Non-Commutative/Non-Associative Geometry and Non …media/math-phys/talks/Luest.pdf1 Workshop on Non-Associativity in Physics and Related Mathematical Structures, PennState, 1st

1Workshop on Non-Associativity in Physics and Related Mathematical Structures,

PennState, 1st . May 2014

Non-Commutative/Non-Associative Geometry and

Non-geometric String Backgrounds

DIETER LÜST (LMU, MPI)

Mittwoch, 7. Mai 14

1Workshop on Non-Associativity in Physics and Related Mathematical Structures,

PennState, 1st . May 2014

O. Hohm, D.L., B. Zwiebach, arXiv:1309.2977;I. Bakas, D.L., arXiv:1309.3172;

R. Blumenhagen, M. Fuchs, F. Hassler, D.L., R. Sun, arXiv:1312.0719;F. Hassler, D.L., arXiv:1401.5068;

A. Betz, R. Blumenhagen, D.L., F. Rennecke, arXiv: 1402.1686

Non-Commutative/Non-Associative Geometry and

Non-geometric String Backgrounds

DIETER LÜST (LMU, MPI)

Mittwoch, 7. Mai 14

Outline:

2

II) Non-commutative/non-associative closed string geometry and non-geometric string backgrounds

I) Introduction

III) Some Remarks on Double Field Theory

☛ Talk by Ralph Blumenhagen

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Two complementary approaches to quantum gravity:

3

I) Introduction

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3

I) Introduction

- Canonical quantum gravity (LQG,CDT) for point-like fields:

Discrete (non-commutative)fuzzy space-time:

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3

I) Introduction



- String theory (finitely extended objects):

Smooth geometry (resolution of singularities)

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3

I) Introduction



What is the relation between these two approaches?

- String theory (finitely extended objects):

Smooth geometry (resolution of singularities)

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As we will see, non-geometric string backgrounds and T-duality will provide a very interesting and new classical relation between fuzzy space and finite extension of the string.

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⇒ Non-commutative and non-associative closed

string geometry:R. Blumenhagen, E. Plauschinn, arXiv:1010.1263; D. Lüst, arXiv:1010.1361;

R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arXiv:1106.0316

C. Condeescu, I. Florakis, D. L., JHEP 1204 (2012), 121, arXiv:1202.6366

D. Andriot, M. Larfors, D.Lüst, P. Patalong, arXiv:1211.6437

A. Bakas, D.Lüst, arXiv:1309.3172

R. Blumenhagen, M. Fuchs, F. Hassler, D. Lüst, R. Sun, arXiv:1312.0719

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„Fundamental“ closed string non-

associativity in WZW-model with H-flux

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Closed string non-commutativity in „tori“ with

non-geometric fluxes and T-duality,„derived“ non-associativity follows

as violation of Jacobi identity

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as violation of Jacobi identityNon-associativity in CFT‘s with

geometric and T-dual non-geometric fluxes

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⇒ New classical uncertainty relations - minimal

volume due to finite string length!












as violation of Jacobi identityNon-associativity in CFT‘s with

geometric and T-dual non-geometric fluxes

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Non-associativity in physics:

• Multiple M2-branes and 3-algebras J. Bagger, N. Lambert (2007)

• Jordan & Malcev algebras, octonionsM. Günaydin, F. Gürsey (1973); M. Günaydin, D. Minic, arXiv:1304.0410.

• Closed string field theory A. Strominger (1987), B. Zwiebach (1993)

• Magnetic monopoles R. Jackiw (1985); M. Günaydin, B. Zumino (1985)

• T-duality and principle torus bundlesP. Bouwknegt, K. Hannabuss, Mathai (2003)

• Nambu dynamicsY. Nambu (1973); D. Minic, H. Tze (2002); M. Axenides, E. Floratos (2008)

• D-branes in curved backgroundsL. Cornalba, R. Schiappa (2001)

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II) Non-geometric backgrounds and non- commutative & non-associative geometry

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II) Non-geometric backgrounds and non- commutative & non-associative geometry Closed string background fields: Gij , Bij , �

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Generalized metric: HMN =✓

Gij �GikBkj

BikGkj Gij �BikGklBlj

◆

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T-duality - O(D,D) transformations:

They contain: Bij ! Bij + 2⇡⇤ij ,

HMN ! ⇤PM HPQ ⇤Q

N , ⇤ 2 O(D,D)R ! L2

s/R


Gij �GikBkj


◆

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N , ⇤ 2 O(D,D)R ! L2

s/R

String length (not hbar!)


Gij �GikBkj


◆

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XM = (Xi, Xi)

Doubling of closed string coordinates and momenta:

- Coordinates: O(D,D) vector

- Momenta: O(D,D) vector

winding momentum

pM = (pi, pi)

(Here D=3)← T-duality →





N , ⇤ 2 O(D,D)R ! L2

s/R

String length (not hbar!)


Gij �GikBkj


◆

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Non-geometric backgrounds are generic within the landscape of string „compactifications“.Several potentially interesting applications in string phenomenology and cosmology.

● They are only consistent in string theory.

● They can be potentially used for the construction of de Sitter vacua

● Left-right asymmetric spaces ⇒ Asymmetric orbifolds(Kawai, Lewellen, Tye, 1986; Lerche, D.L. Schellekens, 1986, Antoniadis, Bachas, Kounnas, 1987; Narain, Sarmadi, Vafa, 1987)

● Make use of string symmetries, T-duality ⇒ T-folds,

● T-dualiy: classical bounce (pre-big bang) models(Brandenberger, Vafa, 1989; Meissner, Veneziano, 1991; Gasperini, Veneziano, 1993)

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(Hellerman, McGreevy, Williams (2002); C. Hull (2004); Shelton, Taylor, Wecht, 2005; Dabholkar, Hull, 2005)

[Xi, Xj ] 6= 0Q-space will become non-commutative:

- Non-geometric Q-fluxes: spaces that are locally still Riemannian manifolds but not anymore globally.

Transition functions between two coordinate patches are given in terms of O(D,D) T-duality transformations:

Di↵(MD) ! O(D,D)C. Hull (2004)

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(Hellerman, McGreevy, Williams (2002); C. Hull (2004); Shelton, Taylor, Wecht, 2005; Dabholkar, Hull, 2005)

- Non-geometric R-fluxes: spaces that are even locally not anymore manifolds.

R-space will become non-associative:

[Xi, Xj , Xk] := [[Xi, Xj ], Xk] + cycl. perm. == (Xi · Xj) · Xk �Xi · (Xj · Xk) + · · · 6= 0

[Xi, Xj ] 6= 0Q-space will become non-commutative:

- Non-geometric Q-fluxes: spaces that are locally still Riemannian manifolds but not anymore globally.

Transition functions between two coordinate patches are given in terms of O(D,D) T-duality transformations:

Di↵(MD) ! O(D,D)C. Hull (2004)

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Example: Three-dimensional flux backgrounds:

Fibrations: 2-dim. torus that varies over a circle:

The fibration is specified by its monodromy properties.

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T 2 :

S1

O(2,2) monodromy:

T 2X1,X2 ,! M3 ,! S1

X3

HMN (X3)Metric, B-field of

HMN (X3 + 2⇡) = ⇤O(2,2) HPQ(X3) ⇤�1O(2,2)

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Example: Three-dimensional flux backgrounds:

Fibrations: 2-dim. torus that varies over a circle:

The fibration is specified by its monodromy properties.

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T 2 :

S1

O(2,2) monodromy:

T 2X1,X2 ,! M3 ,! S1

X3

Complex structure of :

Kähler parameter of :

T 2

T 2 ⇢(X3 + 2⇡) =a0⇢(X3) + b0

c0⇢(X3) + d0

⌧(X3 + 2⇡) =a⌧(X3) + b

c⌧(X3) + d⌧

⇢

HMN (X3)Metric, B-field of

HMN (X3 + 2⇡) = ⇤O(2,2) HPQ(X3) ⇤�1O(2,2)

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Torus

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Torus with non-constant B-field (H-flux), B-field is patched together by a B-field (gauge) transformation: B ! B + 2⇡H

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Non geometric torus, metric is patched together by a T-duality transformation: Gij ! Gij

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Non geometric torus, metric is patched together by a T-duality transformation: Gij ! Gij

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3-dimensional fibration:

Twisted torus with f-flux

S1

⌧(X3 + 2⇡) = � 1⌧(X3)

S1X3

T 2X1X2

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3-dimensional fibration:

Non-geometric space with Q-flux

⇢(X3 + 2⇡) = � 1⇢(X3)

T 2X1X2

S1X3

S1

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Chain of four T-dual spaces:

(a) Geometric space: 3-dimensional torus with H - fluxB12 = HX3

H,

⇢(X3H) = i R1R2 �HX3

H

Gij =

0

@R2

1 0 00 R2

2 00 0 R2

3

1

A, H123 = @3B12 = H

X3H ! X3

H + 2⇡R3 ) gO(2,2) : ⇢(X3H + 2⇡R3) = ⇢(X3

H) + 2⇡HR3

(i) (Non-)geometric backgrounds with parabolic monodromy and constant 3-form fluxes:

(Bianchi I)

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Chain of four T-dual spaces:

(a) Geometric space: 3-dimensional torus with H - fluxB12 = HX3

H,

⇢(X3H) = i R1R2 �HX3

H

Gij =

0

@R2

1 0 00 R2

2 00 0 R2

3

1

A, H123 = @3B12 = H

X3H ! X3

H + 2⇡R3 ) gO(2,2) : ⇢(X3H + 2⇡R3) = ⇢(X3

H) + 2⇡HR3

(i) (Non-)geometric backgrounds with parabolic monodromy and constant 3-form fluxes:

(b) Geometric spaces: twisted 3-torus with f - flux

,

(f ⌘ H)T-duality in X1 :

⌧(X3f ) = i R1R2 � fX3

f

Gij =

0

BB@

1R2

1� fX3

f

R21

0

� fX3f

R21

R22 +

⇣fX3

f

R1

⌘2

00 0 R2

3

1

CCABij = 0

X3f ! X3

f + 2⇡R3 ) gO(2,2) : ⌧(X3f + 2⇡R3) = ⌧(X3

f ) + 2⇡fR3

(Bianchi II)

(Bianchi I)

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(c) Non-geometric space: T-fold with Q-fluxT-duality in X2 :

(Q ⌘ f ⌘ H)

This does not correspond to a standard diffeomorphism but to a T-duality transformation.

Gij =

0

B@

FR2

10 0

0 FR2

20

0 0 R23

1

CA , Bij = F

0

BB@

0 �QX3Q

R21R2

20

QX3Q

R21R2

20 0

0 0 0

1

CCA , F =

0

@1 +

QX3

Q

R1R2

!21

A�1

⇢(X3Q) =

1QX3

Q � iR1R2) gO(2,2) : ⇢(X3

Q + 2⇡R3) =⇢(X3

Q)1 + 2⇡R3Q ⇢(X3

Q)

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(c) Non-geometric space: T-fold with Q-fluxT-duality in X2 :

(Q ⌘ f ⌘ H)

This does not correspond to a standard diffeomorphism but to a T-duality transformation.

T-duality in X3 :(d) Non-geometric space with R-flux

Now the Buscher rules for T-duality cannot be applied.There exist no locally defined metric and B-field.

Gij =

0

B@

FR2

10 0

0 FR2

20

0 0 R23

1

CA , Bij = F

0

BB@

0 �QX3Q

R21R2

20

QX3Q

R21R2

20 0

0 0 0

1

CCA , F =

0

@1 +

QX3

Q

R1R2

!21

A�1

⇢(X3Q) =

1QX3

Q � iR1R2) gO(2,2) : ⇢(X3

Q + 2⇡R3) =⇢(X3

Q)1 + 2⇡R3Q ⇢(X3

Q)

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Tx1 Tx2 Tx3Flat torus with H-flux

Twisted torus with

f-flux

Non-geometric space with

Q-flux


R-flux

Summary:

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Twisted torus with

f-flux


Q-flux


R-flux

[XiH,f , Xj

H,f ] = 0

Summary:

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Twisted torus with

f-flux


Q-flux


R-flux

[XiH,f , Xj

H,f ] = 0 [X1Q, X2

Q] ' Q p3

Summary:

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Twisted torus with

f-flux


Q-flux


R-flux

[XiH,f , Xj

H,f ] = 0

[[X1R, X2

R], X3R] ' R

[X1Q, X2

Q] ' Q p3

Summary:

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D. Andriot, M. Larfors, D. L., P. Patalong, arXiv:1211.6437

They can be computed by

- standard world-sheet quantization of the closed string

- CFT & canonical T-duality I. Bakas, D.L. to appear soon


Twisted torus with

f-flux


Q-flux


R-flux

[XiH,f , Xj

H,f ] = 0

[[X1R, X2

R], X3R] ' R

[X1Q, X2

Q] ' Q p3

Summary:

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Q-flux:

[X1Q(⌧,�), X2

Q(⌧,�0)] =

� i

2Q p3

✓X

n 6=0

1n2

e�in(�0��) � (�0 � �)X

n 6=0

1n

e�in(�0��) +i

2(�0 � �)2

◆

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Q-flux:

[X1Q(⌧,�), X2

Q(⌧,�0)] =

� i

2Q p3

✓X

n 6=0

1n2

e�in(�0��) � (�0 � �)X

n 6=0

1n

e�in(�0��) +i

2(�0 � �)2

◆

winding number

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Q-flux:

The non-commutativity of the torus (fibre) coordinates is determined by the winding in the circle (base) direction.

� ! �0 :

[X1Q(⌧,�), X2

Q(⌧,�)] = �i⇡2

6Q p3

[X1Q(⌧,�), X2

Q(⌧,�0)] =

� i

2Q p3

✓X

n 6=0

1n2

e�in(�0��) � (�0 � �)X

n 6=0

1n

e�in(�0��) +i

2(�0 � �)2

◆

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Corresponding uncertainty relation:

The spatial uncertainty in the - directions grows with the dual momentum in the third direction: non-local strings with winding in third direction.

X1, X2

(�X1Q)2(�X2

Q)2 � L6s Q2 hp3i2

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⇒ For the case of non-geometric R-fluxes one gets:

T-duality in -direction ⇒ R-flux x

3

21

R ⌘ Q

R-flux background:

[X1R, X2

R] = �i⇡2

6R p3

Use Non-associative algebra:[X3R, p3] = i =)

[[X1R(⌧,�), X2

R(⌧,�)], X3R(⌧,�)] + perm. =

⇡2

6R

p3 ! p3 , XQ,3 ⌘ X3R

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⇒ For the case of non-geometric R-fluxes one gets:

T-duality in -direction ⇒ R-flux x

3

21

R ⌘ Q

R-flux background:

[X1R, X2

R] = �i⇡2

6R p3

Use Non-associative algebra:[X3R, p3] = i =)

[[X1R(⌧,�), X2

R(⌧,�)], X3R(⌧,�)] + perm. =

⇡2

6R

p3 ! p3 , XQ,3 ⌘ X3R

Corresponding classical „uncertainty relations“:(�X1

R)2(�X2R)2 � L6

s R2 hp3i2

(�X1R)2(�X2

R)2(�X3R)2 � L6

s R2Volume:(see also: D. Mylonas, P. Schupp, R.Szabo, arXiv:1312.1621)

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The algebra of commutation relation looks different in each of the four duality frames.Non-vanishing commutators and 3-brackets:

T-dual frames Commutators Three-brackets

H-flux [x1, x2] ⇠ Hp3

[x1, x2, x3] ⇠ H

f -flux [x1, x2] ⇠ fp3

[x1, x2, x3] ⇠ f

Q-flux [x1, x2] ⇠ Qp3

[x1, x2, x3] ⇠ Q

R-flux [x1, x2] ⇠ Rp3

[x1, x2, x3] ⇠ R

However: R-flux & winding coordinates: [xi, x

j, x

k] = 0

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The algebra of commutation relation looks different in each of the four duality frames.Non-vanishing commutators and 3-brackets:

T-dual frames Commutators Three-brackets

H-flux [x1, x2] ⇠ Hp3

[x1, x2, x3] ⇠ H

f -flux [x1, x2] ⇠ fp3

[x1, x2, x3] ⇠ f

Q-flux [x1, x2] ⇠ Qp3

[x1, x2, x3] ⇠ Q

R-flux [x1, x2] ⇠ Rp3

[x1, x2, x3] ⇠ R

However: R-flux & winding coordinates: [xi, x

j, x

k] = 0T-duality is mapping these commutators and 3-brackets

onto each other:

⇔ T-duality

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(ii) (Non-)geometric backgrounds with elliptic monodromy and non-geometric fluxes.

They can be described in terms of twisted tori and (a)symmetric freely acting orbifolds.

C. Condeescu, I. Florakis, D. Lüst, JHEP 1204 (2012), 121, arXiv:1202.6366C. Condeescu, I. Florakis, C. Kounnas, D.Lüst, arXiv:1307.0999

D. Lüst, JHEP 1012 (2011) 063, arXiv:1010.1361,

In general not T-dual to a geometric space!(Only consistent in string theory (respectively in DFT).)

The fibre torus depends on the third coordinate in a more complicate way.

A. Dabholkar, C. Hull (2002, 2005)

C. Hull; R. Read-Edwards (2005, 2006, 2007, 2009)

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The corresponding commutators can be explicitly derived in CFT.

More complicate, non-linear commutation relations:

[x1, x

2] = [x1, x

2] = [x1, x

2] = [x1, x

2] = i⇥(p3)

⇥(p3) =

⇡

2

cot(⇡p3R)

R-frame:

This algebra cannot be T-dualized to a commutative algebra!

The string always moves on a non-commutative/non-associative fuzzy space:

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⇒ 3-Cocycles, 2-cochains and star-products

Mathematical framework to describe non-geometric string backgrounds and the non-associative algebras:

● Group theory cohomology - Hochschild; Stasheff; Cartan, Eilenberg, ...

I. Bakas, D. Lüst, arXiv:1309.3172;

D. Mylonas, P. Schupp, R.Szabo, arXiv:1207.0926, arXiv:1312.162, arXiv:1402.7306.

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Open string non-commutativity:

[xi, xj ] = ✓ijConstant Poisson structure:

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Open string non-commutativity:

[xi, xj ] = ✓ijConstant Poisson structure:

Moyal-Weyl star-product:

Non-commutative gauge theories:

(f1 ? f2)(~x) = e

i✓

ij

@

x1i

@

x2j

f1(~x1) f2(~x2)|~x

S ��

dnxTrFab � F ab

(N. Seiberg, E. Witten (1999); J. Madore, S. Schraml, P. Schupp, J. Wess (2000); .... )

Zd

nx (f ? g) =

Zd

nx (g ? f)2-cyclicity:

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Are the similar structures for closed strings?

Deformed theory of gravity?

⇒ Tri-product.

Possibly yes, but only off-shell.

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[xi, p

j ] = i~�

ij, [pi

, p

j ] = 0

Recall: closed string parabolic model in R-flux frame:

[xi, x

j ] ⇠ l

3s~�1

R ✏

ijkpk

[x1, x

2, x

3] := [[x1, x

2], x

3] + cycl. perm. ⇠ l

3sR

Non-associative algebra:

Are the similar structures for closed strings?

Deformed theory of gravity?

⇒ Tri-product.

Possibly yes, but only off-shell.

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3-cocycles in Lie group cohomology

Consider the following group elements (loops):

U(~a, ~b) = ei(~a·~x+~

b·~p)

Now we want to consider the product of two or three group elements in order to derive the non-commutative/non-associative phases in the group

products (BCH formula).

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Group cohomology:

Non-commutativity is determined by the following 2-cochain:

U(~a1, ~b1)U(~a2, ~b2) = e�i ⇡2R12 '(~a1,~a2)U(~a2, ~b2)U(~a1, ~b1).

'2(~a1,~a2) = (~a1 ⇥ ~a2) · ~p

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Product law of three group elements becomes non-associative:

Non-associativity is determined by the 3-cocycle:

3-cocycle: volume of tetrahedon:

Volume(~a1,~a2,~a3) =

1

6

|(~a1 ⇥ ~a2) · ~a3|

⇣U(~a1, ~b1)U(~a2, ~b2)

⌘U(~a3, ~b3) = e�i R

2 (~a1⇥~a2)·~a3U(~a1, ~b1)⇣U(~a2, ~b2)U(~a3, ~b3)

⌘.

'3(~a1,~a2,~a3) = (~a1 ⇥ ~a2) · ~a3

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Derivation of the star productThe multiplication of the group elements and the use of

Weyl‘s correspondence rule lead to star 2- and 3-products for the multiplication of functions .

f(~x, ~p)

(f1 ?

p

f2)(~x, ~p) = e

i2 ✓

IJ (p) @I⌦@J (f1 ⌦ f2)|~x; ~p

6-dimensional Poisson tensor:

✓IJ(p) =

0

@Rijkpk �i

j

��ji 0

1

A ; Rijk =⇡2R

6✏ijk

D. Mylonas, P. Schupp, R.Szabo, arXiv:1207.0926.

(The full phase space including also dual coordinates and dual momenta is 12-dimensional!) I. Bakas, D. Lüst, arXiv:1309.3172

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It leads to the following 3-product:

(f143 f243 f3)(~x) = ((f1 ?p f2) ?p f3) (~x)

This delta-product is non-associative.

(f143 f243 f3)(~x) = e

iR

ijk

@

x1i

@

x2j

@

x3k

f1(~x1) f2(~x2) f3(~x3)|~x

It is consistent with the 3-bracket among the coordinates:

f1 = x

i, f2 = x

j, f3 = x

k :

f143 f243 f3 = [xi, x

j, x

k] = `

4s R

ijk

It obeys the 3-cyclicity property:Z

d

nx (f143 f2)43 f3 =

Zd

nx f143 (f243 f3)

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was already derived in CFT from the multiplication of 3 tachyon vertex operators:43


Scattering of 3 momentum states in R-background:(corresponds to 3 winding states in H-background)

However this non-associative phase is vanishing, when going on-shell in CFT and using momentum conservation:

On-shell CFT amplitudes are associative!

p1 = �(p2 + p3)

⌦V�(1) V�(1) V�(1)

↵R

=

⌦V1 V2 V3

↵R⇥ exp

⇣�i⌘�Rijk p1,ip2,jp3,k

⌘.

Vi(z, z) =: exp

�ipiX

i(z, z)

�:

(⌘� = 0, 1)

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33

Effective field theory description of non-geometric spaces:

IV) Double geometry - double field theoryW. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...)

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33


(i) Geometric (H,f)-spaces: Standard supergravity.


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33



(ii) Non-geometric (Q,R)-spaces: -supergravity, algebroids�

D. Andriot, M. Larfors, D. Lüst, P. Patalong, arXiv:1106.4015,D. Andriot, O. Hohm, M. Larfors, D. Lüst, P. Patalong, arXiv:1202.30,16, arXiv:1204.1979.D. Andriot, A. Betz, arXiv:1306.4381,R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, arXiv:1210.1591, arXiv: 1211.0030,R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, C. Schmid, arXiv:1304.2784.

(still T-dual to geometric spaces)


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33



(iii) Non-geometric (H,f,Q,R)-spaces: Double field theoryO. Hohm, D. Lüst, B. Zwiebach, arXiv:1309.2977,R. Blumenhagen, M. Fuchs, Hassler, D. Lüst, R. Sun, arXiv:1312.0719,F. Hassler, D.Lüst arXiv:1401.5068.

(ii) Non-geometric (Q,R)-spaces: -supergravity, algebroids�

D. Andriot, M. Larfors, D. Lüst, P. Patalong, arXiv:1106.4015,D. Andriot, O. Hohm, M. Larfors, D. Lüst, P. Patalong, arXiv:1202.30,16, arXiv:1204.1979.D. Andriot, A. Betz, arXiv:1306.4381,R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, arXiv:1210.1591, arXiv: 1211.0030,R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, C. Schmid, arXiv:1304.2784.

(still T-dual to geometric spaces)


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34

• O(D,D) invariant effective string action containing momentum and winding coordinates at the same time:

Double field theory:

X

M = (xm, x

m)

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34



X

M = (xm, x

m)• Covariant flux formulation of DFT.

(Geissbuhler, Marques, Nunez, Penas; Aldazabal, Marques, Nunez)

FABC = D[AEBM EC]M , DA = EA

M @M .

Comprises all fluxes (Q,f,Q,R) into one covariant expression:

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34



X

M = (xm, x

m)• Covariant flux formulation of DFT.

(Geissbuhler, Marques, Nunez, Penas; Aldazabal, Marques, Nunez)

FABC = D[AEBM EC]M , DA = EA

M @M .

Comprises all fluxes (Q,f,Q,R) into one covariant expression:

�⇠EAM = L⇠E

AM = ⇠P @P EA

M + (@M⇠P � @P ⇠M )EAP

• The DFT action is invariant under generalized, non-associative diffeomorphisms: ⇠M = (�m, �m)

[⇠1, ⇠2]MC = ⇠N

1 @N⇠M2 �

12⇠1N@M⇠N

2 � (⇠1 $ ⇠2)

[L⇠1 ,L⇠2 ] = L[⇠1,⇠2]C

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35

Generalized diffeomorphisms act on the generalized coordinates in a non-associative way:

The generalized diffeomorphisms contain simultaneous coordinate and B - , - gauge field transformations.�

C. Hull, B. Zwiebach, arXiv:0908.1792;O. Hohm, B. Zwiebach, arXiv:1207.4198;O. Hohm, D. Lüst, B. Zwiebach, arXiv:1309:2977.

The Courant bracket violates the Jacobi identity.

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36

However for generalized functions f(X) (e.g. the background fields) one has to require the strong constraint (string level matching condition):

@M@M · = 0 , @Mf @Mg = DAf DAg = 0

Then the algebra of diffeomorphisms on functions closes and becomes associative.

(CFT origin of the strong constraint: A. Betz, R. Blumenhagen, D. Lüst, F. Rennecke, arXiv:1402.1686)

The strong constraint defines a D-dim. hypersurface (brane) in 2D-dim. double geometry.

Functions must depend only on one kind of coordinates.

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37

Dimensional reduction of double field theory:O. Hohm, D. Lüst, B. Zwiebach, arXiv:1309.2977;F. Hassler, D. Lüst, arXiv:1401.5068.

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37


• Consistent DFT solutions (generalized Scherk-Schwarz compactifications): RMN = 0

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37


• The corresponding backgrounds are in general non- geometric and go beyond dimensional reduction of 10D SUGRA or generalized geometry.


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37




• Killing vectors, , correspond to generalized diffeomorphisms that depend on and .

L⇠EAM = 0

xm x

m

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37




• Killing vectors, , correspond to generalized diffeomorphisms that depend on and .

L⇠EAM = 0

xm x

m

• Patching of coordinate charts correspond to generalized coordinate transformations of the form

X 0M = XM � @M� ,where the gauge functions in general depend on �

x

m

xmand . Mittwoch, 7. Mai 14

38

Consider the following 3+3 dimensional backgrounds:

(i) Parabolic background spaces: Single fluxes:

H123 f123 Q12

3 R123oror or

These backgrounds do not satisfy RMN = 0 .

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38

Consider the following 3+3 dimensional backgrounds:

(i) Parabolic background spaces: Single fluxes:

H123 f123 Q12

3 R123oror or

These backgrounds do not satisfy RMN = 0 .(ii) Elliptic background spaces: Multiple fluxes:

• Single elliptic T-dual, non-geometric space:

• Double elliptic, genuinely non-geometric space:

These backgrounds do satisfy RMN = 0 .

• Single elliptic geometric space: f213 = f1

23 = f

H123 = Q123 = H , f2

13 = f123 = f

H123 = Q123 = H

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39

E.g. double elliptic background:

=) ⌧(2⇡) = � 1⌧(0)

, ⇢(2⇡) = � 1⇢(0)

⌧(x3) =

⌧0 cos(fx3) + sin(fx3)

cos(fx3)� ⌧0 sin(fx3), f 2 1

4

+ Z ,

⇢(x3) =

⇢0 cos(Hx3) + sin(Hx3)

cos(Hx3)� ⇢0 sin(Hx3), H 2 1

4

+ Z .

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39


=) ⌧(2⇡) = � 1⌧(0)

, ⇢(2⇡) = � 1⇢(0)

⌧(x3) =


cos(fx3)� ⌧0 sin(fx3), f 2 1

4

+ Z ,

⇢(x3) =


cos(Hx3)� ⇢0 sin(Hx3), H 2 1

4

+ Z .

�(x1, x

2, x1, x2) =

12

�x

1x

2 + x1x2 � x1x2 � x

1x2

�� 3

2�x1x

1 + x2x2�

� 14

�(x1)2 + (x2)2 + (x1)2 + (x2)2

�.

Patching is generated by the coordinate transformation:

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39


=) ⌧(2⇡) = � 1⌧(0)

, ⇢(2⇡) = � 1⇢(0)

⌧(x3) =


cos(fx3)� ⌧0 sin(fx3), f 2 1

4

+ Z ,

⇢(x3) =


cos(Hx3)� ⇢0 sin(Hx3), H 2 1

4

+ Z .

�(x1, x

2, x1, x2) =

12

�x

1x

2 + x1x2 � x1x2 � x

1x2

�� 3

2�x1x

1 + x2x2�

� 14

�(x1)2 + (x2)2 + (x1)2 + (x2)2

�.


Corresponding Killing vectors of background:

K JI

=

0

BBBBBB@

1 0 0 0 0 00 1 � 1

2 (Hx3 + fx3) 12 (Hx2 + fx2) � 1

2 (fx3 + Hx3) 12 (fx2 + Hx2)

0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

1

CCCCCCA

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39


=) ⌧(2⇡) = � 1⌧(0)

, ⇢(2⇡) = � 1⇢(0)

⌧(x3) =


cos(fx3)� ⌧0 sin(fx3), f 2 1

4

+ Z ,

⇢(x3) =


cos(Hx3)� ⇢0 sin(Hx3), H 2 1

4

+ Z .

�(x1, x

2, x1, x2) =

12

�x

1x

2 + x1x2 � x1x2 � x

1x2

�� 3

2�x1x

1 + x2x2�

� 14

�(x1)2 + (x2)2 + (x1)2 + (x2)2

�.



K JI

=

0

BBBBBB@

1 0 0 0 0 00 1 � 1

2 (Hx3 + fx3) 12 (Hx2 + fx2) � 1

2 (fx3 + Hx3) 12 (fx2 + Hx2)

0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

1

CCCCCCA

Background satisfies strong

constraint

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39


=) ⌧(2⇡) = � 1⌧(0)

, ⇢(2⇡) = � 1⇢(0)

⌧(x3) =


cos(fx3)� ⌧0 sin(fx3), f 2 1

4

+ Z ,

⇢(x3) =


cos(Hx3)� ⇢0 sin(Hx3), H 2 1

4

+ Z .

�(x1, x

2, x1, x2) =

12

�x

1x

2 + x1x2 � x1x2 � x

1x2

�� 3

2�x1x

1 + x2x2�

� 14

�(x1)2 + (x2)2 + (x1)2 + (x2)2

�.



K JI

=

0

BBBBBB@

1 0 0 0 0 00 1 � 1

2 (Hx3 + fx3) 12 (Hx2 + fx2) � 1

2 (fx3 + Hx3) 12 (fx2 + Hx2)

0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

1

CCCCCCA


constraint

Patching does not satisfy strong

constraint

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39


=) ⌧(2⇡) = � 1⌧(0)

, ⇢(2⇡) = � 1⇢(0)

⌧(x3) =


cos(fx3)� ⌧0 sin(fx3), f 2 1

4

+ Z ,

⇢(x3) =


cos(Hx3)� ⇢0 sin(Hx3), H 2 1

4

+ Z .

�(x1, x

2, x1, x2) =

12

�x

1x

2 + x1x2 � x1x2 � x

1x2

�� 3

2�x1x

1 + x2x2�

� 14

�(x1)2 + (x2)2 + (x1)2 + (x2)2

�.



K JI

=

0

BBBBBB@

1 0 0 0 0 00 1 � 1

2 (Hx3 + fx3) 12 (Hx2 + fx2) � 1

2 (fx3 + Hx3) 12 (fx2 + Hx2)

0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

1

CCCCCCA


constraint

Patching does not satisfy strong

constraint

Killing vectors do not satisfy strong constraint.

However their algebra closes!

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V) Outlook & open questions

40

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40

● Non-commutative & non-associative closed string geometry arises in the presence of non-geometric fluxes (like open string non-commutativity on D- branes with gauge flux). This leads to a non-associative tri-product (like the star-product).

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40

● Non-commutative & non-associative closed string geometry arises in the presence of non-geometric fluxes (like open string non-commutativity on D- branes with gauge flux). This leads to a non-associative tri-product (like the star-product). ● However the non-associativity is not visible in on-shell CFT amplitudes.

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40


Non-associativity is an off-shell phenomenon!

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40



● Are there situations, where the strong constraint for the background can be relaxed? - This seems to be the case for certain very asymmetric orbifolds. C. Condeescu, I. Florakis, C. Kounnas, D.Lüst, arXiv:1307.0999

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● Is there a non-commutative (non-associative) theory of gravity?

(A. Chamseddine, G. Felder, J. Fröhlich (1992), J. Madore (1992); L. Castellani (1993)P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp, J. Wess (2005),

L. Alvarez-Gaume, F. Meyer, M. Vazquez-Mozo (2006))


40



● Are there situations, where the strong constraint for the background can be relaxed? - This seems to be the case for certain very asymmetric orbifolds. C. Condeescu, I. Florakis, C. Kounnas, D.Lüst, arXiv:1307.0999

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