G. Pradisi - Non-perturbative...
Non-perturbative transitions among
intersecting brane vacua
Gianfranco Pradisi
Universita' di Roma “Tor Vergata” and
INFN Sezione di Roma “Tor Vergata”
C. Angelantonj, C. Condeescu, E. Dudas, G.P., arXiv:1105.3465 [hep-th]
Stockholm, june 10th, 2011
NORDITA Nordic Institute for Theoretical Physics
G. Pradisi - Non-perturbative...
OUTLINE
Brane Recombination
Geometry
6-d Supersymmetric vacua
6-d BSB vacua
Recombination vs Higgsing
Recombination in the bulk: the effective potential
Stability of BSB vacua under recombination
Classical solutions and brane transmutation
4-d examples
Conclusions
G. Pradisi - Non-perturbative...
Brane Recombination
Homology of the two-torus
Orientifolds
Intersection form
for a four-torus
D.Cremades
L. Ibanez
F. Marchesano
M. Cvetic
G. Shiu
A. Uranga
R. Blumenhagen
L. Goerlich
T. Ott
G. Pradisi - Non-perturbative...
Orbifold homology
bulk 2-cycles
Intersection form
exceptional (collapsed) 2-cycles
localized at the fixed points
A generic 2-cycle
fractional
2-cycles
D.Diaconescu
M. Douglas (1997)
J. Gomis
G. Pradisi - Non-perturbative...
Susy orientifold vacua
2-cycles wrapped by O7-planes
We focus on factorizable cycles
complex charges
Invariant cycles where
A vacuum is determined by the wrapping numbers
and (at least) one crossed fixed point (the origin)
G. Pradisi - Non-perturbative...
Reps and Multiplicities of charged hypermultiplets in T4/Z2
Stack a The intersection numbers determine the multiplicities
G. Pradisi - Non-perturbative...
Tadpole cancellation conditions
untwisted
Twisted (automatic)
Brane Recombination
(RR charge conservation)
Example: U(16) x U(16) [BSGP] modelM. Bianchi
A. Sagnotti (1990)
E. Gimon
J. Polchinski (1996)
G. Pradisi - Non-perturbative...
(Partial) recombination to a single additional factorizable stack
A first solution: complete brane recombination
R. Blumenhagen
V. Braun
B. Koers (2002)
D. Luest
G. Pradisi - Non-perturbative...
a second solution: partial brane recombination
4 D7 branes with all the D7' branes
a model without D5-branes
C. Angelantonj
I. Antoniadis (1999)
E. Dudas
A. Sagnotti
G. Pradisi - Non-perturbative...
Decomposition of representations
This suggests that the Higgs mechanism captures
the brane recombination for the massless spectra
First Solution
vev
G. Pradisi - Non-perturbative...
Second Solution
Here the Higgs mechanism is more involved
One needs a multiple step procedure and also the introduction of suitable
Discrete Wilson Lines.
The conjecture is: brane recombination is at work. There should exist a way to realize
the transition in a single step, not captured by the Higgs mechanism
G. Pradisi - Non-perturbative...
Two step procedure
Add Wilson lines to distribute the branes for U(16)a U(12) x U(4) and U(16)b U(4)4
then give vev's to the scalars to break U(4)5 U(4)diag
G. Pradisi - Non-perturbative...
BSB orientifold vacua
2-cycles wrapped by O7-planes
real charges
Invariant cycles where
s revert the twisted sectors projection
Exotic O-plane: T>0 , Q>0 (It requires anti-D-branes…)
I. Antoniadis
E. Dudas (1999)
A. Sagnotti
G. Pradisi - Non-perturbative...
fractional branes now combine into physical brane charged wrt twisted forms
• the action of orbifold on the Chan Paton charges affects the multiplicities
• D-branes wrapping the same O-plane cycles give rise to O and USp groups
• in d=6 supersymmetry is necessarily broken (in the open sector)
the invariant cycles have now components also along the exceptional cycles
G. Pradisi - Non-perturbative...
G. Pradisi - Non-perturbative...
Additional constraints are required for Brane Recombination
In order to be a consistent cycles:
• the twisted charges must combine to be non-zero on four collapsed cycles
• the twisted and the untwisted cycles must be compatible
This condition supplements the RR charge conservation in brane
recombination
G. Pradisi - Non-perturbative...
the SO(16)2 x Usp(16)2 model8 copies of four stack of (orthogonal) physical branes:
a first solution: complete brane recombination
The resulting cycles seems to be of a "bulk" type. The correct interpretation is as a pair
of fractional cycles providing a U(8) x U(8) gauge group
One cannot combine a pair of them. At least three different stacks must participate
(unless one moves them in the bulk, see later)
Anti-D-branes
I. Antoniadis
E. Dudas (1999)
A. Sagnotti
G. Pradisi - Non-perturbative...
a second solution: partial brane recombination
A choice consistent with all the constraints is as follows
The final configuration is
where
(a model without anti-D5-branes)
C. Angelantonj
I. Antoniadis (1999)
E. Dudas
A. Sagnotti
The gauge group is SO(16) x SO(8) x U(4)
G. Pradisi - Non-perturbative...
decomposition of representations
the first solution
vev
Vector
Multiplet
G. Pradisi - Non-perturbative...
Too many states stay massless
1
The other fields acquire a mass thanks to couplings
Fermions :
Bosons:
G. Pradisi - Non-perturbative...
Recombination in the BulkIn the presence of coincident images, one can also move the branes in the bulk
Do Recombination and Displacement Commute?
1. Recombination of the fractional branes
vev
superHiggs
G. Pradisi - Non-perturbative...
2. Recombination of the bulk branes
?
No way to get 8 L fermions in 28
vev
G. Pradisi - Non-perturbative...
The 1-loop Effective Potential
One-Loop effective potential (free energy) for vev's of positions and WL's
Better the transverse channel
G. Pradisi - Non-perturbative...
• a divergent tadpole contribution
• divergent when branes are on top of the O-planes
For only R1 much bigger than alpha' one gets
G. Pradisi - Non-perturbative...
One should evaluate
The effective action
string frame
Einstein frame
expanding
G. Pradisi - Non-perturbative...
Is it meaningful?
• massive state contributions are suppressed
• the tadpole is subtle but we know from similar calculation that the
resummation produces no corrections to the gaugino mass E. Dudas
M. Nicolosi
G.P. (2004)
A. Sagnotti
G. Pradisi - Non-perturbative...
Classical Solutions
and Brane Transmutation
Now we go back to the T-dual D9-D5 system
1. The two-dimensional case (not really D5-D9)
We want to describe solutions in which the
D5 brane dissolves into the D9 brane
The vev of the scalars can be modeled by
a Fayet-Iliopoulos term in the effective action
It corresponds to a d=6 SYM compactified on a two-torus
N. Arkani-Hamed
T. Gregoire (2001)
J. Wacker
G. Pradisi - Non-perturbative...
Localized source
The solution generates, as expected, a constant flux
eom
Ansatz
Zero-mode of the Laplacian
Choosing
G. Pradisi - Non-perturbative...
2. The four-dimensional case (D5-D9)
eom
Ansatz
Solution
G. Pradisi - Non-perturbative...
3. A closer look
violates anti-self-duality
locally
A constant magnetic field
!!!
G. Pradisi - Non-perturbative...
4. Can non-linear corrections soften the singularity?
NO
Using the Ansatz in the Born-Infeld eom
and looking for perturbative corrections
Singularity not affected. Large distance ok. [see ]
M. Billo'
M. Frau
A. Lerda (2005)
S. Sciuto
G. Vallone
4. Can higher derivative terms soften the singularity?
Hopefully yes! however
The abelian part of the SU(2) Instanton solution
cannot be the constant magnetic field !
G. Pradisi - Non-perturbative...
Four dimensional examples
Homology similar to the 6d case
bulk 3-cycles
x x
G. Pradisi - Non-perturbative...
exceptional 3-cycles
generic 3-cycle
O6-planes
discrete torsion
Our choice
Can we connect susy and non-susy models?
G. Pradisi - Non-perturbative...
1. a BSB model with orthogonal 6-branes/antibranes:
gauge group
2. a susy model with rotated 6-branes/antibranes:
gauge group
3. another susy model with rotated 6-branes/antibranes
gauge group
C. Angelantonj
I. Antoniadis
D'Appollonio (2000)
E. Dudas
A. Sagnotti
E. Dudas
C. Timirgaziu (2005)
P. Camara
C. Condeescu (2010)
E. Dudas
M. Lennek
G. Pradisi - Non-perturbative...
• 1. 2. OK
• 1. 3. Not allowed. There are not enough D6 in the h-direction.
One possibility is to add brane-antibrane pairs.
• 2. 3. ???
Higgsing?
• alpha' corrections needed: BI?
• What are these bound states in the LEEA?
G. Pradisi - Non-perturbative...
Conclusions
Brane Recombination is a mechanism that connects (non-perturbatively)
orientifold vacua differing for the brane configuration. It works also for
vacua that are non supersymmetric but classically stable (BSB).
At the level of effective action, in the simplest supersymmetric cases, it is
captured by an Higgs mechanism. In the non-susy cases the low energy
description is subtle and requires additional ingredients.
It is T dual to inverse "small-instanton" transition (brane transmutation)
but the magnetic field flux is due to finite volume effects.
It connects non-perturbatively non-susy to susy orientifold vacua.
Thank you