Let me first try to conceptually motivate the entire set-up. Consider a closed D5-brane surrounding
D3-branes, i.e., such that the D3-branes thread the D5-brane. The Hanany–Witten (HW) effect [282] allows
us to argue that each of these D3-branes will be connected to the D5-brane by a fundamental type IIB
string. Consequently, the lowest energy configuration should not allow the D5-brane to contract to a single
point, but should describe these D3-branes with strings attached to them allowing one to
connect the D3 and D5-branes. In the large limit, one can replace the D3-branes by their
supergravity backreaction description. The latter has an AdS_{5} × S^{5} near horizon. One can
think of the D5-brane as wrapping the 5-sphere and the strings emanating from it can be
pictured as having their endpoints on the AdS_{5} boundary. This is the original configuration
interpreted in [500, 265] as a baryon-vertex of the super-Yang–Mills (SYM)
theory.

At a technical level and based on our previous discussions regarding BIons, one can describe the baryon
vertex as a single D5-brane carrying units of world volume electric charge [315, 125] to account for the
type IIB strings. If one assumes all the electric charge is concentrated at one point, then one expects
the minimum energy configuration to preserve the rotational invariance around it. Such
configuration will be characterised by the radial position of the D5-brane in AdS_{5} as a function of
the co-latitude angle on S^{5}. This is the configuration studied in [315, 125, 152]. Since it is, a
priori, not obvious whether the requirement of minimal energy forces the configuration to be
invariant, one can relax this condition and look for configurations where the charge
is distributed through different points. One can study whether these configurations preserve
supersymmetry and saturate some energy bound. This is the approach followed in [248], where the
term baryonic branes was coined for all these kinds of configurations, and the one I will follow
below.

Plugging the ansatz into the kappa matrix (374), the supersymmetry preserving condition (214) reduces, after some algebra, to

where and .Given the physical interpretation of the sought solitons, one imposes two supersymmetry projections on the constant Killing spinors :

These are expected from the local preservation of 1/2 supersymmetry by the D5-brane and the IIB string in the radial direction, respectively. These projections imply Using these relations, one can rewrite the right-hand side of Eq. (382) as where . The coefficients of and in Eq. (386) vanish when Furthermore, the ones of and also do. I will eventually interpret Eq. (387) as the BPS equation for a world volume BIon. One concludes that Eq. (382) is satisfied as a consequence of Eq. (387) provided that It can be checked that this is indeed the case whenever Eq. (387) holds.

For a discussion of the first-order equations (398) and (399) for , see [126, 133]. Here, I will focus on the near horizon geometry corresponding to a=0. The Hamiltonian density bound (397) allows us to establish an analogous one for the total energy

While the first inequality is saturated under the same conditions as above, the second requires to not change sign within the integration region. For this configuration to describe a baryonic brane, one must identify this region with a 5-sphere having some number of singular points removed. Assuming the second inequality is saturated when the first one is, the total energy equals where is a 4-ball of radius having the ’th singular point as its center. This expression suggests that one interpret the ’th term in the sum as the energy of the IIB string(s) attached to the ’th singular point. No explicit solutions to Eq. (400) with these boundary conditions are known though.Consider invariant configurations (for a discussion of less symmetric configurations, see [248]). In this case ,

and . The BPS condition (399) reduces to [315, 125, 152] where , while the Gauss’ law (395) equals Its solution was first found in [125] where is an integration constant restricted to lie in the interval .Given this explicit solution, let me analyse whether the second inequality in Eq. (401) is saturated when the first one is, as I assumed before. Notice

where I used Eq. (404). The sign of is determined by the sign of the denominator. Thus, it will not change if it has no singularities within the region (except, possibly, at the endpoints ). Since one concludes that the denominator for vanishes at the endpoints but is otherwise positive provided is. This condition is only satisfied for , in which case Eq. (406) becomes Integrating the differential equation (404) for after substituting Eq. (409), one finds [125] where is the value of at . It was shown in [125] that this configuration corresponds to N fundamental strings attached to the D5-brane at the point , where diverges.Solutions to Eq. (404) for were also obtained in [125]. The range of the angular variable for which these solutions make physical sense is smaller than because the D5-brane does not completely wrap the 5-sphere. Consequently, the D5 probe captures only part of the five form flux. This suggests that one interpret these spike configurations as corresponding to a number of strings less than . In fact, it was argued in [109, 314] that baryonic multiquark states with quarks in SYM correspond to strings connecting the D5-brane to while the remaining strings connect it to . Since the D5-brane solutions do reach , it is tempting to speculate on whether they correspond to these baryonic multi-quark states.

Living Rev. Relativity 15, (2012), 3
http://www.livingreviews.org/lrr-2012-3 |
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