closed strings and non-commutative/non …outline: 2 ii) non-geometric ﬂux backgrounds...

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)


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


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


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-


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)


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,





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






● Effective gauged supergravities with non-geometric fluxes(Dall‘Agata, Villadoro, Zwirner, 2009; Nicolai, Samtleben,2001; Shelton, Taylor, Wecht, 2005; Dibitetto, Linares, Roest, 2010; ...)







● 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





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)


8

Non-geometric string backgrounds:(Hellerman, McGreevy, Williams (2002);

Shelton, Taylor, Wecht, 2005;Dabholkar, Hull, 2005)


8



[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)


8



- 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)


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:


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


R-flux

The fibre torus depends linearly on the third coordinate.


17

Tx1 Tx2 Tx3



Twisted torus with

f-flux


Q-flux


R-flux

[XiH,f , Xj

H,f ] = 0



17

Tx1 Tx2 Tx3



Twisted torus with

f-flux


Q-flux


R-flux

[XiH,f , Xj

H,f ] = 0


[X1Q, X2

Q] ' Q p̃3


17

Tx1 Tx2 Tx3



Twisted torus with

f-flux


Q-flux


R-flux

[XiH,f , Xj

H,f ] = 0

[[X1R, X2

R], X3R] ' R


[X1Q, X2

Q] ' Q p̃3


17

Tx1 Tx2 Tx3



Twisted torus with

f-flux


Q-flux


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


[X1Q, X2

Q] ' Q p̃3


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


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:


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!


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?


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


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


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


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


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


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)


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


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)


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:


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)

⌘.


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))


23





= 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


23





= 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


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:


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)


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:


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


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


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)


30



30


So far this is only half of the story of the phase space!


30


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


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


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:


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.


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)


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


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


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)


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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).


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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.


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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


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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


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In all other cases, non-associativity is for real, obstructing canonical quantization.

What can be used as substitute?

?p � product ?


V) Outlook & open questions

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● Systematic description of non-associative string geometry in terms of deformation theory of Lie algebras and their cohomology.



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.



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● 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.


● 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))


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● 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?


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closed strings and non-commutative/non …outline: 2 ii) non-geometric ﬂux backgrounds...

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