Supersymmetric Holomorphic Masses in AdS/CFT with Flavour

TL;DR

This paper constructs exact BPS D7-brane probes in the D3/D7 system with worldvolume scalars y that are holomorphic in z or antiholomorphic in z. These solutions preserve 2D N=(4,0) or (0,4) SUSY along the D3-brane directions and correspond holographically to position-dependent, holomorphic/antiholomorphic masses m(z) in the dual N=4 SYM with N_f hypermultiplets. The authors prove SUSY preservation both in field theory (via background vector multiplets and hypermultiplet variations) and in supergravity (via kappa-symmetry and BPS bounds), and show that holographic renormalisation yields zero energy and zero one-point functions for the mass operator. Zeros of m correspond to 8ND D7-branes and give rise to 2D superconformal defects with chiral fermions, signaling SUSY enhancement to (8,0) or (0,8) at defects. The work provides a broad framework for studying translational symmetry breaking, defects, and holographic mass holonomies in strongly coupled systems, with potential extensions to more general probe-brane intersections and back-reacted solutions.

Abstract

In type IIB supergravity (SUGRA), in the extremal background of a number N_c of D3-branes we consider a number N_f of probe D7-branes extended along all four D3-brane directions, (x_0,x_1,x_2,x_3). Using (x_2,x_3) to define complex coordinates (z,\bar{z}) and defining the D7-branes' worldvolume scalars as (y,\bar{y}), we prove that any holomorphic function y(z) or antiholomorphic function y(\bar{z}) is a BPS solution of the D7-branes' equations of motion preserving N=(4,0) or (0,4) supersymmetry (SUSY) along (x_0,x_1), respectively. In the near-horizon geometry, five-dimensional Anti-de Sitter (AdS) spacetime times a five-sphere, the AdS/Conformal Field Theory correspondence states that type IIB SUGRA is holographically dual to four-dimensional N=4 supersymmetric SU(N_c) Yang-Mills theory at large N_c and large 't Hooft coupling, and the N_f D7-branes are dual to N_f N=2 hypermultiplets in the fundamental representation, i.e. flavour fields. Our D7-brane solutions are dual to a position-dependent hypermultiplet mass m that is holomorphic, m(z), or antiholomorphic, m(\bar{z}). We provide field theory proofs, valid for any N_c, N_f, and 't Hooft coupling, that such m preserve N=(4,0) or (0,4) SUSY along (x_0,x_1). We also perform holographic calculations with probe D7-branes to show that the theory with such m has zero energy and zero expectation value of the mass operator, and that a zero of m is dual to a superconformal defect described by chiral fermions along (x_0,x_1).

Figures (1)

• Figure : Figure: Cartoon of the probe D7-branes' worldvolume for the linear solution, $y(z)=c\,z$ with dimensionless real constant $c >0$. The vertical axis is $r = \sqrt{\rho^2 + 2 |y|^2}$ in arbitrary units, where the Poincaré horizon is at $r=0$ and the boundary is at $r \to \infty$. The horizontal axis is $\textrm{Re}(z)$ in arbitrary units. The shaded triangle depicts the D7-branes' extent inside $AdS_5$. The black circles represent the $S^5$, while the orange circles represent the D7-branes' worldvolume along an $S^3 \subset S^5$, the orange dots represent an $S^3\subset S^5$ that has collapsed to a pole of the $S^5$, and the orange disk represents D7-branes wrapping the entire $S^5$. On the right we plot the angle of latitude that parametrises the $S^3$ fibration, $\theta \in [0,\pi/2]$, versus $|z|$, for three representative values of $r$. When $\theta$ is zero, the $S^3$ is maximal, whereas when $\theta$ reaches $\pi/2$ the $S^3$ collapses to a pole of the $S^5$, as shown on the left and right edges of the triangle. If we move down along the left edge of the triangle, through $r=0$, and then up the right edge, then at $r=0$ the D7-branes must "jump" from the south pole to the north pole of the $S^5$ in a single point. At that point they thus extend along the entire range of $\theta \in [0,\pi/2]$, as we depict on the bottom right, so at that point they wrap the entire $S^5$.