# An unusual BPS equation

Constantin P. Bachas, Lorenzo Bianchi, Zhongwu Chen

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Constantin P. Bachas, Lorenzo Bianchi, Zhongwu Chen. An unusual BPS equation. Journal of High Energy Physics, 2025, 2025 (04), pp.082. 10.1007/JHEP04(2025)082. hal-04931074

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# An unusual BPS equation

Constantin Bachas<sup>1, a, b</sup> Lorenzo Bianchi<sup>1, c</sup> and Zhongwu Chen<sup>1, a</sup>

E-mail: costas.bachas@phys.ens.fr, lorenzo.bianchi@unito.it,
zhongwu.chen@phys.ens.fr

**ABSTRACT:** We prove a conjectured relation between the energy-momentum and the displacement norm of superconformal defects. The proof completes earlier results, and shows that supersymmetry identifies two natural notions of brane tension in Anti-de Sitter gravity. As a byproduct we show that a modification of the energy-momentum tensor that removes the stress of static superconformal defects, ensures also that the radiation these emit obeys the Null Energy Condition. This sheds new light on the radiation-reaction problem for moving charges.

**KEYWORDS:** AdS-CFT Correspondence, Brane Dynamics in Gauge Theories, Supersymmetry and Duality, Wilson, 't Hooft and Polyakov loops

**ARXIV ePRINT:** 2501.13197

<footnote><sup>a</sup>Laboratoire de Physique de l'Ecole Normale Supérieure, CNRS,
PSL Research University and Sorbonne Universités,
24 rue Lhomond, 75005 Paris, France</footnote>

<footnote><sup>b</sup>Physics Department, National Technical University of Athens,
15780 Zografou, Athens, Greece</footnote>

<footnote><sup>c</sup>Dipartimento di Fisica, Università di Torino and INFN - Sezione di Torino,
Via P. Giuria 1, 10125 Torino, Italy</footnote>
# Contents

<table><tr><td>1</td><td>Introduction</td><td>1</td></tr><tr><td>1.1</td><td>Definitions and outline</td><td>3</td></tr><tr><td>2</td><td>Minimal superconformal defects</td><td>4</td></tr><tr><td>2.1</td><td>Line defects</td><td>5</td></tr><tr><td>2.2</td><td>Surface defects</td><td>7</td></tr><tr><td>2.3</td><td>Higher-p defects</td><td>8</td></tr><tr><td>3</td><td>An equivalent conjecture</td><td>9</td></tr><tr><td>3.1</td><td>The displacement-stress tensor correlation</td><td>9</td></tr><tr><td>3.2</td><td>Line defects in N = 2 abelian gauge theory</td><td>10</td></tr><tr><td>4</td><td>Stress, radiation and the Schott term</td><td>12</td></tr><tr><td>4.1</td><td>Static stress and the NEC</td><td>12</td></tr><tr><td>4.2</td><td>Energy flux of a moving defect</td><td>13</td></tr><tr><td>5</td><td>Proof of the conjecture</td><td>16</td></tr><tr><td>A</td><td>Real subalgebras of F(4; 2)</td><td>19</td></tr><tr><td>B</td><td>Superconformal algebras</td><td>20</td></tr><tr><td>C</td><td>Supersymmetry transformations</td><td>21</td></tr></table>

## 1 Introduction

Two important observables related to an external probe, or defect, of a Quantum Field Theory are the energy-momentum stored in its fields, and its resistance to deformations. When the theory is conformal these are described by two dimensionless parameters, $a_T$ and $C_D$ [1]. Lewkowycz and Maldacena observed that a linear relation between them ($C_D = -18a_T$) allowed to reconcile two different calculations of the energy emitted by an accelerating half-BPS quark in $N = 4$ super-Yang-Mills [2]. Based on this and some other examples, ref. [3] conjectured that the relation

$$ \frac{C_D}{a_T} = - \frac{2(n-1)(p+2)\Gamma(p+1)}{n \pi^{p-n/2} \Gamma(\frac{p}{2}+1) \Gamma(\frac{n-p}{2})} \quad (1.1) $$

holds for any $p$-dimensional superconformal defect in $n$ spacetime dimensions. We will prove this conjecture in the present paper.
<table><caption>**Table 1.** The minimal supersymmetric DCFTs discussed in the paper. The second column gives the smallest $N$ that a $n$-dimensional SCFT must have to admit $p$-dimensional superconformal defects. The corresponding bulk superalgebras and maximal $p$-embeddings are given in the third and fourth columns. The subscript 'c' denotes bosonic subalgebras that commute with the preserved supercharges (for details see section 2). For missing $(p, n)$ pairs superconformal and rotation-invariant defects do not exist.</caption><thead><tr><th>defect</th><th>(n, N)</th><th>superalgebra</th><th>p-embedding</th></tr></thead><tbody><tr><td rowspan="3">line</td><td>(3,2)</td><td>osp(2|4; R)</td><td>su(1, 1|1) &oplus; u(1)<sub>c</sub></td></tr><tr><td>(4,2)</td><td>su(2, 2|2)</td><td>osp(4*|2)</td></tr><tr><td>(5,1)</td><td>F(4; 2)</td><td>D(2, 1; 2; 0) &oplus; su(2)<sub>c</sub></td></tr><tr><td rowspan="3">surface</td><td>(4,1)</td><td>su(2, 2|1)</td><td>su(1, 1|1) &oplus; su(1, 1)<sub>c</sub> &oplus; u(1)<sub>c</sub></td></tr><tr><td>(5,1)</td><td>F(4; 2)</td><td>D(2, 1; 2; 0) &oplus; so(2, 1)<sub>c</sub></td></tr><tr><td>(6,1)</td><td>osp(8*|2)</td><td>osp(4*|2) &oplus; so(2, 1)<sub>c</sub> &oplus; so(3)<sub>c</sub></td></tr><tr><td>p = 3</td><td>(5,1)</td><td>F(4; 2)</td><td>osp(2|4; R) &oplus; u(1)<sub>c</sub></td></tr><tr><td>p = 4</td><td>(6,1)</td><td>osp(8*|2)</td><td>su(2, 2|1) &oplus; u(1)<sub>c</sub></td></tr></tbody></table>

A physical interpretation of (1.1) was given in ref. [4] which pointed out that in AdS/CFT there are two ways to assign an invariant tension to a $p$-brane dual of a conformal defect. There is a gravitational tension $\sigma_{gr} = \gamma_{gr}|_{a_T}$ that extends the notion of ADM mass,<fnref n="1" /> but also an inertial tension or stiffness $\sigma_{in} = \gamma_{in}C_D$. The coefficients $\gamma_{gr}$ and $\gamma_{in}$ were fixed in [4] by requiring that both tensions reduce to the parameter of the Nambu-Goto action in the limit of a classical probe brane coupled to gravity. The conjecture (1.1) turns out to imply that $\sigma_{gr} = \sigma_{in}$, even when the brane is quantized and/or back-reacts strongly. A BPS equation usually relates the mass or tension of an object to its charge. Here it relates two tensions, whence the title of our paper.

Clearly $C_D$ is not defined when $p=0$ (i.e. for local operators), and we will see that $a_T$ is not defined for $p=n-1$ (i.e. for interfaces or boundaries). Thus (1.1) applies to $0 < p < n-1$. Our proof works for all superconformal defects that preserve the transverse-rotation symmetry $so(n-p)$. It is guided by, but supersedes earlier results for special cases [3, 6–9]. We only need to prove (1.1) for the minimal (mutually-compatible) number of bulk and defect supersymmetries. For each pair $(n, p)$ there is at most one such minimal embedding of the defect superalgebra into the bulk superalgebra. The list is given in the table above.

Our proof relies on reformulating (1.1) as a property of the two-point function $\langle T^{\mu\nu}D^j \rangle$, where $D^j$ is the displacement operator. This will shed light on the role of supersymmetry for defect dynamics. We mentioned already that (1.1) was originally motivated by studies of the radiation from an accelerating ‘quark’ [10–20]. What the authors of [2] observed was that in $N=4$ super-Yang-Mills one can reconcile two calculations of the emitted energy, proportional respectively to $C_D$ and $|a_T|$, by changing the bulk energy-momentum tensor $T^{\mu\nu} \rightarrow \tilde{T}^{\mu\nu}$. They suggested that this works because $\tilde{T}^{\mu\nu}$ separates radiation from the quark self-energy, but the precise reason remained unclear. It was later pointed out in ref. [19] that for a free scalar field such a modification of $T^{\mu\nu}$ removes the conformal-improvement term which is known to violate the Null Energy Condition (NEC) [21].

<footnote n="1">¹For earlier discussions of the gravitational or ADM-like tension see [5] and references therein.</footnote>
We will show that the two effects are related. The key point is that the BPS equation (1.1) guarantees that the same modification which restores the NEC in the radiation flux also removes the stress stored in the fields of a static defect. Zero stress is necessary if one wants to absorb the leading singularity of $T^{\mu\nu}$ by renormalizing the (infinite) bare mass of the defect. This argument suggests that supersymmetry may help solve some time-honoured puzzles of radiation reaction for moving charges in electrodynamics [22–27]. We defer this problem to future work.

In two-dimensional CFT the 2-point function $\langle T^{\mu\nu}D^j \rangle$ encodes universal aspects of energy transport across a conformal defect [28]. Since $a_T$ is not defined in this case, the energy transport depends only on $C_D$, or in the dual AdS$_3$ gravity on $\sigma_{in}$ [29–32]. Part of our motivation in the present work was to extend these studies to higher dimensions. Let us finally mention that for even-dimensional defects eq. (1.1) relates two anomaly coefficients [33–37].

## 1.1 Definitions and outline

We now define more precisely the two main characters in (1.1). A Defect Conformal Field Theory (DCFT) has in addition to the bulk stress tensor $T^{\mu\nu}$, a displacement operator $D^\nu$ living on the defect. It is defined through the Ward identity of broken translation invariance

$$ \partial_{\mu}T^{\mu\nu} = \delta_{\text{def}}(x) D^{\nu}, \qquad (1.2) $$

where $\delta_{\text{def}}$ is the delta function localized on the defect.<fnref n="2" /> For a (hyper)planar defect in $\mathbb{R}^{(1,n-1)}$ the only non-vanishing components of $D^\nu$ are transverse.

In the background of such a (hyper)planar defect the one-point function $\langle T^{\mu\nu} \rangle$ and the two point-function $\langle D^j D^k \rangle$ are fixed by the unbroken $\mathfrak{so}(2, p) \oplus \mathfrak{so}(n-p)$ symmetry modulo two free parameters $a_T$ and $C_D$, [1]

$$ \langle T^{\alpha\beta}(x) \rangle = a_T \left( \frac{n-p-1}{n} \right) \frac{\eta^{\alpha\beta}}{|x_{\perp}|^n}, \qquad \langle T^{\alpha j}(x) \rangle = 0, \qquad (1.3a) $$

$$ \langle T^{ij}(x) \rangle = a_T \left[ \frac{x^i x^j}{|x_{\perp}|^{n+2}} - \left( \frac{p+1}{n} \right) \frac{\delta^{ij}}{|x_{\perp}|^n} \right], \qquad (1.3b) $$

$$ \text{and} \qquad \langle D^j(y) D^k(0) \rangle = C_D \frac{\delta^{jk}}{|y|^{2p+2}}. \qquad (1.4) $$

We here separated parallel and transverse directions, $x^\mu = (x^\alpha, x^j)$ and $y^\mu = (y^\alpha, 0)$, and denoted the distance of $x$ from the defect $|x_\perp|$. Our notation is described in more detail in section 3.1. Note that the conservation equation (1.2) fixes the normalization of $D^j$, so that $C_D$ is a piece of DCFT data. Note also that the right-hand sides in (1.3) vanish identically when $p = n-1$, which is why $a_T$ is not defined for interfaces and boundaries.

Unitarity implies $C_D > 0$, while positivity of $T^{00}$ implies that $a_T < 0$.<fnref n="3" /> The two pieces of DCFT data are otherwise unconstrained. They are in general different functions of

<footnote n="2">²When Lorentzian our defects are always timelike. The word ‘defect’ is used for both the defect and the defect’s worldvolume. The meaning should be clear from the context.</footnote>

<footnote n="3">³The spacetime signature is mostly plus. Some of the literature, including ref. [3], uses instead of $a_T$ the parameter $h = -a_T(n-p-1)/n$.</footnote>
the marginal couplings and other parameters, such as the ranks of gauge groups. This is illustrated by the free-field example of section 3.2.

The paper is structured as follows. In section 2 we explain table 1. For each pair $(p, n)$ we find the minimal bulk supersymmetry $\mathcal{N}$ that admits rotation-invariant superconformal defects. We do this by examining all superalgebra embeddings, along the lines of [38–40]. A subtle point about real forms of the 5d $\mathcal{N} = 1$ superalgebra is discussed in appendix A.

In section 3 we introduce the two-point function $\langle T^{\mu\nu}D^j \rangle$, which is determined in terms of $a_T$ and $C_D$ by conformal Ward identities [1]. The relation (1.1) takes a particularly simple form when expressed as a relation between the coefficients of various tensor structures in $\langle T^{\mu\nu}D^j \rangle$. This is the starting point of our proof. We put some flesh into these abstract formulae with the free-field example of line defects in the 4d $\mathcal{N} = 2$ abelian gauge theory.

Section 4 explains how the modified stress tensor introduced in [2] can be tuned to cancel either the stress of the static fields, or the NEC-violating radiation from an accelerating defect. The BPS equation (1.1) ensures that both are simultaneously cancelled. The conclusion only depends on the structure of $\langle T^{\mu\nu}D^j \rangle$, and is valid for an arbitrary DCFT. Sections 3 and 4 are autonomous, readers only interested in the problem of radiation reaction can skip the rest of the paper.

Section 5 presents the proof of (1.1). The BPS equation is formulated as a special property of $\langle T_{zz}D_z \rangle$, where $z$ is a complex transverse direction. This property follows then from supersymmetric Ward identities that have the same form for all $(n, p)$. The superalgebras of table 1 are summarized for completeness in appendix B. The proof uses the supersymmetry transformations of the displacement and stress-tensor multiplets, which are summarized in appendix C.

## 2 Minimal superconformal defects

This section explains the entries of table 1. We adapt to our purposes and extend the work of refs. [38–40] that classified various superconformal defects.<fnref n="4" />

We are interested in defects for which both $C_D$ and $a_T$ are defined. This excludes local operators and domain walls, so $1 \le p \le n - 2$. Since superconformal theories (SCFTs) only exist in $n \le 6$ dimensions [42], we only need to consider $n = 3, 4, 5, 6$. The corresponding superconformal algebras $\mathfrak{G}_s^{(n)}$, consistent with the existence of a unitary conserved stress-tensor multiplet, are well known [43, 44]. We list them below for the reader’s convenience

$$ \begin{array}{l@{\quad}l@{\quad}l@{\quad}l} n=3: & \mathbf{osp}(\mathcal{N}|4; \mathbb{R}) & \stackrel{\text{bos}}{\supset} & \mathbf{so}(2,3) \oplus \mathbf{so}(\mathcal{N})_R, & \mathcal{N}=1, \dots, 6, 8; \\ n=4: & \mathbf{su}(2,2|\mathcal{N}) & \stackrel{\text{bos}}{\supset} & \mathbf{so}(2,4) \oplus \mathbf{u}(\mathcal{N})_R, & \mathcal{N}=1, 2, 3; \\ & \mathbf{psu}(2,2|\mathcal{N}) & \stackrel{\text{bos}}{\supset} & \mathbf{so}(2,4) \oplus \mathbf{su}(4)_R, & \mathcal{N}=4; \hfill (2.1) \\ n=5: & F(4;2) & \stackrel{\text{bos}}{\supset} & \mathbf{so}(2,5) \oplus \mathbf{su}(2)_R, & \mathcal{N}=1; \\ n=6: & \mathbf{osp}(8^*|2\mathcal{N}) & \stackrel{\text{bos}}{\supset} & \mathbf{so}(2,6) \oplus \mathbf{usp}(2\mathcal{N})_R, & \mathcal{N}=1, 2. \end{array} $$

<footnote n="4">⁴For related work from the holographic perspective see also [41].</footnote>
As usual, $\mathcal{N}$ is the number of irreducible $so(1, n-1)$ spinor supercharges. There is one superalgebra for each pair $(\mathcal{N}, n)$. Also given in (2.1) are the maximal bosonic (even-degree) subalgebras $so(2, n) \oplus \mathfrak{R}$, where $so(2, n)$ is the conformal group and $\mathfrak{R}$ is the compact R-symmetry.<fnref n="5" />

A planar superconformal $p$-dimensional defect breaks $\mathfrak{G}_s^{(n)} \hookrightarrow \mathfrak{g}_s \oplus \mathfrak{g}_c$, where $\mathfrak{g}_s$ is the defect superalgebra and $\mathfrak{g}_c$ a bosonic algebra that commutes with all preserved supercharges. Most of the time $\mathfrak{g}_c$ is a compact symmetry that may, but need not, be broken by the superconformal defect. But for surface defects it may include a chiral $so(2, 1)$ half of the conformal symmetry.

We restrict attention to defects invariant under both worldvolume and transverse rotations. This condition was already implicit in eqs. (1.3) and (1.4). It implies that the displacement operator sits in an irreducible multiplet of the defect algebra [40]. It also implies (the last of) the following chain of inclusions

$$ so(2, n) \oplus \mathfrak{R} \supset so(2, p) \oplus so(n-p) \oplus \mathfrak{R} \supset \mathfrak{g}_s^{\text{bos}} \oplus \mathfrak{g}_c \supset so(2, p) \oplus so(n-p)_{\text{def}}. \quad (2.2) $$

Here $\mathfrak{g}_s^{\text{bos}} = so(2, p) \oplus \mathfrak{r}$ is the bosonic subalgebra of $\mathfrak{g}_s$, with $\mathfrak{r}$ the defect R-symmetry. The subscript 'def' indicates that if a simple component of $so(n-p)$ fits in $\mathfrak{R}$, the preserved rotations could be accompanied by bulk R-transformations.<fnref n="6" />

We call the inclusion of $\mathfrak{g}_s \oplus \mathfrak{g}_c$ in $\mathfrak{G}_s^{(n)}$ a 'p-embedding' if the non-compact unbroken symmetry is $so(2, p)$. A p-embedding is maximal if it cannot be properly included in any other p-embedding. Note that a p-embedding may be included in a higher-p embedding, as befits the symmetry structure of a defect inside a defect. Such composite defects break the worldvolume and/or transverse-rotation symmetry, and we do not discuss them here.

Our problem is now to classify p-embeddings in the bulk superalgebras (2.1), for $p \le n-2$. Fortunately, maximal real subalgebras of the osp and su series and of $F(4; 2)$ have been classified [38–40], and are collected conveniently in table 2 of ref. [40].<fnref n="7" /> If there is no maximal p-embedding for some $\mathfrak{G}_s^{(n)}$ in the list, we conclude that the corresponding SCFT does not admit stand-alone p-dimensional superconformal defects.

## 2.1 Line defects

We begin with the line defects, $p = 1$, which were classified (without assuming rotation symmetry) in table 12 of ref. [40].

Inspection of this table shows, first of all, that bulk theories with minimal, $\mathcal{N} = 1$, supersymmetry in $n = 3, 4$ and $6$ dimensions do not admit any superconformal line defects. One can understand why using elementary spinor properties.<fnref n="8" /> Indeed, the $\mathcal{N} = 1$ R-symmetry is too small to contain the group of transverse rotations, so the preserved $so(n-1)_{\text{def}}$ is canonically embedded in the ambient Lorentz group $so(1, n-1)$. It follows that the bulk

<footnote n="5">⁵We will use the special inclusion symbol $\supset \supset^{\text{bos}}$ for the bosonic subalgebra of a superalgebra.</footnote>

<footnote n="6">⁶More generally, if the bulk theory has a flavour symmetry, the unbroken rotations could act non-trivially in flavour space. Since both the stress-tensor and the displacement supermultiplets are ‘flavour blind’, this will not affect our proof.</footnote>

<footnote n="7">⁷A bug in this table will be corrected in section 2.2.</footnote>

<footnote n="8">⁸Our argument uses rotation invariance, but breaking it does not help, see [40].</footnote>
Poincaré supercharges, $Q$, must transform as an irreducible spinor of both $so(1, n-1)$, since supersymmetry is minimal, and of $so(n-1)_{\text{def}}$. But for $n = 3, 4$ and $6$ the irreducible spinors of $so(n-1)$ have the same dimension as those of $so(1, n-1)$ (see e.g. appendix B of ref. [45]). Since the defect must break some supersymmetries, this proves that the above $N = 1$ SCFTs have no rotation-invariant superconformal line defects.

This simple counting argument does not exclude line defects in the six-dimensional $N = (2, 0)$ theory. The bulk Poincaré supercharges transform in this case in the $(4, 2, 1) \oplus (4, 1, 2)$ representation of $so(1, 5) \oplus su(2) \oplus su(2)'$, where $su(2) \oplus su(2)' \subset so(5)_R \simeq u\mathfrak{sp}(4)_R$ [45]. A half-BPS line defect could, a priori, reduce these to one irreducible spinor of $so(5)_{\text{def}}$, but the rigidity of superconformal algebras forbids it. Indeed, none of the maximal subalgebras of $osp(8^*|4)$ can describe the symmetries of an odd-dimensional defect [40]. The best one can do is to consider maximal 1-embeddings in some higher-$p$ subalgebra of $osp(8^*|4)$, but this breaks the $so(5)$ rotation symmetry as we have explained.

The case $n = 5$ is special because a Weyl-Majorana spinor of $so(4)$ has half as many components as an irreducible spinor of $so(1, 4)$. Rotation-invariant line defects are in this case allowed. They are described by the maximal 1-embedding

$$F(4; 2) \supset D(2, 1; 2; 0) \oplus su(2)_c \supset so(2, 1) \oplus so(4) \oplus su(2)_c. \quad (2.3)$$

Note that the bosonic subalgebra contains three $su(2)$ factors, one coming from the bulk R-symmetry, $su(2)_R$, and the others from $so(4) \simeq su(2) \oplus su(2)'$. The defect R-symmetry is $so(4) \simeq su(2) \oplus su(2)_R$ while $su(2)_c = su(2)'$ commutes with the supercharges (see appendix A). Line defects of the five-dimensional SCFT have been analyzed in refs. [46–49].

Superconformal line defects also exist in three- and four-dimensional SCFTs but with $N = 2$ supersymmetry. The relevant maximal 1-embeddings are

$$n = 3: \quad osp(2|4; \mathbb{R}) \supset su(1, 1|1) \oplus u(1)_c \supset so(2, 1) \oplus so(2)_{\text{def}} \oplus u(1)_c; \quad (2.4)$$

$$n = 4: \quad su(2, 2|2) \supset osp(4^*|2) \supset so(2, 1) \oplus so(3) \oplus u\mathfrak{sp}(2). \quad (2.5)$$

In three dimensions the preserved rotation symmetry is a mixture of space rotations and R-symmetry transformations, and the commuting $u(1)_c$ may (but need not) be broken (for an overview on explicit constructions of 3d line defects see [50] and references therein). In four dimensions, on the other hand, both $so(3)$ and $u\mathfrak{sp}(2)_R$ are part of the defect superalgebra that has no commutant in $\mathfrak{G}_s^{(4)}$.

Extended $N > 2$ supersymmetry allows a large variety of BPS line defects. These were classified in ref. [40] where the reader can also find references to explicit constructions. The defect superalgebras in three dimensions are $su(1, 1|m)$ with $m = 1, 3, 4$, or $psu(1, 1|2)$, and in four dimensions $osp(4^*|2m)$ with $m = 1, 2$ (modulo bosonic commutants $g_c$). In all cases the unbroken symmetry contains the minimal defect superalgebras $su(1, 1|1)$ for $n = 3$ and $osp(4^*|2)$ for $n = 4$. These are sufficient, as we will see, for proving the conjecture (1.1).

To summarize our discussion up to here, minimal superconformal line defects exist in $n = 3, 4$ and 5-dimensional SCFTs with $N = 2, 2$ and 1 supersymmetries. The corresponding defect superalgebras are $su(1, 1|1)$, $osp(4^*|2)$ and $D(2, 1; 2; 0)$, and the unbroken supercharges transform as spinors of $so(n-1)$. Simple counting shows that these defects are half-BPS, and that there exist no other, less supersymmetric line defects.
## 2.2 Surface defects

We move next to $p = 2$ and $n = 4, 5, 6$. The conformal group of a surface defect factorizes into a left- and a right-moving piece, $so(2, 2) \simeq so(2, 1)_+ \oplus so(2, 1)_-$. Each can be extended separately to a one-dimensional superconformal algebra.

Consider first the $\mathcal{N} = 1$ four-dimensional SCFTs, whose Lie superalgebra is $su(2, 2|1) \supset so(2, 4) \oplus u(1)_R$. Among its maximal subalgebras (see table 2 of [40]) there is a unique 2-embedding<fnref n="9" />

$$ su(2, 2|1) \supset su(1, 1|1) \oplus su(1, 1)_c \oplus u(1)_c \supset so(2, 1) \oplus so(2)_{\text{def}} \oplus so(2, 1)_c \oplus u(1)_c. \quad (2.6) $$

Since a chiral half of the conformal algebra commutes with the surviving supercharges, these defects have (2,0) worldsheet supersymmetry. As was the case for line defects in three dimensions, the preserved $so(2)_{\text{def}}$ is in general non-trivially embedded in the bulk $so(2) \oplus u(1)_R$, while the residual $u(1)_c$ may be broken. Surface defects of this type have been discussed in refs. [51–53], and the conjecture (1.1) was proved for this case in ref. [3].

Extended $\mathcal{N} > 1$ supersymmetry allows several different maximal 2-embeddings,

$$ su(2, 2|\mathcal{N}) \supset su(1, 1|\mathcal{N}_+) \oplus su(1, 1|\mathcal{N}_-) \oplus u(1)_c \quad (2.7) $$

where $\mathcal{N}_+ + \mathcal{N}_- = \mathcal{N}$ and $su(1, 1|0) = su(1, 1)$.<fnref n="10" /> The corresponding superconformal surface defects have $2\mathcal{N}_+$ left-moving and $2\mathcal{N}_-$ right-moving supercharges. All these share a common $su(1, 1|1)$ subalgebra which is sufficient for the proof of the conjecture (1.1). There is a large literature treating various aspects of such defects, for a partial list of references see [54–59].

Consider next the six-dimensional SCFTs whose symmetry is $osp(8^*|2\mathcal{N})$, where $\mathcal{N} = 1$ or 2. The relevant maximal real subalgebras are<fnref n="11" />

$$ osp((8-2m)^*|2\mathcal{N}_1) \oplus osp(2m^*|2\mathcal{N}_2) \quad \text{with} \quad \mathcal{N}_1 + \mathcal{N}_2 = \mathcal{N}, \quad 0 \le m \le 4. \quad (2.8) $$

For $\mathcal{N} = 1$ there is a unique maximal 2-embedding

$$ osp(4^*|2) \oplus osp(4^*)_c \supset so(2, 1) \oplus su(2) \oplus u\mathfrak{sp}(2) \oplus so(2, 1)_c \oplus su(2)_c. \quad (2.9) $$

A chiral half of $so(2, 2) \simeq so(2, 1)_+ \oplus so(2, 1)_-$ commutes in this embedding with the supercharges, so the half-BPS surface defects have (4,0) worldsheet supersymmetry. The defect R-symmetry is $so(4)_r \simeq su(2) \oplus u\mathfrak{sp}(2)$, while the (unbroken) rotation group is $so(4) \simeq su(2) \oplus su(2)_c$.

The $\mathcal{N} = 2$ superalgebra offers more possibilities with (8,0), (4,0) or (4,4) worldsheet supersymmetry. Examples of (4,4) and (4,0) defects are self-dual strings or M2-branes ending

<footnote n="9">⁹The other maximal subalgebra in the list is osp(1|4; $\mathbb{R}$). Its bosonic component is so(2, 3), so it corresponds to $p = 3$ superconformal defects, i.e. interfaces or boundaries.</footnote>

<footnote n="10">¹⁰In the special case $(\mathcal{N}_+, \mathcal{N}_-) = (2, 2)$ the maximal 2-embedding is actually centrally extended as follows: $psu(2, 2|4) \supset [u(1) \times psu(1, 1|2) \times u(1)] \oplus [u(1) \times psu(1, 1|2) \times u(1)]'$.⁹</footnote>

<footnote n="11">¹¹When one of the even-integer entries in osp(2m*|2k) is zero the algebra is purely bosonic, osp(0|2k) ≅ u\mathfrak{sp}(2k) and osp(2k*|0) = so(2k*). The star in so(2k*) indicates the quaternionic real form of the complex Lie algebra so(2k, $\mathbb{C}$). For small $k$ one has the following equivalences: so(2*) ≅ so(2); so(4*) ≅ so(2, 1) ⊕ so(3); so(6*) ≅ su(1, 3) and so(8*) ≅ so(2, 6). Note also in passing the equivalences osp(2*|2) ≅ su(2|1) and osp(4*|2) ≅ D(2, 1; -2; 0) [38].</footnote>
on M5-branes (for a partial list of references see [60–65]). The (8,0) embedding, on the other hand, is not realized by non-trivial surface defects, i.e. defects that interact with the bulk SCFT. The reason is that the putative displacement operator should transform in the $(j, j') = (\frac{1}{2}, \frac{1}{2})$ representation of $so(4)_{\text{def}} \simeq \mathfrak{su}(2) \oplus \mathfrak{su}(2)'$ and have scaling dimension $\Delta = 3$. None of the unitary representations of $osp(4^*|4)$ (see table 8 of ref. [40]) includes such an operator as its top component.

What is in any case important for us here is that all superconformal surface defects in six dimensions share an $osp(4^*|2) \simeq D(2, 1; -2; 0)$ superalgebra which is sufficient for proving the conjecture (1.1).

The last theory to consider is the $N = 1$ SCFT in five dimensions. It admits the maximal 2-embedding

$$F(4; 2) \supset D(2, 1; 2; 0) \oplus \mathfrak{so}(2, 1)_c \supset \mathfrak{so}(2, 1) \oplus \mathfrak{so}(4) \oplus \mathfrak{so}(2, 1)_c, \quad (2.10)$$

which describes surface defects with (4,0) worldsheet supesymmetry. There seems to be some confusion in the literature concerning real subalgebras of $F(4; 2)^{12}$ so we work out this case in detail in appendix A. Holographic surface defects of this type were found in gauged supergravity in ref. [66].

To summarize, half-BPS superconformal and rotation-invariant surface defects exist for $N = 1$ theories in $n = 4, 5$ and $6$ dimensions. The preserved superalgebras, modulo bosonic commutants, are $\mathfrak{su}(1, 1|1)$, $D(2, 1; 2; 0)$ and $\mathfrak{osp}(4^*|2)$.

## 2.3 Higher-$p$ defects

Finally we consider $p = 3, 4$. Stand-alone, superconformal $p = 3$ defects do not exist in $n = 6$ dimensions because none of the maximal subalgebras of $osp(8^*|2N)$ is an odd-$p$ embedding [40]. A maximal 3-embedding does exist in $n = 5$ dimensions

$$F(4; 2) \supset \mathfrak{osp}(2|4; \mathbb{R}) \oplus \mathfrak{u}(1)_c \supset \mathfrak{so}(2, 3) \oplus \mathfrak{so}(2)_{\text{def}} \oplus \mathfrak{u}(1)_c. \quad (2.11)$$

As with other codimension-two defects, the preserved transverse rotations are non-trivially embedded in the bulk $\mathfrak{so}(2) \oplus \mathfrak{u}(1)_R$. Holographic defects of this type have been analyzed in refs. [67, 68].$^{13}$

The last case is $p = 4, n = 6$ corresponding to the maximal 4-embedding

$$\mathfrak{osp}(8^*|2N) \supset \mathfrak{su}(2, 2|N) \oplus \mathfrak{u}(1)_c \supset \mathfrak{so}(2, 4) \oplus \mathfrak{so}(2)_{\text{def}} \oplus \mathfrak{su}(N) \oplus \mathfrak{u}(1)_c. \quad (2.12)$$

These defects are half-BPS, with the transverse rotations once again non-trivially embedded in the bulk R-symmetry. The gravity duals of such holographic defects for $N = 2$ are expected to belong to the general class of M-theory solutions found in refs. [69, 70]. The solutions were mainly considered as AdS$_5$ compactifications, i.e. as autonomous four-dimensional SCFTs, and it would be very interesting to explore them further as $n = 6$ DCFTs along the lines of refs. [71–73].

<footnote>$^{12}$In particular, the first $F(4; 2)$ entry in table 2 of [40] cannot be correct, since the fermionic generators in $\mathfrak{su}(2|1) \oplus \mathfrak{su}(1, 2)_c$ would commute with the dilatation operator.</footnote>

<footnote>$^{13}$Despite the title of [67], the defects analyzed in this paper are three-dimensional.</footnote>
This completes our derivation of table 1. The last thing we need to check is that it is enough to prove (1.1) for the entries of this table. One may have worried, for example, that 4d $\mathcal{N} = 4$ super Yang-Mills has 1/4-BPS defects that cannot be expressed as half-BPS defects of a $\mathcal{N} = 2$ theory. For line defects this possibility is ruled out by inspection of the complete list of ref. [40]. For surface defects in four and six dimensions there exist no non-trivial 2-embeddings in the extended superconformal algebras $\mathcal{G}_s^{(n)}$ that are not also embedded in a $\mathcal{N} = 1$ subalgebra of $\mathcal{G}_s^{(n)}$. Finally, the only 4-embeddings in the 6d $\mathcal{N} = (2, 0)$ theory are $\mathfrak{su}(2, 2|2)$, or its unique up to isomorphisms subalgebra $\mathfrak{su}(2, 2|1)$.

### 3 An equivalent conjecture

The conjecture (1.1) relates the one-point function of the stress tensor $\langle T^{\mu\nu} \rangle$ to the two-point function of the displacement $\langle D^i D^j \rangle$. In this section we will reexpress it as a condition on the bulk-to-defect two-point function $\langle T^{\mu\nu} D^j \rangle$. Here and in what follows $\langle \cdots \rangle$ stands for the normalized correlation functions in the background of a static (hyper)planar defect.

#### 3.1 The displacement-stress tensor correlation

The two-point function of any bulk conformal primary with the displacement is strongly constrained by conformal invariance [1]. Consider as a warmup a scalar operator $\mathcal{O}$ with scaling dimension $\Delta_{\mathcal{O}}$, and with one-point function

$$ \langle\langle \mathcal{O}(x) \rangle\rangle = \frac{a_{\mathcal{O}}}{|x_{\perp}|^{\Delta_{\mathcal{O}}}}. \qquad (3.1) $$

Its two-point function with the displacement is completely fixed by this data

$$ \langle\langle \mathcal{O}(x) D^j(y) \rangle\rangle = b_{\mathcal{O}} \frac{x^j |x_{\perp}|^{p-\Delta_{\mathcal{O}}}}{|x-y|^{2(p+1)}}, \quad \text{with} \quad b_{\mathcal{O}} = 2^p \Gamma\left(\frac{p+1}{2}\right) \pi^{-(p+1)/2} \Delta_{\mathcal{O}} a_{\mathcal{O}}. \qquad (3.2) $$

Our notation is as follows: $x = (x^\alpha, x^j)$ and $y = (y^\alpha, 0)$ where early Greek letters label directions along the defect, $\alpha \in \{0, \dots, p-1\}$, while middle Latin letters label the directions transverse to the defect, $j \in \{p, \dots, n-1\}$. Furthermore $|x|^2 = \eta_{\mu\nu}x^\mu x^\nu$ and $|x_\perp|^2 = \delta_{ij}x^i x^j$.

The general form (3.2) is fixed by the unbroken $\mathfrak{so}(2, p) \oplus \mathfrak{so}(n-p)$ invariance. The coefficient $b_{\mathcal{O}}$ follows from the identity

$$ \int d^p y \langle\langle \mathcal{O}(x) D^j(y) \rangle\rangle = - \frac{\partial}{\partial x^j} \langle\langle \mathcal{O}(x) \rangle\rangle \qquad (3.3) $$

which expresses the fact that the operator $\exp(ia \int d^p y D^j(y))$ translates the defect in the $j$th direction by an amount $a$.<fnref n="14" /> This identity is valid for any operator, scalar or tensor, descendant or primary.

Consider next a symmetric traceless tensor $T^{\mu\nu}$. The unbroken conformal and transverse-rotation symmetry, $\mathfrak{so}(2, p) \oplus \mathfrak{so}(n-p)$, determines $\langle T^{\mu\nu} D^j \rangle$ in terms of three parameters $b_{1,2,3}$. Explicitly [1, 4]

$$ \langle\langle T^{\mu\nu}(x) D^j(y) \rangle\rangle = \frac{|x_{\perp}|^{p-\Delta_T}}{|x-y|^{2p+2}} G^{\mu\nu; j}(x, y) \qquad (3.4) $$

<footnote n="14">¹⁴The integral in (3.3) is well-defined in the Euclidean theory.</footnote>
with

$$G^{\alpha\beta;j} = \frac{1}{n}((n-p-1)\mathfrak{b}_2 - \mathfrak{b}_1)\eta^{\alpha\beta}x^j + 4\mathfrak{b}_1 \frac{(x-y)^\alpha(x-y)^\beta x^j|x_\perp|^2}{|x-y|^4}; \quad (3.5a)$$

$$G^{i\beta;j} = -\mathfrak{b}_3\delta^{ij}\frac{|x_\perp|^2(x-y)^\beta}{|x-y|^2} + (\mathfrak{b}_3 - 2\mathfrak{b}_1)\frac{x^i x^j(x-y)^\beta}{|x-y|^2} + 4\mathfrak{b}_1\frac{x^i x^j(x-y)^\beta|x_\perp|^2}{|x-y|^4}; \quad (3.5b)$$

$$G^{ik;j} = -\frac{1}{n}((p+1)\mathfrak{b}_2 + \mathfrak{b}_1)\delta^{ik}x^j + \frac{\mathfrak{b}_3}{2}(\delta^{ji}x^k + \delta^{jk}x^i)\left(1 - \frac{2|x_\perp|^2}{|x-y|^2}\right) \\ + (\mathfrak{b}_1 + \mathfrak{b}_2 - \mathfrak{b}_3)\frac{x^i x^k x^j}{|x_\perp|^2} + (2\mathfrak{b}_3 - 4\mathfrak{b}_1)\frac{x^i x^k x^j}{|x-y|^2} + 4\mathfrak{b}_1\frac{x^i x^k x^j|x_\perp|^2}{|x-y|^4}. \quad (3.5c)$$

The identity (3.3) with $\mathcal{O}$ replaced by $T^{\mu\nu}$ fixes two of the parameters,

$$(p+1)\mathfrak{b}_2 + \mathfrak{b}_1 = \frac{\Delta_T}{2}\mathfrak{b}_3 \quad \text{and} \quad \mathfrak{b}_3 = 2^{p+2}\pi^{-(p+1)/2}\Gamma\left(\frac{p+3}{2}\right) a_T, \quad (3.6)$$

so one parameter stays free. But in the special case where $T^{\mu\nu}$ is the SCFT energy-momentum tensor, $\Delta_T = n$ and the broken conservation law $\partial_\mu T^{\mu j} = \delta(x_\perp)D^j$ gives one more equation that relates the remaining parameter to the displacement norm, [1]

$$2p \mathfrak{b}_2 - (2n - p - 2) \mathfrak{b}_3 = \frac{(n - p) \Gamma\left(\frac{n-p}{2}\right)}{\pi^{(n-p)/2}} C_D. \quad (3.7)$$

Taken together the relations (3.6) and (3.7) can now be used to determine $\langle T^{\mu\nu}D^j \rangle$ completely in terms of the DCFT data $a_T$ and $C_D$.

Since the conjecture (1.1) relates $C_D$ to $a_T$, we can re-express it as a relation between the parameters $\mathfrak{b}_{1,2,3}$. It takes the elegant form

$$2n\mathfrak{b}_1 = (p+2)\mathfrak{b}_3. \quad (3.8)$$

We have thus succeeded in reformulating the original conjecture as a condition on the bulk-to-defect correlation function $\langle T^{\mu\nu}D^j \rangle$ which, as we will prove in section 5, follows from superconformal Ward identities.

## 3.2 Line defects in $\mathcal{N} = 2$ abelian gauge theory

To put some flesh into these abstract formulae, and to prepare the ground for a discussion of the underlying physics, let us consider the example of superconformal line defects in the $\mathcal{N} = 2$ abelian gauge theory in four dimensions.

The $\mathcal{N} = 2$ vector multiplet contains the gauge field $A_\mu$, two Weyl photinos and two real scalar fields $\phi_{I=1,2}$. The general (electrically-charged) conformal line defect is described by the operator

$$W = \exp \left( \int ds (ieA_\mu \dot{y}^\mu + g |\dot{y}| \hat{n}^I \phi_I) \right), \quad (3.9)$$

where $\hat{n}^I \hat{n}_I = 1$, dots are s-derivatives and the bulk fields are evaluated at $y^\mu(s)$. The unit vector $n^I$ could in principle depend on $s$, but we here take it constant and write for short $\hat{n}^I \phi_I = \phi$.
The stress tensor of the bosonic fields is $T_{\mu\nu} = T_{\mu\nu}^{(s)} + T_{\mu\nu}^{(v)}$ where

$$T_{\mu\nu}^{(s)} = \partial_{\mu}\phi\partial_{\nu}\phi - \frac{1}{2}\eta_{\mu\nu}|\partial\phi|^2 + \frac{1}{6}(\eta_{\mu\nu}\Box - \partial_{\mu}\partial_{\nu})\phi^2, \qquad (3.10a)$$

$$\text{and} \quad T_{\mu\nu}^{(v)} = F_{\mu}^{\rho} F_{\nu\rho} - \frac{1}{4} \eta_{\mu\nu} |F|^2. \qquad (3.10b)$$

The scalar stress tensor $T_{\mu\nu}^{(s)}$ includes the total-derivative term that makes it traceless. To find the displacement operator we parametrise the worldline in static gauge, $(y^0, y^j) = (s, y^j(s))$, and expand (3.9) to linear order in $y^j$. The result is

$$D^j = eF^{0j} + g \partial^j \phi. \qquad (3.11)$$

Finally, the classical background fields created by the static defect read

$$\phi_{\text{class}} = \frac{g}{4\pi|x_{\perp}|}, \quad \text{and} \quad F_{\text{class}}^{0j} = \frac{ex^j}{4\pi|x_{\perp}|^3}. \qquad (3.12)$$

Since the DCFT at hand is free, eqs. (3.10) - (3.12) is all that we need to calculate any correlation function.

The one-point function $\langle T^{\mu\nu} \rangle$ is given entirely by the classical fields. Comparing to the general form eq. (1.3) we find

$$a_{T}^{(s)} = -\frac{g^2}{48\pi^2}; \quad a_{T}^{(v)} = -\frac{e^2}{16\pi^2}. \qquad (3.13)$$

From the scalar propagator $\Delta(x-y) = 1/4\pi^2|x-y|^2$ we furthermore get

$$C_D^{(s)} = \frac{g^2}{2\pi^2}; \quad C_D^{(v)} = \frac{e^2}{\pi^2}. \qquad (3.14)$$

For $p=1, n=4$ the conjecture (1.1) reads $C_D = -18a_T$. This is not valid separately for the photon or the scalar. But the half-BPS line defect has $e = g$,<fnref n="15" /> and one can check that the conjecture is indeed satisfied.

To calculate $\langle T^{\mu\nu} D^j \rangle$ we contract the displacement with one of the two free fields in the stress tensor, and replace the remaining field by its classical value. In the Lorentzian theory $\Delta(x-y)$ is the Feynman propagator if $y^j(s)$ is treated as a quantum field, or the retarded one if it is treated as a classical background. Since the general form (3.4) - (3.5) relied only on conformal symmetry, the result has this form separately for the scalar and the vector. One finds

$$\mathfrak{b}_1^{(s)} = -\frac{g^2}{4\pi^3}, \quad \mathfrak{b}_2^{(s)} = -\frac{g^2}{24\pi^3}, \quad \mathfrak{b}_3^{(s)} = -\frac{g^2}{6\pi^3}; \qquad (3.15a)$$

$$\mathfrak{b}_1^{(v)} = 0, \quad \mathfrak{b}_2^{(v)} = -\frac{e^2}{2\pi^3}, \quad \mathfrak{b}_3^{(v)} = -\frac{e^2}{2\pi^3}. \qquad (3.15b)$$

The conjecture (3.8), which is equivalent to (1.1), reads $8\mathfrak{b}_1 = 3\mathfrak{b}_3$. Not surprisingly, it is not valid for the scalar and the vector separately, but if $e = g$ it is indeed satisfied, $8(\mathfrak{b}_1^{(s)} + \mathfrak{b}_1^{(v)}) = 3(\mathfrak{b}_3^{(s)} + \mathfrak{b}_3^{(v)})$.

<footnote n="15">¹⁵There also exist 1/4-BPS defects but these break the transverse-rotation symmetry [74].</footnote>
## 4 Stress, radiation and the Schott term

Despite its simplicity, the new form (3.8) of the conjecture cannot yet be exploited for a proof. Furthermore, its physical meaning is obscure. In search of inspiration let us return to the original argument of Lewkowycz and Maldacena [2] that motivated the relation between $C_D$ and $a_T$.

These authors wanted to reconcile two different calculations<fnref n="16" /> of the energy emitted by an accelerating half-BPS ‘quark’ in the $N = 4$ super Yang-Mills theory. One calculation used the Bremstrahlung function which is proportional to $C_D$ [15], while the other computes the energy flux directly in the background of a uniformly-accelerating quark which is conformal to a static one [16]. The discrepancy between the two results was attributed to the difficulty of separating the radiation from the self-energy of the defect. This is a time-honoured problem even in classical electrodynamics [22–27].

Noting that the $N = 4$ theory has a scalar operator $O$ with scaling dimension $\Delta_O = n-2$ which couples to the defect, Lewkowycz and Maldacena proposed that the emitted radiation should be computed by the modified stress tensor

$$ \tilde{T}^{\mu\nu} = T^{\mu\nu} + \xi (\eta^{\mu\nu} \Box - \partial^{\mu} \partial^{\nu}) O \quad (4.1) $$

for an appropriate value of $\xi$. The new stress tensor is conserved but not traceless. The intuition in ref. [2], further elaborated in [17–20], was that choosing $\xi$ so as to remove the leading short-distance singularity of $\tilde{T}^{\mu\nu}$ near the defect would also neatly separate the defect’s self-energy.

The intuition is indeed correct, but the role of supersymmetry has remained unclear. In this section we propose a more detailed and physical explanation.

### 4.1 Static stress and the NEC

The leading singularity of $\tilde{T}^{\mu\nu}$ near the defect is given by its one-point function. Using the general form (3.1) for $\langle O(x) \rangle$ we find

$$ \langle \tilde{T}^{\alpha\beta} \rangle = \langle T^{\alpha\beta} \rangle + \xi a_O p(n-2) \frac{\eta^{\alpha\beta}}{|x_{\perp}|^n}, \quad \langle \tilde{T}^{i\beta} \rangle = \langle T^{i\beta} \rangle = 0 \quad (4.2a) $$

$$ \text{and} \quad \langle \tilde{T}^{ik} \rangle = \langle T^{ik} \rangle + \xi a_O (n-2) \left[ (p+1) \frac{\delta^{ik}}{|x_{\perp}|^n} - n \frac{x^i x^k}{|x_{\perp}|^{n+2}} \right]. \quad (4.2b) $$

Comparing to expression (1.3) for $\langle T^{\mu\nu} \rangle$ shows that it is impossible to remove the singularity in both the worldvolume and transverse components. Since the energy better stay positive, we should try to remove the latter singularity. This is indeed possible with the choice $\xi = \xi_{\text{def}}$, where

$$ \xi_{\text{def}} a_O = \frac{a_T}{n(n-2)}. \quad (4.3) $$

The subscript ‘def’ is here to remind us that the above choice of $\xi$ depends on the defect via the DCFT data $a_O$ and $a_T$.

<footnote n="16">¹⁶For earlier calculations of radiation from the holographic-dual string in AdS see refs. [10–14].</footnote>
Inserting (4.3) in (4.2) one finds

$$ \langle \tilde{T}^{\alpha\beta} \rangle = a_T \left(1 - \frac{1}{n}\right) \frac{\eta^{\alpha\beta}}{|x_{\perp}|^n}, \qquad \langle \tilde{T}^{i\beta} \rangle = \langle \tilde{T}^{ik} \rangle = 0. \qquad (4.4) $$

The key point here is that, with the above choice of $\xi$, the singularity of $\tilde{T}^{\mu\nu}$ at the defect can be absorbed by a redefinition of the defect's tension or mass. For a line defect with proper-time $\tau$ for example

$$ T_{\text{tot}}^{\mu\nu}(x) = m \int d\tau \delta^{(n)}(x-y) \frac{dy^\mu}{d\tau} \frac{dy^\nu}{d\tau} + \tilde{T}_{\text{reg}}^{\mu\nu}(x) \qquad (4.5) $$

where $y^\mu(\tau)$ is the worldline of the defect, $m$ its renormalised mass and $\tilde{T}_{\text{reg}}^{\mu\nu}$ has no $|x-y|^{-n}$ singularity. This would not have worked with $T^{\mu\nu}$ because the singular stress $\langle \tilde{T}^{ik} \rangle$ cannot be absorbed in $m$.

Two questions arise immediately. Who has ordered the modification of the bulk stress tensor (4.1), and why do we need supersymmetry? The free-field example of the previous section provides the answers. In this example the scalar operator $O$ is $\phi^2$ and its one-point function is $\langle \phi^2 \rangle = g^2/16\pi^2|x_{\perp}|^2$, so to cancel the transverse stress we must choose

$$ \xi_{\text{def}} = \frac{a_T}{8a_O} = -\frac{g^2 + 3e^2}{24g^2}. \qquad (4.6) $$

This can be done for arbitrary $e$ and $g$. But something special happens when $e = g$, i.e. for supersymmetric defects. In this case $\xi = -1/6$ which is precisely the value needed to remove the total-derivative term in (3.10a). This term comes from the conformal $R\phi^2$ coupling of the scalar field and is known to violate even the weakest of local energy conditions, the Null Energy Condition (NEC), see e.g. [21]. It was indeed shown in ref. [19] that for $\xi \neq -1/6$ an accelerating defect coupling to $\phi$ emits NEC-violating radiation.<fnref n="17" />

The above free-field example motivates the following:

*The subtraction that restores the NEC for the bulk energy-momentum tensor also removes the transverse stress of static planar superconformal defects.*

We will now explain why this assertion is generally valid.

## 4.2 Energy flux of a moving defect

Ever since Dirac's seminal paper [22] the question of radiation reaction for a moving charge has been the subject of controversy. The problem can be summarized as follows: the energy flux in the fields created by the charge is the sum of a radiation and a 'bound-field' part [25, 27]. These are separately conserved. Integrating the radiation flux in electrodynamics gives the standard Larmor formula which is proportional to the acceleration squared. In extensions of Maxwell's theory, such as the $N = 2$ supersymmetric theory of section 3.2, the charge may also couple to scalar fields. In this case there can be a NEC-violating radiation flux proportional to the derivative of the acceleration [19].

<footnote n="17">¹⁷But the A(verage) NEC is satisfied if the defect stops accelerating at $\tau \to \pm\infty$.</footnote>
The ‘bound-energy’ flux, on the other hand, is singular near the defect. The lore is that this singular flux can be absorbed by redefining the particle mass, and that the leftover is the Schott term in the Lorentz-Dirac equation of motion. This term is also proportional to the derivative of the acceleration. Its necessity in the Lorentz-Dirac equation follows from a heuristic argument of Landau and Lifshitz [24]. But calculating it has remained a challenge, see e.g. ref. [27].

As anticipated above, the two problems are related. The operator that sets the defect in motion is $\exp(i \int ds y^j(s) D^j(y))$ where $s = y^0$ in static gauge. The effects of interest are linear in the acceleration, so they can be extracted from $\int \langle T^{i0} D^j \rangle y^j$. The bulk-to-defect 2-point function is given by (3.4) - (3.5b), we copy it here for the reader’s convenience:

$$ \langle T^{i0} D^j \rangle = \frac{(x^0 - y^0) |x_\perp|^{p-n}}{|x-y|^{2p+4}} \left( -\mathfrak{b}_3 \delta^{ij} |x_\perp|^2 + (\mathfrak{b}_3 - 2\mathfrak{b}_1) x^i x^j + 4\mathfrak{b}_1 \frac{x^i x^j |x_\perp|^2}{|x-y|^2} \right). \quad (4.7) $$

It has singularities when $x-y$ is null, or when $|x_\perp| = 0$.

The NEC-violating radiation comes from the leading $|x-y|$ singularity which we can write as follows

$$ \langle T^{i0}(x) D^j(y) \rangle = - \frac{32 \mathfrak{b}_1 |x_\perp|^{p+2-n}}{p(p+1)(p+2)} \partial^0 \partial^i \partial^j \left( \frac{1}{|x-y|^{2p}} \right) + \text{subleading.} \quad (4.8) $$

Note that this term is absent in Maxwell’s theory, which obeys the NEC and for which $\mathfrak{b}_1 = 0$. In the free-field example of section 3.2 such a term can be seen to arise when we contract the displacement with one of the two scalar fields in $T^{i0} \sim \partial^i \partial^0 \phi^2$. The relevant Lorentzian Green function is the retarded one

$$ \frac{1}{4\pi^2|x-y|^2} \rightarrow \Delta_R(x-y) = \frac{i\theta(x^0-y^0)}{2\pi} \delta(|x-y|^2), \quad (4.9) $$

and all three derivatives act on it. Converting $\partial^i \partial^j$ to derivatives with respect to $s$ as in the familiar calculation of the Liénard-Wiechert potential, and integrating by parts gives the NEC-violating radiation found in [19].

Generalising to an interacting theory sounds daunting, but fortunately conformal invariance comes here to the rescue. What we want is to replace the Euclidean correlator (4.7) by the retarded one,

$$ \langle T^{i0}(x) D^j(y) \rangle \rightarrow \langle \theta(x^0 - y^0) [T^{i0}(x), D^j(y)] \rangle. \quad (4.10) $$

In a conformal theory this can be done by a simple analytic continuatio of Euclidean time.<fnref n="18" /> Explicitly, the ordered n-point function $\langle \mathcal{O}_1(x_1) \cdots \mathcal{O}_n(x_n) \rangle$ is given by the analytic continuation $x_r^0 \rightarrow x_r^0 - i\epsilon_r$ with $\epsilon_1 > \epsilon_2 > \cdots > \epsilon_n > 0$. For line defects ($p=1$) this prescription leads precisely to the replacement of (4.9) in (4.8). This shows that, apart from the unknown coefficient $\mathfrak{b}_1$, the NEC-violating radiation is universal — the same in all dimensions, and in free as well as strongly-interacting DCFTs.<fnref n="19" /> This agrees with the holographic calculation of ‘heavy quark’ radiation in $\mathcal{N} = 4$ super Yang-Mills at strong coupling [13].

<footnote n="18">¹⁸For reviews of Lorentzian CFT see [75, 76].</footnote>

<footnote n="19">¹⁹The reader may wonder why the propagator of a free scalar field $\phi$ in dimensions $n \neq 4$ does not enter in the correlator (4.8). The reason is that conformal defects do not couple linearly to $\phi$ in other dimensions.</footnote>
Having identified the term responsible for the NEC-violating radiation in the 2-point correlator (4.7) let us try now to remove it from $\langle T^{i0}(x)D^j(y) \rangle$. Using the expression (3.2) for $\langle O(x)D^j(y) \rangle$, with $\Delta_O = n-2$, gives

$$ \begin{aligned} -\xi \partial^i \partial^0 \langle OD^j \rangle &= \xi b_O (2p+2) \partial^i \left( x^0 x^j \frac{|x_\perp|^{p-n+2}}{|x|^{2p+4}} \right) = \\ &= \xi b_O (2p+2) \frac{x^0 |x_\perp|^{p-n+2}}{|x|^{2p+4}} \left( \delta^{ij} + (p-n+2) \frac{x^i x^j}{|x_\perp|^2} - (2p+4) \frac{x^i x^j}{|x|^2} \right). \end{aligned} \quad (4.11) $$

Adding this to (4.7) shows that to cancel the NEC-violating term we must choose

$$ \xi b_O(p+1)(p+2) = b_1. \quad (4.12) $$

To remove the static stress, on the other hand, we had to choose $\xi$ as in eq. (4.3). When combined with (3.2) this reads

$$ \xi_{\text{def}} b_O(p+1) = \frac{a_T}{n} (p+1) 2^p \Gamma\left(\frac{p+1}{2}\right) \pi^{-(p+1)/2} = \frac{b_3}{2n}. \quad (4.13) $$

Lo and behold, the two values of $\xi$ agree if and only if the conjectured BPS relation (3.8) is satisfied! This proves the boxed assertion of section 4.1.

Adding (4.7) and (4.11), and using the value (4.12) of $\xi b_O$ that restores the NEC we find

$$ \langle \tilde{T}^{i0}(x)D^j(y) \rangle = -b_3 \left(1 - \frac{1}{n}\right) \frac{|x_\perp|^{p-n+2}}{|x-y|^{2p+4}} (x^0 - y^0) \delta^{ij} + [\text{non susy}], \quad (4.14) $$

where [non susy] vanishes iff (3.8) is obeyed. The above $\propto \delta^{ij}$ correlation function is required by relativistic covariance. The reason is that the energy flux of a defect moving at constant speed in the jth direction can be computed either by a Lorentz boost of the energy-momentum $\langle \tilde{T}^{\mu\nu} \rangle$ at rest, eq. (4.4), or equivalently by inserting the operator $\exp(iv \int dy^0 y^0 D^j)$. Agreement of the two results fixes uniquely the residual correlator (4.14).

It would be interesting to understand whether supersymmetry allows a proper derivation of the Schott term in the Lorentz-Dirac equation. Absorbing the leading singularities of $T^{\mu\nu}$ by redefining the mass looks at odds, as we have explained, with relativistic invariance when the fields carry non-zero stress in the particle's rest frame. This is the case for the Coulomb field, and it could be the source of the encountered difficulties in standard electrodynamics. We defer the question to future work.

A last remark concerns the modified energy-momentum tensor. Since dynamical defects break anyway the conformal symmetry, the fact that $\tilde{T}_\mu^\mu \neq 0$ is not vexing. But the modification (4.1) relies on the existence of an R-singlet scalar operator of dimension $n-2$, coupling appropriately to all superconformal defects. Candidates exist in the stress-tensor multiplet of most bulk SCFTs, see ref. [44], but with two notable exceptions: $N=1$ in four dimensions and $N=2$ in three dimensions. The BPS equation (1.1) continues to hold in these cases, but $O$ is just a fictitious intermediate device of the proof.
## 5 Proof of the conjecture

Let us come then to the proof of the conjecture (3.8). All we will use from the previous section is eq. (4.14) which implies, in particular, that $\langle \tilde{T}^{i0}(x)D^j(y) \rangle = 0$ for $i \neq j$. Pick two transverse coordinates, say $x^p$ and $x^{p+1}$, and define $z = x^p + ix^{p+1}$. We have then equivalently

$$ \langle T_{\alpha z}(x) D_z(y) \rangle \propto \frac{\partial}{\partial x^{\alpha}} \frac{\partial}{\partial z} \left( \frac{\bar{z} |x_{\perp}|^{p-n+2}}{|x-y|^{2p+2}} \right) \quad \text{iff} \quad \frac{\mathfrak{b}_1}{\mathfrak{b}_3} = \frac{p+2}{2n}. \qquad (5.1) $$

We used here the rotation symmetry to get rid of the $i=j$ components on the left, plus the fact that the last two terms of (4.7) combine as in the right-hand side above if and only if $\mathfrak{b}_1/\mathfrak{b}_3$ is fixed as in (3.8). Note that time can be replaced by any $x^\alpha$ along the defect worldvolume, and that the only role of the operator $\mathcal{O}$ is to determine the right-hand side of (5.1).

This form of the conjecture is appealing because all superconformal defects share a common unbroken $\mathfrak{su}(1, 1|1) \supset \mathfrak{so}(2, 1) \oplus \mathfrak{so}(2)_{\text{def}}$ symmetry, as one can verify from table 1. The $\mathfrak{so}(2, 1)$ subgroup of this superalgebra is the conformal group in the worldvolume time $x^0$ (or $x^+$ for surface defects), the $\mathfrak{so}(2)_{\text{def}}$ rotates the transverse $z$-plane (and in certain cases also the R-charges), and the two supercharges $Q, \bar{Q}$ obey $\{Q, \bar{Q}\} = P^0$ or $P^+$. Since $T_{z0}$ is a top component of the full bulk superalgebra, it is also the top component of an $\mathfrak{su}(1, 1|1)$ multiplet whose superprimary $\mathcal{A}$ has dimension $\Delta_{\mathcal{A}} = n-1$. One can then try to prove the conjecture in a single go from the Ward identity $0 = \bar{Q}Q \langle AD_z \rangle$, using the fact that the displacement operator is also an $\mathfrak{su}(1, 1|1)$ top component [40].

The strategy works nicely for line defects of a three-dimensional $\mathcal{N}=2$ SCFT, but computing the general form of $\langle AD_z \rangle$ turned out to be arduous. We have thus followed an alternative route that uses a single, judiciously-chosen supercharge $Q$, and the following slight reformulation of the conjecture

$$ \langle T_{zz}(x) D_z(y) \rangle \propto \frac{\partial^2}{\partial z^2} \left( \frac{\bar{z} |x_{\perp}|^{p-n+2}}{|x-y|^{2p+2}} \right) \quad \text{iff} \quad \frac{\mathfrak{b}_1}{\mathfrak{b}_3} = \frac{p+2}{2n}. \qquad (5.2) $$

This can be derived in the same way as (5.1).

To avoid clattering, we will change completely our notation in this final section, hoping this will not confuse the reader. We work in Euclidean signature and let the transverse coordinate $z$ be $x_1 + ix_2,^{20}$ and the imaginary time be one of $x_{3,4,5}$. We also now use early Greek letters as spinor indices. The index structure of the gamma matrices in $n=3,5$ dimensions is $\gamma_\mu = (\gamma_\mu)_\alpha^\beta$, while in $n=4,6$ it is $\gamma_\mu = (\gamma_\mu)_\alpha^{\dot{\beta}}$ and $\bar{\gamma}_\mu = (\bar{\gamma}_\mu)_{\dot{\alpha}}^\beta$. Indices are raised and lowered with the charge-conjugation matrices $(C^{(n)})_{\alpha\beta}$ in $n=3,5$ dimensions, $(C^{(4)})_{\alpha\beta} = (C^{(4)})_{\dot{\alpha}\dot{\beta}}$ in four dimensions, and $(C^{(6)})_{\alpha\beta}$ in six dimensions where the conjugate spinor representations are inequivalent. All the bulk superalgebras of table 1 have $\mathfrak{u}(1)$ and/or $\mathfrak{su}(2)$ R-symmetries. We denote their generators by $r$ and $(R_I)_A^B$ with $I=1,2,3$ and $A,B=1,2$. The index structure of the supercharges is thus $Q_\alpha^A$ or $\bar{Q}_\alpha^A$. The $\mathfrak{su}(2)$ R-symmetry index is raised by multiplying with $\epsilon^{AB}$ on the left where $\epsilon^{12}=1$.

In order to unify the proofs we also choose the following bases of $\gamma$-matrices: in three dimensions we use the $2 \times 2$ Pauli matrices $\sigma_i$. In four dimensions $\gamma_\mu = (\sigma_i, i\mathbb{I}_2)$ and $\bar{\gamma}_\mu =$

<footnote>$^{20}$So that a lower z-index stands for $V_z = \frac{1}{2}(V_1 - iV_2)$.</footnote>
<table><caption>**Table 2.** The preserved supercharges for all the DCFTs of table 1. The directions 1,2 are always transverse to the defect worldvolume, while the parallel directions are shown in the third column above. For $(n,p) = (4,1)$, the supercharges are complex and only the real parts shown in the table are unbroken. Our proof uses Ward identities of the first preserved supercharge in each case.</caption><thead><tr><th>n</th><th>p</th><th>defect directions</th><th>preserved supercharges</th></tr></thead><tbody><tr><td rowspan="2">6</td><td>2</td><td>3,4</td><td>Q<sub>1</sub><sup>1</sup>, Q<sub>3</sub><sup>1</sup>, Q<sub>1</sub><sup>2</sup>, Q<sub>3</sub><sup>2</sup></td></tr><tr><td>4</td><td>3,4,5,6</td><td>Q<sub>1</sub><sup>1</sup>, Q<sub>2</sub><sup>1</sup>, Q<sub>3</sub><sup>2</sup>, Q<sub>4</sub><sup>2</sup></td></tr><tr><td rowspan="3">5</td><td>1</td><td>5</td><td>Q<sub>1</sub><sup>1</sup>, Q<sub>4</sub><sup>1</sup>, Q<sub>1</sub><sup>2</sup>, Q<sub>4</sub><sup>2</sup></td></tr><tr><td>2</td><td>3,4</td><td>Q<sub>1</sub><sup>1</sup>, Q<sub>3</sub><sup>1</sup>, Q<sub>1</sub><sup>2</sup>, Q<sub>3</sub><sup>2</sup></td></tr><tr><td>3</td><td>3,4,5</td><td>Q<sub>1</sub><sup>1</sup>, Q<sub>2</sub><sup>1</sup>, Q<sub>3</sub><sup>2</sup>, Q<sub>4</sub><sup>2</sup></td></tr><tr><td rowspan="2">4</td><td>1</td><td>4</td><td>Re{Q<sub>1</sub><sup>1</sup>, Q<sub>2</sub><sup>1</sup>, Q<sub>1</sub><sup>2</sup>, Q<sub>2</sub><sup>2</sup>}</td></tr><tr><td>2</td><td>3,4</td><td>Q<sub>1</sub>, Q&#773;<sub>2</sub></td></tr><tr><td>3</td><td>1</td><td>3</td><td>Q<sub>1</sub>, Q&#773;<sub>2</sub></td></tr></tbody></table>

($\sigma_i, -i\mathbb{I}_2$). In five dimensions we choose the following basis$^{21}$

$$ \gamma_1 = \sigma_1 \otimes I_2, \quad \gamma_2 = \sigma_2 \otimes I_2, \quad \gamma_3 = \sigma_3 \otimes \sigma_1, \quad \gamma_4 = \sigma_3 \otimes \sigma_2, \quad \gamma_5 = \sigma_3 \otimes \sigma_3. \qquad (5.3) $$

In six dimensions we add $\gamma_6 = i\mathbb{I}_4$ with $\bar{\gamma}_6 = -i\mathbb{I}_4$ while all other $(\bar{\gamma}_i)_{\hat{\alpha}}^{\beta}$ are the same as $(\gamma_i)_{\alpha}^{\dot{\beta}}$. The charge conjugation matrices are $C^{(3)} = C^{(4)} = i\sigma_2$ and $C^{(5)} = C^{(6)} = \sigma_1 \otimes i\sigma_2$.

Choosing the $\gamma$-matrices as above, and orienting the static (hyper)planar defects in an appropriate way, allows us to list the preserved supercharges for all the DCFTs of table 1 in a unified way, as shown in table 2 above. All the defects are half-BPS, so there are four unbroken supercharges in the 5d and 6d $\mathcal{N}=1$ and 4d $\mathcal{N}=2$ cases. The 4d $\mathcal{N}=1$ and 3d $\mathcal{N}=2$ DCFTs have two unbroken supercharges.

We want to show that the special form of $\langle T_{zz}(x)D_z(y) \rangle$ in (5.2) follows from super-conformal Ward identities. To this end we need the transformations of various fields in the stress-tensor multiplets. We list these transformation in appendix C. They are obtained from the Jacobi identities for the superconformal algebras which are recorded for completeness in appendix B. The transformation of $T_{\mu\nu}$ under the $Q$-generators has the universal form

$$ Q_{\alpha}^{A}(T_{\mu\nu}) = \frac{1}{4}(\partial_{\rho}J_{\mu\beta}^{A})(\gamma^{\rho})_{\alpha}^{\beta} + (\mu \leftrightarrow \nu), \qquad (5.4) $$

where the index $A$ can be dropped if the R-symmetry is $u(1)$. We will need a single preserved supercharge: $Q_1^1$ in 6d and 5d, Re($Q_1^1$) in 4d $\mathcal{N}=2$, and $Q_1$ for the last two rows of the table. Our previous painstaking choices were designed to simplify the action of this preserved supercharge on $T_{zz}$.

A second fact that we will need is that there exists a fermionic defect operator $\Lambda$ in the displacement multiplet such that $Q(\Lambda) = D_z$. Note indeed that there is always a broken supercharge $Q_{br}$ such that $\{Q, Q_{br}\} = P_z$. If we choose $\Lambda$ to be the corresponding goldstino then $Q(\Lambda) = D_z$ up to a derivative of a lower component. We show that this derivative-term is absent by working out explicitly the transformation of the displacement multiplets in appendix C.

<footnote>$^{21}$This basis is not the same as the one in appendix A.</footnote>
We are now ready for the proof. For the 4d surface defects and the 3d line defects the result follows from Ward identities of the supercharge $Q = Q_1$. Its action on $T_{zz}$ and the $u(1)$ R-symmetry current $j_z$ reads

$$Q(T_{zz}) = \frac{1}{2} \partial_z J_{z1}, \quad Q(j_z) = \frac{1}{2} J_{z1}, \qquad (5.5)$$

where $J_{\mu\alpha}$ is the supercurrent. Then $Q\langle T_{zz}(x)\Lambda(y)\rangle = Q\langle j_z(x)\Lambda(y)\rangle = 0$ gives

$$\langle T_{zz}(x)D_z(y)\rangle = \frac{\partial}{\partial z}\langle j_z(x)D_z(y)\rangle \propto \frac{\partial^2}{\partial z^2}\left(\frac{\bar{z}|x_{\perp}|^{p-n+2}}{|x-y|^{2p+2}}\right). \qquad (5.6)$$

We also used here the fact that the kinematic structure of $\langle j_{\mu}(x)D_{j}(y)\rangle$ is fixed as above by conformal symmetry.<fnref n="22" />

In 5d and 6d we need the transformation of $T_{zz}$, $j_z^3$ and $O$ under the preserved supercharge $Q = Q_1^1$, where $j_{\mu}^3$ is $R^{I=3}$ component of the $\mathfrak{su}(2)$ R-symmetry current, and $O$ is the scalar in the stress-tensor multiplet. One finds

$$Q(T_{zz}) = \frac{1}{2} \partial_z J_{z1}^1, \quad Q(j_z^3) = \frac{1}{2} J_{z1}^1 + \eta \partial_z \chi_1^1, \quad Q(O) = \chi_1^1. \qquad (5.7)$$

The factor $\eta$ is $\frac{1}{5}$ in 6d and $\frac{1}{4}$ in 5d. Supersymmetry implies that $Q\langle T_{zz}(x)\Lambda(y)\rangle$, $Q\langle j_z^3(x)\Lambda(y)\rangle$ and $Q\langle O(x)\Lambda(y)\rangle$ all vanish. Combining these Ward identities and using the rigid kinematic structure of $\langle j_z^3(x)D_z(y)\rangle$ gives the desired result

$$\langle T_{zz}(x)D_z(y)\rangle = \frac{\partial}{\partial z}\langle j_z^3(x)D_z(y)\rangle - \eta \frac{\partial^2}{\partial z^2}\langle O(x)D_z(y)\rangle \propto \frac{\partial^2}{\partial z^2}\left(\frac{\bar{z}|x_{\perp}|^{p-n+2}}{|x-y|^{2p+2}}\right). \qquad (5.8)$$

Finally, for 4d line defects we need the transformation of $T_{zz}$, $j_z^3$ and $O$ under $Q = Q_1^1+\bar{Q}_1^1$, where $j_{\mu}^3$ is the $R^{I=3}$ component of the $\mathfrak{su}(2)$ R-symmetry current, and $O$ is the scalar superprimary in the stress-tensor multiplet,<fnref n="23" />

$$Q(T_{zz}) = \frac{1}{2} \partial_z (J_{z1}^1 + \bar{J}_{z1}^1), \quad Q(j_z^3) = \frac{1}{2} (J_{z1}^1 + \bar{J}_{z1}^1) - \frac{1}{3} \partial_z (\chi_1^1 - \bar{\chi}_1^1), \quad Q(O) = \chi_1^1 - \bar{\chi}_1^1. \qquad (5.9)$$

The vanishing of $Q\langle T_{zz}(x)\Lambda(y)\rangle$, $Q\langle j_z^3(x)\Lambda(y)\rangle$ and $Q\langle O(x)\Lambda(y)\rangle$ then gives

$$\langle T_{zz}(x)D_z(y)\rangle = \frac{1}{3}\frac{\partial^2}{\partial z^2}\langle O(x)D_z(y)\rangle \propto \frac{\partial^2}{\partial z^2}\left(\frac{\bar{z}|x_{\perp}|^{p-n+2}}{|x-y|^{2p+2}}\right). \qquad (5.10)$$

This completes the proof of (5.2) in all the cases.

## Acknowledgments

We thank Nadav Drukker, Luc Frappat, Marco Meineri and Paul Sorba for discussions. We also thank the Pollica Physics Center for hosting the workshop “Defects, from condensed matter to quantum gravity”, where this project started. CB acknowledges the hospitality of the physics department at the National Technical University of Athens, and thanks the greek ministry of education for support through grant no. ΥΠ2TA-0559198.

<footnote n="22">²²This vanishes for codimension other than two.</footnote>

<footnote n="23">²³These transformations can be also found in [3, 18]. Our $j_{\mu}^3$ is the $t_{\mu 1}$ of this reference.</footnote>
## A Real subalgebras of $F(4; 2)$

The bosonic algebra of $F(4; 2)$ that generates the symmetries of the $N = 1$ SCFT in five dimensions is $so(2, 5) \oplus su(2)_R$. The fermionic generators are a (spinor, doublet) of this algebra, see e.g. [77]. They are split by their dilatation charge in raising and lowering operators, viz. Poincaré supercharges $Q$ and conformal supercharges $S$.

To understand their reality properties it is convenient to start with a Weyl spinor of $so(2, 8)$ which also transforms as (spinor, doublet) of $so(2, 5) \oplus su(2)_R$. The $Q$ and $S$ supercharges are $so(8)$ spinors with opposite chirality. Imposing the Majorana condition leaves eight real Poincaré and as many conformal supercharges, which is precisely the content of $F(4; 2)$. In short $Q \in (8_s, 2)$ and $S \in (8_c, 2)$ where $8_s$ and $8_c$ are the two inequivalent spinors of $so(8)$.

Note that from the viewpoint of $so(2, 5) \oplus su(2)_R$, the $so(8)$ Majorana condition is a symplectic-Majorana condition.<fnref n="24" /> One can understand why using the standard basis of the Clifford algebra in $2k$ dimensions in terms of Pauli matrices

$$k = 1: \quad \gamma^1 = \sigma^2, \quad \gamma^2 = \sigma^3;$$

$$k = 2: \quad \gamma^1 = \sigma^2 \otimes \sigma^1, \quad \gamma^2 = \sigma^3 \otimes \sigma^1, \quad \gamma^3 = \mathbf{1} \otimes \sigma^2, \quad \gamma^4 = \mathbf{1} \otimes \sigma^3;$$

and so on till $k = 4$. In this basis the Majorana condition for $so(8)$ spinors is $\Psi^* = B\Psi$, where $B = \gamma^1\gamma^3\gamma^5\gamma^7$ is the product of all imaginary $\gamma$-matrices [45]. Writing $\Psi$ as a doublet of $so(5)$ spinors gives $\Psi_a^* = i\epsilon_{ab}(C\Psi)^b$ where $C = i\gamma^1\gamma^3\gamma^5$. Note that $CC^* = -1$ which is why $so(5)$ spinors are pseudoreal, and why there are no Weyl-Majorana $so(6)$ spinors.

Consider now the real subalgebras of $F(4; 2)$ corresponding to superconformal line and surface defects. For line defects the unbroken bosonic symmetry is

$$\mathfrak{so}(2, 1) \oplus \mathfrak{so}(4)_{\text{def}} \oplus \mathfrak{su}(2)_R \simeq \mathfrak{so}(2, 1) \oplus \mathfrak{so}(3) \oplus \mathfrak{so}(3)' \oplus \mathfrak{su}(2)_R, \quad (A.1)$$

with the $so(2, 8)$ spinor transforming in the $(2, 2, 0, 2) \oplus (2, 0, 2, 2)$ representation. It is clearly compatible to keep only half of this spinor by imposing the projection $\gamma^2\gamma^3\gamma^4\gamma^5 = +$. This gives four real Poincaré and as many conformal supercharges, which is precisely the fermionic content of $D(2, 1; 2; 0)$. The residual commuting $su(2)_c$ in eq. (2.3) is identified with $so(3)'$.

For surface defects, on the other hand, the unbroken bosonic symmetry is

$$\mathfrak{so}(2, 2) \oplus \mathfrak{so}(3)_{\text{def}} \oplus \mathfrak{su}(2)_R \simeq \mathfrak{so}(2, 1)_+ \oplus \mathfrak{so}(2, 1)_- \oplus \mathfrak{so}(3)_{\text{def}} \oplus \mathfrak{su}(2)_R, \quad (A.2)$$

and the $so(2, 8)$ spinor transforms in the $(2, 0, 2, 2) \oplus (0, 2, 2, 2)$ representation. The compatible projection is now a chiral projection on the string worldsheet, and the commuting subalgebra is $so(2, 1)_-$ or $so(2, 1)_+$. This agrees with eq. (2.10)

To confirm this analysis, we have also constructed explicitly the real superalgebra embeddings for the above line and surface defects.

<footnote n="24">²⁴A nice general discussion of this condition can be found in [78].</footnote>
## B Superconformal algebras

The common part of all superconformal algebrs is the conformal algebra so(2, d),

$$ 
\begin{align} 
[D, P_{\mu}] &= P_{\mu}, \quad [D, K_{\mu}] = -K_{\mu}, \nonumber \\ 
[M_{\mu\nu}, M_{\rho\sigma}] &= \eta_{\mu\rho}M_{\nu\sigma} + \eta_{\nu\sigma}M_{\mu\rho} - \eta_{\nu\rho}M_{\mu\sigma} - \eta_{\mu\sigma}M_{\nu\rho}, \tag{B.1} \\ 
[M_{\mu\nu}, P_{\rho}] &= \eta_{\mu\rho}P_{\nu} - \eta_{\nu\rho}P_{\mu}, \quad [M_{\mu\nu}, K_{\rho}] = \eta_{\mu\rho}K_{\nu} - \eta_{\nu\rho}K_{\mu}, \nonumber \\ 
[P_{\mu}, K_{\nu}] &= 2M_{\mu\nu} - 2\eta_{\mu\nu}D. \nonumber 
\end{align} 
$$

We work in the real form where $(P_\mu)^\dagger = K^\mu$. When there is a $\mathfrak{su}(2)$ symmetry its generators are normalized by $[R^I, R^J] = i\epsilon^{ijk}R_k$.

The fermionic generators $Q$ and $S$ carry a spinor and an R-symmetry index which determine their commutators with the bosonic generators. The $M_{\mu\nu}$ generators in the spinor representation are $\frac{1}{2}(\gamma_{\mu\nu})_{\alpha}{}^{\beta}$ or $\frac{1}{2}(\bar{\gamma}_{\mu\nu})_{\dot{\alpha}}{}^{\dot{\beta}}$, so for example $[M_{\mu\nu}, Q_{\alpha}^{A}] = \frac{1}{2}(\gamma_{\mu\nu})_{\alpha}{}^{\beta}Q_{\beta}^{A}$ etc. The $\mathfrak{su}(2)$ R-symmetry generators are $\frac{1}{2}\sigma^I$. When there is also a $\mathfrak{u}(1)$ the generators $Q$ have R-charge $+\frac{1}{2}$ while the generators $S$ have R-charge $-\frac{1}{2}$. Finally we always have

$$ [D, Q] = \frac{1}{2}Q, \quad [D, S] = -\frac{1}{2}S \quad \text{and} \quad [P_\mu, Q] = [K_\mu, S] = 0. \qquad (B.2) $$

The anticommutators $\{Q, Q\}$ and $\{S, S\}$ are also fixed up to an overall factor. The only non-trivial anticommutator is $\{Q, S\}$. Schematically $\{Q, S\} \sim M + D + R$, and one determines the coefficient of each term by imposing the super-Jacobi identities. We now give the remaining (anti-)commutators for the superalgebras of table 1. The classical superalgebras can be found in ref. [79], the exceptional $F(4; 2)$ in section 20.2.1 of ref. [77], and all line-defect superalgebras in ref. [40]. The conventions in these references differ however from ours.

### osp(8*|4)

This is the superconformal algebra of 6d $\mathcal{N} = (1, 0)$ with $\mathfrak{su}(2)$ R-symmetry:

$$ 
\begin{align} 
[K_{\mu}, Q_{\alpha}^{A}] &= (\gamma_{\mu})_{\alpha}{}^{\dot{\beta}} S_{\dot{\beta}}^{A}, \quad [P_{\mu}, S_{\dot{\alpha}}^{A}] = -(\bar{\gamma}_{\mu})_{\dot{\alpha}}{}^{\dot{\beta}} Q_{\dot{\beta}}^{A}, \nonumber \\ 
\{Q_{\alpha}^{A}, Q_{\beta}^{B}\} &= 2 \epsilon^{AB} (\gamma^{\mu})_{\alpha\beta} P_{\mu}, \quad \{S_{\dot{\alpha}}^{A}, S_{\dot{\beta}}^{B}\} = -2 \epsilon^{AB} (\bar{\gamma}^{\mu})_{\dot{\alpha}\dot{\beta}} K_{\mu}, \tag{B.3} \\ 
\{Q_{\alpha}^{A}, S_{\beta}^{B}\} &= \epsilon^{AB} (\gamma^{\mu\nu})_{\alpha\dot{\beta}} M_{\mu\nu} - 2 \epsilon^{AB} (C^{(6)T})_{\alpha\dot{\beta}} D - 8 (\sigma_{I})^{AB} (C^{(6)T})_{\alpha\dot{\beta}} R^{I}. \nonumber 
\end{align} 
$$

### F(4; 2)

This is the superconformal algebra of 5d $\mathcal{N} = 1$ with $\mathfrak{su}(2)$ R-symmetry:

$$ 
\begin{align} 
[K_{\mu}, Q_{\alpha}^{A}] &= (\gamma_{\mu})_{\alpha}{}^{\dot{\beta}} S_{\dot{\beta}}^{A}, \quad [P_{\mu}, S_{\dot{\alpha}}^{A}] = -(\bar{\gamma}_{\mu})_{\dot{\alpha}}{}^{\dot{\beta}} Q_{\dot{\beta}}^{A}, \nonumber \\ 
\{Q_{\alpha}^{A}, Q_{\beta}^{B}\} &= 2 \epsilon^{AB} (\gamma^{\mu})_{\alpha\beta} P_{\mu}, \quad \{S_{\dot{\alpha}}^{A}, S_{\dot{\beta}}^{B}\} = 2 \epsilon^{AB} (\bar{\gamma}^{\mu})_{\alpha\beta} K_{\mu}, \tag{B.4} \\ 
\{Q_{\alpha}^{A}, S_{\beta}^{B}\} &= -\epsilon^{AB} (\gamma^{\mu\nu})_{\alpha\beta} M_{\mu\nu} + 2 \epsilon^{AB} (C^{(5)})_{\alpha\beta} D + 6 (\sigma_{I})^{AB} (C^{(5)})_{\alpha\beta} R^{I}. \nonumber 
\end{align} 
$$

### su(2, 2|2)

This is the superconformal algebra of 4d $\mathcal{N} = 2$ with R-symmetry $\mathfrak{su}(2) \oplus \mathfrak{u}(1)$:

$$ 
\begin{align} 
[K_{\mu}, Q_{\alpha}^{A}] &= (\gamma_{\mu})_{\alpha}{}^{\dot{\beta}} \bar{S}_{\dot{\beta}}^{A}, \quad [K_{\mu}, \bar{Q}_{\dot{\alpha}}^{A}] = (\bar{\gamma}_{\mu})_{\dot{\alpha}}{}^{\dot{\beta}} S_{\dot{\beta}}^{A}, \nonumber \\ 
[P_{\mu}, S_{\alpha}^{A}] &= -(\gamma_{\mu})_{\alpha}{}^{\dot{\beta}} \bar{Q}_{\dot{\beta}}^{A}, \quad [P_{\mu}, \bar{S}_{\dot{\alpha}}^{A}] = -(\bar{\gamma}_{\mu})_{\dot{\alpha}}{}^{\dot{\beta}} Q_{\dot{\beta}}^{A}, \nonumber \\ 
\{Q_{\alpha}^{A}, \bar{Q}_{\beta}^{B}\} &= 2 \epsilon^{AB} (\gamma^{\mu})_{\alpha\dot{\beta}} P_{\mu}, \quad \{S_{\alpha}^{A}, \bar{S}_{\beta}^{B}\} = 2 \epsilon^{AB} (\bar{\gamma}^{\mu})_{\alpha\dot{\beta}} K_{\mu}, \nonumber 
\end{align} 
$$
$$
\begin{align}
\{Q_{\alpha}^{A}, S_{\beta}^{B}\} &= \varepsilon^{AB}(\gamma^{\mu\nu})_{\alpha\beta}M_{\mu\nu} - 2\varepsilon^{AB}(C^{(4)})_{\alpha\beta}(D-r) - 4(\sigma_{I})^{AB}(C^{(4)})_{\alpha\beta}R^{I}, \nonumber \\
\{\bar{Q}_{\dot{\alpha}}^{A}, \bar{S}_{\dot{\beta}}^{B}\} &= -\varepsilon^{AB}(\bar{\gamma}^{\mu\nu})_{\dot{\alpha}\dot{\beta}}M_{\mu\nu} + 2\varepsilon^{AB}(C^{(4)})_{\dot{\alpha}\dot{\beta}}(D+r) + 4(\sigma_{I})^{AB}(C^{(4)})_{\dot{\alpha}\dot{\beta}}R^{I}. \tag{B.5}
\end{align}
$$

su(2, 2|1)

This is the superconformal algebra of 4d $\mathcal{N} = 1$ with $\mathfrak{u}(1)$ R-symmetry:

$$
\begin{align}
[K_{\mu}, Q_{\alpha}] &= (\gamma_{\mu})_{\alpha}{}^{\dot{\beta}} \bar{S}_{\dot{\beta}}, & [K_{\mu}, \bar{Q}_{\dot{\alpha}}] &= (\bar{\gamma}_{\mu})_{\dot{\alpha}}{}^{\beta} S_{\beta}, \nonumber \\
[P_{\mu}, S_{\alpha}] &= -(\gamma_{\mu})_{\alpha}{}^{\dot{\beta}} \bar{Q}_{\dot{\beta}}, & [P_{\mu}, \bar{S}_{\dot{\alpha}}] &= -(\bar{\gamma}_{\mu})_{\dot{\alpha}}{}^{\beta} Q_{\beta}, \nonumber \\
\{Q_{\alpha}, \bar{Q}_{\dot{\beta}}\} &= 2 (\gamma^{\mu})_{\alpha\dot{\beta}} P_{\mu}, & \{S_{\alpha}, \bar{S}_{\dot{\beta}}\} &= -2 (\gamma^{\mu})_{\alpha\dot{\beta}} K_{\mu}, \tag{B.6} \\
\{Q_{\alpha}, S_{\beta}\} &= (\gamma^{\mu\nu})_{\alpha\beta} M_{\mu\nu} - 2 (C^{(4)})_{\alpha\beta} (D - 3r), \nonumber \\
\{\bar{Q}_{\dot{\alpha}}, \bar{S}_{\dot{\beta}}\} &= (\bar{\gamma}^{\mu\nu})_{\dot{\alpha}\dot{\beta}} M_{\mu\nu} - 2 (C^{(4)})_{\dot{\alpha}\dot{\beta}} (D + 3r). \nonumber
\end{align}
$$

osp(2|4; R)

This is the superconformal algebra of 3d $\mathcal{N} = 2$ with $\mathfrak{u}(1)$ R-symmetry:

$$
\begin{align}
[K_{\mu}, Q_{\alpha}] &= (\gamma_{\mu})_{\alpha}{}^{\dot{\beta}} \bar{S}_{\dot{\beta}}, & [K_{\mu}, \bar{Q}_{\dot{\alpha}}] &= (\gamma_{\mu})_{\dot{\alpha}}{}^{\beta} S_{\beta}, \nonumber \\
[P_{\mu}, S_{\alpha}] &= -(\gamma_{\mu})_{\alpha}{}^{\dot{\beta}} \bar{Q}_{\dot{\beta}}, & [P_{\mu}, \bar{S}_{\dot{\alpha}}] &= -(\gamma_{\mu})_{\dot{\alpha}}{}^{\beta} Q_{\beta}, \nonumber \\
\{Q_{\alpha}, \bar{Q}_{\dot{\beta}}\} &= 2 (\gamma^{\mu})_{\alpha\dot{\beta}} P_{\mu}, & \{S_{\alpha}, \bar{S}_{\dot{\beta}}\} &= -2 (\gamma^{\mu})_{\alpha\dot{\beta}} K_{\mu}, \tag{B.7} \\
\{Q_{\alpha}, S_{\beta}\} &= (\gamma^{\mu\nu})_{\alpha\beta} M_{\mu\nu} - 2 (C^{(3)})_{\alpha\beta} (D - 2r), \nonumber \\
\{\bar{Q}_{\dot{\alpha}}, \bar{S}_{\dot{\beta}}\} &= (\bar{\gamma}^{\mu\nu})_{\dot{\alpha}\dot{\beta}} M_{\mu\nu} - 2 (C^{(3)})_{\dot{\alpha}\dot{\beta}} (D + 2r). \nonumber
\end{align}
$$

D(2, 1; λ, 0)

This is a 1d superconformal algebra with four supercharges. The R-symmetry algebra is $\mathfrak{su}(2)_r \times \mathfrak{su}(2)_R$. The respective generators $r^I$ and $R^I$. are normalized as above. We use $a, b = 1, 2$ for the $\mathfrak{su}(2)_r$ index, and $A, B = 1, 2$ for the $\mathfrak{su}(2)_R$ index. The (anti-) commutators read

$$
\begin{align}
&[K, Q^{Aa}] = S^{Aa}, \quad [P, S^{Aa}] = -Q^{Aa}, \nonumber \\
&\{Q^{Aa}, Q^{Bb}\} = 2 \varepsilon^{AB} \varepsilon^{ab} P, \quad \{S^{Aa}, S^{Bb}\} = 2 \varepsilon^{AB} \varepsilon^{ab} K, \tag{B.8} \\
&\{Q^{Aa}, S^{Bb}\} = -2\lambda \varepsilon^{AB} (\sigma_I)^{ab} r^I + 2 \varepsilon^{AB} \varepsilon^{ab} D + 2(\lambda+1) (\sigma_I)^{AB} \varepsilon^{ab} R^I. \nonumber
\end{align}
$$

The special case $\lambda = 1$ is isomorphic to $\mathfrak{osp}(4^*|2)$, and the case $\lambda = 0$ is isomorphic to $\mathfrak{psu}(1, 1|2) \oplus \mathfrak{su}(2)$ which contains $\mathfrak{su}(1, 1|1)$ as a subalgebra.

C Supersymmetry transformations

In this appendix we present the transformation under the action of the Poincaré supercharges $Q$ of the stress-tensor and displacement multiplets.

The stress tensor multiplets for all bulk superconformal algebras in $n \ge 3$ dimensions
can be found in [44]. We restrict to the minimal SCFTs of table 1. The four-dimensional
case was worked out, in a slightly different notation, in [3, 18]. For the six-dimensional
N = (2, 0) SCFT the Q-actions can be found in [80]. We used the technique of this latter reference to work out all other cases.

To be more precise, let $Q(A)$ be the action of $Q$ on some operator $A$. We begin with the most general ansatz for $Q(A)$ that is consistent with the field content of the multiplet and the bosonic symmetries. We also impose the conservation and zero-trace conditions of $A$, if any. Finally we fix the coefficients for each tensor structure by requiring that $\{Q, Q\} = 2P$, that is $\{Q, Q\}(A) = 2\partial A$.

$$osp(8^*|4)$$

The (40 + 40) stress tensor multiplet contains a scalar operator $O$, a spinor $\chi_\alpha^A$, a self-dual 3-form $H_{[\mu\nu\rho]}$, the $\mathfrak{su}(2)$ R-symmetry currents $j_\mu^I$, the supersymmetry currents $J_{\mu\alpha}^A$ and the stress tensor $T_{\mu\nu}$.<fnref n="25" /> Schematically

$$O \xrightarrow{Q} \chi_{\alpha}^{A} \xrightarrow{Q} j_{\mu}^{I} \oplus H_{\mu\nu\rho} \xrightarrow{Q} J_{\mu\alpha}^{A} \xrightarrow{Q} T_{\mu\nu}. \quad (C.1)$$

The explicit transformations are

$$
\begin{align*}
Q_{\alpha}^{A}(O) &= \chi_{\alpha}^{A}, \\
Q_{\alpha}^{A}(\chi_{\beta}^{B}) &= j_{\mu}^{I}(\gamma^{\mu})_{\alpha\beta}(\sigma_{I})^{AB} + H_{\mu\nu\rho}(\gamma^{\mu\nu\rho})_{\alpha\beta}\varepsilon^{AB} + (\partial_{\mu}O)(\gamma^{\mu})_{\alpha\beta}\varepsilon^{AB}, \\
Q_{\alpha}^{A}(j_{\mu}^{I}) &= \frac{1}{2}J_{\mu\alpha}^{B}(\sigma^{I})_{B}{}^{A} - \frac{1}{5}(\partial_{\nu}\chi_{\beta}^{B})(\sigma^{I})_{B}{}^{A}(\gamma_{\mu}{}^{\nu})_{\alpha}{}^{\beta}, \\
Q_{\alpha}^{A}(H_{\mu\nu\rho}) &= -\frac{1}{48}J_{[\mu|\beta]}^{A}(\gamma_{\nu\rho})_{\alpha}{}^{\beta} + \frac{1}{30}(\partial_{\sigma}\chi_{\beta}^{A})(\gamma^{\sigma}\bar{\gamma}_{\mu\nu\rho})_{\alpha}{}^{\beta}, \tag{C.2} \\
Q_{\alpha}^{A}(J_{\mu\beta}^{B}) &= 2T_{\mu\nu}(\gamma^{\nu})_{\alpha\beta}\varepsilon^{AB} - \frac{2}{5}(\partial_{\nu}j_{\rho}^{I})(\gamma_{\mu}{}^{\nu\rho} - 4\delta_{\mu}{}^{\rho}\gamma^{\nu})_{\alpha\beta}\varepsilon^{AC}(\sigma_{I})_{C}{}^{B} \\
&\quad + \frac{2}{5}(\partial_{\nu}H_{\rho\sigma\lambda})(\delta_{\mu}{}^{\nu}\gamma^{\rho\sigma\lambda} - 6\delta_{\mu}{}^{\rho}\gamma^{\nu\sigma\lambda} - 18\delta_{\mu}{}^{\rho}\eta^{\nu\sigma}\gamma^{\lambda})_{\alpha\beta}\varepsilon^{AB}, \\
Q_{\alpha}^{A}(T_{\mu\nu}) &= \frac{1}{4}(\partial_{\rho}J_{\mu\beta}^{A})(\gamma_{\nu}{}^{\rho})_{\alpha}{}^{\beta} + (\mu \leftrightarrow \nu).
\end{align*}
$$

$$F(4; 2)$$

The (32 + 32) stress tensor multiplet contains a scalar $O$, a spinor $\chi_\alpha^A$, a 2-form $B_{\mu\nu}$, the $\mathfrak{su}(2)$ R-symmetry currents $j_\mu^I$, the supercurrents $J_{\mu\alpha}^A$ and $T_{\mu\nu}$,

$$O \xrightarrow{Q} \chi_{\alpha}^{A} \xrightarrow{Q} j_{\mu}^{I} \oplus B_{\mu\nu} \xrightarrow{Q} J_{\mu\alpha}^{A} \xrightarrow{Q} T_{\mu\nu}. \quad (C.3)$$

The explicit transformations are

$$
\begin{align*}
Q_{\alpha}^{A}(O) &= \chi_{\alpha}^{A}, \\
Q_{\alpha}^{A}(\chi_{\beta}^{B}) &= j_{\mu}^{I}(\gamma^{\mu})_{\alpha\beta}(\sigma_{I})^{AB} + B_{\mu\nu}(\gamma^{\mu\nu})_{\alpha\beta}\varepsilon^{AB} + (\partial_{\mu}O)(\gamma^{\mu})_{\alpha\beta}\varepsilon^{AB}, \\
Q_{\alpha}^{A}(j_{\mu}^{I}) &= \frac{1}{2}J_{\mu\alpha}^{B}(\sigma^{I})_{B}{}^{A} - \frac{1}{4}(\partial_{\nu}\chi_{\beta}^{B})(\sigma^{I})_{B}{}^{A}(\gamma_{\mu}{}^{\nu})_{\alpha}{}^{\beta}, \\
Q_{\alpha}^{A}(B_{\mu\nu}) &= -\frac{1}{4}J_{\rho\beta}^{A}(\gamma_{\mu\nu}{}^{\rho})_{\alpha}{}^{\beta} + \frac{1}{8}(\partial_{\rho}\chi_{\beta}^{A})(2\gamma_{\mu\nu}{}^{\rho} + \delta_{[\mu}{}^{\rho}\gamma_{\nu]})_{\alpha}{}^{\beta}, \tag{C.4} \\
Q_{\alpha}^{A}(J_{\mu\beta}^{B}) &= 2T_{\mu\nu}(\gamma^{\nu})_{\alpha\beta}\varepsilon^{AB} - \frac{1}{2}(\partial_{\nu}j_{\rho}^{I})(\gamma_{\mu}{}^{\nu\rho} - 3\delta_{\mu}{}^{\rho}\gamma^{\nu})_{\alpha\beta}(\sigma_{I})^{AB} \\
&\quad + \frac{1}{2}(\partial_{\nu}B_{\rho\sigma})(\gamma_{\mu}{}^{\nu\rho\sigma} - 2\delta_{\mu}{}^{\rho}\gamma^{\nu\sigma} + 2\eta^{\nu\rho}\gamma_{\mu}{}^{\sigma} - 6\delta_{\mu}{}^{\rho}\eta^{\nu\sigma})_{\alpha\beta}\varepsilon^{AB}, \\
Q_{\alpha}^{A}(T_{\mu\nu}) &= \frac{1}{4}(\partial_{\rho}J_{\mu\beta}^{A})(\gamma_{\nu}{}^{\rho})_{\alpha}{}^{\beta} + (\mu \leftrightarrow \nu).
\end{align*}
$$

<footnote n="25">²⁵The self-duality condition is $(^*H)^{\mu\nu\rho} = iH^{\mu\nu\rho}$ and the supercurrents satisfy $(\tilde{\gamma}^{\mu})_{\alpha}{}^{\beta}J_{\mu\beta}^{M} = 0$.</footnote>
su(2, 2|2)

The (24 + 24) stress tensor multiplet contains a scalar O, a spinor $\chi^A_\alpha$ (and $\bar{\chi}^A_\alpha$), a symmetric bispinor $H_{\alpha\beta}$ (and $\bar{H}_{\dot{\alpha}\dot{\beta}}$), the $\mathfrak{u}(1)$ and $\mathfrak{su}(2)$ R-symmetry currents $j_\mu$ and $j_\mu^I$, the supercurrent $J_{\mu\alpha}^A$ (and $\bar{J}_{\mu\dot{\alpha}}^A$) and $T_{\mu\nu}$,

$$O \xrightarrow{Q} \chi_{\alpha}^{A} \oplus \bar{\chi}_{\dot{\alpha}}^{A} \xrightarrow{Q} H_{\alpha\beta} \oplus j_{\mu} \oplus j_{\mu}^{I} \oplus \bar{H}_{\dot{\alpha}\dot{\beta}} \xrightarrow{Q} J_{\mu\alpha}^{A} \oplus \bar{J}_{\mu\dot{\alpha}}^{A} \xrightarrow{Q} T_{\mu\nu}. \quad (C.5)$$

The explicit transformations are

$$
\begin{align*}
Q_{\alpha}^{A}(O) &= \chi_{\alpha}^{A}, & \bar{Q}_{\dot{\alpha}}^{A}(O) &= -\chi_{\dot{\alpha}}^{A}, \\
Q_{\alpha}^{A}(\chi_{\beta}^{B}) &= H_{\alpha\beta} \varepsilon^{AB}, & \bar{Q}_{\dot{\alpha}}^{A}(\bar{\chi}_{\beta}^{B}) &= \bar{H}_{\dot{\alpha}\dot{\beta}} \varepsilon^{AB}, \\
\bar{Q}_{\alpha}^{A}(\chi_{\beta}^{B}) &= [-j_{\mu} \varepsilon^{AB} + j_{\mu}^{I} (\sigma_{I})^{AB} - (\partial_{\mu} O) \varepsilon^{AB}] (\bar{\gamma}^{\mu})_{\dot{\alpha}\dot{\beta}}, & & \\
Q_{\alpha}^{A}(\bar{\chi}_{\beta}^{B}) &= [j_{\mu} \varepsilon^{AB} + j_{\mu}^{I} (\sigma_{I})^{AB} - (\partial_{\mu} O) \varepsilon^{AB}] (\gamma^{\mu})_{\alpha\dot{\beta}}, & & \\
\bar{Q}_{\dot{\alpha}}^{A}(H_{\beta\gamma}) &= \left[\frac{1}{2} J_{\mu\beta}^{A} + \frac{2}{3} \partial_{\mu} \chi_{\beta}^{A}\right] (\bar{\gamma}^{\mu})_{\dot{\alpha}\gamma} + (\beta \leftrightarrow \gamma), & & \\
Q_{\alpha}^{A}(\bar{H}_{\dot{\beta}\dot{\gamma}}) &= \left[\frac{1}{2} \bar{J}_{\mu\dot{\beta}}^{A} - \frac{2}{3} \partial_{\mu} \bar{\chi}_{\dot{\beta}}^{A}\right] (\gamma^{\mu})_{\alpha\dot{\gamma}} + (\dot{\beta} \leftrightarrow \dot{\gamma}), & & \\
Q_{\alpha}^{A}(j_{\mu}) &= \frac{1}{2} J_{\mu\alpha}^{A} - \frac{2}{3} (\partial_{\nu} \chi_{\beta}^{A}) (\gamma_{\mu}^{\nu})_{\alpha}{}^{\beta}, & \bar{Q}_{\dot{\alpha}}^{A}(j_{\mu}) &= -\frac{1}{2} \bar{J}_{\mu\dot{\alpha}}^{A} - \frac{2}{3} (\partial_{\nu} \bar{\chi}_{\dot{\beta}}^{A}) (\bar{\gamma}_{\mu}^{\nu})_{\dot{\alpha}\dot{\beta}}, & \\
Q_{\alpha}^{A}(j_{\mu}^{I}) &= \left[\frac{1}{2} J_{\mu\alpha}^{B} + \frac{1}{3} (\partial_{\nu} \chi_{\beta}^{B}) (\gamma_{\mu}^{\nu})_{\alpha}{}^{\beta}\right] (\sigma^{I})_{B}{}^{A}, & & \\
\bar{Q}_{\dot{\alpha}}^{A}(j_{\mu}^{I}) &= \left[\frac{1}{2} \bar{J}_{\mu\dot{\alpha}}^{B} - \frac{1}{3} (\partial_{\nu} \bar{\chi}_{\dot{\beta}}^{B}) (\bar{\gamma}_{\mu}^{\nu})_{\dot{\alpha}\dot{\beta}}\right] (\sigma^{I})_{B}{}^{A}, & & \\
Q_{\alpha}^{A}(J_{\mu\beta}^{B}) &= (\partial_{\nu} H_{\gamma\beta}) \varepsilon^{AB} (\gamma_{\mu}^{\nu})_{\alpha}{}^{\gamma} + \frac{1}{3} (\partial_{\nu} H_{\gamma\alpha}) \varepsilon^{AB} (\gamma_{\mu}^{\nu})_{\beta\gamma}, & & \\
\bar{Q}_{\dot{\alpha}}^{A}(\bar{J}_{\mu\dot{\beta}}^{B}) &= (\partial_{\nu} \bar{H}_{\dot{\gamma}\dot{\beta}}) \varepsilon^{AB} (\bar{\gamma}_{\mu}^{\nu})_{\dot{\alpha}\dot{\gamma}} - \frac{1}{3} (\partial_{\nu} \bar{H}_{\dot{\gamma}\dot{\alpha}}) \varepsilon^{AB} (\bar{\gamma}_{\mu}^{\nu})_{\dot{\beta}\dot{\gamma}}, & & \\
\bar{Q}_{\dot{\alpha}}^{A}(J_{\mu\beta}^{B}) &= -2 T_{\mu\nu} \varepsilon^{AB} (\bar{\gamma}^{\nu})_{\dot{\alpha}\dot{\beta}} + \frac{1}{3} [(\partial_{\nu} j_{\rho}) \varepsilon^{AB} + 2 (\partial_{\nu} j_{\rho}^{I}) (\sigma_{I})^{AB}] (\bar{\gamma}_{\mu}^{\nu\rho} - 2 \delta_{\mu}^{\rho} \bar{\gamma}_{\nu})_{\dot{\alpha}\dot{\beta}}, & & \\
Q_{\alpha}^{A}(\bar{J}_{\mu\beta}^{B}) &= 2 T_{\mu\nu} \varepsilon^{AB} (\gamma^{\nu})_{\alpha\dot{\beta}} + \frac{1}{3} [(\partial_{\nu} j_{\rho}) \varepsilon^{AB} - 2 (\partial_{\nu} j_{\rho}^{I}) (\sigma_{I})^{AB}] (\gamma_{\mu}^{\nu\rho} - 2 \delta_{\mu}^{\rho} \gamma^{\nu})_{\alpha\dot{\beta}}, & & \\
Q_{\alpha}^{A}(T_{\mu\nu}) &= \frac{1}{4} (\partial_{\rho} J_{\mu\beta}^{A}) (\gamma^{\rho})_{\alpha}{}^{\beta} + (\mu \leftrightarrow \nu), & & \\
\bar{Q}_{\dot{\alpha}}^{A}(T_{\mu\nu}) &= \frac{1}{4} (\partial_{\rho} \bar{J}_{\mu\dot{\beta}}^{A}) (\bar{\gamma}^{\rho})_{\dot{\alpha}\dot{\beta}} + (\mu \leftrightarrow \nu). & & \tag{C.6}
\end{align*}
$$

su(2, 2|1)

This is a subalgebra of the previous case, but we give it separately for the reader's convenience. The (8 + 8) stress-tensor multiplet contains the $\mathfrak{u}(1)$ R-symmetry current $j_\mu$, the supercurrents $J_{\mu\alpha}$, $\bar{J}_{\mu\dot{\alpha}}$ and $T_{\mu\nu}$,

$$j_{\mu} \xrightarrow{Q} J_{\mu\alpha} \oplus \bar{J}_{\mu\dot{\alpha}} \xrightarrow{Q} T_{\mu\nu}. \quad (C.7)$$

Notice that this multiplet contains no scalar, the super-primary is the R-symmetry current. The explicit transformations are

$$
\begin{align*}
Q_{\alpha}(j_{\mu}) &= \frac{1}{2} J_{\mu\alpha}, & \bar{Q}_{\dot{\alpha}}(j_{\mu}) &= -\frac{1}{2} \bar{J}_{\mu\alpha}, \\
\bar{Q}_{\dot{\alpha}}(J_{\mu\beta}) &= 2 (\bar{\gamma}^{\nu})_{\dot{\alpha}\beta} T_{\mu\nu} - (\partial_{\nu} j_{\rho}) (\bar{\gamma}_{\mu}^{\nu\rho} - 2 \delta_{\mu}^{\rho} \bar{\gamma}^{\nu})_{\dot{\alpha}\beta}, & & \\
Q_{\alpha}(\bar{J}_{\mu\beta}) &= 2 (\gamma^{\nu})_{\alpha\dot{\beta}} T_{\mu\nu} + (\partial_{\nu} j_{\rho}) (\gamma_{\mu}^{\nu\rho} - 2 \delta_{\mu}^{\rho} \gamma^{\nu})_{\alpha\dot{\beta}}, & &
\end{align*}
$$
$$ 
\begin{aligned}
Q_{\alpha}(T_{\mu\nu}) &= \frac{1}{4}(\partial_{\rho}J_{\mu\beta})(\gamma^{\rho})_{\alpha}{}^{\beta} + (\mu \leftrightarrow \nu), \\
\bar{Q}_{\dot{\alpha}}(T_{\mu\nu}) &= \frac{1}{4}(\partial_{\rho}\bar{J}_{\mu\dot{\beta}})(\bar{\gamma}^{\rho})_{\dot{\alpha}}{}^{\dot{\beta}} + (\mu \leftrightarrow \nu).
\end{aligned} 
\qquad (C.8) $$

osp(2|4; R)

The (4 + 4) stress-tensor multiplet contains the u(1) R-symmetry current $j_{\mu}$, the supercurrents $J_{\mu\alpha}$, $\bar{J}_{\mu\alpha}$ and $T_{\mu\nu}$,

$$ j_{\mu} \xrightarrow{Q} J_{\mu\alpha} \oplus \bar{J}_{\mu\alpha} \xrightarrow{Q} T_{\mu\nu}. \qquad (C.9) $$

The explicit transformations are

$$ 
\begin{aligned}
Q_{\alpha}(j_{\mu}) &= \frac{1}{2}J_{\mu\alpha}, & \bar{Q}_{\alpha}(j_{\mu}) &= -\frac{1}{2}\bar{J}_{\mu\alpha}, \\
\bar{Q}_{\alpha}(J_{\mu\beta}) &= 2(\gamma^{\nu})_{\alpha\beta}T_{\mu\nu} - 2(\partial_{\nu}j_{\rho})(\gamma^{\nu})^{\rho}_{\mu} - \delta_{\mu}^{\rho}\gamma^{\nu})_{\alpha\beta}, \\
Q_{\alpha}(\bar{J}_{\mu\beta}) &= 2(\gamma^{\nu})_{\alpha\beta}T_{\mu\nu} + 2(\partial_{\nu}j_{\rho})(\gamma^{\nu})^{\rho}_{\mu} - \delta_{\mu}^{\rho}\gamma^{\nu})_{\alpha\beta}, \qquad (C.10) \\
Q_{\alpha}(T_{\mu\nu}) &= \frac{1}{4}(\partial_{\rho}J_{\mu\beta})(\gamma^{\rho})_{\alpha}{}^{\beta} + (\mu \leftrightarrow \nu), \\
\bar{Q}_{\alpha}(T_{\mu\nu}) &= \frac{1}{4}(\partial_{\rho}\bar{J}_{\mu\beta})(\gamma^{\rho})_{\alpha}{}^{\beta} + (\mu \leftrightarrow \nu).
\end{aligned}
 $$

The transformation of the displacement multiplets under the preserved supercharges is much simpler. The generic multiplet contains the displacement vector $D$, a fermion $\Lambda$, and in some of the cases a scalar $\Phi$. This is the structure under the transverse-rotation group $so(n-p)_{\text{def}}$. The fermion and the preserved supercharges $Q$ are also worldvolume spinors, and they may carry an extra $r$-symmetry index $A$. The defect superalgebras of table 1 were recorded in appendix B. The transformation laws are fixed by imposing the super-Jacobi identities.

We label the cases by $(n, p)$ and group them according to the codimension and the number of preserved supercharges. What is required for our proof is that there always exists a supercharge $Q$ and a fermion $\Lambda$ such that $Q(\Lambda) = D_z$.

(4, 2) and (3, 1)

This is the simplest multiplet, generated by two supercharges and without a scalar component $\Phi$. Complexifying the transverse $so(2)_{\text{def}}$ index gives

$$ \{Q, \Lambda\} = D, \quad [\bar{Q}, D] = 2 \partial \Lambda, \qquad (C.11) $$

and the conjugate relations. Here $\partial$ stands for $\partial_t$ when $p=1$ and for $\partial_+$ when $p=2$.

(6, 4) and (5, 3)

For $(n, p) = (6, 4)$ the non-vanishing (anti-)commutators are

$$ 
\begin{aligned}
&[Q_{\alpha}, \Phi] = \Lambda_{\alpha}, \quad [\bar{Q}_{\dot{\alpha}}, D] = -2 (\bar{\gamma}^{i})_{\dot{\alpha}}{}^{\beta} \partial_{i} \Lambda_{\beta}, \\
&\{Q_{\alpha}, \Lambda_{\beta}\} = C_{\alpha\beta}^{(p)} D, \quad \{\bar{Q}_{\dot{\alpha}}, \Lambda_{\beta}\} = 2 (\bar{\gamma}^{i})_{\dot{\alpha}} \partial_{i} \Phi.
\end{aligned} 
\qquad (C.12) $$

The index $i$ runs from 1 to $p$, and we have again complexified the transverse $so(2)_{\text{def}}$ indices. For $(n, p) = (5, 3)$ one must replace $\bar{\gamma}^i \to \gamma^i$ and $\dot{\alpha} \to \alpha$.
(6, 2) and (5, 1)

Here $Q$ and $\Lambda$ are complex $50(4)$ spinors and they have an additional $r$-symmetry index. The displacement can be written as a bispinor $D^{ab}$ and there is no scalar $\Phi$. The non-vanishing (anti-)commutators read

$$ \{Q^{Aa}, \Lambda^{Bb}\} = \varepsilon^{AB} D^{ab}, \quad [Q^{Aa}, D^{b\dot{c}}] = -2 \varepsilon^{ab} \partial\Lambda^{A\dot{c}}, \qquad (C.13) $$

together with the conjugate relations.

(5, 2) and (4, 1)

Here $Q$ and $\Lambda$ are transverse-$50(3)$ spinors and have an additional $r$-symmetry index. The displacement is a symmetric bispinor $D^{ab} = (\sigma_j)^{ab} D^j$ ($j = 1, 2, 3$). The non-vanishing (anti-)commutators read

$$ \begin{gathered} [Q^{Aa}, \Phi] = \Lambda^{Aa}, \quad [Q^{Aa}, D^{bc}] = -(\varepsilon^{ac} \partial\Lambda^{Ab} + \varepsilon^{ab} \partial\Lambda^{Ac}), \qquad (C.14) \\ \{Q^{Aa}, \Lambda^{Bb}\} = \varepsilon^{AB}(D^{ab} + \varepsilon^{ab}\partial\Phi), \end{gathered} $$

This is the only tricky case because $\{Q, \Lambda\} \sim D + \partial\Phi$. But with the choices made in section 5 one finds $D_z \propto D^{22}$, and the $\partial\Phi$ term drops out in the anticommutator $\{Q^{12}, \Lambda^{22}\}$. This is sufficient for our proof.

**Data Availability Statement.** This article has no associated data or the data will not be deposited.

**Code Availability Statement.** This article has no associated code or the code will not be deposited.

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

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