5.8 Baryon vertex
As a first example of a supersymmetric soliton in a non-trivial background, I will review the baryon
vertex [500
, 265
]. Technically, this will provide an example of how to deal with non-constant Killing
spinors. Conceptually, it is a nice use of the tools explained in this review having an interesting AdS/CFT
interpretation.
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 AdS5 × S5 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 AdS5 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 AdS5 as a function
of
the co-latitude angle
on S5. 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.
Set-up:
One is interested in solving the equations of motion of a single D5-brane in the background of
D3-branes carrying some units of electric charge to describe type IIB strings. The background is
described by a constant dilaton, a non-trivial metric and self-dual 5-form field strength
[195]
where
is the
-invariant metric on the unit 5-sphere,
is its volume 5-form and
its
Hodge dual. The function
is
Notice
corresponds to the full D3-brane background solution, whereas
to its near-horizon
limit.
Consider a probe D5-brane of unit tension wrapping the 5-sphere. Let
be the world volume
coordinates, so that
(
) are coordinates for the worldspace 5-sphere. This will be achieved
by the static gauge
Since one is only interested in radial deformations of the world space carrying electric charge, one considers
the ansatz
Even though the geometry will be curved, it can give some intuition to think of this system in terms of the
array
viewing the 9-direction as the radial one.
Supersymmetry analysis:
Given the electric nature of the world volume gauge field, the kappa symmetry
matrix reduces to
Given the ansatz (373) and the background (369), the induced world volume metric equals
where
and
stands for the
-invariant metric on the unit 5-sphere. Taking into account the non-trivial
vielbeins, the induced gamma matrices equal
where the matrices
are defined as
in terms of the fünfbein
in the 5-sphere. Thus,
.
To solve the kappa symmetry preserving condition (214), one requires the background Killing spinors.
These are of the form
where
is a covariantly constant spinor on
subject to the projection condition
describing the D3-branes in the background. Importantly,
is not constant when using polar coordinates
in
. Indeed, covariantly constant spinors on
were constructed explicitly in [359
] for a
sphere parameterisation obtained by iteration of
. The result can be
written in terms of the
angles
and the antisymmetrised products of pairs of the
constant
Clifford matrices
. For
, defining
, these equal
where
satisfies Eq. (380). Even though there are additional Killing spinors in the near-horizon limit,
the associated extra supersymmetries will be broken by the baryonic D5-brane probe configuration I am
about to construct, so these can be ignored.
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.
Hamiltonian analysis:
Solving the Hamiltonian constraint
in Eq. (224) allows to write the
Hamiltonian density for static configurations as [248
]
where
is a covariantised electric field density related to
by
For the ansatz (373), this reduces to
It was shown in [248
] that one can rewrite the energy density (389) as
where
indicates contraction with
, and
To achieve this, the 5-sphere metric was written as
where
is the
invariant metric on the 4-sphere, which one takes to have coordinates
. In
this way, all primes above refer to derivatives with respect to
and
are the
components of the
inverse S5 metric
.
Using the Gauss’ law constraint
which has a non-trivial source term due to the RR 5-form flux background, one can show that
where
has components
From Eq. (392), and the divergent nature of
, one deduces the bound
The latter is saturated when
Combining Eqs. (398) and (399) with the Gauss law (395) yields the equation
Any solution to this equation gives rise to a 1/4 supersymmetric baryonic brane.
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.
Related work:
There exists similar work in the literature. Besides the study of non-
invariant
baryonic branes in AdS5 × S5, [248] also carried the analysis for baryonic branes in M-theory. Similar BPS
bounds were found for D4-branes in D4-brane backgrounds or more generically, for D-branes in a D-brane
background [126, 133] and D3-branes in
5-branes [452, 357]. Baryon vertex configurations have also
been studied in AdS5 × T1,1 [19], AdS5 × Yp,q [134] and were extended to include the presence of
magnetic flux [319]. For a more general analysis of supersymmetric D-brane probes either in AdS or its
pp-wave limit, see [458].