Generalizations of AdS/CFT: Quarks and mesons

34 Chapter 2. The AdS/CFT correspondence

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89 0123

4567

D3 N

R4

AdS₅

open/closed string duality

7−7

AdS₅

brane flavour open/open

string duality conventional

3−7 quarks

3−3 SYM

N probe D7_f

Figure 2.1: The figure sketches the original AdS/CFT correspondence between open and closed strings and its extension to fundamental matter relating open strings to each other. On the left side the geometry of a stack of coincidentN D3-branes (repre-sented by the thick vertical line) and a small number of coincidentNf D7-branes is shown. This is the setup within which the full string theory description is re-duced to the effective Dirac-Born-Infeld description on the world volume of the D7-branes. On the left side of the figure the geometry ofAdS5×S⁵is outlined on which the classical supergravity description is defined. At each point on the disc representingAdS5 anS⁵ exists but is not drawn for simplicity. The curved lines with labelsp−qrepresent strings starting at the stack of Dpbranes and ending on the stack of Dq-branes. This figure is taken from [25].

0 1 2 3 4 5 6 7 8 9

D3 x x x x

D7 x x x x x x x x

Figure 2.2: Coordinate directions in which the Dp-branes extend are marked by ’x’. D3- and D7-branes always share the four Minkowski directions and may be separated in the8,9-directions which are orthogonal to both brane types.

2.3. Generalizations of AdS/CFT: Quarks and mesons 35

gauge symmetry. We will call the strings starting on the stack of Dp-branes and ending on the stack of Dq-branes p− q strings. The original 3− 3 strings are unchanged while the 3−7- or equivalently7−3strings are interpreted as quarks on the gauge theory side of the correspondence. This can be understood by looking at the3−3strings again. They come in the adjoint representation of the gauge group which can be interpreted as the decomposition of a bifundamental representation (N² −1)⊕1 = N ⊗N¯. So the two string ends on the D3-brane are interpreted as one giving the fundamental, the other giving the anti-fundamental representation in the gauge theory. In contrast to this the3−7string has only one end on the D3-brane stack corresponding to a single fundamental representation which we interpret as a single quark in the gauge theory.

We can also give mass to these quarks by seperating the stack of D3-branes from the D7-branes in a direction orthogonal to both D7-branes. Now3−7strings are forced to have a finite length Lwhich is the minimum distance between the two brane stacks. On the other hand a string is an object with tension and if it assumes a minimum length, it needs to have a minimum energy being the product of its length and tension. The dual gauge theory object is the quark and it now also has a minimum energy which we interpret as its massM_q=L/(2πα^′).

The 7 −7 strings decouple from the rest of the theory since their effective coupling is suppressed by Nf/N. In the dual gauge theory this limit corresponds to neglecting quark loops which is often called the quenched approximation. Nevertheless, they are important for the description of mesons as we will see below.

Let us be a bit more precise about the fundamental matter introduced by3−7strings. The gauge theory introduced by these strings (in addition to the original setup) gives a N = 2 supersymmetricU(N)gauge theory containingNf fundamental hypermultiplets.

D7 embeddings & meson excitations Mesons correspond to fluctuations of the D7-branes ⁵ embedded in the AdS5 × S⁵-background generated by the D3-branes. From the string-point of view these fluctuations are fluctuations of the hypersurface on which the7−7 strings can end, hence these are small oscillations of the7−7string ends. The7−7strings again lie in the adjoint representation of the flavor gauge group for the same reason which we employed above to argue that3−3strings are in the adjoint of the (color) gauge group.

Mesons are the natural objects in the adjoint flavor representation. Vector mesons correspond to fluctuations of the gauge field on the D7-branes.

Before we can examine mesons as D7-fluctuations we need to find out how the D7-branes are embedded into the 10-dimensional geometry without any fluctuations. Such a stable con-figuration needs to minimize the effective action. The effective action to consider is the world volume action of the D7-branes which is composed of a Dirac-Born-Infeld as given in (2.21) and a topological Chern-Simons part

SD7 =−TD7

Z

d⁸σe^−Φ q

−det{P[g+B]αβ + (2πα^′)Fαβ}+(2πα^′)² 2 TD7

Z

P[C4]∧F∧F . (2.71) The preferred coordinates to examine the fluctuations of the D7 are obtained from the coordi-nates given in (2.50) by the transformation̺² =w12+· · ·+w42, r² =̺²+w52+w62. Then

5To be precise the fluctuations correspond to the mesons with spins 0, 1/2 and 1 [38, 69].

36 Chapter 2. The AdS/CFT correspondence

the metric reads

ds² = r²

R²d~x² +R²

r²(d̺²+̺²dΩ32

+ dw52

+ dw62

), (2.72)

where ~x is a four vector in Minkowski directions 0,1,2,3 and R is the AdS radius. The coordinater is the radial AdS coordinate while ̺ is the radial coordinate on the coincident D7-branes. For a static D7 embedding with vanishing field strengthF on the D7 world volume the equations of motion are

0 = d d̺



 ̺³ q

1 +w^′₅²+w₆^′2 dw_5,6

d̺



 , (2.73)

wherew_5,6denotes that these are two equations for the two possible directions of fluctuation.

Since (2.73) is the same type of equation as for the motion of a supergravity field in the bulk which was considered in (2.66), also the solution takes a form resembling (2.67) near the boundary

w5,6 =L+ c

̺² +. . . , (2.74)

with L being the quark mass acting as a source and c being the expectation value of the operator which is dual to the field w_5,6. While ccan be related to the scaled quark conden-satec∝ hqq¯ i(2πα^′)³.

If we now separate the D7-branes from the stack of D3-branes the quarks become massive and the radius of the S³ on which the D7 is wrapped becomes a function of the radial AdS coordinate r. The separation of stacks by a distance L modifies the metric induced on the D7 P[g] such that it contains the term R²̺²/(̺² +L²)dΩ32. This expression vanishes at a radius̺² =r²−L² = 0such that theS³ shrinks to zero size at a finite AdS radius.

Fluctuations about thesew5 andw6 embeddings give scalar and pseudoscalar mesons. We take

w5 = 0 + 2πα^′χ , w6 =L+ 2πα^′ϕ (2.75) After plugging these into the effective action (2.71) and expanding to quadratic order in fluc-tuations we can derive the equations of motion forϕandχ. As an example we consider scalar fluctuations using an Ansatz

ϕ=φ(̺)e^i~k·~xYl(S³), (2.76) whereYl(S³)are the scalar spherical harmonics on theS³,φsolves the radial part of the equa-tion and the exponential represents propagating waves with real momentum~k. We additionally have to assume that the mass-shell condition

M² =−~k² (2.77)

is valid. Solving the radial part of the equation we get the hypergeometric function φ ∝ F(−α, −α+l+ 1; (l+ 2); ^−̺_L2²)and the parameter

α=−1− q

1−~k²R⁴/L²

2 (2.78)