Myers Effect in D-Brane Polarization

• Myers effect is a phenomenon in which D0-branes polarize into fuzzy higher-dimensional branes via dielectric expansion, bridging dual descriptions.
• This effect is realized in the ABJM framework, where wrapped D-brane configurations with magnetic flux yield stable baryon-vertex-like states with dissolved charges.
• The dielectric description equates macroscopic wrapped-brane actions with microscopic D0-brane matrix models, offering insights into finite ’t Hooft coupling dynamics.

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 (Lozano et al., 2011).

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 (Lozano et al., 2011).

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 (Lozano et al., 2011).

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 (Lozano et al., 2011).

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

$L=\left(\frac{32\pi^2 N}{k}\right)^{1/4},$

$CP^n$0

with $CP^n$1 the Kähler form on $CP^n$2, normalized by

$CP^n$3

The flat $CP^n$4-field required by the Freed-Witten anomaly on $CP^n$5 is

$CP^n$6

These ingredients determine the wrapped-brane sectors in which the dielectric interpretation is realized (Lozano et al., 2011).

The relevant particle-like D$CP^n$7-branes wrap cycles $CP^n$8 with $CP^n$9. In this setup, a D6 wrapped on all of $p$0 is the baryon vertex dual to $p$1 external quarks, a D2 wrapped on $p$2 is a monopole-like object dual to $p$3 strings, and a D4 wrapped on $p$4 is the dibaryon; the D4 requires the Freed-Witten $p$5 background (Lozano et al., 2011).

The worldvolume magnetic flux is taken as

$p$6

and the gauge-invariant combination entering the DBI action is

$p$7

The DBI action is

$p$8

For $p$9, the flux effectively appears as $CP^{p/2}$0 because of the $CP^{p/2}$1 shift, while for the D4 the $CP^{p/2}$2 cancels the Freed-Witten worldvolume flux so that $CP^{p/2}$3 (Lozano et al., 2011).

3. Flux, dissolved charges, and reduced-quark baryon vertices

The DBI energy of the magnetized wrapped brane takes the form

$CP^{p/2}$4

with

$CP^{p/2}$5

and

$CP^{p/2}$6

This is the macroscopic energetic input that the microscopic D0-brane dielectric description later reproduces (Lozano et al., 2011).

The magnetic flux dissolves lower-dimensional brane charges. For the D4 one has

$CP^{p/2}$7

so the D4 carries D2 charge, and also

$CP^{p/2}$8

so it carries D0 charge. More generally, the number of D$CP^{p/2}$9-branes dissolved in a D$p$0-brane is

$p$1

This dissolved D0 charge is precisely what enables the dielectric reinterpretation (Lozano et al., 2011).

The flux also modifies the fundamental-string tadpole. The D4 and D6 acquire F-string charge

$p$2

Explicitly,

$p$3

These tadpoles control the baryon-vertex-like configurations with a reduced number $p$4 of external strings (Lozano et al., 2011).

The generalized configuration consists of a wrapped D$p$5-brane at $p$6, $p$7 strings stretching from $p$8 to the boundary, and $p$9 strings stretching from $p$0 to the center. Writing

$p$1

the existence condition becomes

$p$2

and implies the lower bound

$p$3

When $p$4, the strings become radial and the configuration degenerates into free quarks. For allowed flux, one can still have $p$5, so the state is a genuine bound baryon-vertex-like configuration. For the D2 and D6, $p$6 has an upper bound; for the D4 there is both a lower and upper allowed flux window, and the stable “best” point occurs at

$p$7

All of these statements are part of the macroscopic side of the dielectric construction (Lozano et al., 2011).

4. Microscopic dielectric description by D0-branes

The microscopic formulation starts from $p$8 coincident D0-branes governed by the Myers DBI action

$p$9

with

$\mathcal N=6$0

The number of D0-branes is related to the magnetic flux by

$\mathcal N=6$1

The dielectric statement is then that these D0-branes expand into a fuzzy $\mathcal N=6$2 (Lozano et al., 2011).

The fuzzy $\mathcal N=6$3 is described by matrices $\mathcal N=6$4 in the symmetric representation $\mathcal N=6$5 of $\mathcal N=6$6, satisfying

$\mathcal N=6$7

with

$\mathcal N=6$8

The fuzzy Kähler form is

$\mathcal N=6$9

This provides the noncommutative geometry into which the D0-branes polarize (Lozano et al., 2011).

In the large-$$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k},$$0 limit, the D0-brane DBI action becomes

$$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k},$$1

Using $$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k},$$2, one obtains the same macroscopic energy,

$$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k},$$3

This exact matching is the central technical evidence that the magnetized wrapped D$$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k},$$4-brane is equivalently a dielectric D0-brane state (Lozano et al., 2011).

The microscopic Chern-Simons action,

$$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k},$$5

reproduces the required F-string charges. In the fuzzy $$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k},$$6/D2 case, a coupling of the form

$$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k},$$7

gives the correct $$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k},$$8-charge. In the fuzzy $$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k},$$9/D6 case, the coupling

$AdS_4\times CP^3$0

reproduces the $AdS_4\times CP^3$1 external strings in the large-$AdS_4\times CP^3$2 limit. The paper also proposes new dielectric couplings not included in the original Myers action, such as

$AdS_4\times CP^3$3

for the D4 case and

$AdS_4\times CP^3$4

for the D6 case, together with a higher-curvature dielectric term

$AdS_4\times CP^3$5

These terms are introduced to match the flux-induced tadpoles and the higher-curvature cancellation structure seen macroscopically (Lozano et al., 2011).

5. Stability, binding energy, and finite-coupling regime

The reduced-quark configurations are solutions to the classical equations of motion, but existence is not identical to stability. The fluctuation analysis shows that only longitudinal string fluctuations can destabilize the solution. Imposing the boundary condition at $AdS_4\times CP^3$6 yields the numerical critical value

$AdS_4\times CP^3$7

This strengthens the classical existence bound to the stability bound

$AdS_4\times CP^3$8

Stable baryon-vertex-like configurations therefore exist only in a restricted region of $AdS_4\times CP^3$9-space, tighter than the classical solution bound, while the D$L=\left(\frac{32\pi^2 N}{k}\right)^{1/4},$0-brane itself is stable against fluctuations (Lozano et al., 2011).

The geometric size of the baryon configuration is

$L=\left(\frac{32\pi^2 N}{k}\right)^{1/4},$1

Its binding energy is always negative: $L=\left(\frac{32\pi^2 N}{k}\right)^{1/4},$2 The force is therefore attractive and the state is bound (Lozano et al., 2011).

The significance of the dielectric picture is not limited to the existence of a dual description. Because the microscopic construction is in terms of D0-branes expanding into fuzzy projective spaces, it provides a way to analyze these baryon-vertex-like states in the region of finite ’t Hooft coupling. This suggests that the Myers effect, in this setting, is a bridge between the wrapped-brane description valid at large scales and a matrix description that remains meaningful away from the strict supergravity limit (Lozano et al., 2011).

6. Scope, terminology, and common confusions

The term “Myers effect” is frequently confused with several unrelated arXiv literatures that share the surname “Myers” but do not involve dielectric D-brane polarization. The most common confusion is with Myers–Perry black holes, which are higher-dimensional rotating vacuum black holes; that literature explicitly does not discuss a dielectric-brane mechanism, nonabelian scalar polarization of D-branes, or “Myers effect” in the string-theory sense (Myers, 2011). The same applies to work on excitations and $L=\left(\frac{32\pi^2 N}{k}\right)^{1/4},$3-form separability in Myers–Perry backgrounds, which concerns field propagation on rotating black-hole geometries rather than D-brane dielectric expansion (Lunin, 2019).

A second major source of ambiguity is the Horowitz–Myers literature. Horowitz–Myers metrics, the Horowitz–Myers conjecture, and the Horowitz–Myers geon are asymptotically hyperbolic geometries tied to negative cosmological constant, mass inequalities, and rigidity statements. They concern positive-energy-type problems and geometric inequalities, not dielectric D0-branes expanding into fuzzy projective spaces (Barzegar et al., 2019). Likewise, the many Bonnet–Myers results in Riemannian geometry and graph curvature study compactness, diameter bounds, Bakry–Émery curvature, or Ollivier Ricci curvature; these are geometrical generalizations of Myers’ theorem and are unrelated to the string-theoretic dielectric phenomenon (Wan, 2017).

Within the present technical usage, the Myers effect should therefore be reserved for the D-brane dielectric polarization mechanism. In the explicit ABJM realization discussed above, its hallmark is the equivalence between a magnetized wrapped D$L=\left(\frac{32\pi^2 N}{k}\right)^{1/4},$4-brane carrying dissolved D0-brane charge and a nonabelian D0-brane system expanded into fuzzy $L=\left(\frac{32\pi^2 N}{k}\right)^{1/4},$5, together with the associated charge, stability, and finite-coupling structure (Lozano et al., 2011).