11institutetext:  Department of Physics and Astronomy
Uppsala University, Box 516, SE-751 20 Uppsala, Sweden 22institutetext:
I.N.F.N – sezione di Roma Tor Vergata and Dipartimento di Fisica
Via della Ricerca Scientifica, I-00133 Roma, Italy

# OPE coefficients in Argyres-Douglas theories

A. Bissi 2 F. Fucito 1 A. Manenti 2 J.F. Morales 2 and R. Savelli
{agnese.bissi<EMAIL_ADDRESS>
{fucito, morales<EMAIL_ADDRESS>
###### Abstract

The calculation of physical quantities in certain quantum field theories such
as those of the Argyres-Douglas type is notoriously hard, due to the lack of a
Lagrangian description. Here we tackle this problem following two alternative
approaches. On the one hand, we use localization on the four-sphere to compute
two-correlators and OPE coefficients in Argyres-Douglas superconformal
theories. On the other hand, we use the conformal bootstrap machinery to put
stringent bounds on such coefficients, only relying on the knowledge of
central charge and conformal dimension of the operators. We compare the
results obtained with these two methods and find good agreement for all rank-
one cases and for the rank-two Argyres-Douglas theories $(A_{1},A_{4})$ and
$(A_{1},A_{5})$, in the moduli space of pure $SU(5)$ and $SU(6)$ super Yang-
Mills. We also apply our results from localization to obtain bounds on the
dimensions of the lightest neutral unprotected operators of the CFTs.

††preprint:  UUITP-65/21

##  1 Introduction and summary

This paper deals with $\mathcal{N}=2$ four-dimensional superconformal field
theories (SCFT) and with the computation of the OPE coefficients of the chiral
primary operators parametrizing the Coulomb branch (CB) of their moduli space.
Research in the last two decades has enormously increased the landscape of
such theories, which features points lacking marginal deformations, hence
constituting isolated non-Lagrangian theories with an intrinsically strong
interaction. The archetypes of such theories are the Argyres-Douglas (AD) and
Minahan-Nemeschansky (MN) theories Argyres:1995jj ; Argyres:1995xn ;
Eguchi:1996vu ; Eguchi:1996ds ; Minahan:1996cj . More recently Argyres:2015ffa
; Argyres:2015gha ; Argyres:2016xmc ; Argyres:2016xua  this group of theories
has been further enlarged in the attempt to compile a complete list of
$\mathcal{N}=2$ SCFT classifying their moduli space of vacua, characterized by
their Coulomb branches, Higgs branches and possibly mixed branches. The main
properties of such theories have been derived either with field theory methods
thanks to their connection to gauge theories through dualities and RG-flows
Argyres:2010py ; Argyres:2007tq ; Maruyoshi:2016aim ; Agarwal:2016pjo , with
geometric techniques via class ${\cal S}$ constructions and engineering in
string theory Gaiotto:2009we ; Bonelli:2011aa ; Xie:2012hs ; Cecotti:2012jx ;
Cecotti:2013lda ; Chacaltana:2014nya ; Wang:2015mra ; Chacaltana:2016shw ;
Giacomelli:2017ckh  or by studying the superconformal index Buican:2015ina ;
Cordova:2015nma ; Agarwal:2018zqi ; Xie:2021omd ; Song:2021dhu ;
Buican:2021elx .

It is well known that the dynamics of superconformal theories is strongly
constrained. Indeed, superconformal symmetry is powerful enough to determine
all correlators of half-BPS operators of the theory out of the two- and three-
point functions of superconformal primaries. Denoting by $\\{{\cal O}_{a}\\}$
a basis of primary operators, a SCFT is therefore specified by the conformal
dimensions $\Delta_{a}$, the metric $g_{ab}$ and the OPE coefficients
$\lambda_{abc}$, respectively defined as

$G_{ab}(x)=\langle\mathcal{O}_{a}(x)\mathcal{O}_{b}(0)\rangle={g_{ab}\over|x|^{\Delta_{a}+\Delta_{b}}}\,,\qquad\qquad{\cal
O}_{a}(x){\cal O}_{b}(0)=\sum_{c}{\lambda_{abc}{\cal
O}_{c}(0)\over|x|^{\Delta_{a}+\Delta_{b}-\Delta_{c}}}+\cdots\,,$ (1)

where the dots stand for subleading terms in the limit $x\to 0$. A
particularly interesting class of operators are the scalar _chiral_ primaries
$\\{{\cal O}_{i}\\}$. This set of operators form a commutative ring ${\cal
O}_{i}(x){\cal O}_{j}(0)\sim{\cal O}_{i}(0){\cal O}_{j}(0)+\ldots$, which
makes the OPE coefficients either zero or one. The chiral/anti-chiral two-
point functions
$G_{ij}=\langle\mathcal{O}_{i}(x)\bar{\mathcal{O}}_{j}(0)\rangle$, instead,
are non-trivial. Alternatively, in an orthonormal basis where
$\hat{g}_{ij}=\delta_{ij}$, the SCFT data is encoded in the OPE coefficients
$\hat{\lambda}_{ijk}$ which can be expressed in terms of the metric
coefficients $G_{ij}$. The OPE coefficients $\hat{\lambda}_{ijk}$ will be the
focus of this paper.111OPE coefficients will always be written in the
orthonormal basis, so the hat will be omitted from now on. We will compute
them using two different approaches: the conformal bootstrap and localization.

The starting point of the conformal bootstrap is the writing of the four-point
function as an infinite sum of conformal blocks associated to the exchange of
operators of arbitrary weight and spin, with coefficients given by quadratic
combinations of the unknown OPE coefficients. Crossing relations are then used
to bound the region of allowed values for the OPE coefficients after a
specific numerical optimization procedure. This approach has been pioneered in
Rattazzi:2008pe  and detailed in the supersymmetric $\mathcal{N}=2$ case in
Beem:2014zpa ; Cornagliotto:2017snu ; Gimenez-Grau:2020jrx . See Simmons-
Duffin:2016gjk ; Poland:2018epd  for recent reviews.

The second approach we use is based on localization techniques, introduced in
Nekrasov:2002qd ; Flume:2002az ; Bruzzo:2002xf  for gauge theories on
$\mathbb{R}^{4}$, and later in Pestun:2007rz  for $S^{4}$. Two-point
correlators of chiral/anti-chiral primary operators have been computed using
localization on the four-sphere, with operator insertions at the North and
South poles Baggio:2014ioa ; Baggio:2014sna ; Baggio:2015vxa ; Baggio:2016skg
; Gerchkovitz:2016gxx ; Rodriguez-Gomez:2016ijh ; Rodriguez-Gomez:2016cem ;
Billo:2017glv ; Billo:2019job ; Beccaria:2020hgy ; Beccaria:2021hvt ;
Billo:2021rdb . Results at weak coupling were shown to reproduce those
obtained via Feynman-diagram techniques. Instanton corrections for $SU(2)$
with $N_{f}=4$ massless flavours were shown to be in complete agreement with
those obtained by the conformal bootstrap method Beem:2014zpa .

In this paper, we are interested in two-point correlators of those SCFT’s of
the AD type, lying in the CB moduli space of $\mathcal{N}=2$ gauge theories
with matter, at points where mutually non-local degrees of freedom become
simultaneously massless. Consequently these theories lack a Lagrangian
description.222Some of these theories, although they have no ${\mathcal{N}}=2$
conformal manifold, may be reached from ${\mathcal{N}}=1$ Lagrangian theories
in the UV by an RG flow Maruyoshi:2016tqk .

The work is inspired by the observation Russo:2014nka ; Russo:2019ipg  that,
for specific choices of the masses, the partition function on $S^{4}$ of
$SU(N)$ gauge theories with fundamental matter is dominated at large radius by
saddle points, some of which are precisely the AD points, suggesting that the
large-radius physics is described by a superconformal field theory. Here we
exploit this observation to extract the two-point correlators of the CB
operators and, consequently, their OPE coefficients for interacting non-
Lagrangian conformal field theories, from the large-radius limit of gauge
theories with fundamental matter.

Following Billo:2017glv ; Billo:2019job ; Gerchkovitz:2016gxx , we derive a
localization formula for the two-point correlators on $S^{4}$, and cast the
result in the form

$G_{ij}(x)=\langle\mathcal{O}_{i}(x)\bar{\mathcal{O}}_{j}(0)\rangle=C_{ij}-C_{im}C^{mn}C_{nj}\,,$
(2)

where $C_{ij}$ is the matrix-model two-point function

$C_{ij}={1\over{Z}_{S^{4}}}\int
d{a}\,O_{i}({a},q)\bar{O}_{j}({a},\bar{q})\big{|}Z_{\mathbb{R}^{4}}({a},\epsilon_{1},\epsilon_{2},q)\big{|}^{2}\,,$
(3)

and $x=2\pi R$, with $R$ the radius of the four sphere. The integral is
performed over the imaginary axes for ${a}=\\{a_{1},\ldots,a_{N-1}\\}$.
$Z_{\mathbb{R}^{4}}$ and $O_{i}$ are respectively the partition function and
the one-point function on $\mathbb{R}^{4}$ in an $\Omega$-background where
$\epsilon_{1}=\epsilon_{2}=R^{-1}$, whereas ${Z}_{S^{4}}$ is the partition
function on the four-sphere, $q=\Lambda^{2N-N_{f}}/4$, $N_{f}$ is the number
of flavours and $\Lambda$ is the renormalization-group invariant scale. The
sum over $m,n$ in (2) runs over all operators with dimensions lower than
$\Delta_{i}$ and gets rid of the operator mixing via Gram-Schmidt
orthogonalization.

In the large-radius limit, the integral (3), for specific choices of the
masses, is dominated by a saddle point corresponding to an AD conformal theory
in the moduli space of the gauge theory, and $G_{ij}$ will therefore give rise
to the two-point correlator of that conformal theory.

There are two main difficulties in evaluating the saddle-point integral.
First, the saddle point is located at strong coupling, so one has to rely on
non-perturbative formulae for the prepotential, where the entire tower of
instanton contributions has been resummed. Second, the correlators receive
contributions from higher-order terms in the radius expansion, codified by the
generalized prepotentials ${\cal F}_{g\geq 2}$, which survive the limit
${a}\to{a}_{*}$, $R\to\infty$ with $({a}-{a}_{*})R$ finite, where $a_{*}$ is
the saddle-point value of $a$. In the case of $SU(2)$ gauge theories with
matter, and the AD theories describing their strong coupling regimes, exact
formulae for the SW periods are known and the contributions of the SW
prepotential ${\cal F}_{0}$ and its first gravitational correction ${\cal
F}_{1}$ can be evaluated explicitly. For higher-rank AD theories, explicit
formulae for the SW periods are, to our knowledge, not known. Here we compute
the SW periods for some rank-two AD theories describing the strong coupling
regimes of pure $SU(5)$ and $SU(6)$ gauges theories (see Masuda:1996np  for
studies of the SW periods in the underlying SQCD theories). The SW periods are
explicitly evaluated and summed up to generalized hypergeometric functions.

While there is a priori no reason to expect that the higher gravitational
terms encoded in the functions ${\cal F}_{g>1}$ are suppressed, we find
remarkable evidence that they give mild contributions to the OPE coefficients:
Indeed, we show that, in almost all cases we analyzed, results obtained
including only the contributions of ${\cal F}_{0}$ and ${\cal F}_{1}$ are
inside the bootstrap windows. Similar conclusions were reached in
Grassi:2019txd , where the results for the OPE coefficients of the rank-one AD
theories have been extrapolated from matrix-model formulae derived in the
large-R-charge limit. We show similar agreement for the rank-two AD theories,
known as $(A_{1},A_{4})$ and $(A_{1},A_{5})$, which lie in the moduli space of
pure $SU(5)$ and $SU(6)$ super Yang-Mills. Thanks to the relatively low values
of the conformal dimensions and the central charges, we are able to compute,
using the bootstrap machinery, stringent bounds for the OPE coefficients in
these theories, providing remarkable tests of the localization formula.

Finally, we observe that the localization formula for the OPE coefficients
makes reference only to intrinsic data of the SCFT, and can therefore be
extended to other non-Lagrangian SCFT’s, not necessarily connected to gauge
theories, like the MN ones. Unfortunately, the conformal dimensions in these
theories are not so small, and the windows obtained from the conformal
bootstrap are too wide to qualify as a valid test.

This is the plan of the paper: In Section 2 we discuss localization and
compute the OPE coefficients. In Section 3 we discuss the results obtained
with the bootstrap. In Section 4 we verify that our formulae for computing the
OPE coefficients are compatible with those valid at large R-charge available
in the literature. Finally in Section 5 we draw our conclusions and discuss
some open questions. Technical details are deferred to the appendices.

##  2 OPE coefficients from localization

This section is divided into five subsections. In 2.1 we introduce the
localization formula for two-point correlators on $S^{4}$, and discuss the
large-radius limit in 2.2 and the integration measure in 2.3. In 2.4 and 2.5
we compute the OPE coefficients for the AD theories lying in the CB moduli
space of $SU(2)$ SQCD and $SU(5),SU(6)$ pure super Yang-Mills respectively.

###  2.1 Localization formula

####  2.1.1 One-point functions on $\mathbb{R}^{4}$

The partition function on ${\mathbb{R}}^{4}$ for a supersymmetric
$\mathcal{N}=2$ $SU(N)$ gauge theory with $N_{f}$ flavours in an $\Omega$
background with parameters $\epsilon_{1},\epsilon_{2}$, can be written as the
product of a tree level, a one-loop, and an instanton contribution as
Nekrasov:2002qd

$Z_{\mathbb{R}^{4}}(a,\epsilon_{1},\epsilon_{2},q)=Z_{\rm
tree}(a,\epsilon_{1},\epsilon_{2},q)Z_{\rm one-
loop}(a,\epsilon_{1},\epsilon_{2})Z_{\rm
inst}(a,\epsilon_{1},\epsilon_{2},q)\,,$ (4)

where Alday:2009aq ; Billo:2013fi ; Fucito:2013fba

$\displaystyle Z_{\rm tree}(a,q)$ $\displaystyle=$ $\displaystyle
q^{\sum_{u}a_{u}^{2}\over 2\epsilon_{1}\epsilon_{2}}$ $\displaystyle Z_{\rm
one-looop}(a)$ $\displaystyle=$
$\displaystyle{\prod_{u=1}^{N}\prod_{f=1}^{N_{f}}\Gamma_{2}\left(a_{u}-m_{i}+{\epsilon\over
2}\right)\over\prod_{u<v}^{N}\Gamma_{2}(a_{uv})\Gamma_{2}(a_{uv}+\epsilon)}$
$\displaystyle Z_{\mathbb{R}^{4},\rm inst}(a,q)$ $\displaystyle=$
$\displaystyle\sum_{Y}q^{|Y|}Z_{Y}=\sum_{Y}q^{|Y|}{\prod_{u=1}^{N}\prod_{f=1}^{N_{f}}z_{Y_{u},0}\left(a_{u}-m_{i}+{\epsilon\over
2}\right)\over\prod_{u,v=1}^{N}z_{Y_{u},Y_{v}}(a_{uv})}$ (5)

with $\epsilon=\epsilon_{1}+\epsilon_{2}$, $q=e^{2\pi{\rm
i}\tau}\mu^{2N-N_{f}}$ the instanton-counting parameter ($\mu$ being the
renormalization reference scale) and

$\tau={\theta\over 2\pi}+{4\pi{\rm i}\over g^{2}}\,.$ (6)

The sum over $Y=\\{Y_{u}\\}_{u=1,\ldots,N}$ runs over the $N$-tuplets of Young
tableaux with a total number $|Y|$ of boxes and

$\displaystyle z_{Y_{u},Y_{v}}$ $\displaystyle=$ $\displaystyle\prod_{(i,j)\in
Y_{u}}\left[x+\epsilon_{1}(i-k_{vj})-\epsilon_{2}(j-1-\tilde{k}_{ui})\right]$
(7) $\displaystyle\times\prod_{(i,j)\in
Y_{v}}\left[x-\epsilon_{1}(i-1-k_{uj})-\epsilon_{2}(j-\tilde{k}_{vi})\right]\,,$

where $k_{uj}$ and $\tilde{k}_{ui}$ denote the lengths of the $j^{\rm th}$ row
and $i^{\rm th}$ column of $Y_{u}$.

We are interested in correlators involving chiral primary operators made out
of the scalar field $\varphi$ in the vector multiplet. We write

${\cal O}_{J_{1},J_{2}\ldots J_{n}}(x)={\rm tr\,}\varphi^{J_{1}}(x)\,{\rm
tr\,}\varphi^{J_{2}}(x)\,\ldots{\rm tr\,}\varphi^{J_{n}}(x)\,.$ (8)

Sometimes we will find it convenient to use a collective index $i$ (which
refers to the total R-charge) to denote the operators in (8) as ${\cal
O}_{i}(x)$ and their dimension as $\Delta_{i}$. The one-point function of such
operators on ${\mathbb{R}^{4}}$ is given by

$O_{i}(a,q)=\langle\,{\cal O}_{J_{1},J_{2}\ldots
J_{n}}(x)\rangle=\sum_{Y}q^{|Y|}Z_{Y}O_{J_{1},Y}O_{J_{2},Y}\ldots
O_{J_{n},Y}\,,$ (9)

where

$O_{J,Y}=\sum_{s=1}^{|Y|}\left[(\chi_{s}{+}\epsilon_{1})^{J}{+}(\chi_{s}{+}\epsilon_{2})^{J}{-}\chi_{s}^{J}{-}(\chi_{s}{+}\epsilon)^{J}\right]\,,$
(10)

and

$\chi_{s}=\chi_{u,(i,j)\in
Y_{u}}=a_{u}+(i-1)\epsilon_{1}+(j-1)\epsilon_{2}\,.$ (11)

####  2.1.2 Two-point correlators on $S^{4}$

The partition function on $S^{4}$ of a supersymmetric $\mathcal{N}=2$ $SU(N)$
gauge theory can be written as Pestun:2007rz 333Notice that here we have
reabsorbed the Vandermonde determinant $\prod_{u<v}^{N}a_{uv}^{2}$ into the
modulus square of the one-loop partition function (5) using the identity
$\Gamma(x+\epsilon_{1})\Gamma(x+\epsilon_{2})=x\Gamma(x)\Gamma(x+\epsilon)$

${Z}_{S^{4}}=\int
d{a}\,\big{|}Z_{\mathbb{R}^{4}}({a},\epsilon_{1},\epsilon_{2},q)\big{|}^{2}\,,$
(12)

where ${a}=\\{a_{u}\in{\mathbb{C}}\\}_{u=1,\ldots,N}$, with $\sum_{u}a_{u}=0$,
and the integral is performed over the imaginary axes. $Z_{\mathbb{R}^{4}}$ is
the partition function on ${\mathbb{R}}^{4}$ in an $\Omega$ background with
parameters

$\epsilon_{1}\epsilon_{2}={1\over R^{2}}\,,\qquad
b=\sqrt{\epsilon_{1}\over\epsilon_{2}}\,,$ (13)

with $R$ the radius of the sphere and $b$ the squashing parameter that here
will always be set to one.

We now introduce the two-point matrix-model integral

$C_{ij}={1\over{Z}_{S^{4}}}\int
d{a}\,O_{i}({a},q)\bar{O}_{j}({a},\bar{q})\big{|}Z_{\mathbb{R}^{4}}({a},\epsilon_{1},\epsilon_{2},q)\big{|}^{2}\,,$
(14)

where $O_{i}({a},q)$ are the one-point functions of the chiral primary
operators on $\mathbb{R}^{4}$, defined in (9).

Following Gerchkovitz:2016gxx ; Billo:2017glv ; Billo:2019job , the two-point
correlators $G_{ij}$ on the sphere can be obtained as

$G_{ij}=\langle{\cal O}_{i}(2\pi R)\bar{\cal
O}_{j}(0)\rangle=C_{ij}-C_{im}C^{mn}C_{nj}\,,$ (15)

where the second term “subtracts” the mixing contributions between operators
of different dimensions. Indeed, the sum over $m,n$ runs over all the
operators with dimensions lower than $\Delta_{i}$ and $C^{mn}$ is the inverse
of the mixing matrix restricted to the space of such operators. It follows
from (15) that $G_{in}=0$, i.e. there is no mixing between operators of
different dimensions, as expected.

Let us consider the OPE coefficients. Since we are dealing with chiral
operators, there are no contractions between them, and therefore

${\cal O}_{i}(x){\cal O}_{j}(0)={\cal O}_{i}(0){\cal O}_{j}(0)+\ldots\,,$ (16)

i.e. the only non-trivial OPE coefficients are $\hat{\lambda}_{i,j,k_{ij}}=1$
with $k_{ij}$ the index label for the operator ${\cal O}_{i}{\cal O}_{j}$. One
can introduce the canonically-normalized operators $\hat{\cal O}_{i}$

${\cal O}_{i}(x)=\sqrt{G_{ii}}\,{\cal\hat{O}}_{i}(x)\,.$ (17)

In the new basis $\hat{G}_{ij}\sim\delta_{ij}/R^{2\Delta_{i}}$ and

$\hat{\lambda}_{i,j,k_{ij}}=\sqrt{\frac{G_{k_{ij},k_{ij}}}{G_{ii}G_{jj}}}\,.$
(18)

For the first few non-zero two-point correlators one finds

$\displaystyle G_{11}$ $\displaystyle=$ $\displaystyle C_{11}{-}C_{01}C_{10}$
$\displaystyle G_{22}$ $\displaystyle=$ $\displaystyle
C_{22}{-}\frac{C_{01}C_{12}C_{20}{+}C_{02}C_{10}C_{21}{-}C_{02}C_{11}C_{20}{-}C_{12}C_{21}}{C_{01}C_{10}{-}C_{11}}\,,$

Alternatively one can write Gerchkovitz:2016gxx

$G_{nn}={{\rm det}_{i,j\leq n}C_{ij}\over{\rm det}_{i,j\leq n-1}C_{ij}}\,.$
(20)

Using these formulae one can evaluate the two-point correlators and the OPE
coefficients for any SCFT with a known prepotential. In the case of the
$\mathcal{N}=2$ SCFT with $SU(2)$ gauge group and four fundamental massless
flavours, the computation of $\lambda_{112}$ has been performed in
Beem:2014zpa  numerically, keeping into account the first few instantonic
contributions, and matched against the result obtained via the conformal
bootstrap. Note that formula (20) cannot be used for rank higher than one,
because of the mixing of operators of the same dimension.444In the large-$N$
limit of gauge theories, the mixing of multi-trace operators can be neglected,
and a formula like (20) can still be used Beccaria:2020hgy ; Beccaria:2021hvt
; Billo:2021rdb . Formula (15), instead, can be used for any rank, leading in
general to a non-diagonal matrix on the space of operators of the same
dimension.

###  2.2 The flat-space limit

As we hinted in the Introduction, an important class of $\mathcal{N}=2$ SCFT
Argyres:1995xn ; Argyres:1995jj ; Minahan:1996cj  live in the strong coupling
regime of an asymptotically-free gauge theory. They are isolated and do not
admit a Lagrangian formulation. Can localization be used in these cases? The
answer is yes. The crucial observation was put forward in Russo:2014nka ;
Russo:2019ipg , where the partition function $Z_{S^{4}}$ of the $SU(N)$ gauge
theories with fundamental matter was studied in the limit where the radius of
the four-sphere is large. In this limit the gauge prepotential ${\cal F}$ can
be written as

${\cal F}(a,R)\approx\sum_{g=0}^{\infty}{\cal F}_{g}(a)\,R^{-2g}\,,$ (21)

where ${\cal F}_{0}(a)$ is the SW prepotential characterizing the low energy
dynamics of the gauge theory in flat space and ${\cal F}_{g}$ account for the
gravitational corrections arising from the $\Omega$-curvature of the
spacetime. The integral (12) is dominated by a saddle point at the extremum
${a}={a}_{*}$ of the SW prepotential ${\cal F}_{0}(a)$, i.e. where

$a^{s}_{D}(a_{*})=-{1\over 2\pi{\rm i}}{\partial{\cal F}_{0}\over\partial
a_{s}}\Big{|}_{a=a_{*}}=0\,,\qquad\quad s=1,\ldots,N-1\,,$ (22)

leading to $Z_{S^{4}}\approx\big{|}e^{\Lambda^{2}R^{2}f_{*}}\big{|}^{2}$ with
${\cal F}_{0}(a_{*})=\Lambda^{2}\,f_{*}$ and $f_{*}$ a complex number. More
precisely, based on dimensional analysis, the expansion for large $\Lambda R$
of the prepotential can be written as555We have taken all masses to be of the
order of $\Lambda$, and they will eventually be tuned to specific values
corresponding to the AD points.

$R^{2}\,{\cal F}(a,\Lambda,R)\approx(\Lambda R)^{2}f_{*}+F[(a-a_{*})R]\,,$
(23)

where we discarded terms suppressed in the limit $\Lambda R\to\infty$.
Plugging this into (14) one finds

$C_{ij}\approx{1\over Z_{S^{4}}}\int
d{a}\,O_{i}({a})\,O_{j}({a})e^{F[(a-a_{*})R]+{\rm h.c.}}\,,$ (24)

with

$Z_{S^{4}}\approx\int d{a}\,e^{F[(a-a_{*})R]+{\rm h.c.}}\,.$ (25)

It is important to observe that the physics near a generic saddle point is
better described in terms of the dual periods $a^{s}_{D}$. However, if the
saddle point corresponds to an AD conformal point, either $a$ or $a_{D}$ can
be used, because they are related by

$a^{s}_{D}\approx\tau^{ss^{\prime}}\left(a-a_{*}\right)_{s^{\prime}}\,,$ (26)

with $\tau^{ss^{\prime}}$ a symmetric matrix with positive-definite imaginary
part, whose entries depend homogeneously on $a_{s}$ with degree zero.

###  2.3 The u-plane integral

The SW periods $a_{s}(u)$ provide a relation between the variables
$a=\\{a_{s}\\}_{s=1,\ldots N-1}$ and the curve parameters
$u=\\{u_{n}\\}_{n=2,\ldots,N}$ representing the gauge-invariant coordinates of
the CB moduli space. One can use these relations to write the partition
function as an integral over the u-plane. The SW prepotential ${\cal
F}_{0}(u)$ can be obtained from $a_{D}(u)$ upon u-integration

${\partial{\cal F}_{0}(u)\over\partial u_{n}}=-2\pi{\rm
i}\,a_{D}^{s}(u){\partial a_{s}(u)\over\partial u_{n}}\,.$ (27)

The first gravitational correction ${\cal F}_{1}$ can also be written in terms
of the SW periods and the discriminant $\Delta({u})$ of the SW geometry
Nakajima:2003uh ; Witten:1995gf ; Moore:1997pc ; Manschot:2019pog ;
Shapere:2008zf

${\cal F}_{1}(u)=-\log\left[\Lambda^{\frac{N(N-1)}{2}}{\rm
det}\left({\partial{a_{s}}\over\partial{u_{n}}}\right)\right]^{1\over
2}+\left(b^{2}+{1\over
b^{2}}\right)\log\left[{\Delta(u)\over\Lambda^{2N(2N-1)}}\right]^{1\over
24}\\!\\!.$ (28)

The above formula can be checked perturbatively in $q$ using the Nekrasov
partition function reviewed in Appendix A, or verified using modular
invariance of the path integral. Plugging (28) into (14) and setting the
squashing parameter $b=1$ one finds

$C_{ij}={1\over{Z}_{S^{4}}}\int d{u}\,{\cal O}_{i}({u})\,{\cal
O}_{j}({u})\,\left|\Delta({u})^{1\over 6}e^{2R^{2}{\cal
F}_{0}({u})}\right|\,+\ldots\,,$ (29)

and

${Z}_{S^{4}}=\int d{u}\,\left|\Delta({u})^{1\over 6}e^{2R^{2}{\cal
F}_{0}({u})}\right|+\ldots\,,$ (30)

where the dots are the contribution of higher gravitational terms ($g\geq
2$).666The inclusion of ${\cal F}_{1}$ is crucial in order to reconstruct a
modular covariant u-integration measure Shapere:2008zf . These contributions
are in general not known in a closed form. In this paper we will compute the
correlators and the OPE coefficients including only the contributions of
${\cal F}_{0}$ and ${\cal F}_{1}$. Surprisingly, the results already fall, in
most of the cases, inside the narrow windows determined by the conformal
bootstrap techniques, showing small discrepancies only for few operators of
low dimensions.

###  2.4 $\boldsymbol{SU(2)}$ gauge theory with fundamentals

In this section we apply the method we previously discussed to the computation
of the OPE coefficients for the $SU(2)$ gauge theory with $N_{f}=1,2,3$
fundamental flavors (setting equal masses for all the flavors). The SW
geometry of these theories is given by

$w(x)^{2}=\prod_{i=1}^{4}(x-e_{i})=P(x)^{2}-\Lambda^{4-N_{f}}\,(x-m)^{N_{f}}\,,$
(31)

with

$P(x)=\left\\{\begin{array}[]{ll}x^{2}-u&N_{f}=1\,,\\\
x^{2}-u+{\Lambda^{2}\over 4}&N_{f}=2\,,\\\ x^{2}-u+{\Lambda\over
4}(x-3m)&N_{f}=3\,.\\\ \end{array}\right.$ (32)

The SW prepotential ${\cal F}_{0}$ can be obtained from the SW periods via

${\cal F}_{0}(u)=-2\pi{\rm i}\int^{u}du^{\prime}a_{D}(u^{\prime}){\partial
a(u^{\prime})\over\partial u^{\prime}}\,,$ (33)

with $a(u)$ and $a_{D}(u)$ the SW periods, whose explicit expressions are
given in Appendix A.

We are interested in the physics near the AD conformal point, obtained by
requiring that three branch points collide, i.e.

$w^{2}(x)=(w^{2})^{\prime}(x)=(w^{2})^{\prime\prime}(x)=0\,,$ (34)

that can be solved for $x$, $u$, and $m$.

$\begin{array}[]{|c|cccccc|}\hline\cr N_{f}&d&c&x_{*}&u_{*}&m_{*}&\alpha\\\
\hline\cr
1&{\textstyle\frac{6}{5}}&{\textstyle\frac{11}{30}}&{\textstyle\frac{\Lambda}{2}}&{\textstyle\frac{3\Lambda^{2}}{4}}&-{\textstyle\frac{3\Lambda}{4}}&2\\\
2&{\textstyle\frac{4}{3}}&{\textstyle\frac{1}{2}}&{\Lambda\over
2}&{\Lambda^{2}\over 2}&{\Lambda\over 2}&3\\\
3&{\textstyle\frac{3}{2}}&{\textstyle\frac{2}{3}}&{\Lambda\over
8}&{5\Lambda^{2}\over 64}&-{\Lambda\over 8}&4\\\ \hline\cr\end{array}$

Table 1: Summary of the data identifying the AD theories in the CB of $SU(2)$
SQCD. $c$ denotes the central charge, $d$ the CB-operator conformal dimension,
and $\alpha$ the power at which the CB coordinate appears in the discriminant.

Expanding the discriminant, given in Eq. (98), and the prepotential in Eq.
(33) around the AD saddle point one finds

$\displaystyle\Delta(u)$ $\displaystyle=$ $\displaystyle
c_{\Delta}\,\Lambda^{12}\,\left({u-u_{*}\over\Lambda^{2}}\right)^{\alpha}\,,$
$\displaystyle{\cal F}_{0}(u)$ $\displaystyle=$
$\displaystyle\Lambda^{2}\,\left[f_{*}-{c^{2}_{\cal F}\over
2}\,\left({u-u_{*}\over\Lambda^{2}}\right)^{2\over d}\right]\,,$ (35)

with $c_{\Delta}$ and $c^{2}_{\cal F}$ some real dimensionless constants,
whose value is irrelevant for our purposes,777The minus sign in front of
$c^{2}_{\cal F}$, though, is crucial in order for the integrals in (29) to
converge. But, as can be seen by combining (35) with (26), this sign is
guaranteed by the positive-definiteness of the gauge coupling ${\rm
Im}\,\tau$. and $d,\alpha$ rational numbers characterizing the AD theory. The
results for $N_{f}=1,2,3$ are displayed in Table 1. The dynamics of the gauge
theory near these points is described by the rank-$1$ AD conformal field
theories known as $\mathcal{H}_{0}$, $\mathcal{H}_{1}$, and $\mathcal{H}_{2}$
respectively. In particular, one finds

$\alpha=N_{f}+1=12\hskip 1.0pt\frac{d-1}{d}\,.$ (36)

Now let us consider the two-point correlators in these theories. The chiral
ring is generated by a single operator that can be taken to be

${\cal O}(x)=\Lambda^{d-2}\left({\textstyle\frac{1}{2}}{\rm
tr}\varphi^{2}(x)-u_{*}\right)=\tilde{u}\,,$ (37)

with the shift chosen such that $\langle{\cal O}\rangle=0$ at the SCFT point
$u=u_{*}$, and the normalization chosen such that all the dependence on the
scale $\Lambda$ drops out in two point functions $C_{ij}$. The remaining
operators in the chiral ring are obtained as powers ${\cal O}_{i}(x)={\cal
O}(x)^{i}$ of the chiral ring generator. The one-point function factorizes as

$O_{i}(u)=\langle{\cal O}(x)^{i}\rangle=\tilde{u}^{i}\,.$ (38)

Plugging (35) and (38) into (29) one finds888Here we use
$\int_{0}^{\infty}e^{-cx^{\beta}}x^{\gamma}=\beta^{-1}c^{-{1+\gamma\over\beta}}\Gamma\left({1+\gamma\over\beta}\right)\,.$

$\displaystyle C_{ij}$ $\displaystyle\approx$
$\displaystyle{\int_{0}^{\infty}d\tilde{u}\,e^{-R^{2}c^{2}_{\cal
F}\,\tilde{u}^{\frac{2}{d}}}\,\tilde{u}^{i+j+{\alpha\over
6}}\over\int_{0}^{\infty}d\tilde{u}\,e^{-R^{2}c^{2}_{\cal
F}\,\hat{u}^{\frac{2}{d}}}\,\tilde{u}^{\alpha\over 6}}={1\over(Rc_{\cal
F})^{d(i+j)}}{\Gamma\left(\frac{d}{2}\left(1+i+j+{\textstyle\frac{\alpha}{6}}\right)\right)\over\Gamma\left(\frac{d}{2}\left(1+\frac{\alpha}{6}\right)\right)}\,.$
(39)

Notice that $C_{ij}\sim R^{-{(i+j)d}}$ as expected from conformal invariance.

The two-point correlators follow then from (15) or (20), and the OPE
coefficients from (18). The results are displayed in Table 2.

Interestingly our results depend only on two numbers, $d$ and $\alpha$,
characterizing the dimension of the CB operator and the degree of the
discriminant of the curve. In particular, the latter is related to the central
charge $c$ of the theory. These two numbers are codified in the SW curves and
can be easily computed for non-lagrangian theories like the MN theories with
flavour symmetry $E_{6}$, $E_{7}$, and $E_{8}$.

This data will be compared in Section 3 against the results obtained using the
bootstrap approach.

$\begin{array}[]{|c|cccccc|}\hline\cr{\rm
SCFT}&\mathcal{H}_{0}&\mathcal{H}_{1}&\mathcal{H}_{2}&E_{6}&E_{7}&E_{8}\\\
\hline\cr{\rm
SW}&y^{2}=x^{3}{+}{u}&y^{2}=x^{3}{+}{u}x&y^{2}=x^{3}{+}{u}^{2}&y^{2}=x^{3}{+}{u}^{4}&y^{2}=x^{3}{+}{u}^{3}x&y^{2}=x^{3}{+}{u}^{5}\\\
\hline\cr d&\frac{6}{5}&\frac{4}{3}&\frac{3}{2}&3&4&6\\\
c&\frac{11}{30}&\frac{1}{2}&\frac{2}{3}&{\textstyle\frac{13}{6}}&{\textstyle\frac{19}{6}}&{\textstyle\frac{31}{6}}\\\
\alpha&2&3&4&8&9&10\\\ \hline\cr\lambda^{2}_{u\hskip 1.0ptu\hskip
1.0ptu^{2}}&2.09823&2.24125&2.42063&4.51365&6.75467&15.1158\\\
\lambda^{2}_{u\hskip 1.0ptu^{2}\hskip
1.0ptu^{3}}&3.30002&3.67408&4.17529&12.0469&24.011&95.3327\\\
\lambda^{2}_{u^{2}\hskip 1.0ptu^{2}\hskip
1.0ptu^{4}}&7.20621&8.62414&10.7157&67.008&222.212&2443.47\\\
\hline\cr\end{array}$

Table 2: Some OPE coefficients for the rank-$1$ AD theories
$\mathcal{H}_{0},\mathcal{H}_{1},\mathcal{H}_{2}$, and for the rank-$1$ MN
theories $E_{6},E_{7},E_{8}$. We have removed the $\tilde{}$ from $u$ in order
not to clutter the notation.

###  2.5 AD conformal points in pure $\boldsymbol{SU(N)}$ gauge theories

The SW geometry describing the low energy dynamics of pure $SU(N)$ gauge
theories can be written as

$w(x)^{2}=P_{N}(x)^{2}-\Lambda^{2N}\,,$ (40)

where

$P_{N}=x^{N}-u_{2}x^{N-2}-u_{3}x^{N-3}\ldots-u_{N}$ (41)

and $u_{n}$ the CB parameters. These theories contain an AD conformal point
Eguchi:1996ds , usually called $(A_{1},A_{N})$, at

$u_{2}=u_{3}=\ldots u_{N-1}=0\,,\qquad u_{N}=\Lambda^{N}\,,$ (42)

which is obtained by setting

$x=\tilde{x}\nu\,,\qquad\qquad w=\tilde{w}\,{\rm
i}\sqrt{2}\,(\Lambda\nu)^{N/2}\,,\qquad\qquad
u_{n}=\tilde{u}_{n}\,\nu^{n}+\delta_{n,N}\Lambda^{N}\,,$ (43)

and keeping the leading order in the limit $\nu\to 0$. This leads to the AD
curve

$\tilde{w}(x)^{2}=\tilde{x}^{N}-\tilde{u}_{2}\,\tilde{x}^{N-2}-\tilde{u}_{3}\tilde{x}^{N-3}\ldots-\tilde{u}_{N}\,,$
(44)

and the SW differential

$\lambda={\tilde{w}d\tilde{x}}\,.$ (45)

The conformal dimensions of the parameters $\tilde{u}_{n}$ entering the curve
is determined by requiring that the SW differential has dimension one
$d(\lambda)=1$, leading to

$d(\tilde{x})={2\over N+2}\,,\qquad\qquad d(\tilde{w})={N\over
N+2}\,,\qquad\qquad d(\tilde{u}_{n})={2n\over N+2}\,.$ (46)

These parameters are interpreted as Coulomb branch parameters, masses and
couplings depending on whether dimensions are bigger, equal or lower than 1
respectively. In particular Coulomb branch parameters $\tilde{u}_{n}$
correspond to the choices

$N\geq n>{N\over 2}+1\,.$ (47)

There are $r=\left[{\textstyle\frac{N-1}{2}}\right]$ of such operators, with
$r$ often referred to as the rank of the CFT and $[A]$ denoting the integral
part of $A$. Finally the central charge of the CFT is given by Shapere:2008zf

$c={d(\Delta)\over 12}+{r\over 6}={N(N-1)\over
6(N+2)}+{\textstyle\frac{1}{6}}\left[\frac{N-1}{2}\right]\,.$ (48)

$\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr N&d(\tilde{u}_{n})&{\rm rank}&{\rm
curve}&{\rm CFT}&\Delta&c\\\ \hline\cr 3&{4\over 5},{\bf{6\over
5}}&1&\tilde{w}^{2}=\tilde{x}^{3}-v&\mathcal{H}_{0}\equiv(A_{1},A_{2})&-3^{3}\,v^{2}&{\textstyle\frac{11}{30}}\\\
4&{2\over 3},1,{\bf{4\over
3}}&1&\tilde{w}^{2}=\tilde{x}^{4}-v&\mathcal{H}_{1}\equiv(A_{1},A_{3})&-4^{4}\,\,v^{3}&{\textstyle\frac{1}{2}}\\\
5&{4\over 7},{6\over 7},{\bf{8\over 7}},{\bf{10\over
7}}&2&\tilde{w}^{2}=\tilde{x}^{5}-u\,x-\,v&(A_{1},A_{4})&-4^{4}\,u^{5}+5^{5}\,v^{4}&{\textstyle\frac{17}{21}}\\\
6&~{}~{}{1\over 2},{3\over 4},1,{\bf{5\over 4}},{\bf{3\over
2{}{}}}&2&~{}~{}\tilde{w}^{2}=\tilde{x}^{6}-u\,x-v{}{}&(A_{1},A_{5})&5^{5}\,u^{6}+6^{6}\,v^{5}&{\textstyle\frac{23}{24}}\\\
\hline\cr\end{array}\ $

Table 3: CB conformal dimensions and SW curves for the $(A_{1},A_{N})$
theories with $N=3,\ldots,6$. We set masses and couplings to zero.

In Table 3 we list the conformal dimensions $d(\tilde{u}_{n})$ of the
parameters entering the SW curve for the theories of rank up to two,
indicating in boldface those operators spanning the Coulomb branch. We display
also the curves (and their determinant) obtained by setting to zero masses and
couplings, and renaming the CB operators as $u$, $v$. For $N=3,4$, one finds
the rank-one CFTs $\mathcal{H}_{0}$, $\mathcal{H}_{1}$ studied in the previous
section. For $N=5,6$ one finds the rank-two AD CFTs known as $(A_{1},A_{4})$
and $(A_{1},A_{5})$.

In a rank-two theory, one can consider correlators involving a single Coulomb
branch operator or mixed ones. In the former case, the correlator can be
naively computed, by integrating over the one-dimensional slice defined by
deformations from the conformal point involving only the CB operator we are
interested in. The two-point correlators $G_{nn}$ are then given (20) with
$C_{ij}$ given by the rank-one formula (39). The resulting OPE coefficients,
say for correlators involving only $u$-insertions, depend then only on the
dimension $d(\tilde{u})$ and the exponent $\alpha_{u}$ defining the vanishing
rate of the discriminant $\Delta\sim u^{\alpha_{u}}$. The results obtained
from this one-dimensional truncation are displayed in Table 4. While we do not
know how to physically justify such a truncation, we will see that it
nevertheless gives results in good agreement with the bootstrap.

$\begin{array}[]{|c|cc|cc|c|c|c|}\hline\cr&~{}d(u){}&~{}\alpha_{u}{}&~{}d(v){}&~{}\alpha_{v}{}&~{}c{}&\lambda^{2}_{u\,u\,u^{2}}&\lambda^{2}_{v\,v\,v^{2}}\\\
\hline\cr(A_{1},A_{4})&{\textstyle\frac{8}{7}}&5&{\textstyle\frac{10}{7}}&4&{\textstyle\frac{17}{21}}&1.981&2.327\\\
(A_{1},A_{5})&{\textstyle\frac{5}{4}}&6&{\textstyle\frac{3}{2}}&5&{\textstyle\frac{23}{24}}&2.077&2.378\\\
\hline\cr\end{array}$

Table 4:  OPE coefficients for the rank-$2$ AD theories of type
$(A_{1},A_{N})$ obtained by truncating the Coulomb branch integral to a single
operator slice.

In the next two sections we will derive the OPE coefficients (including mixed
ones) from a full-fledged two-dimensional integration over the CB moduli
space.

####  2.5.1 $(A_{1},A_{5})$ Argyres-Douglas theory

After setting the masses and couplings to zero, the low energy dynamics of
$(A_{1},A_{5})$ at a generic point in the Coulomb branch is described by the
curve999This analysis generalizes the results of Eguchi:1996vu  in this case.

$\tilde{w}^{2}=\tilde{x}^{6}-u\tilde{x}-v\,.$ (49)

The SW periods are defined by

$a_{s}=\oint_{\alpha_{s}}\lambda\,,\quad\qquad
a_{D}^{s}=\oint_{\beta_{s}}\lambda\,,$ (50)

with

$\lambda=d\tilde{x}\sqrt{\tilde{x}^{6}-u\tilde{x}-v}\,,$ (51)

$(\alpha_{s},\beta_{s})$ is a basis of cycles with intersection matrix
$\alpha_{s}\cap\beta_{s^{\prime}}=\delta_{ss^{\prime}}$ and
$s,s^{\prime}=1,2$. The integrals (50) can be explicitly performed in the
limit of small $u$. In this limit the branch points are located around the
sixth roots

$e_{i}=v^{1\over 6}\,w^{i}$ (52)

with $w=e^{\pi{\rm i}\over 3}$. The SW differential can be expanded as

$\lambda=d\tilde{x}\sum_{n=0}^{\infty}{\Gamma({\textstyle\frac{3}{2}})\,(-u\tilde{x})^{n}\over
n!\Gamma({\textstyle\frac{3}{2}}-n)}\,(\tilde{x}^{6}-v)^{{1\over 2}-n}$ (53)

leading to the integrals

$\Pi_{i}=\oint_{\gamma_{i}}\lambda=2\int_{e_{i}}^{e_{i+1}}\lambda={\rm
i}\,w^{i}\,V^{2\over 3}A(w^{i}\kappa)\,,$ (54)

where

$A(\kappa)=\sqrt{\pi}\sum_{n=0}^{\infty}\frac{\Gamma\left(\frac{n+7}{6}\right)\kappa^{n}\,\sin\left({(n+1)\pi\over
6}\right)}{(n{+}1)!\Gamma\left(\frac{5}{6}(2{-}n)\right)}$ (55)

and

${\rm i}V^{2\over 3}=v^{2\over 3}w^{{1\over
2}}\qquad,\qquad\kappa=uv^{-{5\over 6}}w^{1\over 2}\,.$ (56)

The variables $V$ and $\kappa$ are chosen such that the $a_{s}$’s are purely
imaginary for $V$ and $\kappa$ real and positive. This choice follows from
Pestun:2007rz  and is dictated by the mathematical consistency of the integral
over $S^{4}$ and the gluing of the two charts. The sum in the right hand side
of (55) can be carried out and written in terms of hypergeometric functions
but the explicit formula is not very illuminating, so we will omit it here.
The periods can then be written as

$a_{1}=\Pi_{0}\,,\qquad\quad a_{2}=\Pi_{3}\,,\qquad\quad
a_{D}^{1}=\Pi_{1}\,,\qquad\quad a_{D}^{2}=\Pi_{4}\,.$ (57)

As a consistency check of our choices, we have verified that the Riemann
bilinear identities are satisfied, leading to a symmetric $\tau$ matrix, with
positive imaginary part. Furthermore we notice that the $a_{s}$’s are purely
imaginary for $\kappa$ and $V$ real and positive.

The SW prepotential ${\cal F}_{0}(a)$ can be computed in terms of the SW
periods. Indeed, using the fact that ${\cal F}_{0}(a)$ is a local homogeneous
function of degree two in the $a$’s, one finds

${\cal F}_{0}(a)=\frac{1}{2}\sum_{s=1}^{2}a_{s}{\partial\over\partial
a_{s}}{\cal F}_{0}(a)=-\pi{\rm i}\sum_{s=1}^{2}a_{s}\,a_{D}^{s}\,.$ (58)

We introduce the function

$f(\kappa)=-2\,V^{-{4\over 3}}\,{\rm Re}\,{\cal F}_{0}=-{\rm
Re}\left\\{2\pi{\rm
i}\,w\,\left[A(\kappa)A(w\kappa)+A(-\kappa)A(-w\kappa)\right]\right\\}\,.$
(59)

One can check that $f(\kappa)$ is a positive, monotonously increasing function
for $\kappa\geq 0$. The matrix-model type integral (29) reduces to

$\displaystyle C_{u^{m}v^{n}}$ $\displaystyle=$
$\displaystyle\frac{1}{Z_{S^{4}}}\int
d\kappa\,dV\left|6^{6}{-}5^{5}\kappa^{6}\right|^{1\over 6}\,e^{-V^{4\over
3}\,f(\kappa)}\,V^{{5\over 3}+{5i\over 6}+n}\kappa^{m}$ (60) $\displaystyle=$
$\displaystyle\frac{1}{Z_{S^{4}}}\int
d\kappa\left|6^{6}{-}5^{5}\kappa^{6}\right|^{1\over
6}\,\Gamma\left(2{+}{\textstyle\frac{5m}{8}}{+}{\textstyle\frac{3n}{4}}\right)\left[f(\kappa)\right]^{-2-{\textstyle\frac{5m}{8}}-{\textstyle\frac{3n}{4}}}\,\kappa^{m}\,.$

The last integral is performed numerically. The results are displayed in Table
5.

$\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr&~{}[u]{}&~{}[v]{}&~{}c{}&\lambda^{2}_{u\,\hskip
0.5ptu\,\hskip 0.5ptu^{2}}&\lambda^{2}_{u\,\hskip 0.5ptv\,\hskip
0.5ptuv}&\lambda^{2}_{v\,\hskip 0.5ptv\,\hskip 0.5ptv^{2}}\\\
\hline\cr(A_{1},A_{4})&{\textstyle\frac{8}{7}}&{\textstyle\frac{10}{7}}&{\textstyle\frac{17}{21}}&1.878&1.043&2.231\\\
(A_{1},A_{5})&{\textstyle\frac{5}{4}}&{\textstyle\frac{3}{2}}&{\textstyle\frac{23}{24}}&1.929&1.039&2.203\\\
\hline\cr\end{array}$

Table 5:  Some OPE coefficients for the rank-$2$ AD theories of type
$(A_{1},A_{N})$.

####  2.5.2 $(A_{1},A_{4})$ Argyres-Douglas theory

After setting masses and couplings to zero, the low energy dynamic of
$(A_{1},A_{4})$ at a generic point in the CB is described by the curve

$\tilde{w}^{2}=\tilde{x}^{5}-u\tilde{x}-v\,.$ (61)

The SW differential is now

$\lambda=d\tilde{x}\sqrt{\tilde{x}^{5}-u\tilde{x}-v}\,.$ (62)

Again we computed the integrals as an expansion for $u$ small. In this limit
the branch points are located around the fifth roots

$e_{i}=v^{1\over 5}\,w^{i}$ (63)

with $w=e^{2\pi{\rm i}\over 5}$. The SW differential can be expanded as

$\lambda=d\tilde{x}\sum_{n=0}^{\infty}{\Gamma({\textstyle\frac{3}{2}})\,(-u\tilde{x})^{n}\over
n!\Gamma({\textstyle\frac{3}{2}}-n)}\,(\tilde{x}^{5}-v)^{{1\over 2}-n}$ (64)

leading to the integrals

$\Pi_{i}=\oint_{\gamma_{i}}\lambda=2\int_{e_{i}}^{e_{i+1}}\lambda={\rm
i}\,w^{i}\,V^{7\over 10}A(w^{i}\kappa)\,,$ (65)

where

$A(\kappa)=\sqrt{\pi}\sum_{n=0}^{\infty}\frac{\Gamma\left(\frac{n+6}{5}\right)\kappa^{n}\,\sin\left({(n+1)\pi\over
5}\right)}{(n{+}1)!\Gamma\left(\frac{1}{10}(17{-}8n)\right)}$ (66)

and

${\rm i}V^{7\over 10}=v^{7\over 10}w^{{1\over
2}}\,,\quad\qquad\kappa=uv^{-{4\over 5}}w^{1\over 2}\,.$ (67)

The periods are now given by

$a_{1}=\Pi_{0}\,,\qquad\quad a_{2}=\Pi_{4}+\Pi_{1}\,,\qquad\quad
a_{D}^{1}=\Pi_{1}\,,\qquad\quad a_{D}^{2}=-\Pi_{3}\,.$ (68)

The SW prepotential ${\cal F}_{0}(a)$ is again computed from (58) leading to

$f(\kappa)={-}2\,V^{-{7\over 5}}\,{\rm Re}\,{\cal F}_{0}=-{\rm
Re}\left\\{2\pi{\rm
i}\,\left[w\,A(\kappa)A(w\kappa){-}w^{2}\,A(w^{3}\kappa)A(w^{4}\kappa){-}w^{4}\,A(w^{3}\kappa)A(w\kappa)\right]\right\\}\,.$
(69)

Once again $f(\kappa)$ is a positive, monotonously increasing function for
$\kappa\geq 0$. The matrix-model integral (29) reduces to

$\displaystyle C_{u^{m}v^{n}}$ $\displaystyle=$
$\displaystyle\frac{1}{Z_{S^{4}}}\int
d\kappa\,dV\left|5^{5}{+}4^{4}\kappa^{5}\right|^{1\over 6}\,e^{-V^{7\over
5}\,f(\kappa)}\,V^{{22\over 15}+{4m\over 5}+n}\kappa^{m}$ (70)
$\displaystyle=$ $\displaystyle\frac{1}{Z_{S^{4}}}\int
d\kappa\left|5^{5}{+}4^{4}\kappa^{5}\right|^{1\over
6}\,\Gamma\left({\textstyle\frac{37}{21}}{+}{\textstyle\frac{4m}{7}}{+}{\textstyle\frac{5n}{7}}\right)\left[f(\kappa)\right]^{{-}{{\textstyle\frac{37}{21}}}{-}{\textstyle\frac{4m}{7}}{-}{\textstyle\frac{5n}{7}}}\,\kappa^{m}\,.$

The last integral is performed numerically. The results are displayed in Table
5.

##  3 Conformal bootstrap approach

In this section we would like to obtain upper and lower bounds on the OPE
coefficients that have been computed in the previous section and summarised in
Table 2. The approach that we follow is the conformal bootstrap that allows us
to find bounds for the dimensions and OPE coefficients of intermediate
operators by exploiting the symmetries of the specific CFT. In our setup, such
consistency conditions come from superconformal symmetry, unitarity and
crossing symmetry of the four-point functions. The numerical study of these
conditions give bounds on the CFT data. While we defer the details to the
following sections, we would like to emphasize that the bounds that we find
are fully non-perturbative and they do not refer to any Lagrangian
description.

This section is divided into seven subsections. In Subsection 3.1 we show the
multiplets exchanged in the OPE of two Coulomb branch operators. In Subsection
3.2 we write down the crossing equations and then we discuss their numerical
implementation in Subsection 3.3. Subsections 3.4 and 3.5 explain the setup
for obtaining bounds on the OPE coefficients and operator dimensions,
respectively, while Subsections 3.6 and 3.7 show the results we obtained.

###  3.1 Superconformal multiplets exchanged

The OPE of two chiral operator has been investigated in Fitzpatrick:2014oza ;
Beem:2014zpa ; Lemos:2015awa ; Gimenez-Grau:2020jrx . In this section, we
review their results using the notation introduced in Cordova:2016emh . A
superconformal multiplet is fully specified by the Cartan eigenvalues of its
superconformal primary. The Cartan eigenvalues consist in the conformal
dimension $\Delta$, the left and right spins $j,{\bar{\jmath}}$, the $SU(2)$
Cartan $R$ and the $U(1)$ charge $r$. This information is encoded as

$L\mkern 3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu[j\hskip
1.0pt;{\bar{\jmath}}]^{(R;r)}_{\Delta}\,.$ (71)

In principle one can infer any possible shortening condition by the values of
$\Delta,R,r,j,{\bar{\jmath}}$. For convenience, however, shortening conditions
will be indicated by replacing $L$ or $\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu$ with the symbols
$A_{1}$, $A_{2}$, $B_{1}$ or $\bar{A}_{1}$, $\bar{A}_{2}$, $\bar{B}_{1}$. The
meaning of these symbols has to do with the states that become null and are
thus factored out of the multiplet.

The chiral and antichiral operators are the superconformal primaries of the
following multiplets101010In Cordova:2016emh ’s conventions $Q$ has
$\mathfrak{u}(1)$ charge $-1$ so chiral operators must have charge $\pm
2\Delta$. For simplicity, we rescale the R-charge by $-\tfrac{1}{2}$ so that
$r$ will denote both the conformal dimension and the charge of $\phi_{r}$. The
multiplet notation, however, will stay consistent with its original work. In
addition we emphasize that the $SU(2)$ R-charge is expressed with the Dynkin
label (i.e. R-spin $\frac{1}{2}$ means $R=1$.)

$\phi_{r}\in B_{1}\mkern 3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern
1.5mu[0\hskip 1.0pt;0]_{r}^{(0;-2r)}\,,\qquad{\bar{\phi}}_{r}\in L\mkern
3.5mu\overline{\mkern-3.5muB\mkern-1.5mu}\mkern 1.5mu_{1}[0\hskip
1.0pt;0]_{r}^{(0;2r)}\,.$ (72)

For example for a SCFT gauge theory with ${\cal N}=2$ supersymmetry,
$\phi_{r}$ can represent the primary ${\rm tr}\,\varphi^{r}(x)$ (or a multi-
trace operator made of $r$ scalars) and the same for ${\bar{\phi}}_{r}$ with
chiral fields replaced by anti-chiral ones.

The OPE of a chiral and an antichiral operator depends on the sign of the two
charges. Assuming first that $r_{1}\geq r_{2}+1$ and letting
$r_{12}=r_{1}-r_{2}$ one has Lemos:2015awa

$\phi_{r_{1}}\times{\bar{\phi}}_{r_{2}}=B_{1}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu[0\hskip
1.0pt;0]_{r_{12}}^{(0;-2r_{12})}+A_{t}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu[\ell\hskip
1.0pt;\ell]^{(0;-2r_{12})}_{\ell+2+r_{12}}+L\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu[\ell\hskip
1.0pt;\ell]^{(0;-2r_{12})}_{\Delta}\,,$ (73)

with $\Delta>\ell+2+r_{12}$ and $t=1$ if $\ell>0$ and $t=2$ if $\ell=0$. Since
we do not have permutation symmetry, $\ell$ can assume all non-negative even
and odd values. If instead $0<r_{1}-r_{2}<1$ the $B_{1}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu$ multiplet is absent,
while the rest of the OPE remains the same. If $r_{2}>r_{1}$ we have the same
conclusion with the conjugate multiplets while if $r_{1}=r_{2}\equiv r$ we
have Fitzpatrick:2014oza ; Beem:2014zpa

$\phi_{r}\times{\bar{\phi}}_{r}=A_{2}\mkern
3.5mu\overline{\mkern-3.5muA\mkern-1.5mu}\mkern 1.5mu_{2}[0\hskip
1.0pt;0]^{(0;0)}_{2}+L\mkern 3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern
1.5mu[\ell\hskip 1.0pt;\ell]^{(0;0)}_{\Delta}\,,$ (74)

with, again, $\Delta>\ell+2$. Also here $\ell$ can be even or odd.

The interesting operators that appear are the chiral operators themselves in
the OPE with different R-charges and the stress tensor multiplet $A_{2}\mkern
3.5mu\overline{\mkern-3.5muA\mkern-1.5mu}\mkern 1.5mu_{2}[0\hskip
1.0pt;0]_{2}^{(0;0)}$. There are no higher spin analogs $A_{1}\mkern
3.5mu\overline{\mkern-3.5muA\mkern-1.5mu}\mkern 1.5mu_{1}[\ell\hskip
1.0pt;\ell]$ as they contain higher spin currents that appear only in free
theories Maldacena:2011jn ; Boulanger:2013zza ; Alba:2013yda .

The superconformal blocks encoding the contributions of the above multiplets,
including also the whole towers of superdescendants, to a given four-point
functions are all expressed in terms of a single function. This function has
been denoted $\mathcal{G}^{t}_{\Delta,\ell}$ for the $t$-channel correlator
and $\mathcal{G}^{u}_{\Delta,\ell}$ $u$-channel correlator. In Section 3.2 we
give the explicit definitions. In Figure 1 we depict the ranges of $\Delta$
and $\ell$ that correspond to a given multiplet. The fact that, for example,
the long multiplet half-line starts at the stress tensor point means that,
from this four-point functions’ point of view, the stress tensor multiplet is
indistinguishable from a long multiplet at dimension very close to two. The
same phenomenon will appear also elsewhere. This is important when applying
numerical bootstrap techniques, and it is the reason why it is necessary to
input a small gap in the long sector whenever one wants to assume something
about a protected operator which sits at the unitarity bound.

Now let us move to the OPE of two chiral operators. The list of exchanged
superconformal multiplets is now much richer. In Figure 2 we depict the range
of spins and conformal dimensions of the blocks exchanged in this channel. The
superconformal blocks reduce to usual conformal blocks $g_{\Delta,\ell}$ in
this case because the selection rules allow the exchange of only one operator
per multiplet. For notational consistency, however, we will denote them as
$\mathcal{G}^{s}_{\Delta,\ell}$. In the equation below all superconformal
multiplets, with the exception of the first one, contribute to the OPE through
a superdescendant rather than a superprimary. For example, in a long multiplet
$\mathcal{O}$ only $Q^{4}\mathcal{O}$ will contribute as it is the only chiral
operator in it. The OPE reads Lemos:2015awa

$\displaystyle\phi_{r_{1}}\times\phi_{r_{2}}$ $\displaystyle=B_{1}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu[0\hskip
1.0pt;0]^{(0;-2(r_{1}+r_{2}))}_{r_{1}+r_{2}}+B_{1}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu[0\hskip
1.0pt;0]^{(2;-2(r_{1}+r_{2})+2)}_{r_{1}+r_{2}+1}+$ (75)
$\displaystyle\;\quad+B_{1}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu[0\hskip
1.0pt;1]^{(1;-2(r_{1}+r_{2})+1)}_{r_{1}+r_{2}+\frac{1}{2}}+A_{t}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu[\ell\hskip
1.0pt-2;\ell]^{(0;-2(r_{1}+r_{2})+2)}_{r_{1}+r_{2}-1+\ell}+$
$\displaystyle\;\quad+A_{t}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu[\ell-1\hskip
1.0pt;\ell]^{(1;-2(r_{1}+r_{2})+3)}_{r_{1}+r_{2}+\frac{1}{2}+\ell}+L\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu[\ell\hskip
1.0pt;\ell]_{\Delta-2}^{(0;-2(r_{1}+r_{2})+4)}\,,$

where $t=2$ if the first spin label is zero and $t=1$ otherwise. The first
multiplet appearing in (75) is the one whose OPE coefficients we want to
extremize. The second one is necessarily present if there is a mixed branch in
the theory. Assuming its absence requires in practice to assume a gap on the
scalar long multiplets.111111The numerical results are weakly affected by
assuming the absence of this class of operators.

$\phi_{r_{1}}\times{\bar{\phi}}_{r_{2}}$$\ell=0$$\Delta$$B_{1}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu$$r_{12}$$A_{2}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu$$r_{12}+2$$L\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern
1.5mu$$\ell>0$$\Delta$$A_{1}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu$$\ell+r_{12}+2$$L\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu$

$\phi_{r}\times{\bar{\phi}}_{r}$$\ell=0$$\Delta$$0$$A_{2}A_{2}$$2$$L\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern
1.5mu$$\ell>0$$\Delta$$\ell+2$$L\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu$

Figure 1: OPE in the $\phi_{r_{1}}\times{\bar{\phi}}_{r_{2}}$ channel depicted
according to the conformal blocks $\mathcal{G}^{t}_{\Delta,\ell}$ or
$\mathcal{G}^{u}_{\Delta,\ell}$ exchanged. The white dot implies that there is
no operator there.
$\phi_{r_{1}}\times\phi_{r_{2}}$$\ell=0$$\Delta$$B_{1}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern
1.5mu$$r_{1}+r_{2}$$B_{1}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern
1.5mu^{(2)}$$r_{1}+r_{2}+2$$L\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern
1.5mu^{(4)}$$\ell=1$$\Delta$$B_{1}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern
1.5mu^{(1)}$$r_{1}+r_{2}+1$$A_{2}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern
1.5mu^{(3)}$$r_{1}+r_{2}+3$$L\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern
1.5mu^{(4)}$$\ell>1$$\Delta$$A_{t}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern
1.5mu^{(2)}$$r_{1}+r_{2}+\ell$$A_{1}\mkern
3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern 1.5mu^{(3)}$$\quad
r_{1}+r_{2}+\ell+2$$L\mkern 3.5mu\overline{\mkern-3.5muL\mkern-1.5mu}\mkern
1.5mu^{(4)}$ Figure 2: OPE in the $\phi_{r_{1}}\times\phi_{r_{2}}$ channel
depicted according to the conformal blocks $\mathcal{G}^{s}_{\Delta,\ell}$
exchanged. When present, the superscript denotes (twice) the difference
between the R-charge of the superconformal primary and $r_{1}+r_{2}$, which
also corresponds to the level of the superdescendant that enters the OPE. The
light blue dot implies that the operator is there only if a mixed branch is
present. We also framed in red the the operator whose OPE coefficient we want
to extremize.

###  3.2 Bootstrap equations

In order to find the relations constraining the OPE coefficients that we are
interested in, we need to study four-point correlators. In this section we
will review the crossing equations of the system of correlators involving only
$\phi_{r}$ and ${\bar{\phi}}_{r}$, which were first obtained in Beem:2014zpa .
In appendix C we will also show the crossing equations of the mixed correlator
system involving two pairs of chiral fields $\phi_{r_{1}}$, $\phi_{r_{2}}$
which first appeared in Lemos:2015awa .121212In the context of superconformal
field theories, mixed correlators have been studied in Lemos:2015awa ;
Gimenez-Grau:2020jrx ; Agmon:2019imm ; Bissi:2020jve  The latter setup is
needed to obtain OPE bounds on theories of higher rank or on coefficients of
the type $\lambda_{u\,u^{2}\,u^{3}}^{2}$.

Let us start with four-point functions involving only operators $\phi_{r}$ and
${\bar{\phi}}_{r}$. The only nonzero ones have an equal number of chiral and
antichiral fields. We will use different OPEs to define the functions
$f_{s,t,u}$ as follows

$\langle{\bar{\phi}}_{r}(x_{1})\phi_{r}(x_{2})\phi_{r}(x_{3}){\bar{\phi}}_{r}(x_{4})\rangle=\frac{f_{t}(z,{\bar{z}})}{(x_{12}^{2})^{r}(x_{34}^{2})^{r}}=\,\frac{f_{s}(1-z,1-{\bar{z}})}{(x_{23}^{2})^{r}(x_{14}^{2})^{r}}=\frac{f_{u}\big{(}\frac{z}{z-1},\frac{{\bar{z}}}{{\bar{z}}-1}\big{)}}{(x_{12}^{2})^{r}(x_{34}^{2})^{r}}\,,$
(76)

with $x_{ij}^{2}=|x_{i}-x_{j}|^{2}$ and

$\frac{x_{12}^{2}\hskip 1.0ptx_{34}^{2}}{x_{13}^{2}\hskip
1.0ptx_{24}^{2}}=u=z{\bar{z}}\,,\qquad\frac{x_{14}^{2}\hskip
1.0ptx_{23}^{2}}{x_{13}^{2}\hskip 1.0ptx_{24}^{2}}=v=(1-z)(1-{\bar{z}})\,.$
(77)

The functions $f_{s,t,u}(z,{\bar{z}})$ must satisfy these crossing constraints

$\displaystyle((1-z)(1-{\bar{z}}))^{r}\,f_{t}(z,{\bar{z}})=(z{\bar{z}})^{r}\,f_{s}(1-z,1-{\bar{z}})\,,$
(78a)
$\displaystyle((1-z)(1-{\bar{z}}))^{r}\,f_{u}(z,{\bar{z}})=(z{\bar{z}})^{r}\,f_{u}(1-z,1-{\bar{z}})\,.$
(78b)

In order to write these constraints in a form amenable to numerical
computations we must expand the functions $f_{s,t,u}$ in conformal blocks.
Based on the analysis of section 3.1 we have131313We remind the reader that
$\lambda_{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}$ stands for the
coefficient of $\mathcal{O}_{3}$ in the OPE of
$\mathcal{O}_{1}\times\mathcal{O}_{2}$, which means that it is computed by the
three-point function $\langle\mathcal{O}_{1}\mathcal{O}_{2}{\mkern
3.5mu\overline{\mkern-3.5mu\mathcal{O}\mkern-1.5mu}\mkern 1.5mu}_{3}\rangle$.

$\displaystyle f_{t}(z,{\bar{z}})$
$\displaystyle=\mathcal{G}^{t}_{0,0}(z,{\bar{z}})+\frac{r^{2}}{6\hskip
0.5ptc}\mathcal{G}^{t}_{2,0}(z,{\bar{z}})+\sum_{\ell=0}^{\infty}\sum_{\Delta>\ell+2}|\lambda_{\phi_{r}{\bar{\phi}}_{r}\mathcal{O}_{\Delta,\ell}}|^{2}(-1)^{\ell}\,\mathcal{G}^{t}_{\Delta,\ell}(z,{\bar{z}})\,,$
(79a) $\displaystyle f_{s}(z,{\bar{z}})$
$\displaystyle=|\lambda_{\phi_{r}\phi_{r}\phi_{2r}}|^{2}\mathcal{G}^{s}_{2r,0}(z,{\bar{z}})+\sum_{\begin{subarray}{c}\ell=0\\\
\mathrm{even}\end{subarray}}^{\infty}\sum_{\begin{subarray}{c}\Delta=\ell+2r\\\
\Delta\geq\ell+2r+2\end{subarray}}|\lambda_{\phi_{r}\phi_{r}\mathcal{O}_{\Delta,\ell,2r}}|^{2}\mathcal{G}^{s}_{\Delta,\ell}(z,{\bar{z}})\,,$
(79b) $\displaystyle f_{u}(z,{\bar{z}})$
$\displaystyle=\mathcal{G}^{u}_{0,0}(z,{\bar{z}})+\frac{r^{2}}{6\hskip
0.5ptc}\mathcal{G}^{u}_{2,0}(z,{\bar{z}})+\sum_{\ell=0}^{\infty}\sum_{\Delta>\ell+2}|\lambda_{\phi_{r}{\bar{\phi}}_{r}\mathcal{O}_{\Delta,\ell}}|^{2}\,\mathcal{G}^{u}_{\Delta,\ell}(z,{\bar{z}})\,,$
(79c)

where $c$ is the central charge. It is not necessary to input $c$ explicitly:
it can be left arbitrary just like the other OPE coefficients. However we will
fix it to a specific values on all our computations. In the first line we used
$\lambda_{{\bar{\phi}}\phi\mathcal{O}}=(-1)^{\ell}\lambda_{\phi{\bar{\phi}}\mathcal{O}}$
and in the second we used $\lambda_{{\bar{\phi}}{\bar{\phi}}{\mkern
3.5mu\overline{\mkern-3.5mu\mathcal{O}\mkern-1.5mu}\mkern
1.5mu}}=\lambda_{\phi\phi\mathcal{O}}^{*}$. We denote the operators as
$\mathcal{O}_{\Delta,\ell}$ if they are R-neutral, otherwise as
$\mathcal{O}_{\Delta,\ell,r}$, with $r$ denoting the R-charge (in units where
$\phi_{r}$ has R-charge $r$). Notice that from this point on, the subscripts
labeling $\lambda$ refer to the $r$-charge of the corresponding operators.
These conventions are more suitable in this description, being also consistent
with previous literature.

The blocks $\mathcal{G}^{s},\mathcal{G}^{t}$ and $\mathcal{G}^{u}$ are
explicit functions given by

$\mathcal{G}^{I}_{\Delta,\ell}(z,{\bar{z}})=\frac{z{\bar{z}}}{z-{\bar{z}}}\bigl{(}k^{I}_{\Delta+\ell}(z)k^{I}_{\Delta-\ell-2}({\bar{z}})-z\leftrightarrow{\bar{z}}\bigr{)}\,,\qquad
I=s,t,u\,,$ (80)

with

$\displaystyle k^{s}_{\beta}(z)$
$\displaystyle=z^{\frac{\beta}{2}}\,{}_{2}F_{1}\left(\frac{\beta}{2},\frac{\beta}{2};\beta;z\right)\,,$
(81a) $\displaystyle k^{t}_{\beta}(z)$
$\displaystyle=z^{\frac{\beta}{2}}\,{}_{2}F_{1}\left(\frac{\beta}{2},\frac{\beta}{2};\beta+2;z\right)\,,$
(81b) $\displaystyle k^{u}_{\beta}(z)$
$\displaystyle=z^{\frac{\beta}{2}}\,{}_{2}F_{1}\left(\frac{\beta}{2},\frac{\beta+4}{2};\beta+2;z\right)\,.$
(81c)

With all these ingredients in place, we can finally write down the crossing
equation as follows

$\displaystyle\sum_{\Delta,\ell}|\lambda_{\phi_{r}\phi_{r}\mathcal{O}_{\Delta,\ell,2r}}|^{2}\,\vec{V}^{\mathrm{charged}}_{\Delta,\ell}+\sum_{\Delta,\ell}(\lambda_{\phi_{r}{\bar{\phi}}_{r}\mathcal{O}_{\Delta,\ell}})^{2}\,\vec{V}^{\mathrm{neutral}}_{\Delta,\ell}=-\vec{V}^{\mathrm{neutral}}_{0,0}-\frac{r^{2}}{6\hskip
0.5ptc}\;\vec{V}^{\mathrm{neutral}}_{2,0}\,,$ (82)

with

$\vec{V}^{\mathrm{charged}}_{\Delta,\ell}\equiv\begin{bmatrix}-(-1)^{\ell}\mathcal{F}^{s}_{+,\Delta,\ell}\\\
(-1)^{\ell}\mathcal{F}^{s}_{-,\Delta,\ell}\\\
0\end{bmatrix}\,,\quad\vec{V}^{\mathrm{neutral}}_{\Delta,\ell}\equiv\begin{bmatrix}(-1)^{\ell}\mathcal{F}^{\hskip
0.5ptt}_{+,\Delta,\ell}\\\ (-1)^{\ell}\mathcal{F}^{\hskip
0.5ptt}_{-,\Delta,\ell}\\\ \mathcal{F}^{u}_{-,\Delta,\ell}\end{bmatrix}\,,$
(83)

where we introduced the (anti)symmetric combinations (which implicitly depend
on $r$)

$\mathcal{F}^{I}_{\pm,\Delta,\ell}(z,{\bar{z}})=v^{r}\mathcal{G}^{I}_{\Delta,\ell}(z,{\bar{z}})\pm
u^{r}\mathcal{G}^{I}_{\Delta,\ell}(1-z,1-{\bar{z}})\,,\qquad I=s,t,u\,.$ (84)

The crossing equation (82) can be thought of as a vector equation in an
infinite dimensional space (the space of 3-vector valued functions of $z$ and
${\bar{z}}$). In the next section we will review the standard numerical
methods adopted to study it.

###  3.3 Numerical implementation

In this section, we will give a quick review on the numerical implementation
of the crossing equation (82). The general strategy is to find contradictions
to the bootstrap equations that stem from the existence of a functional
$\alpha$ satisfying some properties. Schematically, $\alpha$ should satisfy
some inequalities that take the form

$\exists\;\alpha\quad\mathrm{s.t}\quad\forall\;(\Delta,\ell)\in\mathrm{spectrum}\quad\alpha[\vec{V}_{\Delta,\ell}]\geq
0\quad\mathrm{and}\quad\alpha[\vec{V}_{0,0}^{\mathrm{neutral}}]=1\,.$ (85)

This would make the left hand side of (82) non-negative and the right hand
side strictly negative, hence a contradiction, meaning that the assumptions
made on the spectrum were inconsistent. There are essentially two obstacles
that one needs to overcome in order to translate this problem in something
that can be made easily accessible by computers

  1. 1.

Searching in a space of functionals which is infinite dimensional (i.e. the
dual of a space of functions)

  2. 2.

Imposing an inequality over an infinite, and in any case not fully specified,
set of $\Delta$’s and $\ell$’s

To overcome 1. we can restrict ourselves to a finite dimensional subspace of
functionals. Empirically it turns out that the following set works
particularly well141414Here we show the action of $\alpha$ on a single entry
of the vector $\vec{V}$. The actual $\alpha$ we are looking for is a vector of
functionals, each acting on a different entry of $\vec{V}$.

$\alpha[F]=\sum_{n\leq
m}^{n+m\leq\Lambda}\alpha_{n,m}\frac{\partial^{n}}{\partial
z^{n}}\frac{\partial^{m}}{\partial{\bar{z}}^{m}}F(z,{\bar{z}})\big{|}_{z={\bar{z}}=\frac{1}{2}}\,.$
(86)

This way, if $\Lambda$ is odd, the space of functionals has dimension

$\mathrm{dim}(\Lambda)=\frac{1}{4}(\Lambda+1)(\Lambda+3)\,.$ (87)

The choice of $z={\bar{z}}=1/2$ is motivated by the fact that it is the point
where both the $s$ channel OPE ($z,{\bar{z}}\sim 0$) and $t$ channel OPE
($z,{\bar{z}}\sim 1$) converge optimally. In principle one could evaluate the
derivatives in slightly different points as well, but this has shown no
significant improvements in the past.

To overcome 2. we need to work more. In principle one could discretize
$\Delta$ and put a cutoff on both $\Delta$ and $\ell$ and impose positivity on
all the points in this two-dimensional lattice. Despite being possibile, this
is not computationally very convenient. A better approach takes advantage of
the analytic structure of the conformal blocks. It is a known fact that
conformal blocks and their derivatives in $z,{\bar{z}}$ can be systematically
approximated as a positive function times a polynomial in $\Delta$
Poland:2011ey

$\partial_{z}^{n}\partial_{\bar{z}}^{m}\,{\cal
G}_{\Delta,\ell}\big{(}\tfrac{1}{2},\tfrac{1}{2}\big{)}=\chi(\Delta)\,p^{m,n}_{\ell}(\Delta)\,,\qquad\chi(\Delta)>0\quad\forall\;\Delta\geq\ell+2\,.$
(88)

The actual expression for $\chi(\Delta)$ will be shown later (142). Because of
this, we can ignore $\chi(\Delta)$ for the purpose of imposing
$\alpha[g_{\Delta,\ell}]\geq 0$. Therefore, we only have to study positivity
of the polynomial $p^{m,n}_{\ell}$ for all $\Delta\geq\ell+2$. This is a known
problem in mathematics that can be mapped to imposing positive-
semidefiniteness on a pair of matrices.151515Very roughly, any polynomial
$P(x)$ which is positive for $x>x_{*}$ can be written as
$P(x)=\vec{u}^{T}(x)\cdot M\cdot\vec{u}(x)+(x-x_{*})\,\vec{u}^{T}(x)\cdot
N\cdot\vec{u}(x)\,,$ where $u(x)$ is a vector of independent polynomials,
whose length is the degree of $P$ divided by two, and $M,N$ are positive
definite matrices. We refer the reader to Appendix B for details.

To summarize, we consider a finite subspace of functionals $\alpha$ given by
(86) and we take rational approximations of the blocks. Next we put a cutoff
$\ell_{\mathrm{max}}$ on the number of spins and impose (85) on all spins up
to it. The rational approximation of the blocks is what allows us to impose
positivity for all $\Delta$’s in one go.

###  3.4 Bounds on the OPE coefficient

In this specific instance of the bootstrap we want to put upper and lower
bounds on the OPE coefficient of $\lambda_{\phi_{r}\phi_{r}\phi_{2r}}$. It is
possible to get lower bounds only if the operator is isolated. Indeed for the
theories that we are considering this is the case, as one can easily see from
section 3.1. The bootstrap problem that we want to study is

> Fix $c$, then maximize (minimize)
> $B_{\pm}\equiv\alpha\left[\vec{V}^{\mathrm{neutral}}_{0,0}+\frac{r^{2}}{6c}\,\vec{V}_{2,0}^{\mathrm{neutral}}\right]$
> subject to
>
>   1. 1.
>
> $\alpha[\vec{V}_{\Delta,\ell}^{\mathrm{charged}}]\geq
> 0\quad\forall\;\ell\geq
> 0\;\mathrm{even}\,,\quad\forall\;\Delta=\ell+2r\;\;\mathrm{or}\;\;\Delta\geq\ell+2r+2$
>
>   2. 2.
>
> $\alpha[\vec{V}_{\Delta,\ell}^{\mathrm{neutral}}]\geq
> 0\quad\forall\;\ell>0\,,\quad\forall\;\Delta\geq\ell+2$
>
>   3. 3.
>
> $\alpha[\vec{V}_{\Delta,0}^{\mathrm{neutral}}]\geq 0\quad\forall\;\Delta\geq
> 2+\epsilon$
>
>   4. 4.
>
> $\alpha[\vec{V}_{4,0}^{\mathrm{charged}}]=\pm 1$
>
>

The parameter $\epsilon$ can be taken to be $\sim 0.1$ and it is needed to
make $V_{2,0}^{\mathrm{neutral}}$ isolated so that it is meaningful to impose
a value of the central charge. The final result will be the two-sided bound

$B_{-}\leq|\lambda_{\phi_{r}\phi_{r}\phi_{2r}}|^{2}\leq B_{+}\,.$ (89)

There will be such a bound for every value of $r$ and $c$, so we can compute
it for any rank-one $\mathcal{N}=2$ SCFT or any other higher-rank SCFT if we
focus only on a single Coulomb branch operator. Bounds involving mixed
correlators are also possible but we do not show the list of bootstrap
assumptions for brevity.

###  3.5 Bound on the dimension of the lightest neutral unprotected operator

If we know the OPE coefficient of the chiral scalar of dimension $2r$ we can
put it in the right hand side of the bootstrap equation. The assumptions that
we need are as follows

> Fix the value of $\lambda_{\phi_{r}\phi_{r}\phi_{2r}}$ and of $c$. Then find
> $\alpha$ such that
>
>   1. 1.
>
>
> $\alpha\left[\vec{V}^{\mathrm{neutral}}_{0,0}+|\lambda_{\phi_{r}\phi_{r}\phi_{2r}}|^{2}\hskip
> 1.0pt\vec{V}^{\mathrm{charged}}_{2r,0}+\frac{r^{2}}{6c}\,\vec{V}_{2,0}^{\mathrm{neutral}}\right]=1$
>
>   2. 2.
>
> $\alpha[\vec{V}_{\Delta,\ell}^{\mathrm{neutral}}]\geq
> 0\quad\forall\;\ell>0\,,\quad\forall\;\Delta\geq\ell+2$
>
>   3. 3.
>
> $\alpha[\vec{V}_{\Delta,\ell}^{\mathrm{charged}}]\geq
> 0\quad\forall\;\ell\geq
> 2\;\mathrm{even}\,,\quad\forall\;\Delta=\ell+2r\;\;\mathrm{or}\;\;\Delta\geq\ell+2r+2$
>
>   4. 4.
>
> $\alpha[\vec{V}_{2r+2,0}^{\mathrm{charged}}]\geq 0$
>
>   5. 5.
>
> $\alpha[\vec{V}_{\Delta,0}^{\mathrm{neutral}}]\geq
> 0\quad\forall\;\Delta\geq\Delta_{\mathrm{gap}}>2$
>
>

The gap $\Delta_{\mathrm{gap}}$ can be taken to be inside an interval
$[\Delta_{\mathrm{min}},\Delta_{\mathrm{max}}]$ and we can run a binary search
by checking
$\Delta_{\mathrm{gap}}=\frac{1}{2}(\Delta_{\mathrm{min}}+\Delta_{\mathrm{max}})$
and updating the upper or lower limit according to whether an $\alpha$
satisfying the above assumptions is found or not.

###  3.6 Results for the OPE coefficient

In Tables 6 and 7 we report our results for the OPE bounds obtained via the
numerical bootstrap. The first and last line use the single correlator setup
described in Section 3.2. The second line uses the mixed correlator setup
instead, whose crossing vectors are shown in appendix C.

In order to obtain these bounds we set up a bootstrap problem as explained in
section 3.4. We then use a newly developed framework, sailboot,161616Available
at gitlab.com/maneandrea/sailboot. to translate crossing equations and
assumptions into numerical vectors that can be fed to sdpb.171717Available at
github.com/davidsd/sdpb Landry:2019qug .

The upper and lower bounds for the simplest OPE coefficient in the
$\mathcal{H}_{0}$ theory, namely $\lambda_{\phi_{r}\,\phi_{r}\phi^{2}_{r}}$ at
$r=\frac{6}{5}$, were previously obtained in Cornagliotto:2017snu  while
bounds for the same OPE coefficient but in the $\mathcal{H}_{1}$ and
$\mathcal{H}_{2}$ theories first appeared in Gimenez-Grau:2020jrx . We also
did the same computation for Minahan-Nemeschansky theories but, unfortunately,
the bounds were either extremely weak or non-existent. This is generally
expected when one considers correlators of operators with large external
dimensions. A possible cause of this is that when the external dimensions are
large there is a big gap between the unitarity bound and the generalized free
theory spectrum, leading to many “fake” approximate solutions to crossing that
spoil the numerics.181818We thank Alessandro Vichi for discussions about this
point.

theory | $\mathcal{H}_{0}$ | $\mathcal{H}_{1}$ | $\mathcal{H}_{2}$
---|---|---|---
$(r,c)$ | $(\frac{6}{5},\frac{11}{30})$ | $(\frac{4}{3},\frac{1}{2})$ | $(\frac{3}{2},\frac{2}{3})$
$\lambda^{2}_{u\,u\,u^{2}}$ | 2.167 | 2.359 | 2.698
2.142 | 2.215 | 2.298
$\lambda^{2}_{u\,u^{2}\,u^{3}}$ | 3.637 | 4.445 |
3.192 | 3.217 |
Table 6: Bootstrap bounds on OPE coefficients of some rank-one theories
obtained with the single correlator setup at $\Lambda=43$
($\lambda^{2}_{u\,u\,u^{2}}$) and the mixed correlator setup at $\Lambda=19$
($\lambda^{2}_{u\,u^{2}\,u^{3}}$).

Next we considered some theories of rank two. In Table 5 we show the results
for some theories of interest. The OPE coefficients that we considered are
either $\lambda_{u\,v\,uv}^{2}$, where $u\equiv\phi_{r_{1}}$ and
$v\equiv\phi_{r_{2}}$ are the two Coulomb branch operators, or
$\lambda_{\phi_{i}\,\phi_{i}\,\phi_{i}^{2}}^{2}$ with $\phi_{i}$ either $u$ or
$v$. For the former one needs to use the mixed correlator setup, described in
appendix C, whereas for the latter the single correlator setup is sufficient.
It should be noted that when studying
$\lambda_{\phi_{i}\,\phi_{i}\,\phi_{i}^{2}}^{2}$, the information that the
theory is of rank two is not fed into the bootstrap equations, if not
indirectly through its central charge. Some more results for the OPE
coefficient bounds of heavier operators in the pure $SU(5)$ theory can be
found in Table 8.

theory | $(A_{1},A_{4})$ | $(A_{1},A_{5})$
---|---|---
$(r_{u},r_{v},c)$ | $(\frac{8}{7},\frac{10}{7},\frac{17}{21})$ | $(\frac{5}{4},\frac{3}{2},\frac{23}{24})$
$\lambda^{2}_{u\,u\,u^{2}}$ | 2.102 | 2.231
2.024 | 2.055
$\lambda^{2}_{u\,v\,uv}$ | 1.125 | 1.233
0.981 | 0.960
$\lambda^{2}_{v\,v\,v^{2}}$ | 2.533 | 2.709
2.181 | 2.195
Table 7: Bootstrap bounds on OPE coefficients of some rank-two theories obtained with the single correlator setup at $\Lambda=41$ ($\lambda^{2}_{u\,u\,u^{2}}$ and $\lambda^{2}_{v\,v\,v^{2}}$) and the mixed correlator setup at $\Lambda=19$ ($\lambda^{2}_{u\,v\,uv}$).  coefficient | $\lambda^{2}_{u\,u^{2}\,u^{3}}$ | $\lambda^{2}_{u^{2}\,u^{2}\,u^{4}}$ | $\lambda^{2}_{v^{2}\,v^{2}\,v^{4}}$
---|---|---|---
bounds | 3.512 | 12.42 | 142.6
0.890 | 2.871 | 0.855
Table 8: Bootstrap bounds on OPE other coefficients of the rank-two theory
$(A_{1},A_{4})$ with $c=17/21$, $r_{1}=8/7$ and $r_{2}=10/7$ obtained with the
single correlator setup at $\Lambda=41$ ($\lambda^{2}_{u^{2}\,u^{2}\,u^{4}}$
and $\lambda^{2}_{v^{2}\,v^{2}\,v^{4}}$) and the mixed correlator setup at
$\Lambda=19$ ($\lambda^{2}_{u\,u^{2}\,u^{3}}$).

The pure $SU(5)$ theory belongs to a family of theories of incresing rank all
obtained by going to a special locus of a pure gauge theory with gauge group
$SU(N)$ Martone:2021ixp ; Eguchi:1996vu ; Eguchi:1996ds . The central charges
and Coulomb branch dimensions are summarized in equations (48) and (46),
respectively. In Figure 3 we plot upper and lower bounds for the OPE
coefficients $\lambda_{u\,u\,u^{2}}^{2}$ of the theories above for $N$ ranging
from $5$ to $12$ where $r$ is the dimension of the heaviest Coulomb branch
generator, namely $r=2N/(N+2)$.

$\lambda_{r\,r\,2r}^{2}$$N$ Figure 3: Upper and lower bounds for the OPE
coefficients $\lambda_{r\,r\,2r}^{2}$ in the pure $SU(N)$ theories where $r$
is the conformal dimension of the heaviest Coulomb branch generator. The
shaded area is disallowed.

For rank-two, we can carve out the cube of OPE bounds by scanning over the
values of $\lambda^{2}_{u\,u\,u^{2}}$, $\lambda^{2}_{v\,v\,v^{2}}$ and
obtaining for each point an upper and a lower bound on
$\lambda^{2}_{u\,v\,uv}$. This can be done by putting the vectors
$\vec{V}^{2r_{i}}_{2r_{i},0}$ of (147) to the right of the crossing equations,
just like the first point in Subsection 3.5. The results for the
$(A_{1},A_{n})$ theories for $n=4,5$ can be found in Figure 4.

(a) $(A_{1},A_{4})$

(b) $(A_{1},A_{5})$

Figure 4: Upper and lower bounds on $\lambda_{u\,v\,uv}$ as a function of the
values of $\lambda^{2}_{u\,u\,u^{2}}$ and $\lambda^{2}_{v\,v\,v^{2}}$. The
base of the plot is bounded by the values found in the single correlator
bootstrap in Table 7.

###  3.7 Results for the lightest scalar neutral unprotected operator

In this subsection we present upper bounds on the dimension of the lightest
unprotected neutral operator appearing in the OPE
$\phi_{r}\times{\bar{\phi}}_{r}$. The definition of the bootstrap problem to
consider was given in subsection 3.5. The following plots will present a scan
on several values of the OPE coefficient $\lambda_{u\,u\,u^{2}}^{2}$ ranging
between the bounds obtained in the previous subsection. The result is an upper
bound on the lightest conformal dimension, which we denote as
$\Delta_{\mathrm{gap}}$. It is interesting to notice that the upper end of the
OPE window requires a very low $\Delta_{\mathrm{gap}}$, suggesting that
probably the true value of $\lambda_{u\,u\,u^{2}}^{2}$ is located towards the
lower end. This conclusion stems from the expectation that a strongly
interacting theory is not expected to have operator dimensions too close to
the free theory value. The plots are presented in Figure 5.

$\Delta_{\mathrm{gap}}$$\lambda^{2}_{u\,u\,u^{2}}$ (a) Theory
$\mathcal{H}_{0}$, $r=\tfrac{6}{5}$

$\Delta_{\mathrm{gap}}$$\lambda^{2}_{u\,u\,u^{2}}$ (b) Theory
$\mathcal{H}_{1}$, $r=\tfrac{4}{3}$

$\Delta_{\mathrm{gap}}$$\lambda^{2}_{u\,u\,u^{2}}$ (c) Theory
$\mathcal{H}_{2}$, $r=\tfrac{3}{2}$

$\Delta_{\mathrm{gap}}$$\lambda^{2}_{u\,u\,u^{2}}$ (d) Theory $D_{4}$, $r=2$

Figure 5: Upper bound on the dimension of the lightest scalar neutral
unprotected operator $\Delta_{\mathrm{gap}}$ as a function of the OPE
coefficient $\lambda_{u\,u\,u^{2}}^{2}$ for various rank-one theories. The
upper area (shaded in red) is disallowed. The blue vertical line marks the
value found by localization (see Table 2).

##  4 Large charge asymptotics

The two-point functions $G_{nn}$ behave in a universal way as the charge (and,
consequently, the dimension) of the operators is sent to infinity. More
precisely, if we denote $nr\equiv\mathcal{J}$, we have the following
asymptotic behavior Hellerman:2017sur ; Hellerman:2020sqj ; Hellerman:2021duh
191919After completion of the present paper, we learned that in
Hellerman:2018xpi  this asymptotic formula has been improved, resumming the
large-charge expansion. See also Beccaria:2020azj  for the higher-rank
generalization.

$G_{nn}\;\underset{n\to\infty}{\sim}\;\tilde{\mathcal{Y}}\,\Gamma(\mathcal{J}+1)\left(\frac{N_{\mathcal{O}}}{2\pi
R}\right)^{2\mathcal{J}}\mathcal{J}^{\alpha}\,,$ (90)

where $\tilde{\mathcal{Y}}$ and $N_{\mathcal{O}}$ are constants and $\alpha$
is a constant related to the $a$-anomaly of the SCFT considered.202020Not to
be confused with the $\alpha$ of Table 1. The values $\alpha$ for the theories
of interest are summarized in Table 9.

In this section we want to compare our results with this asymptotic expansion.
This will provide a nontrivial check of our method and will also allow us to
estimate the values of $\tilde{\mathcal{Y}}$ and $N_{\mathcal{O}}$. The
formula (20), with $C_{ij}$ given by (39), can in principle be computed
exactly for all $n$, but since we want to study the behavior at very large $n$
it is easier to simply evaluate it numerically because this will significantly
speed up the computation of the determinants. We found that the agreement with
the formula (90) is excellent already at $n\sim 100$. In Figure 6 we plot the
values of $G_{nn}$ (in blue) together with the asymptotic form (in red). For
brevity, we only show $\mathcal{H}_{0}$ and MN with flavor group $E_{6}$. The
other AD or MN theories look similar.. The small inset instead shows the
percent error in green. Then in Table 9 we show our best estimates for
$\tilde{\mathcal{Y}}$ and $N_{\mathcal{O}}$.

(a) $r=\frac{6}{5}$ (b) $r=3$

Figure 6: Comparison of $G_{nn}$ with the asymptotic form (90) where $c_{F}$ and $R$ have been set to $1$. $r$ | $\frac{6}{5}$ | $\frac{4}{3}$ | $\frac{3}{2}$ | $2$ | $3$ | $4$ | $6$
---|---|---|---|---|---|---|---
$\alpha$ | $\frac{3}{10}$ | $\frac{1}{2}$ | $\frac{3}{4}$ | $\frac{3}{2}$ | $3$ | $\frac{9}{2}$ | $\frac{15}{2}$
$\sqrt{c_{F}}N_{\mathcal{O}}$ | $2.197$ | $2.411$ | $2.641$ | $3.1416$ | $3.676$ | $3.937$ | $4.176$
$\tilde{\mathcal{Y}}$ | $1.164$ | $1.111$ | $1.010$ | $0.6347$ | $0.137$ | $1.619\times 10^{-2}$ | $6.530\times 10^{-5}$
Table 9: In the second row we show the values of the parameter $\alpha$ of
(90). In the last two rows we show the best fit estimates for
$\tilde{\mathcal{Y}}$ and $N_{\mathcal{O}}$ coming from the plots of Figure 6.

##  5 Conclusions and Outlook

In this paper we have computed the OPE coefficients of some four-point
correlators for $\mathcal{N}=2$ SCFT theories of the AD type. We have
performed the computations using both localization and the bootstrap method as
summarized in Table 10. We find surprising agreement between the two methods,
except for a few values. We remark that our results for the OPE coefficient
$\lambda_{u\,u\,u^{2}}$ of the rank-one AD theories were also found in
Grassi:2019txd , extrapolating from the large-charge expansion, and in
Cornagliotto:2017snu ; Gimenez-Grau:2020jrx  using the bootstrap method.

OPE | method | $\mathcal{H}_{0}$ | $\mathcal{H}_{1}$ | $\mathcal{H}_{2}$
---|---|---|---|---
$\lambda^{2}_{u\,u\,u^{2}}$ | Boot.$\;\Big{\\{}$ | 2.167 | 2.359 | 2.698
2.142 | 2.215 | 2.298
Localiz. | 2.098 | 2.241 | 2.421
$\lambda^{2}_{u\,u^{2}\,u^{3}}$ | Boot.$\;\Big{\\{}$ | 3.637 | 4.445 |
3.192 | 3.217 |
Localiz. | 3.300 | 3.674 | 4.175

OPE | method | $(A_{1},A_{4})$ | $(A_{1},A_{5})$
---|---|---|---
$\lambda^{2}_{u\,u\,u^{2}}$ | Boot.$\;\Big{\\{}$ | 2.102 | 2.231
2.024 | 2.055
Localiz. | 1.878 | 1.929
$\lambda^{2}_{u\,v\,uv}$ | Boot.$\;\Big{\\{}$ | 1.125 | 1.233
0.981 | 0.960
Localiz. | 1.043 | 1.039
$\lambda^{2}_{v\,v\,v^{2}}$ | Boot.$\;\Big{\\{}$ | 2.533 | 2.709
2.181 | 2.195
Localiz. | 2.231 | 2.203

Table 10: Summary of the results from localization and numerical bootstrap,
reproduced here for convenience and for easing the comparison between the two
methods.

Probably the most mysterious point, at the conceptual level, is why the
localization formula, though truncated at $\mathcal{F}_{1}$, gives so accurate
values for the OPE coefficients. Indeed, lacking a parametrically-small
quantity that regulates the genus expansion, higher-genus corrections are in
principle not expected to give negligible contributions. While it would be
very interesting to have a deeper understanding of this point, it would also
be important to have a handle on the functions $\mathcal{F}_{g}$ with $g\geq
2$ for the AD theories, to try and estimate their effect on the OPE
coefficients. We hope to report on this matter in the near future.

On the more technical level, the localization formula for two-point
correlators we used in this paper relies on knowledge of the SW prepotential
$\mathcal{F}_{0}$, and hence of the SW periods. For $SU(N)$ gauge theories
with fundamental matter of rank higher than one, explicit expressions for the
SW periods of AD theories are, to our knowledge, not known. Here we tackle
this problem, by considering directly the SW curve in the vicinity of the
conformal points and computing explicitly the SW periods for the rank-two AD
appearing in the moduli space of pure $SU(5)$ and $SU(6)$ theories. It would
be nice to extend our computation to more theories of the AD type. Also, for
higher-rank theories, we have been elusive regarding the integration contours
of the matrix-model integral, and only paid attention to pick those which
makes the integral over the CB parameters convergent. While this seems a
reasonable guiding principle, further clarifications are needed on this point.

###### Acknowledgements.

We are particularly grateful to R. Poghossian for illuminating discussions on
the periods of Riemann surface. We are also grateful to A. Grassi, J. Minahan,
D. Orlando, J. Russo, and L. Tizzano for discussions. AM would like to thank
G. Fardelli and A. Gimenez-Grau for discussions. The work of AB and AM is
supported by Knut and Alice Wallenberg Foundation under grant KAW 2016.0129
and by VR grant 2018-04438. AM and AB would like to thank the INFN and
University of Rome Tor Vergata for their hospitality. The computations in this
work were enabled by resources in project SNIC 2020/15-320 and SNIC
2021/22-500 provided by the Swedish National Infrastructure for Computing
(SNIC) at UPPMAX, partially funded by the Swedish Research Council through
grant agreement no. 2018-05973.

##  Appendix A The Seiberg-Witten prepotential for $\boldsymbol{SU(2)}$ gauge
theory

###  A.1 Elliptic geometry

In the quartic form, an elliptic geometry can be written as

$w^{2}=\prod_{i=1}^{4}(x-e_{i})=x^{4}+d_{3}\,x^{3}+d_{2}\,x^{2}+d_{1}\,x+d_{0}\,.$
(91)

The first period is

$w_{1}={{\rm i}\over\pi}\int_{e_{1}}^{e_{2}}{dx\over
w}=\int_{0}^{1}{dz\over\pi\sqrt{e_{13}e_{24}\,z(1-z)(1-z\zeta)}}={{}_{2}F_{1}({\textstyle\frac{1}{2}},{\textstyle\frac{1}{2}},1,\zeta)\over\sqrt{e_{13}e_{24}}}\,,$
(92)

with $z={(x-e_{1})e_{24}\over(x-e_{4})e_{21}}$ and

$\zeta={e_{12}e_{34}\over e_{13}e_{24}}\,.$ (93)

The second period is obtained permuting the roots $(123)\to(231)$ leading to

$w_{2}={{\rm i}\over\pi}\int_{e_{2}}^{e_{3}}{dx\over
w}={{}_{2}F_{1}\left({\textstyle\frac{1}{2}},{\textstyle\frac{1}{2}},1,{e_{23}e_{14}\over
e_{21}e_{34}}\right)\over\sqrt{e_{12}e_{43}}}={{\rm
i}\,_{2}F_{1}({\textstyle\frac{1}{2}},{\textstyle\frac{1}{2}},1,1-\zeta)\over\sqrt{e_{13}e_{24}}}\,,$
(94)

where in the last equation we used the identity

${}_{2}F_{1}\left({\textstyle\frac{1}{2}},{\textstyle\frac{1}{2}},1,x\right)=(1-x)^{-{1\over
2}}{}_{2}F_{1}\left({\textstyle\frac{1}{2}},{\textstyle\frac{1}{2}},1,{x\over
x-1}\right)\,.$ (95)

Ordering the roots

$e_{1}<e_{2}<e_{3}<e_{4}$ (96)

one finds that $\zeta^{-1}>1$, so $\zeta^{-1}\notin[0,1]$ as required. The
torus complex structure is defined as the ratio of the periods

$\tau={w_{2}\over w_{1}}={{\rm
i}\,{}_{2}F_{1}({\textstyle\frac{1}{2}},{\textstyle\frac{1}{2}},1,1-\zeta)\over{}_{2}F_{1}({\textstyle\frac{1}{2}},{\textstyle\frac{1}{2}},1,\zeta)}\,.$
(97)

Alternatively, the periods of the curve can be expressed in terms of
permutation invariant quantities $D$, $\Delta$ given in terms of the expansion
coefficients $d_{n}$ rather than the roots. The two invariants are defined as

$\displaystyle D$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{16}}\sum_{i\neq j\neq k\neq
l}e_{ij}^{2}\,e_{kl}^{2}=d_{2}^{2}-3\,d_{1}\,d_{3}+12d_{0}$
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\prod_{i<j}e_{ij}^{2}=d_{2}^{2}-3\,d_{1}\,d_{3}+12d_{0}=-27d_{1}^{4}-4d_{3}^{3}d_{1}^{3}+18d_{2}d_{3}d_{1}^{3}-4d_{2}^{3}d_{1}^{2}+d_{2}^{2}d_{3}^{2}d_{1}^{2}\,,$
(98)
$\displaystyle-6d_{0}d_{3}^{2}d_{1}^{2}+144d_{0}d_{2}d_{1}^{2}+18d_{0}d_{2}d_{3}^{3}d_{1}-192d_{0}^{2}d_{3}d_{1}-80d_{0}d_{2}^{2}d_{3}d_{1}+16d_{0}d_{2}^{4}$
$\displaystyle-27d_{0}^{2}d_{3}^{4}+256d_{0}^{3}-128d_{0}^{2}d_{2}^{2}-4d_{0}d_{2}^{3}d_{3}^{2}+144d_{0}^{2}d_{2}d_{3}^{2}\,.$

In terms of these variables, we can define the following two functions

$\displaystyle w_{3}$ $\displaystyle=$ $\displaystyle D^{-{1\over
4}}\,_{2}F_{1}\left({\textstyle\frac{1}{12}},{\textstyle\frac{5}{12}},1,J^{-1}\right)\,,$
(99) $\displaystyle w_{4}$ $\displaystyle=$ $\displaystyle{\rm i}D^{-{1\over
4}}\,\left[-_{2}F_{1}\left({\textstyle\frac{1}{12}},{\textstyle\frac{5}{12}},1,J^{-1}\right)+{\Gamma({\textstyle\frac{5}{12}})\Gamma({\textstyle\frac{1}{12}})\over
2\pi^{3\over
2}}\,_{2}F_{1}\left({\textstyle\frac{1}{12}},{\textstyle\frac{5}{12}},{\textstyle\frac{1}{2}},1{-}J^{-1}\right)\right]\,,$

with

$J(\zeta)={4D^{3}\over 27\,\Delta}={4(1-\zeta+\zeta^{2})\over
27\zeta^{2}(1-\zeta)^{2}}\,.$ (100)

Using quadratic and cubic transformation identities of hypergeometric
functions one can check that for

$\displaystyle\zeta\ll 1:$ $\displaystyle\qquad w_{3}=w_{1}\quad,\quad
w_{4}=w_{2}\,.$ (101)

These formulae can be analytically continued away from $\zeta$ small, where
the periods $w_{1}$, $w_{2}$ can be written as linear combinations of $w_{3}$,
$w_{4}$, related to those in the original region by modular transformations.

The same formulae are obtained by writing the curve in the cubic Weiertrass
form

$w^{2}=\prod_{i=1}^{3}(x-e_{i})=x^{3}+a_{2}x+a_{3}\,,$ (102)

where now

$\zeta={e_{12}\over e_{13}}\,,\ \qquad J={4a^{3}\over 4a^{3}+27b^{2}}\,.$
(103)

###  A.2 SU(2) gauge theories with matter

The curves of SU(2) gauge theories with fundamental matter of equal masses
will be written as

$y^{2}+P(x)y+{\Lambda^{4-N_{f}}\over 4}\,(x-m)^{N_{f}}=0\,.$ (104)

The branch cuts are located at

$w(x)^{2}=P(x)^{2}-\Lambda^{4-N_{f}}\,(x-m)^{N_{f}}=\prod_{i=1}^{4}(x-e_{i})=\sum_{n=0}^{4}d_{n}\,x^{n}\,,$
(105)

The function $P(x)$ for $N_{f}\leq 3$ is given by

$P(x)=\left\\{\begin{array}[]{cc}x^{2}-u&N_{f}=0,1\\\
x^{2}-u+{\Lambda^{2}\over 4}&N_{f}=2\\\ x^{2}-u+{\Lambda\over
4}(x-3m)&N_{f}=3\,,\\\ \end{array}\right.$ (106)

We will compute the periods using the permutation invariant variables $D$ and
$\Delta$, so we write

$\displaystyle{\partial a(u)\over\partial u}$ $\displaystyle=$
$\displaystyle\oint_{\alpha}{\partial\lambda\over\partial u}={1\over 2\pi
i}\oint_{\alpha}{dx\over w(x)}=w_{1}\,,$ $\displaystyle{\partial
a_{D}(u)\over\partial u}$ $\displaystyle=$
$\displaystyle\oint_{\alpha}{\partial\lambda\over\partial u}={1\over 2\pi
i}\oint_{\beta}{dx\over w(x)}=w_{2}\,,$ (107)

with $w_{1}$, $w_{2}$ given by (92) and (94). The parameter $u$ is chosen such
that the SW prepotential $\lambda$ behaves at large $x$ as

$-2\pi{\rm i}\lambda=x{d\log y(x)\over
dx}\approx\sum_{n=0}^{\infty}{\left\langle{\rm
tr}\varphi^{n}\right\rangle\over x^{n}}\approx 2+{2u\over x^{2}}+\ldots$ (108)

leading to $u={\textstyle\frac{1}{2}}{\rm tr}\varphi^{2}$. The weak coupling
expansion of periods can alternatively be obtained via localization with
instanton-counting parameter $q$ related to $\Lambda$ via

$4q=\Lambda^{4-N_{f}}\,.$ (109)

####  A.2.1 $\boldsymbol{N_{f}=0}$ fundamentals

Let us consider first the pure gauge theory. The SW curve becomes

$w(x)^{2}=(x^{2}-u)^{2}-\Lambda^{4}\,,$ (110)

giving the invariants

$\displaystyle D$ $\displaystyle=$ $\displaystyle 4(4u^{2}-3\Lambda^{4})\,,$
$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
256\Lambda^{8}(u^{2}-\Lambda^{4})\,,$ (111)

Combining (107) with (101) and (99), one finds the weak coupling expansion

$\displaystyle{\partial a\over\partial u}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{u}}\left(\frac{1}{2}+{3\Lambda^{4}\over
32u^{2}}+{105\Lambda^{8}\over 2048u^{4}}+\ldots\right)\,.$ (112)

Integrating over $u$, inverting to find $u(a)$, and evaluating the second
period one finds

$\displaystyle a(u)$ $\displaystyle=$ $\displaystyle\sqrt{u}-{\Lambda^{4}\over
16u^{3\over 2}}-{15\Lambda^{8}\over 1024u^{7\over 2}}-{105\Lambda^{12}\over
16384u^{11\over 2}}+\ldots\,,$ $\displaystyle u(a)$ $\displaystyle=$
$\displaystyle a^{2}+{\Lambda^{4}\over 2^{3}a^{2}}+{5\Lambda^{8}\over
2^{9}a^{6}}+{9\Lambda^{12}\over 2^{12}\,a^{10}}+\ldots\,,$ $\displaystyle{\cal
F}_{0}(a)$ $\displaystyle=$ $\displaystyle
a^{2}\log\left(64{a^{4}\over\Lambda^{4}}\right)-6a^{2}-{\Lambda^{4}\over
8a^{2}}-{5\Lambda^{8}\over 1024\,a^{6}}-{3\Lambda^{12}\over
4096\,a^{10}}+\ldots\,.$ (113)

The same formula follows from the Nekrasov partition function

${\cal F}_{\rm
loc}={-}\frac{2q}{\left(4a^{2}{-}\epsilon^{2}\right)}{-}\frac{q^{2}\left(20a^{2}{+}7\epsilon_{1}^{2}{+}7\epsilon_{2}^{2}{+}16\epsilon_{1}\epsilon_{2}\right)}{\left(4a^{2}{-}\epsilon^{2}\right){}^{2}\left(4a^{2}{-}(2\epsilon_{1}{+}\epsilon_{2})^{2}\right)\left(4a^{2}{-}(\epsilon_{1}{+}2\epsilon_{2})^{2}\right)}+\ldots$
(114)

sending $\epsilon_{1,2}$ to zero and setting $4q=\Lambda^{4}$.

Using the expansions above one can check the relations Nakajima:2003uh

$\displaystyle u$ $\displaystyle=$ $\displaystyle\frac{\theta_{3}^{4}(q_{\rm
IR})+\theta_{2}^{4}(q_{\rm IR})}{\theta_{3}^{2}(q_{\rm
IR})\theta_{2}^{2}(q_{\rm IR})}\Lambda^{2}\,,$ $\displaystyle\frac{\partial
u}{\partial a}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}\Lambda}{\theta_{3}(q_{\rm IR})\theta_{2}(q_{\rm
IR})}\,,$ (115)

where the $\theta_{i}(q_{\rm IR})$ are the standard theta functions and
$q_{\rm IR}=e^{\pi{\rm i}\partial a_{D}/\partial a}$.

####  A.2.2 $\boldsymbol{N_{f}=1}$ fundamentals

Next we consider the $N_{f}=1$ theory. The curve becomes

$w(x)^{2}=\left(x^{2}-u\right)^{2}-\Lambda^{3}(x-m)\,.$ (116)

The invariants read

$\displaystyle D$ $\displaystyle=$ $\displaystyle 4(4u^{2}+3m\Lambda^{3})\,,$
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\Lambda^{6}\left(-27\Lambda^{6}+256\Lambda^{3}m^{3}+256m^{2}u^{2}-288\Lambda^{3}mu-256u^{3}\right)\,.$
(117)

At weak coupling, combining (107) with (101) and (99), one finds

$\displaystyle a(u)$ $\displaystyle=$
$\displaystyle\sqrt{u}+\frac{\Lambda^{3}m}{16u^{3/2}}+\frac{3\Lambda^{6}\left(u-5m^{2}\right)}{1024u^{7/2}}-\frac{35\Lambda^{9}\left(mu-3m^{3}\right)}{16384u^{11/2}}+\ldots\,,$
$\displaystyle u(a)$ $\displaystyle=$ $\displaystyle
a^{2}-\frac{\Lambda^{3}m}{8a^{2}}+\frac{\Lambda^{6}\left(5m^{2}-3a^{2}\right)}{512a^{6}}+\frac{\Lambda^{9}\left(7a^{2}m-9m^{3}\right)}{4096a^{10}}{+}\ldots\,,$
$\displaystyle{\cal F}_{0}(a)$ $\displaystyle=$ $\displaystyle
a^{2}\log\left(-64{\rm i}{a^{4}\over\Lambda^{4}}\right)-{9a^{2}\over
2}-{\textstyle\frac{1}{2}}(a+m)^{2}\log\frac{(a+m)}{\Lambda}-{\textstyle\frac{1}{2}}(a-m)^{2}\log\frac{(a-m)}{\Lambda}$
(118)
$\displaystyle+\frac{\Lambda^{3}m}{8a^{2}}+\frac{\Lambda^{6}\left(3a^{2}-5m^{2}\right)}{1024a^{6}}+\frac{\Lambda^{9}m\left(9m^{2}-7a^{2}\right)}{12288a^{10}}+\ldots\,.$

The same formula follows from the partition function computed using
localization

${\cal F}_{\rm
loc}=\frac{2mq}{4a^{2}-\epsilon^{2}}+\frac{q^{2}\left(3\left(4a^{2}-\epsilon^{2}\right)^{2}-4m^{2}\left(20a^{2}+7\epsilon_{1}^{2}+7\epsilon_{2}^{2}+16\epsilon_{1}\epsilon_{2}\right)\right)}{4\left(4a^{2}-\epsilon^{2}\right)^{2}\left(4a^{2}-\left(2\epsilon_{1}+\epsilon_{2}\right){}^{2}\right)\left(4a^{2}-\left(\epsilon_{1}+2\epsilon_{2}\right){}^{2}\right)}$
(119)

sending $\epsilon_{1,2}$ to zero and setting $4q=\Lambda^{3}$.

At strong coupling, near the AD conformal point $u_{*}={3\Lambda^{2}\over 4}$,
$m_{*}=-{3\Lambda\over 4}$, one finds

$\Delta\approx-432\Lambda^{8}(u-u_{*})^{2}\,\qquad,\qquad
D=24\Lambda^{2}\,(u-u_{*})\qquad,\qquad J=-{128(u-u_{*})\over
27\Lambda^{2}}\,.$ (120)

Expanding (99) in this limit we get

$f_{i}\sim D^{-{1\over 4}}\,J^{1\over 12}\sim(u-u_{*})^{-\frac{1}{6}}\,.$
(121)

Integrating over $u$ one finds

$\displaystyle a_{D}\sim a-a_{*}$ $\displaystyle\sim$
$\displaystyle(u-u_{*})^{5\over 6}\qquad,\qquad\tau={a_{D}\over
a-a_{*}}=e^{\pi{\rm i}\over 3}$ (122)

where we used the fact that $J=0$ for $\tau=e^{\pi{\rm i}\over 3}$ up to
SL$(2,\mathbb{Z})$ transformations.

####  A.2.3 $\boldsymbol{N_{f}=2}$ fundamentals

Next we consider the $N_{f}=2$ theory with equal masses. The curve becomes

$w(x)^{2}=\left(x^{2}-u+{\Lambda^{2}\over
4}\right)^{2}-\Lambda^{2}(x-m)^{2}\,.$ (123)

The invariants read

$\displaystyle D$ $\displaystyle=$ $\displaystyle
16u^{2}-12m^{2}\Lambda^{2}-4u\Lambda^{2}+\Lambda^{4}\,,$ $\displaystyle\Delta$
$\displaystyle=$ $\displaystyle
16\Lambda^{4}(u^{2}-m^{2}\Lambda^{2})(4u-4m^{2}-\Lambda^{2})^{2}\,.$ (124)

At weak coupling, combining (107) with (101) and (99), one finds

$\displaystyle a(u)$ $\displaystyle=$
$\displaystyle\sqrt{u}{-}\frac{\Lambda^{2}\left({u}+m^{2}\right)}{16u^{3/2}}{-}\frac{3\Lambda^{4}\left(5m^{4}{+}2m^{2}u{+}u^{2}\right)}{1024u^{7/2}}{-}\frac{5\Lambda^{6}\left(21m^{6}{+}7m^{4}u{+}3m^{2}u^{2}{+}u^{3}\right)}{16384u^{11/2}}+\ldots\,,$
$\displaystyle u(a)$ $\displaystyle=$ $\displaystyle
a^{2}{+}{a^{2}{+}m^{2}\over
8a^{2}}\Lambda^{2}+{a^{4}{-}6a^{2}m^{2}{+}5m^{4}\over
512a^{6}}\Lambda^{4}{+}{5a^{4}{-}14a^{2}m^{2}{+}9m^{4}\over
4096a^{10}}m^{2}\Lambda^{6}{+}\ldots\,,$ $\displaystyle{\cal F}_{0}(a)$
$\displaystyle=$ $\displaystyle
a^{2}\log\left(64{a^{4}\over\Lambda^{4}}\right)-3a^{2}-(a+m)^{2}\log\frac{(a+m)}{\Lambda}-(a-m)^{2}\log\frac{(a-m)}{\Lambda}$
(125) $\displaystyle{-}{\Lambda^{2}(a^{2}{+}m^{2})\over
8a^{2}}{-}{\Lambda^{4}(5m^{4}{-}6a^{2}m^{2}{+}a^{4})\over
1024\,a^{6}}+\ldots\,.$

The same formula follows from the partition function computed using
localization

${\cal F}_{\rm
loc}=-\frac{q\left(4\left(a^{2}{+}m^{2}\right){-}\epsilon^{2}\right)}{2\left(4a^{2}{-}\epsilon^{2}\right)}{-}\frac{q^{2}\left(\left(4a^{2}{-}\epsilon^{2}\right)^{2}\left({-}4a^{2}{+}24m^{2}{+}\epsilon^{2}\right){-}16m^{4}\left(20a^{2}{+}7\epsilon_{1}^{2}{+}7\epsilon_{2}^{2}{+}16\epsilon_{1}\epsilon_{2}\right)\right)}{16\left(4a^{2}{-}\epsilon^{2}\right)^{2}\left(4a^{2}{-}\left(2\epsilon_{1}{+}\epsilon_{2}\right){}^{2}\right)\left(4a^{2}{-}\left(\epsilon_{1}{+}2\epsilon_{2}\right){}^{2}\right)}$
(126)

sending $\epsilon_{1,2}$ to zero and setting $4q=\Lambda^{2}$.

At strong coupling, near the AD conformal point $u_{*}={\Lambda^{2}\over 2}$,
$m_{*}={\Lambda\over 2}$, one finds

$\Delta\approx 256\,\Lambda^{6}\,(u-u_{*})^{3}\,\qquad,\qquad
D=12\Lambda^{2}\,(u-u_{*})\qquad,\qquad J=1\,.$ (127)

Expanding (99) in this limit we get

$f_{i}\sim D^{-{1\over 4}}\,J^{1\over 12}\sim(u-u_{*})^{-\frac{1}{4}}\,.$
(128)

Integrating over $u$ one finds

$\displaystyle a_{D}\sim a-a_{*}$ $\displaystyle\sim$
$\displaystyle(u-u_{*})^{3\over 4}\qquad,\qquad\tau={a_{D}\over a-a_{*}}={\rm
i}$ (129)

where we used the fact that $J=1$ for $\tau={\rm i}$ up to SL$(2,\mathbb{Z})$
transformations.

####  A.2.4 $\boldsymbol{N_{f}=3}$ fundamentals

Finally we consider the $N_{f}=3$ theory with equal masses. The curve becomes

$w(x)^{2}=\left(x^{2}-u+{\Lambda\over 4}(x-3m)\right)^{2}-\Lambda(x-m)^{3}\,.$
(130)

The invariants read

$\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{4}}{256}-\frac{3\Lambda^{3}m}{8}+\frac{1}{2}\Lambda^{2}\left(9m^{2}-2u\right)+12\Lambda
m\left(m^{2}+u\right)+16u^{2}\,,$ (131) $\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{2}}{8}\left(2m^{2}-\Lambda
m-2u\right)^{3}\left(256\Lambda m^{3}-3\Lambda^{2}m^{2}+96\Lambda
mu+256u^{2}-\Lambda^{2}u\right)\,.$

At weak coupling, combining (107) with (101) and (99), one finds

$\displaystyle a(u)$ $\displaystyle=$ $\displaystyle\sqrt{u}+\frac{\Lambda
m\left(m^{2}+3u\right)}{16u^{3/2}}-\frac{\Lambda^{2}\left(15m^{6}+27m^{4}u+21m^{2}u^{2}+u^{3}\right)}{1024u^{7/2}}+\ldots\,,$
$\displaystyle u(a)$ $\displaystyle=$ $\displaystyle a^{2}-\frac{\Lambda
m\left(3a^{2}+m^{2}\right)}{8a^{2}}+\frac{\Lambda^{2}\left(a^{2}-m^{2}\right)^{2}\left(a^{2}+5m^{2}\right)}{512a^{6}}{+}\ldots\,,$
$\displaystyle{\cal F}_{0}(a)$ $\displaystyle=$ $\displaystyle
a^{2}\log\left(-64{{\rm i}a^{4}\over\Lambda^{4}}\right)-{3a^{2}\over
2}-{\textstyle\frac{3}{2}}(a+m)^{2}\log\frac{(a+m)}{\Lambda}-{\textstyle\frac{3}{2}}(a-m)^{2}\log\frac{(a-m)}{\Lambda}$
(132) $\displaystyle+\frac{\Lambda
m^{3}}{8a^{2}}-\frac{\Lambda^{2}m^{2}\left(3a^{4}-9a^{2}m^{2}+5m^{4}\right)}{1024a^{6}}+\ldots\,.$

The same formula follows from the partition function computed using
localization

${\cal F}_{\rm
loc}=q\frac{(3m-\epsilon)(4a^{2}-\epsilon^{2})+4m^{3}}{2\left(4a^{2}{-}\epsilon^{2}\right)}+\ldots$
(133)

sending $\epsilon_{1,2}$ to zero and setting $4q=\Lambda$.

At strong coupling, near the AD conformal point $u_{*}={5\Lambda^{2}\over
64}$, $m_{*}=-{\Lambda\over 8}$, one finds

$\Delta\approx-27\Lambda^{4}(u-u_{*})^{4}\,\qquad,\qquad
D=16\,(u-u_{*})^{2}\qquad,\qquad J=-{16384(u-u_{*})^{2}\over
729\Lambda^{4}}\,.$ (134)

Expanding (99) in this limit we get

$f_{i}\sim D^{-{1\over 4}}\,J^{1\over 12}\sim(u-u_{*})^{\frac{1}{3}}\,.$ (135)

Integrating over $u$ one finds

$\displaystyle a_{D}\sim a-a_{*}$ $\displaystyle\sim$
$\displaystyle(u-u_{*})^{2\over 3}\qquad,\qquad\tau={a_{D}\over
a-a_{*}}=e^{\pi{\rm i}\over 3}$ (136)

where we used the fact that $J=0$ for $\tau=e^{\pi{\rm i}\over 3}$ up to
SL$(2,\mathbb{Z})$ transformations.

##  Appendix B Conformal block recursion relations

In equation (88) we mentioned that the derivatives of conformal blocks can be
written as a positive factor times a polynomial. This property can be checked
rather easily. First let us take the explicit expression of $g_{\Delta,\ell}$
in terms of hypergeometric functions from (80) with (81a). The derivatives
with order higher than two acting on $k^{s}_{\beta}(z)$ can be recast as a
first derivative or the function itself times polynomials, thanks to the
hypergeometric equation. Then it suffices to do a Pade approximation of
$k^{s}_{\beta}(1/2)$ and $\partial_{z}k^{s}_{\beta}(1/2)$. After this, it can
be explicitly checked that the denominators will turn out to be positive for
all unitary values of $\Delta$. See for example Poland:2011ey .

Let us now motivate the existence of such an approximation in general.
Conformal blocks can be seen as meromorphic functions of $\Delta$ and
representation theory can be used to predict their pole structure. Indeed
their poles correspond to null states that arise in the degenerate Verma
modules $\mathcal{V}_{\Delta,\ell}$ when $\Delta$ assumes specific non-unitary
values. The presence of these poles can be intuitively understood from the
schematic definition of a conformal block as a projector onto a given
conformal multiplet inserted inside a four-point function

$\frac{g_{\Delta,\ell}(z,{\bar{z}})}{(x_{12}^{2})^{\Delta_{\phi}}(x_{34}^{2})^{\Delta_{\phi}}}=\sum_{n,m=0}^{\infty}\big{\langle}\phi(x_{1})\phi(x_{2})\big{|}\Psi_{n}^{\Delta,\ell}\big{\rangle}\,\left(\big{\langle}\Psi_{n}^{\Delta,\ell}\big{|}\Psi_{m}^{\Delta,\ell}\big{\rangle}\right)^{-1}\,\big{\langle}\Psi_{m}^{\Delta,\ell}\big{|}\phi(x_{3})\phi(x_{4})\big{\rangle}\,,$
(137)

where $|\Psi_{0}^{\Delta,\ell}\rangle$ is the state corresponding to the
conformal primary of dimension $\Delta$ and spin $\ell$ and $n,m$ enumerate
its descendants. The factor in the middle is the inverse of the matrix of two-
point functions. If a certain Verma module $\mathcal{V}_{\Delta,\ell}$ has a
null descendant — which can only happen for non-unitary values of $\Delta$ and
$\ell$ — the matrix of two-point functions will have a zero eigenvalue, thus
leading to a pole in (137). These poles have been classified in
Penedones:2015aga . The residue of each pole is a sum over all the descendants
of the singular state, which means that it can be written as a conformal
blocks with shifted parameters, times possibly a coefficient $R_{*}$

$g_{\Delta,\ell}(z,{\bar{z}})\quad\underset{\Delta\to\Delta^{*}}{\longrightarrow}\quad\frac{R_{*}\,g_{\Delta^{*},\ell^{*}}(z,{\bar{z}})}{\Delta-\Delta^{*}}\,.$
(138)

With this property one can approximate the blocks very efficiently using a
recursion relation. First we remove the essential singularity in $\Delta$ by
defining a block $h_{\Delta,\ell}$ as follows

$g_{\Delta,\ell}(z,{\bar{z}})=r^{\Delta}\,h_{\Delta,\ell}(z,{\bar{z}})\,,$
(139)

where $r$ is related to the cross ratios $z$ and ${\bar{z}}$ as follows

$r=\sqrt{|\rho\bar{\rho}|}\,,\qquad\rho=\frac{z}{(1+\sqrt{1-z})^{2}}\,,\qquad\bar{\rho}=\frac{{\bar{z}}}{(1+\sqrt{1-{\bar{z}}})^{2}}\,.$
(140)

Now $h_{\Delta,\ell}$, if regarded as a complex function of $\Delta$, is
meromorphic and it approaches a constant at infinity, which means that it is
fully specified by its poles

$h_{\Delta,\ell}(z,{\bar{z}})=h_{\infty,\ell}(z,{\bar{z}})+\sum_{\mathrm{poles}\,\Delta_{A}^{*}}\frac{R_{A}\,r^{n_{A}}\,h_{\Delta^{*}_{A},\ell^{*}_{A}}(z,{\bar{z}})}{\Delta-\Delta^{*}_{A}}+\sum_{\mathrm{double\,poles}\,\Delta_{B}^{*}}\frac{R_{B}\,r^{n_{B}}\,h_{\Delta^{*}_{B},\ell^{*}_{B}}(z,{\bar{z}})}{(\Delta-\Delta^{*}_{B})^{2}}\,.$
(141)

The sum over double poles appears only for CFTs in even spacetime dimensions.
Triple and higher poles never appear. The precise expressions for
$h_{\infty,\ell}$, $R_{A}$, $\Delta_{A}^{*}$, $\ell_{A}^{*}$ and $n_{A}$ are
given in Penedones:2015aga . The values of the poles $\Delta^{*}_{B}$ are also
available in Penedones:2015aga  (where it can be seen that two types of poles
overlap) but the residues are unpublished. The detailed recursion relation is
nevertheless implemented in an openly available software for all
dimensions.212121Available at gitlab.com/bootstrapcollaboration/scalar_blocks.
One can then truncate the sum by keeping only a finite number of poles. Noting
that, since $n_{A}$ is always at least $2$, the sum is suppressed by a factor
of $r^{2}$, which, at the crossing symmetric point, is about $0.029$. This
allows us to solve the system iteratively and obtain a rapidly converging
power series in $r$. This type of recursion relation first appeared in the
context of two-dimensional CFTs Zamolodchikov:1984eqp ; Zamolodchikov:1987eqp
but was later extended to general dimensions in Kos:2013tga . See also
Kos:2014bka  for a more detailed exposition.

Now that we have such a rational approximation of the conformal blocks, it is
easy to take derivatives. Letting $r_{*}=3-2\sqrt{2}$ being the value of $r$
at $z={\bar{z}}=1/2$ we have

$\displaystyle\partial_{z}^{n}\partial_{\bar{z}}^{m}\,g_{\Delta,\ell}\big{(}\tfrac{1}{2},\tfrac{1}{2}\big{)}$
$\displaystyle=r_{*}^{\Delta}\Bigg{(}q^{m,n}_{\ell}(\Delta)+\sum_{\mathrm{poles}\,\Delta_{A}^{*}}\frac{a_{\ell,A}^{m,n}}{\Delta-\Delta^{*}_{A}}\Bigg{)}$
(142)
$\displaystyle=\frac{r_{*}^{\Delta}}{\prod_{\mathrm{poles}\,\Delta_{A}^{*}}(\Delta-\Delta_{A}^{*})}\,p^{m,n}_{\ell}(\Delta)\,,$

for some polynomial $q^{m,n}_{\ell}$ and some coefficients $a_{\ell,A}^{m,n}$.
In the second line we simply took a common denominator. As we have explained
earlier, the origin of the poles $\Delta_{A}^{*}$ guarantees that, for unitary
values of $\Delta$, the denominator is always a positive function.

In order to have very precise expressions for the blocks, one would like to
keep many poles. If the polynomials are too big, however, the semidefinite
programming problem becomes more expensive. A good trade-off between precision
and performance is obtained by computing the blocks at very high recursion
order and then Padé approximating the obtained expression to a rational
function with fewer poles. This is known as the pole-shifting procedure and it
was introduced in Kos:2013tga .

##  Appendix C Crossing equations of the system of mixed correlators

In this appendix we show the crossing equations for the system of correlators
involving two distinct chiral fields $\phi_{r_{1}}$, $\phi_{r_{2}}$ and their
antichiral partners. Before writing the crossing vectors we need to introduce
the blocks for unequal external operators. In order not to clutter the
equations with too many subscripts, we depart from the notation in the main
text and use $g$ in place of $\mathcal{G}^{s}$, $\tilde{\mathcal{G}}$ in place
of $\mathcal{G}^{t}$ and $\mathcal{G}$ in place of $\mathcal{G}^{u}$. This is
also consistent with previous works Lemos:2015awa . With this notation, the
generalizations of (80) read

$\displaystyle g^{(r,r^{\prime})}_{\Delta,\ell}(z,{\bar{z}})$
$\displaystyle=\frac{z{\bar{z}}}{z-{\bar{z}}}\bigl{(}k^{s}_{\Delta+\ell}(z)k^{s}_{\Delta-\ell-2}({\bar{z}})-z\leftrightarrow{\bar{z}}\bigr{)}\,,$
(143) $\displaystyle\mathcal{G}^{(r)}_{\Delta,\ell}(z,{\bar{z}})$
$\displaystyle=\frac{z{\bar{z}}}{z-{\bar{z}}}\bigl{(}k^{u}_{\Delta+\ell}(z)k^{u}_{\Delta-\ell-2}({\bar{z}})-z\leftrightarrow{\bar{z}}\bigr{)}\,,$
$\displaystyle\tilde{\mathcal{G}}^{(r)}_{\Delta,\ell}(z,{\bar{z}})$
$\displaystyle=\frac{z{\bar{z}}}{z-{\bar{z}}}\bigl{(}k^{t}_{\Delta+\ell}(z)k^{t}_{\Delta-\ell-2}({\bar{z}})-z\leftrightarrow{\bar{z}}\bigr{)}\,,$

where now the $k$ functions are defined as

$\displaystyle k^{s}_{\beta}(z)$
$\displaystyle=z^{\frac{\beta}{2}}\,{}_{2}F_{1}\left(\frac{\beta-r}{2},\frac{\beta+r^{\prime}}{2};\beta;z\right)\,,$
(144) $\displaystyle k^{u}_{\beta}(z)$
$\displaystyle=z^{\frac{\beta}{2}}\,{}_{2}F_{1}\left(\frac{\beta-r}{2},\frac{\beta-r+4}{2};\beta+2;z\right)\,,$
$\displaystyle k^{t}_{\beta}(z)$
$\displaystyle=z^{\frac{\beta}{2}}\,{}_{2}F_{1}\left(\frac{\beta-r}{2},\frac{\beta+r}{2};\beta+2;z\right)\,.$

Next we construct the crossing functions which generalize (84)

$\displaystyle F^{ijkl}_{\pm,\Delta,\ell}(z,{\bar{z}})$
$\displaystyle=v^{\frac{r_{j}+r_{k}}{2}}g^{(r_{ij},r_{kl})}_{\Delta,\ell}(z,{\bar{z}})\pm
u^{\frac{r_{j}+r_{k}}{2}}g^{(r_{ij},r_{kl})}_{\Delta,\ell}(1-z,1-{\bar{z}})\,,$
(145a) $\displaystyle\mathcal{F}^{ijkl}_{\pm,\Delta,\ell}(z,{\bar{z}})$
$\displaystyle=v^{\frac{r_{j}+r_{k}}{2}}\mathcal{G}^{(r_{ij})}_{\Delta,\ell}(z,{\bar{z}})\pm
u^{\frac{r_{j}+r_{k}}{2}}\mathcal{G}^{(r_{ij})}_{\Delta,\ell}(1-z,1-{\bar{z}})\,,$
(145b)
$\displaystyle\tilde{\mathcal{F}}^{ijkl}_{\pm,\Delta,\ell}(z,{\bar{z}})$
$\displaystyle=v^{\frac{r_{j}+r_{k}}{2}}\tilde{\mathcal{G}}^{(r_{ij})}_{\Delta,\ell}(z,{\bar{z}})\pm
u^{\frac{r_{j}+r_{k}}{2}}\tilde{\mathcal{G}}^{(r_{ij})}_{\Delta,\ell}(1-z,1-{\bar{z}})\,.$
(145c)

with $r_{ij}=r_{i}-r_{j}$ and $r_{i}$ is both the R-charge and the dimension
of $\phi_{r_{i}}$. The blocks $\mathcal{G}$ and $\tilde{\mathcal{G}}$ have
only one superscript because they will only appear with $r_{ij}=-r_{kl}$ and
$r_{ij}=r_{kl}$ respectively.

Let us abbreviate $\phi_{i}=\phi_{r_{i}}$ and
${\bar{\phi}}_{i}={\bar{\phi}}_{r_{i}}$. The crossing equation can then be
written as follows

$\displaystyle\sum_{\mathcal{O}\in\phi_{1}\times\phi_{2}}|\lambda_{\phi_{1}\phi_{2}\mathcal{O}}|^{2}\,\vec{V}^{r_{1}+r_{2}}_{\Delta,\ell}+\sum_{i=1,2}\sum_{\mathcal{O}\in\phi_{i}\phi_{i}}|\lambda_{\phi_{i}\phi_{i}\mathcal{O}}|^{2}\,\vec{V}^{2r_{i}}_{\Delta,\ell}$
(146)
$\displaystyle+\sum_{\mathcal{O}\in\phi_{1}\times{\bar{\phi}}_{2}}|\lambda_{\phi_{1}{\bar{\phi}}_{2}\mathcal{O}}|^{2}\,\vec{V}^{r_{1}-r_{2}}_{\Delta,\ell}+\sum_{\mathcal{O}\in\phi_{i}\times{\bar{\phi}}_{i}}(\lambda_{\phi_{1}{\bar{\phi}}_{1}\mathcal{O}}^{*}\;\lambda_{\phi_{2}{\bar{\phi}}_{2}\mathcal{O}}^{*})\cdot\vec{\mathcal{V}}^{\mathrm{neutral}}_{\Delta,\ell}\cdot\begin{pmatrix}\lambda_{\phi_{1}{\bar{\phi}}_{1}\mathcal{O}}\\\
\lambda_{\phi_{2}{\bar{\phi}}_{2}\mathcal{O}}\end{pmatrix}=$
$\displaystyle=-\vec{V}_{\mathds{1}}-\frac{1}{6\hskip 0.5ptc}\vec{V}_{T}\,,$

with the crossing vectors
$\vec{V}^{q}_{\Delta,\ell},\vec{V}^{\mathrm{neutral}}_{0,0},\vec{V}_{T}$ and
the crossing matrix $\vec{\mathcal{V}}^{\mathrm{neutral}}_{\Delta,\ell}$
defined below

$\displaystyle\small\vec{V}^{r_{1}+r_{2}}_{\Delta,\ell}=\left[\begin{array}[]{c}\mp(-1)^{\ell}F_{\pm,\Delta,\ell}^{2121}\\\
0_{2}\\\ \mp F_{\pm,\Delta,\ell}^{1221}\\\
0_{6}\end{array}\right]\,,\qquad\vec{V}^{2r_{1}}_{\Delta,\ell}=\left[\begin{array}[]{c}0_{6}\\\
\mp(-1)^{\ell}F_{\pm,\Delta,\ell}^{1111}\\\ 0_{4}\end{array}\right]\,,$ (147h)
$\displaystyle\vec{V}^{2r_{2}}_{\Delta,\ell}=\left[\begin{array}[]{c}0_{9}\\\
\mp(-1)^{\ell}F_{\pm,\Delta,\ell}^{2222}\\\
0\end{array}\right]\,,\qquad\vec{V}^{r_{1}-r_{2}}_{\Delta,\ell}=\left[\begin{array}[]{c}(-1)^{\ell}\tilde{\mathcal{F}}_{\pm,\Delta,\ell}^{2121}\\\
\mathcal{F}_{\pm,\Delta,\ell}^{1221}\\\ 0_{8}\end{array}\right]\,,$ (147o)
$\displaystyle\vec{\mathcal{V}}^{\mathrm{neutral}}_{\Delta,\ell}=\left[\begin{array}[]{c}\boldsymbol{0}_{2}\\\
\begin{pmatrix}0&\mp\frac{1}{2}\mathcal{F}_{\pm,\Delta,\ell}^{1122}\\\
\mp\frac{1}{2}\mathcal{F}_{\pm,\Delta,\ell}^{1122}&0\end{pmatrix}\\\
\begin{pmatrix}0&\frac{(-1)^{\ell}}{2}\tilde{\mathcal{F}}_{\pm,\Delta,\ell}^{1122}\\\
\frac{(-1)^{\ell}}{2}\tilde{\mathcal{F}}_{\pm,\Delta,\ell}^{1122}&0\end{pmatrix}\\\
\begin{pmatrix}(-1)^{\ell}\tilde{\mathcal{F}}_{\pm,\Delta,\ell}^{1111}&0\\\
0&0\end{pmatrix}\\\ \begin{pmatrix}\mathcal{F}_{-,\Delta,\ell}^{1111}&0\\\
0&0\end{pmatrix}\\\ \begin{pmatrix}0&0\\\
0&(-1)^{\ell}\tilde{\mathcal{F}}_{\pm,\Delta,\ell}^{2222}\end{pmatrix}\\\
\begin{pmatrix}0&0\\\ 0&\mathcal{F}_{-,\Delta,\ell}^{2222}\end{pmatrix}\\\
\end{array}\right]\,,\hskip 38.00008pt$ (147w)
$\displaystyle\vec{V}_{\mathds{1}}=\left[\begin{array}[]{c}0_{2}\\\
\mp\mathcal{F}^{1122}_{\pm,0,0}\\\ \tilde{\mathcal{F}}^{1122}_{\pm,0,0}\\\
\tilde{\mathcal{F}}^{1111}_{\pm,0,0}\\\ \mathcal{F}^{1111}_{-,0,0}\\\
\tilde{\mathcal{F}}^{2222}_{\pm,0,0}\\\
\mathcal{F}^{2222}_{-,0,0}\end{array}\right]\,,\qquad\vec{V}_{T}=\left[\begin{array}[]{c}0_{2}\\\
\mp r_{1}r_{2}\,\mathcal{F}^{1122}_{\pm,2,0}\\\
r_{1}r_{2}\,\tilde{\mathcal{F}}^{1122}_{\pm,2,0}\\\
r_{1}^{2}\,\tilde{\mathcal{F}}^{1111}_{\pm,2,0}\\\
r_{1}^{2}\,\mathcal{F}^{1111}_{-,2,0}\\\
r_{2}^{2}\,\tilde{\mathcal{F}}^{2222}_{\pm,2,0}\\\
r_{2}^{2}\,\mathcal{F}^{2222}_{-,0,0}\end{array}\right]\,.\hskip 22.0pt$
(147al)

In the above vectors we defined as $0_{n}$ a sequence of $n$ zeros and as
$\boldsymbol{0}_{n}$ a sequence of $n$ null two-by-two matrices. Entries with
a $\pm$ span two rows, one for each choice of sign.

For the allowed values of $\Delta$ and $\ell$ in each OPE refer to section
3.1. The vector $\vec{V}_{\mathds{1}}$ represents the contribution of the
identity and the vector $\vec{V}_{T}$ represents the contribution of the
stress tensor multiplet, namely $A_{2}\mkern
3.5mu\overline{\mkern-3.5muA\mkern-1.5mu}\mkern 1.5mu_{2}[0;0]_{2}^{(0;0)}$.
They are both particular cases of
$\vec{\mathcal{V}}^{\mathrm{neutral}}_{\Delta,\ell}$ dotted with the
appropriate OPE coefficients.

##  Appendix D Normalization of the operators

In this appendix, we show the operator normalization that we chose for out
computations. This will make the meaning of $\lambda_{ijk}$ unambiguous.
Scalar operators are normalized as

$\langle\mathcal{O}_{i}(x_{1}){\mkern
3.5mu\overline{\mkern-3.5mu\mathcal{O}\mkern-1.5mu}\mkern
1.5mu}_{j}(x_{2})\rangle=\frac{\delta_{ij}}{(x_{12}^{2})^{\Delta_{i}}}\,.$
(148)

Their OPE is fixed by defining the conformal blocks. Namely, we expand a four-
point function as

$\langle\mathcal{O}_{1}(x_{1})\cdots\mathcal{O}_{4}(x_{4})\rangle=\frac{\left(\frac{x_{24}^{2}}{x_{14}^{2}}\right)^{\frac{1}{2}\Delta_{12}}\left(\frac{x_{14}^{2}}{x_{13}^{2}}\right)^{\frac{1}{2}\Delta_{34}}}{(x_{12}^{2})^{\frac{1}{2}(\Delta_{1}+\Delta_{2})}(x_{34}^{2})^{\frac{1}{2}(\Delta_{3}+\Delta_{4})}}\,\sum_{\Delta,\ell}\lambda_{12(\Delta,\ell)}\lambda_{34(\Delta,\ell)}^{*}\,g^{\Delta_{12},\Delta_{34}}_{\Delta,\ell}(z,{\bar{z}})\,,$
(149)

where the function $g^{\Delta_{12},\Delta_{34}}_{\Delta,\ell}(z,{\bar{z}})$ is
defined in (143). A property that this block satisfies is

$g^{\Delta_{12},\Delta_{34}}_{\Delta,\ell}(z,z)\;\overset{z\to
0}{\longrightarrow}\;z^{\Delta}\hskip 1.0pt(\ell+1)\,.$ (150)

The choices (148) and (150) are enough to unambiguously determine the meaning
of $\lambda_{ijk}$.

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