5.8 Baryon vertex

As a first example of a supersymmetric soliton in a non-trivial background, I will review the baryon vertex [500Jump To The Next Citation Point, 265Jump To The Next Citation Point]. 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 N D3-branes, i.e., such that the D3-branes thread the D5-brane. The Hanany–Witten (HW) effect [282Jump To The Next Citation Point] allows us to argue that each of these N 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 N D3-branes with N strings attached to them allowing one to connect the D3 and D5-branes. In the large N 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 N 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 𝒩 = 4 d = 4 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 N units of world volume electric charge [315Jump To The Next Citation Point, 125Jump To The Next Citation Point] to account for the N 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 SO (5) rotational invariance around it. Such configuration will be characterised by the radial position of the D5-brane in AdS5 as a function r(πœƒ) of the co-latitude angle πœƒ on S5. This is the configuration studied in [315Jump To The Next Citation Point, 125Jump To The Next Citation Point, 152Jump To The Next Citation Point]. Since it is, a priori, not obvious whether the requirement of minimal energy forces the configuration to be SO (5) 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 [248Jump To The Next Citation Point], where the term baryonic branes was coined for all these kinds of configurations, and the one I will follow below.

One is interested in solving the equations of motion of a single D5-brane in the background of N 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 R5 [195]

2 −1βˆ•2 2 (1,3) 1βˆ•2 [ 2 2 2] ds10 = U ds (𝔼 ) + U dr + r d Ω5 (369 ) R = 4R4 [ω + ⋆ω ] (370 ) 5 5 5
where dΩ25 is the SO (6)-invariant metric on the unit 5-sphere, ω5 is its volume 5-form and ⋆ω5 its Hodge dual. The function U is
(L4 )4 ( 4 ′ 2) U = a + --- L 4 = 4πgsN (α ) . (371 ) r
Notice a = 1 corresponds to the full D3-brane background solution, whereas a = 0 to its near-horizon limit. Consider a probe D5-brane of unit tension wrapping the 5-sphere. Let ξμ = (t,πœƒi) be the world volume coordinates, so that i πœƒ (i = 1,...,5) are coordinates for the worldspace 5-sphere. This will be achieved by the static gauge
X0 = t, Θi = πœƒi. (372 )
Since one is only interested in radial deformations of the world space carrying electric charge, one considers the ansatz
X1 = X2 = X3 = 0 , r = r(πœƒi) , F = 1-F0i(πœƒi)dt ∧ dπœƒi. (373 ) 2
Even though the geometry will be curved, it can give some intuition to think of this system in terms of the array
D3 : 1 2 3 x x x x x x background D5 : x x x 4 5 6 7 8 x probe F1 : x x x x x x x x 9 soliton
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

1--------1------- μ1...μ6 Γ κ = 6!∘ -------------πœ€ [γμ1...μ6τ1 + 15 F μ1μ2γμ3...μ6(iτ2)] . (374 ) − det(𝒒 + F )
Given the ansatz (373View Equation) and the background (369View Equation), the induced world volume metric equals
( ) 𝒒 = − U −1βˆ•2 0 (375 ) μν 0 gij
( ) gij = U 1βˆ•2 r2¯gij + ∂ir∂jr , (376 )
and ¯gij stands for the SO (6)-invariant metric on the unit 5-sphere. Taking into account the non-trivial vielbeins, the induced gamma matrices equal
γ = U − 1βˆ•4Γ , γ = U 1βˆ•4rˆγ + U1βˆ•4∂ rΓ , (377 ) 0 0 i i i r
where the matrices ˆγi are defined as
γˆi = eiaΓ a , (378 )
in terms of the fünfbein a ei in the 5-sphere. Thus, {ˆγi,ˆγj} = 2g¯ij. To solve the kappa symmetry preserving condition (214View Equation), one requires the background Killing spinors. These are of the form
1 πœ– = U− 8χ, (379 )
where χ is a covariantly constant spinor on (1,3) 6 𝔼 × π”Ό subject to the projection condition
Γ i τχ = χ , (380 ) 0123 2
describing the D3-branes in the background. Importantly, χ is not constant when using polar coordinates in 𝔼6. Indeed, covariantly constant spinors on Sn were constructed explicitly in [359Jump To The Next Citation Point] for a sphere parameterisation obtained by iteration of ds2= dπœƒ2 + sin2 πœƒ ds2 n n n n−1. The result can be written in terms of the n angles i ˆΔ± πœƒ = (πœƒ,πœƒ ) and the antisymmetrised products of pairs of the constant d = 10 Clifford matrices Γ a = (Γ πœƒ,Γ ˆΔ±). For n = 5, defining Γ ˆ5 ≡ Γ πœƒ, these equal
πœƒ ∏4 πœƒˆΘ· χ = e 2 Γ rπœƒ e− 2-Γ ˆΘ·ˆΘ·+1 πœ–0, (381 ) ˆΘ·=1
where πœ–0 satisfies Eq. (380View Equation). 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 (374View Equation), the supersymmetry preserving condition (214View Equation) reduces, after some algebra, to

∘ --------------------------- U det [r2¯gij + ∂ir∂jr − F0iF0j]πœ– = 5√ ----- 3√----- ij [U r det¯g Γ 0√γ∗τ1-− U r det ¯gF0j ∂irγˆ γ∗Γ r(iτ2) + U r4 detg¯ˆγiγ∗(F0i(iσ2) + ∂irΓ 0rτ1) ]πœ– (382 )
where ˆγi = ¯gijˆγj and γ∗ = Γ 45678.

Given the physical interpretation of the sought solitons, one imposes two supersymmetry projections on the constant Killing spinors πœ–0:

Γ 0γ∗τ1πœ–0 = πœ–0 , (383 ) Γ τ πœ– = πœ– . (384 ) 0r 3 0 0
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
Γ τ πœ– = [cosπœƒ − sinπœƒ Γ ] πœ–, 0r 3 rπœƒ Γ 0γ∗ τ1πœ– = [cosπœƒ − sinπœƒ Γ rπœƒ ] πœ–, Γ iγ∗iτ2πœ– = − Γ riπœ–, − πœƒΓ rπœƒ Γ iγ∗Γ 0r τ1πœ– = Γ rie πœ–, γ∗Γ r iτ2πœ– = − πœ–. (385 )
Using these relations, one can rewrite the right-hand side of Eq. (382View Equation) as
[ ′ ′ ˆΔ± Δ5 (r sin πœƒ) + Γ rπœƒ ((r cosπœƒ) − F0 πœƒ) + Γ rˆγ (∂ˆΔ±rcosπœƒ − F0ˆΔ±) ] +ˆγˆΔ±ˆΘ· 1r (∂ˆΔ±rF0 ˆΘ· − ∂Θ·ˆrF0ˆΔ±) + ˆγˆΔ±Γ πœƒ 1r (∂ˆΔ±rF0πœƒ − r′F0ˆΔ± + r∂ˆΔ±rsinπœƒ ) , (386 )
where √ ----- Δ5 = U r4 det¯g. The coefficients of Γ rπœƒ and Γ rˆγˆΔ± in Eq. (386View Equation) vanish when
F0i = ∂i(r cos πœƒ). (387 )
Furthermore, the ones of ˆΔ±ˆΘ· ˆγ and ˆΔ± ˆγ Γ πœƒ also do. I will eventually interpret Eq. (387View Equation) as the BPS equation for a world volume BIon. One concludes that Eq. (382View Equation) is satisfied as a consequence of Eq. (387View Equation) provided that
∘ --------------------------- U det[r2¯gij + ∂ir∂jr − F0iF0j ] = Δ5 (rsinπœƒ )′. (388 )
It can be checked that this is indeed the case whenever Eq. (387View Equation) holds.

Hamiltonian analysis:
Solving the Hamiltonian constraint β„‹ = 0 in Eq. (224View Equation) allows to write the Hamiltonian density for static configurations as [248Jump To The Next Citation Point]

1[ ] β„‹2 = U− 2 E&tidle;i &tidle;Ejgij + det g , (389 )
where &tidle;Ei is a covariantised electric field density related to F0i by
∘ -------------- j (det g)F0i = − det(𝒒 + F )E&tidle; gij. (390 )
For the ansatz (373View Equation), this reduces to
√ ----- i 1βˆ•4 detg ij E&tidle; = U ∘----------------------g F0j. (391 ) 1 − U 1βˆ•2gmn F0m F0n
It was shown in [248Jump To The Next Citation Point] that one can rewrite the energy density (389View Equation) as
2 2 [ ′ &tidle; i ]2 ˆΔ±ˆΘ· &tidle;ˆΔ± 2 β„‹ = 𝒡 5 + Δ5 (rcos πœƒ) − E ∂i (r sin πœƒ) + |Δ5 ¯g ∂ ˆΘ·r − r E | , (392 )
where ||2 indicates contraction with g ˆΔ±ˆΘ·, and
′ &tidle;i 𝒡5 = Δ5 (r sin πœƒ) + E ∂i(rcosπœƒ ). (393 )
To achieve this, the 5-sphere metric was written as
ds2 = dπœƒ2 + sin2 πœƒd Ω2, (394 ) 4
where 2 dΩ4 is the SO (5) invariant metric on the 4-sphere, which one takes to have coordinates ˆΔ± πœƒ. In this way, all primes above refer to derivatives with respect to πœƒ and ¯gˆΔ±Θ·ˆ are the ˆΔ±ˆΘ· components of the inverse S5 metric ¯gij. Using the Gauss’ law constraint
&tidle;i 4∘ ----- ∂iE = − 4R det¯g , (395 )
which has a non-trivial source term due to the RR 5-form flux background, one can show that 𝒡 = ∂ 𝒡i 5 i 5 where βƒ— 𝒡5 has components
∘ ----- ( r5 ) 𝒡 πœƒ5 = E&tidle;πœƒ r cosπœƒ + det¯g sinπœƒ a -- + rR4 , 5 𝒡ˆΔ± = E&tidle;ˆΔ±r cosπœƒ . (396 ) 5
From Eq. (392View Equation), and the divergent nature of 𝒡5, one deduces the bound
β„‹ ≥ |𝒡5|. (397 )
The latter is saturated when
¯gˆΔ±ˆΘ·∂ r E&tidle;ˆΔ± = Δ5 ---ˆΘ·-, (398 ) r ( ) &tidle;πœƒ ---Δ5--- ′ ¯gˆΔ±ˆΘ·∂ˆΔ±r∂-ˆΘ·(rsinπœƒ) E = (rsinπœƒ)′ (rcos πœƒ) − r . (399 )
Combining Eqs. (398View Equation) and (399View Equation) with the Gauss law (395View Equation) yields the equation
( ) [ ( ) ] ∘ ----- ∂ˆΔ± Δ5 ¯gˆiˆj∂ˆΘ·r + ∂πœƒ ---Δ5--- (rcos πœƒ)′ − ¯gˆiˆj∂ˆΔ±r ∂ˆΘ·r sin-πœƒ = − 4R4 det ¯g. (400 ) r (rsinπœƒ )′ r
Any solution to this equation gives rise to a 1/4 supersymmetric baryonic brane.

For a discussion of the first-order equations (398View Equation) and (399View Equation) for a = 1, see [126Jump To The Next Citation Point, 133Jump To The Next Citation Point]. Here, I will focus on the near horizon geometry corresponding to a=0. The Hamiltonian density bound (397View Equation) allows us to establish an analogous one for the total energy E

∫ |∫ | 5 || 5 || E ≥ d σ|𝒡5 | ≥ | d σ𝒡5 |. (401 )
While the first inequality is saturated under the same conditions as above, the second requires 𝒡5 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
∑ ∫ E = lim dβƒ—S ⋅𝒡⃗, (402 ) δ→0 k Bk
where Bk is a 4-ball of radius δ having the k’th singular point as its center. This expression suggests that one interpret the k’th term in the sum as the energy of the IIB string(s) attached to the k’th singular point. No explicit solutions to Eq. (400View Equation) with these boundary conditions are known though.

Consider SO (5) invariant configurations (for a discussion of less symmetric configurations, see [248Jump To The Next Citation Point]). In this case &tidle;ˆΔ± E = 0,

∘ ------- &tidle;Eπœƒ = det g(4)E&tidle;(πœƒ), (403 )
and r = r(πœƒ). The BPS condition (399View Equation) reduces to [315, 125Jump To The Next Citation Point, 152]
r′ Δ sin πœƒ + &tidle;E cosπœƒ --= ----------&tidle;------, (404 ) r Δ cosπœƒ − E sin πœƒ
where Δ = R4 sin4πœƒ, while the Gauss’ law (395View Equation) equals
E&tidle;′ = − 4 R4 sin4πœƒ. (405 )
Its solution was first found in [125Jump To The Next Citation Point]
&tidle; 1- 4 [ 3 ] E = 2 R 3 (νπ − πœƒ) + 3sinπœƒ cosπœƒ + 2 sin πœƒcos πœƒ , (406 )
where ν is an integration constant restricted to lie in the interval [0,1].

Given this explicit solution, let me analyse whether the second inequality in Eq. (401View Equation) is saturated when the first one is, as I assumed before. Notice

∘ ------- ( &tidle; )2 ( &tidle; )2 𝒡 = detg(4)𝒡 (πœƒ) with 𝒡 (πœƒ) = r -Δ-cosπœƒ-−-E(-sin-πœƒ--+---Δ-sin-πœƒ)-+-E-cos-πœƒ--, (407 ) 5 Δ cos πœƒ − &tidle;E sin πœƒ
where I used Eq. (404View Equation). The sign of 𝒡 is determined by the sign of the denominator. Thus, it will not change if it has no singularities within the region πœƒ ∈ [0,π ] (except, possibly, at the endpoints πœƒ = 0, π). Since
&tidle; 3- 4 Δ cos πœƒ − E sin πœƒ = 2 R sinπœƒ η(πœƒ) with η(πœƒ) ≡ πœƒ − νπ − sinπœƒ cosπœƒ, (408 )
one concludes that the denominator for 𝒡 vanishes at the endpoints πœƒ = 0,π but is otherwise positive provided η (πœƒ) is. This condition is only satisfied for ν = 0, in which case Eq. (406View Equation) becomes
[ ] E&tidle; = 1-R4 3(sin πœƒ cosπœƒ − πœƒ) + 2sin3πœƒ cosπœƒ . (409 ) 2
Integrating the differential equation (404View Equation) for r(πœƒ) after substituting Eq. (409View Equation), one finds [125Jump To The Next Citation Point]
( 6) 13 1 r = r0 -- (cosec πœƒ)(πœƒ − sin πœƒcosπœƒ )3 , (410 ) 5
where r0 is the value of r at πœƒ = 0. It was shown in [125Jump To The Next Citation Point] that this configuration corresponds to N fundamental strings attached to the D5-brane at the point πœƒ = π, where r(πœƒ) diverges.

Solutions to Eq. (404View Equation) for ν ⁄= 0 were also obtained in [125]. The range of the angular variable πœƒ for which these solutions make physical sense is smaller than [0,π ] 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 N. In fact, it was argued in [109, 314] that baryonic multiquark states with k < N quarks in 𝒩 = 4 d = 4 SYM correspond to k strings connecting the D5-brane to r = ∞ while the remaining N − k strings connect it to r = 0. Since the ν ⁄= 0 D5-brane solutions do reach r = 0, 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-SO (5) 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 (p,q)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].

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