--- title: Myers Effect in D-Brane Polarization url: https://www.emergentmind.com/topics/myers-effect type: topic --- # Myers Effect in D-Brane Polarization The Myers effect, in the D-brane sense, is the dielectric polarization phenomenon in which a collection of D0-branes expands into a higher-dimensional fuzzy brane in background flux. In the realization studied in the ABJM setting, magnetically charged baryon-vertex-like configurations in \(AdS_4\times CP^3\) admit both a macroscopic description by wrapped D-branes with magnetic worldvolume flux and a microscopic description by D0-branes expanding into fuzzy \(CP^n\) spaces by the Myers dielectric effect; this microscopic description also makes it possible to explore the region of finite ’t Hooft coupling [1105.0939]. ## 1. Definition and conceptual content In the literature under consideration, the defining mechanism of the Myers effect is that magnetic flux on a wrapped D\(p\)-brane dissolves D0-brane charge, and the same configuration can then be reinterpreted microscopically as many D0-branes expanding into a fuzzy \(CP^{p/2}\). The effect is therefore “dielectric” in the sense that lower-dimensional branes polarize into a higher-dimensional configuration rather than remaining pointlike [1105.0939]. The construction is explicitly nonperturbative in the worldvolume description. On the macroscopic side, the relevant object is a wrapped D\(p\)-brane carrying magnetic flux. On the microscopic side, the same state is encoded by matrix-valued D0-brane degrees of freedom, whose noncommutative configuration space realizes a fuzzy projective space. In the ABJM example, this correspondence is not merely qualitative: the D0-brane action reproduces the same energy and charge data as the wrapped D\(p\)-brane description [1105.0939]. A central point is that the effect is not an isolated kinematical curiosity. In the ABJM background it organizes the structure of baryon-vertex-like states with a reduced number of attached quarks, determines flux-induced dissolved charges, and provides a controlled microscopic interpretation of the corresponding wrapped-brane configurations [1105.0939]. ## 2. ABJM background and wrapped brane configurations The explicit realization discussed here is the \(\mathcal N=6\) ABJM theory with gauge group \[ U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k}, \] whose Type IIA dual is string theory on \(AdS_4\times CP^3\). The curvature scale and background fluxes are \[ L=\left(\frac{32\pi^2 N}{k}\right)^{1/4}, \] \[ F_2=\frac{2L}{g_s}J,\qquad F_4=\frac{3L^3}{8g_s}\,d{\rm Vol}(AdS_4),\qquad F_6=\frac{6L^5}{g_s}\,d{\rm Vol}(CP^3), \] with \(J\) the Kähler form on \(CP^3\), normalized by \[ \int_{CP^1}J=\pi,\qquad \int_{CP^2}J\wedge J=\pi^2,\qquad \int_{CP^3}J^3=\pi^3. \] The flat \(B\)-field required by the Freed-Witten anomaly on \(CP^2\) is \[ B_2=-2\pi J. \] These ingredients determine the wrapped-brane sectors in which the dielectric interpretation is realized [1105.0939]. The relevant particle-like D\(p\)-branes wrap cycles \(CP^{p/2}\subset CP^3\) with \(p=2,4,6\). In this setup, a D6 wrapped on all of \(CP^3\) is the baryon vertex dual to \(N\) external quarks, a D2 wrapped on \(CP^1\) is a monopole-like object dual to \(k\) strings, and a D4 wrapped on \(CP^2\) is the dibaryon; the D4 requires the Freed-Witten \(B_2\) background [1105.0939]. The worldvolume magnetic flux is taken as \[ F=\mathcal N J, \qquad \mathcal N\in 2\mathbb Z, \] and the gauge-invariant combination entering the DBI action is \[ \mathcal F = F+\frac{1}{2\pi}B_2. \] The DBI action is \[ S_p=-T_p\int d^{p+1}\xi\,e^{-\phi}\sqrt{\left|\det P[g+2\pi \mathcal F]\right|}, \qquad T_p=\frac{1}{(2\pi)^p}. \] For \(p=2,6\), the flux effectively appears as \((\mathcal N-1)\) because of the \(B_2\) shift, while for the D4 the \(B_2\) cancels the Freed-Witten worldvolume flux so that \(\mathcal F=\mathcal N J\) [1105.0939]. ## 3. Flux, dissolved charges, and reduced-quark baryon vertices The DBI energy of the magnetized wrapped brane takes the form \[ S_{DBI}^{Dp}=-Q_p\int d\tau\,\frac{2\rho}{L}, \] with \[ Q_p=\frac{T_p}{g_s}{\rm Vol}(CP^{p/2})\left(L^4+(2\pi)^2(\mathcal N-1)^2\right)^{p/4}\quad (p=2,6), \] and \[ Q_4=\frac{T_4}{g_s}{\rm Vol}(CP^2)\left(L^4+(2\pi \mathcal N)^2\right). \] This is the macroscopic energetic input that the microscopic D0-brane dielectric description later reproduces [1105.0939]. The magnetic flux dissolves lower-dimensional brane charges. For the D4 one has \[ S_{CS}^{D4}=2\pi T_4\int_{\mathbb R\times CP^2} C_3\wedge F =\frac{\mathcal N}{2}\,T_2\int C_3, \] so the D4 carries D2 charge, and also \[ S_{CS}^{D4}=\frac12(2\pi)^2T_4\int_{\mathbb R\times CP^2} C_1\wedge F\wedge F =\frac{\mathcal N^2}{8}\,T_0\int C_1, \] so it carries D0 charge. More generally, the number of D\(s\)-branes dissolved in a D\(p\)-brane is \[ n=\frac{\mathcal N^{(p-s)/2}}{2^{(p-s)/2}\,\left(\frac{p-s}{2}\right)!}. \] This dissolved D0 charge is precisely what enables the dielectric reinterpretation [1105.0939]. The flux also modifies the fundamental-string tadpole. The D4 and D6 acquire F-string charge \[ q=k\,\frac{\mathcal N^{\frac p2-1}}{2^{\frac p2-1}\left(\frac p2-1\right)!}. \] Explicitly, \[ q_{D4}=k\frac{\mathcal N}{2}, \qquad q_{D6}=N+k\,\frac{\mathcal N(\mathcal N-2)}{8}. \] These tadpoles control the baryon-vertex-like configurations with a reduced number \(l\) of external strings [1105.0939]. The generalized configuration consists of a wrapped D\(p\)-brane at \(\rho=\rho_0\), \(l\) strings stretching from \(\rho_0\) to the boundary, and \(q-l\) strings stretching from \(\rho_0\) to the center. Writing \[ a\equiv \frac{l}{q}, \qquad \sqrt{1-\beta^2}\equiv \frac{2Q_p}{L\,q\,T_{F1}}, \] the existence condition becomes \[ \frac{2Q_p}{L\,q\,T_{F1}}\le 1, \] and implies the lower bound \[ l\ge l_{\min}=\frac{q}{2}\left(1+\sqrt{1-\beta^2}\right). \] When \(\beta=0\), the strings become radial and the configuration degenerates into free quarks. For allowed flux, one can still have \(l