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On unquenched N=2 holographic flavor

Angel Paredes

TL;DR

The paper constructs a holographic model for ${\cal N}=2$ SYM with unquenched fundamental matter by embedding $N_f$ flavor D5-branes in the wrapped-D5 background and solving the backreacted first-order equations in the Veneziano limit $N_f/N_c\sim O(1)$. The authors derive how the flavor backreaction alters the holomorphic gauge coupling, reproduce expected field-theory running and monodromies, and demonstrate that the meson spectrum forms a tower of ${\cal N}=2$ massive vector multiplets within a regularized UV framework. The approach relies on smeared flavor branes forming a domain-wall-like shell to avoid pathologies, and the resulting BPS system reduces to a PDE for a master function $z(\rho,\sigma)$ that ties the geometry to gauge-theory data. Despite IR curvature singularities and UV dilaton divergence in the underlying background, the analysis yields consistent holomorphic decoupling and coupling-constant behavior, and shows only modest shifts in meson masses due to unquenched flavors, providing a controlled avenue to study holographic unquenched flavors in ${\cal N}=2$ theories.

Abstract

The addition of fundamental degrees of freedom to a theory which is dual (at low energies) to N=2 SYM in 1+3 dimensions is studied. The gauge theory lives on a stack of Nc D5 branes wrapping an S^2 with the appropriate twist, while the fundamental hypermultiplets are introduced by adding a different set of Nf D5-branes. In a simple case, a system of first order equations taking into account the backreaction of the flavor branes is derived (Nf/Nc is kept of order 1). From it, the modification of the holomorphic coupling is computed explicitly. Mesonic excitations are also discussed.

On unquenched N=2 holographic flavor

TL;DR

The paper constructs a holographic model for SYM with unquenched fundamental matter by embedding flavor D5-branes in the wrapped-D5 background and solving the backreacted first-order equations in the Veneziano limit . The authors derive how the flavor backreaction alters the holomorphic gauge coupling, reproduce expected field-theory running and monodromies, and demonstrate that the meson spectrum forms a tower of massive vector multiplets within a regularized UV framework. The approach relies on smeared flavor branes forming a domain-wall-like shell to avoid pathologies, and the resulting BPS system reduces to a PDE for a master function that ties the geometry to gauge-theory data. Despite IR curvature singularities and UV dilaton divergence in the underlying background, the analysis yields consistent holomorphic decoupling and coupling-constant behavior, and shows only modest shifts in meson masses due to unquenched flavors, providing a controlled avenue to study holographic unquenched flavors in theories.

Abstract

The addition of fundamental degrees of freedom to a theory which is dual (at low energies) to N=2 SYM in 1+3 dimensions is studied. The gauge theory lives on a stack of Nc D5 branes wrapping an S^2 with the appropriate twist, while the fundamental hypermultiplets are introduced by adding a different set of Nf D5-branes. In a simple case, a system of first order equations taking into account the backreaction of the flavor branes is derived (Nf/Nc is kept of order 1). From it, the modification of the holomorphic coupling is computed explicitly. Mesonic excitations are also discussed.

Paper Structure

This paper contains 14 sections, 71 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction and summary of results
  2. A review of the dual of ${\cal N}=2$ SYM
  3. The solution
  4. Rewriting the solution and Killing spinors
  5. The dual with flavor
  6. Flavor branes in the solution
  7. Computing the flavor brane backreaction
  8. Gauge theory features
  9. Mesonic excitations
  10. Fluctuation of the scalar fields
  11. Fluctuation of the gauge field
  12. Analysis of the spectrum
  13. Estimate of the mass spectrum
  14. Appendix: WKB approximation, useful formulae

Figures (4)

  • Figure 1: A scheme of the configuration on the $\rho,\phi_2$ plane, which is the complex $u$-plane up to a factor of $\Lambda$. The inner circle at radius $\rho_0$ is the distribution of $N_c$ vevs for the adjoint scalars. The outer circle represents the $N_f$ masses of the fundamental hypermultiplets. The associated flavor branes are also extended along the transverse $\sigma,\phi_1$ directions. These smeared distributions restore the $U(1)_R$ broken by each single vev or mass. The point at a scale $\rho_q$ represents a flavor probe brane whose dynamics will be studied in section \ref{['sect:mesons']}.
  • Figure 2: Numerical approximations to the function $z(\rho,\sigma)$ that enters the metric. Both plotted solutions correspond to fixing $c=0$ ($\rho_0=\sqrt{e}$) and $\rho_Q=5$. It is apparent from the graphics the qualitative change of behavior at such values of $\rho$. On the left $N_f = N_c$. On the right, $N_f = 6 N_c$, a non-asymptotically free field theory that eventually hits a Landau pole. This corresponds to $z$ becoming zero at some value of $\rho$, as shown in the graph.
  • Figure 3: The figure in the left represents the potential (\ref{['pot']}) with fixed $\hbox{$\frac{N_f}{N_c}=0$},\ \hbox{$l=2$}, \ \hbox{$\bar{M}=3$},\ c=-1$ for different values of $\rho_q=1,2,3,4$ (the biggest the $\rho_q$ the upper is the line). The thick line corresponds to (\ref{['flatpot']}). The figure on the right is for fixed $\frac{N_f}{N_c}=0,\ \hbox{$l=2$},\ \hbox{$\bar{M}=3$},\ \rho_q=3$ for $c=-1,0,1$. The different lines are almost coincident and the meson masses should not strongly depend on $c$.
  • Figure 4: On the left, $\xi^{-1}$ versus $\rho_q$ for $c=0$. Starting from above, the three lines correspond to $\sigma_{cutoff}=50,70,90$. On the right, $\xi^{-1}$ versus $\rho_q$ with fixed $\sigma_{cutoff}=50$ for $c=0,2,4$. In both plots, $\frac{N_f}{N_c}\rightarrow 0$.