Miscegenation and Cultural Hybridity

t1 + iF 2π



=

N

X

j=0

c_{N −j}(F )t^j , (2.2.20)

where the group G is U (N ) (or any other N dimensional compact group) and F can be regarded as an N × N matrix, forgetting that it is a 2 form. The coefficients of the expansion are the Chern classes and can be easily computed.

The Chern character has an expansion ch(F ) = N + c1(F ) +1

2(c1(F )²− 2c2(F )) + . . . + 1

n!c_N(F ) . (2.2.21) The first class is c1, the class Tr F² is actually the combination c²₁− 2c2, the N th class is just det F .

There are also anomaly terms related to the curvature. They topopogical invariants built from powers of the Riemann tensor R^λ_µνρσ. Since we will not need them in the following we refer the reader to [52].

2.2.3 p–brane solutions

We now would like to present an aspect that apparently has not much in common with D–branes, but it will reveal to be strictly related to them, resulting funda-mental in the derivation of the AdS /CFT correspondence: p–brane solutions to supergravity. It is better to take a few steps back and talk about charged black holes, also known as Reissner–Nordstr¨om black holes.

6Which is the set of closed form modulo exact forms, equipped with the sum as a group structure.

The Reissner–Nordstr¨om black hole is a spherically symmetric solution to the Einstein– The parameters of the solution are the total mass M of the black hole and the electrical charge Q. The charge is found obviously integrating the electric field over a sphere at infinity and dividing by its area. In more general terms we can express this condition in the language of differential forms

Q = lim

Also, keep in mind that F = dA and the electromagnetic coupling is given by ∼ QR dtA.

The solution is given by

ds²= −

where Ω2 is the line element of S². The limit Q → 0 is the Schwarzschild metric. The electric field is simply the point charge:

F = Q

r²dt ∧ dr . (2.2.25)

If the sign of M − Q is negative we do not have null surfaces (i.e. surfaces with a normal vector n^µ of null length nµn^µ = 0), which means that there is no horizon. We exclude this case because it is believed that no naked singularities exist in nature (r = 0 is a singularity of this solution, a “naked” singularity is a singularity which is not surrounded by an event horizon). If M − Q is non negative then we have two null surfaces (one if M = Q)

r = r_±, r_±= M ±p

M²− Q². (2.2.26)

The situation concerning M = Q is very interesting: in this case a bound called Bo-gomol’nyi –Prasad –Sommerfield (BPS) bound is saturated and this means that the symmetry of the solution is enhanced. We will not prove here this statement, the argu-ment consists in finding a Killing spinor^aof the metric. Some evidence of this additional symmetry can be seen in the near horizon limit, choosing R = r − Q and taking R → 0 we have part). This space has a much larger symmetry than regular Minkowski space.

aKilling spinors are the spinorial generalization of the concept of Killing vectors. A Killing vector is the generator of a symmetry of the metric under a certain group of diffeomorphisms.

As we have showed in Section 2.1, the massless excitations in string theory

always contain a graviton. However, the theory describing the degrees of freedom at low energies is not pure Einstein gravity because the metric interacts with the p forms and with a spin 3/2 field (the gravitino). The specific form of these couplings is fixed by a local supersymmetry, this is why the theory is called Supergravity. Type IIA and Type IIB superstring theory reduce, at low energies, to Type IIA and Type IIB supergravity respectively. A possible way to build these theories from the world sheet action is to consider the Weyl invariance of the Polyakov action. First of all one has to rewrite the Polyakov action with the backreaction of the fields gµν, Bµν and φ (respectively the metric, the Kalb–

Ramond and the Dilaton arising from the massless NSNS sector). It takes the form

where ε^αβ is the antisymmetric tensor and R⁽²⁾ is the Ricci scalar in the two dimensional worldsheet. Now the theory is not automatically invariant under Weyl rescalings, but we can write the beta functions of gµν, B_µν and φ, regarding them as couplings of the action SP. The equations

β(G) = µ∂g(µ)

encode the Weyl invariance of the Polyakov action. It turns out that they can be obtained as a minimization of a Lagrangian, which is precisely the Lagrangian of Supergravity. The bosonic parts of the actions of Type IIA and Type IIB supergravity are, respectively,

where we have defined for brevity

The constant κ10 is the Newton constant, related to the string tension as

2κ10 = (2π)⁷α⁰⁴. (2.2.33)

For Type IIA we have the following definitions

F_(p) = dC(p−1), Fe₍₄₎ = dC(3)− C₍₁₎∧ F₍₃₎ . (2.2.34) While for Type IIB we have

F_(p) = dC(p−1), H₍₃₎ = dB(2) , Fe₍₃₎ = F(3)− C₍₀₎H₍₃₎ , Fe₍₅₎ = F(5)− 1

2C₍₂₎∧ H₍₃₎+ 1

2B₍₂₎∧ F₍₃₎ , (2.2.35)

and eF₍₅₎ must satisfy a self–duality condition ^?Fe₍₅₎= eF₍₅₎.

It is interesting to mention that the theory Type IIA can be obtained as a dimensional reduction of eleven dimensional supergravity over a circle of radius g_sl_s (gs being the vacuum value of the dilaton φ). The bosonic part of eleven dimensional supergravity is where F(4) = dC(3) and κ11 is the eleven dimensional Newton constant.

The solution showed in the insert is the forerunner of many other solutions is Supergravity⁷. In fact the following generalization has been studied (see refs. 94, 95 of [52]): a solution rotationally symmetric on a 9−p dimensional spacelike sub-space of R^1,9, with a RR C(p+1)form acting as an electric source and a dilaton ap-pearing in the Einstein–Hilbert coupling in the standard way SEH ∼R √g e^−2φR.

The gravitino and the Kalb–Ramond are set to zero in these solutions. The solution, called 10 dimensional black p–brane solution, reads

ds² = Z_p^−1/2(r) −K(r)dt²+

The parameter rH is the horizon radius and the other parameters are given by

Here N as a free parameter representing the units of charge of the black brane.

The charge is given, in analogy as before, by the following condition 1

(2π)^7−p(α⁰)^7−p2 Z

S^8−p

?dC(p+1)= αpN , (2.2.40)

where S^8−p is a sphere surrounding the black p–brane (here the normalization of the RR form is the standard one, in the following we will use a different normal-ization where the unpleasant α⁰ and π factors disappear). The BPS condition analogous to M = Q here is realized by imposing αp = 1. The physical inter-pretation of this class of solutions is very interesting: it is the warped geometry in presence of a p dimensional extended object in the origin of space charged under N units of a RR p + 1 form, or simply called p–brane. We argued before that D–branes do warp space time and they are charged under RR forms via the standard coupling R

DpC_(p+1) on the p + 1 dimensional world volume. So this solution is a good candidate to study, in the low energy limit, the behaviour of space time in presence of D–branes. The charge N is interpreted as the number of D–branes. In the same spirit of the Reissner–Nordstr¨om solution we would like to study the near horizon limit of this metric. Sadly for generic p the horizon located at r = 0 shrinks to a point and it is a singularity, however for p = 3 a special feature appears: the horizon is a sphere with finite volume (there is a cancellation between the divergent warp factor Zp and the sphere metric ∝ r²).

Not by chance the metric in this limit is again an AdS metric times a compact manifold, namely AdS5× S⁵. This feature is not only interesting as it is: it fur-nishes a bridge between string theory/D–branes and classical gravity and it will be crucial in the formulation of the AdS /CFT correspondence, as we shall see shortly.