Towards the Double Field Theory for Double D-branesweb.phys.ntu.edu.tw › string › files2010Mar...
Transcript of Towards the Double Field Theory for Double D-branesweb.phys.ntu.edu.tw › string › files2010Mar...
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Towards the Double Field Theory
for Double D-branes
Shou-Huang Dai
(NTNU)
Based on arXiv: 1107.0876 by C Albertsson, SHD, PW Kao, FL Lin
2011/11/25 @ NTU String Seminar
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� “Double“ means the string coordinates and their T-dualized ones are both
included in the background grometry. The background becomes a
“doubled space“.
� T-duality (or, O(n,n) transformation) is realized as a symmetry.
� Goal: to derive the O(n,n) invariant effective action for the D-brane in the
doubled formalism.
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doubled formalism.
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� Introduction
Outline of this talk
� Doubled formalism of string worldsheet action, and double D-branes
� Effective action for the Double D-branes
� Discussion, and open issues
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� Dualities relate different theories:
- AdS/CFT: gravity QFT
- S-duality: strongly weakly coupled theories
Introduction
- S-duality: strongly weakly coupled theories
- T-duality: IIA string theory on a circle with radius R
IIB string theory on a dual circle with radius 1/R
� Question: Is it possible to construct a unified theory under a duality?
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� For T-duality, such a theory implies, at very fundamental level, the
spacetime may not be treated as fundamental constituents.
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� Important as it relates different types of string theories and their excitations.
T-duality
� For closed strings, T-duality swaps the KK-momentum ni on a circle with
radius R to the winding mode wi on the dual circle with radius 1/R.
� For open strings, T-duality swaps the Dirichlet/Neumann b.c. to Neumann/
Dirichlet b.c. of the dual string. This can be seen more easily in B=0 case,
where T-duality swaps to , and to . Thus T-duality relates Xτ∂− X~
σ∂ Xσ∂− X~
τ∂
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where T-duality swaps to , and to . Thus T-duality relates
D-branes with different # of dimensions.
� The open string boundary gauge field A is dualized to D-brane position X
(D-brane worldvolume Higgs) in the dual theory.
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Xτ∂− Xσ∂ Xσ∂− Xτ∂
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� X = (XL + XR)/2 as the string coordinate on a circle with radius R
X~
T-duality
= (XL - XR)/2 as the coordinate on the dual circle with radius 1/R.
� The theory which includes X and is under the chiral constraint such
that XL is left-moving and XR is right-moving:
dXL = *dXL , dXR = -*dXR
so that the physical degrees of freedom are not doubled.
X~
X~
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so that the physical degrees of freedom are not doubled.
� It’s natural to consider vertex operators ekXL and ekXR in the string theory,
therefore to include X and its dual . The first one to consider such theory
is Tseytlin (Nucl.Phys.B350 (1991) 395-440 & Phys. Lett. B242, 163, ’92).
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X~
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T-duality as O(n,n;ZZZZ) transformations
[Giveon, hep-th/9401139 ]
� The Hamiltonian
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� In the basis of (wi,ni), the Hamiltonian can be written as
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� Taking into account both the L and R sectors, the moduli space for
toroidal compactifications is isomorphic to O(n,n;Z)/ O(n;Z) × O(n;Z),
which preserves the “Lorentz length“ , for n , wi Z. ∈which preserves the “Lorentz length“ , for ni, wi Z.
But in general the “Euclidean length“ is not preserved.
� H is O(n,n) covariant:
∈
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� Equivalently, define the nonlinear action of h on a n× n matrix E=B+G:
→ Buscher’s rule: mix G and B
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� T-duality on a single cycle:
Examples of O(2,2;ZZZZ) transformations
1
2,12,1
−→ RR
� Shift of B-field: B = B + N
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� Geometric GL(n,Z) transformation of Tn:
where A is a 2x2 matrix with integer entries. This correspond to the
basis change on Tn and doesn’t change the physics.
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� Since T-duality mixes G and B, it generates non-geometric string theory
background. As a result, the T-duality-patched background is locally
Hull‘s doubled formalism
geometric but globally not.
� The effect is same as regarding T-duality as a transition function between
patches of the background manifold. This non-geometric background is
called a T-fold.
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called a T-fold.
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� Toy example of T-fold:
- Start with T3 with H-flux: (x,y,z) ~ (x+1,y,z) ~ (x,y+1,z) ~ (x,y,z+1)
[hep-th/0211182, 0508133, 0602025]
- T-dualizing along x-direction yields a twisted T3:
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- T-dualizing along y-direction yields a characterized by a non-geometric
Q-flux = N:
i.e. the volume of T2 shrinks as one moves along S1 in z.
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� Hull geometrizes the T-fold on the Tn torus fibration over a base manifold
N by doubling the torus bundle to T2n which contains the original torus and
the T-dualized one. → doubled geometrythe T-dualized one. → doubled geometry
� G and B are represented geometrically by a “generalized metric”, and thus
become part of the doubled geometry.
� The O(n,n;Z) transition functions between patches in the base space N is
also the diffeomorphism of the T2n fibre.
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also the diffeomorphism of the T2n fibre.
� A self-duality constraint is imposed to eliminate half of d.o.f.’s and reduce
to the physical space.
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Double field string worldsheet model [Hull, hep-th/0406102, 0605149]
� H is a 2n x 2n, symmetric “generalized metric“; X is a 2n-dimensional vector
whose components parameterize a T2n fibred over N.
� the double torusT2n contains the original n-torus with coordinates and
the dual n-torus with coordinates .
⊂
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� invariant under O(n,n;Z) ⊂ GL(2n):
for h ∈ O(n,n;Z)
(i.e. Invariant under T-duality transformation ⊂ diffeomorphism of T2n )
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2n � One can define a natural O(n,n;Z) invariant metric LIJ on the T2n fibre:
� Impose self-duality constraint at the on-shell level to eliminate half of d.o.f.
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self-duality constraint: chiral constraint, half of the d.o.f.’s are left-moving
while the other half are right-moving. GL(2n) broken to O(n,n;Z) due to LIJ .
� manifest worldsheet Lorentz symmetry
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� Choose a polarization (i.e. a local choice of the physical subspace Tn ⊂ T2n)
such that:
This is equivalent to picking out a GL(n) ⊂ O(n,n) and breaks O(n,n) down
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⊂
to GL(n) × GL(n).
� After choosing a polarization and imposing the self-duality condition, the
O(n,n) transformation reduces to the standard Buscher’s rule.
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� In fact, Tseytlin already found the expressions for L and H in his pioneering
work (Nucl. Phys. B350, 395, ’91 & Phys. Lett. B242, 163, ’92) .
� Tseytlin considered a Floreanini-Jackiw action for the chiral scalars X & .
This action doesn‘t have manifest worldsheet Lorence symmetry. By
requring on-shell Lorentz invriance, Tseytlin found the explicit expression for
L, H, and the condition:
X~
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� a p-dimensional object to which the end points
of the open strings attach.
Dp-branes in string theory (conventional)
N
Open
String
(part)
of the open strings attach.
� Defined by the open string boundary conditions
(Neumann)
or (Dirichlet)
N
D
D-brane
00
=∂=σ
µσ X
00
==σ
µδX 00
=∂=σ
µτ X
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� The D-brane worldvolume effective action for the massless sector is the
DBI action:
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)'2(det ~ 1
ababab
p
DBI FBGedS παξ ++−Φ−+
∫ µννµ
µννµ
BXXB
GXXG
baab
baab
∂∂=
∂∂=
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� Defined by O(n,n) covariant projectors ΠN and ΠD on the worldsheet
D-branes in double field theory (Double D-branes)
[Hull, hep-th/0406102]
[Lawrence et al., hep-th/0605149]
boundary:
� From the boundary term of double field worldsheet action variation
arises the boundary conditions
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arises the boundary conditions
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(Neumann) (Dirichlet)
(SD constraint)
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� This implies ΠN and ΠD each projects out n-dimensions out of 2n-dimensional
double space.
� The double D-brane is an n-dimensional object in the 2n-dimensional doubled
space. The conventional sense of a Dp-brane is in fact the p-dimensional
intersection of the double D-brane with the physical Tn ⊂ T2n .
� “T-dualizing a Dp-brane” is to change the orientation of the double D-brane in
the double space by O(n,n) rotation.
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the double space by O(n,n) rotation.
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� Projector condition:
Conditions for the projectors
� Null conditions: b.c should be compatible with the self-duality constraint
� Can also impose the orthogonality condition
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� Can also impose the orthogonality condition
i.e. the Neumann direction can be completely specified by N-b.c.
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� D-brane configurations are specified by ΠN (or equiv. ΠD), by solving those
conditions.
Comments
conditions.
� The boundary conditions specified by ΠN and ΠD describe a T-dualized D-
brane pair.
� There are many solutions (i.e. D-brane configurations) whose physical
interpretations are subtle.
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interpretations are subtle.
� Orthogonality is imposed for physical reasons. It simplifies the calculation,
but also excludes some simple configuration such as {D2 / D0} pair with
non-vanishing B-field.
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D0-D2 pair: Consider the Dirichlet projector
Examples: n=2 case D-brane pair
which gives rise to the Neumann b.c.
after imposing the self-duality constraint the b.c. becomes
(D0-brane in {X,Y})
or (D2-brane in{ })YX~
,~
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or (D2-brane in{ })
One can T-dualize {X,Y} and { } s.t. the D2 in physical space is produced
by
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YX ,
YX~
,~
Similar case for the D1-D1 pair.
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One of the projectors satisfying all three conditions for B 0 is
Examples: n=2 projector
≠
which implies the boundary condition in physical space
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This is similar to the Neumann b.c. with B-field, but with an extra factor b1/(1-a1)
enhancing B.
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What’s this D-brane configuration? Get back to the double space Dirichlet
condition:
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The physical B-field projected from double space is
= Double geometry properties + orientation of the double D-brane!
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1
1
1 a
bB
−+
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� The doubled theory is equivalent to the standard worldsheet theory at the
on-shell level once the self-duality condition is imposed.
Effective action of double D-branes
[Tseytlin, Nucl. Phys. B350, 395, ’91 & Phys. Lett. B242, 163, ’92; Hull, hep-th/0406102 ]
� The equivalence hold also at the quantum level:
the one-loop beta functions from the doubled worldsheet action of the
closed can be reduced to those from the standard worldsheet formalism.
� Alternatively, Hull and Zwiebach derive the closed string double field theory
[Berman et al., 0708.2267]
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� Alternatively, Hull and Zwiebach derive the closed string double field theory
from T-dual covariant closed sting field theory.
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[0904.4664, 0908.1792, 1003.5027,
1006.4823]
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� Quick glance at Hull-Zwiebach‘s closed string double field theory:
Action:
[1006.4823]
with the strong constraint (from level matching condition):
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with the strong constraint (from level matching condition):
(i.e. all fields only depends on physical coordinates)
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� This action is invariant under
where [ , ]C is call the C-bracket, which does not satisfy Jacobi identity but
some specific Jacobiator relation.
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(C-bracket is an O(n,n) covariant extension of the Courant bracket for
doubled fields)
� NB: Hull‘s double field worldsheet action is NOT invariant under δξ.
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[0908.1792]
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� Question: What about the double field theory for the open strings?
� We are going to use background field method to calculate the 1-loop beta � We are going to use background field method to calculate the 1-loop beta
function for the boundary gauge coupling Sb of the open string.
� Assuming the worldsheet conformal symmetry holds at the quantum level
implies the beta function for Sb to vanish
E.O.M for the gauge field
[Callan et al., Nucl.Phys.B280,599,’87]
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double D-brane Effective action
� Question: how to quantize the self-dual double fields?
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� We are going to quantize a chiral-constrained theory.
It’s well-established by Pasti-Sorokin-Tonin (PST), by incorporating the self-
duality condition into the action via Lagrange multipliers non-linearly, and duality condition into the action via Lagrange multipliers non-linearly, and
introducing new gauge symmetry (PST symmetry) to gauge away the non-
chiral d.o.f.’s. As a result, only the fields obeying the chiral constraint are
physical. The physical degrees of freedom are not doubled.
� 2 options to quantize the PST action:
[PST, hep-th/9506109, 9509052, 9611100]
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Covariant: need to introduce ghosts to deal with the PST gauge symmetry.
Non-covariant: gauge-fix the auxiliary fields but break manifest Lorentz symm.
Floreanini-Jackiw (FJ) action
� We follow the non-covariant method.
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[FJ, PRL59,1873,’87; Tseytlin‘91]
[Berman et al., 0708.2267]
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Our model
� we choose a very simple doubled geometry: the doubled flat space, with
constant G & B.
� In Sb, AI contains gauge fields and their dual Higgs. But the Higgs are in the
SFJ Sb
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Dirichlet directions and never contribute in the boundary action.
� However, the unwanted Higgs coupling can be removed after inserting
and imposing the Dirichlet b.c. .
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� The Neumann-projected quantities have no inverse in the full double
space, but they do in the space projected out by Π (i.e Neumann
The reduced notation for the extended Neumann Subspace
space, but they do in the space projected out by ΠN (i.e Neumann
subspace N) and T. It is convenient to introduce a reduced notation for
both the spatial and temporal Neumann directions
where Ap and Xp
denote the Neumann components of AI and XI such
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that
� The boundary action becomes
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� The background field method:
and work in Euclidean worldsheet signature for convenience.and work in Euclidean worldsheet signature for convenience.
� The background field expanded action:
1st order terms in ξ: e.o.m and b.c. for the background fields
2nd order terms in ξ: e.o.m and b.c. for ξ, and the boundary interaction term
where
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where
where Fab arise from Aa = Aa(Xa), or
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� The boundary interaction term corresponds
to the only one-loop graph contributing to the
β-function, and gives rise to the counter term
� The propagator satisfies
e.o.m:
[Callan et al., Nucl.Phys.B280,599,’87]
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e.o.m:
b.c:
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� The Neumann Green’s function solution is given by
wherewhere
The result is of the same form as that for the conventional open string
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The result is of the same form as that for the conventional open string
worldsheet formalism, except that ours are double fields.
� Note that the inverse is taken within the extended Neumann subspace {Xp,T}.
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� In the end, the equation of motion for the double D-brane is to demand
vanishing β-function:
which can be derived from a DBI-like action:
by assuming that Fab also satisfy the Bianchi identity. Here the determinant
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ab
is taken within the extended Neumann subspace.
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� Manifest O(n,n) invariant master action is conjectured by recovering the
N- and D-projectors while the orthogonality condition is imposed:
Discussion and Open issues
where Voln[ΠD] denotes the O(n, n) covariant volume of the space of
Dirichlet projectors, and
� Reduction to non-double DBI:
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� Reduction to non-double DBI:
The double effective DBI-like action can be reduced to the
conventional DBI when B=0, but not in the B ≠ 0 case.
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� This may be due to that the B-field that appears in the doubled metric
does not necessarily coincide with the B-field in physical space. The
eventual physical B-field after we project our theory down from the eventual physical B-field after we project our theory down from the
doubled space is a combination of the doubled geometry properties (i.e.,
the component B of the doubled metric) and the orientation of double D-
branes.
� This may cause more complicated B-field dependence in the reduced
[0712.1026; 0806.1783; 0902.4032; 0904.0380]
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� This may cause more complicated B-field dependence in the reduced
action from double formalism than the standard DBI.
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� In the conventional worldsheet action, the boundary gauge transformation
for A is related to the bulk gauge transformation of B. But this is not clear in
the doubled formalism because O(n,n) mixes the gauge symmetry and the doubled formalism because O(n,n) mixes the gauge symmetry and
diffeomorphism.
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Thank YouThank YouThank YouThank You
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