Continuous symmetries and charge measurement of boundary operators in holography

Abstract

We study holographic charge measurement for continuous internal symmetries. Charged boundary operators are characterized by Wilson lines of bulk gauge fields ending on the boundary, while charge measurement is performed using U-shaped defects hanging from the boundary. We derive universal features of this process from a low-energy point of view, and show how the hanging defect picture mimics the thickening regularization of continuous symmetry operators in field theory. Furthermore, we provide explicit top-down realizations in AdS/CFT setups in Type IIB string theory and M-theory, featuring Abelian as well as non-Abelian symmetries. In the case of Type IIB constructions, we analyze the brane dynamics underlying the charge measurement process. Along the way, we also characterize how hanging brane configurations can be regarded as being topological, and demonstrate how tachyon dynamics account for their fusion rules.

Figures (10)

• Figure 1: On the left: plot of the bump function $f_\epsilon(y)$. On the right: a regularized symmetry operator $U_{\rm reg}$ supported on the green strip is gradually moved past a $\phi$ insertion. In the process, a phase is acquired cumulatively by integrating $a_1$ according to equation \ref{['eq_cumulative_phase']}. This is depicted pictorially as the green line. Once $U_{\rm reg}$ is moved completely past the $\phi$ insertion, the phase stabilizes.
• Figure 2: A symmetry operator for a continuous symmetry is realized as hanging brane with worldvolume $\gamma^1 \times M^{d-1}$, with $M^{d-1}$ a codimension-1 submanifold of $\partial AdS_{d+1}$ and $\gamma^1$ an arc extending into the bulk of $AdS_{d+1}$. If the hanging brane is moved past a Wilson line ending on $\partial AdS_{d+1}$, a Hanany-Witten transition takes place. A new brane is created, depicted as the filled area in light blue. The new brane couples to the gauge field $a_1$ on the hanging brane along the portion $\mathcal{C}^1$ (depicted in red) of the full arc $\gamma^1$.
• Figure 3: Two hanging branes supported on $\gamma_L \cup \overline{\gamma}_R$ and $\gamma^\prime_L \cup \overline{\gamma}^\prime_R$ respectively split and recombine into two hanging branes on $\gamma_L \cup \overline{\gamma}^\prime_R$ and $\gamma^\prime_L \cup \overline{\gamma}_R$ which are now sitting on top of each other.
• Figure 4: The fusion of the operators $U_{\theta} \equiv U(\theta; M^{d-1})$ is associative, i.e. $U_\theta\otimes (U_{\theta^\prime}\otimes U_{\theta^{\prime\prime}})= (U_\theta\otimes U_{\theta^\prime})\otimes U_{\theta^{\prime\prime}}=U_{\theta+\theta^\prime+\theta^{\prime\prime}}$. Here we assume that $\theta,\theta',\theta"$ point in the same "direction" in the Lie algebra $\mathfrak{g}$.
• Figure 5: A D5-brane meets a $\overline{\text{D5}}$-brane at $z=z_0$, forming the hanging brane profile. The downward and upward arrows respectively label a brane and an antibrane.