such ambiguity, and hence all $\omega_{-,n}^{F}$ are present. Now recall from
(2.2) that $\omega_{+}$ functionals can bootstrap anti-crossing symmetric
solutions with a fall-off $O(z)$ . It turns out that there is no anticrossing
‘contact’ diagram i.e. a crossing antisymmetric solution which decomposes into
blocks and their derivatives of dimenisions $\Delta_{n,1}$ (only), and also
has the Regge fall-off $O(z)$.333In section 3.1 we will see that the
anticrossing contact diagram with the strictest Regge behavior is
$\stackrel{{\scriptstyle s\to\infty}}{{\sim}}O(s^{3})$ in Mellin
representation. Typically for an $s^{2j}$ behavior in Mellin space we have in
position space  $\mathcal{G}(z)\stackrel{{\scriptstyle z\to
i\infty}}{{\sim}}z^{2j-1}\,.$ (2.26) This can be proved by assuming a vertex
of the form $(\partial^{j}\Phi)^{4}$ in AdS2 and computing a 4-point contact
diagram [23, 24].  This is consistent with the fact that fermionic $+$ type
functionals $\omega_{+,n}^{F}$ exist for all $n\geq 0$. If there was such an
anticrossing contact diagram, we would have had to subtract it from (2.3.1) to
set the coefficient of one of the blocks to zero.

####  2.3.2  Bosonic case

Now let us consider the other case, bosonic functionals. One can guess that
they require Witten diagrams with fermionic external legs. In [25] the general
$d$ fermionic Witten diagrams with a scalar exchange was computed (see also
[26]). The $s$-channel scalar exchange Witten diagram in the AdS2 case is
given in terms of the bosonic Witten diagram as follows

$W^{(s),F}_{\Delta,0}(z|\Delta_{\phi})=zW^{(s),B}_{\Delta,0}\big{(}z|\Delta_{\phi}+\frac{1}{2}\big{)}\,.$
(2.27)

The factor of $z$ is to account for the $z^{-2\Delta_{\phi}}$ included in our
definitions. In a similar way let us define the following object:

$W^{(s),F}_{\Delta,1}(z|\Delta_{\phi})=\frac{1}{z}W^{(s),B}_{\Delta,1}\big{(}z|\Delta_{\phi}-\frac{1}{2}\big{)}\,.$
(2.28)

We will refer to this as the ‘fermionic Witten diagram’ with an $s$-channel
spin 1 exchange. Note that computing the Witten diagram with four boundary-
bulk fermionic propagators and bulk-bulk spin 1 propagator is a non-trivial
task which we do not perform. However it should be related to the above object
by addition of a suitable contact diagram. The diagram we defined above
already has everything we need for our discussion.

If the fermionic Witten $s$-channel diagram is defined as above, the
corresponding crossed channel diagrams would be given by

$W_{\Delta,1}^{(t),F}(z|\Delta_{\phi})=W_{\Delta,1}^{(s),F}(1-z)\,,\hskip
28.45274ptW_{\Delta,1}^{(u),F}(z|\Delta_{\phi})=(1-z)^{-2\Delta_{\phi}}\text{Re}W_{\Delta,1}^{(s),F}\big{(}\frac{1}{1-z}\big{)}\,.$
(2.29)

Once again we have

$W_{\Delta,1}^{(u),F}(1-z|\Delta_{\phi})=-W_{\Delta,1}^{(u),F}(z|\Delta_{\phi})\,.$
(2.30)

The block decomposition of the above objects are as follows:

$\displaystyle W_{\Delta,1}^{(s),F}(z)$
$\displaystyle=G_{\Delta}(z)+\sum_{n}\big{(}a_{n,1}^{(s),F}G_{\Delta_{n,0}}(z)+b_{n,1}^{(s),F}\partial_{\Delta}G_{\Delta_{n,0}}(z)\big{)}\,.$
$\displaystyle W_{\Delta,1}^{(-),F}(z)$
$\displaystyle=\sum_{n}\big{(}a_{n,1}^{(t),F}G_{\Delta_{n,1}}(z)+b_{n,1}^{(t),F}\partial_{\Delta}G_{\Delta_{n,1}}(z)\big{)}\,,$
$\displaystyle W_{\Delta,1}^{(+),F}(z)$
$\displaystyle=\sum_{n}\big{(}\bar{a}_{n,1}^{(t),F}G_{\Delta_{n,0}}(z)+\bar{b}_{n,1}^{(t),F}\partial_{\Delta}G_{\Delta_{n,0}}(z)\big{)}\,,$
(2.31)

where $W_{\Delta,1}^{(\pm),F}=\frac{1}{2}\big{(}W_{\Delta,1}^{(t),F}\pm
W_{\Delta,1}^{(u),F}\big{)}$ .

With this we define an anticrossing object as follows:

$\displaystyle\mathcal{P}_{+}^{\text{exc}}(z)$
$\displaystyle=W_{\Delta,1}^{(s),F}(z)-W_{\Delta,1}^{(t),F}(z)-W_{\Delta,1}^{(u),F}(z)\,.$
$\displaystyle=W_{\Delta,1}^{(s),F}(z)-2W_{\Delta,1}^{(+),F}(z)\,.$ (2.32)

The ‘exc’ stands for ‘exchange’. Under block decompositon
$\mathcal{P}_{+}^{\text{exc}}$ gives a $G_{\Delta}(z)$ and
$G_{\Delta_{n,0}}(z)$ $\forall n\in\mathbb{Z}_{+,0}$ .

However $\mathcal{P}_{+}^{\text{exc}}$ is not the Polyakov block yet. In fact
there exists another antisymmetric object which has a similar property. It is
the analogue of a contact diagram, and is given by

$\mathcal{P}_{+}^{\text{Con}}(z)=\mathcal{C}_{+}^{(s),F}(z)-\mathcal{C}_{+}^{(t),F}(z)-\mathcal{C}_{+}^{(u),F}(z)\,.$
(2.33)

Here the first term on r.h.s. can be written in terms of a general dimension
Mellin representation as follows:

$\mathcal{C}_{+}^{(s),F}(z)=z^{-2\Delta_{\phi}}\Bigg{[}\int_{-i\infty}^{i\infty}ds\,dt\,z^{2s}\,(1-z)^{2t}\Gamma^{2}(\Delta_{\phi}+\frac{1}{2}-s)\Gamma^{2}(-t)\Gamma^{2}(s+t)\big{(}s+2t\big{)}\Bigg{]}\,.$
(2.34)

The other two terms (crossed channels) are given by 444Here
$\mathcal{C}_{+}^{(t),F}$ and $\mathcal{C}_{+}^{(u),F}$ can be obtained by
replacing $\Delta_{\phi}+\frac{1}{2}-s\leftrightarrow-t$ and
$\Delta_{\phi}+\frac{1}{2}-s\leftrightarrow s+t$ in the Mellin integral
respectively. Rescaling the prefactor leads to the respective extra factors
$\frac{1-z}{z}$ and $\frac{1}{z}$ in the Mellin amplitudes.:

$\mathcal{C}_{+}^{(t),F}(z)=\mathcal{C}_{+}^{(s),F}(1-z)\,,\hskip
28.45274pt\mathcal{C}_{+}^{(u),F}(z)=(1-z)^{-2\Delta_{\phi}}\text{Re}\,\mathcal{C}_{+}^{(s),F}\big{(}\frac{1}{1-z}\big{)}\,.$
(2.35)

It is easy to verify that

$\mathcal{C}_{+}^{(u),F}(1-z)=-\mathcal{C}_{+}^{(u),F}(z)\,.$ (2.36)

Furthermore we have the block decomposition

$\displaystyle\mathcal{C}_{+}^{(s),F}(z)$
$\displaystyle=\sum_{n}\big{(}a_{n,C}^{(s),F}G_{\Delta_{n,0}}(z)+b_{n,C}^{(s)}\partial_{\Delta}G_{\Delta_{n,0}}(z)\big{)}\,.$
$\displaystyle\mathcal{C}_{+}^{(-),F}(z)$
$\displaystyle=\sum_{n}\big{(}a_{n,C}^{(t),F}G_{\Delta_{n,1}}(z)+b_{n,C}^{(t)}\partial_{\Delta}G_{\Delta_{n,1}}(z)\big{)}\,,$
$\displaystyle\mathcal{C}_{+}^{(+),F}(z)$
$\displaystyle=\sum_{n}\big{(}\bar{a}_{n,C}^{(t),F}G_{\Delta_{n,0}}(z)+\bar{b}_{n,C}^{(t)}\partial_{\Delta}G_{\Delta_{n,0}}(z)\big{)}\,,$
(2.37)

where
$\mathcal{C}_{+}^{(\pm),F}=\frac{1}{2}\big{(}\mathcal{C}_{+}^{(t),F}\pm\mathcal{C}_{+}^{(u),F}\big{)}$
.

At large $z$ the anticrossing contact diagram
$\mathcal{P}_{+}^{\text{Con}}(z)\sim O(z)$. So any solution obtained from
analytic $+$ type bosonic functionals can be deformed by it. This is an
ambiguity that should be taken into account in order to have the correct
Polyakov block. Under block decomposition the Polyakov block must have one
coefficient zero, so to reflect this ambiguity in the sum rules i.e.
functional actions (see the last paragraph of previous subsection).

Hence we define the _bosonic_ anti-crossing symmetric Polyakov block as
follows

$\displaystyle\mathcal{P}_{+}(z)$
$\displaystyle=\mathcal{P}_{+}^{\text{exc}}(z)+k\,\mathcal{P}_{+}^{\text{Con}}(z)\,$
$\displaystyle=G_{\Delta}(z)-\sum_{n}\Big{[}\alpha^{B}_{+,n}(\Delta)G_{\Delta_{n,0}}(z)+\beta^{B}_{+,n}(\Delta)\partial_{\Delta}G_{\Delta_{n,0}}(z)\Big{]}\,,$
(2.38)

which implies the following relation

$F_{+,\Delta}(z)=\sum_{n}\Big{[}\alpha_{+,n}^{B}(\Delta|\Delta_{\phi})F_{+,\Delta_{n,0}}(z)+\beta_{+,n}^{B}(\Delta|\Delta_{\phi})\partial_{\Delta}F_{+,\Delta_{n,0}}(z)\Big{]}\,.$
(2.39)

The constant $k$ in (2.3.2) is chosen such that
$\beta_{+,0}^{B}(\Delta|\Delta_{\phi})=0$. This ensures that all the
coefficients $\alpha_{+,n}^{B}(\Delta|\Delta_{\phi})$ and
$\beta_{+,n}^{B}(\Delta|\Delta_{\phi})$ are actions of the bosonic + type
functionals $\alpha_{+,n}^{B}$ and $\beta_{+,n}^{B}$ respectively. This can be
easily verified.

To summarize this section we have shown the existence of two + type Polyakov
blocks, ‘bosonic’ and ‘fermionic’. They are built from known Witten diagrams
in AdS2. Through their conformal block decomposition shown in (2.3.1) and
(2.3.2) they compute the actions of + type analytic functionals.

##  3 General $d$ Polyakov blocks and dispersion relation

In this section we inititate a new formulation in general dimension to analyse
crossing antisymmetric functions. This is analogous to the usual Polyakov
Bootstrap for the crossing symmetric case. We propose a crossing antisymmetric
basis built from Witten diagrams. This is then explicitly demonstrated by
setting up a dispersion relation for Mellin amplitudes of the functions. Most
of the discussions in this section is formulated in Mellin space.

Let us define a new Polyakov block $\mathcal{P}_{+,\Delta,\ell}(u,v)$ in terms
of which an anticrossing correlator can be expanded. We would like to have
$\mathcal{P}_{+,\Delta,\ell}(u,v)$ to be a completely crossing antisymmetric
object. By this we mean the following:

$\mathcal{P}_{+,\Delta,\ell}(u,v)=-\mathcal{P}_{+,\Delta,\ell}(v,u)=-u^{-\Delta_{\phi}}\mathcal{P}_{+,\Delta,\ell}(1/u,v/u)=-v^{-\Delta_{\phi}}\mathcal{P}_{+,\Delta,\ell}(u/v,1/v)\,.$
(3.1)

It is possible to write such an object in terms of Witten exchange diagrams
$W_{\Delta,\ell}(u,v)$ for odd $\ell$ in the following way (see Figure 1)

$\mathcal{P}_{+,\Delta,\ell}(u,v)=W_{\Delta,\ell}^{(s)}(u,v)-W_{\Delta,\ell}^{(t)}(u,v)-W_{\Delta,\ell}^{(u)}(u,v)+\text{contacts}\,.$
(3.2)

Figure 1:  The Polyakov block $\mathcal{P}_{+,\Delta,\ell}$ is a crossing
antisymmetric combination of Witten exchange diagrams $W_{\Delta,\ell}(u,v)$
(with odd spin $\ell$) and antisymmetric ‘contact’ diagrams’
$\mathcal{C}_{+}(u,v)$.

These diagrams are all defined in AdSd+1 with identical external scalars of
dimension $\Delta_{\phi}$ . It is convenient to write them in the Mellin
representation as follows:

$W_{\Delta,\ell}(u,v)=\int
ds_{1}ds_{2}\,u^{s_{1}+\frac{2\Delta_{\phi}}{3}}v^{s_{2}-\frac{\Delta_{\phi}}{3}}\Bigg{[}\prod_{i=1}^{3}\Gamma^{2}\big{(}\frac{\Delta_{\phi}}{3}-s_{i}\big{)}\Bigg{]}M_{\Delta,\ell}(s_{1},s_{2})\,,$
(3.3)

where $s_{3}=-s_{1}-s_{2}$. The $M_{\Delta,\ell}(s_{1},s_{2})$ is a Mellin
amplitude defined in Appendix A. The ‘contacts’ denote crossing antisymmetric
functions $\mathcal{C}_{+}(u,v)$ with polynomial Mellin amplitudes which we
discuss in a moment. We follow the conventions of [16, 15]. The $s_{i}$ are
related to the $s,t,u$ variables of previous section by the simple shifts
$s_{1}=s-\frac{2\Delta_{\phi}}{3}$ and $s_{2}=\frac{\Delta_{\phi}}{3}+t$.

The totally crossing antisymmetric property (3.1) directly follows from the
odd spin Mack polynomials in the Mellin amplitudes, and is easily seen from
the meromorphic form (A.3). The exchange Witten diagrams have the following
conformal block decomposition:

$\displaystyle W_{\Delta,\ell}^{(s)}(u,v)$
$\displaystyle=G_{\Delta,\ell}(z)+\sum_{n}\big{[}a_{n,\ell}^{(s)}(\Delta|\Delta_{\phi})G_{\Delta_{n,\ell},\ell}(u,v)+b_{n,\ell}^{(s)}(\Delta|\Delta_{\phi})\partial_{\Delta}G_{\Delta_{n,\ell},\ell}(u,v)\big{]}\,,$
$\displaystyle W_{\Delta,\ell}^{(-)}(u,v)$
$\displaystyle=\sum_{n}\sum_{\ell^{\prime}\text{even}}\big{[}a_{n,\ell^{\prime}|\ell}^{(t)}(\Delta|\Delta_{\phi})G_{\Delta_{n,\ell^{\prime}},\ell}(u,v)+b_{n,\ell^{\prime}|\ell}^{(t)}(\Delta|\Delta_{\phi})\partial_{\Delta}G_{\Delta_{n,\ell^{\prime}},\ell^{\prime}}(u,v)\big{]}\,,$
$\displaystyle W_{\Delta,\ell}^{(+)}(u,v)$
$\displaystyle=\sum_{n}\sum_{\ell^{\prime}\text{odd}}\big{[}\bar{a}_{n,\ell^{\prime}|\ell}^{(t)}(\Delta|\Delta_{\phi})G_{\Delta_{n,\ell^{\prime}},\ell^{\prime}}(u,v)+\bar{b}_{n,\ell^{\prime}|\ell}^{(t)}(\Delta|\Delta_{\phi})\partial_{\Delta}G_{\Delta_{n,\ell^{\prime}},\ell^{\prime}}(u,v)\big{]}\,.$
(3.4)

Here $W_{\Delta,\ell}^{(\pm)}=\frac{1}{2}\big{(}W_{\Delta,\ell}^{(t)}\pm
W_{\Delta,\ell}^{(u)}\big{)}$ and $\Delta_{n,\ell}=2\Delta_{\phi}+2n+\ell$ .

Then antisymmetry of (3.2) implies the following equation:

$F_{+,\Delta,\ell}(u,v)=\sum_{n}\sum_{\ell^{\prime}\text{odd}}\Big{[}\alpha_{n,\ell|\ell^{\prime}}F_{+,\Delta_{n,\ell^{\prime}},\ell}(u,v)+\beta_{n,\ell|\ell^{\prime}}\partial_{\Delta}F_{+,\Delta_{n,\ell^{\prime}},\ell}(u,v)\Big{]}\,.$
(3.5)

The $\alpha$ and $\beta$ are given by

$\displaystyle\alpha_{n,\ell^{\prime}|\ell}(\Delta|\Delta_{\phi})$
$\displaystyle=-a_{n,\ell}^{(s)}(\Delta|\Delta_{\phi})\delta_{\ell\ell^{\prime}}+2\bar{a}_{n,\ell^{\prime}|\ell}^{(t)}(\Delta|\Delta_{\phi})+\cdots\,,$
(3.6) $\displaystyle\beta_{n,\ell^{\prime}|\ell}(\Delta|\Delta_{\phi})$
$\displaystyle=-b_{n,\ell}^{(s)}(\Delta|\Delta_{\phi})\delta_{\ell\ell^{\prime}}+2\bar{b}_{n,\ell^{\prime}|\ell}^{(t)}(\Delta|\Delta_{\phi})+\cdots\,.$
(3.7)

Here $\cdots$ are contact diagram contributions discussed below.

Similar to the usual crossing symmetric case this implies a basis of expansion
for $F_{+,\Delta,\ell}(u,v)$. If the anticrossing correlator has a conformal
block decomposition in $G_{\Delta,\ell}(u,v)$ like (2.2) we should be able to
write the equations:

$\sum_{\Delta,\ell}a_{\Delta,\ell}\alpha_{n,\ell|\ell^{\prime}}(\Delta|\Delta_{\phi})=\sum_{\Delta,\ell}a_{\Delta,\ell}\beta_{n,\ell|\ell^{\prime}}(\Delta|\Delta_{\phi})=0\,.$
(3.8)

These are analogous to the ‘Polyakov Bootstrap equations’ or ‘Polyakov
conditions’ in the usual crossing symmetric case. They are equivalent to a set
of functionals for crossing antisymmetry equations just like its 1d
counterpart, the $+$ type (fermionic) analytic functional of section 2.3.1. We
return to these conditions in a moment.

###  3.1 Contact diagrams

We now address the missing piece of this formulation: the contact diagrams.
For a contact diagram $\mathcal{C}(u,v)$ the corresponding Mellin amplitude
$M_{\mathcal{C}}(s_{1},s_{2})$ is a polynomial. A completely antisymmetric
polynomial Mellin amplitude may be written in terms of a crosing symetric one
in the following way:

$M_{\mathcal{C},+}(s_{1},s_{2})=(s_{1}-s_{2})(s_{2}-s_{3})(s_{3}-s_{1})M_{\mathcal{C},-}(s_{1},s_{2})\,.$
(3.9)

The superscripts ‘$+$’ and ‘$-$’ respectively denote crossing symmetric and
antisymmetric as usual. It is known that $M_{\mathcal{C},-}$ are polynomials
of the invariants $x=-(s_{1}s_{2}+s_{2}s_{3}+s_{3}s_{1})$ and
$y=-s_{1}s_{2}s_{3}$ . For convenience let us also denote
$w=(s_{1}-s_{2})(s_{2}-s_{3})(s_{3}-s_{1})$. So we get

$M_{\mathcal{C},+}(s_{1},s_{2})=w\sum_{p,q}c_{p,q}x^{p}y^{q}\,.$ (3.10)

Note that $M_{\mathcal{C},+}(s_{1},s_{2})$ can have the lowest degree 3 in the
variables. It is followed by $5,6,$ etc .

We may now define the Polyakov block as follows:

$\mathcal{P}_{+,\Delta,\ell}(u,v)=W_{\Delta,\ell}^{(s)}(u,v)-W_{\Delta,\ell}^{(t)}(u,v)-W_{\Delta,\ell}^{(u)}(u,v)+\mathcal{C}_{+}(u,v)(u,v)\,.$
(3.11)

The contact diagram $\mathcal{C}_{+}(u,v)$ has the conformal block
decomposition:

$\mathcal{C}_{+}(u,v)=\sum_{n}\sum_{\ell^{\prime}\text{odd}}\big{[}a_{n,\ell^{\prime}|\mathcal{C}}(\Delta_{\phi})G_{\Delta_{n,\ell^{\prime}}}(u,v)+b_{n,\ell^{\prime}|\mathcal{C}}(\Delta_{\phi})\partial_{\Delta}G_{\Delta_{n,\ell^{\prime}}}(u,v)\big{]}\,.$
(3.12)

It is straightforward to obtain the coefficients (see Appendix A) . The
expressions (3.6) and (3.7) are now modified with the additional terms
$a_{n,\ell^{\prime}|\mathcal{C}}$ and $b_{n,\ell^{\prime}|\mathcal{C}}$
respectively.

###  3.2 Dispersion relation

We would like to show that an expansion of an anticrossing correlator in
$\mathcal{P}_{+,\Delta,\ell}$ is indeed valid, thereby fixing the coefficients
$c_{p,q}$ . For this we follow the framework set up for the crossing symmetric
case in [16, 15]. The idea in those papers was to write down a dispersion
relation for the correlator, which is manifestly crossing symmetric. This
leads to introduction of new spurious poles in the Mellin amplitude, that are
called ‘nonlocal’ terms. Subtracting out these terms one is left with crossing
symmetric sums of exchange Witten diagrams and polynomial terms. The latter
fixes the required contact term ambiguities. We review the details briefly in
Appendix B.

Consider the Mellin amplitude $\mathcal{M}_{+}(s_{1},s_{2})$ of a crossing
antisymmetric function. We will drop the ‘$+$’ subscript below since we
consider only the antisymmetric case in this subsection. We will assume the
usual Regge bounded condition 555Interestingly the final dispersion formula
(3.16) allows a more relaxed Regge behavior of $O(s_{1}^{3-\epsilon})$. Also
we will see that the associated sum rules will reduce in 1d to the ones of
analytic functionals discussed in section 2. Indeed the allowed Regge behavior
is, quite nicely, the same with both methods.

$\mathcal{M}(s_{1},s_{2})\stackrel{{\scriptstyle
s_{1}\to\infty}}{{\sim}}O(s_{1}^{2-\epsilon})\,\ \text{with}\
\epsilon>0\,,\text{ $s_{2}$ fixed}\,.$ (3.13)

We will write down a twice subtracted dispersion relation for this in a
manifestly crossing antisymmetric way, following a strategy similar to [16,
15]. To write such a relation we first use a change of variables:

$s_{i}=a-\frac{a(z-z_{i})^{3}}{z^{3}-1}\,,$ (3.14)

where $z_{i}$ denotes cube roots of 1, and

$a=\frac{s_{1}s_{2}s_{3}}{s_{1}s_{2}+s_{2}s_{3}+s_{3}s_{1}}=\frac{y}{x}\,.$
(3.15)

The variable $a$ is crossing symmetric. 666The reader should not confuse this
$z$ with the cross ratios. In this section we use only $u,v$ in position
space. The ‘physical cuts’ (analogous to QFT scattering amplitudes) in each
$s_{i}$ plane can be mapped to the $z$-plane. Instead of the usual technique
of integrating along $s_{1}$ keeping $s_{2}$ fixed, our dispersion relation is
in the $z$ variable keeping $a$ fixed. We impose the condition of antisymmetry
on the discontinuity of the Mellin amplitude accross the cut, and rewrite the
relation in terms of the original variables $s_{i}$. We show the steps in
detail in Appendix B. The result is the crossing antisymmetric dispersion
relation:

$\mathcal{M}(s_{1},s_{2})=\frac{1}{\pi}\int_{\tau^{(0)}}^{\infty}\frac{ds_{1}^{\prime}}{s_{1}^{\prime}}\mathcal{A}(s_{1}^{\prime};s_{2}^{+}(s_{1}^{\prime},a))H(s_{1}^{\prime};s_{1},s_{2},s_{3})\,.$
(3.16)

In the above $\mathcal{A}(s_{1},s_{2})$ is the $s$-channel discontinuity. The
integration is over the cut $s_{1}\geq\tau^{(0)}$ which for a CFT correlator
is basically a series of poles starting at $\tau^{(0)}$. Here the crossing
antisymmetric kernel $H$ given by

$H(s;s_{1},s_{2},s_{3})=\Big{(}{\frac{s-a}{s+3a}}\Big{)}^{\frac{1}{2}}\Bigg{[}\frac{s_{2}-s_{3}}{s-s_{1}}+\frac{s_{3}-s_{1}}{s-s_{2}}+\frac{s_{1}-s_{2}}{s-s_{3}}\Bigg{]}\,.$
(3.17)

Finally in the absorptive part we also have:

$\displaystyle s_{2}^{+}(s,a)$
$\displaystyle=-\frac{s}{2}\Big{[}{1-\sqrt{\frac{s+3a}{s-a}}}\Big{]}\,.$
(3.18)

We now demand that the Mellin amplitude has an expansion in the crossing
symmetric invariants $a$ and $x$ as follows

$\mathcal{M}(s_{1},s_{2})=w\sum_{p,q=0}^{\infty}\mathcal{M}_{p,q}x^{p+q}a^{q}\,,$
(3.19)

Replacing $a=y/x$ we get an expansion of the form $x^{p}y^{q}$. For an
arbitrary amplitude from (3.16) one can get negative powers of $x$ i.e. $p<0$.
These are unphysical (nonlocal) powers which we set to 0 by imposing the
‘locality constraints’ :

$\mathcal{M}_{p<0,q}=0\,.$ (3.20)

The counterparts of these conditions in the crossing symmetric case were shown
in [15] to be equivalent to the ‘null constraints’ [27, 28] that lead to two
sided bounds on Wilson coefficients in effective field theories. The new
conditions (3.20) are hence a set of independent constraints when the EFT
scattering amplitude has an antisymmetric sector. However in this paper we
only consider the CFT case.

Now let us see how the dispersion relation reproduces the Polyakov block
$\mathcal{P}_{\Delta,\ell}$. If the antisymmetric correlator decomposes in the
$s$-channel into operators of dimension $\Delta$ and spin $\ell$ then the
discontinuity $\mathcal{A}$ can be written as a sum over the ‘partial waves’:

$\mathcal{A}(s_{1},s_{2})=\pi\sum_{\Delta,\ell,k}c_{\Delta,\ell}^{(k)}P_{\Delta,\ell}(\tau_{k},s_{2})\delta(\tau_{k}-s_{1})\,.$
(3.21)

Here we have $\tau_{k}=\frac{\Delta-\ell}{2}+k-\frac{2\Delta_{\phi}}{3}$ and
$k\in\mathbb{Z}_{\geq 0}$ . Also $P_{\Delta,\ell}(s_{1},s_{2})$ is a shifted
Mack polynomial, and $c^{(k)}_{\Delta,\ell}$ is the squared OPE coefficient
$a_{\Delta,\ell}$ times a normalization. We define them in Appendix A.

Then we get from (3.16)

$\mathcal{M}(s_{1},s_{2})=\sum_{\Delta,\ell,k}c_{\Delta,\ell}^{(k)}\mathcal{M}_{\Delta,\ell,k}(s_{1},s_{2})\,,$
(3.22)

where

$\mathcal{M}_{\Delta,\ell,k}(s_{1},s_{2})=\frac{1}{\tau_{k}}Q_{\ell,k}^{(\Delta)}(a)H(\tau_{k},s_{1},s_{2},s_{3})\,,$
(3.23)

and
$Q_{\ell,k}^{(\Delta)}(a)=P_{\Delta,\ell}(\tau_{k},s_{2}^{\prime}(\tau_{k},a))$
.

We would like to show that this reduces to a crossing antisymmetric
combination of Witten diagrams when $\ell$ is odd. For this let us define the
quantity

$\mathcal{M}_{\Delta,\ell,k}^{(0)}(s_{1},s_{2})=\frac{P_{\Delta,\ell}(s_{1},s_{2})}{s_{1}-\tau_{k}}-\frac{P_{\Delta,\ell}(s_{2},s_{1})}{s_{2}-\tau_{k}}-\frac{P_{\Delta,\ell}(s_{3},s_{2})}{s_{3}-\tau_{k}}\,,$
(3.24)

with $s_{3}=-s_{1}-s_{2}$ . It can be checked for every odd $\ell$ that this
is antisymmetric under $s_{i}\leftrightarrow s_{j}$ $(i\neq j)$. Each term in
$\mathcal{M}_{\Delta,\ell,k}^{(0)}(s_{1},s_{2})$ is a pole of the Mellin
amplitude of exchange Witten diagram $W_{\Delta,\ell}^{(i)}$ when we write it
in its meromorphic form (see (A.3) in Appendix A).

We now expand $\mathcal{M}_{\Delta,\ell,k}$ and
$\mathcal{M}^{(0)}_{\Delta,\ell,k}$ in small $x$ and $a$, and look at their
difference.

$\displaystyle\mathcal{M}_{\Delta,\ell,k}(s_{1},s_{2})=$
$\displaystyle\mathcal{M}_{\Delta,\ell,k}^{(0)}(s_{1},s_{2})+w\sum_{p,q=0}^{2p+3q\leq
L}C_{p,q}^{(\ell)}x^{p+q}a^{q}$
$\displaystyle+w\sum_{p<0}\widetilde{C}^{(\ell)}_{p,q}x^{p+q}a^{q}\,.$ (3.25)

The second term is precisely the polynomial pieces that define the contact
terms we wanted to fix in (3.11). For each $\ell$ there is a finite number of
such polynomials up to a maximum degree $L$ (note that $x^{p}y^{q}$ is a
$2p+3q$ degree polynomial). Finally the last term i.e. second line of r.h.s.
vanishes when we collectively impose the locality constraints (3.20) on the
full amplitude .

For $\ell=1$ we get $C_{p,q}^{(\ell)}=\widetilde{C}^{(\ell)}_{p,q}=0$ . For
$\ell=3$ we get $L=0$ and $C_{0,0}^{(\ell)}=\frac{1}{16\tau_{k}}$ . For
$\ell=5$ we have $L=4$ which we work out in Appendix C.1.

To conclude, we have shown that if an anticrossing correlator decomposes into
odd spins it can be expanded in $+$ type Polyakov blocks. For a more general
anticrossing function an expansion like (3.22) remains valid.

###  3.3 Polyakov Conditions

Let us discuss how to bootstrap the Mellin amplitude to obtain the OPE data.
For this let us reinstate the subscripts ($+$)$-$ for crossing (anti)symmetric
correlators. In terms of $\mathcal{M}_{\pm}$ they read

$\mathcal{G}_{\pm}(u,v)=\int
ds_{1}ds_{2}\,u^{s_{1}+\frac{2\Delta_{\phi}}{3}}v^{s_{2}-\frac{\Delta_{\phi}}{3}}\Bigg{[}\prod_{i=1}^{3}\Gamma^{2}\big{(}\frac{\Delta_{\phi}}{3}-s_{i}\big{)}\Bigg{]}\mathcal{M}_{\pm}(s_{1},s_{2})\,,$
(3.26)

Since the only poles in $s_{i}$ that contribute to OPE come from (3.21) the
poles from the Gamma function measure are spurious.777We assume no operator at
the exact locations $\Delta=2\Delta_{\phi}+2p+\ell$. So at these locations the
Mellin amplitude must have double zeroes. These are the ‘Polyakov conditions’.
We write them as follows:

$\displaystyle\mathfrak{F}_{\pm,p}(s_{2})$
$\displaystyle\equiv\mathcal{M}_{\pm}(s_{1}=\frac{\Delta_{\phi}}{3}+p,s_{2})=0\,,$
$\displaystyle\widetilde{\mathfrak{F}}_{\pm,p}(s_{2})$
$\displaystyle\equiv\partial_{s_{1}}\mathcal{M}_{\pm}(s_{1}=\frac{\Delta_{\phi}}{3}+p,s_{2})=0\,,$
(3.27)

for all $p\in\mathbb{Z}_{\geq 0}$. Let us focus on the first set of equations
i.e. $\mathfrak{F}_{\pm,p}(s_{2})=0$. Using (3.22) and summing over $k$ we can
write
$\mathfrak{F}^{(r)}_{\pm,p}(s_{2})=\sum_{\Delta,\ell}a_{\Delta,\ell}\,\mathfrak{F}^{(r)}_{\pm,p,\Delta,\ell}(s_{2})$.
We may Taylor expand this around $s_{2}=0$ to get a set of infinite sum rules
as follows: 888 For the $-$ case it was shown in [16] that with a special
combination  $\displaystyle\Omega_{\pm,p_{1},p_{2},p_{3}}(s_{2})=$
$\displaystyle-\Big{[}\frac{\mathfrak{F}_{\pm,p_{1}}}{(p_{1}-p_{2})(p_{1}+p_{3}+s_{2}+\frac{\Delta_{\phi}}{3})}+(p_{1}\leftrightarrow
p_{2})\Big{]}-\frac{\mathfrak{F}_{\pm,p_{3}}}{(p_{1}+p_{3}+s_{2}+\frac{\Delta_{\phi}}{3})(p_{2}+p_{3}+s_{2}+\frac{\Delta_{\phi}}{3})}\,$
(3.28) the sum rules (3.3) are equivalent to those obtained from
nonperturbative Mellin amplitudes [29].

$\mathfrak{F}_{\pm,p,\Delta,\ell}(s_{2})=\sum_{r=0}^{\infty}(s_{2})^{r}\mathfrak{F}^{(r)}_{\pm,p,\Delta,\ell}\,\
\implies\
\sum_{\Delta,\ell}a_{\Delta,\ell}\mathfrak{F}^{(r)}_{\pm,p,\Delta,\ell}=0\,.$
(3.29)

A similar set of sum rules are obtained also from
$\widetilde{\mathfrak{F}}_{\pm,p}(s_{2})$. Note that these sum rules are
equivalent to the conditions (3.8) since they are related by a Mellin
integral.

We point out that the description of $\mathcal{M}_{-}$ ($\mathcal{M}_{+}$) in
terms of Witten diagrams is not possible when $\ell$ is odd (even) as the
crossing (anti)symmetric properties are not well-defined. However the
dispersion relation representation holds for any spin. In particular for the
$-$ case it can be checked that they lead to identical sum rules as [29] for
any spin (see footnote 8). This is useful e.g. in global symmetry problems
where the (anti)crossing equation may involve all spins (see next section).

##### Numerical checks:

We have numerically tested our proposals in two different examples. First with
a fictitious Mellin amplitude, we have checked that an expansion in $x$ and
$a$ indeed works and locality conditions (3.20) are satisfied. In the second
example we have worked with an antisymmetric sector of a 2d Wess-Zumino-Witten
(WZW) model. For this we have shown the working of Polyakov conditions (3.3).
Both analyses are discussed in Appendix C.2.

##  4 Applications to functionals

The $+$ type Polyakov blocks can be useful in a number of Bootstrap
applications. The most immediate examples are computing the ‘product
functionals’ in higher $d$, and bootstrapping CFTs with global symmetries. In
this section we discuss them in turn.

###  4.1 Product Functionals in even $d$

We begin by reviewing the product functionals that were introduced in [20] as
a set of efficient functionals in even $d$. In 2d it was shown that they have
nice positivity properties making them ideal for numerical applications. The
main advantage of these functionals is that they are built of 1d analytic
functionals $\omega_{\pm}$. Through the Polyakov blocks we now understand how
to compute their action in a simple way.

In this subsection we only focus on $d=2$. We review the cases of $d=4,6$ in
Appendix D. The 2d conformal blocks are given by:

$G^{d=2}_{\Delta,\ell}(z,\bar{z})=\frac{1}{2}\Big{[}G_{\frac{\tau}{2}}(z|\text{\tiny$\frac{\Delta_{\phi}}{2}$})G_{\frac{\rho}{2}}(\bar{z}|\text{\tiny$\frac{\Delta_{\phi}}{2}$})+(z\leftrightarrow\bar{z})\Big{]}\,,\
\ \tau=\Delta-\ell\,,\ \rho=\Delta+\ell\,.$ (4.1)

The crossing vector is given by:

$F^{d=2}_{-,\Delta,\ell}(z,\bar{z}|\Delta_{\phi})=\frac{1}{4}\big{[}F_{-,\frac{\tau}{2}}(z|\text{\tiny$\frac{\Delta_{\phi}}{2}$})F_{+,\frac{\rho}{2}}(\bar{z}|\text{\tiny$\frac{\Delta_{\phi}}{2}$})+F_{-,\frac{\rho}{2}}(z|\text{\tiny$\frac{\Delta_{\phi}}{2}$})F_{+,\frac{\tau}{2}}(\bar{z}|\text{\tiny$\frac{\Delta_{\phi}}{2}$})+(z\leftrightarrow\bar{z})\big{]}\,.$
(4.2)

Notice that the $z$ and $\bar{z}$ dependencies have separated out. If we
ensure that the $z$ dependence is only through $F_{-}$, and $\bar{z}$ through
$F_{+}$ then we can write a functional for $F^{d=2}_{-,\Delta,\ell}$ as
products of $\omega_{-}$ and $\omega_{+}$. Therefore we choose the 2d
functionals as follows:

$\omega^{(1)}_{-}\otimes\omega^{(2)}_{+}(\Delta,\ell)=2\int_{1}^{\infty}\frac{dzd{\bar{z}}}{\pi^{2}}h^{(1)}_{-}(z)h^{(2)}_{+}(\bar{z})\big{[}\mathcal{I}_{z}\mathcal{I}_{\bar{z}}F_{-,\Delta,\ell}(z,\bar{z})+\mathcal{I}_{z}\mathcal{I}_{\bar{z}}F_{-,\Delta,\ell}(z,1-\bar{z})\big{]}\,.$
(4.3)

The structure is symmetrized in $\bar{z}\to 1-\bar{z}$ to have the above
mentioned feature. All kernels $h^{(1)}_{-},h^{(2)}_{+}$ are chosen from 1d
functional kernels such that we have the following four types of functionals
(suppressing $B,F$ superscripts as any bosonic/fermionic functional should
work):

$\displaystyle\omega^{(1)}_{-}\omega^{(2)}_{+}\in\Big{\\{}\alpha_{-,n}\alpha_{+,m}\,,\
\alpha_{-,n}\beta_{+,m}\,,\ \beta_{-,n}\alpha_{+,m}\,,\
\beta_{-,n}\beta_{+,m}\Big{\\}}\,.$ (4.4)

We have simplified the notation
$\omega^{(1)}_{-}\omega^{(2)}_{+}\equiv\omega^{(1)}_{-}\otimes\omega^{(2)}_{+}$
. Each functional action is given by

$\omega^{(1)}_{-}\omega^{(2)}_{+}(\Delta,\ell)=\frac{1}{2}\big{[}\omega^{(1)}_{-}(\tau|\Delta_{\phi}|2)\omega^{(2)}_{+}(\rho|\Delta_{\phi}|2)+\omega^{(1)}_{-}(\rho|\Delta_{\phi}|2)\omega^{(2)}_{+}(\tau|\Delta_{\phi}|2)\big{]}\,.$
(4.5)

Here we defined for convenience
$\omega_{\pm}(\Delta|\Delta_{\phi}|2)=\omega_{\pm}(\frac{\Delta}{2}|\frac{\Delta_{\phi}}{2})$.
Using the results from section 2.3 these are entirely determined from
conformal block decomposition of Witten diagrams. The $+$ sign in the
combination (4.5) ensures the positive behavior of product functional actions.
This was used in [20] with $\beta_{-,0}^{F}\alpha_{+,0}^{F}$, for instance, to
obtain an upper bound on the lowest non-identity operator.

Let us also explore the case of crossing antisymmetric vector
$F^{d=2}_{+,\Delta,\ell}(z,\bar{z})$ which can be written as

$F^{d=2}_{+,\Delta,\ell}(z,\bar{z}|\Delta_{\phi})=\frac{1}{4}\big{[}F_{-,\frac{\tau}{2}}(z|\text{\tiny$\frac{\Delta_{\phi}}{2}$})F_{-,\frac{\rho}{2}}(\bar{z}|\text{\tiny$\frac{\Delta_{\phi}}{2}$})+F_{+,\frac{\rho}{2}}(z|\text{\tiny$\frac{\Delta_{\phi}}{2}$})F_{+,\frac{\tau}{2}}(\bar{z}|\text{\tiny$\frac{\Delta_{\phi}}{2}$})+(z\leftrightarrow\bar{z})\big{]}\,.\,.$
(4.6)

As before we have to symmetrize/antisymmetrize in $z\to 1-z$ to have the
necessary variable separation. So we choose the functionals as follows:

$\omega^{(1)}_{\pm}\omega^{(2)}_{\pm}(\Delta,\ell)=2\int_{1}^{\infty}\frac{dzd{\bar{z}}}{\pi^{2}}h^{(1)}_{\pm}(z)h^{(2)}_{\pm}(\bar{z})\big{[}\mathcal{I}_{z}\mathcal{I}_{\bar{z}}F_{+,\Delta,\ell}(z,\bar{z})\pm\mathcal{I}_{z}\mathcal{I}_{\bar{z}}F_{+,\Delta,\ell}(z,1-\bar{z})\big{]}\,.$
(4.7)

The crossing antisymmetric product functionals and their respective functional
actions are as follows

$\displaystyle\omega^{(1)}_{\pm}\omega^{(2)}_{\pm}\in\Big{\\{}\alpha_{\pm,n}\alpha_{\pm,m}\,,\
\alpha_{\pm,n}\beta_{\pm,m}\,,\ \beta_{\pm,n}\beta_{\pm,m}\Big{\\}}\,.$ (4.8)
$\displaystyle\omega^{(1)}_{\pm}\omega^{(2)}_{\pm}(\Delta,\ell)=\omega^{(1)}_{\pm}(\tau|\Delta_{\phi}|2)\omega^{(2)}_{\pm}(\rho|\Delta_{\phi}|2)+\omega^{(2)}_{\pm}(\tau|\Delta_{\phi}|2)\omega^{(1)}_{\pm}(\rho|\Delta_{\phi}|2)\,.$
(4.9)

###  4.2 Global symmetry - Simple functionals

####  4.2.1 $d=1$

An important application for crossing antisymmetric functionals is in the
problem of global symmetries. Let us recall from section 1 that if we write
the correlator of charged scalar fields $\phi_{i}$ in terms of the components
$\mathcal{G}^{\mathfrak{a}}$ for each irrep $\mathfrak{a}$ appearing in the
OPE, then under crossing we get the relation:

$\mathcal{G}^{\mathfrak{a}}(z)=C^{\mathfrak{a}\mathfrak{b}}\mathcal{G}^{\mathfrak{b}}(1-z)\,,$
(4.10)

where $C^{\mathfrak{b}\mathfrak{a}}$ is the crossing matrix. We can define the
projector matrices
$P^{\mathfrak{b}\mathfrak{a}}_{\pm}=\frac{1}{2}(\delta^{\mathfrak{b}\mathfrak{a}}+C^{\mathfrak{b}\mathfrak{a}})$
in terms of which one may write the crossing vectors

$\mathcal{F}^{\mathfrak{b}|\mathfrak{a}}(z)=P^{\mathfrak{b}\mathfrak{a}}_{+}F_{-,\Delta}(z)+P^{\mathfrak{b}\mathfrak{a}}_{-}F_{+,\Delta}(z)\,.$
(4.11)

With the above crossing vectors one can write the crossing equation (4.10) as
follows:

$\sum_{\mathfrak{b}}\sum_{\Delta}a_{\Delta}^{\mathfrak{b}}\mathcal{F}^{\mathfrak{b}|\mathfrak{a}}(z)=0\,.$
(4.12)

In [14] a set of global symmetry functionals, called ‘simple functionals’ were
introduced for the above problem. They have the following structure:

$\omega^{\mathfrak{b}|\mathfrak{a}}=P^{\mathfrak{b}\mathfrak{a}}_{+}\omega_{-}+P^{\mathfrak{b}\mathfrak{a}}_{-}\omega_{+}\,.$
(4.13)

Here $\omega_{\pm}\in\\{\alpha_{\pm,n}^{B,F},\beta_{\pm,n}^{B,F}\\}$ i.e. the
1d functionals of uncharged case. In the above notation for
$\omega^{\mathfrak{b}|\mathfrak{a}}$ the superscript $\mathfrak{b}$ denotes a
label and $|\mathfrak{a}$ a component. The action of these functionals on the
crossing vectors $\mathcal{F}^{\mathfrak{c}|\mathfrak{a}}$ (4.11) is given by

$\omega^{\mathfrak{b}}(\mathfrak{c},\Delta)=P^{\mathfrak{b}\mathfrak{c}}_{+}\omega_{-}(\Delta)+P^{\mathfrak{b}\mathfrak{c}}_{-}\omega_{+}(\Delta)\,.$
(4.14)

The action of these functionals on crossing symmetry (4.12) are entirely
determined by our knowledge of the $\omega_{\pm}(\Delta)$, which are given by
Witten diagram decomposition coefficients (section 2.3). For numerical
applications simple functionals are useful as they retain the positivity
properties of the uncharged functionals. For instance any bound obtained using
1d functionals (say $\omega_{-}$) can be easily generalized to global
symmetries if one can write a linear combination of $\omega^{\mathfrak{b}}$
whose action is proportional to only $\omega_{-}$ with positive coefficients.

####  4.2.2 $d=2$ with product functionals

The idea of simple functionals is not restricted to 1d. Indeed for any $d$ one
can consider (4.13) choosing $\omega_{\pm}$ as appropriate in that dimension.
In 2d we may choose them to be the product functionals

$\omega_{-}\to\omega_{-}^{(1)}\omega_{+}^{(2)}\,,\ \
\omega_{+}\to\omega_{\pm}^{(3)}\omega_{\pm}^{(4)}\,.$ (4.15)

The action of 2d simple product functionals is given by

$\omega^{\mathfrak{b}}(\mathfrak{c},\Delta,\ell)=P_{+}^{\mathfrak{b}\mathfrak{c}}\big{[}\omega_{-}^{(1)}\omega_{+}^{(2)}(\Delta,\ell)\big{]}+P_{-}^{\mathfrak{b}\mathfrak{c}}\big{[}\omega_{\pm}^{(3)}\omega_{\pm}^{(4)}(\Delta,\ell)\big{]}\,.$
(4.16)

This allows us to translate any bound found with product functionals in CFTs
without global symmetry to bounds for global symmetry problems. A simple
demonstration is done in Appendix D.3 where we extend an analytic bound for
OPE density for 2d uncharged CFTs to the charged case. As mentioned in section
4.1 the positivity properties of product functionals can be useful to obtain
sharp numerical bounds for global symmetry problems.

Of course the $d=2$ simple product functionals case can easily be generalized
to $d=4$ and $6$ (see Appendix D).

####  4.2.3 General $d$

To obtain simple functionals for general dimensions we may use the Polyakov
condition sum rules (3.3). In this case we may use
$\omega_{\pm}\to\mathfrak{F}_{\pm,p},\widetilde{\mathfrak{F}}_{\pm,p}$ so that
simple functional actions are given by

$\omega^{\mathfrak{b}}({\mathfrak{c},\Delta,\ell})\in\big{\\{}P_{+}^{\mathfrak{b}\mathfrak{c}}\mathfrak{F}^{(r)}_{-,p,\Delta,\ell}+P_{-}^{\mathfrak{b}\mathfrak{c}}\mathfrak{F}^{(r)}_{+,p,\Delta,\ell};r\in\mathbb{Z}_{\geq
0}\big{\\}}\,,$ (4.17)

and similarly with $\widetilde{\mathfrak{F}}$. The quantities
$\mathfrak{F}^{(r)}_{-,p,\Delta,\ell}$ were defined in (3.29). It should be
noted that these simple functionals are a variation of those discussed in
[14]. As we use only bosonic Witten diagrams, in the language of 1d
functionals it is like using ‘bosonic functionals’ for the $-$ case and
‘fermionic functionals’ for the $+$ case. They are not dual to any interesting
CFT e.g. GFF, although they are perfectly good functionals for global symmetry
problems.

The idea of simple functionals can be extended to QFTs with global symmetries.
Here the $\pm$ functional actions should be replaced with the locality
constraints of the $\mathcal{M}_{\pm}(s_{1},s_{2})$. We leave this for future
work.

##  5 Conclusion

In this paper we show that a crossing antisymmetric correlator can be expanded
in a manifestly crossing antisymmetric basis. Similar to Polyakov’s original
proposal for crossing symmetric correlators, this basis is mapped to AdS
Witten diagrams. We explore its relation to analytic functionals, and set up a
dispersion relation that proves it.

We call the building blocks of the basis the $+$ type Polyakov blocks. In 1d
the simplest of these, the fermionic $+$ type Polyakov blocks, are constructed
from a manifestly crossing antisymmetric sum of AdS2 spin 1 exchange bosonic
Witten diagrams. Their conformal block decomposition coefficients compute 1d
fermionic $+$ type functional actions. The $+$ type functionals allow a more
relaxed Regge behavior compared to the $-$ type. This is used to obtain the
bosonic Polyakov block by suitably combining spin 1 fermionic Witten diagrams
and a manifestly anticrossing function which we call a ‘contact diagram’. Once
again under OPE decomposition this exactly reproduces the bosonic $+$
functional action.

In general dimensions the $+$ type Polyakov blocks are built from odd spin
Witten exchange diagrams and crossing antisymmetric contact diagrams. We
formulate this in Mellin space. To obtain the exact structure of the blocks we
introduce a crossing antisymmetric dispersion relation for a general crossing
antisymmetric Mellin amplitude. This utilises a change of variables that
results in nonlocal singularities. Imposing the locality constraints i.e.
setting these nonlocal terms to zero each odd spin sector of the dispersion
relation reproduces a sum of odd spin exchange Witten diagrams and finite
number of contact diagrams. Finally we give a set of sum rules corresponding
to the ‘Polyakov conditions’ i.e. the residue at spurious double trace poles
in Mellin variables should be zero. Using the dispersion relation the sum
rules can be implemented for any spin. We have numerical checked with a simple
example how locality conditions work, and how the Polyakov conditions are
satisfied for the antisymmetric sector of a 2d WZW model, elaborately
discussed in the Appendix.

Our results imply a number of simplifications for bootstrapping 4-point
functions where crossing antisymmetry becomes important. The most significant
ones being product functionals in even $d$ and various simple functionals for
global symmetry problems in general $d$. There are many other interesting
directions that one can pursue:

  1. 1.

Constraints on QFTs: A version of crossing symmetric dispersion relation has
been used in QFT context [15] to derive constraints on effective field
theories (equivalent to [27, 28]) and scattering cross sections. They key
element in that analysis was the locality constraints. These are similar to
our conditions (3.20) which are independent and should lead to new constraints
on scattering amplitudes when they have a crossing antisymmetric component
e.g. when global symmetries are present.

  2. 2.

Geometric Function Theory: An interesting new direction in constraining
scattering amplitudes using ideas of Geometric Function theory (GFT) has been
initiated in [30, 31]. The crossing symmetric dispersion relation allows a
formulation in terms of ‘typically real functions’ where GFT constraints (e.g.
Bieberbach-Rogonski inequalities) can be applied. In [32] this approach was
extended to $O(N)$ theories building on the work of [33]. It will be
interesting to see how the crossing antisymmetric dispersion relation can be
tied into the GFT framework and if the associated constraints/sum rules are
connected to [33, 32] or independent ones.

  3. 3.

Correlator bounds/Master functionals: An interesting problem of obtaining
bounds on CFT correlators was addressed in [34, 14]. To prove e.g. the
minimization of 1d correlators one can construct a ‘master functional’ which
packages analytic functionals in a certain way. Interestingly the master
functional action is equivalent to a Polyakov block. It would be interesting
to see how a crossing (anti)symmetric Polyakov block in higher dimensions is
encoded in bounds (e.g. the numerical bounds obtained in [35] in 3d) on CFT
correlators.

  4. 4.

Fermionic/Multiple correlator bootstrap: Crossing antisymmetry also appears in
interesting problems like bootstrapping correlators of fermions in higher $d$
or a set of correlators with unequal scalars [36, 37, 38]. Our results are
hence applicable to these cases (see e.g. [39]). However when fermions or
unequal scalars are involved the conformal blocks are different, and also we
may lose full crossing (anti)symmetry. So a modification of analytic
functionals or Polyakov Bootstrap may be necessary.

  5. 5.

Nonperturbative Mellin amplitude/dispersive sum rules: As shown in [16] the
Polyakov conditions from crossing symmetric dispersion relation are equivalent
to the analogous constraints proposed in [40, 29] for nonperturbative Mellin
amplitudes. These conditions can also be mapped to the “dispersive sum rules”
from position space dispersion relation or general $d$ analytic functionals
[8]. It would be interesting to see if there is a similar picture for
anticrossing correlators and if the associated sum rules are related to the
crossing antisymmetric dispersion relation.

## Acknowledgements

The author acknowledges Subham Dutta Chowdhury, Kausik Ghosh, Rajesh
Gopakumar, Miguel Paulos, Junchen Rong, Volker Schomerus, Aninda Sinha and
Ahmadullah Zahed for many useful discussions. The author is thankful to Rajesh
Gopakumar, Volker Schomerus and Aninda Sinha for their comments on the draft
and various helpful suggestions. This work is supported by the German Research
Foundation DFG under Germany’s Excellence Strategy – EXC 2121 “Quantum
Universe” – 390833306.

##  Appendix A Details of Witten diagrams

In this section we give the details of the Witten diagrams used in the main
text. The exchange Witten diagram $W_{\Delta,\ell}^{(s)}(u,v)$ with scalar
external legs in general dimension i.e. AdSd+1 is conveniently expressed in
Mellin space as follows [21]

$W_{\Delta,\ell}^{(s)}(u,v)=\int_{-i\infty}^{i\infty}[ds][dt]u^{s}v^{t}\Gamma^{2}(\Delta_{\phi}-s)\Gamma^{2}(-t)\Gamma^{2}(s+t)M^{(s)}_{\Delta,\ell}(s,t)\,,$
(A.1)

where $s,t$ are related to $s_{i}$ variables of section 3 by
$s_{1}=s-\frac{2\Delta_{\phi}}{3}$ and $s_{2}=\frac{\Delta_{\phi}}{3}+t$, and

$\displaystyle M_{\Delta,\ell}(s,t)$
$\displaystyle=\widehat{P}_{\Delta-h,\ell}^{(s)}(s,t)\frac{\Gamma^{2}(\frac{\Delta+\ell}{2}+\Delta_{\phi}-h)}{(\frac{\Delta-\ell}{2}-s)\Gamma(\Delta-h+1)}$
$\displaystyle\times{}_{3}F_{2}\Big{[}\frac{\Delta-\ell}{2}-s,1+\frac{\Delta-\ell}{2}-\Delta_{\phi},1+\frac{\Delta-\ell}{2}-\Delta_{\phi};1+\frac{\Delta-\ell}{2}-s,\Delta-h+1;1\Big{]}\,.$
(A.2)

Here $h=d/2$ and we define $\widehat{P}_{\Delta-h,\ell}^{(s)}(s,t)$ below. The
Mellin amplitude $M_{\Delta,\ell}(s,t)$ allows the decomposition

$M_{\Delta,\ell}(s,t)=\Gamma^{2}(\frac{\Delta+\ell}{2}+\Delta_{\phi}-h)\widehat{P}_{\Delta-h,\ell}^{(s)}(s,t)\sum_{r=0}^{\infty}\frac{(1+\frac{\Delta-\ell}{2}-\Delta_{\phi})_{r}^{2}(\frac{\Delta-\ell}{2}-s)}{r!\Gamma(\Delta-h+1+r)}\frac{1}{\frac{\Delta-\ell}{2}-s+r}\,.$
(A.3)

The above form was used in e.g. (3.2) to match the dispersion relation with
the Witten diagrams. The coefficient $c_{\Delta,\ell}^{(k)}$ in (3.21) is
given by

$c_{\Delta,\ell}^{(k)}=a_{\Delta,\ell}\,\mathcal{N}_{\Delta,\ell}\mathcal{R}_{\Delta,\ell}^{(k)}$
(A.4)

where $a_{\Delta,\ell}$ is the squared OPE coefficient and

$\mathcal{N}_{\Delta,\ell}=\frac{(-2)^{\ell}(\Delta+\ell-1)\Gamma(\Delta-h+1)\Gamma(\ell+\Delta-1)^{2}}{\Gamma(\Delta-1)\Gamma^{4}\left(\frac{\ell+\Delta}{2}\right)\Gamma^{2}\left(\Delta_{\phi}-\lambda_{2}\right)\Gamma^{2}\left(\Delta_{\phi}+\lambda_{1}-h\right)}\,,\
\mathcal{R}_{\Delta,\ell}^{(k)}=\frac{\Gamma^{2}(\lambda_{1}+\Delta_{\phi}-h)(1+\lambda_{2}-\Delta_{\phi})_{k}^{2}}{k!\Gamma(\Delta-h+1+k)}\,.$
(A.5)

Here $\lambda_{1}=(\Delta+\ell)/2$ and $\lambda_{2}=(\Delta-\ell)/2$ .

The $s$-channel Mack Polynomial $\widehat{P}_{\Delta-h,\ell}^{(s)}(s,t)$ is
given by

$\widehat{P}_{\Delta-h,\ell}^{(s)}(s,t)=\sum_{m+n\leq\ell}\mu^{(\ell)}_{m,n}\big{(}\frac{\Delta-\ell}{2}-s\big{)}_{m}(-t)_{n}$
(A.6)

where

$\displaystyle\mu^{(\ell)}_{m,n}$
$\displaystyle=\frac{2^{-\ell}\ell!(-1)^{m+n}}{m!n!(\ell-
m-n)!}\left(\lambda_{1}-m\right)_{m}\left(\lambda_{2}+n\right)_{\ell-n}(h+\ell-1)_{-m}(\ell+\Delta-1)_{n-\ell}\left(\lambda_{2}+m+n\right)_{\ell-
m-n}$
$\displaystyle\times\,_{4}F_{3}\left(-m,\lambda_{2}-h+1,\lambda_{2}-h+1,\Delta+n-1;\lambda_{1}-m,\lambda_{2}+n,\lambda_{2}-2h+2;1\right)\,.$
(A.7)

To define the discontinuity and dispersion relation in section 3 we have also
used the following shifted polynomials:

$P_{\Delta,\ell}(s_{1},s_{2})=\widehat{P}_{\Delta-h,\ell}^{(s)}(s_{1}+\frac{2\Delta_{\phi}}{3},s_{2}-\frac{\Delta_{\phi}}{3})\,.$
(A.8)

To obtain expressions for 1d CFTs one should replace $u=z\bar{z}$ and
$v=(1-z)(1-\bar{z})$ in (A.1) .To obtain the deomposition into double trace
blocks (see (2.19) and (2.3.1)) of these exchange Witten diagrams one has to
compute the residues of poles at $s=\Delta_{\phi}+n$ . In [21] it was shown
that a convenient way of doing this is via the following expansion
($i=s,t,u$):

$M^{(i)}_{\Delta,\ell}(s,t)=\sum_{\ell^{\prime}}q^{(i)}_{\Delta,\ell^{\prime}|\ell}(s)Q_{\ell^{\prime},0}^{(2s+\ell^{\prime})}(t)\,.$
(A.9)

Here $Q_{\ell^{\prime},0}^{(2s+\ell^{\prime})}(t)$ is a continuous Hahn
polynomial defined by

$Q_{\ell^{\prime},0}^{(2s+\ell^{\prime})}(t)=\frac{2^{\ell}((s)_{\ell})^{2}}{(2s+\ell-1)_{\ell}}{}_{3}F_{2}[-\ell,2s+\ell-1,s+t;s,s;1]\,.$
(A.10)

The coefficients $q^{(s)}_{\Delta,\ell^{\prime}|\ell}(s)$ are given by
rational functions, and computed using orthonormality of the Hahn polynomials.
In the crossed channels we have $q^{(t)}_{\Delta,\ell^{\prime}|\ell}(s)$ that
is given by a ${}_{7}F_{6}$ hypergoemetric function. See equations (3.6) and
(3.28) of [21]. See also [14] Appendix D on how to use recursively compute the
coefficients of decomposition in double trace blocks using these relations.
The decomposition (A.9) holds for contact diagrams i.e.
$M_{\mathcal{C}}(s_{1},s_{2})$ (from (3.9)) as well and the corresponding
coefficients $q_{\ell^{\prime}}$ are easy to compute rational functions.

##  Appendix B Derivation of the dispersion relation

In this appendix we show the derivation of the crossing antisymmetric
dispersion relation while also reviewing the crossing symmetric case following
[15] (supplementary material).

###  B.1 General case - review

We first briefly review the derivation of a general dispersion relation
without assuming crossing symmetry or antisymmetry. Let us recall the
transformation of Mellin variables ($i=1,2,3$):

$\displaystyle s_{i}$ $\displaystyle=a-\frac{a(z-z_{i})^{3}}{z^{3}-1},$ (B.1)
$\displaystyle a$
$\displaystyle=\frac{s_{1}s_{2}s_{3}}{s_{1}s_{2}+s_{2}s_{3}+s_{1}s_{3}}\,,$
(B.2)

with $z_{i}$ being cube roots of unity. The Mellin amplitude has a sequence of
poles in $s_{i}\geq\tau^{(0)}$ that correspond to the physical operator
content. 999This is analogous to taking the branch cut
$s_{i}\geq\frac{8m^{2}}{3}$ in a QFT scattering problem where $s_{i}$ denotes
Mandelstam variables and $m$ the mass of external particles. In terms of the
new independent variables $z$ and $a$, these ‘physical cuts’ correspond in the
$z$ plane to the region $V_{1}(a)\cup V_{2}(a)\cup V_{3}(a)$ , as shown in
Fig. 2, where

$\displaystyle V_{1}(a)$
$\displaystyle=\begin{cases}&\\{z;|z|=1,\frac{2}{3}\pi\leq|\text{arg
}z|\leq\phi_{0}(a)\\}\ \ \ \ \text{if }-\frac{\tau^{(0)}}{3}<a<0\,,\\\
&\\{z;|z|=1,\phi_{0}(a)\leq|\text{arg }z|\leq\frac{2}{3}\pi\\}\ \ \ \ \text{if
}0<a<\tau^{(0)}\,,\\\ &\\{z;|z|=1,\frac{2}{3}\pi\leq|\text{arg
}z|\leq\pi\\}\cup\\{z;\rho_{-}(a)\leq|z|\leq\rho_{+}(a),\text{arg }z=\pi\\}\ \
\text{if }a<-\frac{\tau^{(0)}}{3}\end{cases}$ $\displaystyle V_{2}(a)$
$\displaystyle=\exp\Big{[}\frac{2\pi i}{3}\Big{]}\,V_{1}(a)$ (B.3)
$\displaystyle V_{3}(a)$ $\displaystyle=\exp\Big{[}\frac{4\pi
i}{3}\Big{]}\,V_{1}(a)\,,$

and we have defined

$\displaystyle\phi_{0}(a)$
$\displaystyle=\tan^{-1}\bigg{[}\frac{\big{[}(\tau^{(0)}-a)(a+\frac{\tau^{(0)}}{3})\big{]}^{\frac{1}{2}}}{a-\frac{\tau^{(0)}}{3}}\bigg{]}$
$\displaystyle\rho_{\pm}(a)$
$\displaystyle=\frac{3}{2\tau^{(0)}}\Bigg{[}\bigg{(}\frac{\tau^{(0)}}{3}-a\bigg{)}\pm\bigg{[}\big{(}\tau^{(0)}-a\big{)}\Big{(}-a-\frac{\tau^{(0)}}{3}\Big{)}\bigg{]}^{\frac{1}{2}}\Bigg{]}\,.$
(B.4)

Figure 2:  A schematic picture of the physical cuts in $z$-plane. The colours
indicate $V_{1}$ (red), $V_{2}$ (green) and $V_{3}$ (blue).

Notice that the cuts are defined only at $|z|=1$ or $\text{arg }z=\pi\text{
mod }\frac{2\pi}{3}$. These conditions correspond to $\text{Im }s_{i}=0$. Let
us call the Mellin ampltude
$\mathcal{M}(s_{1},s_{2})=\overline{\mathcal{M}}(z,a)$ in $z,a$ variables.
Then discontinuities of $\overline{\mathcal{M}}(z,a)$ across the cut $V(a)$
are defined as

$\overline{\mathcal{A}}(z,a)=\begin{cases}\lim\limits_{\epsilon\to
0}\frac{1}{2i\epsilon}\big{(}\overline{\mathcal{M}}((1+\epsilon)z,a)-\overline{\mathcal{M}}((1-\epsilon)z,a)\big{)}\
\ \text{if }|z|=1\,,\\\ \\\ \lim\limits_{\epsilon\to
0}\frac{1}{2i\epsilon}\big{(}\overline{\mathcal{M}}(ze^{i\epsilon},a)-\overline{\mathcal{M}}(ze^{-i\epsilon},a)\big{)}\
\ \text{if }\text{arg }z=\pi\text{ mod }\frac{2\pi}{3}\,.\end{cases}$ (B.5)

In terms of the $\mathcal{A}_{i}(s_{1},s_{2})$ which is the usual
discontinuity of $\mathcal{M}(s_{1},s_{2})$ defined across the cut in $s_{i}$,
we have

$\overline{\mathcal{A}}(z,a)=\begin{cases}\pm\mathcal{A}_{i}(s_{1},s_{2})\ \
\forall\ z\in V_{i}^{\pm}(a)\ \ \text{ if }a<0\,,\\\
\mp\mathcal{A}_{i}(s_{1},s_{2})\ \ \forall\ z\in V_{i}^{\pm}(a)\ \ \text{ if
}a>0\,.\end{cases}$ (B.6)

Here we defined $V_{i}(a)=V_{i}^{+}(a)\cup V_{i}^{-}(a)$ such that
$V_{i}^{\pm}=\\{z;z\in V_{i},\,\pm\text{Im\,}(z/z_{i})\text{ or
}\pm(|z|-1)<0\\}$ (shown in Fig. 2) .

To proceed one assumes the following Regge behavior of the Mellin amplitude:
${\mathcal{M}}(s_{1},s_{2})\sim O(s_{i}^{2-\epsilon})$ (with $\epsilon>0$) at
large $s_{i}$ [41] which translates to $\overline{\mathcal{M}}(z,a)\sim
O((z-z_{i})^{-2+\epsilon})$ as $z\to z_{i}$. This implies that upon Taylor
expansion $\overline{\mathcal{M}}(z,a)$ will have the form:

$\overline{\mathcal{M}}(z,a)=\sum_{n=0}^{\infty}f_{n}(a)\,z^{n}\,.$ (B.7)

that should converge for $|z|<\rho_{-}$ or $|z|<1$.

Let us also add here the relation
$\overline{\mathcal{M}}(z,a)=\overline{\mathcal{M}}^{*}(1/z^{*},a)$ which
simply follows from
$\mathcal{M}(s_{1},s_{2})=\mathcal{M}^{*}(s_{1}^{*},s_{2}^{*})$ . Note also
that as $z\to 0$ or $z\to\infty$ one has $s_{i}\to 0$. Therefore
$\overline{\mathcal{M}}(0,a)=\overline{\mathcal{M}}(\infty,a)=\overline{\mathcal{M}}^{*}(0,a)=\mathcal{M}(s_{i}=0)=f_{0}$,
implying $f_{0}$ must be real and independent of $a$ .

Now one considers the following contour integrals

$\displaystyle I_{-}$ $\displaystyle=\frac{1}{2\pi
i}\oint_{\mathcal{I}}dz^{\prime}\frac{z^{\prime 3}-1}{z^{\prime
3}(z^{\prime}-z)}\overline{\mathcal{M}}(z^{\prime},a)=\frac{z^{3}-1}{z^{3}}\overline{\mathcal{M}}(z,a)+\frac{f_{0}}{z^{3}}+\frac{f_{1}(a)}{z^{2}}+\frac{f_{2}(a)}{z}\,,$
(B.8) $\displaystyle I_{+}$ $\displaystyle=\frac{1}{2\pi
i}\oint_{\mathcal{E}}dz^{\prime}\frac{z^{\prime 3}-1}{z^{\prime
3}(z^{\prime}-z)}\overline{\mathcal{M}}(z^{\prime},a)=f_{0}\,.$ (B.9)

The kernels are chosen to keep the integral finite near $z\to z_{i}$ and have
poles at desired places. Here $\mathcal{I}$ denotes the interior of the ‘cut’
$V(a)$ where one computes the residues at $z^{\prime}=z$ (the first term of
$I_{-}$) and $z^{\prime}=0$ (for the other three terms). Also $\mathcal{E}$
denotes the exterior of $V(a)$ where one considers only the pole at
$z^{\prime}=\infty$. Then as we push $\mathcal{I},\mathcal{E}$ towards $V(a)$
performing $I_{+}-I_{-}$ gives a dispersion relation with the discontinuity
$\overline{\mathcal{A}}(z,a)$ across $V(a)$, which may be written as

$\overline{\mathcal{M}}(z,a)=f_{0}+f_{1}(a)\frac{z}{1-z^{3}}+f_{2}(a)\frac{z^{2}}{1-z^{3}}+\frac{z^{3}}{1-z^{3}}\frac{1}{\pi}\int_{V(a)}dz^{\prime}\frac{z^{\prime
3}-1}{z^{\prime 3}(z^{\prime}-z)}\overline{\mathcal{A}}(z^{\prime},a)\,.$
(B.10)

###  B.2 Imposing crossing (anti)symmetry

So far we have not used any assumption on full crossing symmetry or
antisymmetry. In the notations of section 3 a crossing symmetric amplitude is
always an expansion in $x^{p}y^{q}$ and for an anticrossing amplitude it
should be in $wx^{p}y^{q}$ ($p,q\in\mathbb{Z}_{\geq 0}$). This implies the
following behavior under the map (B.1)

$\overline{\mathcal{M}}_{-}(z,a)\propto\bigg{[}\frac{{z^{3}}}{(z^{3}-1)^{2}}\bigg{]}^{n}\,,\
\
\overline{\mathcal{M}}_{+}(z,a)\propto\frac{{z^{3}(1+z^{3})}}{(z^{3}-1)^{3}}\bigg{[}\frac{{z^{3}}}{(z^{3}-1)^{2}}\bigg{]}^{n}\,,\
n\geq 0\,.$ (B.11)

where $+$ ($-$) denotes crossing (anti)symmetric case. Notice that both
expression are functions of $z^{3}$. Hence the discontinuity
$\overline{\mathcal{A}}_{+}(z,a)$ should be such that the general dispersion
relation (B.10) under small $z$ expansion would allow only the powers
$z^{3m},\,m\in\mathbb{Z}_{\geq 0}$ . Therefore we may replace it with

$\overline{\mathcal{M}}_{\pm}(z,a)=f_{\pm,0}+\frac{z^{3}}{(1-z^{3})\pi}\int_{V(a)}dz^{\prime}\frac{z^{\prime
3}-1}{z^{\prime}(z^{\prime
3}-z^{3})}\overline{\mathcal{A}}_{\pm}(z^{\prime},a)\,,$ (B.12)

where $f_{+,0}=0$. The above equation has an expansion in $z^{3m}$ and is
equal to (B.10) up to those powers. Indeed it can be checked that in (B.8) and
(B.9) if we began with the kernel of (B.12) we would end up with the same
result.

Now let us map this formula back to the variables $s_{i}$. For this we use
(B.1) to solve $z^{\prime}$ in terms of $s_{1}^{\prime}$ which gives two
solutions $z^{\prime}(s_{1}^{\prime})$ and $z^{\prime*}(s_{1}^{\prime})$
related by complex conjugation. 101010Using any $s^{\prime}_{i}$ ($i=1,2,3$)
leads to the same result using croossing (anti)symmetry.) We also focus on the
case $-\frac{\tau^{(0)}}{3}<a<0$ where we have
$|z^{\prime}(s_{1}^{\prime})|=1$, which further implies
$z^{\prime*}(s_{1}^{\prime})=1/z(s_{1}^{\prime})$. Under this transformation
the integrand of (B.12) becomes

$\frac{dz^{\prime}(s_{1}^{\prime})}{ds_{1}^{\prime}}\,\frac{z^{\prime}(s_{1}^{\prime})^{3}-1}{z^{\prime}(s_{1}^{\prime})(z^{\prime}(s_{1}^{\prime})^{3}-z^{3})}\,\overline{\mathcal{A}}_{\pm}(z^{\prime},a)+\frac{dz^{\prime*}(s_{1}^{\prime})}{ds_{1}^{\prime}}\,\frac{z^{\prime*}(s_{1}^{\prime})^{3}-1}{z^{\prime*}(s_{1}^{\prime})(z^{\prime*}(s_{1}^{\prime})^{3}-z^{3})}\,\overline{\mathcal{A}}_{\pm}(z^{\prime*},a)\,$
(B.13)

which is to be integrated over $s_{1}^{\prime}$ and on
$s_{1}^{\prime}\geq\tau^{(0)}$. We now also solve $s_{2}^{\prime}$ in terms of
$s_{1}^{\prime}$ and $a$ using (B.2) (recall that $a$ is fixed) which gives
two solutions $s_{2}^{\pm}(s_{1}^{\prime},a)$. We choose to use $s_{2}^{+}$ .
111111Choosing between $s_{2}^{+}$ and $s_{2}^{-}$ corresponds to the
convention of writing the branch cut. E.g. in writing (3.21) for odd $\ell$
the correct choice is $s_{2}^{+}$.

Now we point out that we must have (using (B.6))

$\overline{\mathcal{A}}_{\pm}(z^{\prime}(s_{1}^{\prime}),a)=\overline{\mathcal{A}}_{\pm}(z^{\prime*}(s_{1}^{\prime}),a)=\mp{\mathcal{A}}_{\pm}(s_{1}^{\prime},s_{2}^{+}(s_{1},a))\,.$
(B.14)

Note the ‘$-$’ sign in the second equality for the crossing antisymmetric
case. The change/invariance under $z\to z^{*}=1/z$ is simple to see using
(B.11) (as the $z$ dependencies are also shared by $\overline{\mathcal{A}}$).
Now putting in the factor $\frac{z^{3}}{1-z^{3}}$ in (B.13) we get for the
crossing symmetric case (writing $f_{-,0}=\alpha_{0}$)

$\mathcal{M}_{-}(s_{1},s_{2})=\alpha_{0}+\frac{1}{\pi}\int
ds_{1}^{\prime}\,\mathcal{A}_{-}(s_{1}^{\prime},s_{2}^{+}(s_{1}^{\prime},a))\,H_{-}(s_{1}^{\prime},s_{1},s_{2},s_{3})\,,$
(B.15)

with

$H_{-}(s_{1}^{\prime},s_{1},s_{2},s_{3})=\frac{s_{1}}{s_{1}^{\prime}-s_{1}}+\frac{s_{2}}{s_{1}^{\prime}-s_{2}}+\frac{s_{3}}{s_{1}^{\prime}-s_{3}}\,.$
(B.16)

This is the relation presented in [15, 16]. For the crossing antisymmetric
case we get our result (3.16).

##  Appendix C Checks with the dispersion relation

###  C.1 Derivation of Witten diagrams for $\ell=5$

The polynomial $P_{\Delta,\ell}(s_{1},s_{2})$ from (3.21) for $\ell=5$ may be
written as

$P_{\Delta,\ell=5}(s_{1},s_{2})=\sum_{m+n\leq 5}b_{m,n}s_{1}^{m}s_{2}^{n}\,.$
(C.1)

This form is easily seen by evaluating the polynomial at simple values e.g.
$\Delta_{\phi}=1,\Delta=2,d=4$ . With the above one can write the $\ell=5$
component of the Mellin amplitude in (3.22) as

$\mathcal{M}_{\Delta,\ell=5,k}(s_{1},s_{2})=H(\tau_{k},s_{1},s_{2},s_{3})\frac{1}{\tau_{k}}\sum_{m+n\leq
5}b_{m,n}(\tau_{k})^{m}s_{2}^{+}(s,a)^{n}=w\sum_{p=0}^{\infty}\sum_{q=0}^{\infty}M_{p,q,k}x^{p}a^{q}\,.$
(C.2)

In the last equality we have expanded in all powers of $x$ and $a$.

Let us also define the following quantity

$\mathcal{M}^{(0)}_{\Delta,\ell=5,k}(s_{1},s_{2})=\frac{P_{\Delta,5}(s_{1},s_{2})}{s_{1}-\tau_{k}}-\frac{P_{\Delta,5}(s_{2},s_{1})}{s_{2}-\tau_{k}}-\frac{P_{\Delta,5}(s_{3},s_{2})}{s_{3}-\tau_{k}}\,,$
(C.3)

with $s_{3}=-s_{1}-s_{2}$. This is equivalent to the meromorphic part (A.3) of
exchange Witten diagrams in all three channels. It is easy to see that the
above can be written as

$\mathcal{M}^{(0)}_{\Delta,\ell=5,k}(s_{1},s_{2})=w\sum_{p=0}^{2}\sum_{q=0}^{1}M^{(0)}_{p,q,k}x^{p+q}a^{q}\,.$
(C.4)

We then find that the difference

$\mathcal{M}_{\Delta,5,k}(s_{1},s_{2})-\mathcal{M}^{(0)}_{\Delta,5,k}(s_{1},s_{2})=w\Bigg{[}\Big{\\{}M_{0}+\frac{x}{64\tau_{k}}+\frac{ax}{64\tau^{2}_{k}}\Big{\\}}\,+\,\sum_{m>0}\widetilde{M}^{(1)}_{m}a^{m}+\sum_{n>1}\widetilde{M}^{(2)}_{n}a^{n}x\Bigg{]}\,.$
(C.5)

Here $M_{0}=M_{0,0,k}-M^{(0)}_{0,0,k}$. The last two terms in (C.5) are
identified as nonlocal terms that should be set to zero. The terms in curly
brackets are crossing antisymmetric contact diagrams that should be added to
$\mathcal{M}^{(0)}_{\Delta,\ell=5,k}(s_{1},s_{2})$ to have the correct
Polyakov block $\mathcal{P}_{\Delta,\ell}(u,v)$.

###  C.2 Numerical checks

#### Locality conditions

Here we discuss a numerical check of our discussions in section 3 with an
explicit crossing antisymmetric solution. For this we consider the following
artifical Mellin amplitude:

$\mathcal{M}_{0}(s_{1},s_{2})=\frac{\left(s_{1}-s_{2}\right)\left(2s_{1}+s_{2}\right)\left(s_{1}+2s_{2}\right)\Gamma\left(-2s_{1}-\frac{7}{8}\right)\Gamma\left(-2s_{2}-\frac{7}{8}\right)\Gamma\left(2s_{1}+2s_{2}-\frac{7}{8}\right)}{\Gamma^{2}\big{(}\frac{\Delta_{\phi}}{3}-s_{1}\big{)}\Gamma^{2}\big{(}\frac{\Delta_{\phi}}{3}-s_{2}\big{)}\Gamma^{2}\big{(}s_{1}+s_{2}+\frac{\Delta_{\phi}}{3}\big{)}}\,,$
(C.6)

for $\Delta_{\phi}=\frac{21}{16}$ and $d=2$. It can be checked that at large
$s_{1}$ we have $\mathcal{M}_{0}\sim s_{1}^{2/3}$ ($s_{2}$ fixed). The poles
at $s_{1}=-\frac{7}{16}+m$, $m\in\mathbb{Z}_{\geq 0}$ gives operators at
positions $\Delta=\frac{7}{8}+n+\ell$ with $\ell$ odd.

We will verify the locality conditions (3.20) which can be done by assuming an
expansion as follows:

$\mathcal{M}_{0}^{\prime}(s_{1},s_{2})=\sum_{p=0}^{p_{\text{max}}}\sum_{q=0}^{q_{\text{max}}}\Bigg{[}\frac{s_{2}-s_{3}}{(-\frac{7}{16}+\frac{p}{2})-s_{1}}+\frac{s_{3}-s_{1}}{(-\frac{7}{16}+\frac{p}{2})-s_{2}}+\frac{s_{1}-s_{2}}{(-\frac{7}{16}+\frac{p}{2})-s_{3}}\Bigg{]}a^{q}c_{p,q}\,,$
(C.7)

with $s_{3}=-s_{1}-s_{2}$ . The term in parentheses is an expansion in
$wx^{i}y^{j}$. However the powers in $a$ introduce nonlocal terms. We can
focus on the $p$-th pole of $s_{1}$ in
$\mathcal{M}_{0}-\mathcal{M}^{\prime}_{0}$ and expand around $s_{2}=0$ .
Solving them order by order one can obtain $c_{p,q}$ .

In Table 1 below we compare $\mathcal{M}_{0}(s_{1},s_{2})$ with
$\mathcal{M}_{0}^{\prime}(s_{1},s_{2})$ for $p_{\text{max}}=q_{\text{max}}=2$
and $p_{\text{max}}=q_{\text{max}}=8$ :

$s_{1}$ |  $s_{2}$ |  $\mathcal{M}_{0}$ |  $\mathcal{M}^{\prime}_{0}\,[p_{\max}=q_{\max}=2]$ |  $\mathcal{M}^{\prime}_{0}\,[p_{\max}=q_{\max}=8]$
---|---|---|---|---
$\frac{i}{2}$ | $\frac{1}{14}$ | $-0.00899-0.18137i$ | $-0.01089-0.19530i$ | $-0.00915-0.18246i$
$\frac{1}{10}$ | $\frac{1}{20}$ | $-0.056362$ | $-0.056168$ | $-0.056351$
$\frac{1}{19}+\frac{i}{30}$ | $\frac{1}{10}(1+i)$ | $-0.00536+0.01651i$ | $-0.00505+0.01645i$ | $-0.00532+0.01650i$
Table 1: Comparing $\mathcal{M}_{0}$ and $\mathcal{M}^{\prime}_{0}$ for
different choices of $s_{1},s_{2}$ while increasing $p_{\max}$ and $q_{\max}$.
The $s_{1},s_{2}$ values were chosen to have a fast convergence.

This shows that it makes sense to work with variable $a$. Indeed the nonlocal
singularities in (C.7) must go to 0 which confirms the locality conditions.

In order to check the precise disprsion formula (3.16) one needs to extract
the OPE coefficients $a_{\Delta,\ell}$ from $\mathcal{M}_{0}$ using the Mellin
integral (3.26). For a good match however one has to include operators with
$n,\ell\sim 100$ (see [16]). We do not carry out this check. Instead we
demonstrate the validity of Polyakov conditions (3.3) on a more physically
relevant correlator.

#### Polyakov conditions in WZW models

We will consider $SU(2)_{k}$ WZW models in 2d. These theories have the global
symmetry $SO(4)\cong\frac{SU(2)_{L}\times SU(2)_{R}}{Z_{2}}$. The 4-point
function of scalars in a non-linear sigma model with a level-$k$ WZW term is
known [42]. In [43] the $O(4)$ sector of the correlator was studied for $k=1$.
The same can be similarly obtained for any $k$.

We will work with $k=2$ because it is simple and also our equations converge
faster. The corresponding $O(4)$ sector is given by

$\displaystyle\frac{\mathcal{G}_{ijkl}(z,\bar{z})}{(z\bar{z})^{-\Delta_{\phi}}}=$
$\displaystyle
P(\xi_{\pm},\bar{\xi}_{\pm})\Bigg{[}\bigg{[}\frac{\left(\bar{\xi}_{-}+\bar{\xi}_{+}\right){}^{2}}{\bar{\xi}_{-}\bar{\xi}_{+}}-\frac{\left(\xi_{+}-\xi_{-}\right)\left(\bar{\xi}_{+}-\bar{\xi}_{-}\right)\left(\bar{\xi}_{-}\bar{\xi}_{+}+\xi_{-}\xi_{+}\right)}{\xi_{-}\xi_{+}\sqrt{\xi_{-}\xi_{+}+1}\bar{\xi}_{-}\bar{\xi}_{+}\sqrt{\bar{\xi}_{-}\bar{\xi}_{+}+1}}+\frac{\xi_{+}}{\xi_{-}}+\frac{\xi_{-}}{\xi_{+}}-2\bigg{]}\delta_{ij}\delta_{kl}$
$\displaystyle+\bigg{[}\frac{\left(\xi_{+}-\xi_{-}\right)\left(\bar{\xi}_{+}-\bar{\xi}_{-}\right)\left(\frac{1}{\bar{\xi}_{-}\bar{\xi}_{+}}+\frac{1}{\xi_{-}\xi_{+}}+2\right)}{\sqrt{\xi_{-}\xi_{+}+1}\sqrt{\bar{\xi}_{-}\bar{\xi}_{+}+1}}-\frac{\bar{\xi}_{+}}{\bar{\xi}_{-}}-\frac{\bar{\xi}_{-}}{\bar{\xi}_{+}}-\frac{\xi_{-}}{\xi_{+}}-\frac{\xi_{+}}{\xi_{-}}+4\bigg{]}\delta_{ik}\delta_{jl}$
$\displaystyle+\bigg{[}\frac{\left(\xi_{-}+\xi_{+}\right)\left(\bar{\xi}_{-}-\bar{\xi}_{+}\right)^{2}+\left(\xi_{-}+\xi_{+}\right)\left(\xi_{-}-\xi_{+}\right)^{2}+\frac{2\sqrt{z}\left(\bar{\xi}_{+}-\bar{\xi}_{-}\right)\left(\bar{\xi}_{-}\bar{\xi}_{+}+\xi_{-}\xi_{+}+2\right)}{\sqrt{\xi_{-}\xi_{+}+1}\sqrt{\bar{\xi}_{-}\bar{\xi}_{+}+1}}}{\xi_{-}\xi_{+}\left(\xi_{-}+\xi_{+}\right)\bar{\xi}_{-}\bar{\xi}_{+}}\bigg{]}\delta_{il}\delta_{kj}\Bigg{]}\,,$
(C.8)

where $\Delta_{\phi}=\frac{3}{8}$ and

$P(\xi_{\pm},\bar{\xi}_{\pm})=\frac{1}{16}\left(\xi_{-}+\xi_{+}\right)\left(\bar{\xi}_{-}+\bar{\xi}_{+}\right)({\xi_{-}\xi_{+}\bar{\xi}_{-}\bar{\xi}_{+}})^{\frac{1}{4}}\,,\
\xi_{\pm}=(1\pm\sqrt{z})^{\frac{1}{2}}\,,\
\bar{\xi}_{\pm}=(1\pm\sqrt{\bar{z}})^{\frac{1}{2}}\,.$ (C.9)

As discussed around (1.3) we can use the parity odd eigenvector of the $O(N)$
crossing matrix (see [14]) to obtain a crossing antisymmetric solution as
follows:

$\displaystyle\frac{\mathcal{G}_{+}(z,\bar{z})}{(z\bar{z})^{-\Delta_{\phi}}}=\frac{\left(\xi_{-}+\xi_{+}\right)\left(\bar{\xi}_{-}+\bar{\xi}_{+}\right)}{16\left(\xi_{-}\xi_{+}\bar{\xi}_{-}\bar{\xi}_{+}\right){}^{3/4}}$
$\displaystyle\bigg{[}2\xi_{-}\xi_{+}\left(\bar{\xi}_{-}^{2}+\bar{\xi}_{+}^{2}\right)-\left(\xi_{-}^{2}+\xi_{+}^{2}\right)\left(\bar{\xi}_{-}-\bar{\xi}_{+}\right)^{2}$
$\displaystyle-\frac{2\left(\xi_{-}-\xi_{+}\right)\left(\bar{\xi}_{-}\bar{\xi}_{+}+\xi_{-}\xi_{+}+1\right)\left(\bar{\xi}_{-}-\bar{\xi}_{+}\right)}{{\left(\xi_{-}\xi_{+}+1\right)^{1/2}\left(\bar{\xi}_{-}\bar{\xi}_{+}+1\right)}^{1/2}}\bigg{]}\,.$
(C.10)

This has a conformal block decomposition with dimensions $\Delta=n+\ell$ and
spin $\ell$ for $n,\ell\in\mathbb{Z}_{\geq 0}$ .

Figure 3:  The plot of Mellin amplitude $\mathcal{M}_{+}(s_{1},s_{2}=0)$ for
$m_{\max}=\ell_{\max}=2$ (left) and $m_{\max}=4\,,\ell_{\max}=14$ (right). The
double zero structure at $s_{1}=\frac{1}{8}+p$ become prominent on adding more
operators.

We will test the conditions
$\mathfrak{F}_{+,p}(0)\equiv\mathcal{M}_{+}(s_{1}=\frac{1}{8}+p,s_{2}=0)=0$
from (3.3) . For this we use the above $\Delta,\ell$ to write an approximate
anticrossing Mellin amplitude as follows

$\mathcal{M}_{+}(s_{1},s_{2})=\sum_{\ell=0}^{\ell_{\text{max}}}\sum_{m=0}^{m_{\text{max}}}H_{+}\Big{(}\text{\small$\frac{m}{2}-\frac{1}{4}$},s_{1},s_{2},-s_{1}-s_{2}\Big{)}\sum_{n=0}^{2m}\frac{c^{(m-\frac{n}{2})}_{\Delta,\ell}}{(\text{\small$\frac{m}{2}-\frac{1}{4}$})}Q_{\ell,m-\frac{n}{2}}^{(\Delta)}(a)\,.$
(C.11)

Here $\ell_{\text{max}}$ is a cutoff in spin and $m_{\text{max}}$ is the
highest pole in $s_{i}$ . In Table 2 we show $\mathfrak{F}_{+,p}(0)$ for
diferent values of $\ell_{\text{max}}$, $m_{\text{max}}$ .

$\ell_{\max}$ |  $m_{\max}$ |  $\mathfrak{F}_{+,0}(0)$ |  $\mathfrak{F}_{+,1}(0)$ |  $\mathfrak{F}_{+,2}(0)$
---|---|---|---|---
4 | 2 | -0.00024 | -0.173489 | -0.37102
12 | 2 | -0.00622 | -0.08865 | -0.25913
12 | 12 | -0.00621 | -0.08770 | -0.21302
Table 2: Values of $\mathfrak{F}_{+,p}$ with $p=0,1,2$ for different choices
of upper limits $\ell_{\max},m_{\max}$.

As we add operators $\mathfrak{F}_{+,p}$ becomes smaller except
$\mathfrak{F}_{+,0}$ which grows with $\ell_{\text{max}}$. The latter is not
very surprising as we are evaluating $\mathcal{M}_{+}(s_{1},s_{2})$ for low
values of $s_{1},s_{2}$, which always has a trivial zero for $s_{1}=s_{2}=0$
(from the antisymmetric term $w$). On adding operators the different zeroes
must decouple.

In Figure 3 we have plotted $\mathcal{M}_{+}(s_{1},s_{2}=0)$ for $0\leq
s_{1}\leq 2.2$ . The nontrivial double zeroes are expected at $s_{1}=0.125+p$
and poles at $\frac{m}{2}-0.25$ . The double zero locations get more accurate
as we add more operators. As expected the convergence gets poorer for larger
values of $s_{1}$. We carried out the above analysis also for level $k=3$,
which yields similar results.

##  Appendix D Product functionals in $d=4$ and $d=6$

In this appendix we review how to use the product functionals in $d=4$ and
$d=6$. We then use the simple functionals discussed in section 4.2 for 2d to
obtain analytic bounds on OPE coefficients.

###  D.1 $d=4$

In $d=4$ we have the conformal blocks:

$\displaystyle G^{d=4}_{\Delta,\ell}(z,\bar{z})$
$\displaystyle=\frac{1}{1+\ell}\frac{z\bar{z}}{z-\bar{z}}\big{[}G_{\frac{\rho}{2}}(z|\text{\tiny$\frac{\Delta_{\phi}}{2}$})G_{\frac{\tau-2}{2}}(\bar{z}|\text{\tiny$\frac{\Delta_{\phi}}{2}$})-(z\leftrightarrow\bar{z})\big{]}\,,$
$\displaystyle=\frac{1}{1+\ell}\frac{1}{z-\bar{z}}\big{[}G_{\frac{\rho}{2}}(z|\text{\tiny$\frac{\Delta_{\phi}-1}{2}$})G_{\frac{\tau-2}{2}}(\bar{z}|\text{\tiny$\frac{\Delta_{\phi}-1}{2}$})-(z\leftrightarrow\bar{z})\big{]}\,.$
(D.1)

The crossing vector becomes

$F^{d=4}_{-,\Delta,\ell}(z,\bar{z}|\Delta_{\phi})=\frac{\big{[}F_{+,\frac{\rho}{2}}(z|\text{\tiny$\frac{\Delta_{\phi}-1}{2}$})F_{+,\frac{\tau-2}{2}}(\bar{z}|\text{\tiny$\frac{\Delta_{\phi}-1}{2}$})+F_{-,\frac{\rho}{2}}(z|\text{\tiny$\frac{\Delta_{\phi}-1}{2}$})F_{-,\frac{\tau-2}{2}}(\bar{z}|\text{\tiny$\frac{\Delta_{\phi}-1}{2}$})-(z\leftrightarrow\bar{z})\big{]}}{2(1+\ell)(z-\bar{z})}\,.$
(D.2)

We can combine
$\mathcal{I}_{z}\mathcal{I}_{\bar{z}}F_{\Delta,\ell}(z,\bar{z})$ and
$\mathcal{I}_{z}\mathcal{I}_{\bar{z}}F_{\Delta,\ell}(z,1-\bar{z})$ to get
expressions involving only $F_{+}$ or only $F_{-}$. So the product functionals
must involve either both $+$ type functionals or both $-$ type. We choose the
functional to be

$[\omega^{(1)}_{\pm}\otimes\omega^{(2)}_{\pm}]^{d=4}(\Delta,\ell|\Delta_{\phi})=2\int_{1}^{\infty}\frac{dzd{\bar{z}}}{\pi^{2}}h^{(1)}_{\pm}(z)h^{(2)}_{\pm}(\bar{z})(z-\bar{z})\big{[}\mathcal{I}_{z}\mathcal{I}_{\bar{z}}F_{-,\Delta,\ell}(z,\bar{z})\pm\mathcal{I}_{z}\mathcal{I}_{\bar{z}}F_{-,\Delta,\ell}(z,1-\bar{z})\big{]}\,.$
(D.3)

The kernels are chosen from $\alpha_{\pm,n},\beta_{\pm,n}$ . However notice
the $z-\bar{z}$ factor in the inegral above. It means we have to pick a
combination of functional that have a stronger Regge falloff and cancels this
factor. Let us work with fermionic functionals
$\alpha^{F}_{\pm,n},\beta^{F}_{\pm,n}$ and pick

$\displaystyle\tilde{\beta}^{F}_{-,n}$
$\displaystyle=\beta^{F}_{-,n}-b_{n}\beta^{F}_{-,0}\,,\hskip
28.45274pt\text{$b_{n}$ chosen such that}\hskip
14.22636ptf_{\tilde{\beta}^{F}_{-,n}}=O(z^{-4})\,,$
$\displaystyle\tilde{\alpha}^{F}_{-,n}$
$\displaystyle=\alpha^{F}_{-,n}-a_{n}\beta^{F}_{-,0}\,,\hskip
28.45274pt\text{$a_{n}$ chosen such that}\hskip
14.22636ptf_{\tilde{\alpha}^{F}_{-,n}}=O(z^{-4})\,.$ (D.4)

Note that for bosonic functions (recall $\beta_{-,0}^{B}=0$) we should
subtract $\alpha^{B}_{-,0}$ instead. The product functionals are then as
follows (denoting
$\omega^{(1)}\omega^{(2)}\equiv[\omega^{(1)}\otimes\omega^{(2)}]^{d=4}$ and
suppressing $F$ superscripts)

$\displaystyle\beta_{+,m}\beta_{+,m}\hskip 28.45274ptm,n\geq 0\,,m\neq
n\,,\hskip 28.45274pt\alpha_{+,m}\alpha_{+,m}\hskip 28.45274ptm,n\geq
0\,,m\neq n\,,$ $\displaystyle\beta_{+,m}\alpha_{+,m}\hskip 28.45274ptm,n\geq
0\,,\hskip 68.28644pt\tilde{\beta}_{-,m}\tilde{\beta}_{-,m}\hskip
28.45274ptm,n\geq 1\,,m\neq n\,,$
$\displaystyle\tilde{\alpha}_{-,m}\tilde{\beta}_{-,m}\hskip 28.45274ptm,n\geq
0\,,m\neq n\,,\hskip 28.45274pt\tilde{\beta}_{-,m}\tilde{\beta}_{-,m}\hskip
28.45274ptm\geq 1\,,n\geq 0\,.$ (D.5)

The functional action is given by

$\omega^{(1)}_{\pm}\omega^{(2)}_{\pm}(\Delta,\ell|\Delta_{\phi})=\frac{1}{\ell+1}\big{[}\omega_{\pm}^{(1)}(\rho|\Delta_{\phi}|4)\omega_{\pm}^{(2)}(\tau-2|\Delta_{\phi}|4)-\omega_{\pm}^{(2)}(\rho|\Delta_{\phi}|4)\omega_{\pm}^{(1)}(\tau-2|\Delta_{\phi}|4)\big{]}\,.$
(D.6)

Here we defined
$\omega_{\pm}(\Delta|\Delta_{\phi}|4)=\omega_{\pm}(\text{\tiny$\frac{\Delta}{2}$}|\text{\tiny$\frac{\Delta_{\phi}-1}{2}$})$
. As in 2d the action is entirely determined from the $\pm$ type Polyakov
blocks.

However one should be careful as subtractions (D.1) are involved in the $-$
case. We saw in section 2.3.2 (for the $+$ case) that to obtain Polyakov
blocks related to functionals with stricter large $z$ fall-off one has to add
more contact diagrams. Therefore to obtain
$\tilde{\beta}_{-,n},\tilde{\alpha}_{-,n}$ one has to add a contact diagram to
$\mathcal{P}_{-,\Delta}(z)$ such that one coefficient in its conformal block
decomposition is zero [3].

###  D.2 $d=6$

The $d=6$ conformal blocks are given by

$\displaystyle G_{\Delta,\ell}^{d=6}(z,\bar{z})$
$\displaystyle=K\frac{z^{2}\bar{z}^{2}}{(z-\bar{z})^{2}}\Big{[}G_{\Delta-2,\ell}^{d=4}(z,\bar{z})-G_{\Delta-2,\ell+2}^{d=4}(z,\bar{z})$
$\displaystyle-\frac{(\Delta-4)(\tau-4)^{2}}{16(\Delta-2)\left(\tau^{2}-8\tau+15\right)}G_{\Delta,\ell}^{d=4}(z,\bar{z})+\frac{(\Delta-4)\rho^{2}}{16(\Delta-2)\left(\rho^{2}-1\right)}G_{\Delta,\ell+2}^{d=4}(z,\bar{z})\Big{]}\,.$
(D.7)

where

$K=-\frac{6\left(\rho^{2}-1\right)\left(\tau^{2}-8\tau+15\right)}{(\rho-1)(\rho+1)(\tau-5)(\tau-3)(\ell+2)}\,.$
(D.8)

The crossing vector is given by

$\displaystyle F_{\Delta,\ell}^{d=6}(z,\bar{z}|\Delta_{\phi})$
$\displaystyle=\frac{K}{\left(z-\bar{z}\right)^{2}}\Big{[}F_{\Delta-2,\ell}^{d=4}(z,\bar{z}|\Delta_{\phi}-2)-F_{\Delta-2,\ell+2}^{d=4}(z,\bar{z}|\Delta_{\phi}-2)$
$\displaystyle-\frac{(\Delta-4)(\tau-4)^{2}}{16(\Delta-2)\left(\tau^{2}-8\tau+15\right)}F_{\Delta,\ell}^{d=4}(z,\bar{z}|\Delta_{\phi}-2)+\frac{(\Delta-4)\rho^{2}}{16(\Delta-2)\left(\rho^{2}-1\right)}F_{\Delta,\ell+2}^{d=4}(z,\bar{z}|\Delta_{\phi}-2)\Big{]}\,.$
(D.9)

Similar to the $d=4$ case we choose the functional kernels in the following
way:

$[\omega^{(1)}_{\pm}\otimes\omega^{(2)}_{\pm}]^{d=6}(\Delta,\ell|\Delta_{\phi})=2\int_{++}\frac{dzd{\bar{z}}}{\pi^{2}}h^{(1)}_{\pm}(z)h^{(2)}_{\pm}(\bar{z})(z-\bar{z})^{3}\big{[}\mathcal{I}_{z}\mathcal{I}_{\bar{z}}F_{\Delta,\ell}(z,\bar{z})\pm\mathcal{I}_{z}\mathcal{I}_{\bar{z}}F_{\Delta,\ell}(z,1-\bar{z})\big{]}\,.$
(D.10)

The functional action is given by
($\omega^{(1)}\omega^{(2)}\equiv\big{[}\omega^{(1)}\otimes\omega^{(2)}\big{]}$)

$\displaystyle[\omega^{(1)}_{\pm}\omega^{(2)}_{\pm}]^{d=6}(\Delta,\ell|\Delta_{\phi})=K\Big{[}[\omega^{(1)}_{\pm}\omega^{(2)}_{\pm}]^{d=4}(\Delta-2,\ell|\Delta_{\phi})-[\omega^{(1)}_{\pm}\omega^{(2)}_{\pm}]^{d=4}(\Delta-2,\ell+2|\Delta_{\phi})$
$\displaystyle-\frac{(\Delta-4)(\tau-4)^{2}[\omega^{(1)}_{\pm}\omega^{(2)}_{\pm}]^{d=4}(\Delta,\ell|\Delta_{\phi})}{16(\Delta-2)\left(\tau^{2}-8\tau+15\right)}+\frac{(\Delta-4)\rho^{2}[\omega^{(1)}_{\pm}\omega^{(2)}_{\pm}]^{d=4}(\Delta,\ell+2|\Delta_{\phi})}{16(\Delta-2)\left(\rho^{2}-1\right)}\Big{]}$
(D.11)

Since now we have a $(z-\bar{z})^{3}$ in the integral of (D.10) we have to
subtract more functionals while choosing the kernels, so that the falloff is
at least $O(z^{-5})$. To have a Regge bounded integrand we should subtract one
functionals from the $+$ type ones and 2 from the $-$ type. Working with
fermionic functionals we pick:

$\displaystyle\hat{\beta}_{-,n}$
$\displaystyle=\beta_{-,n}-b_{n}^{(1)}\beta_{-,0}-b_{n}^{(2)}\alpha_{-,0}\,,\hskip
28.45274pt\text{$b_{n}^{(1,2)}$ chosen such that}\hskip
14.22636ptf_{\hat{\beta}_{-,n}}=O(z^{-6})\,,$
$\displaystyle\hat{\alpha}_{-,n}$
$\displaystyle=\alpha_{-,n}-a_{n}^{(1)}\beta_{-,0}-a_{n}^{(2)}\alpha_{-,0}\,,\hskip
28.45274pt\text{$a_{n}^{(1,2)}$ chosen such that}\hskip
14.22636ptf_{\hat{\alpha}_{-,n}}=O(z^{-6})\,,$
$\displaystyle\hat{\beta}_{+,n}$
$\displaystyle=\beta_{+,n}-c_{n}\beta_{+,0}\,,\hskip 85.35826pt\text{$c_{n}$
chosen such that}\hskip 14.22636ptf_{\hat{\beta}_{+,n}}=O(z^{-5})\,,$
$\displaystyle\hat{\alpha}_{+,n}$
$\displaystyle=\alpha_{+,n}-d_{n}\beta_{+,0}\,,\hskip 85.35826pt\text{$d_{n}$
chosen such that}\hskip 14.22636ptf_{\hat{\alpha}_{+,n}}=O(z^{-5})\,.$ (D.12)

To obtain their actions from Witten diagrams, we should appropriately add two
(crossing symmetric) contact diagrams to $\mathcal{P}_{-,\Delta}(z)$ and one
(crossing antisymmetric) contact diagram to $\mathcal{P}_{+,\Delta}(z)$.

The $d=6$ product functionals are hence chosen as follows

$\displaystyle\hat{\beta}_{+,m}\hat{\beta}_{+,m}\hskip 28.45274ptm,n\geq
1\,,m\neq n\,,\ \hskip 28.45274pt\hat{\alpha}_{+,m}\hat{\alpha}_{+,m}\hskip
28.45274ptm,n\geq 0\,,m\neq n\,,$
$\displaystyle\hat{\beta}_{+,m}\hat{\alpha}_{+,m}\hskip 28.45274ptm\geq
1\,,n\geq 0\,,\ \hskip 42.67912pt\hat{\beta}_{-,m}\hat{\beta}_{-,m}\hskip
28.45274ptm,n\geq 2\,,m\neq n\,,$
$\displaystyle\hat{\alpha}_{-,m}\hat{\beta}_{-,m}\hskip 28.45274ptm,n\geq
0\,,m\neq n\,,\ \hskip 28.45274pt\hat{\beta}_{-,m}\hat{\beta}_{-,m}\hskip
28.45274ptm\geq 2\,,n\geq 0\,.$ (D.13)

###  D.3 Bounds with product functionals

In this subsection we use the simple functionals built with the 2d product
functionals (see section 4.2.2) to extend the bounds on OPE density obtained
in $d=2$ [20] to a CFT with general global symmetry. For the uncharged case
the bound was obtained by choosing the following combination of 1d
functionals:

$\tilde{\alpha}_{\pm,m}=\alpha_{\pm,m}+c_{\pm}\beta_{\pm,m},\ \
\text{$c_{\pm}$ chosen such that}\ \ f_{\alpha_{\pm,m}}\sim
O(z^{\frac{1}{2}(9\pm 1)})\,.$ (D.14)

These modified functionals have the following positivity and asymptotic
properties respectively:

$\displaystyle\tilde{\alpha}_{+,m}(\tau)\geq 0,\ \ \forall\ \tau\geq 0\,,\ \ \
\ \tilde{\alpha}_{-,m}(\tau)\geq 0,\ \ \forall\
\tau\geq\tau_{0}(\Delta_{\phi})\,,$ (D.15)
$\displaystyle\tilde{\alpha}_{\pm,m}(0)\stackrel{{\scriptstyle
m\to\infty}}{{=}}\pm a_{h_{m}}^{\text{free}}\,,$ (D.16)

where

$a_{h}^{\text{free}}=\frac{2\Gamma(\frac{h}{2})^{2}}{\Gamma(h-1)\Gamma(\Delta_{\phi})^{2}}\frac{\Gamma(\frac{h+2\Delta_{\phi}-2}{2})}{\Gamma(\frac{h+2\Delta_{\phi}+2}{2})}\,,\
\ \ \ h_{m}=2+2\Delta_{\phi}+4m\,.$ (D.17)

For $h,h_{m}\gg 1$ with $h-h_{m}$ fixed we have

$\alpha_{\pm,m}(h)\stackrel{{\scriptstyle
h,h_{m}\to\infty}}{{=}}\left(\frac{a_{h_{m}}^{\text{free}}}{a_{h}^{\text{free}}}\right)\left(\frac{4}{\pi}\frac{\sin(\frac{\pi}{4}(h-h_{m}))}{h-h_{m}}\right)^{2}\,.$
(D.18)

Now the action of the product functional
$\tilde{\alpha}_{-,m}\tilde{\alpha}_{+,n}$ on the bootstrap equation implies
the following bound:

$\sum_{\stackrel{{\scriptstyle|\Delta-\Delta^{\text{prod}}_{m,\ell}|\leq
2}}{{\ell=2(m-n)}}}(1+\delta_{\ell,0})a_{\Delta,\ell}\tilde{\alpha}_{-,m}(\tau)\tilde{\alpha}_{+,n}(\rho)\leq-\sum_{\stackrel{{\scriptstyle
0\leq\tau\leq\tau_{0}}}{{\ell=0,2,\cdots}}}a_{\Delta,\ell}\big{[}\tilde{\alpha}_{-,m}(\tau)\tilde{\alpha}_{+,n}(\rho)+\tilde{\alpha}_{+,n}(\tau)\tilde{\alpha}_{-,m}(\rho)\big{]}\,.$
(D.19)

Here $\Delta_{m,\ell}^{\text{prod}}=2+2\Delta_{\phi}+4m+\ell$ . Using the
above properties of $\tilde{\alpha}_{\pm,m}$ we get the following upper bound
on OPE density:

$\lim\limits_{m,n\to\infty}\sum_{\stackrel{{\scriptstyle|\Delta-\Delta^{\text{prod}}_{m,\ell}|\leq
2}}{{\ell=2(m-n)}}}\left(\frac{a_{\Delta,\ell}}{a_{\Delta,\ell}^{\text{prod}}}\right)\left(\frac{4}{\pi}\frac{\sin(\frac{\pi}{4}(\Delta-\Delta_{m,\ell}))}{\Delta-\Delta_{m,\ell}}\right)^{4}\leq
1\,,$ (D.20)

where
$a_{\Delta,\ell}^{\text{prod}}=\frac{2}{1+\delta_{\ell,0}}a_{\rho}^{\text{free}}a_{\tau}^{\text{free}}$
.

Recall from section 4.2 that the global symmetry problem is formulated in
terms of a crossing matrix $C^{\mathfrak{ab}}$. It has the eigenvalues $+1$
with $r_{+}$ degeneracy and $-1$ with $r_{-}$ degeneracy. It was shown in [14]
that there can be $r_{+}-r_{-}$ eigenvectors with $+1$ eigenvalue, that have
positive even parity components and zero parity odd components. These are
found by taking linear combinations of them and (S : singlet):

$E_{+}^{\mathfrak{a}}=\begin{cases}\frac{2+d_{\mathfrak{r}}}{2},&\mathfrak{a}=\textbf{S}\\\
\sqrt{d_{\mathfrak{r}}},&\eta^{\mathfrak{a}}=1\\\
0,&\eta^{\mathfrak{a}}=-1\end{cases}\,,\ \
\tilde{E}_{+}^{\mathfrak{a}}=\begin{cases}\frac{d_{\mathfrak{r}}}{2},&\mathfrak{a}=\textbf{S}\\\
\sqrt{d_{\mathfrak{r}}},&\eta^{\mathfrak{a}}=1\\\
0,&\eta^{\mathfrak{a}}=-1\end{cases}\,,$ (D.21)

which are also $+1$ eigenvectors (and $\eta$ is defined by the tensor
$T^{\mathfrak{a}}_{ij,kl}=\eta^{\mathfrak{a}}T^{\mathfrak{a}}_{ji,kl}$, see
(1.1)). In fact the other linearly independent $r_{-}$ combinations can also
be chosen to have non-negative components.

Now we note that for any eigenvector $e_{\pm}$ we must have
$\sum_{\mathfrak{b}}e_{\pm}^{\mathfrak{b}}P_{\mp}^{\mathfrak{a}\mathfrak{b}}=0$.
Choosing a non-negative eigenvectors as above (lets say
$E^{\mathfrak{b}}_{+,s}$), we can consider the combination of simple
functionals $E_{+,s}^{\mathfrak{b}}\omega^{\mathfrak{b}|\mathfrak{c}}$ whose
action would be proportional to
$\omega_{-}^{(1)}\omega_{-}^{(2)}(\Delta,\ell)$. Choosing
$\omega_{-}^{(1)}\omega_{-}^{(2)}=\tilde{\alpha}_{-,m}\tilde{\alpha}_{+,n}$
its action translates the OPE denisity upper bound (D.20) to the following
bound for global symmetry:

$\lim\limits_{m,n\to\infty}\sum_{\stackrel{{\scriptstyle|\Delta-\Delta^{\text{prod}}_{m,\ell}|\leq
2}}{{|\ell-2(m-n)|\leq
1}}}\left(\frac{\sum_{\mathfrak{a}}E^{\mathfrak{a}}_{+,s}a^{\mathfrak{a}}_{\Delta,\ell}}{a_{\Delta,\ell}^{\text{prod}}}\right)\left(\frac{4}{\pi}\frac{\sin(\frac{\pi}{4}(\Delta-\Delta_{m,\ell}))}{\Delta-\Delta_{m,\ell}}\right)^{4}\leq\sum_{\mathfrak{b}}E^{\mathfrak{b}}_{+,s}a_{0}^{\mathfrak{b}}\,.$
(D.22)

Here $a_{0}^{\mathfrak{a}}$ denotes the OPE of $\Delta=0,\ell=0$ operator in
each irrep $\mathfrak{a}$ (with $a^{\textbf{S}}_{0}=1$). Since all components
$E_{+,s}^{\mathfrak{a}}$ are positive this implies the following individual
bound:

$\lim\limits_{m,n\to\infty}\sum_{\stackrel{{\scriptstyle|\Delta-\Delta^{\text{prod}}_{m,\ell}|\leq
2}}{{|\ell-2(m-n)|\leq
1}}}\left(\frac{E^{\mathfrak{a}}_{+,s}a^{\mathfrak{a}}_{\Delta,\ell}}{a_{\Delta,\ell}^{\text{prod}}}\right)\left(\frac{4}{\pi}\frac{\sin(\frac{\pi}{4}(\Delta-\Delta_{m,\ell}))}{\Delta-\Delta_{m,\ell}}\right)^{4}\leq\frac{1}{E_{+,s}^{\mathfrak{a}}}\sum_{\mathfrak{b}}E^{\mathfrak{b}}_{+,s}a_{0}^{\mathfrak{b}}\,.$
(D.23)

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