non-commutative/non-associative geometry and non …media/math-phys/talks/luest.pdf1 workshop on...
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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
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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
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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
Mittwoch, 7. Mai 14
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Two complementary approaches to quantum gravity:
3
I) Introduction
Mittwoch, 7. Mai 14
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Two complementary approaches to quantum gravity:
3
I) Introduction
- Canonical quantum gravity (LQG,CDT) for point-like fields:
Discrete (non-commutative)fuzzy space-time:
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Two complementary approaches to quantum gravity:
3
I) Introduction
- Canonical quantum gravity (LQG,CDT) for point-like fields:
Discrete (non-commutative)fuzzy space-time:
- String theory (finitely extended objects):
Smooth geometry (resolution of singularities)
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Two complementary approaches to quantum gravity:
3
I) Introduction
- Canonical quantum gravity (LQG,CDT) for point-like fields:
Discrete (non-commutative)fuzzy space-time:
What is the relation between these two approaches?
- String theory (finitely extended objects):
Smooth geometry (resolution of singularities)
Mittwoch, 7. Mai 14
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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.
Mittwoch, 7. Mai 14
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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.
⇒ 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
Mittwoch, 7. Mai 14
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4
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.
⇒ 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
„Fundamental“ closed string non-
associativity in WZW-model with H-flux
Mittwoch, 7. Mai 14
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4
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.
⇒ 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
„Fundamental“ closed string non-
associativity in WZW-model with H-flux
Closed string non-commutativity in „tori“ with
non-geometric fluxes and T-duality,„derived“ non-associativity follows
as violation of Jacobi identity
Mittwoch, 7. Mai 14
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4
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.
⇒ 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
„Fundamental“ closed string non-
associativity in WZW-model with H-flux
Closed string non-commutativity in „tori“ with
non-geometric fluxes and T-duality,„derived“ non-associativity follows
as violation of Jacobi identityNon-associativity in CFT‘s with
geometric and T-dual non-geometric fluxes
Mittwoch, 7. Mai 14
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4
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.
⇒ New classical uncertainty relations - minimal
volume due to finite string length!
⇒ 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
„Fundamental“ closed string non-
associativity in WZW-model with H-flux
Closed string non-commutativity in „tori“ with
non-geometric fluxes and T-duality,„derived“ non-associativity follows
as violation of Jacobi identityNon-associativity in CFT‘s with
geometric and T-dual non-geometric fluxes
Mittwoch, 7. Mai 14
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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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II) Non-geometric backgrounds and non- commutative & non-associative geometry Closed string background fields: Gij , Bij , �
Generalized metric: HMN =✓
Gij �GikBkj
BikGkj Gij �BikGklBlj
◆
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II) Non-geometric backgrounds and non- commutative & non-associative geometry Closed string background fields: Gij , Bij , �
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
Generalized metric: HMN =✓
Gij �GikBkj
BikGkj Gij �BikGklBlj
◆
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II) Non-geometric backgrounds and non- commutative & non-associative geometry Closed string background fields: Gij , Bij , �
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
String length (not hbar!)
Generalized metric: HMN =✓
Gij �GikBkj
BikGkj Gij �BikGklBlj
◆
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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 →
II) Non-geometric backgrounds and non- commutative & non-associative geometry Closed string background fields: Gij , Bij , �
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
String length (not hbar!)
Generalized metric: HMN =✓
Gij �GikBkj
BikGkj Gij �BikGklBlj
◆
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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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16
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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17
(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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17
(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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17
Tx1 Tx2 Tx3Flat torus with H-flux
Twisted torus with
f-flux
Non-geometric space with
Q-flux
Non-geometric space with
R-flux
Summary:
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17
Tx1 Tx2 Tx3Flat torus with H-flux
Twisted torus with
f-flux
Non-geometric space with
Q-flux
Non-geometric space with
R-flux
[XiH,f , Xj
H,f ] = 0
Summary:
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17
Tx1 Tx2 Tx3Flat torus with H-flux
Twisted torus with
f-flux
Non-geometric space with
Q-flux
Non-geometric space with
R-flux
[XiH,f , Xj
H,f ] = 0 [X1Q, X2
Q] ' Q p3
Summary:
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17
Tx1 Tx2 Tx3Flat torus with H-flux
Twisted torus with
f-flux
Non-geometric space with
Q-flux
Non-geometric space with
R-flux
[XiH,f , Xj
H,f ] = 0
[[X1R, X2
R], X3R] ' R
[X1Q, X2
Q] ' Q p3
Summary:
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17
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
Tx1 Tx2 Tx3Flat torus with H-flux
Twisted torus with
f-flux
Non-geometric space with
Q-flux
Non-geometric space with
R-flux
[XiH,f , Xj
H,f ] = 0
[[X1R, X2
R], X3R] ' R
[X1Q, X2
Q] ' Q p3
Summary:
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19
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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19
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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19
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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20
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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22
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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22
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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17
(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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24
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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25
⇒ 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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25
⇒ 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.
Open string non-commutativity:
[xi, xj ] = ✓ijConstant Poisson structure:
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25
⇒ 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.
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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26
Are the similar structures for closed strings?
Deformed theory of gravity?
⇒ Tri-product.
Possibly yes, but only off-shell.
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26
[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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27
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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28
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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29
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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30
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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31
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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32
was already derived in CFT from the multiplication of 3 tachyon vertex operators:43
R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arXiv:1106.0316
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
Effective field theory description of non-geometric spaces:
(i) Geometric (H,f)-spaces: Standard supergravity.
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
Effective field theory description of non-geometric spaces:
(i) Geometric (H,f)-spaces: Standard supergravity.
(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)
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
Effective field theory description of non-geometric spaces:
(i) Geometric (H,f)-spaces: Standard supergravity.
(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)
IV) Double geometry - double field theoryW. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...)
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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
• O(D,D) invariant effective string action containing momentum and winding coordinates at the same time:
Double field theory:
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
• O(D,D) invariant effective string action containing momentum and winding coordinates at the same time:
Double field theory:
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
Dimensional reduction of double field theory:O. Hohm, D. Lüst, B. Zwiebach, arXiv:1309.2977;F. Hassler, D. Lüst, arXiv:1401.5068.
• Consistent DFT solutions (generalized Scherk-Schwarz compactifications): RMN = 0
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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.
• The corresponding backgrounds are in general non- geometric and go beyond dimensional reduction of 10D SUGRA or generalized geometry.
• Consistent DFT solutions (generalized Scherk-Schwarz compactifications): RMN = 0
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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.
• The corresponding backgrounds are in general non- geometric and go beyond dimensional reduction of 10D SUGRA or generalized geometry.
• Consistent DFT solutions (generalized Scherk-Schwarz compactifications): RMN = 0
• Killing vectors, , correspond to generalized diffeomorphisms that depend on and .
L⇠EAM = 0
xm x
m
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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.
• The corresponding backgrounds are in general non- geometric and go beyond dimensional reduction of 10D SUGRA or generalized geometry.
• Consistent DFT solutions (generalized Scherk-Schwarz compactifications): RMN = 0
• 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
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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 .
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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
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 .
�(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
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 .
�(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:
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
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 .
�(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:
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
Background satisfies strong
constraint
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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 .
�(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:
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
Background satisfies strong
constraint
Patching does not satisfy strong
constraint
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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 .
�(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:
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
Background satisfies strong
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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V) Outlook & open questions
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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V) Outlook & open questions
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.
Mittwoch, 7. Mai 14
![Page 87: 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](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a87a1f02d8a6b4c61ac6b/html5/thumbnails/87.jpg)
V) Outlook & open questions
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.
Non-associativity is an off-shell phenomenon!
Mittwoch, 7. Mai 14
![Page 88: 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](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a87a1f02d8a6b4c61ac6b/html5/thumbnails/88.jpg)
V) Outlook & open questions
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.
Non-associativity is an off-shell phenomenon!
● 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
![Page 89: 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](https://reader034.vdocument.in/reader034/viewer/2022042909/5f3a87a1f02d8a6b4c61ac6b/html5/thumbnails/89.jpg)
● 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))
V) Outlook & open questions
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.
Non-associativity is an off-shell phenomenon!
● 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