$κ$-symmetric M5 brane web for defects in $AdS_7 / CFT_6$ holography

Technical Summary: $\kappa$-symmetric M5 brane web for defects in AdS$_7$/CFT$_6$ holography

Problem Statement
This work investigates probe M5 brane solutions embedded in the $AdS_7 \times S^4$ spacetime, which are holographically dual to codimension-2 defects in the boundary 6d $N=(2,0)$ superconformal field theory. Building upon previous general solutions (arXiv:2109.08551) where these branes were codimension-2 in $AdS_7$ and preserved at least two supercharges with zero worldvolume 3-form flux, the author analyzes the system when the self-dual 3-form flux field strength ($h$) is turned on. The primary objective is to determine how non-zero flux modifies the embedding conditions, breaks supersymmetry, and deforms the geometry of the M5 brane worldvolume.

Methodology
The analysis utilizes the covariant PST (Pasti-Sorokin-Tonin) action for a single probe M5 brane. The core methodology involves imposing the $\kappa$-symmetry constraint, $\Gamma_\kappa \epsilon = \pm \epsilon$, which relates the brane's embedding and flux to the Killing spinors of the 11-dimensional background.

1. General Derivation: The author derives general embedding conditions for M5 branes by solving the $\kappa$-symmetry equation with a non-zero self-dual flux $h$. The flux is assumed to be proportional to worldvolume forms, $h = F(X^m) \epsilon_{abc} e^a \wedge e^b \wedge e^c$.
2. Projection Conditions: The analysis employs specific projection conditions on the constant spinor $\epsilon_0$ to isolate solutions with varying degrees of preserved supersymmetry (from 1/32 BPS down to 1/2 BPS).
3. Specific Examples: To interpret the general results, the author selects specific holomorphic embedding constraints (e.g., $\Phi_1=0, Z_2=0$) that yield simpler worldvolume topologies ($AdS_5 \times S^1$) when flux is zero. These specific cases are then analyzed with non-zero flux to observe deformations.
4. On-shell Action: The regularized on-shell action is computed for the deformed configurations to determine their physical value.

Key Contributions and Results

• Modified Embedding Conditions: The introduction of a non-zero flux field $h$ modifies the embedding conditions derived in previous work. The general solution is constrained by two arbitrary holomorphic functions $F^{(I)}(\Phi_0, \dots, Z_2) = 0$ satisfying specific scaling differential equations. Crucially, the flux function $F(X^m)$ is determined to be:
  $$F(X^m) = \frac{\cos(\theta/2) - \sin(\theta/2)}{\cos(\theta/2) + \sin(\theta/2)}$$
  This implies that the flux vanishes only when the coordinate $\theta = \pi/2$. Consequently, any solution with non-zero flux must shift away from the $\theta = \pi/2$ location.

• Supersymmetry Breaking:

  • 1/2 BPS to 1/8 BPS: For specific ansatzes (e.g., $\Phi_1=0, Z_2=0$) that are 1/2 BPS with zero flux, turning on the flux $h = (e_0 \wedge e_1 \wedge e_4 + e_3 \wedge e_6 \wedge e_{10})F$ breaks an additional factor of 1/4 of supersymmetry, resulting in a 1/8 BPS configuration. This requires three independent projection conditions.
  • 1/16 BPS Configurations: By turning on different components of the flux field (e.g., $h = (e_0 \wedge e_3 \wedge e_6 + e_1 \wedge e_4 \wedge e_{10})F$), the supersymmetry is further broken to 1/16 BPS.

• Worldvolume Deformations and Brane Webs:

  • The breaking of supersymmetry by the flux necessitates a deformation of the M5 brane worldvolume. The author proposes that these deformations manifest as "2d ridge-like spiked shapes" emerging from the brane in directions orthogonal to the worldvolume.
  • The analysis suggests that these deformed branes do not exist in isolation but form a complex "brane web." For instance, a 1/8 BPS solution with $\Phi_1=0$ can be viewed as the intersection of spikes connecting it to other codimension-4 M5 brane solutions (analyzed in previous work [16]) that share the same residual supersymmetry.
  • The flux field $h$ is sourced by the endpoints of these spikes (interpreted as open M2 branes ending on the M5).

• On-shell Action: The regularized on-shell action for the deformed configurations (with non-zero flux) is calculated to be zero ($S_{PST}^{reg} = 0$), consistent with the expectation that these are BPS states.

• Holographic Duals:

  • In the boundary limit ($r \to \infty$), the dual defects correspond to codimension-2 objects in the 6d theory.
  • For the specific case $\Phi_1=0, Z_2=0$ (where the coupling parameter $c_0=0$), the dual defect is identified as a "rigid" surface defect. Unlike non-rigid defects, these preserve an enhanced bosonic symmetry subgroup ($SO(2,4) \times SO(3)_R \times SO(2) \times SO(2)_R$) and exhibit a $1/\log x$ singularity rather than a $1/x$ pole.
  • The deformation of the brane worldvolume (spikes in $AdS_7$ and $S^4$ directions) is hypothesized to determine how the dual defect couples to the bosonic fields (scalars and the 2-form potential $B_{mn}$) of the boundary theory.

Significance
The paper establishes that non-zero worldvolume flux in codimension-2 M5 brane probes necessitates a departure from the $\theta = \pi/2$ locus and breaks additional supersymmetry. The primary significance lies in the geometric interpretation of this breaking: the flux induces a deformation of the brane into a web-like structure of intersecting spikes. This provides a classical, $\kappa$-symmetric framework for understanding how codimension-2 defects in the 6d $N=(2,0)$ theory can be deformed and how they couple to the bulk fields, offering a potential bridge to analyzing defect-CFTs via the bootstrap program and understanding the effective action of these defects including quantum corrections.