figurec

#  Ironing out the crease

Nadav Drukker<EMAIL_ADDRESS>Department of Mathematics, King’s
College London,
London, WC2R 2LS, United Kingdom Maxime Trépanier<EMAIL_ADDRESS>Institute for Theoretical and Mathematical Physics (ITMP),
Moscow 119991, Russia

###### Abstract

The crease is a surface operator folded by a finite angle along an infinite
line. Several realisations of it in the 6d ${\cal N}=(2,0)$ theory are studied
here. It plays a role similar to the generalised quark-antiquark potential, or
the cusp anomalous dimension, in gauge theories. We identify a finite quantity
that can be studied despite the conformal anomalies ubiquitous with surface
operators and evaluate it in free field theory and in the holographic dual. We
also find a subtle difference between the infinite crease and its conformal
transform to a compact observable comprised of two glued hemispheres,
reminiscent of the circular Wilson loop. We prove by a novel application of
defect CFT techniques for the $SO(2,1)$ symmetry along the fold that the near-
BPS behaviour of the crease is determined as the derivative of the compact
observable with respect to its angle, as in the bremsstrahlung function. We
also comment about the lightlike limit of the crease in Minkowski space.

###### Contents

  1. 1 Introduction and summary
  2. 2 Free theory
    1. 2.1 The spherical crease
    2. 2.2 The crease singularity
    3. 2.3 The infinite crease
  3. 3 Holography
    1. 3.1 The setup
    2. 3.2 BPS case
    3. 3.3 Non-BPS case
    4. 3.4 Near-BPS expansion
    5. 3.5 Antiparallel planes limit
  4. 4 Lightlike crease
    1. 4.1 Analytic continuation
    2. 4.2 Solutions in lorentzian $AdS$
    3. 4.3 Global solutions
  5. 5 Defect CFT
    1. 5.1 Near sphere expansion
    2. 5.2 Near-BPS expansion
  6. 6 Conclusion
  7. A Infinite crease in the Poincaré patch
    1. A.1 Solution
    2. A.2 Calibration equations

##  1 Introduction and summary

In order to make progress on the 6d ${\mathcal{N}}=(2,0)$ theory we need to
either find avatars of the theory with lagrangian descriptions or use whatever
limited tools there are in the absence of a lagrangian to study the theory.
For examples of recent attempts in the former approach, see e.g. [1, 2, 3].
Here instead we follow the latter approach, extending our programme [4, 5, 6,
7, 8] focused on the most natural observables on the theory—the 2d surface
operators.

The simplest possible surfaces are the plane and sphere [9] which can preserve
the maximal amount of residual supersymmetry, 16 supercharges, as well as
conformal symmetry and overall the algebra $\mathfrak{osp}(4^{*}|2)^{2}$.
Those are very natural starting points for exploring general surface
operators, but contain little information on their own. The sphere suffers
from a conformal anomaly so its expectation value depends on its radius $R$

$\left\langle V_{S^{2}}\right\rangle\propto R^{4c-2a_{1}}\,,$ (1.1)

where $a_{1}$ and $c$ are the anomaly coefficients given below in (1.10).

One can view the plane as the worldsurface of an infinite straight string. A
natural question is to evaluate the potential between two strings, or the
plane-antiplane configuration, as was studied already in [10]. Amongst other
things, in this paper we enrich this configuration by allowing each plane to
associate to a different scalar field, which gives a continuous parameter
$\theta$ interpolating between the anti-parallel planes with $\theta=0$ and
the BPS parallel planes at $\theta=\pi$.

A further generalisation of the system is the crease: two half-planes joined
at an angle $\phi$ along a line (see figure 1). When $\phi=0$ this is a single
1/2 BPS plane and when $\phi\to\pi$ it approaches the antiparallel planes.

(a) The infinite crease

(b) The compact crease

Figure 1: On the left, the infinite crease with $\phi=2\pi/3$. On the right, a
representation of the same crease after a conformal transformation. Each of
the half planes is mapped to a hemisphere, here represented by the two arcs on
$S^{2}$ (since we cannot draw $S^{3}$). Each point along the arc represents a
circle, which shrinks at the equator ($z=0$ in the image) and is of maximal
radius 1 at the north pole, where it is also glued to the second hemisphere.

This is a clear analogue of the cusped Wilson loop or “generalised quark-
antiquark” system studied in [11, 12, 13, 14, 15, 16]. Another natural 2d
analogue of the cusp is a cone. Under a conformal transformation to
$S^{5}\times\mathbb{R}$ the cone becomes a cylinder which is a circle in
$S^{5}$ extending along the $\mathbb{R}$ direction. BPS versions of such
operators were studied in [17] and the limit of a BPS cylinder in
$\mathbb{R}^{6}$ in [8].

The crease configuration with the additional angle $\theta$ and several
related surfaces are the subject of this paper. Our goal is to define and
evaluate a meaningful physical quantity $U(\phi,\theta)$, analogous to the
generalised quark-antiquark potential or cusp anomalous dimension. This
quantity can be viewed as the potential between the two half planes, but on a
more abstract level, it is a finite observable of the ${\mathcal{N}}=(2,0)$
theory that can be (at least partially) evaluated. As we now have a good
handle on the anomaly coefficients, we are behoven to find more refined
quantities associated to surface operators and this is exactly such an
observable.

In the remainder of the introduction we define the quantity $U(\phi,\theta)$
and discuss its salient features. We then proceed to evaluate it in the
subsequent sections.

The observable we study has several realisations. First, as an infinite flat
crease between two half-planes parametrised by $\tau,\sigma\in\mathbb{R}$

$x^{\mu}=\begin{cases}(\tau,\sigma,0,0,0,0)\,,&\sigma\leq 0\,,\\\
(\tau,\sigma\cos\phi,\sigma\sin\phi,0,0,0)\,,&\sigma\geq 0\,.\end{cases}$
(1.2)

The surface also breaks the R-symmetry. It is natural to have along each half-
plane a uniform breaking, expressed by a unit 5-vector $n^{I}$ as

$n^{I}=\begin{cases}(1,0,0,0,0)\,,&\sigma\leq 0\,,\\\
(\cos\theta,\sin\theta,0,0,0)\,,&\sigma\geq 0\,.\end{cases}$ (1.3)

The 1/2 BPS plane is clearly at $\theta=\phi=0$. If $\theta=\pm\phi\neq 0$
this operator is 1/4 BPS and according to the classification of BPS surface
operators in [7] it is of Type-$\mathbb{R}$, since it has translation
invariance (it is also of Type-$\mathbb{H}$, which is utilised in Appendix
A.2). For $\theta\neq\pm\phi$ no supersymmetry is preserved, but still in all
cases the surface operator preserves the 1d conformal group $SO(2,1)$, the
transverse rotation group $SO(3)$ as well as $SO(3)$ R-symmetry. We refer to a
surface operator with this geometry as an _infinite crease_.

Under a conformal transformation a plane is mapped to a sphere and a half-
plane to a hemisphere. The crease is mapped into a pair of hemispheres inside
$S^{3}$ joined along a large circle. Explicitly we can express
$S^{3}\subset\mathbb{R}^{4}$ as

$\displaystyle x^{1}$ $\displaystyle=R\cos u\cos v\,,\qquad x^{2}=R\cos u\sin
v\,,\qquad$ (1.4) $\displaystyle x^{3}$ $\displaystyle=R\sin u\cos w\,,\qquad
x^{4}=R\sin u\sin w\,,$

with $u\in[0,\pi/2]$ and $v,w\in[0,2\pi)$. Each hemisphere is specified by a
fixed $w$, which we take to be $w=\phi/2$ and $w=\pi-\phi/2$. They are clearly
joined at an angle $\pi-\phi$ along the $u=0$ circle. In the following we
refer to this as the _spherical crease_.

Again we can associate to each of the hemispheres a different scalar coupling
separated by an angle $\theta$. A simple way to write it is by extending the
domain of $u$ to $[-\pi/2,\pi/2]$ with $x^{4}=R\sin|u|\sin(\phi/2)$ and

$\displaystyle n^{1}=\cos\frac{\theta}{2}\,,\qquad
n^{2}=\operatorname{sign}{u}\sin\frac{\theta}{2}\,.$ (1.5)

As with the infinite crease, when $\theta=\phi$ this is 1/4 BPS, but now it is
not invariant under Poincaré supercharges but only combinations with special
conformal transformations, making it of Type-S, in the classification of [7].

Translation invariance of the infinite crease suggests that any observable we
calculate would be extensive in its length, but in fact any calculation
preserving the conformal symmetry should be extensive in the area of the half-
planes when endowed with a hyperbolic metric. It is thus natural to
conformally transform $\mathbb{R}^{6}$ to $AdS_{2}\times S^{4}$ as111By
$AdS_{2}$ we always mean Euclidean $AdS_{2}$, i.e. $\mathbb{H}_{2}$.

$\sum_{\mu=1}^{6}\mathrm{d}x^{\mu}\mathrm{d}x^{\mu}=r^{2}\left(\frac{\mathrm{d}x^{1}\mathrm{d}x^{1}+\mathrm{d}r\mathrm{d}r}{r^{2}}+\mathrm{d}\Omega_{4}^{2}\right)\,,\qquad
r^{2}=\sum_{\mu=2}^{6}(x^{\mu})^{2}\,.$ (1.6)

The crease (1.2) is the union of two copies of $AdS_{2}$ at points on $S^{4}$
separated by the angle $\pi-\phi$. In this picture, the crease singularity is
replaced by the jump at the boundary of the hyperbolic spaces.

Likewise for the spherical crease, if we start with the theory on
$\mathbb{R}^{6}$, stereographically project to $S^{6}\subset\mathbb{R}^{7}$
given by (1.4) with $R=1$ and the $w$ circle extended to $S^{4}$, we can
rewrite the $S^{6}$ metric as

$\mathrm{d}u^{2}+\cos^{2}u\,\mathrm{d}v^{2}+\sin^{2}u\,\mathrm{d}\Omega^{2}_{4}=\frac{1}{\cosh^{2}\rho}\left(\mathrm{d}\rho^{2}+\sinh^{2}\rho\,\mathrm{d}v^{2}+\mathrm{d}\Omega^{2}_{4}\right).$
(1.7)

This is accomplished with $\cosh\rho=1/\sin u$.

The area of $AdS_{2}$ diverges so any quantity we evaluate should also suffer
from those divergences. The natural regularisation of the area gives zero for
the upper half plane model in (1.6) and $-2\pi$ for the disc model in (1.7).

In the latter case we can therefore unambiguously define the potential
$U(\phi,\theta)$ to be the density that arises in the calculation, such that
when using the disc model of $AdS_{2}$, the regularised expectation value of
the spherical crease is $\exp[2\pi U(\phi,\theta)]$. In the case of the
infinite crease, where the regularised area vanishes, we can still try to
define the potential via the density, but it is more subtle, as it is
multiplied by zero in the final expression.

There is one more subtlety though, which is the conformal anomaly arising in
calculating surface operators. The expectation value of a general surface
operator suffers from logarithmic divergences proportional to the integral of
the anomaly density [18, 4]

${\mathcal{A}}=\frac{1}{4\pi}\left(a_{1}{\mathcal{R}}+a_{2}(H^{2}+4\operatorname{tr}P)+b\operatorname{tr}W+c(\partial
n)^{2}\right),$ (1.8)

where ${\mathcal{R}}$ is the Ricci scalar of the induced metric $h_{ab}$ on
the surface, $H^{\mu}$ the mean curvature vector, $P=h^{ab}P_{ab}$ is the
trace of the pullback of the Schouten tensor, $W$ is the pullback of the Weyl
tensor and $\partial n$ is the gradient of the scalar coupling $n^{I}$. The
prefactors are known as anomaly coefficients and are intrinsic properties of
the theory and type of surface operator. It is known [6] that for this theory
$b=0$ and $c=-a_{2}$. In the abelian theory the coefficients are [4]

$a_{1}^{(1)}=\frac{1}{2}\,,\qquad c^{(1)}=\frac{1}{2}\,,$ (1.9)

and for a surface operator in the fundamental representation in the nonabelian
theory with algebra $A_{N-1}$ they are [19, 20, 21, 22]

$a_{1}^{(N)}=\frac{1}{2}-\frac{1}{2N}\,,\qquad
c^{(N)}=N-\frac{1}{2}-\frac{1}{2N}\,.$ (1.10)

The classical holographic calculation [19] just captures $a_{1}\sim 0$ and
$c\sim N$.

One may be concerned that all the creases that we study have an anomaly
arising from the curvature singularity at the crease itself where $H$ and
$\partial n$ have delta-function contributions. The formula (1.8) above does
not apply for singular cases, and as we confirm in Section 2.2, indeed the
singularity does not lead to an anomaly. Thus we see that the infinite crease
has no anomaly and we can therefore try to define $U(\phi,\theta)$ as the
potential density, but there is still the issue that its regularised integral
vanishes. The spherical crease is anomalous, but since the anomaly is local
and does not get a contribution from the crease singularity, it is independent
of the angles, and the same as for the sphere with $\phi=\theta=0$ and equal
to $2a_{1}-4c$ (1.1) We then can define

$U(\phi,\theta)=\frac{1}{2\pi}\big{(}\log{\left\langle
V_{\phi,\theta}\right\rangle}-\log{\left\langle
V_{S^{2}}\right\rangle}\big{)}\,,$ (1.11)

with the $2\pi$ factor arises from the area of the Poincaré disc.

It is most obvious that the anomaly does not depend on $\phi$ and $\theta$ in
the $AdS_{2}\times S^{4}$ picture. Here there are two disconnected $AdS_{2}$
factors with vanishing extrinsic curvature, so by the Gauss-Codazzi equation,
the second fundamental form is twice the Ricci scalar, which is a constant
$-2$, so the anomaly density is $(a_{1}+2a_{2}){\mathcal{R}}/4\pi$ which after
integration (and with the appropriate boundary term) gives for each $AdS_{2}$
either zero or $a_{1}-2c$, depending on the topology.

Having defined $U(\phi,\theta)$ we proceed in the rest of the paper to
evaluate it: First in the free abelian version of the ${\mathcal{N}}=(2,0)$
theory in Section 2 and then in the holographic dual in terms of M2-branes in
$AdS_{7}\times S^{4}$ in Section 3. The results of the two calculations are
not identical, so this observable is an interesting quantity that depends on
the rank of the algebra underlying the theory (as well as a representation of
that algebra to which the surface operator is related) and $\phi$ and
$\theta$.

If we focus on the BPS case, when $\phi=\theta$ the expressions do not vanish,
as one may have expected. Instead we find

$U(\phi,\phi)=\frac{C}{\pi}\log{\cos\frac{\phi}{2}}\,.$ (1.12)

In both the free theory and at leading order at large $N$, the constant $C$ is
the same as the anomaly coefficient $c$ in (1.10). Interestingly, we find the
same functional dependence on $\phi$ in both calculations.

Like in the recently studied cases of the torus and cylinder [8] or those in
[17], this is a finite nonzero quantity associated to a BPS observable,
similar in that regard to the circular Wilson loop in ${\mathcal{N}}=4$ SYM in
4d [23, 24, 25].

This can be extended to the near-BPS limit, where

$U(\phi,\theta)=\frac{C}{\pi}\log{\cos\frac{\phi}{2}}-\frac{C}{2\pi}(\phi-\theta)\tan\frac{\phi}{2}+\mathcal{O}((\theta-\phi)^{2})\,.$
(1.13)

For small $\phi$ and $\theta$, we can show that there is no $\phi\theta$ term
and find

$U(\phi,\theta)=-\frac{C}{8\pi}(2\phi^{2}-\theta^{2})+\dots$ (1.14)

This expression is similar in spirit to the bremsstrahlung formula of [13,
26], which arises from the nearly straight cusped Wilson loop. The expression
(1.13), which can also be written as

$U(\phi,\theta)=\left(1+(\phi-\theta)\frac{\mathrm{d}}{\mathrm{d}\phi}\right)U(\phi,\phi)+{\mathcal{O}}((\phi-\theta)^{2})\,,$
(1.15)

is then akin to the “generalised bremsstrahlung formula” valid for near-BPS
cusps in ${\mathcal{N}}=4$ SYM. In Section 5 we reproduce these results from a
defect CFT analysis and show that indeed $C=c$, the anomaly coefficient.

One more special example of the crease, when the two half-planes approach
lightlike surfaces, is studied in Section 4. This is very similar to the case
of the lightlike cusp [27] as well as its extension to the 4-cusp solution of
Alday and Maldacena [28]. In our case the analogue of the 4-cusp solution
involves 2 lightcones joined at a circular crease. We discuss some properties
of those solutions and evaluate their action.

We comment about possible applications and extensions of this work in Section
6.

##  2 Free theory

###  2.1 The spherical crease

We start with the abelian version of the ${\mathcal{N}}=(2,0)$ theory [29,
30]. The field content of the theory is

  * •

$B_{\mu\nu}$, a two-form with self-dual field strength.

  * •

$\Phi^{I}$, five scalar fields.

  * •

$\Psi$, eight symplectic Majorana fermions.

The surface operators of this theory are constructed explicitly in [4, 31].
They are built out of the fields $B$ and $\Phi$ and are written as

$V_{\phi,\theta}=\exp\int_{\Sigma}\sqrt{h}\left[\frac{i}{2}B_{\mu\nu}\partial_{a}x^{\mu}\partial_{b}x^{\nu}\varepsilon^{ab}-n^{I}\Phi^{I}\right]\mathrm{d}^{2}\sigma\,.$
(2.1)

Here $a,b$ are the indices of the coordinates $\sigma^{a}$ parametrising the
surface; in our definition of the crease (1.4), we take $\sigma^{a}=(u,v)$.
$h_{ab}$ is the induced metric, $\varepsilon^{ab}$ is the antisymmetric tensor
normalised with a factor of $1/\sqrt{h}$ and $n^{I}$ are the scalar couplings,
which in our examples are given in (1.5).

Despite the subtleties related to the self-duality of $B$, one can derive
propagators for the bosonic fields in flat space. Including a short distance
regulator $\epsilon$ they are [32, 33, 31, 4]

$\displaystyle\left\langle\Phi^{I}(x)\Phi^{J}(y)\right\rangle$
$\displaystyle=\frac{\delta^{IJ}}{2\pi^{2}(\left|x-y\right|^{2}+2\epsilon^{2})^{2}}\,,$
(2.2a) $\displaystyle\left\langle B_{\mu\nu}(x)B_{\rho\sigma}(y)\right\rangle$
$\displaystyle=\frac{\delta_{\mu\rho}\delta_{\nu\sigma}-\delta_{\mu\sigma}\delta_{\nu\rho}}{2\pi^{2}(\left|x-y\right|^{2}+2\epsilon^{2})^{2}}\,.$
(2.2b)

We can use these propagators to calculate the expectation value of the
spherical crease. It is easy to show that because the theory is free, the
expectation value is given by (the exponential of) the propagator integrated
over 2 copies of the surface. There are two contributions, coming from
integrating the propagator over two copies of the same hemisphere and the
opposites hemispheres. Using the explicit parametrisation (1.4), (1.5), the
log of the expectation value can be expressed as

$\log{\left\langle V_{\phi,\theta}\right\rangle}=\frac{1}{2}\int
R^{4}\cos{u}\cos
u^{\prime}\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}u^{\prime}\,\mathrm{d}v^{\prime}\left(2\Delta_{\pi,\pi}(u,v;u^{\prime},v^{\prime})+2\Delta_{\phi,\theta}(u,v;u^{\prime},v^{\prime})\right)\,,$
(2.3)

where the combined propagator is

$\Delta_{\phi,\theta}(u,v;u^{\prime},v^{\prime})=\frac{1}{8\pi^{2}}\frac{\cos{\theta}-\cos
u\cos{u^{\prime}}\cos(v-v^{\prime})\cos{\phi}+\sin{u}\sin{u^{\prime}}}{R^{4}\left(1-\cos{u}\cos{u^{\prime}}\cos{(v-v^{\prime})}+\sin{u}\sin{u^{\prime}}\cos{\phi}+\epsilon^{2}/R^{2}\right)^{2}}\,.$
(2.4)

We can perform the $v$ and $v^{\prime}$ integral using equation (2.554) of
[34]

$\int_{0}^{2\pi}\mathrm{d}v\,\mathrm{d}v^{\prime}\frac{A+B\cos(v-v^{\prime})}{(a+b\cos(v-v^{\prime}))^{2}}=2\pi\int_{0}^{2\pi}\mathrm{d}v\frac{A+B\cos{v}}{(a+b\cos{v})^{2}}=4\pi^{2}\frac{aA-
bB}{\left[a^{2}-b^{2}\right]^{3/2}}\,,\qquad a>b\,.$ (2.5)

With the appropriate $A$, $B$, $a$ and $b$ and replacing $s=\sin u$ and
$s^{\prime}=\sin u^{\prime}$ this is (with $R=1$)

$\displaystyle\int\mathrm{d}v\,\mathrm{d}v^{\prime}\,\Delta_{\phi,\theta}$
$\displaystyle=\frac{1}{2}\frac{(\cos{\theta}-\cos{\phi})(1+\epsilon^{2}-ss^{\prime}\cos{\phi})}{\left[(1+\epsilon^{2}+ss^{\prime}\sin\phi)^{2}-(1-s^{2})(1-s^{\prime
2})\right]^{3/2}}$ (2.6)
$\displaystyle\quad+\frac{1}{2}\frac{\cos{\phi}(s^{2}+s^{\prime
2}+\epsilon^{2})+ss^{\prime}(1+\epsilon^{2}+\cos^{2}{\phi})}{\left[(1+\epsilon^{2}+ss^{\prime}\sin\phi)^{2}-(1-s^{2})(1-s^{\prime
2})\right]^{3/2}}\,.$

Integrating this gives

$\int\mathrm{d}s\,\mathrm{d}v\,\mathrm{d}s^{\prime}\,\mathrm{d}v^{\prime}\,\Delta_{\phi,\theta}=(\cos{\theta}-\cos{\phi})\left[\frac{R\phi}{2^{3/2}\,\epsilon\sin{\phi}}-\frac{1}{4\cos^{2}(\phi/2)}\right]+\log{\left(2\cos\frac{\phi}{2}\right)}+\mathcal{O}(\epsilon)\,.$
(2.7)

The contribution from two copies of the same hemisphere, for which
$\phi=\theta=\pi$, has further UV divergences and we cannot rely on the
regularised result of the integral above. Instead, after substracting
$\epsilon^{2}$ from the numerator in (2.4) we find that (2.6) is now

$\int\mathrm{d}v\,\mathrm{d}v^{\prime}\Delta_{\pi,\pi}=\frac{1}{2\sqrt{\epsilon^{4}+2\epsilon^{2}(1-ss^{\prime})+(s-s^{\prime})^{2}}}\,,$
(2.8)

and

$\int\mathrm{d}s\,\mathrm{d}v\,\mathrm{d}s^{\prime}\,\mathrm{d}v^{\prime}\,\Delta_{\pi,\pi}=-\log\frac{\sqrt{2}\,\epsilon}{R}+\mathcal{O}(\epsilon)\,.$
(2.9)

Combining with (2.7) and removing the linear divergence we find

$\log{\left\langle
V_{\phi,\theta}\right\rangle}=\log{\frac{\sqrt{2}\,R\cos(\phi/2)}{\epsilon}}+\frac{\cos\phi-\cos\theta}{4\cos^{2}(\phi/2)}+\mathcal{O}(\epsilon)\,.$
(2.10)

Setting $\phi=\theta=0$ indeed reproduces the result for the sphere (1.1) with
the anomaly coefficients (1.9).

We can then take the difference to the regular sphere case to read off the
potential (1.11)

$U^{(1)}(\phi,\theta)=\frac{1}{2\pi}\log{\cos\frac{\phi}{2}}+\frac{\cos\phi-\cos\theta}{8\pi\cos^{2}(\phi/2)}\,.$
(2.11)

###  2.2 The crease singularity

In the above derivation we found a linear $R/\epsilon$ divergence. More
confusing is the relation of this result to the anomaly formula (1.8). Along
either the infinite or the compact crease, the mean curvature vector $H$ has a
delta function singularity, invalidating the formula. One may worry that under
regularisation we would get a term similar to $\epsilon^{-1}\log\epsilon$,
which we did not find above. We examine this apparent contradiction here.

For simplicity we choose to analyse the case of the sphere $\phi=0$ with
$\theta\neq 0$. $n^{I}$ is given in (1.5) and clearly is singular at $u=0$; we
can regularise it by replacing the step function $\operatorname{sign}(u)$ by a
continuous function $f_{l}(u)$ with a regulator $l$. We can take for instance

$f_{l}(u)=\begin{cases}u/l\,,&-l\leq u\leq l\,,\\\
\operatorname{sign}(u)\,,&\text{otherwise}\,.\end{cases}$ (2.12)

As $l\to 0$ this reduces to the step function.

It is easy to check that now $|\partial n|\sim 1/l$ for $|u|\leq l$ and the
anomaly formula indeed gives a $l^{-1}\log{\epsilon}$ divergence. Any other
regulator $f_{l}$ would give the same behaviour up to a numerical factor.

The mechanism by which this discrepancy is resolved is easy to understand in
this example. Using (2.4), the expectation value for the regularised crease in
the free theory is

$\frac{1}{16\pi^{2}}\int\cos{u}\cos
u^{\prime}\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}u^{\prime}\,\mathrm{d}v^{\prime}\frac{n^{I}(u)n^{I}(u^{\prime})-\cos{u}\cos{u^{\prime}}\cos{(v-v^{\prime})}-\sin{u}\sin{u^{\prime}}}{(1-\cos{u}\cos{u^{\prime}}\cos{(v-v^{\prime})}-\sin{u}\sin{u^{\prime}}+\epsilon^{2}/R^{2})^{2}}\,.$
(2.13)

Instead of evaluating the exact expression, we expand in a power series in
$\theta$. For $\theta=0$ this is the regular sphere, so there is no
singularity, only the usual $-\log\epsilon$ anomaly. The first nontrivial term
is of order $\theta^{2}$ and it is enough to capture the
$l^{-1}\log{\epsilon}$ divergence. After expanding, the $v,v^{\prime}$
integrals are performed as before and substituting again $s=\sin{u}$ and
$s^{\prime}=\sin{u^{\prime}}$ we are left with

$-\frac{1}{32}\int_{-1}^{1}\mathrm{d}s\,\mathrm{d}s^{\prime}\,\frac{(1-ss^{\prime})(f_{l}(s)-f_{l}(s^{\prime}))^{2}}{\left((s-s^{\prime})^{2}+(1-ss^{\prime})\epsilon^{2}+\epsilon^{4}\right)^{3/2}}\,.$
(2.14)

This integral can be performed and expanding the result first in $\epsilon$
with fixed $l$ and then expanding in small $l$ gives

$\frac{1}{16l}+\frac{1}{8l}\log\frac{\epsilon}{4l}+\text{finite}.$ (2.15)

For fixed $l$ and $\epsilon\to 0$ we indeed find the result
$l^{-1}\log\epsilon$ predicted by the anomaly formula. The $1/l$ and
$l^{-1}\log l$ terms are then regarded as finite contributions.

But this expression does not lead to a $\epsilon^{-1}\log\epsilon$ divergence,
as when we take $l\to\epsilon$, the $\log l$ compensates for the
$\log\epsilon$ and all we are left with is a linear divergence $1/l\to
1/\epsilon$, in agreement with the explicit calculations. Furthermore, there
are no new $\log\epsilon$ terms for $l\sim\epsilon$, and all the anomaly for
the sphere with $\theta\neq 0$ is the same as the usual sphere, as indeed we
found in (2.10).

Of course in this example we only look at the term of order $\theta^{2}$ for a
simple choice of $f_{l}$. We did not calculate the regularised crease for any
$\phi$ and $\theta$ but we expect the same mechanism to reconcile the anomaly
formula prediction with the results for the singular crease. Indeed $\epsilon$
in the logarithm must appear with some other scale, and the scale relevant for
the singularity is $l$.

The observation that the crease does not give additional $\log^{2}{\epsilon}$
or $\log{\epsilon}$ divergences was also made in the context of holographic
entanglement entropy in [35, 36, 37].

###  2.3 The infinite crease

We defined the potential (1.11) in terms of the spherical crease and in all
our calculations it is identical also to its $AdS_{2}$ realisation. Here we
examine what can be calculated for the infinite crease.

Repeating the calculation with this geometry, the interesting contribution
comes from propagators between different half-planes in (1.2). Using (2.2a),
(2.2b) it reads

$\displaystyle\log{\left\langle
V_{\phi,\theta}^{\mathrm{inf}}\right\rangle}=\frac{\cos{\theta}-\cos{\phi}}{2\pi^{2}}\int_{-\infty}^{\infty}\mathrm{d}\tau\,\mathrm{d}\tau^{\prime}\int_{0}^{\infty}\frac{\mathrm{d}\sigma\,\mathrm{d}\sigma^{\prime}}{[(\tau-\tau^{\prime})^{2}+(\sigma-\sigma^{\prime})^{2}+4\sigma\sigma^{\prime}\cos^{2}(\phi/2)+2\epsilon^{2}]^{2}}\,.$
(2.16)

Performing the integrals over $\tau^{\prime}$, $\sigma^{\prime}$ and $\sigma$,
we obtain

$\displaystyle\log\left\langle
V_{\phi,\theta}^{\text{inf}}\right\rangle=\frac{\cos\theta-\cos\phi}{2^{3/2}\pi\sin\phi}\phi\int_{-\infty}^{\infty}\frac{\mathrm{d}\tau}{\epsilon}\,.$
(2.17)

The resulting integral gives the overall length $L/\epsilon$, which as a
linear divergence can be removed. But we wish to view it instead as the volume
of $AdS_{2}$ in the Poincaré patch

$\int_{-\infty}^{\infty}\frac{\mathrm{d}\tau}{\epsilon}=\int_{-\infty}^{\infty}\mathrm{d}\tau\int_{\epsilon}^{\infty}\frac{\mathrm{d}\sigma}{\sigma^{2}}\,.$
(2.18)

Then we view the prefactor in (2.17) as representing the potential between the
two planes

$U^{\text{inf}}_{\phi,\theta}=\frac{\cos\theta-\cos\phi}{2^{3/2}\pi\sin\phi}\phi\,,$
(2.19)

where we did not divide by $1/2\pi$ as in (1.11), since here we divide by the
divergent area of the upper half plane instead of the finite regularised area
of the Poincaré disc.

Interestingly, the answer is different from the case of the spherical crease
(2.11), and here it vanishes in the BPS case $\phi=\theta$. Also, it has the
same functional form as the cusped Wilson loop in ${\mathcal{N}}=4$ SYM at
one-loop order [12].

A naive integration of (2.16) without the $\epsilon$ regulator is clearly
divergent. But for fixed $\sigma$ and $\tau$ one can perform the full
$\sigma^{\prime}$, $\tau^{\prime}$ integral to find

$\frac{\cos\theta-\cos\phi}{8\pi\cos^{2}(\phi/2)}\,,$ (2.20)

which is now the same as (2.11) except for the $1/(2\pi)\log\cos(\phi/2)$
term. The remaining $\tau$, $\sigma$ integrals are as in (2.18).

These two calculations are consistent with using two different cutoffs
$\epsilon$ and $\epsilon^{\prime}$ related by

$\epsilon=\frac{2^{3/2}\cos^{2}(\phi/2)\phi}{\sin\phi}\epsilon^{\prime}\,.$
(2.21)

Note that to reproduce the missing term from (2.11), we would need to include
a $\theta$ dependence in cutoff, which is not very natural as a field theory
cutoff. So we conclude that the result is rather ambiguous and scheme
dependent, but in any case vanishes in the BPS case.

We look at the holographic dual of the infinite crease in Appendix A and find
similar ambiguities.

##  3 Holography

###  3.1 The setup

The spherical crease is inside $S^{3}$ and involves two scalar fields. It can
therefore be described by an M2-brane inside $AdS_{4}\times S^{1}\subset
AdS_{7}\times S^{4}$. Writing the $AdS_{4}$ factor as an $AdS_{2}\times S^{1}$
foliation, this is

$\mathrm{d}s^{2}=4L^{2}\left[\mathrm{d}\nu^{2}+\cosh^{2}{\nu}(\mathrm{d}\rho^{2}+\sinh^{2}\rho\,\mathrm{d}v^{2})+\sinh^{2}\nu\,\mathrm{d}\varphi^{2}\right]+L^{2}\mathrm{d}\vartheta^{2}\,.$
(3.1)

Defining $\sinh\nu=1/t$ the metric becomes

$\mathrm{d}s^{2}=\frac{4L^{2}}{t^{2}}\left[\frac{\mathrm{d}t^{2}}{1+t^{2}}+(1+t^{2})(\mathrm{d}\rho^{2}+\sinh^{2}\rho\,\mathrm{d}v^{2})+\mathrm{d}\varphi^{2}\right]+L^{2}\mathrm{d}\vartheta^{2}\,.$
(3.2)

The M2-brane can be parametrised by the coordinates of $AdS_{2}$: $\rho,v$,
and in addition $t$. Because the crease has $AdS_{2}$ symmetry, we look for
brane embeddings depending only on $t$, so we should solve for $\varphi(t)$
and $\vartheta(t)$. This description is not single valued; $\varphi(t)$ and
$\vartheta(t)$ have two branches corresponding to each hemisphere of the
spherical crease. At the boundary of $AdS$ ($t\to 0$), $\varphi$ and
$\vartheta$ should approach their asymptotic values (shifted to be symmetric
around $\pi/2$ and 0)

$\varphi(0)=\phi/2\,,\ \pi-\phi/2\,,\qquad\vartheta(0)=\pm\theta/2\,.$ (3.3)

A single branch of the solution has $t\in(0,t_{\text{max}}]$ and both branches
meet at $t_{\text{max}}$ (see for example Figure 2)

$\varphi(t_{\text{max}})=\pi/2\,,\qquad\vartheta(t_{\text{max}})=0\,.$ (3.4)

The action is

$S_{\text{M2}}=8T_{\text{M2}}L^{3}\int\sinh{\rho}\,\mathrm{d}\rho\,\mathrm{d}v\,\mathrm{d}t\,\frac{1+t^{2}}{t^{3}}\sqrt{\frac{1}{1+t^{2}}+\varphi^{\prime
2}+\frac{t^{2}}{4}\vartheta^{\prime 2}}\,.$ (3.5)

Here $T_{\text{M2}}$ is the M2-brane tension, which by the $AdS$/CFT
dictionary [38] is $N/4\pi L^{3}$.

This lagrangian has two conserved quantities: the canonical momenta conjugate
to $\varphi$ and $\vartheta$

$J_{\varphi}=\frac{(1+t^{2})\varphi^{\prime}}{t^{3}\sqrt{\frac{1}{1+t^{2}}+\varphi^{\prime
2}+\frac{t^{2}}{4}\vartheta^{\prime 2}}}\,,\qquad
J_{\vartheta}=\frac{(1+t^{2})\vartheta^{\prime}}{4t\sqrt{\frac{1}{1+t^{2}}+\varphi^{\prime
2}+\frac{t^{2}}{4}\vartheta^{\prime 2}}}\,.$ (3.6)

Inverting these relations gives the first order equations of motion

$\displaystyle\vartheta^{\prime}(t)$
$\displaystyle=\frac{4J_{\vartheta}t}{\sqrt{(1+t^{2})(1+2t^{2}+(1-4J_{\vartheta}^{2})t^{4}-J_{\varphi}^{2}t^{6})}}\,,$
(3.7) $\displaystyle\varphi^{\prime}(t)$
$\displaystyle=\frac{J_{\varphi}t^{3}}{\sqrt{(1+t^{2})(1+2t^{2}+(1-4J_{\vartheta}^{2})t^{4}-J_{\varphi}^{2}t^{6})}}\,.$

In order for the brane to be smooth, we need that at $t_{\text{max}}$ both
$\varphi^{\prime},\vartheta^{\prime}\to\infty$ together. This determines
$t_{\text{max}}$ as one of the roots of the polynomial in the denominator of
(3.7). More precisely, because both $J_{\varphi}$ and $J_{\vartheta}$ are
real, it is simple to show that the polynomial only has a single positive
root. It can be factorised as

$1+2t^{2}+(1-4J_{\vartheta}^{2})t^{4}-J_{\varphi}^{2}t^{6}=-J_{\varphi}^{2}(t^{2}-\tau_{1})(t^{2}-\tau_{2})(t^{2}-t_{\text{max}}^{2})\,.$
(3.8)

The roots $\tau_{1}$, $\tau_{2}$ can be expressed explicitly in terms of
$J_{\varphi}$, $J_{\vartheta}$ using the cubic formula, but we leave them
implicit. They are either complex or real and negative. The root
$t_{\text{max}}$ is real and positive.

###  3.2 BPS case

As we show in Appendix A.2, in the BPS case, the set of equations obtained in
[7] relate the two momenta

$2J_{\vartheta}=\pm J_{\varphi}\,,$ (3.9)

or equivalently $\vartheta^{\prime}=\pm 2\varphi^{\prime}/t^{2}$. On the first
branch, we expect $\varphi$ to be monotonously increasing and $\vartheta$ to
be decreasing, and because their derivatives are proportional to the conserved
momenta (3.7), we should take the solution with the negative sign.

In particular this means that the polynomial appearing in the denominator of
(3.7)

$(1+t^{2})\left[(1+t^{2})^{2}-(4J_{\vartheta}^{2}+J_{\varphi}^{2}t^{2})t^{4}\right],$
(3.10)

has degenerate roots at $t^{2}=-1$ and simplifies to

$(1+t^{2})^{2}\left(1+t^{2}-J_{\varphi}^{2}t^{4}\right).$ (3.11)

The factorisation of an additional factor of $1+t^{2}$ implies that in this
case, the equations of motion (3.7) contain the square root of a quadratic
polynomial, so their solution is trigonometric, rather than elliptic.

Comparing with (3.8), we find a simple expression for $t_{\text{max}}$

$\displaystyle
t_{\text{max}}^{2}=\frac{1+\sqrt{1+4J_{\varphi}^{2}}}{2J_{\varphi}^{2}}\,.$
(3.12)

Integrating $\vartheta^{\prime}$ in (3.7) and imposing the boundary conditions
(3.4) at $t_{\text{max}}$, we obtain

$\displaystyle\vartheta=\arctan\left(\frac{2J_{\varphi}\sqrt{1+t^{2}-J_{\varphi}^{2}t^{4}}}{1+(1+2J_{\varphi}^{2})t^{2}}\right)\,,$
(3.13)

and similarly for $\varphi$

$\displaystyle\varphi=\vartheta+\arcsin\left(\frac{J_{\varphi}t^{2}}{\sqrt{1+t^{2}}}\right)\,.$
(3.14)

We can read the value of $\phi$, $\theta$ from the solution by setting $t=0$:
(3.14) gives $\phi=\theta$ as expected (the case $\phi=-\theta$ is obtained by
picking the positive sign in (3.9) instead), and (3.13) expresses
$J_{\varphi}$ in term of the angle $\phi$

$\displaystyle J_{\varphi}=\frac{1}{2}\tan\frac{\phi}{2}\,.$ (3.15)

Finally, with these solutions in hand we can evaluate the action (3.5). The
integral over $AdS_{2}$ diverges but is regularised to $-2\pi$. Using the
equations of motion and the BPS condition (3.9), the action reduces to

$\displaystyle
S_{\text{M2}}=\frac{4N}{\pi}(-2\pi)\int_{\epsilon}^{t_{\text{max}}}\frac{(1+t^{2})\mathrm{d}t}{t^{3}\sqrt{1+t^{2}-J_{\varphi}^{2}t^{4}}}\,.$
(3.16)

The integral over $t$ gives

$-\frac{\sqrt{1+t^{2}-J_{\varphi}^{2}t^{4}}}{2t^{2}}-\frac{1}{4}\operatorname{arctanh}\frac{2\sqrt{1+t^{2}-J_{\varphi}^{2}t^{4}}}{2+t^{2}}\,.$
(3.17)

This vanishes at $t=t_{\text{max}}$. At $t=\epsilon$ the first term gives an
$\epsilon^{-2}$ divergence corresponding to the usual area divergence; it can
be safely discarded (or removed by an appropriate counterterm). The second
term gives an important $\log\epsilon$ divergence

$\displaystyle S_{\text{M2}}$
$\displaystyle=-2N\operatorname{arctanh}\left(1-\frac{1}{8}(1+4J_{\varphi}^{2})\epsilon^{4}+\dots\right)=N\log\frac{(1+4J_{\varphi}^{2})\epsilon^{4}}{16}+\dots$
(3.18) $\displaystyle=N\log\frac{\epsilon^{4}}{16\cos^{2}(\phi/2)}+\dots$

Note that the coefficient of $\log\epsilon$ matches the expected conformal
anomaly for the sphere $4N+\mathcal{O}(N^{0})=4c_{1}-2a_{2}$ (1.10). It must
also appear in the form of $\log{\epsilon/R}$, and reproduces the expected
behavior (1.1). Subtracting the action for the sphere then gives a finite
quantity, from which we can read the BPS potential

$\displaystyle U^{(N)}(\phi,\phi)=\frac{N}{\pi}\log{\cos\frac{\phi}{2}}\,.$
(3.19)

###  3.3 Non-BPS case

For generic $J_{\varphi}$ and $J_{\vartheta}$, the equations of motion (3.7)
can be brought into canonical elliptic form by performing the change of
variables

$\displaystyle
y^{2}(t)=\frac{(t_{\text{max}}^{2}-t^{2})(1+\tau_{2})}{(t_{\text{max}}^{2}-\tau_{2})(1+t^{2})}\,,$
(3.20)

after which the equations are

$\displaystyle\mathrm{d}\vartheta$
$\displaystyle=\frac{4J_{\vartheta}}{J_{\varphi}\sqrt{(t_{\text{max}}^{2}-\tau_{1})(1+\tau_{2})}}\frac{\mathrm{d}y}{\sqrt{(1-y^{2})(1-y^{2}/y^{2}(\tau_{1}))}}\,,$
(3.21) $\displaystyle\mathrm{d}\varphi$
$\displaystyle=-\frac{1}{\sqrt{(t_{\text{max}}^{2}-\tau_{1})(1+\tau_{2})}}\frac{\mathrm{d}y}{\sqrt{(1-y^{2})(1-y^{2}/y^{2}(\tau_{1}))}}\left[1-\frac{(1+\tau_{2})(1+t_{\text{max}}^{2})}{1+\tau_{2}-(\tau_{2}-t_{\text{max}}^{2})y^{2}}\right].$

These match the definition of elliptic integrals of the first and third kind

$\displaystyle\mathbf{F}(\psi|k^{2})$
$\displaystyle=\int_{0}^{\sin{\psi}}\frac{\mathrm{d}y}{\sqrt{(1-y^{2})(1-k^{2}y^{2})}}\,,$
(3.22) $\displaystyle\mathbf{\Pi}(n;\psi|k^{2})$
$\displaystyle=\int_{0}^{\sin{\psi}}\frac{\mathrm{d}y}{(1-ny^{2})\sqrt{(1-y^{2})(1-k^{2}y^{2})}}\,.$

The parameters appearing in their definition can be read from above, they are

$\psi=\arcsin\sqrt{\frac{(t_{\text{max}}^{2}-t^{2})(1+\tau_{2})}{(t_{\text{max}}^{2}-\tau_{2})(1+t^{2})}}\,,\qquad
k^{2}=\frac{(t_{\text{max}}^{2}-\tau_{2})(1+\tau_{1})}{(t_{\text{max}}^{2}-\tau_{1})(1+\tau_{2})}\,,\qquad
n=\frac{\tau_{2}-t_{\text{max}}^{2}}{1+\tau_{2}}\,.$ (3.23)

Imposing the gluing condition (3.4), we obtain

$\vartheta(t)=-\frac{4J_{\vartheta}}{J_{\varphi}}\frac{\mathbf{F}(\psi|k^{2})}{\sqrt{(t_{\text{max}}^{2}-\tau_{1})(1+\tau_{2})}}\,,\qquad\varphi(t)=\frac{\pi}{2}+\frac{\mathbf{F}(\psi|k^{2})-(1+t_{\text{max}}^{2})\mathbf{\Pi}(n;\psi|k^{2})}{\sqrt{(t_{\text{max}}^{2}-\tau_{1})(1+\tau_{2})}}\,.$
(3.24)

At $t=t_{\text{max}}$, the elliptic integrals vanish ($\psi=0$), so the
boundary conditions are manifestly respected.

What is not manifest in these expressions is that $\varphi$ and $\vartheta$
are real functions. The behavior of the roots $\tau_{1},\tau_{2}$ are captured
by the discriminant, they are real and negative if

$\displaystyle
J_{\varphi}^{2}\leq-\frac{2}{27}\left[1-36J_{\vartheta}^{2}-(1+12J_{\vartheta}^{2})^{3/2}\right]\,,$
(3.25)

which in particular includes the BPS case $J_{\varphi}=\pm 2J_{\vartheta}$,
otherwise they are complex conjugate. When they are real, taking
$\tau_{1}<\tau_{2}$ makes all the quantities real, including $\varphi$,
$\vartheta$. When they are complex conjugate, exchanging $\tau_{1}$ and
$\tau_{2}$ is a symmetry so $\varphi$ and $\vartheta$ must be real as well.

Evaluating (3.24) at $t=0$ relates $\phi$ and $\theta$ to $J_{\varphi}$ and
$J_{\vartheta}$, but unlike the BPS case, these relations cannot be inverted
in terms of fundamental functions. Instead we do this perturbatively below
around $\phi=\theta$ (near-BPS limit) and around $\phi\sim\pi$ and $\theta=0$
(antiparallel limit). Of course, it can also be done numerically as in Figure
2.

Figure 2: Example of solution to the equations of motion (3.7) for
$\phi=\pi/3$ and $\theta=\pi/2$. $\vartheta(t)$ and $\varphi(t)$ are given
analytically in (3.24), they depend on the parameters $J_{\varphi}=0.262$,
$J_{\vartheta}=-0.210$ which are found numerically. In this example
$t_{\text{max}}=3.76$.

We now turn to calculating the action. Using the equations of motion (3.7) and
again regularising the volume of $AdS_{2}$, the action (3.5) reduces to

$\displaystyle
S_{\text{M2}}=\frac{4N}{\pi}(-2\pi)\int_{\epsilon}^{t_{\text{max}}}\frac{(1+t^{2})^{3/2}\mathrm{d}t}{t^{3}\sqrt{(1+t^{2})^{2}-4J_{\vartheta}^{2}t^{4}-J_{\varphi}^{2}t^{6}}}\,.$
(3.26)

Using the change of variables (3.20), it too can be expressed in terms of
elliptic integrals

$\displaystyle\int\frac{\mathrm{d}y}{(1-n^{\prime}y^{2})^{2}\sqrt{(1-y^{2})(1-k^{2}y^{2})}}=\frac{n^{\prime
2}y}{1-n^{\prime}y^{2}}\frac{\sqrt{(1-y^{2})(1-k^{2}y^{2})}}{2(k^{2}-n^{\prime})(n^{\prime}-1)}$
(3.27) $\displaystyle\hskip
28.45274pt-\frac{n^{\prime}\mathbf{E}(\psi|k^{2})+(k^{2}-n^{\prime})\mathbf{F}(\psi|k^{2})+\left(n^{\prime}(n^{\prime}-2)+k^{2}(2n^{\prime}-3)\right)\mathbf{\Pi}(n^{\prime};\psi|k^{2})}{2(k^{2}-n^{\prime})(1-n^{\prime})}\,,$

where $n^{\prime}=1/y^{2}(0)$ differs from $n$ in (3.23) and $\mathbf{E}$ is
the elliptic integral of the second kind

$\displaystyle\mathbf{E}(\psi|k^{2})=\int_{0}^{\sin{\psi}}\mathrm{d}y\sqrt{\frac{1-k^{2}y^{2}}{1-y^{2}}}\,.$
(3.28)

At $t=0$ both the first term and $\mathbf{\Pi}$ diverge. This is the same
divergence we encountered in the BPS case (3.18): to see it, it is easier to
go back to (3.26) and expand for small $t$, which gives

$\displaystyle-8N\int_{\epsilon}^{t_{\text{max}}}\frac{\mathrm{d}t}{t^{3}}\left[1+\frac{t^{2}}{2}+\mathcal{O}(t^{4})\right]=-8N\left[\frac{1}{2\epsilon^{2}}-\frac{1}{2}\log{\epsilon}\right]+\text{finite}\,.$
(3.29)

Notice that both terms are independent of $J_{\vartheta},J_{\varphi}$, so they
are independent of the angles and do not contribute to the potential
$U(\phi,\theta)$.

The rest of the integral (3.27) is UV finite and calculates the potential
$U(\phi,\theta)$ (up to the prefactor $4N/\pi$). The dependence on the angles
is implicit, since all the variables and roots of the cubic polynomial (3.8)
depend on $J_{\varphi},J_{\vartheta}$, and these are related to $\phi$,
$\theta$ by imposing the boundary conditions at $t=0$ on the solution (3.24).

It is easy to evaluate the potential for specific $\phi$ and $\theta$
numerically, as in Figure 3.

Figure 3: In blue, the potential $U(\phi,\theta)/N$ at large $N$ obtained
numerically from (3.26) for different values of $\theta$: $\theta=0$ (solid),
$2\pi/3$ (dashed) and $\pi$ (dotted). Notice the divergence near $\phi=\pi$,
which is captured by the antiparallel planes limit (3.41). The potential at
$N=1$ for the same values of $\theta$ (2.11) is in red.

The analysis thus far restricted to solutions with $0<\vartheta<\theta/2$, but
there are solutions that are multiply wound around the $S^{1}_{\vartheta}$
circle, where for each branch of the solution $0<\vartheta<\pi n+\theta/2$,
see for instance Figure 4. Those solutions can be described analytically, but
they also arise when searching numerically for $J_{\varphi}$, $J_{\vartheta}$.

We should compare their action to that of the unwound solution. They all have
the same behaviour and divergence at small $t$, which is removed by the same
counterterm. Looking at the action density, the difference between the
solutions with different windings is finite, so they can be viewed as
instantonic corrections. But it is more appropriate to treat the entire brane
including the factor of the area of $AdS_{2}$ in the action (3.5). One may be
tempted to replace the area by $-2\pi$ as in (3.26), but this is inconsistent.
Firstly, this amounts to different subtractions for different solutions, so
does not form a consistent regularisation scheme. Furthermore, because of the
negative sign, the more the solution is wound, the more dominant it would be.
If we are to use a consistent regularisation scheme that renders the unwound
solution finite, the wound solutions have infinite action and are therefore
infinitely suppressed.

(a) $\theta=\pi/3$

(b) $\theta=7\pi/3$

(c) $\theta=13\pi/3$

Figure 4: Example of solutions to the equations of motion with nontrivial
windings for $\phi=\pi/2$ and $\theta=(\pi/3\mod 2\pi)$. The radial direction
is $(1+t^{2})^{-1/2}$, with the black circle corresponding to the conformal
boundary. The blue curve is $\varphi(t)$ and the red curve is $\vartheta(t)$.

###  3.4 Near-BPS expansion

Here we study the solutions in a systematic expansion beyond the BPS case in
Section 3.2.

Expanding (3.7) around $J_{\varphi}=-2J_{\vartheta}$ gives a series of the
form

$\vartheta^{\prime}=\sum_{n=1}^{\infty}\frac{p_{n}(t,J_{\varphi})}{(1+t^{2})^{n+1}(1+t^{2}-J_{\varphi}t^{4})^{n+1/2}}(J_{\varphi}+2J_{\vartheta})^{n}\,,$
(3.30)

with some polynomials $p_{n}$ and likewise for $\varphi^{\prime}$.

The resulting integrals are all trigonometric functions, which is simply the
small $k^{2}$ (3.23) expansion of the elliptic integrals. Integrating between
$t=0$ and $t_{\text{max}}$ (3.12), we find

$\displaystyle\phi$
$\displaystyle=2\arctan{2J_{\varphi}}+\frac{1}{J_{\varphi}^{2}}\left[-\frac{3+8J_{\varphi}^{2}}{1+4J_{\varphi}^{2}}+\frac{3}{2J_{\varphi}}\arctan{2J_{\varphi}}\right]\left(J_{\varphi}+2J_{\vartheta}\right)+\mathcal{O}((J_{\varphi}+2J_{\vartheta})^{2})\,,$
(3.31) $\displaystyle\theta$
$\displaystyle=2\arctan{2J_{\varphi}}+\frac{1}{J_{\varphi}^{2}}\left[-\frac{6+20J_{\varphi}^{2}}{1+4J_{\varphi}^{2}}+\frac{3}{J_{\varphi}}\arctan{2J_{\varphi}}\right]\left(J_{\varphi}+2J_{\vartheta}\right)+\mathcal{O}((J_{\varphi}+2J_{\vartheta})^{2})\,.$

These series can be inverted to give

$\displaystyle J_{\varphi}$
$\displaystyle=\frac{\tan(\phi/2)}{2}+\left[\frac{1}{4\cos^{2}(\phi/2)}+\frac{\tan^{3}(\phi/2)}{6(\phi-2\tan(\phi/2))}\right]\left(\phi-\theta\right)+\mathcal{O}((\phi-\theta)^{2})\,,$
(3.32) $\displaystyle J_{\vartheta}$
$\displaystyle=-\frac{\tan(\phi/2)}{4}-\left[\frac{1}{8\cos^{2}(\phi/2)}+\frac{\tan^{3}(\phi/2)}{6(\phi-2\tan(\phi/2))}\right]\left(\phi-\theta\right)+\mathcal{O}((\phi-\theta)^{2})\,.$

Expanding the action in series also gives trigonometric functions. The lowest
order is the BPS case (3.18), and higher-order terms in the
$J_{\varphi}-2J_{\vartheta}$ expansion calculate the near-BPS corrections to
the potential. As we already proved, no further $\log\epsilon$ terms arise and
we only find new power-like divergences, which can be discarded. The potential
is then

$\displaystyle U^{(N)}(\phi,\theta)=\frac{N}{\pi}\bigg{\\{}$
$\displaystyle\log{\cos\frac{\phi}{2}}-\frac{\tan(\phi/2)}{2}(\phi-\theta)-\left[\frac{1}{\cos^{2}(\phi/2)}+\frac{4\tan^{3}(\phi/2)}{3(\phi-2\tan(\phi/2))}\right]\frac{(\phi-\theta)^{2}}{8}$
(3.33)
$\displaystyle{}-\frac{\tan(\phi/2)}{\cos^{2}(\phi/2)}\left[\frac{1}{3}+\frac{2\tan(\phi/2)}{\phi-2\tan(\phi/2)}+\frac{10(13+\cos{\phi})\tan^{2}(\phi/2)}{27(\phi-2\tan(\phi/2))^{2}}\right.$
$\displaystyle\hskip
72.26999pt+\left.\frac{280\sin^{2}(\phi/2)\tan^{3}(\phi/2)}{81(\phi-2\tan(\phi/2))^{3}}\right](\phi-\theta)^{3}+\cdots\Big{\\}}\,.$

###  3.5 Antiparallel planes limit

The limit $\pi-\phi=\delta\ll 1$ is not much simpler than the general case but
is interesting because it calculates the potential between antiparallel
planes. It corresponds to taking $J_{\varphi}\to\infty$. To retain nonzero
$\theta$ we should take a double scaling limit $J_{\vartheta}\to\infty$
keeping $q=J_{\vartheta}/J_{\varphi}^{2/3}$ fixed. The polynomial (3.8) then
becomes (with $t=s/J_{\varphi}^{1/3}$)

$\displaystyle
1-4q^{2}s^{4}-s^{6}=-(s^{2}-\tau_{1})(s^{2}-\tau_{2})(s^{2}-s_{\text{max}}^{2})\,.$
(3.34)

To leading order in $J_{\varphi}$, we have

$\displaystyle\vartheta=-4q\int_{0}^{s_{\text{max}}}\frac{s\,\mathrm{d}s}{\sqrt{1-4q^{2}s^{4}-s^{6}}}+\mathcal{O}(J_{\varphi}^{-2/3})\,.\qquad$
(3.35)

Expressing the integral in terms of elliptic integrals (3.22) gives

$\displaystyle\theta=\frac{8q}{\sqrt{s_{\text{max}}^{2}-\tau_{1}}}\mathbf{F}\left(\arcsin\frac{s_{\text{max}}}{\sqrt{s_{\text{max}}^{2}-\tau_{2}}}\,\bigg{|}\,\frac{s_{\text{max}}^{2}-\tau_{2}}{s_{\text{max}}^{2}-\tau_{1}}\right)+\mathcal{O}(J_{\varphi}^{-2/3})\,.$
(3.36)

This is an implicit relation between $q$ and $\theta$. Similarly for $\phi$,
expanding (3.7) to leading order in $J_{\varphi}$ gives (where all the
elliptic integrals have the same arguments as in (3.36))

$\displaystyle\delta=\pi-\phi=2J_{\varphi}^{-1/3}\left[\frac{\tau_{1}}{\sqrt{s_{\text{max}}^{2}-\tau_{1}}}\mathbf{F}+\sqrt{s_{\text{max}}^{2}-\tau_{1}}\mathbf{E}\right]+\mathcal{O}(J_{\varphi}^{-1})\,.$
(3.37)

The action at leading order is

$\displaystyle
S_{\text{M2}}=4NJ_{\varphi}^{2/3}\left(\frac{\tau_{1}}{\sqrt{s_{\text{max}}^{2}-\tau_{1}}}\mathbf{F}+\sqrt{s_{\text{max}}^{2}-\tau_{1}}\mathbf{E}\right)+\mathcal{O}(J_{\varphi}^{0})\,.$
(3.38)

Unlike (3.27), here there are no divergences to treat because the
$\log{\epsilon}$ divergence is subleading in $J_{\varphi}$. Using (3.37) to
solve for $J_{\varphi}$, we find

$\displaystyle
U^{(N)}(\pi-\delta,\theta)=-\frac{8N}{\pi\delta^{2}}\left(\frac{\tau_{1}}{\sqrt{s_{\text{max}}^{2}-\tau_{1}}}\mathbf{F}+\sqrt{s_{\text{max}}^{2}-\tau_{1}}\mathbf{E}\right)^{3}\,.$
(3.39)

The dependence on $\theta$ is implicit. For small $\theta$, $q$ is small and
we can invert (3.36) explicitly

$\displaystyle\theta=\frac{4\sqrt{\pi}\Gamma(1/3)}{3\Gamma(5/6)}q-\frac{16q^{3}}{3}+\mathcal{O}(q^{5})\,.$
(3.40)

Expanding (3.39) at small $q$ and inverting $q(\theta)$ gives

$\displaystyle
U^{(N)}(\pi-\delta,\theta)=-\frac{8N\sqrt{\pi}\Gamma({2}/{3})^{3}}{\Gamma(1/6)^{3}\delta^{2}}$
$\displaystyle\left[1-\frac{3\sqrt{3}}{8\pi}\theta^{2}+\left(1-\frac{3^{5/2}\Gamma(2/3)\Gamma(5/6)^{4}}{2^{7/3}\pi^{3}}\right)\frac{9\theta^{4}}{64\pi^{2}}\right.$
(3.41) $\displaystyle\ +\dots\Big{]}+\frac{2N}{\pi}\log\delta+\dots\,.$

For $\theta=0$, this matches the original calculation of the antiparallel
planes (6.4) [10]. Here we have a perturbative expansion that can easily be
continued to higher orders.

The free field result (2.11) is exact in $\phi$ and $\theta$, but expanding it
around $\phi=\pi$ yields

$U^{(1)}(\pi-\delta,\theta)=\frac{1}{2\pi}\log\sin\delta/2-\frac{\cos\delta+\cos\theta}{8\pi\sin^{2}\delta/2}=-\frac{1+\cos\theta}{2\pi\delta^{2}}+\frac{1}{2\pi}\log\delta+\dots$
(3.42)

Like (3.41), this too has a double pole at $\delta=0$ and a logarithmic
correction.

Note the coefficient of the $\log\delta$ term in the holographic expression
(3.41) is double that of the $\log\cos\phi/2$ part for the BPS expression in
(3.33). That implies that for $\theta=\pi-\delta$, the series above sums up to
$\sim\delta^{2}\log\delta$, to cancel the power law divergence and fix the
prefactor of the log.

##  4 Lightlike crease

A famous limit of the cusped Wilson loop is when the two rays approach the
lightcone in Minkowski space. The resulting anomaly is proportional to the
boost parameter between the lines, with a coefficient known as the universal
cusp anomalous dimension. It governs hard processes in QCD [39, 40] and also
plays a crucial role in ${\mathcal{N}}=4$ SYM, where it is captured by the
spinning string solution [41], the cusped Wilson loop in Minkowski space [27],
and served as the cornerstone for the analysis of scattering amplitudes in
that theory including the BDS ansatz [42] and the Wilson loop scattering
amplitude duality [43].

We present here analog calculations for a lightlike crease, starting by the
analytic continuation of the solution in Section 3 to imaginary angle $\phi$
and then writing down solutions in lorentzian $AdS$ including the analogue of
the 4-cusp solution, which turns out to be two lightcones joined along a
circular lightlike crease.

A different adaptation of the 4-cusp solution to surface operators was studied
in [44]. We comment on the differences below.

###  4.1 Analytic continuation

We first look at the solution in Section 3 and find a limit where $\phi$
diverges. The equation for $\varphi$ (3.7) can be integrated over the zeros of
the polynomial (3.10) in the denominator, and its integral converges unless
the zeros are degenerate. The latter is determined by the vanishing of the
discriminant (3.25)

$\displaystyle
J_{\varphi}^{2}=-\frac{2}{27}\left[1-36J_{\vartheta}^{2}\pm(1+12J_{\vartheta}^{2})^{3/2}\right]\,.$
(4.1)

We want to get imaginary $\phi$, so should take the negative sign and as the
result should not depend on $\theta$, we can take $J_{\vartheta}=0$ and

$\displaystyle J_{\varphi}^{2}=-\frac{4}{27}\,.$ (4.2)

Now we get that $\varphi$ is given by (3.7)

$\displaystyle\varphi^{\prime}=\pm\frac{2it^{3}}{(t^{2}+3)\sqrt{(1+t^{2})(3+4t^{2})}}\,,$
(4.3)

and its integral up to the root $t^{2}=-3$ gives $(\pi-\phi)/2$. With a
regulator $\delta$, we have

$\displaystyle\phi=\pi+4i\int_{0}^{\sqrt{3}i-\delta
i}\frac{t^{3}\mathrm{d}t}{(t^{2}+3)\sqrt{(1+t^{2})(3+4t^{2})}}\,.$ (4.4)

The real part of the integral cancels the factor $\pi$, while the imaginary
part diverges logarithmically as $\delta\to 0$. Its coefficient is the residue
at $t=\sqrt{3}i$, so we get

$\displaystyle\phi=-{i}{\sqrt{2}}\log|\delta|+\dots\to i\infty\,.$ (4.5)

To evaluate the action in that limit, we take the analytic continuation of
(3.26). The integral becomes

$\displaystyle
S_{\text{M2}}=\frac{4N}{\pi}(-2\pi)\int_{\epsilon}^{i\sqrt{3}-i\delta}\frac{3^{3/2}(1+t^{2})^{3/2}\mathrm{d}t}{t^{3}(3+t^{2})\sqrt{3+4t^{2}}}\,.$
(4.6)

Again the pole gives a logarithmic divergence. Taking the residue and
comparing against $\phi$ we find

$\displaystyle
S_{\text{M2}}=8N\frac{\sqrt{2}}{3^{3/2}}\log|\delta|+\dots\to-\frac{8N|\phi|}{3^{3/2}}\,,\qquad\phi\to
i\infty\,.$ (4.7)

As in the case of the lightlike cusp [27], we find a result proportional to
the boost angle $|\phi|$ and the coefficient may be seen as the analogue of
the universal cusp anomalous dimension.

The linear growth of the expectation value at large boost is also present in
the free theory. There the analytic continuation is trivial, simply take
(2.11) in the limit $\phi\to i\infty$ to obtain

$\displaystyle U^{(1)}\to\frac{|\phi|}{2\pi}+\dots$ (4.8)

We do not know whether there is really a meaning to this expression as a
potential and to what extent we should be analytically continuing the solution
for the compact crease rather than the infinite one. To address these points
we turn now to study the system directly in lorentzian signature.

###  4.2 Solutions in lorentzian $AdS$

We now look for solutions to represent the infinite crease in Minkowski space.
We use the metric of lorentzian $AdS_{4}$ subspace of $AdS_{7}\times S^{4}$

$\mathrm{d}s^{2}=\frac{4L^{2}}{z^{2}}(\mathrm{d}z^{2}+(\mathrm{d}x^{1})^{2}+(\mathrm{d}x^{2})^{2}-\mathrm{d}t^{2})=\frac{4L^{2}}{z^{2}}(\mathrm{d}z^{2}+(\mathrm{d}x^{1})^{2}+\mathrm{d}\xi^{2}-\xi^{2}\,\mathrm{d}\chi^{2})\,.$
(4.9)

As world-volume coordinates we take (for $|x^{2}|>|t|$)
$\xi=\sqrt{(x^{2})^{2}-t^{2}}$, $\chi=\operatorname{arccoth}(x^{2}/t)$ and $v$
and the ansatz

$z=u(\chi)\xi\,,\qquad x^{1}=\sqrt{1+u(\chi)^{2}}\,v\,.$ (4.10)

The scaling of $z$ with $\xi$ is natural because of dilatation invariance of
the crease. The scaling of $x^{1}$ with $\sqrt{1+u^{2}}$ may seem unnatural at
first, but it is required to have a full 1d conformal symmetry $SO(2,1)$ as is
discussed in Appendix A. This seems to be one difference from the solution in
[44] and there should be more solutions interpolating between the two.

Given this, the lagrangian is

${\mathcal{L}}=8T_{\text{M2}}L^{3}\frac{\sqrt{(1+u^{2})(1+u^{2}-u^{\prime
2})}}{\xi^{2}u^{3}}\,.$ (4.11)

Ignoring $8L^{3}/\xi^{2}$, the conserved energy is

$E=\frac{(1+u^{2})^{3/2}}{u^{3}\sqrt{1+u^{2}-u^{\prime 2}}}\,.$ (4.12)

Inverting this gives

$\chi=\int\frac{Eu^{3}\,\mathrm{d}u}{\sqrt{(1+u^{2})(E^{2}u^{6}-(1+u^{2})^{2})}}\,.$
(4.13)

In these coordinates we can find the solutions that are the analytic
continuation of those in Appendix A for finite boost angle. The solutions we
get do not cover the entire world volume and they require continuation to
$|x^{2}|\leq|t|$, where we take instead the metric

$\mathrm{d}s^{2}=\frac{4L^{2}}{z^{2}}(\mathrm{d}z^{2}+(\mathrm{d}x^{1})^{2}-\mathrm{d}\xi^{2}+\xi^{2}\,\mathrm{d}\chi^{2})\,,$
(4.14)

where now $\xi=\sqrt{t^{2}-(x^{2})^{2}}$ and
$\chi=\operatorname{arccoth}(t/x^{2})$.

To study solutions in this patch (4.14), the ansatz is

$z=u(\chi)\xi\,,\qquad x^{1}=\sqrt{u(\chi)^{2}-1}\,v\,,$ (4.15)

and the lagrangian becomes

${\mathcal{L}}=8T_{\text{M2}}L^{3}\frac{\sqrt{(u^{2}-1)(u^{2}-1-u^{\prime
2})}}{\xi^{2}u^{3}}\,.$ (4.16)

The simplest solution to the equations of motion has constant $u$, where
imposing that in the Euler-Lagrange equation for (4.16) gives the relation
$u^{2}=3$. In the original coordinate patch this is $u=\pm i\sqrt{3}$, which
also arises near the singularity of (4.4), where the solution becomes
stationary.

For this solution in the new patch, the action becomes

$S_{\text{M2}}=\frac{4N}{3\sqrt{3}\,\pi}\int\frac{\mathrm{d}v\,\mathrm{d}\xi\,\mathrm{d}\chi}{\xi^{2}}\,.$
(4.17)

This is proportional to the area of $AdS_{2}$ times the extent of the
coordinate $\chi$. Comparing to (4.7), we see that they agree, if we replace
the $v$, $\xi$ integral with the integral over global euclidean $AdS_{2}$ with
regularised area $-2\pi$.

###  4.3 Global solutions

The lightlike crease solution above is obtained in the Poincaré patch and can
be embedded in global $AdS$. For the lightlike cusp solution of [27], the
analysis of the global structure of the solution revealed that the lightlike
cusp is in fact related by a conformal transformation to the lightlike Wilson
loop with 4-cusps [28]. Here we find that the single lightlike crease is
conformally related to a surface consisting of two lightcones glued along a
crease.

Following [27, 28], we can embed our solution into global $AdS$ by writing it
in terms of the embedding coordinates

$\displaystyle X_{-1}$
$\displaystyle=\frac{1}{2z}(L^{2}+z^{2}-t^{2}+(x^{i})^{2})\,,\quad$
$\displaystyle X_{0}$ $\displaystyle=\frac{Lt}{z}\,,$ (4.18) $\displaystyle
X_{6}$ $\displaystyle=-\frac{1}{2z}(-L^{2}+z^{2}-t^{2}+(x^{i})^{2})\,,\quad$
$\displaystyle X_{i}$ $\displaystyle=\frac{Lx^{i}}{z}\,,\quad$

as

$X_{0}^{2}-X_{2}^{2}=\frac{L^{2}}{3}\,,\qquad X_{3}=X_{4}=X_{5}=0\,,$ (4.19)

and then $X_{-1}^{2}-X_{1}^{2}-X_{6}^{2}=2L^{2}/3$. This can be written
homogenously as

$X_{-1}^{2}-X_{1}^{2}-X_{6}^{2}=2X_{0}^{2}-2X_{2}^{2}\,,\qquad
X_{3}=X_{4}=X_{5}=0\,.$ (4.20)

Conformal transformations are given by rotations of the embedding coordinates,
and we can now consider the rotated solution

$X_{0}^{2}-X_{1}^{2}-X_{2}^{2}=2X_{-1}^{2}-2X_{6}^{2}\,,\qquad
X_{3}=X_{4}=X_{5}=0\,.$ (4.21)

In the new Poincaré patch, this new solution is

$z=\sqrt{\frac{3}{2}(t^{2}-r^{2})}\,,$ (4.22)

with $r=\sqrt{(x^{1})^{2}+(x^{2})^{2}}$. For $z=0$, this is a two-dimensional
lightcone $t=r$.

We can write its action as

$\displaystyle
S_{\text{M2}}=\frac{4N}{3\pi}\sqrt{\frac{2}{3}}\int\frac{\sinh\rho}{z}\mathrm{d}z\,\mathrm{d}\rho\,\mathrm{d}v\,,$
(4.23)

where $\rho=\operatorname{arccoth}(t/r)$ and $v$ the angle in the
$(x^{1},x^{2})$ plane parametrising an $AdS_{2}$ at fixed $z$. The integral
over $AdS_{2}$ is regularised to $-2\pi$ and the integral over $z$ diverges as
$\log{z}$. This agrees with the analytic continuation (4.7) if we identify the
$\log{z}$ divergence with $|\phi|/\sqrt{2}$.

To obtain a compact surface, we start again with (4.21), swap $X_{6}$ with
$X_{3}$ and map $X_{0}\to(X_{0}+X_{-1})/\sqrt{2}$ and
$X_{-1}\to(X_{-1}-X_{0})/\sqrt{2}$. We find

$2X_{-1}^{2}-X_{0}^{2}\to\frac{X_{-1}^{2}+X_{0}^{2}-6X_{-1}X_{0}}{2}=2X_{3}^{2}-X_{1}^{2}-X_{2}^{2}\,,\qquad
X_{4}=X_{5}=X_{6}=0\,.$ (4.24)

This can be recast as

$\frac{z^{2}}{3}=\frac{1}{2}(t-L)^{2}-(x^{3})^{2}\,,\qquad
r^{2}=4L^{2}+2(x^{3})^{2}-\frac{1}{2}(t-3L)^{2}\,.$ (4.25)

Setting $z=0$ gives the surface on the boundary parametrised by $t$ (and angle
$v$) with two branches

$\displaystyle x^{3}=\pm\frac{L-t}{\sqrt{2}}\,,\qquad
r=\frac{L+t}{\sqrt{2}}\,.$ (4.26)

The physical domain is $-L<t<L$, so $r,|x^{3}|<\sqrt{2}L$. Each branch
parametrises a lightcone emanating from $t=-L$ and $x^{3}=\pm\sqrt{2}L$ and
merging along a crease at $t=L$, $x^{3}=0$ and $r=\sqrt{2}L$.

It’s convenient to parametrise the solution by $z$, $x_{3}$ and $v$, such that
the action is

$\displaystyle
S_{\text{M2}}=\frac{4LN}{\sqrt{3}\pi}\int\frac{\mathrm{d}z\,\mathrm{d}x_{3}\,\mathrm{d}v}{z^{2}\sqrt{z^{2}+3x_{3}^{2}}}\,.$
(4.27)

There are two sources of divergence. First when $z$ goes to $0$, which is the
usual divergence from the volume of $AdS$, and then in addition when also
$x_{3}\to 0$, which is associated with the crease singularity. Introducing a
cutoff $z>\epsilon$ regularises both. To perform the integral, notice that the
M2-brane closes in $AdS$ when $r=0$, which sets the range of $x_{3}$ and $z$
to be

$\displaystyle x_{3}^{2}<\frac{(z-2\sqrt{3}L)^{2}-6L^{2}}{3}\,,\qquad
z<\sqrt{6}(\sqrt{2}-1)L\,.$ (4.28)

Integrating we find

$\displaystyle
S_{\text{M2}}=-\frac{16N}{3}\frac{L}{\epsilon}\log{\epsilon}+\text{finite}.$
(4.29)

This result is peculiar and is in contradiction to our assertion in Section
2.2 that creases do not suffer from $\epsilon^{-1}\log{\epsilon}$ divergences.
It deserves further exploration.

An alternative regularisation which preserves the $AdS_{2}$ symmetry is to
write the lorentzian $AdS_{7}$ as a foliation over $AdS_{2}\times dS_{4}$, as
in the case of the compact crease in Section 3.1. This is the analytic
continuation in $\nu$ of (3.1). Keeping only the timelike coordinate from the
de Sitter component, the metric is

$ds^{2}=L^{2}\left[d\nu^{2}+\cos^{2}\nu(\mathrm{d}\rho^{2}+\sinh^{2}\rho\,\mathrm{d}v^{2})-\sin^{2}\nu\,\mathrm{d}\varphi^{2}\right].$
(4.30)

The boundary of space is at $\varphi\to\pm\infty$ or $\rho\to\infty$. Taking
$\nu(\alpha)$ the action is

${\mathcal{L}}=8T_{\text{M2}}L^{3}\cosh\rho\cos^{2}\nu\sqrt{\sin^{2}\nu-\nu^{\prime
2}}\,.$ (4.31)

Looking for constant $\nu$ solutions gives $\sin^{2}\nu=1/3$ and regularising
the $AdS_{2}$ area leads to the action

$\displaystyle S_{\text{M2}}=-\frac{8N}{3\sqrt{3}}\int d\varphi,$ (4.32)

which again agrees with the analytic continuation of the potential (4.7) and
the action of the lightlike crease (4.17).

##  5 Defect CFT

As the crease preserves part of the conformal group, it is natural to study it
from a defect CFT perspective [45, 46, 47, 48, 49, 50]. In particular, we use
these techniques to find expressions for the potential as an expansion around
BPS configurations, including quadratic order around the plane, producing
(1.14) and linear order around the $\theta=\phi$ cusp, resulting in (1.15).
This approach does not rely on a particular realisation of the
${\mathcal{N}}=(2,0)$ theory and the surface operators, so is applicable to
theories with any $ADE$ algebra and surface operators in any representation of
the algebra.

The principle at action is very similar to that of the bremsstrahlung function
$B(\lambda)$ and “generalised bremsstrahlung function” $B(\phi,\lambda)$ for
cusped Wilson loops in ${\mathcal{N}}=4$ SYM (with multiple generalisations)
[13, 16, 51, 52].

A superconformally invariant surface has a distinguished set of operators
restricted to the defect, known as the displacement operator multiplet, with
conformal primaries: $\mathbb{D}^{m}$ ($m=3,\dots,6$) for broken translations,
$\mathbb{O}^{i}$ ($i=2,\dots,5$) for broken R-symmetries and $\mathbb{Q}$ for
broken supercharges [53, 54, 55, 56, 6]. For the spherical surface operator,
they can be expressed as contact terms for the conservation equations for the
energy momentum tensor $T^{\mu\nu}$ and R-current $j^{\mu IJ}$ as

$\displaystyle\partial_{\mu}T^{\mu\nu}(x_{\parallel},x_{\perp})V_{S^{2}}$
$\displaystyle=e^{\nu}_{n}V_{S^{2}}[\mathbb{D}^{n}(x_{\parallel})]\delta^{(4)}(x_{\perp})\,,$
(5.1) $\displaystyle\partial_{\mu}j^{\mu
i1}(x_{\parallel},x_{\perp})V_{S^{2}}$
$\displaystyle=V_{S^{2}}[\mathbb{O}^{i}(x_{\parallel})]\delta^{(4)}(x_{\perp})\,.$
(5.2)

There is also a similar equation for broken supersymmetries $\mathbb{Q}$. Here
$x_{\parallel}$ are the coordinates along the surface and $x_{\perp}$ are
normal. Furthermore, we use the notation $V[\mathbb{D}]$ to denote the
insertion of the operator $\mathbb{D}$ on the defect. $e^{\nu}_{n}$ are the
vielbein restricted to the normal space at the point $x_{\parallel}$.

These operators are special because, as we review below, their correlators
capture the expectation value of small deformations of the conformal defect.
We start with the case of deformations of the sphere to find (1.14), and then
proceed to study the deformations away from the BPS crease at finite
$\phi=\theta$ to derive (1.15).

###  5.1 Near sphere expansion

Consider a surface which is a small deformation of a sphere. This can be
obtained by a local deformation along a normal vector
$\xi_{\nu}(x_{\parallel})$. It can be expressed as a six dimensional integral
over the derivative of the current $\partial_{\mu}T^{\mu\nu}$, and including
also a local R-symmetry deformation $\omega^{i}$ leads to a formal expression
defining the operator $V_{\xi,\omega}$

$\displaystyle
V_{\xi,\omega}=\exp\left[\int\mathrm{d}^{6}x\left(\xi_{\nu}\partial_{\mu}T^{\mu\nu}+\omega^{i}\partial_{\mu}j^{\mu
1i}\right)\right]V_{S^{2}}\,.$ (5.3)

Clearly if $\xi$ and $\omega$ are constants, the exponential simply expresses
a global translation/R-symmetry rotation. As we allow $\xi$, $\omega$ to vary
along the sphere, it describes a local deformations that can affect the
expectation value. We adopt spherical coordinates for $\mathbb{R}^{6}$

$\displaystyle\mathrm{d}s^{2}=\mathrm{d}r^{2}+r^{2}\left(\mathrm{d}u^{2}+\cos^{2}{u}\,\mathrm{d}v^{2}+\sin^{2}{u}\left(\mathrm{d}\theta_{3}^{2}+\cos^{2}{\theta_{3}}\left(\mathrm{d}\theta_{2}^{2}+\cos^{2}{\theta_{2}}\,\mathrm{d}\theta_{1}^{2}\right)\right)\right),$
(5.4)

and the sphere is located at $\theta_{1,2,3}=0$, $r=R$. $\xi$, $\omega$ are
now functions of $u$, $v$, and since $\xi$ is a normal vector, it can have
components $\partial_{\theta_{1}},\partial_{\theta_{2}},\partial_{\theta_{3}}$
and $\partial_{r}$.

We can calculate the expectation value of $V_{\xi,\omega}$ for small $\xi$,
$\omega$ by expanding the exponential in (5.3). At the linear order, using
(5.2), we get insertions of displacement operators into the surface operator

$\displaystyle
R^{2}\int\cos{u}\,\mathrm{d}u\,\mathrm{d}v\,\left(\xi_{\nu}(u,v)e^{\nu}_{n}V_{S^{2}}[\mathbb{D}^{n}(u,v)]+\omega^{i}(u,v)V_{S^{2}}[\mathbb{O}^{i}(u,v)]\right).$
(5.5)

The factor $R^{2}\cos{u}$ comes from the metric on the sphere of radius $R$.

These terms do not contribute to the expectation value since 1-point functions
vanish by symmetry. The first nontrivial contribution is instead at quadratic
order, where we find pairs of displacement operators from the currents acting
on $V$. In addition there are contact terms from the currents acting on defect
operators. This can be seen as delta function contributions to the
$\mathbb{D}\mathbb{D}$ and $\mathbb{O}\mathbb{O}$ OPEs, giving

$\displaystyle\partial_{\mu}j^{\mu
i1}(u,v,x^{\perp})V_{S^{2}}[\mathbb{O}^{j}(u^{\prime},v^{\prime})]$
$\displaystyle=V_{S^{2}}[\mathbb{O}^{i}(u,v)\mathbb{O}^{j}(u^{\prime},v^{\prime})]\delta^{(4)}(x^{\perp})$
$\displaystyle\quad+\frac{\alpha}{R^{2}}V_{S^{2}}[\mathds{1}]\delta^{ij}\delta^{(4)}(x^{\perp})\delta(\sin(u-u^{\prime}))\delta(v-v^{\prime})\,,$
(5.6)
$\displaystyle\partial_{\mu}T^{\mu\nu}(u,v,x^{\perp})V_{S^{2}}[\mathbb{D}^{n}(u^{\prime},v^{\prime})]$
$\displaystyle=V_{S^{2}}[e^{\nu}_{m}\mathbb{D}^{m}(u,v)\mathbb{D}^{n}(u^{\prime},v^{\prime})]\delta^{(4)}(x^{\perp})$
$\displaystyle\quad+\frac{\beta}{R^{4}}V_{S^{2}}[\mathds{1}]e^{\nu}_{m}\delta^{mn}\delta^{(4)}(x^{\perp})\delta(\sin(u-u^{\prime}))\delta(v-v^{\prime})\,.$
(5.7)

The contact terms come with unknown prefactors $\alpha$ and $\beta$ and only
the identity part of the singular OPE is retained, since all other 1-point
functions vanish. These terms come with appropriate powers fo $R$, on
dimensional grounds, so they don’t appear in the dCFT description of the plane
in [6].

Using (5.1), (5.1) we obtain the change in expectation value for arbitrary
$\xi(u,v)$, $\omega(u,v)$

$\displaystyle\left\langle V_{\xi,\omega}\right\rangle-\left\langle
V_{S^{2}}\right\rangle=$ $\displaystyle\ \frac{R^{4}}{2}\int\cos
u\,\mathrm{d}u\,\mathrm{d}v\cos
u^{\prime}\,\mathrm{d}u^{\prime}\,\mathrm{d}v^{\prime}\,\omega^{i}\omega^{j}\left\langle
V_{S^{2}}[\mathbb{O}^{i}(u,v)\mathbb{O}^{j}(u^{\prime},v^{\prime})]\right\rangle$
(5.8) $\displaystyle+\frac{R^{4}}{2}\int\cos u\,\mathrm{d}u\,\mathrm{d}v\cos
u^{\prime}\,\mathrm{d}u^{\prime}\,\mathrm{d}v^{\prime}\,\xi^{\mu}e_{\mu}^{m}\xi^{\nu}e_{\nu}^{n}\left\langle
V_{S^{2}}[\mathbb{D}^{m}(u,v)\mathbb{D}^{n}(u^{\prime},v^{\prime})]\right\rangle$
$\displaystyle+\frac{1}{2}\left\langle
V_{S^{2}}\right\rangle\int\left(\alpha\omega^{2}+\frac{\beta\xi^{2}}{R^{2}}\right)\cos
u\,\mathrm{d}u\,\mathrm{d}v+\dots$

The 2-point functions of $\mathbb{D}$ and $\mathbb{O}$ that enter this
expression are fixed by their dimensions 3 and 2 respectively and for the
spherical surface are

$\displaystyle\frac{\left\langle
V_{S^{2}}[\mathbb{D}^{m}(u,v)\mathbb{D}^{n}(u^{\prime},v^{\prime})]\right\rangle}{\left\langle
V_{S^{2}}\right\rangle}$
$\displaystyle=\frac{C_{\mathbb{D}}}{8\pi^{2}R^{6}}\frac{\delta^{mn}}{(1-\sin
u\sin u^{\prime}-\cos u\cos u^{\prime}\cos(v-v^{\prime}))^{3}}\,,$ (5.9)
$\displaystyle\frac{\left\langle
V_{S^{2}}[\mathbb{O}^{i}(u,v)\mathbb{O}^{j}(u^{\prime},v^{\prime})]\right\rangle}{\left\langle
V_{S^{2}}\right\rangle}$
$\displaystyle=\frac{C_{\mathbb{O}}}{4\pi^{2}R^{4}}\frac{\delta^{ij}}{(1-\sin
u\sin u^{\prime}-\cos u\cos u^{\prime}\cos(v-v^{\prime}))^{2}}\,.$ (5.10)

The normalisation of $\mathbb{D}$ is fixed by (5.2), so $C_{\mathbb{D}}$ and
$C_{\mathbb{O}}$ on the right hand side are not arbitrary and are part of the
dCFT data. In fact, for any surface operator in a CFT, it was shown in [57]
that $C_{\mathbb{D}}$ is related to the coefficient $a_{2}$ entering the
conformal anomaly (1.8) as $C_{\mathbb{D}}=-16a_{2}$. For the
${\mathcal{N}}=(2,0)$ theory, supersymmetry implies that $a_{2}=-c$ and the
normalisation of the 2-point function of $\mathbb{O}$ is given by
$C_{\mathbb{O}}=c$ [6].

It is now a simple matter to fix $\alpha$ and $\beta$. Taking $\xi$ to be any
conformal Killing vector, for example $\partial_{\theta_{1}}$, leads to a
global rotation which should not change the expectation value. This fixes the
constant $\beta$. We can also fix $\alpha$ by taking $\omega^{i}=\delta^{i2}$
to generate a constant R-symmetry rotation. Performing the integrals (5.8)
with these values of $\xi$, $\omega$, we find

$\displaystyle\alpha=\frac{c}{4\pi}\,,\qquad\beta=-\frac{3c}{2\pi}\,.$ (5.11)

With $\alpha$, $\beta$ in hand, we now evaluate (5.8) for the case of the
crease of angle $\phi$, $\theta$ as defined in (1.4) and (1.3). We take

$\displaystyle\omega^{i}=\begin{cases}\theta\delta^{i2}\,,&u>0\\\ 0\,,&u<0\\\
\end{cases}\,,\qquad\xi^{\mu}e_{\mu}^{m}=\begin{cases}\phi
e_{\theta_{1}}^{m}=\phi R\sin{u}\delta^{6m}\,,&u>0\\\ 0\,,&u<0\\\
\end{cases}\,,$ (5.12)

which rotates one of the hemisphere by an angle $\phi$ and the scalar coupling
by $\theta$. Plugging these in (5.8) and performing the integrals we obtain

$\displaystyle\frac{\left\langle V_{\phi,\theta}\right\rangle}{\left\langle
V_{S^{2}}\right\rangle}=1+\frac{c}{4}(\theta^{2}-2\phi^{2})+\dots$ (5.13)

from which (1.14) follows directly. This is also in agreement with the
expansion of the explicit results (2.11) and (3.19).

###  5.2 Near-BPS expansion

The previous subsection relied on the defect CFT description of the
deformation of the sphere. To study the crease with finite angles $\phi$,
$\theta$, we now turn to the formulation of the defect CFT description of
deformations of the crease itself.

It is easiest to derive the formalism for the infinite crease and then use a
conformal transformation to apply it to the spherical crease. The symmetries
of the infinite crease (1.2) are
$\mathfrak{so}(2,1)\oplus\mathfrak{so}(3)\oplus\mathfrak{so}(3)$,
corresponding respectively to the 1d conformal group acting on both half-
planes, transverse rotations in $x^{4,5,6}$ and transverse R-symmetry
rotations of $n^{3,4,5}$. Note that from the point of view of symmetries, this
setup is identical to inserting a line operator inside the plane. According to
the classification of [58], this is a supersymmetric line defect breaking the
transverse rotational symmetry.

On the sphere or plane, $\mathbb{D}^{m}$ and $\mathbb{O}^{i}$ are quartets of
a pair of $\mathfrak{so}(4)$ symmetries, broken to $\mathfrak{so}(3)$ by the
crease. This singles out one displacement $\mathbb{D}^{3}$ (for the crease in
the $(2,3)$ plane extended in the $x^{1}$ direction) and one R-rotation
$\mathbb{O}^{2}$ that are singlets and can therefore have expectation values.
Another way to say this is that these operators are no longer primaries and
they mix with the identity. Explicitly we have

$\displaystyle\left\langle
V_{\phi,\phi}[\mathbb{D}^{3}(x_{2})]\right\rangle=\frac{h_{\mathbb{D}}(\phi)}{x_{2}^{3}}\,,$
(5.14)

with the power in the denominator fixed by the conformal dimension of
$\mathbb{D}$ and $h_{\mathbb{D}}$ is an unknown constant. Likewise,

$\displaystyle\left\langle
V_{\phi,\phi}[\mathbb{O}^{2}(x_{2})]\right\rangle=\frac{h_{\mathbb{O}}(\phi)}{x_{2}^{2}}\,.$
(5.15)

In addition to these bosonic symmetries, when $\phi=\theta$ the crease
preserves $4{\mathsf{Q}}$’s which offer more relations. Using $\gamma$ for the
6d gamma matrices and $\rho$ for the R-symmetry matrices (see [7] for
notations), the unbroken supersymmetries are those satisfying

$\displaystyle(1+i\rho_{1}\gamma_{12}){\mathsf{Q}}_{+}=(1+i\rho_{2}\gamma_{13}){\mathsf{Q}}_{+}=0\,.$
(5.16)

These are 2 independent constraints, so they select 4 out of the 16
supersymmetries of the ${\mathcal{N}}=(2,0)$ theory. Similarly, there are also
4 preserved ${\mathsf{S}}$ conformal supercharges. Note that in this equation
and below we suppress the spinor indices.

Along with the bosonic part, the symmetries assemble into
$\mathfrak{osp}(4^{*}|2)$ and impose further constraints on the correlators
(5.14) and (5.15) relating $h_{\mathbb{D}}$ and $h_{\mathbb{O}}$. To derive
them, consider acting with one of the supersymmetry ${\mathsf{Q}}_{+}$ on the
one-point function

$\displaystyle\left\langle{\mathsf{Q}}_{+}V_{\phi,\phi}[\mathbb{Q}]\right\rangle=0\,.$
(5.17)

Away from $x_{2}=0$, the transformations under supersymmetry of $\mathbb{Q}$
are the same as on the plane, which were obtained in [6]. They read (with
$a=1,2$)

$\displaystyle\begin{split}{\mathsf{Q}}\mathbb{D}_{m}&=\frac{1}{2}\gamma_{am}\partial^{a}\mathbb{Q}\,,\\\
{\mathsf{Q}}\mathbb{Q}&=2\gamma_{m}\mathbb{D}^{m}+2\rho_{1i}\gamma_{a}\partial^{a}\mathbb{O}^{i}\,,\\\
{\mathsf{Q}}\mathbb{O}_{i}&=\frac{1}{2}\rho_{1i}\mathbb{Q}\,,\end{split}$
(5.18)

for any of the supersymmetries ${\mathsf{Q}}$ preserved by the plane.
Restricting these transformations to the subset ${\mathsf{Q}}_{+}$ also
preserved by the crease, we can evaluate (5.17) to find

$\displaystyle\left\langle{\mathsf{Q}}_{+}V_{\phi,\phi}[\mathbb{Q}]\right\rangle=2\left[\gamma_{3}\left\langle
V_{\phi,\phi}[\mathbb{D}^{3}]\right\rangle-\gamma_{23}\gamma_{a}\partial^{a}\left\langle
V_{\phi,\phi}[\mathbb{O}^{2}]\right\rangle\right]=0\,.$ (5.19)

To obtain this, we used (5.16) to eliminate $\rho$ in favor of $\gamma$
matrices. Clearly the derivative is only nonzero along $x_{2}$. We are left
with

$\displaystyle\left\langle
V_{\phi,\phi}[\mathbb{D}^{3}(x_{2})]\right\rangle=-\partial_{2}\left\langle
V_{\phi,\phi}[\mathbb{O}^{2}(x_{2})]\right\rangle\qquad\Leftrightarrow\qquad
h_{\mathbb{D}}(\phi)=2h_{\mathbb{O}}(\phi)\,.$ (5.20)

We can now use this constraint to study small deformations of the spherical
crease. A conformal transformation of the above relations now gives

$\displaystyle\frac{\left\langle
V_{\phi,\phi}[\mathbb{D}^{3}(u,v)]\right\rangle}{\left\langle
V_{\phi,\phi}\right\rangle}=\frac{h_{\mathbb{D}}(\phi)}{R^{3}\sin^{3}{u}}\,,\qquad\frac{\left\langle
V_{\phi,\phi}[\mathbb{O}^{2}(u,v)]\right\rangle}{\left\langle
V_{\phi,\phi}\right\rangle}=\frac{h_{\mathbb{O}}(\phi)}{R^{2}\sin^{2}{u}}\,.$
(5.21)

Following the same logic as in Section 5.1, we can relate the constants
$h_{\mathbb{D}}$ and $h_{\mathbb{O}}$ to derivatives of the potential. For
instance, integrating the one-point function of $\mathbb{O}^{2}$ calculates
the small angle expansion around the BPS case

$\displaystyle\left\langle V_{\phi,\theta}\right\rangle=\left\langle
V_{\phi,\phi}\right\rangle-R^{2}(\theta-\phi)\int_{0}^{\pi/2}\cos{u}\,\mathrm{d}u\int_{0}^{2\pi}\mathrm{d}v\left\langle
V_{\phi,\phi}[\mathbb{O}^{2}(u)]\right\rangle+\dots$ (5.22)

Plugging (5.21) and integrating we find

$\displaystyle\frac{1}{\pi}\frac{\mathrm{d}}{\mathrm{d}\theta}\exp\left(2\pi
U(\phi,\theta)\right)\bigg{|}_{\theta=\phi}=-2h_{\mathbb{O}}(\phi)\int_{0}^{\pi/2}\frac{\cos{u}\,\mathrm{d}u}{\sin^{2}{u}}\int_{0}^{2\pi}\mathrm{d}v\,.$
(5.23)

The right-hand side contains the integral over $AdS_{2}$ in the disc topology.
After regularisation this integrates to $-2\pi$. We then obtain

$\displaystyle
h_{\mathbb{O}}(\phi)=\frac{1}{4\pi^{2}}\frac{\mathrm{d}}{\mathrm{d}\theta}\exp\left(2\pi
U(\phi,\theta)\right)\bigg{|}_{\theta=\phi}\,.$ (5.24)

In a similar way $h_{\mathbb{D}}$ is related to the derivative with respect to
$\phi$. Integrating the one-point function gives

$\displaystyle\left\langle V_{\phi,\theta}\right\rangle=\left\langle
V_{\theta,\theta}\right\rangle+R^{2}(\phi-\theta)\int_{0}^{\pi/2}\cos{u}\,\mathrm{d}u\int_{0}^{2\pi}\mathrm{d}v\,e_{\theta_{1}}^{3}\left\langle
V_{\phi}[\mathbb{D}^{3}(u)]\right\rangle+\dots$ (5.25)

Again the integral gives the volume of $AdS_{2}$ and leads to

$\displaystyle
h_{\mathbb{D}}(\phi)=-\frac{1}{4\pi^{2}}\frac{\mathrm{d}}{\mathrm{d}\phi}\exp\left(2\pi
U(\phi,\theta)\right)\bigg{|}_{\theta=\phi}\,.$ (5.26)

Finally, comparing (5.24) with (5.26) and using (5.20) we obtain a relation
between the first derivatives of the potential

$\displaystyle\frac{\mathrm{d}U(\phi,\theta)}{\mathrm{d}\phi}\bigg{|}_{\phi=\theta}=-2\frac{\mathrm{d}U(\phi,\theta)}{\mathrm{d}\theta}\bigg{|}_{\theta=\phi}\,.$
(5.27)

Therefore, expanding the potential in Taylor series, we find

$\displaystyle U(\phi,\theta)$
$\displaystyle=U(\phi,\phi)+(\theta-\phi)\frac{\mathrm{d}U(\phi,\theta)}{\mathrm{d}\theta}\bigg{|}_{\theta=\phi}+\dots$
(5.28)
$\displaystyle=U(\phi,\phi)+(\phi-\theta)\frac{dU(\phi,\phi)}{\mathrm{d}\phi}+\dots$

This proves (1.15).

The explicit expressions in (1.13) with $C=c$ then gives

$h_{\mathbb{D}}=2h_{\mathbb{O}}=\frac{c}{2\pi^{2}}\cos^{2c}\frac{\phi}{2}\tan\frac{\phi}{2}=-\partial_{\phi}\left[\frac{1}{4\pi^{2}}\cos^{2c}\frac{\phi}{2}\right].$
(5.29)

It would be interesting to try to derive this relation between
$h_{\mathbb{D}}$ and $c$, which would then prove that the leading form of the
potential is indeed (1.13).

##  6 Conclusion

In this paper we defined and studied several realisations of the crease
surface observable in the ${\mathcal{N}}=(2,0)$ theory. We were able to define
a quantity $U(\phi,\theta)$, which can be thought of as a surface-antisurface
potential, and calculated it in the abelian theory and using classical
M2-branes in holography.

For $\theta=\phi$ the surfaces preserve at least eight supercharges and the
function $U(\phi,\phi)$ takes the simple form (1.12) proportional to
$\log\cos(\phi/2)$ in both calculations. Furthermore, the proportionality
constant $C$ is equal in both cases to the anomaly coefficient $c$ in (1.9),
(1.10), leading us to conjecture that it is true at any $N$.

It is natural to draw an analogy with cusped Wilson loops in supersymmetric
gauge theories and in particular ${\mathcal{N}}=4$ SYM. One clear point of
deviation are the anomalies—Wilson loop are anomalous only in the presence of
a cusp, while surface observables are anomalous also for the spherical
geometry. Furthermore, the natural quantity to associate to the cusp is its
anomaly which depends on its angle, while the surface anomaly is independent
of the angle meaning that the ratio $\left\langle
V_{\phi,\phi}\right\rangle/\left\langle V_{S^{2}}\right\rangle$ is well-
defined and finite.

The comparison is most straightforward then in the BPS case when the cusped
Wilson loop is not anomalous, but still can provide a finite expectation
value, when taking the two longitude configuration on $S^{2}$ (or a compact
cusp), for which the expectation value is related to that of the circular
Wilson loop $W_{\circ}(N,\lambda)$ [59, 60, 61, 62] as

$\left\langle W_{\phi,\phi}(N,\lambda)\right\rangle=\left\langle
W_{0,0}(N,\tilde{\lambda})\right\rangle\equiv\left\langle
W_{\circ}(N,\tilde{\lambda})\right\rangle,\qquad\tilde{\lambda}=\left(1-\frac{\phi^{2}}{\pi^{2}}\right)\lambda\,,$
(6.1)

and the expectation value of the circular Wilson loop is [23, 24, 25]

$\left\langle
W_{\circ}(N,\lambda)\right\rangle=\frac{1}{N}L^{1}_{N-1}\left(-\frac{\lambda}{4N}\right)e^{\frac{\lambda}{8N}}\,.$
(6.2)

For the surface operator, we have the potential as calculated from the
spherical crease in equation (1.12)
$U(\phi,\phi)=\frac{c}{\pi}\log\cos(\phi/2)$ with the proportionality constant
$c$ being the anomaly coefficient (1.10).

The BPS Wilson loop depends on three parameters: $N$, $\lambda$ and $\phi$,
but the latter are combined into the effective coupling $\tilde{\lambda}$.
Still the resulting expression has a rich interplay of $N$ and
$\tilde{\lambda}$, while in the absence of a marginal coupling in the
${\mathcal{N}}=(2,0)$ theory, the BPS potential is a factorised function of
the integer $N$ and continuous angle $\phi$. For surfaces in higher
representations, the expression (1.10) gets modified accordingly [20, 21, 22],
and the same is true for the Wilson loop (6.2) [63, 64, 65, 66, 67, 68, 69,
70].

Although we couldn’t prove (1.12) for any $N$, we did prove rigorously that
for small $\phi$ the potential has the form (1.14), with $C=c$, as well as the
relation (1.15) for the near-BPS behaviour.

This is also completely analogous to the cusped Wilson loop. While the
functions that appear are different, in both cases the near-BPS expansion is
given by the derivative with respect to the angle $\phi$ of the BPS result.
For the cusped Wilson loop the generalised bremsstrahlung function [13] can be
written as

$\displaystyle B(\phi,\lambda,N)=-\frac{1}{2}\partial_{\phi}\log\left\langle
W_{\phi,\phi}(N,\lambda)\right\rangle\,,$ (6.3)

and the result for the crease (1.13), (1.15) is identical. This relation for
the near BPS expansion is guaranteed by the defect CFT analysis in Section
5.2, based on the residual supersymmetry preserved by the crease. In the
presence of the crease, there is a displacement operator $\mathbb{D}$ that
develops a 1-point function, and likewise for its superpartner $\mathbb{O}$.
Their expectation value calculates first derivatives of the potential, and the
relations between them allows to derive (1.15).

Going beyond the quadratic order around the plane, or beyond linear deviations
from the crease requires far more complicated calculations. In the former, it
requires the 4-point function of displacements and in the latter the 2-point
function of the singlet displacement. The 4-point function of the displacement
operator in the Wilson loop is well studied [71, 72, 73, 74] and nontrivial
relations among them were found in [75, 76]. The same was done for the surface
operators in [5, 76], but it certainly deserves further study.

The potential $U(\phi,\theta)$ is well-defined also away from the BPS limit
and we calculated it both in the free theory and at large $N$. Here one would
wish to invoke the analogy to the generalised cusp anomalous dimension of
[12], with the caveat, that the latter is an anomaly while the divergence in
the potential is regularised either to zero in the case of the infinite
crease, or to a finite $-2\pi$ (multiplying $U$) in the case of the spherical
crease.

Regardless of this analogy, $U(\phi,\theta)$ allows to interpolate between the
plane and the antiparallel planes (or more properly between the sphere and to
coincident hemispheres). For planes separated a distance $D$ and with opposite
orientation, it was found in [38] that the potential density is

$\mathcal{E}=-\frac{8N\sqrt{\pi}\,\Gamma(2/3)}{\Gamma(1/6)^{3}}\frac{1}{D^{2}}\,.$
(6.4)

The dependence on the distance is fixed by conformal symmetry but the
prefactor, which is not, was one of the first predictions of the $AdS$/CFT
correspondence. Among other things, we generalise this expression to include
an R-symmetry angle $\theta$ between the planes (3.41). We also obtained the
explicit M2-brane for any angle and can calculate the potential for arbitrary
$\phi$ and $\theta$, although the dependence on the angles is implicit.

Another natural limit is $\phi\to i\infty$, for which the potential grows
linearly in $|\phi|$ (4.7), with no obvious relation between the prefactors at
$N=1$ and $N\to\infty$. This limit can be interpreted as calculating the
action for the lighlike crease in Minkowski space, which we verify by
obtaining the minimal M2-brane in lorentzian $AdS$ and comparing its action to
the potential. By embedding the lightlike crease solution in global $AdS$ we
also obtain a solution describing two lightcones glued along a crease, which
is the analogous to the 4-cusp lightlike Wilson loop of [27, 28]. It would be
interesting to see if the lightlike cones and creases have a physical
interpretation similar to the lightlike Wilson loops, which calculate
scattering amplitudes in ${\mathcal{N}}=4$ SYM. The lightlike cusp is also
related by an analytic continuation [77] to the rotating string [41], and it
would be interesting to check whether the solutions presented in Section 4 are
related to the rotating membrane solutions obtained in [78, 79].

In this paper we found finite, well-defined observables associated to surface
operators in the ${\mathcal{N}}=(2,0)$ theory, including some BPS protected
observables and some that are not. There are many more BPS protected
observables found in [7] that merit further study. And many of them may have
natural non-BPS extensions. Unraveling them offers a window into the dynamics
of this mysterious theory.

To point out just one example, there is a family of 1/4 BPS cones
interpolating between the plane and the cylinder. The BPS cylinder which was
studied in [8] has a finite nonzero expectation value. The classical M2-brane
solutions for the cones of finite angles, whether BPS or not have not been
found yet. It should be noted that they are related under dimensional
reduction to BPS “latitude” Wilson loops of 5d SYM [17] on $S^{5}$.

Another natural direction is to study creases in surface operators in
different theories in varied dimensions.

## Acknowledgements

We would like to thank Lorenzo Di Pietro for illuminating discussions, and
Arkady Tseytlin, Yifan Wang for their helpful comments on the manuscript. MT
gratefully acknowledges support from King’s College London, Université Laval,
the Simons Center for Geometry and Physics, Stony Brook University and New
York University, where part of this project was realised. His visit to KCL was
supported by ERC Consolidator Grant No. 681908, “Quantum black holes: A
microscopic window into the microstructure of gravity.” This research was
supported in part by Perimeter Institute for Theoretical Physics. Research at
Perimeter Institute is supported by the Government of Canada through the
Department of Innovation, Science and Economic Development and by the Province
of Ontario through the Ministry of Research and Innovation. ND’s research is
supported by the Science Technology & Facilities council under the grants
ST/T000759/1 and ST/P000258/1.

##  Appendix A Infinite crease in the Poincaré patch

###  A.1 Solution

Most of the paper is focused on the compact crease and the holographic section
(Section 3) used the $AdS_{2}\times S^{1}$ picture. It is interesting to redo
the calculation for the infinite crease whose holographic description should
be in the Poincaré patch.

As before, we can focus on an $AdS_{4}\times S^{1}$ subspace of $AdS_{7}\times
S^{4}$ and now use the metric

$\mathrm{d}s^{2}=\frac{y}{L}\left((\mathrm{d}x^{1})^{2}+\mathrm{d}r^{2}+r^{2}\,\mathrm{d}\varphi^{2}\right)+\frac{L^{2}}{y^{2}}\left(\mathrm{d}y^{2}+y^{2}\mathrm{d}\vartheta^{2}\right).$
(A.1)

Here $r$ and $\varphi$ are polar coordinates in the $(x^{2},x^{3})$ plane, $y$
is the radial coordinate in $AdS$, and if we change coordinates to
$y=4L^{3}/z^{2}$ we get the metric in the form

$\mathrm{d}s^{2}=\frac{4L^{2}}{z^{2}}(\mathrm{d}z^{2}+(\mathrm{d}x^{1})^{2}+\mathrm{d}r^{2}+r^{2}\,\mathrm{d}\varphi^{2})+L^{2}\mathrm{d}\vartheta^{2}\,.$
(A.2)

As world-volume coordinates we take $u$, $v$ and $r$ and the ansatz

$z=ur\,,\qquad
x^{1}=\sqrt{1+u^{2}}\,v\,,\qquad\varphi=\varphi(u)\,,\qquad\vartheta=\vartheta(u)\,.$
(A.3)

The scaling of $z$ with $r$ is natural because of dilatation invariance of the
crease as in the case of the cusp [11]. The scaling of $x^{1}$ with
$\sqrt{1+u^{2}}$ may seem unnatural at first, but it is required to have a
full 1d conformal symmetry $SO(2,1)$. To see that we write the induced metric

$4L^{2}\left[\frac{1+u^{2}}{u^{2}}\frac{\mathrm{d}r^{2}+\mathrm{d}v^{2}}{r^{2}}+\left(r^{2}+\frac{u^{2}v^{2}}{1+u^{2}}+r^{2}\varphi^{\prime
2}+\frac{r^{2}u^{2}}{4}\vartheta^{\prime
2}\right)\frac{\mathrm{d}u^{2}}{r^{2}u^{2}}+\frac{2ur\,\mathrm{d}u\,\mathrm{d}r+2uv\,\mathrm{d}u\,\mathrm{d}v}{r^{2}u^{2}}\right].$
(A.4)

Fixed $u$ slices are clearly $AdS_{2}$, and hence the required symmetry.

It is possible to solve the equations of motion also without this extra factor
of $\sqrt{1+u^{2}}$ in $x^{1}$, but the solution does not correspond to the
conformally invariant crease that we are studying here.

Comparing to (3.2), we see that we should identify $u$ with $t$ and indeed the
resulting lagrangian is then identical to (3.5) with the replacement of the
measure factors

$\sinh\rho\,\mathrm{d}\rho\,\mathrm{d}v\to\frac{\mathrm{d}r\,\mathrm{d}v}{r^{2}}\,.$
(A.5)

The equations of motion are then the same, but the total action vanishes
because the natural regularisation of the area of $AdS_{2}$ in this coordinate
system is zero. One can also implement other regularisations, with a cutoff on
$z$ instead of $r$ and possibly restrict $x^{1}$ to a fixed length, rather
than set a cutoff on $v$. Those calculations yield different (and complicated)
expressions.

###  A.2 Calibration equations

Here we derive the BPS condition (3.9) used in section 3.2 from the
calibration equations obtained in [7].

Of the four main families of BPS operators identified in [7], the infinite
crease falls into two families, type-$\mathbb{R}$ and type-$\mathbb{H}$. For
each of the families there exists a calibration form $\phi$ such that the
surface satisfies the set of equations

$\partial_{m}X^{M}=\frac{1}{2}g_{ml}\varepsilon^{lnp}G^{ML}\phi_{LNP}\partial_{n}X^{N}\partial_{p}X^{P}\,.$
(A.6)

Here $X^{M}$ are the target space coordinates with metric $G_{MN}$, $m$ are
the world-volume indices with induced metric $g_{mn}$ and $\varepsilon_{mnp}$
is the Levi-Civita tensor density.

Using the metric

$\mathrm{d}s^{2}=\frac{y}{L}\mathrm{d}x^{\mu}\mathrm{d}x^{\mu}+\frac{L^{2}}{y^{2}}\mathrm{d}y^{I}\mathrm{d}y^{I}\,,$
(A.7)

with $\mu=1,\dots,6$, $I=1,\dots,5$ and $y=|y|$, all surfaces of type
$\mathbb{R}$ are calibrated with respect to the form

$\phi^{\mathbb{R}}=-\mathrm{d}x^{1}\wedge\sum_{I=1}^{5}(\mathrm{d}x^{I+1}\wedge\mathrm{d}y^{I})\,.$
(A.8)

For the coordinates relevant for the infinite crease (A.1), this reduces to

$\phi^{\mathbb{R}}\big{|}_{AdS_{3}\times
S^{1}}=-\mathrm{d}x^{1}\wedge\left[\cos(\varphi-\vartheta)(\mathrm{d}r\wedge\mathrm{d}y+ry\,\mathrm{d}\varphi\wedge\mathrm{d}\vartheta)+\sin(\varphi-\vartheta)(y\,\mathrm{d}r\wedge\mathrm{d}\vartheta+r\,\mathrm{d}y\wedge\mathrm{d}\varphi)\right].$
(A.9)

Surfaces of type $\mathbb{H}$ have the calibration form

$\phi^{\mathbb{H}}=\frac{1}{2}\eta_{\mu\nu}^{I}\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}\wedge\mathrm{d}y^{I}-\frac{L^{3}}{y^{3}}\mathrm{d}y_{1}\wedge\mathrm{d}y_{2}\wedge\mathrm{d}y_{3}\,,$
(A.10)

with $\eta^{I}_{\mu\nu}$ the ’t Hooft symbol in 4d

$\eta^{3}_{12}=\eta^{3}_{34}=\eta^{2}_{31}=\eta^{2}_{24}=\eta^{1}_{14}=\eta^{1}_{23}=1\,.$
(A.11)

This form is supported on $AdS_{5}\times S^{2}$. The crease solution is in
$AdS_{3}\times S^{1}$, but it is useful to keep $AdS_{3}\times S^{2}$
components. For the orientation of the crease along $x^{1}$ we should replace
$y^{1}\to y^{3},y^{3}\to-y^{1}$ in the above, and then the calibration form is

$\displaystyle\phi^{\mathbb{H}}\big{|}_{AdS_{3}\times S^{2}}$
$\displaystyle=\phi^{\mathbb{R}}\big{|}_{AdS_{3}\times
S^{1}}+\mathrm{d}y^{3}\wedge\left(\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}-\frac{L^{3}}{y^{3}}\mathrm{d}y^{1}\wedge\mathrm{d}y^{2}\right)$
(A.12) $\displaystyle=\phi^{\mathbb{R}}\big{|}_{AdS_{3}\times
S^{1}}+\mathrm{d}y^{3}\wedge\left(r\,\mathrm{d}r\wedge\mathrm{d}\varphi-\frac{L^{3}}{y^{2}}\mathrm{d}y\wedge\mathrm{d}\vartheta\right).$

The extra structure in $\phi^{\mathbb{H}}$ is useful because $y^{3}$ vanishes
identically, so taking $X^{M}=y^{3}$ in (A.6) we find

$\varepsilon^{lnp}\phi_{y^{3}NP}\partial_{n}X^{N}\partial_{p}X^{P}=0\,,$
(A.13)

and in particular with the derivatives with respect to $r$ and $u$ (the latter
written as prime) this is

$\displaystyle
r\varphi^{\prime}-\frac{L^{3}}{y^{2}}\partial_{r}(y)\vartheta^{\prime}=r\left(\varphi^{\prime}+\frac{u^{2}}{2}\vartheta^{\prime}\right)=0\,.$
(A.14)

This reproduces the BPS condition (3.9) and therefore shows that the solution
with $J_{\vartheta}=-2J_{\varphi}$ is indeed BPS. Note that the other
components of (A.6) also imply the rest of the equations of motion (3.7).

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