Double Field Theory and strings at the self-dual radius Mariana - - PowerPoint PPT Presentation

• G. Aldazabal, S. Iguri, M. Mayo, C. Nuñez, A.Rosabal

Mariana Graña CEA / Saclay France In collaboration with Mainz, September 2015

arXiv:1509.xxxx

Double Field Theory and strings at the self-dual radius

Motivation

ls

R

Massless states: Bosonic closed string

gmn

Bmn

, dilaton R ≫ ls

→ gµν gµy gyy

vector scalar

→ Bµν Bµy

vector

g, B, U(1) × U(1) = ls

M 2 = 2 α0(N + ¯ N − 2) + p2 R2 + ˜ p2 ˜ R2

momentum # winding #

˜ R = α0

R

= √ α0

+

¯ N = 1 p = ˜ p = ±1

N = 0

2 vectors 2 scalars

¯ N − N = p˜ p

N = 1 ¯ N = 0 p = −˜ p = ±1

2 vectors 2 scalars

SU(2) × SU(2)

p = ±2

2 scalars

˜ p = ±2

2 scalars 9 scalars 1 scalar

Can we find the effective action using DFT ?

N = ¯ N = 0

describe the physics using DFT ?

Some easy math...

M = Md × S1 gµν Bµν →

d2

dof

6 vectors

→

6d

9 scalars

→

32 (d + 3)2

dim ⇥ O(d + 3, d + 3) O(d + 3) × O(d + 3) ⇤ = (d + 3)2

Outline

• Strings on S1
• Effective action from string theory
• DFT description
• Effective action from DFT
• “Internal double space”

String theory on S1

Momentum state for non-compact coordinate

eik(xL(z)+xR(¯

z))

x = xL + xR

Momentum state for compact coordinate

k 2 R ei(kLyL(z)+kRyR(¯

z))

kL,R = p

R

y = yL + yR ' y + 2πR

Momentum & winding state for compact coordinate

± ˜

p ˜ R

˜ y = yL yR ' y + 2π ˜ R

DFT

Md ⇥ S1 Md ⇥ S1 ⇥ ˜ S1

X(z, ¯ z) = xL(z) + xR(¯ z)

Y (z, ¯ z) = yL(z) + yR(¯ z) ˜ Y (z, ¯ z) = yL(z) yR(¯ z)

String theory on R

Massless states at M 2 = 2(N + ¯ N − 2) + p2 R2 + ˜ p2 ˜ R2

R = ˜ R = 1

α0 = 1

Vectors

¯ Nx = 1

• SU(2)L

Ny = 1

• Ny = 0

p = ˜ p = ±1 (kL = ±2) J3(z) = ∂yL(z)

J±(z) = e±2iyL(z)

: A±

µ

Ji(z) = P Ji

m z−(m+1)

P [Ji

m, Jj n] = m 2 ijm,−n + ✏ijkJk m+n

SU(2)L current algebra Vectors

• SU(2)R

Nx = 1 Ai → ¯ Ai

• Scalars (3,3) Nx = ¯

Nx = 0

(a) Ny = 1, ¯ Ny = 1 (b) Ny = 1, p = −˜ p = ±1 (¯ k = ±2) (c) ¯ Ny = 1, p = ˜ p = ±1 (k = ±2) (d) p = ±2, ˜ p = 0 (k = ¯ k = ±2) (e) p = 0, ˜ p = ±2 (k = −¯ k = ±2)

: M 33 : M 3± : M ±3 : M ±± : M ±⌥ V ∼ J3(z) · (¯ @XµeikX) V ∼ J±(z) · (¯ @XµeikX) Ji(z) → ¯ Ji(¯ z) V ij ⇠ JiJj eikX : Aµ A3 (gµy + Bµy) (gyy)

¯ N − N = p˜ p ,

Level-matching

yL → yR

Effective action from string theory

Computing 3-point functions <V V V> we read off

F i = dAi + ✏ijkAj ∧ Ak DµM ii = @µM ii + f ijkAj

µM ki + f ijk ¯

Aj

µM ik

L = R − 1 12HµνρHµνρ + 1 4F i

µνF iµν + 1

4 ¯ F i

µν ¯

F iµν

H = dB + Ai ∧ F i − ¯ Ai ∧ ¯ F i +1 4M ijF i

µν ¯

F jµν + DµM ijDµM ij − detM

Higgs mechanism

M ij → ✏ ij

33 + M 0ij

A± ¯ A±

acquire mass2

= ✏ M ±±, M ±⌥

acquire mass2

= ✏2

SU(2) x SU(2) → U(1) x U(1)

Gerardo Aldazabal’s talk

Double field theory

In GG/DFT on S1

TS1

@y

1 ⊕ T ⇤S1

+ dy

natural pairing

< @y + dy, @y + dy >= 2◆∂ydy = 2 < V, V >= ηMNV MV N

MN =

 0 1 1  

' ' ∂˜

y

T ˜ S1

< ∂y, ∂˜

y >= 1

Md ⇥ S1 Md ⇥ S1 ⇥ ˜ S1

y ˜ y R ˜ R

Vertex operators: depend on and

yL yR

= =

= y + ˜ y y − ˜ y η

ηLR =

   1 −1  

⇒ to reproduce string theory action we need dependence on and

y − ˜ y

Violating weak / strong constraint ? Yes, as expected: ¯ N − N = p˜ p

Level matching condition in usual massless states =0

∂y ' ∂˜

y

⇒

∂y∂˜

y(

) = 0 ηMN∂M∂N( ) = 0

weak constraint here =0

GG/DFT

Frame on

frame dual frame

Generalized metric H = δABEA ⌦ EB

H ⌦ H =   g−1 g−1B Bg−1 g Bg−1B  

Contains g, B dof:

O(D,D) O(D)×O(D)

TM ⊕ T ∗M

ea ea EA =  ea ιeaB ea  

Circle reduction

EA =  ea ιeaB ea  

eˆ

a

= φ (dy + V1)

y ∼ y + 2π

√gyy = R

gµy

#

φ1(∂y + B1)

TM = TMd ⊕ TS1

Frame on

frame dual frame

TM ⊕ T ∗M

ea ea

Bµy

Circle reduction

Frame on

frame ea dual frame ea

EA =  ea ιeaB ea  

eˆ

a

= φ (dy + V1)

y ∼ y + 2π

√gyy = R

gµy

#

φ1(∂y + B1) Bµy  Ed Ed   =  φ1 φ    ∂y + B1 dy + V1  

ηLR

U ± = 1 2(φ−1 ± φ) A = V1 + B1 ¯ A = V1 − B1 J = @y + dy ¯ J = @y − dy ,

EA(x, y) =) = UA

A0(x)E

x)E0

A0(y)

Scherk-Schwarz reduction

= e

= exp( 1

2M 33).

< >

≈ 1 + 1

2 < M 33 >

✏

Effective action valid at energies

E ⇠

1 p α0 ✏ << 1 p α0

U + ≈ 1 U ≈ 1

2M 33

  1

1 2M 33 1 2M 33

1  

 E

L

E

R

  =   =  U + U − U − U +      J + A ¯ J − ¯ A  

So far, no enhancement of symmetry, no double field theory

TM = TMd ⊕ TS1

TM ⊕ T ∗M

DFT & Enhancement of symmetry

TM ⊕ T ∗M

= TMd ⊕ TS1 ⊕ T ⇤S1

1 ⊕ T ⇤Md

⊕ T ˜ S1

' ' ∂˜

y

dy

J = @y + dy ¯ J = @y − dy ,

= ∂y + ∂˜

y

= ∂y ∂˜

y

= ∂yL = ∂yR

Still, this is formal. No dependence on y or ˜

y

Of course, we have not included momentum/winding modes To include winding modes we need DFT:

⇠ e2iy/e2i˜

y

To account for the enhancement of symmetry, we need to enlarge the generalized tangent space

S1, ˜ S1 = TMd ⊕ TS1

1 ⊕ T ⇤Md

⊕ T ˜ S1 O(3,3)

DFT

TMd ⊕ V2 ⊕ TS1 ⊕ T ˜ S1 ⊕ V ∗

2 ⊕ T ∗Md

  1

1 2M 33 1 2M 33

1  

 E

L

E

R

  =      J + A ¯ J − ¯ A  

= TMd ⊕ TS1

1 ⊕ T ⇤Md

⊕ T ˜ S1

Enhancement of symmetry

       E3 E¯

3

  =   1

1 2M 3¯ 3 1 2M ¯ 33

1    J3 + A3 ¯ J¯

3 − ¯

A¯

3

       

       Ei E¯

ı

  =   1

1 2M i¯ | 1 2M¯ ıj

1    Jj + Aj ¯ J ¯

| − ¯

A¯

|

 

9 scalar fields

M i¯

|(x)

Ai(x)

¯ A¯

ı(x)

6 vector fields

Ji(y, ˜ y) ¯ J¯

ı(y, ˜

y)

Should satisfy SU(2)L algebra Should satisfy SU(2)R algebra under some bracket

TMd ⊕ V2 ⊕ TS1 ⊕ T ˜ S1 ⊕ V ∗

2 ⊕ T ∗Md

Effective action

EA(x, y) =

) = UA

A0(x)E

x)E0

A0(y)

L = R − 1 12HµνρHµνρ + 1 4HIJF IµνF J

µν + (DµH)IJ(DµH)IJ

        Ea E

L

E

R

Ea         =         ea ◆eaA ◆ea ¯ A ◆eaB 1 M M ¯ A M t 1 M tA ea                 1 J ¯ J 1        

1 2 1 2

Generalized Scherk-Schwarz reduction of DFT action

Aldazabal, Baron, Marques, Nuñez 11 Geissbuhler 11

− 1 12fIJKfLMN

• HILHJMHKN − 3 HILηJMηKN + 2 ηILηJMηKN

2

≈ @ 1 M M t 1 1 A

I = i,¯ ı H = dB + F I ∧ AI

F I = dAI + f I

JK AJ ∧ AK

J, ¯ J

✏ijk, ✏ı|k

L = R − 1 12HµνρHµνρ + 1 4F i

µνF iµν + 1

4 ¯ F i

µν ¯

F iµν + 1 4M ijF i

µν ¯

F jµν + DµM ijDµM ij

−detM

Exactly string theory action!

C

[E0

J, E0 K] = f IJKE0 K

J, ¯ J

✏ijk, ✏ı|k

C

[E0

J, E0 K] = f IJKE0 K

C-bracket

[V1, V2]C = 1 2(LV1V2 − LV2V1) (

(LV1V2)I = V J

1 ∂JV I 2 + (∂IV1J ∂JV I 1 )V J 2

generalized Lie derivative

Algebra

The following and do the job

 J +

  ¯ J −

vL

1 , vL 2

vR

1 , vR 2

J =      cos 2yL sin 2yL − sin 2yL cos 2yL 1           vL

1

vL

2

dyL         

    ¯ J =      cos 2yR sin 2yR − sin 2yR cos 2yR 1           vR

1

vR

2

dyR        

+ V2 + TS1 + T ˜ S1 + V ∗

2 +

V2 + TS1 + TS1 + V2

Geometry of the “internal space”

J =      cos 2yL sin 2yL − sin 2yL cos 2yL 1           v1L v2L dyL     

E0

L =

The space is

    E0

R = ¯

J =      cos 2yR sin 2yR − sin 2yR cos 2yR 1           v1R v2R dyR     

S1 × ˜ S1 S1 × S1

But if we want to “geometrize” the O(3,3)

T 2

S1 × M3 T 2 M3 × S1

BUT

 Rt Rt    R R  

E0t

0t E0

  =  1 1  

All dependence on

y ˜ y

dissapears ! HOWEVER

H(y, ˜ y) =

   Rt Rt     1 M M t 1    R R    

    =   1 RtMR RtM tR 1  

Dependence on when considering fluctuations M

T 3

T 3

TM 2 ⊕ TS1 ⊕ TS1 ⊕ TM 2 T2 ⊕ T2 V2 ⊕ TS1 ⊕ TS1 ⊕ V2

H H(x, y, ˜ y) = (UE0)t UE0

EA(x, y) =

) = UA

A0(x)E

x)E0

A0(y)

T 2

S1 × M3 T 2 M3 × S1

• DFT description of strings very close to self-dual radius

Conclusions

• Winding modes → explicit dependence on dual coordinate

violate weak constraint satisfy level-matching

• Enhancement of symmetry → extend the generalized tangent space O(3,3)
• When M=0, “6d double space” is a torus, no dependence on
• Moduli (M≠0) bring in dependence on

y ˜ y

• r

y ˜ y and

• By appropriate generalized Scherk-Schwarz reduction of DFT action we

fully recover string theory action