closed strings and non-commutative/non …outline: 2 ii) non-geometric flux backgrounds...
TRANSCRIPT
1Hong Kong HKUST, 16. December 2013
Closed Strings and Non-commutative/Non-Associative
Geometry
DIETER LÜST (LMU, MPI)
Montag, 16. Dezember 13
1Hong Kong HKUST, 16. December 2013
In collaboration with I. Bakas, arXiv:1309.3172R. Blumenhagen, M. Fuchs, R. Sun, arXiv:1312.0719
Closed Strings and Non-commutative/Non-Associative
Geometry
DIETER LÜST (LMU, MPI)
Montag, 16. Dezember 13
Outline:
2
II) Non-geometric flux backgrounds
Non-commutative & non-associative algebras from the closed string world-sheet
I) Introduction
III) 3-Cocycles, 2-chains and star-producs
Lie algebra & Lie group cohomology
IV) Double geometry - double field theory
V) Magnetic field analogue of NC/NAThe fate of angular symmetry
Montag, 16. Dezember 13
Point particles in classical Einstein gravity „see“ continuous Riemannian manifolds.
Geometry in general depends on, with what kind of objects you test it.
Strings may see space-time in a different way.
At the string scale : Ls
3
We expect the emergence of a new kind of stringy geometry.
I) Introduction
Montag, 16. Dezember 13
Recall standard Riemannian geometry:
- Flat space: Triangle: � + ⇥ + ⇤ = ⌅
- Curved space: Triangle: � + ⇥ + ⇤ > ⌅(< ⌅)
Manifold: need different coordinate charts, which are patched together by coordinates transformations, i.e.group of diffeomorphisms: Di�(M) : f : U � U �
4[Xi, Xj ] = 0-
Montag, 16. Dezember 13
Now we want to understand, how extended closed strings may possibly see the (non)-geometry of space.
5
● Non-commutative closed string geometry
● Non-associative closed string geometry
Here new observation for closed strings on non-geometric backgrounds:
R. Blumenhagen, E. Plauschinn, arXiv:1010.1263;
D.L., arXiv:1010.1361;
R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arXiv:1106.0316
This is in contrast to open string non-commutativity for D-branes with gauge flux ⇒Non-commutative gauge theories (Seiberg, Witten)
C. Condeescu, I. Florakis, D. L., JHEP 1204 (2012), 121, arXiv:1202.6366
(Matrix models: T. Chatzistavrakidis, L. Jonke, arXiv:1207.6412, arXiv:1305.1864)
Montag, 16. Dezember 13
Non-geometric backgrounds are generic within the landscape of string „compactifications“.Several potentially interesting applications in string phenomenology and cosmology.Features:
II) Non-geometric flux backgrounds
Montag, 16. Dezember 13
Non-geometric backgrounds are generic within the landscape of string „compactifications“.Several potentially interesting applications in string phenomenology and cosmology.Features:
● They are only consistent in string theory.
II) Non-geometric flux backgrounds
Montag, 16. Dezember 13
Non-geometric backgrounds are generic within the landscape of string „compactifications“.Several potentially interesting applications in string phenomenology and cosmology.Features:
● They are only consistent in string theory.
● Make use of string symmetries, T-duality ⇒ T-folds,
II) Non-geometric flux backgrounds
Montag, 16. Dezember 13
Non-geometric backgrounds are generic within the landscape of string „compactifications“.Several potentially interesting applications in string phenomenology and cosmology.Features:
● They are only consistent in string theory.
● 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,
II) Non-geometric flux backgrounds
Montag, 16. Dezember 13
Non-geometric backgrounds are generic within the landscape of string „compactifications“.Several potentially interesting applications in string phenomenology and cosmology.Features:
● They are only consistent in string theory.
● Effective gauged supergravities with non-geometric fluxes(Dall‘Agata, Villadoro, Zwirner, 2009; Nicolai, Samtleben,2001; Shelton, Taylor, Wecht, 2005; Dibitetto, Linares, Roest, 2010; ...)
● 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,
II) Non-geometric flux backgrounds
Montag, 16. Dezember 13
Non-geometric backgrounds are generic within the landscape of string „compactifications“.Several potentially interesting applications in string phenomenology and cosmology.Features:
● They are only consistent in string theory.
● Effective gauged supergravities with non-geometric fluxes(Dall‘Agata, Villadoro, Zwirner, 2009; Nicolai, Samtleben,2001; Shelton, Taylor, Wecht, 2005; Dibitetto, Linares, Roest, 2010; ...)
● 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,
II) Non-geometric flux backgrounds
Montag, 16. Dezember 13
7
XM = (X̃i, Xi)
The non-geometric backgrounds are best described by doubling closed string coordinates and momenta:
- Coordinates: O(D,D) vector
- Momenta: O(D,D) vector winding momentum
pM = (p̃i, pi)
T-duality:
⇔
T-duality
(Here D=3)
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8
Non-geometric string backgrounds:(Hellerman, McGreevy, Williams (2002);
Shelton, Taylor, Wecht, 2005;Dabholkar, Hull, 2005)
Montag, 16. Dezember 13
8
Non-geometric string backgrounds:(Hellerman, McGreevy, Williams (2002);
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)(See also recent discussion in terms of generalized coordinate transformations:
O. Hohm, D.L., B. Zwiebach, arXiv:13.09.2977)
Montag, 16. Dezember 13
8
Non-geometric string backgrounds:(Hellerman, McGreevy, Williams (2002);
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)(See also recent discussion in terms of generalized coordinate transformations:
O. Hohm, D.L., B. Zwiebach, arXiv:13.09.2977)
Montag, 16. Dezember 13
8
Non-geometric string backgrounds:(Hellerman, McGreevy, Williams (2002);
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)(See also recent discussion in terms of generalized coordinate transformations:
O. Hohm, D.L., B. Zwiebach, arXiv:13.09.2977)
Montag, 16. Dezember 13
8
Non-geometric string backgrounds:(Hellerman, McGreevy, Williams (2002);
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)(See also recent discussion in terms of generalized coordinate transformations:
O. Hohm, D.L., B. Zwiebach, arXiv:13.09.2977)
Montag, 16. Dezember 13
9
Three-dimensional flux backgrounds:
Fibrations: 2-dim. torus that varies over a circle:
T 2X1,X2 ,! M3 ,! S1
X3
We considered two different types of models:
Montag, 16. Dezember 13
17
Tx1 Tx2 Tx3
(i) (Non-)geometric backgrounds with parabolic monodromy and constant 3-form fluxes:
Flat torus with H-flux
Twisted torus with
f-flux
Non-geometric space with
Q-flux
Non-geometric space with
R-flux
The fibre torus depends linearly on the third coordinate.
Montag, 16. Dezember 13
17
Tx1 Tx2 Tx3
(i) (Non-)geometric backgrounds with parabolic monodromy and constant 3-form fluxes:
Flat 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
The fibre torus depends linearly on the third coordinate.
Montag, 16. Dezember 13
17
Tx1 Tx2 Tx3
(i) (Non-)geometric backgrounds with parabolic monodromy and constant 3-form fluxes:
Flat 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
The fibre torus depends linearly on the third coordinate.
[X1Q, X2
Q] ' Q p̃3
Montag, 16. Dezember 13
17
Tx1 Tx2 Tx3
(i) (Non-)geometric backgrounds with parabolic monodromy and constant 3-form fluxes:
Flat 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
The fibre torus depends linearly on the third coordinate.
[X1Q, X2
Q] ' Q p̃3
Montag, 16. Dezember 13
17
Tx1 Tx2 Tx3
(i) (Non-)geometric backgrounds with parabolic monodromy and constant 3-form fluxes:
Flat 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
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
The fibre torus depends linearly on the third coordinate.
[X1Q, X2
Q] ' Q p̃3
Montag, 16. Dezember 13
11
The algebra of commutation relation looks different in each of the four duality frames.
T-duality is mapping these commutators and 3-brackets onto each other.
Non-vanishing commutators and 3-brackets:
T-dual frames Commutators Three-brackets
H-flux [x̃1, x̃2] ⇠ Hp̃3
[x̃1, x̃2, x̃3] ⇠ H
f -flux [x1, x̃2] ⇠ fp̃3
[x1, x̃2, x̃3] ⇠ f
Q-flux [x1, x2] ⇠ Qp̃3
[x1, x2, x̃3] ⇠ Q
R-flux [x1, x2] ⇠ Rp3
[x1, x2, x3] ⇠ R
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17
(ii) (Non-)geometric backgrounds with elliptic monodromy and non-geometric fluxes.
They can be described in terms of (a)symmetric freely acting orbifolds. C. Condeescu, I. Florakis, D. L., JHEP 1204 (2012), 121, arXiv:1202.6366
C. Condeescu, I. Florakis, C. Kounnas, D.L., arXiv:1307:0999
D. L., JHEP 1012 (2011) 063, arXiv:1010.1361,
In general not T-dual to a geometric space!
⌧(x3) = ⌧0 cos(fx
3)+sin(fx
3)
cos(fx
3)�⌧0 sin(fx
3)
, f 2 1
8
+ Z ,
⇢(x3) = ⇢0 cos(Qx
3)+sin(Qx
3)
cos(Qx
3)�⇢0 sin(Qx
3)
, Q 2 1
8
+ Z .
The fibre torus depends on the third coordinate in a more complicate way.
Complex structure and Kähler parameter of the 2-torus:
Montag, 16. Dezember 13
13
The corresponding commutators can be explicitly derived in CFT.
More complicate, non-linear commutation relations:
[x1, x
2] = [x̃1, x
2] = [x1, x̃
2] = [x̃1, x̃
2] = i⇥(p3)
⇥(p3) =
⇡
2
cot(⇡p3R)
R-frame:
This algebra cannot be T-dualized to a commutative algebra!
Montag, 16. Dezember 13
14
● Double (field theory) geometry - Siegel (1993), Hull, Zwiebach (2009), Hohm, Hull, Zwiebach (2010)
⇒ Non-commutative and non-associative algebras.
What is the mathematical framework to describe non-geometric string backgrounds?
● Group theory cohomology - Hochschild (1945), Cartan, Eilenberg (1956)
⇒ 3-cocycles, 2-cochains, - products?
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15
III) 3-Cocycles, 2-cochains and star-products 3.1. Deformations of Lie algebras
[xi, p
j ] = i~�
ij, [pi
, p
j ] = 0
Associator - 3-cocycle
Jacobiator - 2-cochain
This can be (equivalently) called either
Parabolic model in R-flux frame:
[xi, x
j ] ⇠ l
3sR ✏
ijkpk
[x1, x
2, x
3] := [[x1, x
2], x
3] + cycl. perm. ⇠ ~l
3sR
Montag, 16. Dezember 13
Hence this structure can be viewed as 3 kind of (non-associative) deformations resp. contractions:
There are two deformation parameters: & ls ~
(A) Deformation (double contraction) of Abelian algebra:
ls, ~! 0
(B) Deformation (single contraction) of Heisenberg algebra:
finite, ls ! 0~Contraction Structure of commutators Name of Lie algebra
ls = 0, ~ = 0 [xi, x
j ] = 0, [xi, p
j ] = 0 Algebra of translations t6
ls = 0, ~ 6= 0 [xi, x
j ] = 0, [xi, p
j ] = i~�
ij Heisenberg algebra g
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17
3.2 3-cocycles in Lie algebra cohomology(A) Double contraction limit:
Start from Abelian algebra of translations:
with generators[TA, TB ] = 0 TI = x
i, p
i
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17
3.2 3-cocycles in Lie algebra cohomology(A) Double contraction limit:
Start from Abelian algebra of translations:
with generators[TA, TB ] = 0 TI = x
i, p
i
3-cocycle with real-valued coefficients:
Deformation:
c3(x1, x
2, x
3) = 1 c3(TI , TJ , TK) = 0,
Non-trivial cohomology group of this algebra:
otherwise
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18
Associator [x1, x
2, x
3] = �c3(x1, x
2, x
3)
c3([TI , TJ ], TK , TL)� c3([TI , TK ], TJ , TL) + c3([TI , TL], TJ , TK) +c3([TJ , TK ], TI , TL)� c3([TJ , TL], TI , TK) + c3([TK , TL], TI , TJ) = 0
namely
c3 satisfies the 3-coycle condition
dc3(TI , TJ , TK , TL) = 0
Montag, 16. Dezember 13
18
Associator [x1, x
2, x
3] = �c3(x1, x
2, x
3)
c3([TI , TJ ], TK , TL)� c3([TI , TK ], TJ , TL) + c3([TI , TL], TJ , TK) +c3([TJ , TK ], TI , TL)� c3([TJ , TL], TI , TK) + c3([TK , TL], TI , TJ) = 0
namely
c3 satisfies the 3-coycle condition
dc3(TI , TJ , TK , TL) = 0
c3 is a genuine 3-cocycle, it is not a coboundary of a 2-cochain:
c3 6= dc2(TI , TJ , TK)
Montag, 16. Dezember 13
19
(B) Deformation of Heisenberg algebra:
2-cochain taking values in the Heisenberg algebra:
Deformation:
c2(xi, x
j) = ✏
ijkp
k
[xi, x
j ] = ic2(xi, x
j) ⇠ i✏
ijkp
k
[xi, p
j ] = i~�
ij, [pi
, p
j ] = 0
[xi, x
j ] = 0
Montag, 16. Dezember 13
20
dc2(x1, x
2, x
3) = [x1, c2(x2
, x
3)]� [x2, c2(x1
, x
3)] + [x3, c2(x1
, x
2)]
(Difference between (A) and (B): A trivial cocycle in one cohomology might not be trivial in another.)
Now the Jacobiator is determined by the action of the coboundary operator d on , taking the following form in Chevalley-Eilenberg cohomology:
c2
c3 = [x1, x
2, x
3] = �dc2(x1, x
2, x
3)
Montag, 16. Dezember 13
21
3.3 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).
Again this can be described by two different deformations:
Montag, 16. Dezember 13
22
⇒ Product law of three group elements becomes
non-associative:
Non-associativity is determined by the following 3-cocycle: '3(~a1,~a2,~a2) = (~a1 ⇥ ~a2) · ~a3
3-cocycle: volume of tetrahedon:
(A) Deformation by group 3-cocycle:
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)
⌘.
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23
d'3(~a1,~a2,~a3,~a4) = '3(~a2,~a3,~a4)� '3(~a1 + ~a2,~a3,~a4) + '3(~a1,~a2 + ~a3,~a4)�'3(~a1,~a2,~a3 + ~a4) + '3(~a1,~a2,~a2) = 0
Group 3-cocycle condition is satisfied:
This can be also derived from the product of four group elements:
(U1U2)(U3U4) = ei ⇡2R12 [(~a1+~a2)·(~a3⇥~a4)�~a3·(~a1⇥~a2)](U1(U2U3))U4 ,
= e�i ⇡2R12 [(~a3+~a4)·(~a1⇥~a2)�~a2·(~a3⇥~a4)]U1((U2U3)U4) ,
= ei ⇡2R12 (~a1+~a2)·(~a3⇥~a4)((U1U2)U3)U4 ,
= e�i ⇡2R12 (~a3+~a4)·(~a1⇥~a2)U1(U2(U3U4))
Montag, 16. Dezember 13
23
d'3(~a1,~a2,~a3,~a4) = '3(~a2,~a3,~a4)� '3(~a1 + ~a2,~a3,~a4) + '3(~a1,~a2 + ~a3,~a4)�'3(~a1,~a2,~a3 + ~a4) + '3(~a1,~a2,~a2) = 0
Group 3-cocycle condition is satisfied:
This can be also derived from the product of four group elements:
(U1U2)(U3U4) = ei ⇡2R12 [(~a1+~a2)·(~a3⇥~a4)�~a3·(~a1⇥~a2)](U1(U2U3))U4 ,
= e�i ⇡2R12 [(~a3+~a4)·(~a1⇥~a2)�~a2·(~a3⇥~a4)]U1((U2U3)U4) ,
= ei ⇡2R12 (~a1+~a2)·(~a3⇥~a4)((U1U2)U3)U4 ,
= e�i ⇡2R12 (~a3+~a4)·(~a1⇥~a2)U1(U2(U3U4))
Mac Lane‘s pentagon:
(U1U2)(U3U4)
U1(U2(U3U4))((U1U2)U3)U4
U1((U2U3)U4) (U1(U2U3))U4
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23
d'3(~a1,~a2,~a3,~a4) = '3(~a2,~a3,~a4)� '3(~a1 + ~a2,~a3,~a4) + '3(~a1,~a2 + ~a3,~a4)�'3(~a1,~a2,~a3 + ~a4) + '3(~a1,~a2,~a2) = 0
Group 3-cocycle condition is satisfied:
This can be also derived from the product of four group elements:
(U1U2)(U3U4) = ei ⇡2R12 [(~a1+~a2)·(~a3⇥~a4)�~a3·(~a1⇥~a2)](U1(U2U3))U4 ,
= e�i ⇡2R12 [(~a3+~a4)·(~a1⇥~a2)�~a2·(~a3⇥~a4)]U1((U2U3)U4) ,
= ei ⇡2R12 (~a1+~a2)·(~a3⇥~a4)((U1U2)U3)U4 ,
= e�i ⇡2R12 (~a3+~a4)·(~a1⇥~a2)U1(U2(U3U4))
Mac Lane‘s pentagon:
(U1U2)(U3U4)
U1(U2(U3U4))((U1U2)U3)U4
U1((U2U3)U4) (U1(U2U3))U4
However is not a coboundary of a 2-chain!'3
Montag, 16. Dezember 13
24
(B) Deformation by group 2-cochain:
Group cohomology of the Heisenberg-Weyl group:
UW
(g) = ei(~a·~x+~
b·~p+c1)
UW (g1)UW (g2) = e�i ⇡2R12 '2(g1,g2)UW (g1g2)
Violation of Jacobi identity:
3-cocycle: '3 = d'2(g1, g2, g3) = (~a1 ⇥ ~a2) · ~a3
'2(g1, g2) = (~a1 ⇥ ~a2) · ~pNon-commutativity is determined by the following
2-cochain:
Montag, 16. Dezember 13
25
[~a1 · ~x, ~a2 · ~x] = i
⇡
2R
6(~a1 ⇥ ~a2) · ~p , [~a · ~x,
~
b · ~p] = i ~a ·~bCommutator:
Jacobiator:
[~a1 · ~x, ~a2 · ~x, ~a3 · ~x] = i
⇡
2R
2(~a1 ⇥ ~a2) · ~a3
3.4 Derivation of the star product
Recall:
The 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)
Montag, 16. Dezember 13
26
(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
The same expression has already appeared in the work by Mylonas, Schupp, Szabo (arXiv:1207.0926) when considering the quantization of the membrane - model following Kontsevich‘s deformation quantization approach.
�
Here we derived it in a direct way.
(B) Via deformation by group 2-cochain:
Montag, 16. Dezember 13
27
This star product is non-associative:
Consistent with: [xi, x
j, x
k] = 3iR ✏
ijk
((f1 ?
p
f2) ?
p
f3) (~x) = e
i
R
ijk
2 @
x1i
@
x2j
@
x3k
f1(~x1)f2(~x2)f3(~x3)|~x1=~x2=~x3=~x
(f1 ?
p
(f2 ?
p
f3)) (~x) = e
�i
R
ijk
2 @
x1i
@
x2j
@
x3k
f1(~x1)f2(~x2)f3(~x3)|~x1=~x2=~x3=~x
Montag, 16. Dezember 13
28
(A) Via deformation by group 3-cocycle:
Similar procedure leads to the following 3-product:
(This is trivial for the product of two functions.)
agrees with - product of three functions :�3?p f(x)
(f143 f243 f3)(~x) = ((f1 ?p f2) ?p f3) (~x)
This delta-product is again non-associative.
(f143 f243 f3)(~x) = e
iR
ijk
@
x1i
@
x2j
@
x3k
f1(~x1) f2(~x2) f3(~x3)|~x
Montag, 16. Dezember 13
29
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)
⌦Vp1 Vp2 Vp3
↵R
=
⌦Vp1 Vp2 Vp3
↵R=0
⇥
exp
h�iRijk p1,ip2,jp3,k
⇥L
�z12z13
�+ L
�z12z13
�⇤i.
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)
Montag, 16. Dezember 13
30
IV) Double geometry - double field theory
Montag, 16. Dezember 13
30
IV) Double geometry - double field theory
So far this is only half of the story of the phase space!
Montag, 16. Dezember 13
30
IV) Double geometry - double field theory
So far this is only half of the story of the phase space!
Corresponding Lie group
Remember that we have four T-duality frames:
Consider cohomology of Lie algebra: g � g̃
GW ⇥ G̃W
We like to obtain a T-duality covariant description that includes .
x
i, x̃i, pi, p̃
i
Montag, 16. Dezember 13
31
Corresponding group element:
f � flux : '2(g1, g2; g̃1, g̃2) = [(~a1 ⇥ ~c2)� (~a2 ⇥ ~c1)] · ~̃p
H � flux : '2(g1, g2; g̃1, g̃2) = (~c1 ⇥ ~c2) · ~̃p
Q� flux : '2(g1, g2; g̃1, g̃2) = (~a1 ⇥ ~a2) · ~̃p
Four different 2-cochains:
U(~a, ~b;~c, ~d) = ei(~a·~x+~
b·~p+~c·~x̃+~
d·~p̃)
R� flux : '2(g1, g2; g̃1, g̃2) = (~a1 ⇥ ~a2) · ~p
Montag, 16. Dezember 13
32
Construction of double geometry star product:
U(~a1, ~b1; ~c1, ~d1)U(~a2, ~b2; ~c2, ~d2) = e�i2 (~a1·~b2�~a2·~b1+~c1·~d2�~c2·~d1)
U
✓~a1 + ~a2, ~b1 +~b2 �
⇡2R
12(~a1 ⇥ ~a2); ~c1 + ~c2, ~d1 + ~d2
◆
(f1 ?
p,p̃
f2)(~x, ~p; ~x̃,
~
p̃) = e
i
⇡
2R
12 ~p·(~rx1⇥~r
x2 )e
i
2 (~rx1 ·~r
p2�~rx2 ·~r
p1)e
i
2 (~rx̃1 ·~r
p̃2�~rx̃2 ·~r
p̃1)
⇥ f1(~x1, ~p1; ~x̃1,~
p̃1)f2(~x2, ~p2; ~x̃2,~
p̃2)|~x1=~x2=~x,
~
x̃1=~
x̃2=~
x̃, ~p1=~p2=~p,
~
p̃1=~
p̃2=~
p̃
O(3,3) invariant, i.e. independent from the chosen duality frame.
Product of two group elements:
Star product:
Montag, 16. Dezember 13
33
Can be written as:
(f1 ?
p,p̃
f2)(~x,
~
x̃, ~p,
~
p̃) = e
i2⇥IJ (p,p̃) @I⌦@J (f1 ⌦ f2)|
~x,
~
x̃, ~p,
~
p̃
with 12-dimensional doubled Poisson tensor:
⇥IJ =
0
BB@
Rijkpk 0 �ij 0
0 0 0 �ji
��ji 0 0 0
0 ��ij 0 0
1
CCA .
(For elliptic models, which are not T-dual to geometric models, the doubled description is absolutely required.)
I. Bakas, D. L., work in progress.
Montag, 16. Dezember 13
34
Double Field Theory:Provides an O(D,D) invariant effective string action containing momentum and winding coordinates at the same time.
Flux formulation of DFT:
(Siegel; Hohm, Hull, Zwiebach)
Generalized metric: HMN =✓
gij �gikbkj
bikgkj gij � bikgklblj
◆
SDFT =Z
dX e�2d
FABCFA0B0C0
⇣14SAA0
⌘BB0⌘CC0
� 112
SAA0SBB0
SCC0� 1
6⌘AA0
⌘BB0⌘CC0
⌘+ FAFA0
⇣⌘AA0
� SAA0⌘�
Fabc = Habc , Fabc = F a
bc , Fcab = Qc
ab , Fabc = Rabc
(Geissbuhler, Marques, Nunez, Penas; Aldazabal, Marques, Nunez)
Strong constraint: @M@M = 0 , @Mf @Mg = DAf DAg = 0
However the strong constraint could be relaxed for truly non-geometric spaces by required only the closure of the flux symmetry algebra.
(Grana, Marques)
Montag, 16. Dezember 13
35
(R. Blumenhagen, M. Fuchs, D.Lüst, E. Plauschinn, R. Sun, arXiv:1312.0719)
Non-associative deformations in double field theory:
DFT generalization of 3-product:
Imposing the strong constraint the additional term becomes a total derivative and vanishes upon integration in the DFT action.(This is analogous to momentum conservation in on-shell CFT amplitudes.)
Relaxing the strong constraint for truly non-geometric backgrounds leads to a new condition, if one requires the on-shell absence of non-associativity in DFT:
f 43 g43 h = f g h +`4s6FABC DAf DBgDCh + O(`8s)
⇣AB DAf DBg = 0 , ⇣AB = brCFCAB � 2(DCd)FCAB
Montag, 16. Dezember 13
36
V) Magnetic field analogue of NC/NAA spinless point-particle (e,m) in magnetic field background has commutation relations among its coordinates and momenta:
~
B(~x)
[xi, p
j ] = i�
ij, [xi
, x
j ] = 0 , [pi, p
j ] = ie ✏
ijkBk(~x)
leading to non-commutativity of in Maxwell theory withpi
~r · ~B = 0 .Dirac‘s generalization: ~r · ~B 6= 0 .[[pi, pj ], pk] + cyclic ⌘ [pi, pj , pk] = �e ✏ijk ~r · ~B
Associativity is lost in presence of magnetic charges,in analogy to string theory in non-geometric backgrounds with
x
i $ p
i
Montag, 16. Dezember 13
37
Particular solution of the inhomogeneous equation:
Some notable examples are:
● Dirac monopole
Consider a continuous spherically symmetric distribution of magnetic charges, , to study (some of) the implications of NC/NA in classical and quantum theory. We have
⇢(x)
~r · ~
B = ⇢(x) ,
~r⇥ ~
B = 0 (static) (x2 = ~x · ~x)
~
B(~x) =~x
f(x), ⇢(x) =
3f(x)� xf
0(x)f
2
f(x) = x
3/g , ⇢(x) = 4⇡g�(x)
● Constant magnetic charge (cfr. parabolic flux)
f(x) = 3/⇢ = const.
Study the dynamics of point particle for general profile function .f(x)
Montag, 16. Dezember 13
38
Using the Hamiltonian , the Lorentz force on the spinless particle is:
H = ~p · ~p/2m
d
2~x
dt
2= � e
m
1f(x)
✓~x⇥ d~x
dt
◆
d~p
dt= i[H, ~p] =
e
2m(~p⇥ ~B � ~B ⇥ ~p)
which for takes the form~
B(~x) = ~x/f(x)
Lorentz force is proportional to angular momentum and does no work.Energy conservation provides one integral of motion:
E = A/2m , x
2(t) = At
2 + D
D : closest distance to origin (perihelion of trajectory).
Montag, 16. Dezember 13
39
Angular symmetry is broken in the presence of magnetic charges!
d
dt
✓~x⇥ d~x
dt
◆= � e
m
1f(x)
~x⇥✓
~x⇥ d~x
dt
◆=
e
m
x
3
f(x)dx̂
dt
.
For general choices of there are no additional integrals of motion, since
f(x)
f(x) = x
3/g .
The only exemption is the magnetic monopole having
In this case the improved angular momentum is conserved:
~
K = ~x⇥ d~x
dt
� eg
m
x̂
Poincare vector.
Montag, 16. Dezember 13
40
In all other cases, including constant , angular symmetry is broken and the classical motion of the particle appears to be non-integrable.
f(x)
Trajectory of a point particle in the field of a magnetic monopole:
● The magnetic monopole is located at the tip of the cone.
● The Poincare vector provides the axis of the cone . ~K
● The particle precesses with velocity . ~K/(At2 + D)
● determines the opening angle of the cone. ~
K · x̂ = �eg/m
Montag, 16. Dezember 13
41
Non-associativity is responsible for the violation of angular symmetry.
The apparent violation of associativity in a Dirac monopole field is eliminated by imposing the boundary condition
Single valuedness leads to Dirac‘s quantization:
(derivations) are represented by self-adjoint operators, even though that they are defined in patches as
~p
(0) = 0 )
Rotations by an angle are described by✓
R(✓) = e�i✓x̂· ~
J = e�ieg✓
eg = n 2 Z
~p = �i~r� e ~A
Montag, 16. Dezember 13
42
In all other cases, non-associativity is for real, obstructing canonical quantization.
What can be used as substitute?
?p � product ?
Montag, 16. Dezember 13
V) Outlook & open questions
43
Montag, 16. Dezember 13
V) Outlook & open questions
43
● Systematic description of non-associative string geometry in terms of deformation theory of Lie algebras and their cohomology.
Montag, 16. Dezember 13
V) Outlook & open questions
43
● Systematic description of non-associative string geometry in terms of deformation theory of Lie algebras and their cohomology. ● However the non-associativity is not visible in CFT
amplitudes respectively in the effective DFT action upon demanding the string constraint.
Montag, 16. Dezember 13
V) Outlook & open questions
43
● Systematic description of non-associative string geometry in terms of deformation theory of Lie algebras and their cohomology.
● What is the relation to non-associative generalized coordinate transformations in double geometry?
● However the non-associativity is not visible in CFT amplitudes respectively in the effective DFT action upon demanding the string constraint.
Montag, 16. Dezember 13
● Is there are non-commutative (non-associative) theory of gravity? (Non-commutative geometry & gravity: P. Aschieri, M. Dimitrijevic, F. Meyer, J. Wess (2005),
L. Alvarez-Gaume, F. Meyer, M. Vazquez-Mozo (2006))
V) Outlook & open questions
43
● Systematic description of non-associative string geometry in terms of deformation theory of Lie algebras and their cohomology.
● What is the relation to non-associative generalized coordinate transformations in double geometry?
● However the non-associativity is not visible in CFT amplitudes respectively in the effective DFT action upon demanding the string constraint.
Montag, 16. Dezember 13
● Is there are non-commutative (non-associative) theory of gravity? (Non-commutative geometry & gravity: P. Aschieri, M. Dimitrijevic, F. Meyer, J. Wess (2005),
L. Alvarez-Gaume, F. Meyer, M. Vazquez-Mozo (2006))
● What is the generalization of quantum mechanics for this non-associative geometry? How to represent this algebra?
V) Outlook & open questions
43
● Systematic description of non-associative string geometry in terms of deformation theory of Lie algebras and their cohomology.
● What is the relation to non-associative generalized coordinate transformations in double geometry?
● However the non-associativity is not visible in CFT amplitudes respectively in the effective DFT action upon demanding the string constraint.
Montag, 16. Dezember 13