Supersymmetric Holomorphic Masses in AdS/CFT with Flavour

Technical Summary: Supersymmetric Holomorphic Masses in AdS/CFT with Flavour

Problem Statement
The paper addresses the challenge of finding exact solutions for strongly coupled quantum field theories that lack translational and rotational symmetry. While Monte Carlo methods often fail for such systems (particularly at non-zero density due to the sign problem), the authors utilize supersymmetry (SUSY) to construct exact solutions. Specifically, the work investigates the intersection of $N_c$ coincident D3-branes and $N_f$ coincident D7-branes in Type IIB string theory. The goal is to determine if the D7-branes' worldvolume scalars can form non-trivial, position-dependent configurations that preserve a subset of supersymmetry while solving the equations of motion. In the context of the AdS/CFT correspondence, this corresponds to introducing a position-dependent mass $m$ for $N_f$ hypermultiplet "flavour" fields in $\mathcal{N}=4$ Super Yang-Mills (SYM) theory, a scenario where standard translational invariance is broken.

Methodology
The authors employ a dual approach, analyzing the system from both the gravity (supergravity/SUGRA) and field theory (SYM) perspectives.

1. Gravity Side (SUGRA):

   • Setup: The authors consider the extremal D3-brane background, which interpolates between asymptotically flat Minkowski space and the near-horizon $AdS_5 \times S^5$ geometry. They introduce $N_f$ probe D7-branes (where $N_f \ll N_c$) extending along $(x^0, \dots, x^7)$.
   • Ansatz: Using complex coordinates $z = \frac{1}{\sqrt{2}}(x^2 + i x^3)$ on the D3-brane worldvolume and $y = \frac{1}{\sqrt{2}}(x^8 + i x^9)$ transverse to both brane stacks, they propose an ansatz where the D7-brane worldvolume scalar $y$ is a function of $z$ and $\bar{z}$ only, independent of the radial coordinate $\rho$ of the transverse sphere $S^3$.
   • Analysis: They derive the Abelian Dirac-Born-Infeld (DBI) equation of motion for the probe D7-branes. They prove that any holomorphic function $y(z)$ or antiholomorphic function $y(\bar{z})$ satisfies this equation. Furthermore, they demonstrate that these solutions saturate a BPS bound on the action and energy, derived by comparing the D7-brane action to that of trivial 4ND (4 Neumann-Dirichlet) and 8ND (8 Neumann-Dirichlet) D-brane configurations.
   • Supersymmetry: By analyzing the $\kappa$-symmetry projection conditions on the Killing spinors, they determine exactly how many supercharges are preserved by these configurations.

2. Field Theory Side (SYM):

   • Holographic Dictionary: In the near-horizon limit, the system is dual to $\mathcal{N}=4$ SYM with $N_f$ fundamental hypermultiplets. The worldvolume scalar $y$ maps to the hypermultiplet mass $m(z) = \frac{\sqrt{2}}{2\pi\alpha'} y(z)$.
   • Field Theory Proofs: The authors provide two independent proofs within the SYM framework (valid for any $N_c, N_f$, and 't Hooft coupling $\lambda$) that a holomorphic or antiholomorphic mass preserves specific supersymmetries.
     • Proof 1: Introduces a background $\mathcal{N}=2$ vector multiplet with a scalar VEV $\Phi(z)$. They show that if $\Phi$ is holomorphic, the SUSY variations of the gaugino vanish for a specific projection of the supercharges.
     • Proof 2: Directly analyzes the SUSY variations of the massive hypermultiplet action. They show that for position-dependent $m(z)$, the variation vanishes only if $m$ is holomorphic (or antiholomorphic) and the supercharges satisfy specific chirality projections.
   • Holographic Renormalization: They perform holographic renormalization for the probe D7-branes to calculate the renormalized energy and the vacuum expectation value (VEV) of the mass operator $\mathcal{O}$.

Key Contributions and Results

• Existence of Holomorphic Solutions: The paper proves that in the D3-brane background, any holomorphic function $y(z)$ (or antiholomorphic $y(\bar{z})$) is an exact solution to the D7-brane equations of motion. These solutions are BPS states that saturate a bound on the action.
• Supersymmetry Preservation:
  • Generic holomorphic solutions preserve $d=2$ $\mathcal{N}=(4,0)$ supersymmetry along the $(x^0, x^1)$ directions.
  • Generic antiholomorphic solutions preserve $d=2$ $\mathcal{N}=(0,4)$ supersymmetry.
  • This represents a breaking of the original $d=4$ $\mathcal{N}=2$ SUSY down to a chiral $d=2$ subalgebra.
• SUSY Enhancement at Zeros and Poles:
  • At the zeros of $y(z)$, the D7-branes effectively intersect the D3-branes only along $(x^0, x^1)$, transitioning from a 4ND to an 8ND configuration.
  • The authors argue that at these zeros, supersymmetry enhances from $\mathcal{N}=(4,0)$ to $\mathcal{N}=(8,0)$ (or $(0,4) \to (0,8)$).
  • Geometrically, this corresponds to the D7-brane worldvolume interpolating between $AdS_5 \times S^3$ (at generic points) and $AdS_3 \times S^5$ (at the zero), involving a topology change where the trivial cycle $S^3 \subset S^5$ expands to wrap the entire $S^5$.
• Holographic Properties:
  • For any holomorphic or antiholomorphic mass, the renormalized energy of the probe sector vanishes.
  • The vacuum expectation value of the mass operator $\langle \mathcal{O} \rangle$ vanishes, consistent with the operator being a SUSY variation of another operator.
  • Zeros of the mass function $m(z)$ are dual to superconformal defects (specifically, $d=2$ chiral fermions coupled to the $d=4$ SYM), while poles correspond to infinitely massive "scattering centers."
• Field Theory Proofs: The paper establishes that these SUSY-preserving properties hold for the field theory at arbitrary coupling and rank, not just in the large $N_c$, strong coupling limit, by providing direct field-theoretic proofs.

Significance and Claims
The authors claim that this work provides a new class of exact solutions for strongly coupled systems with broken translational and rotational symmetry. By demonstrating that holomorphic dependence on spatial coordinates preserves a specific subset of supersymmetry, the paper offers a controlled framework to study:

• Defects and Quantum Wires: The zeros of the mass function describe $d=2$ conformal defects (or "quantum wires") embedded in $d=4$ SYM.
• Symmetry Breaking: The solutions allow for the study of systems where translational symmetry is broken in a controlled, supersymmetric manner.
• RG Flows: The solutions describe flows from $d=4$ fields to $d=2$ fields at the location of the defects.
• Fusion: The paper notes that the "fusion" of defects (e.g., bringing two zeros together) corresponds to the algebraic addition of the degrees of the zeros in the holomorphic function, allowing for the construction of lattices of defects (e.g., $y(z) \propto \sin(kz)$).

The authors explicitly state that their results are generalizations of known BPS solutions in Minkowski space to the curved $AdS_5 \times S^5$ background, enabling the application of holographic duality to these non-trivial, position-dependent mass configurations. They do not claim to have solved the general sign problem for non-supersymmetric systems but rather provide a solvable, supersymmetric sector that may serve as a benchmark or starting point for understanding more complex symmetry-breaking scenarios.