Defect Soft Theorems in Conformal Defects

• Defect soft theorems are integrated constraints from broken Ward identities that yield sum rules on defect and bulk operator data in conformal field theories.
• They structure operator expansions and correlators, enabling the precise determination of OPE coefficients, spectral bounds, and critical data in defect CFTs.
• Applications span conformal line defects, soft-collinear gravity, and boundary theories, illustrating the versatility of symmetry-breaking analyses in various physical contexts.

Defect soft theorems are integrated constraints implied by broken Ward identities in theories with defects, boundaries, or closely related localized structures. In conformal defect theories, the broken Ward identities imply very general sum rules on the defect CFT data as well as on the DOE data of bulk operators, and these sum rules are explicitly identified as defect soft theorems by Girault–Paulos–van Vliet (Girault et al., 30 Sep 2025). In this usage, the “soft” object is not an external low-momentum particle in flat-space scattering, but a protected defect operator—most notably the tilt, displacement, or displacino—whose integrated insertion is equivalent to acting with a broken symmetry generator on a correlator.

1. Broken Ward identities and the basic defect-soft structure

A conformal defect in a $d$-dimensional CFT is a lower-dimensional locus $\mathcal D$ of dimension $p$, codimension $q=d-p$, along which the dynamics is modified but such that the full system is still invariant under a subgroup of the original conformal symmetry. For a flat defect at $y^a=0$, the bulk conformal symmetry $SO(d+1,1)$ is broken to

$SO(p+1,1)\times SO(d-p),$

possibly together with a reduction $G\to\hat G$ of a bulk global symmetry. The bulk-to-defect operator expansion is

$$|y|^{\Delta_\Psi}\,\Psi(x,y) \;\underset{|y|\to 0}{=}\; \sum_{\hat{\mathcal O}} |y|^{\Delta_{\hat{\mathcal O}}}\,\mu_{\hat{\mathcal O}}^{\Psi}\,\hat{\mathcal O}(x)+\ldots,$$

so DOE coefficients $\mu_{\hat{\mathcal O}}^\Psi$, defect dimensions, and defect OPE coefficients together determine mixed bulk–defect correlators (Girault et al., 30 Sep 2025).

For a broken generator $\mathcal D$0, the corresponding current satisfies a modified Ward identity

$\mathcal D$1

together with the equivalent push relation

$\mathcal D$2

Here $\mathcal D$3 is a defect-local operator with protected dimension. Evaluating the broken charge on the defect vacuum yields

$\mathcal D$4

so the broken charge is represented by an integrated defect insertion.

The elementary single-soft identity is

$\mathcal D$5

This is the direct analogue of a single soft Goldstone theorem: an integrated insertion of the defect Goldstone is traded for the symmetry variation of the remaining operators. The double-soft identity follows from evaluating $\mathcal D$6 in two ways and contains time-ordered products of two defect insertions together with the commutator of broken generators: $\mathcal D$7 In concrete applications, representation theory and kinematics simplify these identities into explicit sum rules for defect spectra, defect OPE coefficients, and DOE coefficients.

2. Tilt operators and broken global symmetries

If the bulk CFT has a global symmetry $\mathcal D$8 and the defect preserves only $\mathcal D$9, then for broken generators the conserved current obeys

$p$0

where $p$1 is the tilt operator. Its protected dimension is

$p$2

so it is marginal on the defect. The two-point function is

$p$3

and the DOE of the broken current is fixed by

$p$4

For the canonical pattern $p$5, broken generators are $p$6, relabeled as $p$7. The single-soft and double-soft identities become

$p$8

and

$p$9

These identities constrain bulk–defect–defect form factors. A general form factor of a bulk scalar $q=d-p$0 and two defect scalars has the structure

$q=d-p$1

with cross-ratio

$q=d-p$2

and block expansion

$q=d-p$3

The soft theorem gives, for example,

$q=d-p$4

$q=d-p$5

and

$q=d-p$6

When the bulk insertions are pushed entirely to the defect, the same Ward identities become sum rules for defect four-point functions. In a standard conformal frame, one obtains the homogeneous tilt soft sum rule

$q=d-p$7

and the inhomogeneous double-soft sum rule

$q=d-p$8

These identities are the direct defect analogues of single- and double-soft Goldstone constraints (Girault et al., 30 Sep 2025).

3. Displacements, broken conformal symmetry, and displacinos

Every flat defect breaks transverse translations $q=d-p$9, transverse special conformal generators $y^a=0$0, and mixed rotations $y^a=0$1. The corresponding Ward identity for the stress tensor is

$y^a=0$2

where $y^a=0$3 is the displacement operator. It transforms as a vector of $y^a=0$4, a scalar of the defect conformal group, and has protected dimension

$y^a=0$5

Its two-point function is

$y^a=0$6

Broken conformal charges act on the defect vacuum as

$y^a=0$7

$y^a=0$8

$y^a=0$9

Accordingly, the single-soft identities become

$SO(d+1,1)$0

$SO(d+1,1)$1

$SO(d+1,1)$2

A compact formulation uses the shadow transform

$SO(d+1,1)$3

for which

$SO(d+1,1)$4

This is especially effective for mixed bulk–defect correlators. The simplest consequence is the exact relation

$SO(d+1,1)$5

which relates the bulk–to–defect OPE coefficient into the displacement to the one-point coefficient of the bulk operator.

For three-point form factors involving one displacement and one defect scalar, the soft theorem yields two independent integral constraints,

$SO(d+1,1)$6

$SO(d+1,1)$7

For displacement–displacement form factors in the singlet channel,

$SO(d+1,1)$8

$SO(d+1,1)$9

The same logic produces homogeneous and inhomogeneous sum rules for defect four-point functions with displacement insertions. In a conformal frame, one finds

$SO(p+1,1)\times SO(d-p),$0

for $SO(p+1,1)\times SO(d-p),$1, together with three independent double-soft sum rules of the schematic form

$SO(p+1,1)\times SO(d-p),$2

There is also a mixed double-soft relation involving two displacements and two tilts,

$SO(p+1,1)\times SO(d-p),$3

Supersymmetric defects admit the same pattern. For a broken supercurrent in a $SO(p+1,1)\times SO(d-p),$4 CFT with a line defect, the projected Ward identity

$SO(p+1,1)\times SO(d-p),$5

defines the displacino $SO(p+1,1)\times SO(d-p),$6, a defect spinor of dimension

$SO(p+1,1)\times SO(d-p),$7

The resulting 1D soft and double-soft sum rules constrain four-point functions involving displacinos and their superpartners. This extends the defect-soft framework from broken internal and conformal symmetries to broken supersymmetry (Girault et al., 30 Sep 2025).

4. Line defects, dispersive functionals, and spectral bounds

For conformal line defects, $SO(p+1,1)\times SO(d-p),$8, the defect conformal group is $SO(p+1,1)\times SO(d-p),$9, defect four-point functions depend on a single cross-ratio $G\to\hat G$0, and the integrated soft constraints can be recast as dispersive sum rules. A generic 1D correlator admits an expansion

$G\to\hat G$1

with

$G\to\hat G$2

Soft sum rules have the schematic form

$G\to\hat G$3

which become linear functionals on the exchanged spectrum after inserting the conformal block decomposition. The dispersive reformulation replaces ordinary blocks by local or double-discontinuity blocks, producing improved functionals with controlled zeros and poles (Girault et al., 30 Sep 2025).

For form factors, the resulting spectral functionals are denoted $G\to\hat G$4. In the tilt–tilt case, a particularly simple result is

$G\to\hat G$5

For four-point functions, the dispersive construction uses the double discontinuity,

$G\to\hat G$6

and yields spectral densities such as

$G\to\hat G$7

for the four-tilt soft sum rule in 1D.

The positivity of OPE coefficients then converts the soft sum rules into rigorous spectral constraints. The central bounds are:

• In any conformal line defect, there must exist an operator, either in the singlet or traceless symmetric representation of transverse rotations, with dimension

$G\to\hat G$8

• If the defect breaks a global symmetry, there must exist an even-parity operator with

$G\to\hat G$9

These bounds are stated to be exact and optimal. They sharpen generic 1D bootstrap bounds by exploiting the broken Ward identities specific to defects.

The formalism is also predictive in perturbation theory. For the pinning line defect of the critical $$|y|^{\Delta_\Psi}\,\Psi(x,y) \;\underset{|y|\to 0}{=}\; \sum_{\hat{\mathcal O}} |y|^{\Delta_{\hat{\mathcal O}}}\,\mu_{\hat{\mathcal O}}^{\Psi}\,\hat{\mathcal O}(x)+\ldots,$$0 model in $$|y|^{\Delta_\Psi}\,\Psi(x,y) \;\underset{|y|\to 0}{=}\; \sum_{\hat{\mathcal O}} |y|^{\Delta_{\hat{\mathcal O}}}\,\mu_{\hat{\mathcal O}}^{\Psi}\,\hat{\mathcal O}(x)+\ldots,$$1,

$$|y|^{\Delta_\Psi}\,\Psi(x,y) \;\underset{|y|\to 0}{=}\; \sum_{\hat{\mathcal O}} |y|^{\Delta_{\hat{\mathcal O}}}\,\mu_{\hat{\mathcal O}}^{\Psi}\,\hat{\mathcal O}(x)+\ldots,$$2

the soft and double-soft sum rules were checked to $$|y|^{\Delta_\Psi}\,\Psi(x,y) \;\underset{|y|\to 0}{=}\; \sum_{\hat{\mathcal O}} |y|^{\Delta_{\hat{\mathcal O}}}\,\mu_{\hat{\mathcal O}}^{\Psi}\,\hat{\mathcal O}(x)+\ldots,$$3, and then used to extract new $$|y|^{\Delta_\Psi}\,\Psi(x,y) \;\underset{|y|\to 0}{=}\; \sum_{\hat{\mathcal O}} |y|^{\Delta_{\hat{\mathcal O}}}\,\mu_{\hat{\mathcal O}}^{\Psi}\,\hat{\mathcal O}(x)+\ldots,$$4 data. In particular,

$$|y|^{\Delta_\Psi}\,\Psi(x,y) \;\underset{|y|\to 0}{=}\; \sum_{\hat{\mathcal O}} |y|^{\Delta_{\hat{\mathcal O}}}\,\mu_{\hat{\mathcal O}}^{\Psi}\,\hat{\mathcal O}(x)+\ldots,$$5

and

$$|y|^{\Delta_\Psi}\,\Psi(x,y) \;\underset{|y|\to 0}{=}\; \sum_{\hat{\mathcal O}} |y|^{\Delta_{\hat{\mathcal O}}}\,\mu_{\hat{\mathcal O}}^{\Psi}\,\hat{\mathcal O}(x)+\ldots,$$6

A plausible implication is that defect soft theorems are not merely kinematic identities: in line-defect bootstrap they function as nonperturbative sum rules with direct quantitative leverage on spectra and OPE data.

5. Defect-like realizations beyond conformal defects

The conformal-defect construction is the sharpest realization of the term, but closely related structures appear in several other settings.

In soft-collinear gravity, the paper does not explicitly use the language of defects, but its formulation is strongly suggestive of a “defect soft theorem” picture. Each energetic particle direction $$|y|^{\Delta_\Psi}\,\Psi(x,y) \;\underset{|y|\to 0}{=}\; \sum_{\hat{\mathcal O}} |y|^{\Delta_{\hat{\mathcal O}}}\,\mu_{\hat{\mathcal O}}^{\Psi}\,\hat{\mathcal O}(x)+\ldots,$$7 defines a null ray $$|y|^{\Delta_\Psi}\,\Psi(x,y) \;\underset{|y|\to 0}{=}\; \sum_{\hat{\mathcal O}} |y|^{\Delta_{\hat{\mathcal O}}}\,\mu_{\hat{\mathcal O}}^{\Psi}\,\hat{\mathcal O}(x)+\ldots,$$8, and soft fields in soft-collinear interactions are evaluated only at that trajectory. The homogeneous background $$|y|^{\Delta_\Psi}\,\Psi(x,y) \;\underset{|y|\to 0}{=}\; \sum_{\hat{\mathcal O}} |y|^{\Delta_{\hat{\mathcal O}}}\,\mu_{\hat{\mathcal O}}^{\Psi}\,\hat{\mathcal O}(x)+\ldots,$$9, vierbein $\mu_{\hat{\mathcal O}}^\Psi$0, and spin connection $\mu_{\hat{\mathcal O}}^\Psi$1 are therefore 1-dimensional fields living on the defect, and the emergent soft gauge symmetries $\mu_{\hat{\mathcal O}}^\Psi$2 become local translations and local Lorentz transformations along the null line. In this interpretation, the leading soft factor $\mu_{\hat{\mathcal O}}^\Psi$3 is the coupling of the first soft gauge field $\mu_{\hat{\mathcal O}}^\Psi$4 to the momentum density, $\mu_{\hat{\mathcal O}}^\Psi$5 is the coupling of the spin connection to angular momentum, and $\mu_{\hat{\mathcal O}}^\Psi$6 arises from the soft Riemann tensor. The same EFT explains why graviton soft theorems are universal through sub-sub-leading order: there is no soft building block of order $\mu_{\hat{\mathcal O}}^\Psi$7, while the first nontrivial soft graviton building block in a source is the Riemann tensor,

$\mu_{\hat{\mathcal O}}^\Psi$8

so source operators containing soft gravitons do not contribute up to $\mu_{\hat{\mathcal O}}^\Psi$9 (Beneke et al., 2022).

A cosmological solid provides a different, medium-based analogue. The background

$\mathcal D$00

breaks spatial translations and rotations to a diagonal $\mathcal D$01, and the defining feature is non-vanishing anisotropic stress at super-Hubble scales. Because of that anisotropic stress, generic long scalar modes are locally observable; only a special isotropic long scalar mode is adiabatic. Consequently, Maldacena’s consistency relation is recovered only upon angular averaging over the long-mode direction,

$\mathcal D$02

while the anisotropic quadrupole remains unconstrained. The same framework also yields mixed tensor–scalar and vector–tensor soft relations. This is not a defect theory, but it is a closely related example in which background structure modifies the mapping between long modes and symmetry generators (Pajer et al., 2019).

A boundary-term realization arises in $\mathcal D$03-dimensional Chern–Simons QED and QCD. The $\mathcal D$04 Chern–Simons term is gauge invariant only up to a boundary term,

$\mathcal D$05

yet the leading soft photon and soft gluon theorems remain unchanged. The corrections appear only at subleading order and are parity-odd. For the single soft photon theorem, the additional term is

$\mathcal D$06

with the sum restricted to external photon legs. For the soft gluon theorem, the correction is

$\mathcal D$07

This provides a concrete case in which boundary/topological data leaves the leading soft theorem intact but modifies the subleading soft factor (Wadhwa, 2 Sep 2025).

6. Recursion, universality, and generalized soft behavior

Soft theorems can also be used as constructive input. For tree amplitudes satisfying a soft expansion

$\mathcal D$08

the all-line soft-BCFW deformation

$\mathcal D$09

together with

$\mathcal D$10

turns soft residues into recursive data. The constructibility criterion is

$\mathcal D$11

if $\mathcal D$12 as $\mathcal D$13. The same paper states that in any context where “defect soft theorems” can be formulated, the same logic suggests a program: construct appropriate complex deformations, identify the analogues of $\mathcal D$14 and the soft operators $\mathcal D$15, and use them to replace unknown large-deformation behavior with controlled defect-soft residues (Luo et al., 2015).

A different amplitude-level lesson comes from hidden Adler zeros in flat-space and decoupling limits of EFTs of inflation. With the soft hierarchy

$\mathcal D$16

the soft theorems depend solely on on-shell data and hold to all orders in perturbation theory. The collection of exchange diagrams whose soft momenta are associated with cubic vertices, which is indeterminate in the soft limit, exhibits an enhanced soft scaling; the enhanced soft scaling explains why the sum of such diagrams does not enter the soft theorems non-trivially. This suggests a broader principle: apparently dangerous off-shell structures can decouple from the soft sector once the correct hierarchy of limits is imposed (Du, 2024).

The EFT organization of generalized soft behavior is equally important. In SCET for gauge theory, the Low–Burnett–Kroll theorem is proved at tree level for well-separated particles, but it is generically spoiled by on-shell corrections, including collinear loops and collinear emissions. The effective theory then replaces the naive theorem by a generalized subleading soft theorem with explicitly factorized soft loops, collinear splitting amplitudes, and fusion terms (Larkoski et al., 2014). In a broader amplitude and string-theory setting, soft theorems remain unmodified for $\mathcal D$17 and $\mathcal D$18 interactions, but $\mathcal D$19 interactions modify the sub-sub-leading soft graviton theorem; in addition, loop corrections are tied to conformal or duality anomalies, while superstring amplitudes at finite $\mathcal D$20 satisfy the same soft theorem as the field-theory counterpart, unlike bosonic closed strings because of $\mathcal D$21 interactions (Bianchi et al., 2014).

A common misconception is that “defect soft theorem” should refer only to exact analogues of soft-particle emission in the presence of a geometric defect. The literature suggests a broader but technically coherent family of phenomena: integrated insertions of protected defect operators in dCFT, soft gauge data localized on null trajectories, symmetry-restricted long modes in ordered media, and boundary-induced subleading soft corrections all share the same organizing principle—soft behavior is controlled by broken or emergent symmetry generators, while the precise range of universality is fixed by the available building blocks, the power counting, and possible anomaly or boundary contributions.