non-commutative/non-associative geometry and non …media/math-phys/talks/luest.pdf1 workshop on...

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:

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:

I) Introduction

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I) Introduction

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

Discrete (non-commutative)fuzzy space-time:

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I) Introduction

- String theory (finitely extended objects):

Smooth geometry (resolution of singularities)

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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

Gij �GikBkj

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N , ⇤ 2 O(D,D)R ! L2

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

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.

O(2,2) monodromy:

T 2X1,X2 ,! M3 ,! S1

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.

O(2,2) monodromy:

T 2X1,X2 ,! M3 ,! S1

Complex structure of :

Kähler parameter of :

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 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

⌧(X3 + 2⇡) = � 1⌧(X3)

T 2X1X2

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3-dimensional fibration:

Non-geometric space with Q-flux

⇢(X3 + 2⇡) = � 1⇢(X3)

T 2X1X2

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Chain of four T-dual spaces:

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

⇢(X3H) = i R1R2 �HX3

1 0 00 R2

2 00 0 R2

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

⇢(X3H) = i R1R2 �HX3

1 0 00 R2

2 00 0 R2

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

1� fX3

� fX3f

⇣fX3

00 0 R2

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.

0 0 R23

CA , Bij = F

0 �QX3Q

CCA , F =

⇢(X3Q) =

Q � iR1R2) gO(2,2) : ⇢(X3

Q + 2⇡R3) =⇢(X3

Q)1 + 2⇡R3Q ⇢(X3

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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.

0 0 R23

CA , Bij = F

0 �QX3Q

CCA , F =

⇢(X3Q) =

Q � iR1R2) gO(2,2) : ⇢(X3

Q + 2⇡R3) =⇢(X3

Q)1 + 2⇡R3Q ⇢(X3

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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)] =

e�in(�0��) � (�0 � �)X

e�in(�0��) +i

2(�0 � �)2

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Q-flux:

[X1Q(⌧,�), X2

Q(⌧,�0)] =

e�in(�0��) � (�0 � �)X

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

[X1Q(⌧,�), X2

Q(⌧,�0)] =

e�in(�0��) � (�0 � �)X

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

R ⌘ Q

R-flux background:

[X1R, X2

R] = �i⇡2

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

[[X1R(⌧,�), X2

R(⌧,�)], X3R(⌧,�)] + perm. =

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

R ⌘ Q

R-flux background:

[X1R, X2

R] = �i⇡2

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

[[X1R(⌧,�), X2

R(⌧,�)], X3R(⌧,�)] + perm. =

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

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

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] = i⇥(p3)

⇥(p3) =

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

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

S ��

dnxTrFab � F ab

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

nx (f ? g) =

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

j ] = 0

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

[xi, x

j ] ⇠ l

3s~�1

[x1, x

3] := [[x1, x

3] + cycl. perm. ⇠ l

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) =

|(~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)

f2)(~x, ~p) = e

i2 ✓

IJ (p) @I⌦@J (f1 ⌦ f2)|~x; ~p

6-dimensional Poisson tensor:

✓IJ(p) =

@Rijkpk �i

��ji 0

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

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

f143 f243 f3 = [xi, x

k] = `

It obeys the 3-cyclicity property:Z

nx (f143 f2)43 f3 =

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)

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

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

Double field theory:

M = (xm, x

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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:

Mittwoch, 7. Mai 14

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

Mittwoch, 7. Mai 14

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.

Mittwoch, 7. Mai 14

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.

Mittwoch, 7. Mai 14

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

Mittwoch, 7. Mai 14

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

Mittwoch, 7. Mai 14

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

Mittwoch, 7. Mai 14

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

L⇠EAM = 0

Mittwoch, 7. Mai 14

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

L⇠EAM = 0

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

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

xmand . Mittwoch, 7. Mai 14

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 .

Mittwoch, 7. Mai 14

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

Mittwoch, 7. Mai 14

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

⇢(x3) =

⇢0 cos(Hx3) + sin(Hx3)

cos(Hx3)� ⇢0 sin(Hx3), H 2 1

Mittwoch, 7. Mai 14

=) ⌧(2⇡) = � 1⌧(0)

, ⇢(2⇡) = � 1⇢(0)

⌧(x3) =

cos(fx3)� ⌧0 sin(fx3), f 2 1

⇢(x3) =

cos(Hx3)� ⇢0 sin(Hx3), H 2 1

�(x1, x

2, x1, x2) =

2 + x1x2 � x1x2 � x

�� 3

2�x1x

1 + x2x2�

� 14

�(x1)2 + (x2)2 + (x1)2 + (x2)2

Patching is generated by the coordinate transformation:

Mittwoch, 7. Mai 14

=) ⌧(2⇡) = � 1⌧(0)

, ⇢(2⇡) = � 1⇢(0)

⌧(x3) =

cos(fx3)� ⌧0 sin(fx3), f 2 1

⇢(x3) =

cos(Hx3)� ⇢0 sin(Hx3), H 2 1

�(x1, x

2, x1, x2) =

2 + x1x2 � x1x2 � x

�� 3

2�x1x

1 + x2x2�

� 14

�(x1)2 + (x2)2 + (x1)2 + (x2)2

Corresponding Killing vectors of background:

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

CCCCCCA

Mittwoch, 7. Mai 14

=) ⌧(2⇡) = � 1⌧(0)

, ⇢(2⇡) = � 1⇢(0)

⌧(x3) =

cos(fx3)� ⌧0 sin(fx3), f 2 1

⇢(x3) =

cos(Hx3)� ⇢0 sin(Hx3), H 2 1

�(x1, x

2, x1, x2) =

2 + x1x2 � x1x2 � x

�� 3

2�x1x

1 + x2x2�

� 14

�(x1)2 + (x2)2 + (x1)2 + (x2)2

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

CCCCCCA

Background satisfies strong

constraint

Mittwoch, 7. Mai 14

=) ⌧(2⇡) = � 1⌧(0)

, ⇢(2⇡) = � 1⇢(0)

⌧(x3) =

cos(fx3)� ⌧0 sin(fx3), f 2 1

⇢(x3) =

cos(Hx3)� ⇢0 sin(Hx3), H 2 1

�(x1, x

2, x1, x2) =

2 + x1x2 � x1x2 � x

�� 3

2�x1x

1 + x2x2�

� 14

�(x1)2 + (x2)2 + (x1)2 + (x2)2

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

CCCCCCA

constraint

Patching does not satisfy strong

constraint

Mittwoch, 7. Mai 14

=) ⌧(2⇡) = � 1⌧(0)

, ⇢(2⇡) = � 1⇢(0)

⌧(x3) =

cos(fx3)� ⌧0 sin(fx3), f 2 1

⇢(x3) =

cos(Hx3)� ⇢0 sin(Hx3), H 2 1

�(x1, x

2, x1, x2) =

2 + x1x2 � x1x2 � x

�� 3

2�x1x

1 + x2x2�

� 14

�(x1)2 + (x2)2 + (x1)2 + (x2)2

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

CCCCCCA

constraint

Patching does not satisfy strong

constraint

Killing vectors do not satisfy strong constraint.

However their algebra closes!

Mittwoch, 7. Mai 14

V) Outlook & open questions

Mittwoch, 7. Mai 14

● 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).

Mittwoch, 7. Mai 14

● 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.

Mittwoch, 7. Mai 14

Non-associativity is an off-shell phenomenon!

Mittwoch, 7. Mai 14

● 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

Mittwoch, 7. Mai 14

● 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))

● 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

Mittwoch, 7. Mai 14

non-commutative/non-associative geometry and non …media/math-phys/talks/luest.pdf1 workshop on...

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