Emergent geometry from String Theory.
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Emergent geometry from String Theory.
Based on:
P. Di Vecchia, M. Frau A. Lerda, I Pesando, R. R., S. Sciuto hep-th/9707068
S. Giusto, F. Morales, R.R. 0912.2270
W. Black, R.R., D. Turton 1007.2856
S. Giusto, R.R., D. Turton 1108.6331
Rodolfo Russo
Queen Mary - London
Torino, 28/10/2011
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 1 / 17Outline of the talk
String theory – world-sheet picture
I Emergent geometry on the world-sheet
String theory – space-time picture
I Black holes, as a prototypical example of emergent geometry
In string theory, (BPS) black holes are naturally constructed by
piling up different types of D-branes
Goal: start from a (BPS) D-brane configuration at gs = 0 and
derive the geometrical backreaction when gs 6= 0
Motivation: test a radical proposal by S. Mathur which requires
string effects to be relevant at the horizon scale
Various cases (with increasing degree of complication)
I Classical p-branes from boundary state (1/2 BPS case)
I 1/4 BPS cases: microscopic black-holes (S 1, A = 0 in 5D)
I 1/8 BPS cases: macroscopic black-holes (S 1 and A 6= 0 in 5D)
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 2 / 17String theory – world-sheet
The vibrations of a (relativistic) string propagating in empty (flat)
space are determined by the free 2D wave equation
The world-sheet
A closed string
spanned by the
at fix time τ0 :
closed string mo-
X µ (τ0 , σ)
tion
In quantum mechanics, the normal modes are mapped into the
creation and annihilation operators if a harmonic oscillator
When strings interact, the world-sheet can be characterised in
terms of the classical theory of Riemann surfaces (number of
handles, boundaries, punctures, . . . )
I A first example of emergent (2D) geometry!
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 3 / 17Strings can also be open: the possible positions of the open string
endpoints define a (new) object in the theory: the D-branes
D-branes are dynamical objects, i.e. they have finite energy and
charge (densities)
For open strings, left and right moving modes are not independent
The identification is fixed by the boundary conditions at the string
endpoints. In this talk, we will be interest in b.c. of the type:
α̃nµ = −R µν α−n
ν
+ ... ,
where α/α̃ are the left/right moving modes and R is a matrix
The explicit form of the reflection matrix R depends on the
particular type of D-branes on which the string world-sheet ends
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 4 / 17Classical p-branes
The simplest type of D-branes have a diagonal reflection matrix:
R ij = δji ( R ij = −δji ) in the Neumann (Dirichlet) case.
When gs 6= 0, the couplings between D-branes and closed string
states are non-trivial.
D-branes act as sources of gravitons (and other fields). These
couplings are captured by the following diagram
Neumann directions
W
δΓboun W
δW =
Dirichlet directions
String world-sheet Space-time picture
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 5 / 17A stack of N Dp-branes produces a large gravitational
backreaction (in the Einstein frame, r 2 = xi2 ):
p−7 p+1
ds2 = (H(r )) 8 ηαβ dx α dx β + (H(r )) 8 (δij dx i dx j ) ,
where H(r ) is a harmonic function in the 9 − p transverse space
7−p √ n+1
Rp 7−p gN(2π α0 )7−p 2π 2
H(r ) = 1 + , Rp = , Ωn = n+1 .
r (7 − p)Ω8−p Γ( 2 )
The large distance limit of the solution determines the conserved
charges of the D-brane configuration (energy, R-R charge).
The on-shell graviton/R-R 1-point functions are also related to the
same charges. Thus matching the supergravity and the string
results determines Rp in terms of α0 , gs , . . .
δΓbound (−1,−1)
= hWNSNS (k = 0)idisk
δWNSNS k =0
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 6 / 17In type II theories, the massless NS-NS sector can be described
by the vertex operators
(−1,−1)
W = Gµν cψ µ e−ϕ e
NSNS c ψeν e−ϕeeik ·x + ghost ,
where Gµν contains the graviton, the dilaton and the B-field
At the linearised level the sugra fields satisfy a harmonic equation
in the transverse space ∇2 φ = δΓbound /δφ
Thus we have φ = ∇−2 δΓbound /δφ , which diagrammatically is
d 9−p k
Z
Propagator
as
= gii (ki ) , gii (x ) = giias (k )eikx
as
(2π)9−p
1 δΓboun
1/k 2 gii (x ) ∼ ηii + giias (x ) + O (Rp /r )2(7−p)
Vp+1 δ gii
where g as is the O (Rp /r )7−p term in the r Rp expansion
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 7 / 17A 1/4-BPS configuration
Let us consider type IIB compactified on Rt × S 1 × R 4 × T 4 . Two
different D1-branes wrapped nw times on S 1 :
directions
directions
Dirichlet
Dirichlet
A 1/4 BPS D1-brane
with a null-like pulse
0 A 1/2 BPS D1-brane 2πR 0 2πR
wrapped on a S 1
The reflection matrix depends on fi (v ) (v = t + y ) determining the
D1’s position in the Dirichlet directions
1 0 0 0
µ 4|ḟ (V )|2 1 −4ḟi (V ) 0
R ν = ,
2ḟ (V ) 0
i −1l 0
0 0 0 −1l
where the indices are ordered (v , u = t − y , i)
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 8 / 17We can go beyond the conserved charges, by taking the closed
string momentum ki 6= 0 along the uncompact Dirichlet directions,
but keeping k 2 = 0. The string correlators now depend on f (v )!
ZLT
κ τ1 nw i
−i d v̂ e−iki f (v̂ ) G µν Rνµ (v̂ )
2LT
0
The Fourier transform gives f -dependent harmonic functions
ZLT ZLT
Q1 −ḟi d v̂ Q1 |ḟ |2 d v̂
ĥvi ≡ Ai = , ĥvv ≡K = ,
2κLT |x i − f i |2 2κLT |x i − f i |2
0 0
This is consistent with the type IIB supergravity solution of Callan
et al. and Dabholkar et al.
−3 1
ds2 = Hf 4 dv − du + Kdv + 2Ai dx i + Hf4 dx I dx I
The R-R fields are also derived in a similar way.
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 9 / 17Main features of the solution in the D1-P frame.
This is a 1/4-BPS solution for any f (v )
The solution is characterised by two charges: the R-R charge
(related to nw ) and the momentum charge (∼ np )
ZLT
gs np α0 nw
= Qp |ḟ |2 d v̂ .
R2 LT
0
The rotational symmetry (SO(4)) in the uncompact transverse
directions is broken. This reflects the nature of the source
(D-strings do not have transverse waves)
The dipoles in K , Ai are determined by disk amplitudes
A configuration with Ai = 0 and K = c/r 2 is a also a 1/4-BPS
solution. . . but it does not correspond to any D-brane bound state!
There is a singularity at x i ∼ f i related to the presence of a source.
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 10 / 17A different duality frame
With a U-duality transformation, we can map the D1 and
momentum charges into D5 and D1 charges
In this duality frame the solution takes a more complicated form.
For instance the (string frame) metric is
s
2 −(dt − A)2 + (dy + B)2 1
2 H1,f 2
ds = + (H1,f H5,f ) 2 dxi + dx ,
1
(H1,f H5,f ) 2 H5,f a
where A = Ai dxi and dB = − ∗4 dA.
This is not the naive superimposition of a D1 and D5 brane – the
two objects form a bound state
The profile f (v ) now determines the vev of the open strings
stretched between the D1 and the D5 branes
We can describe this vev only in perturbation theory
The couplings (δΓbound /δW ) are described by mixed disks
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 11 / 17The first new contribution in the r f expansion is captured by
Twist field Vµ
ϕ ϕ
Vµ = µA e− 2 SA ∆ , ¯,
Vµ̄ = µ̄A e− 2 SA ∆
δΓboun W SA is a Weyl spinor of SO (1, 5)
δW = D1 D5
acting on t , y , xi
W is a massless closed string state
Twist field Vµ̄
I Both RD1 and RD5 are simple and lead to the same result.
I The presence of twist/spin fields make the correlator less-trivial and
1
breaks the SO(4): µ̄A µB = vI (CΓI )[AB] + 3! vIJK (CΓIJK )(AB)
RL
I vtij is proportional to the profile moment fij = L1 0 dv ḟi fj
I Mixed disks diagrams capture the (new) 1/r 3 terms in the solution
If we calculate the couplings for gti and gyi in the same way as
before, we find the the leading order of the 1-forms A and B!
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 12 / 17Higher order terms in the expansion of Ai (∼ f n+1 ) should be
reproduced by disk diagrams with 2n twist fields
Non-linear terms, such as the components gij ∼ Ai Bj , are
reproduced by diagrams with disconnected borders
Twist field Vµ
W
(Super)gravity vertex
At large distances σ
W this amplitude is dominated
by degenerate world-sheets
which look like Two disk
diagrams
Twist field Vµ̄
i.e. we are solving perturbatively ∇2 φ + σφ2 = δΓbound /δφ
These expansions (in the number of twist fields/borders)
reconstruct the full non-linear gravity solution
The non-linear solution turns out to be regular and horizonless
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 13 / 17Three charge solutions from string theory.
We can combine the two ingredients seen so far and construct a
bound state of D1, D5 and momentum charges
The open strings stretched between the two types of D-branes
have a non-trivial vev vtij , vyij (we set vtyi = v ijk = 0 for simplicity).
Each D-brane has a non trivial pulse f (we take fD1 = fD5 , which
makes the string correlators simpler).
There is a nice interplay between world-sheet and space-time
properties: the superghost anomaly implies that the expansion in
v is actually an expansion in 1/r .
We calculate all disk 1-point functions with f exact, but up to O(vtij )
From this result, we derive the (large distance) gravitational
backreaction as before.
We find that all 10D IIB massless fields are non-trivial at O(1/r 4 )
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 14 / 17An example of the new couplings that are generated at O(1/r 4 ) is:
ZLT ZLT
ij
AD1−D5
gra ∼ ĥ vuil ḟj (v̂ ) + . . . , AD1−D5
B ∼ b̂ij vuil ḟj (v̂ ) + . . .
0 0
The solutions studied previously were in a subsector where B = 0
The 4d metric derived from AD1−D5
gra is conformally hyperkhaler at
the linear order in vuij
We checked that the disk amplitudes generate a 1/8-BPS
supergravity solution (at first order in v )
Is it possible to find a full non-linear 1/8-BPS solution where all
fields are present?
Yes! The solution is written as a (complicated) combination of
harmonic forms/functions
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 15 / 17For instance the metric is s
p
α 1 Z1 2
ds2 = √ − d t̂ 2 + Z3 d ŷ 2 + Z1 Z2 ds42 + ds 4 ,
Z1 Z2 Z3 Z2 T
−1
A2
dt + k
d t̂ = dt + k , d ŷ = dy + dt − + a3 , α = 1 −
Z3 Z1 Z2
2
ds4 is exactly hyperkhaler
the 1-form a3 (and a1 , b1 ) has self-dual field strength
there exists a 1-form a2 , with self-dual field strength, such that
dk + ∗4 dk = Z1 da1 + Z2 da2 + Z3 da3 − 2 A db1 ,
the scalars Zi (and A) satisfy
d ∗4 dZ1 = −da2 ∧ da3 , d ∗4 dZ2 = −da1 ∧ da3 ,
d ∗4 dZ3 = −da1 ∧ da2 + db1 ∧ db1 , d ∗4 dA = −db1 ∧ da3 .
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 16 / 17Conclusions and open problems
String amplitudes provide a powerful way to characterise the
couplings between D-brane bound states and closed strings
Disks diagrams contain all necessary information to derive the full
geometrical backreaction when gs 6= 0
Our current work aims to complete this programme for an
interesting (new) class of 1/8-BPS solutions
With the full non-linear solutions one could address questions
such as its regularity in the interior and its interpretation within the
AdS/CFT duality
This approach can be applied more generically, for instance to
four charges systems in 4D
Go beyond systems at equilibrium and describe the interaction
with elementary quanta see Di Vecchia’s talk
Rodolfo Russo (Queen Mary - London ) Emergent Geometry Torino, 28/10/2011 17 / 17You can also read
