ABJM Baryon Stability and Myers effect

We consider magnetically charged baryon vertex like configurations in AdS^4 X CP^3 with a reduced number of quarks l. We show that these configurations are solutions to the classical equations of motion and are stable beyond a critical value of l. Given that the magnetic flux dissolves D0-brane charge it is possible to give a microscopical description in terms of D0-branes expanding into fuzzy CP^n spaces by Myers dielectric effect. Using this description we are able to explore the region of finite 't Hooft coupling.


Baryon potential within AdS/CFT
Heavy baryon potential E (L) is extracted from Wilson loop expectation values W (C ) . Within AdS/CFT, the interaction potential energy of the baryon is given by [prototype by Witten 1998]

Dp-Brane energy
The DBI action of a Dp-brane in string units reads where g is the induced metric and F = F + 1 2π B 2 is the magnetic flux. The metric of a Dp-brane wrapping on CP p/2 cycles (gauge choice is time and the angles of the CP p/2 cycles) reads where F = N J and B 2 = −2πJ, J is the Kähler form (equations of motion are satisfied) and N ∈ 2Z. The energy of the Dp-brane is [Lozano et. al. 2010].
Note: N 2 is comparable to L 4 1.

Dp-brane Charges
The CS action for a Dp-brane reads Both the D4 and D6-branes have CP 1 D2-branes dissolved. Therefore in the presence of a magnetic flux they capture the F 2 flux and develop a tadpole with charge There are three more couplings for D6:

Classical solution
We consider a classical configuration consisting on a Dp-brane wrapped on CP p 2 , located at ρ = ρ 0 , l strings stretching from ρ 0 to the boundary of AdS 4 and (q − l ) straight strings that go from ρ 0 to 0 (Figure). Solving the e.o.m. and imposing the b.c. at the boundary and the baryon vertex we find that the length and the energy of the distribution reads Thus, the binding energy reads [Conformal dependence, non-pertubative and concavity].

Stability analysis
Instabilities can emerge only from the longitudinal fluctuations of the l strings [Sfetsos, K.S. 2008]. Perturbing the embedding according to r = r cl + δr (ρ) and expanding the Nambu-Goto action to quadratic order in the fluctuations, the zero mode solution vanishing in the UV reads imposing the boundary condition at the baryon vertex ρ 0 we find The solution of the transcedental equation is γ c 0.538, thus the bound of F -strings coming from the stability is more restrictive l q 1+γ c (1 + 1 − β 2 ). As for the brane fluctuations they prove to be stable. Note that there are no boundary conditions for these fluctuations, the reason being that the R × CP p 2 space has no boundary.

Microscopical energy
D0-brane charge on the Dp-branes wrapped on (fuzzy) CP p 2 suggests a close analogy with the dielectric effect dielectric [Emparan 97, Myers 99]. The DBI action describing the dynamics of n coincident D0-branes expanded into fuzzy CP p/2 and one has to compute traces of powers of M using the constraints of the construction of CP p/2 space.
For example, for B 2 = 0 we find where r = L 4 m(m+ p 2 +1) . However, in the limit Thus the energy of n D0-branes expanding into a fuzzy CP p 2 is then given to leading order in m by where n = dim(m, 0). For m ∼ N 2 the leading order in m coincides with the macroscopical result. For B 2 = 0 the discussion is more technical and would not be presented here. It turns out, that redefinition of m gets corrected: N = 2m + 2 , p = 2, 4 and N = 2m + 4 , p = 6.

Microscopical charges
Next we shall show how fundamental strings that stretch from the Dp-brane to the boundary of AdS 4 strings arise in the microscopic setup. The CS action for n coincident D0-branes reads

The relevant CS terms in the AdS
Where we have expanded the background potentials on the non-Abelian scalars occurs through the Taylor expansion [Garousi et al 1998] and the pull-backs into the worldline are given in terms of gauge covariant derivatives, In the large m limit we find: , the number of fundamental string charge in each CP 1 , in agreement with the macroscopical result.
S CS 2 N d τA τ , in agreement with the macroscopical result. The third and fourth terms in contribute when we take into account the B 2 field that is necessary to compensate the Freed-Witten worldvolume field of the D4-brane.
To find the total k charge we add the subleading contributions in the large m expansion of S CS 1 . Next we use the corrected redefinition of m we recover precicely the units of F-charge for D2, D4 and D6 brane plus a k/8 contribution for the D6, coming from S CS 4 . Stability analysis goes along the same lines than in the macroscopical set-up; non-singlet classical stable solutions.

Dielectric higher-curvature terms
Generalizing the Chern-Simons action for multiple Dp-branes [Myers 1999] to include higher curvature terms we find for our background where Ω 4 is given in term of the Pontryagin classes of the normal and the tangent bundle of the three CP 2 circles of the CP 3 manifold [Eguchi et al 1980, Bergman et al 2009; Substituting F 2 and Ω 4 we find: Thus this higher curvature coupling cancels the S CS 4 contribution as in the macroscopical case.

Summary-future directions
We have constructed macroscopically various configurations of magnetically charged particle-like branes in ABJM with reduced number of quarks. The stability analysis increases the classical lower bound for each value of the magnetic flux.
We have given an alternative description in terms of D0-branes expanded into fuzzy CP p 2 spaces that allows to explore the finite 't Hooft coupling region, L p n.
We have constructed dielectric higher curvature couplings that to the best of our knowledge have not been considered before in the literature. This new coupling exactly cancels the k/8 contribution to the D6-brane tadpole.
It would be interesting to extend to theories with reduced supersymmetry, like the Klebanov-Strassler backgrounds, where the internal geometry is the T 1,1 conifold. Non-singlet baryon vertex???

Review of the AdS 4 × CP 3 background
In our conventions the AdS 4 × CP 3 metric reads with L the radius of curvature in string units and where we have normalized the two factors such that R µν = −3g µν and 8g αβ for AdS 4 and CP 3 , respectively. The explicit parameterization of AdS 4 we use is For the metric on CP 3 we use the parameterization in [Pope 1984,Warner 1985 Inside CP 3 there is a CP 1 for µ = α = π/2 and fixed χ and ψ and also a CP 2 for fixed θ and φ.
The AdS 4 × CP 3 background fluxes can then be written as where g s = L k . The flux integrals satisfy CP 3 F 6 = 32 π 5 N , The flat B 2 -field that is needed to compensate for the Freed-Witten worldvolume flux in the D4-brane is given by [Aharony et al 2009]