AdS3 BPS Defects: Supersymmetry & Holography

• AdS3 BPS defects are supersymmetric localized structures in AdS3 settings that preserve a fraction of the underlying supersymmetry.
• They are constructed via conical defect geometries and backreacted brane intersections, offering explicit realizations in supergravity and holographic duals.
• Holonomy, Killing spinor conditions, and integrable methods classify these defects, linking higher-spin theories and defect CFTs to insights in quantum gravity.

An AdS$_3$ BPS defect is a supersymmetric codimension-one or codimension-two localized structure—such as a surface, line, or point defect—in a background with local AdS$_3$ geometry that preserves a fraction of the underlying supersymmetries. Such defects are realized both as exact supergravity solutions (either as geometries with conical singularities or as fully backreacted brane intersections) and as operators in holographic dual CFT$_2$ or defect CFTs. They are of central interest in the study of holographic dualities (such as AdS$_3$/CFT$_2$ and AdS$_7$/CFT$_6$), integrability, and the anatomy of supersymmetric states and nonlocal operators in string/M-theory.

1. BPS Defects in AdS$_3$ Supergravity: Geometric and Algebraic Construction

AdS$_3$ supergravity admits supersymmetric defect solutions that are described by geometries with negative mass between the global AdS$_3$ vacuum ($M=-1$) and the zero-mass BTZ black hole ($M=0$). Static and spinning solutions are singular spacetimes with a conical defect at $r=0$: $$ds^2 = -\left( \frac{r^2}{\ell^2} + 1 \right) dt^2 + \frac{dr^2}{\frac{r^2}{\ell^2} + 1} + (1-\alpha)^2 r^2 d\phi^2$$ where $0 < \alpha < 1$ is the angular deficit and $M = -(1-\alpha)^2$ (0904.0475, Giribet et al., 31 Jan 2024). These geometries can be formulated as distributional sources in the Chern–Simons formulation of 3D AdS gravity (0904.0475), and the extremal spinning solutions satisfy the BPS bound $M\ell = -|J|$, admitting globally defined Killing spinors and preserving $1/4$ or $1/2$ of local supersymmetry (Giribet et al., 31 Jan 2024).

From the Chern–Simons perspective, the requirement for smooth conical defects with supersymmetry is encoded in the holonomy of the gauge connection around the spatial circle, and nontrivial spinor monodromies. For higher-spin and extended supersymmetric AdS$_3$ gravities (e.g., $\mathfrak{sl}(N|N-1)$ and $\mathfrak{osp}(2N+1|2N)$ Chern–Simons theories), conical surpluses and BPS defects are classified by holonomy data and are in one-to-one correspondence with highest-weight representations of the corresponding (super)algebras (Chen et al., 2013).

2. Brane Engineering and Supergravity Realizations

AdS$_3$ BPS defects arise as fully backreacted supergravity solutions associated to intersecting brane configurations. For example, in minimal 7D gauged supergravity, one can realize half-BPS Janus flows with an $\mathrm{AdS}_3 \times S^3$ slicing, supported by a dyonic three-form field (Conti et al., 31 Jul 2024): $$ds_7^2 = e^{2U(r)} L^2 ds^2_{\mathrm{AdS}_3} + e^{2W(r)} \kappa^2 ds^2_{S^3} + e^{2V(r)} dr^2, \quad B_3 = k(r) L^3 \mathrm{vol}_{\mathrm{AdS}_3} + l(r) \kappa^3 \mathrm{vol}_{S^3}$$ The BPS equations reduce to a system allowing analytic integration. The geometry is locally AdS$_7$ in the UV and flows to an $\mathrm{AdS}_3 \times \mathbb{R}^4$ cap in the IR, corresponding to a codimension-four defect—physically, an M2-brane intersecting M5-branes, or, in the massive IIA uplift, D2/D4 bound states ending on a D4-D8 system (Conti et al., 31 Jul 2024, Dibitetto et al., 2017).

Similar constructions exist in AdS$_4$ backgrounds with $\mathrm{AdS}_3 \times \mathbb{CP}^3$ foliations describing $(0,6)$-supersymmetric surface defects in ABJM/ABJ theory, with the geometry controlled by a single ODE for the warp factor $h(r)$ and explicit brane-box and quiver gauge theory duals (Lozano et al., 26 Apr 2024).

3. Classification, BPS Conditions, and Holonomy

The BPS nature of AdS$_3$ defects is reflected in the existence of globally well-defined Killing spinors compatible with the holonomy around the defect: $D\epsilon = 0, \qquad \text{(AdS%%%%30%%%% spin connection and vielbein)}$ For static defects in pure supergravity, only specific negative mass values $M=-n^2$ admit Killing spinors, leading to integer angular excesses (Giribet et al., 31 Jan 2024). For higher-spin or extended supergravities, the number and type of preserved supersymmetries depend on the spectrum of eigenvalues (holonomies) of the gauge connection and are directly related to BPS bounds in the corresponding superalgebra (Chen et al., 2013).

The preservation of supersymmetry constrains both the geometry and the boundary CFT representations. In the holographic dual, BPS AdS$_3$ defects correspond to degenerate or non-diagonal representations of the Virasoro algebra or of higher-spin/algebraic generalizations. The semiclassical correspondence is explicit: the defect's mass and spin map to the conformal weights and $U(1)$ quantum numbers of the dual representation (Giribet et al., 31 Jan 2024).

4. Holographic Duality, Defect CFTs, and Central Charges

In AdS$_3$/CFT$_2$ dualities, BPS AdS$_3$ defects are dual to extended defect CFTs: surface, line, or point operators preserving a fraction of the bulk symmetry (Lagares, 17 Sep 2025, Bliard et al., 3 Oct 2024). The preserved symmetry after defect insertion is typically a diagonal subalgebra (e.g., $PSU(1,1|2)_\text{diag}$ for half-BPS lines), and the defect supports protected multiplets such as displacement and tilt operators, with correlators governed by strong-coupling analytic bootstrap constraints (Bliard et al., 3 Oct 2024, Lagares, 17 Sep 2025).

The holographic central charge of the defect is computed via reductions of the supergravity solution, either through Brown–Henneaux for pure AdS$_3$ throats or generalizations in the presence of fluxes and warped geometries. For fully backreacted solutions, the defect central charge $c_\mathrm{def}$ scales as the product of brane charges (e.g., $N_5^3\,\lambda$ for M-theory or $N_2 N_4$ for D2/D4 defects) and counts localized defect degrees of freedom (Conti et al., 31 Jul 2024, Dibitetto et al., 2017).

5. Integrability, Conformal Bootstrap, and Quantum Properties

Many AdS$_3$ BPS defects exhibit integrable structures. The defect CFTs on the defect worldvolumes admit analytic bootstrap methods for correlators, constrained by crossing, superconformal, and topological symmetry, with the solution space reduced to few undetermined OPE data at strong coupling (Bliard et al., 3 Oct 2024, Lagares, 17 Sep 2025). Leading and next-to-leading order (in $1/g$) results for four-point functions, OPE coefficients, and anomalous dimensions are directly reproduced by holographic Witten diagram computations.

In higher-dimensional parent theories, inserting or backreacting BPS AdS$_3$ defects can generate new CFT fixed points in lower dimensions (e.g., 6d $\rightarrow$ 2d, 3d ABJM $\rightarrow$ 2d), and in such cases the brane picture, the RG flow, and the construction of the associated defect quiver gauge theory are explicitly determined. Integrability also plays a notable role in the Wilson loop/cusped-line context, with spectral problems mapped to integrable open spin chains and Y-systems (Lagares, 17 Sep 2025).

6. Higher-Spin Generalizations and Algebraic Classification

In $\mathfrak{sl}(N|N-1)$ and $\mathfrak{osp}(2N+1|2N)$ Chern–Simons supergravities, AdS$_3$ BPS defects generalize to smooth conical surpluses/classical orbits associated to highest-weight modules of the gauge superalgebra. The defect data (holonomy eigenvalues) are mapped to Dynkin labels of the underlying bosonic subalgebra, and the BPS fraction is determined by the number of solutions to the generalized Killing spinor equation respecting the spinor monodromy (Chen et al., 2013). This establishes a detailed algebraic dictionary between smooth defect backgrounds in the bulk and primaries of super–$W$–algebras in the boundary theory.

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AdS$_3$ BPS defects thus represent a unifying theme across lower-dimensional quantum gravity, string/M-theory, higher-spin holography, and integrability, encapsulating the interplay of geometry, algebra, and quantum field theory in the study of supersymmetric defects and their dual conformal structures (0904.0475, Giribet et al., 31 Jan 2024, Chen et al., 2013, Dibitetto et al., 2017, Conti et al., 31 Jul 2024, Bliard et al., 3 Oct 2024, Lagares, 17 Sep 2025, Lozano et al., 26 Apr 2024).