Towards the Double Field Theory for Double D-branesweb.phys.ntu.edu.tw › string › files2010Mar › 20111125Dai.pdf · the one-loop beta functions from the doubled worldsheet action

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

� “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.

� 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


� 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∂


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∂

� 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~


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).

Page 6

X~

T-duality as O(n,n;ZZZZ) transformations

[Giveon, hep-th/9401139 ]

� The Hamiltonian


� In the basis of (wi,ni), the Hamiltonian can be written as

Page 7

� 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:

∈


� Equivalently, define the nonlinear action of h on a n× n matrix E=B+G:

→ Buscher’s rule: mix G and B

Page 8

� 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


� 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.


� 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.


called a T-fold.

Page 11

� 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:


- 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.

Page 12

� 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.


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.

Page 13

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 .

⊂


� invariant under O(n,n;Z) ⊂ GL(2n):

for h ∈ O(n,n;Z)

(i.e. Invariant under T-duality transformation ⊂ diffeomorphism of T2n )

Page 14

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.


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

Page 15

� 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


⊂

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.

Page 16

� 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~


� 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


� The D-brane worldvolume effective action for the massless sector is the

DBI action:

Page 18

)'2(det ~ 1

ababab

p

DBI FBGedS παξ ++−Φ−+

∫ µννµ

µννµ

BXXB

GXXG

baab

baab

∂∂=

∂∂=

� 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


arises the boundary conditions

Page 19

(Neumann) (Dirichlet)

(SD constraint)

� 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.


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


� 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.


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.

Page 22

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~

,~


or (D2-brane in{ })

One can T-dualize {X,Y} and { } s.t. the D2 in physical space is produced

by

Page 23

YX ,

YX~

,~

Similar case for the D1-D1 pair.

One of the projectors satisfying all three conditions for B 0 is

Examples: n=2 projector

≠

which implies the boundary condition in physical space


This is similar to the Neumann b.c. with B-field, but with an extra factor b1/(1-a1)

enhancing B.

Page 24

What’s this D-brane configuration? Get back to the double space Dirichlet

condition:


The physical B-field projected from double space is

= Double geometry properties + orientation of the double D-brane!

Page 25

1

1

1 a

bB

−+

� 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]


� Alternatively, Hull and Zwiebach derive the closed string double field theory

from T-dual covariant closed sting field theory.

Page 26

[0904.4664, 0908.1792, 1003.5027,

1006.4823]

� Quick glance at Hull-Zwiebach‘s closed string double field theory:

Action:

[1006.4823]

with the strong constraint (from level matching condition):


with the strong constraint (from level matching condition):

(i.e. all fields only depends on physical coordinates)

Page 27

� This action is invariant under

where [ , ]C is call the C-bracket, which does not satisfy Jacobi identity but

some specific Jacobiator relation.


(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.

Page 28

[0908.1792]

� 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]


double D-brane Effective action

� Question: how to quantize the self-dual double fields?

Page 29

� 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]


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.

Page 30

[FJ, PRL59,1873,’87; Tseytlin‘91]

[Berman et al., 0708.2267]

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


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. .

Page 31

� 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


that

� The boundary action becomes

Page 32

� 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


where

where Fab arise from Aa = Aa(Xa), or

Page 33

� 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]


e.o.m:

b.c:

Page 34

� The Neumann Green’s function solution is given by

wherewhere

The result is of the same form as that for the conventional open string


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}.

Page 35

� 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


ab

is taken within the extended Neumann subspace.

Page 36

� 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:


� 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.

Page 37

� 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]


� This may cause more complicated B-field dependence in the reduced

action from double formalism than the standard DBI.

Page 38

� 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.


Thank YouThank YouThank YouThank You


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Towards the Double Field Theory for Double D-branesweb.phys.ntu.edu.tw › string › files2010Mar › 20111125Dai.pdf · the one-loop beta functions from the doubled worldsheet action