responsible for eqs. (133) and (134). The required OPE data were presented in
Gaiotto:2013nva :

$\displaystyle\Delta_{\varepsilon}=2-\frac{2\epsilon}{3}+O(\epsilon^{2})\leavevmode\nobreak\
,\qquad
c_{\sigma\sigma\varepsilon}a_{\varepsilon}=-\frac{1}{8}+O(\epsilon)\leavevmode\nobreak\
.$ (137)

In fact, the full result (133), (134) is encoded in the leading transverse
spin correction. Indeed, plugging eq. (137) in eq. (23), we reproduce the
value of the anomalous dimension $\gamma_{s,0}=-\frac{1}{24s}\epsilon$.
Furthermore, the correction $b_{\textup{min}}$ in eq. (24) is
$O(\epsilon^{2})$, and indeed the square root of eq. (10) reduces to (134).
Despite the simplicity of the result, it is not obvious why the large $s$
expansion of the anomalous dimension should truncate at order $1/s$. We can
address the question by means of the inversion formula. By evaluating the
single block contribution eq. (78) with $q=2$, $d=4-\epsilon$ and the CFT data
in eqs. (135) and (137), we indeed get, for $s>0$,

$\gamma_{s,0}=-\frac{\epsilon}{24}\frac{1}{s+1}\,{}_{2}F_{1}(1,1,s+2,1)+O(\epsilon^{2})=-\frac{\epsilon}{24s}+O(\epsilon^{2})\leavevmode\nobreak\
.$ (138)

It is interesting to notice that each bulk collinear block contributes an
infinite series in $1/s$, and the final result emerges from infinitely many
exact cancellations. As before, in principle we must check the small $w$ limit
of the $\braket{\sigma\sigma}$ correlator to confirm that the leading-twist
spectrum (133), as found with the inversion formula, is complete. We could not
obtain this behavior analytically, however, a numerical analysis of the
correlator of Gaiotto:2013nva , suggests that for $w\to 0$ the
$\langle\sigma\sigma\rangle$ correlator behaves like $\sim\alpha_{r}w^{0}$,
where $\alpha_{r}$ is a coefficient that depends on the radius $r$. Thus
$s_{\star}=0$ and the inversion formula (67) only holds for $s>0$. We were not
able to find the contribution of a single scalar block to the OPE coefficient
in closed form. However, from the computation of the OPE coefficient as an
infinite sum, discussed in section 3.3, we can easily check that the same
cancellations are in place: this time, after appropriately including the
Jacobian in eq. (71), no contribution is left at order $\epsilon$. Therefore,
we also recover eq. (134). We can also predict the existence of the higher
transverse twist primaries, with $\widehat{\tau}\to\Delta_{\sigma}+2m$ at
large spin, whose OPE coefficients, for $m\neq 0$, are of order $\epsilon$ as
clear from the fact that (10) vanishes for $m>0$ and $\Delta_{\phi}$ at the
unitarity bound.

Let us conclude with some comments on the $\mathbb{Z}_{2}$ even defect
spectrum. In free theory, the leading transverse twist operators are bilinear
of the $\psi_{s}$, and all operators $\psi_{s_{1}}\psi_{s_{2}}$ with integer
transverse spin $s_{1}+s_{2}=s$ have the same transverse twist
$\widehat{\tau}=2-\epsilon$. This $\lfloor\frac{s+1}{2}\rfloor$-fold
degeneracy is lifted at the Wilson-Fisher fixed point, and we parametrize the
eigenvalues of the matrix of anomalous dimensions as follows:

$\widehat{\tau}_{s,j}=2+\epsilon\left(\delta_{s,j}-1\right)+O(\epsilon^{2})\,.$
(139)

In Gaiotto:2013nva , it was pointed out that the following accumulation points
exist at infinite transverse spin:

$\delta_{\infty,j}=-\frac{1}{12(2j-1)}\leavevmode\nobreak\ ,\qquad
j=1,2,\dots\,.$ (140)

The results of section 2 predict an additional accumulation point: the leading
transverse derivative of the energy operator $\varepsilon$, that is,

$\delta_{\infty,0}=\frac{1}{3}\leavevmode\nobreak\ .$ (141)

In fact, it is not hard to see that this accumulation point exists,323232We
thank D. Mazáč for sharing with us an elegant analytic proof of this fact. and
that furthermore eqs. (140) and (141) comprise all the anomalous dimensions of
this class of operators. As for the accumulation points (140), those are not
transverse derivatives of $\varepsilon$, and therefore we should expect their
OPE coefficient to be subleading at large $s$. We did not check this fact.
Both the lightcone bootstrap and the use of the inversion formula are
complicated by the presence of infinitely many bulk blocks contributing
already at order $\epsilon$, so we leave this analysis for future work.

###  5.6 On the sign of $\braket{T_{\mu\nu}}$

In this subsection, we observe a feature of the one-point function of the
stress tensor in the presence of a defect which is found in many examples. Let
us start from the holographic defect discussed in subsection 5.3, and in
particular from eq. (124). The attractive nature of gravity indicates that the
parameter ${\mathcal{C}}$ should be positive – see _e.g._ , eq. (116). From
eq. (124), we deduce that $a_{T}<0$. The same happens in all unitary defect
conformal field theories we are aware of: in particular, the Wilson lines in
free theories discussed in Billo:2016cpy , the supersymmetric Wilson line
briefly discussed in subsection 5.2, and the Renyi twist operator Hung:2014npa
; Bianchi:2015liz . Let us briefly explain why this is the case for the
latter. In Hung:2011nu , eq. (4.21), the following expression was derived:

$a_{T}(n)=-\frac{d}{d-1}n\left(\mathcal{E}(T_{0})-\mathcal{E}(T_{0}/n)\right)\,.$
(142)

Here, $n$ is the so-called replica index, $\mathcal{E}(T)$ is the energy
density of a thermal state at temperature $T$ on the $(d-1)$-dimensional
hyperbolic space $H^{d-1}$ of unit radius and $T_{0}=1/2\pi$. Stability of the
canonical ensemble implies that the specific heat $\frac{dE}{dT}$ is positive.
Here $E$ is the total energy, but since $\braket{T_{\mu\nu}}$ is constant on
this background, the same property is valid for $\mathcal{E}(T)$. We deduce
that $a_{T}<0$ if $n>1$. On the other hand, exactly at $n=1$ the defect
becomes trivial, and for $n<1$ the theory becomes non unitary: for instance,
the two-point function of the displacement operator Billo:2016cpy  has
negative norm $C_{D}<0$ for $n$ slightly smaller than 1, as it can be deduced
by the following equation mark0 ; Bianchi:2015liz ; Faulkner:2015csl :

$\left.\partial_{n}C_{D}\right|_{n=1}=\frac{2\pi^{2}}{d+1}\frac{C_{T}}{S_{d}^{2}}\,.$
(143)

The evidence put forward so far makes it plausible that $a_{T}\leq 0$ in all
unitary defect conformal field theories. It would be interesting to prove this
statement or look for a counter-example, but we leave this for future work.

##  6 Discussion and outlook

The main technical tool used in this paper is the inversion formula (67). The
formula expresses the defect spectrum and the OPE coefficients as an integral
over the discontinuity of the two-point function. The integral kernel is
analytic in $s$, which means that the spectrum of every defect CFT is
organized in Regge trajectories ${\widehat{\Delta}}(s)$, at least down to some
minimum spin $s_{\star}$. The value of $s_{\star}$ is finite as long as the
correlator is polynomially bounded in a certain double lightcone limit – see
fig. 6.

At large $s$, the generating function (68) localizes close to $\bar{z}=1$, and
is therefore dominated by the bulk-channel OPE limit of the discontinuity.
This fact puts on solid ground the result first obtained in section 2 via the
lightcone bootstrap: in every defect CFT, a subset of the trajectories
asymptote the spectrum of the trivial defect at large $s$ – see eq. (1). We
call transverse derivatives the operators on these trajectories. Since $s$ is
a global charge for the defect spectrum, it is interesting to contrast this
result with the large charge sector of an ordinary CFT Hellerman:2015nra ;
Monin:2016jmo ; Jafferis:2017zna . There, an operator with a large charge
creates in radial quantization a state with homogeneous charge and energy
density. The scaling of the low-lying spectrum with the charge is not linear,
and therefore incompatible with the presence of the transverse derivatives. In
fact, as we saw in section 4, the semiclassical picture in this case is very
different from a condensed matter phase. Since a defect CFT is non local,
there is no notion of energy density on the defect, and once seen from the
bulk, the transverse derivative breaks a spacetime rotational symmetry rather
than a global one.

Finite $s$ corrections to the spectrum and the OPE coefficients can in
principle be computed systematically, by applying the inversion formula block
by block, as a convergent expansion. In contrast to the lightcone bootstrap
where only an asymptotic expansion is obtained, by using the inversion formula
one gets a better and better approximation to the defect spectrum as more bulk
blocks are added. However, in practice we achieve analytic control only by
replacing the bulk-channel blocks with their expansion in powers of $z$.
Although this procedure is in general incorrect, it is sufficient to resum
part of the large $s$ expansion of the anomalous dimensions of the transverse
derivatives. In fact, if the discontinuity receives contributions only by a
finite number of blocks, the procedure yields the full spectrum. This happens
sometimes in a perturbative setup, as we saw in the examples and we further
remark in the next subsection.

The perturbative setting also highlights a difference between this inversion
formula and the inversion formula for the four-point function. The
discontinuity of a single logarithm, unlike the double discontinuity, does not
vanish. This has an immediate consequence for defects embedded in theories
with a slightly broken higher spin symmetry: the transverse derivatives
acquire anomalous dimension at leading order in the breaking parameter. We saw
this happening in subsection 5.5. The opposite is true in the case of the
four-point function, a fact recently exploited in Alday:2017zzv  to
efficiently apply large-spin perturbation theory order by order in the
$\epsilon$ expansion.

Some important aspects concerning the inversion formula remain to be
clarified. Firstly, we did not place an upper bound on $s_{\star}$. Theories
whose correlators are not polynomially bounded in the asymptotic region
mentioned earlier may not display Regge trajectories. It would be very
interesting to prove an upper bound, or to find a counter-example. A bound
cannot be proven in the same way as in Caron-Huot:2017vep , because the bulk
channel OPE is not positive, a fact that is also tied to the appearance of the
discontinuity in the formula, which has no definite sign, unlike the double
discontinuity of the Caron-Huot formula.

Secondly, it would be important to set up a procedure that works order by
order in the small $z$ expansion and at the same time does not spoil the
convergence of the formula. This problem is related to the necessity of
dealing with the towers of double- and multi-twist operators in the bulk
channel OPE. In the case of the four-point function, this issue has been
solved Simmons-Duffin:2016wlq ; Caron-Huot:2017vep .

#### Outlook

The results of this work suggest various future directions of research. It
would be interesting to study the large $s$ spectrum in other examples,
possibly beyond leading order in perturbation theory, or rather in strongly
coupled scenarios. For instance, in section 4 we pointed out that the spectrum
of a Wilson line at large $N$ supports operators whose transverse twist
measures the energy of a heavy-light quark pair – see eq. (87). It would be
interesting to explore this case with the inversion formula, and to study the
transverse derivatives as well. Notice that one-point functions of double
trace operators are suppressed at large $N$, so the inversion formula suggests
that the anomalous dimension of the transverse derivatives receives
contribution only from the exchange of single trace operators at leading order
in $1/N$.

From a technical point of view, the inversion formula (67) can be obtained via
manipulations that are much simpler than in the case of the four-point
function. The formula itself depends on a kernel explicitly known in closed
form. This makes the defect case a good playground, both for applications and
for generalizations of the formula. In particular, deriving the inversion
formula in the case of external spinning operators might be a doable task. For
this purpose, a method similar to the one presented in Karateev:2017jgd  may
be useful. Also in this last case, the presence of the transverse derivatives
at large $s$ is expected from the lightcone bootstrap analysis, which is
essentially unchanged. Their scaling dimensions should again approximate the
trivial defect, _i.e._ ,

$\widehat{\tau}\simeq\tau_{\phi}+2m\leavevmode\nobreak\ ,\qquad
s\to\infty\leavevmode\nobreak\ .$ (144)

Finally, one obvious direction of investigation concerns the inversion of the
bulk OPE, and it is work in progress. Under many respects, this problem is
more similar to the one solved by the inversion formula for the four-point
function Caron-Huot:2017vep . Indeed, the inversion of the bulk OPE would
yield the same spectrum, albeit associated to different OPE coefficients. This
inversion formula would recover bulk information from the defect OPE in the
limit when one of the bulk operators is light-like separated from the defect,
_i.e._ , the opposite limit to the one considered in 2. A preliminary
lightcone bootstrap analysis seems to show that the existence of double twist
operators is sufficient to satisfy the crossing constraint in this limit at
the leading order, so that no obstruction exists at this level for a CFT to
support a conformal defect. An inversion formula for the bulk OPE, combined
with (67), could also allow for an iterative procedure to obtain approximate
solutions to crossing symmetry starting from only a few operators, similarly
to what was done in Simmons-Duffin:2016wlq  for the $3d$ Ising model.
Furthermore, the crossed-channel – which in this case is the defect channel –
is positive, and this may provide a better control over the convergence of the
formula. It would also be interesting to see if the positivity of the defect
OPE can be used to answer the question raised in subsection 5.6 on the sign of
the one-point function of the stress tensor.

###### Acknowledgements.

We have greatly benefited from discussions with L. Bianchi, D. Mazáč, J.
Penedones, B. van Rees, S. Rychkov, V. Schomerus, E. Trevisani, S. Giombi, R.
Rattazzi. M.M. is especially grateful to D. Gaiotto, for crucial suggestions
in the early stage of this project. M.L., P.L. and M.M. thank the Galileo
Galilei Institute for Theoretical Physics for hospitality and the INFN for
partial support during the completion of this work during the workshop
“Conformal field theories and renormalization group flows in dimensions
$d>2$”. M.L., P.L. and M.M. thank ICTP-SAIFR in São Paulo for hospitality
during the Bootstrap 2017 workshop, and the Simons Collaboration on the Non-
perturbative Bootstrap for providing many stimulating workshops and
conferences during which this work carried out. S.S. is thankful to DESY
Hamburg and EPFL for hospitality and support for visits during the completion
of this work. We also thank the participants of the Simons Collaboration’s
kick off, “Bootstrap and Quantum Gravity” and “Numerical bootstrap” meetings
for discussions. M.L. was partially supported by the German Research
Foundation (DFG) via the Emmy Noether program “Exact results in Gauge
theories”. M.M. is supported by the Simons Foundation grant 488649 (Simons
collaboration on the non-perturbative bootstrap) and by the National Centre of
Competence in Research SwissMAP funded by the Swiss National Science
Foundation.

##  Appendix A Conformal blocks

Here we review the kinematics of a two-point function of bulk operators in the
presence of a flat defect of codimension $q$ in $d$ dimensions, following
Billo:2016cpy . Recall that we separate the space time indices
($\mu=0,\dots,d-1$) into two subsets, those orthogonal to the defect are
labeled by $i=0,\dots,q-1$ and those parallel to the defect by
$a=q,\dots,d-1$. The the two-point function of bulk identical scalars depends
on two conformally invariant cross-ratios, and we parametrize it by

$\braket{\phi(x_{1})\phi(x_{2})}=\frac{1}{(\lvert x^{\perp}_{1}\rvert\lvert
x^{\perp}_{2}\rvert)^{\Delta_{\phi}}}g(\chi,\eta)\,,$ (145)

where we use $x^{\perp}=\\{x^{i}\\}$ and $x^{\parallel}=\\{x^{a}\\}$, and the
cross-ratios are given by333333Here $\eta$ is what was called $\cos\phi$ in
Billo:2016cpy .

$\chi=\frac{\lvert x^{\parallel}_{12}\rvert^{2}+\lvert
x^{\perp}_{1}\rvert^{2}+\lvert x^{\perp}_{2}\rvert^{2}}{\lvert
x^{\perp}_{1}\rvert\lvert
x^{\perp}_{2}\rvert}=\frac{1+z\bar{z}}{(z\bar{z})^{1/2}}=\frac{1}{r}+r\,,\qquad\quad\eta=\frac{x_{1i}x_{2}^{i}}{\lvert
x^{\perp}_{1}\rvert\lvert
x^{\perp}_{2}\rvert}=\frac{z+\bar{z}}{2(z\bar{z})^{1/2}}=\frac{1}{2}\left(w+\frac{1}{w}\right)\,.$
(146)

As discussed in section 2 we work in Lorentzian signature with a space-like
defect, and with the geometry shown in fig. 1, so for convenience above we
have shown the relation between the cross-ratios $(\chi,\eta)$ and the
lightcone $(z,\bar{z})$ coordinates, as well as the radial coordinates $r$ and
$w$. The crossing equation reads (or equivalently (4) in $(z,\bar{z})$
coordinates)

$g(\chi,\eta)=\xi^{-\Delta_{\phi}}\sum_{O}c_{\phi\phi
O}\,a_{O}\,f_{\Delta,J}(\chi,\eta)=\sum_{\widehat{O}}(b_{\phi\widehat{O}})^{2}\,\widehat{f}_{\widehat{\tau},s}(\chi,\eta)\,,$
(147)

where we defined the transverse twist $\widehat{\tau}=\widehat{\Delta}-s$, and

$\xi=\chi-2\eta=\frac{(1-z)(1-\bar{z})}{(z\bar{z})^{1/2}}\,.$ (148)

The defect conformal blocks are known exactly Billo:2016cpy 343434The blocks
here differ from the ones of Billo:2016cpy  by a factor of $2^{s}$.

$\widehat{f}_{\widehat{\tau},s}(\chi,\eta)=\binom{s+\frac{q}{2}-2}{\frac{q}{2}-2}^{-1}\,C_{s}^{(q/2-1)}\left(\eta\right)\,_{2}F_{1}\left(\frac{\widehat{\Delta}}{2}+\frac{1}{2},\frac{\widehat{\Delta}}{2};\widehat{\Delta}+1-\frac{p}{2};\frac{4}{\chi^{2}}\right)\,,$
(149)

where $C_{n}^{(m)}\left(x\right)$ is a Gegenbauer polynomial, and changing
variables to $(z,\bar{z})$ the above block can be re-written as (5), after
using the following relation

$\binom{s+\frac{q}{2}-2}{\frac{q}{2}-2}^{-1}C_{s}^{q/2-1}\left(\frac{w}{2}+\frac{1}{2w}\right)=w^{-s}{}_{2}F_{1}\left(-s,\frac{q}{2}-1,2-\frac{q}{2}-s,w^{2}\right)\,.$
(150)

When $q$ is even, an order of limits ambiguity arises in the definition of the
hypergeometric function in (150), with the equality holding if we first take
$s$ to be integer, and then $q$ to be even. This prescription is always
assumed throughout the paper. The normalization is chosen such that, given a
leading contribution to the defect OPE of the kind

$\phi(x^{\mu})\sim b_{\phi\widehat{O}}\,\lvert
x^{\perp}\rvert^{-\Delta_{\phi}+\widehat{\Delta}-s}x^{i_{1}}\cdots
x^{i_{s}}\widehat{O}_{i_{1}\dots i_{s}}(x^{a})+\ldots\,,$ (151)

the contribution of $\widehat{O}$ to the two-point function is as in eq.
(147), provided that the two-point function of defect primaries is normalized
as

$\langle\widehat{O}^{i_{1}...i_{s}}(x_{1}^{a})\widehat{O}^{j_{1}...j_{s}}(x_{2}^{a})\rangle=2^{s}\frac{\mathcal{P}^{i_{1}...i_{s};j_{1}...j_{s}}}{(x_{12}^{2})^{\widehat{\Delta}}}\,.$
(152)

Here $\mathcal{P}$ is the projector onto symmetric and traceless tensors,
which can be defined in terms of the Todorov operator (see Billo:2016cpy  for
more details)

$\mathcal{P}^{i_{1}...i_{s};j_{1}...j_{s}}\equiv\frac{1}{s!\left(\frac{q}{2}-1\right)_{s}}D_{i_{1}}\dots
D_{i_{s}}w_{j_{1}}\dots w_{j_{s}},\\\
D_{i}=\bigg{(}\frac{q-2}{2}+w^{j}\frac{\partial}{\partial
w^{j}}\bigg{)}\frac{\partial}{\partial
w^{i}}-\frac{1}{2}w_{i}\frac{\partial^{2}}{\partial w^{j}\partial w_{j}}\,.$
(153)

Finally, we denote the defect OPE coefficient of the identity,
$b_{\phi\hat{\mathbb{1}}}$, by $a_{\phi}$, such that the one-point function of
a bulk primary of dimension $\Delta$ and spin $J$ is

$\langle\phi(x^{\mu},z^{\mu})\rangle=\frac{a_{\phi}}{|x^{\perp}|^{\Delta_{\phi}}}\left(\frac{(z^{i}x^{i})^{2}}{|x^{\perp}|^{2}}-z^{i}z^{i}\right)^{\frac{J}{2}}\,,$
(154)

where $z^{\mu}$ is an auxiliary null vector used to contract the indices of
the spin $J$ operator (see Billo:2016cpy  for how to recover the index
structure encoded in the polynomial dependence in $z$) .

##  Appendix B Convenient coordinates for the bulk channel OPE

Following Lauria:2017 , let us define the following change of variables:

$\rho=\frac{1-\sqrt{z}}{1+\sqrt{z}}\leavevmode\nobreak\
,\qquad\bar{\rho}=\frac{1-\sqrt{\bar{z}}}{1+\sqrt{\bar{z}}}\leavevmode\nobreak\
.$ (155)

The configuration of the operators in the $\rho$ coordinates is depicted in
fig. 6. The defect is now spherical, and crosses the $(\rho,\bar{\rho})$ plane
in two points. The configuration is of course very similar to the one
customarily used to describe a four-point function of local operators
Hogervorst:2013sma .

$(\rho,\bar{\rho})$$\bar{\rho}=-1$$\rho=1$$\rho=0$$\bar{\rho}=0$

Figure 6: The bulk channel $\rho$ coordinates defined in Lauria:2017 . The
defect is a sphere and crosses the plane at the location of the two red dots.
The operators are drawn in the time-like configuration that corresponds to
$0<z<1/\bar{z}<1$, that is the region of integration in eq. (67).

In the region $0<\rho,\bar{\rho}<1$ the bulk-channel OPE has the following
structure, for a scaling operator of dimension $\Delta$ and $SL(2,\mathbb{R})$
spin $J$:

$g(\rho,\bar{\rho})\sim\bar{\rho}^{\frac{\Delta-J}{2}}\rho^{\frac{\Delta+J}{2}}\leavevmode\nobreak\
.$ (156)

The integral in the Lorentzian inversion formula (67) extends over the region
$(0<\rho<1,-1<\bar{\rho}<0)$, corresponding to the configuration of the
operators drawn in figure 6. The bulk channel OPE in this region still has the
form (156), and is only modified by the phases picked up going past the
singularity at $\bar{\rho}=0$. Since the OPE is absolutely convergent in the
region $\rho,\bar{\rho}<1$, it still converges after the continuation.

The $w\to 0$ limit at fixed $r$ is the limit in which the operators reach the
upper and lower corners of the causal diamond of the defect.

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