Nonassociativity from higher form gauge fields

Leonardo Castellani !

University of Piemonte Orientale and INFN Torino

(Zakopa)(ne)  ≠  (Za)(kopane)      

February  6-­‐12,  2016  

Quantum spacetime ‘16

Geometric theories with p-form gauge fields have a nonassociative symmetry structure, that extends an underlying Lie algebra

This nonassociativity is controlled by Chevalley!cohomology, which classifies free differential algebras, p-form generalizations of Cartan-Maurer eq.s

Possible relations with nonassociative structures in double phase space of flux backgrounds (involving p-forms) in closed string theory, controlled by Chevalley cohomology

Group geometric approach

To interpret all (local) symmetries as ! coordinate transformations!

Thus diff.s, supersymmetry, gauge transformations are! all diffeomorphisms in the (super)group manifold G!

They are invariances of an action invariant under ! group manifold diff.s

Dynamical fields: vielbeins on G

to (super) gauge and gravity theories

Group geometric construction of supergravity theories, Torino group 80’sReview: LC, D’Auria, Fré

• Lie (super)algebra G

• Cartan-Maurer eq.s

Basic steps

[TA, TB ] = CCABTC

On the group manifold G: basis of tangent vectors closes on the same Lie algebra.

tA

Dual (cotangent) basis: left-invariant one-forms , vielbeins of the group manifold

Jacobi id.s

• fundamental fields

• Bianchi identities

More precisely the dynamical fields are the vielbeins of a smooth deformation of , with curvature

measuring the deformation

Example: N=1, D=4 supergravity

• Lie algebra (superPoincaré)

• Dynamical fields

vierbein

spin connection

gravitino

G-coordinates

x

yab

✓↵

• Curvatures

Torsion

Lorentz curvature

gravitino curvature

• Bianchi identities

For example

or

• At this stage the dynamical fields depend on all! (soft) group manifold coordinates

• To remove these extra degrees of freedom

Ansätze on the curvatures:

- horizontality in the Lorentz directions (no legs along )

- components : linear comb. of components

• Bianchi identities become equations for the curvatures

• Solution for the curvatures

with given in terms of , and with

Einstein eq.s

gravitino eq.

• Action

• Having used only diffeomorphic invariant operations ! (exterior products, exterior derivative), the resulting theory! is geometric

• The invariances of the theory are then given by the! diff.s, generated by the Lie derivative along all tangent vectors ! of the (soft) group manifold G

• On the (soft)G vielbeins, the Lie derivative yields:

Invariances

• In the D=4 N=1 supergravity example:

Free differential algebras (FDA)

• generalize Cartan-Maurer eq.s of group manifold G ! 1-form vielbeins , by including p-forms

• convenient algebraic setting for field theories ! with antisymmetric tensors

dBi + CiAj �ABj +

1

(p+ 1)!Ci

A1...Ap+1�A1 ...�Ap+1 = 0

d�A +1

2CA

BC �B�C = 0

rBi

�A Bi

• example : ordinary Cartan-Maurer 1-forms supplemented! by a single p-form in a representation of GDi

j

�A

Bi

• taking of l.h.s. and requiring d d2 = 0

CAB[CC

BDE] = 0

CiAjC

jBk � Ci

BjCjAk = CC

ABCiCk

2 CiA1jC

jA2...Ap+2]

� (p+ 1)CiB[A1...Ap

CBAp+1Ap+2]

= 0

Generalized Jacobi identities

usual Jacobi id.s

representation condition

cocycle condition

• is a (p+1)-cocycle ( )Ci = CiA1...Ap+1

�A1 ...�Ap+1 rCi = 0

• given a FDA, there is a well-defined procedure! to construct a Lagrangian with the p-forms as ! fundamental fields

• To extend a Lie algebra to a FDA: need a ! covariantly closed (p+1)-form !

• given such a form , + covariantly closed (p+1)-form ! still yields a FDA. But if this cov. closed form is cov. exact (= ) ! leads to an equivalent FDA via the redefinition !

Ci

Ci Ci

r�i

Ci +r�i

Bi ! Bi + �i

• Thus inequivalent FDA’s are classified by nontrivial cohomology ! classes of the covariant derivative , i.e. by Chevalley cohomologyr

Example: FDA of D=11 supergravity

d!ab � !ac!

cb = 0 [= Rab]

dV a � !abV

b � i

2 �a = 0 [= Ra]

d � 1

4!ab�ab = 0 [= ⇢]

dA� 1

2 �ab V

aV b = 0 [= R(A)]

�ab �a = 0• the d=11 Fierz identity ensures !

FDA closure ( )!

d2 = 0

• extends the superPoincaré Lie algebra in d=11 with a 3-form A! in the identity representation

CiA1...Ap+1• C ↵�ab = �12(C�ab)↵�

nontrivial 4-cocycle

• The lagrangian of D=11 supergravity can be written as a 11-form,! made out of (exterior) products of fields and curvatures, therefore! invariant by construction under diff.s. !

• Infinitesmal diff.s are generated by Lie derivatives along all ! G directions

`✏AtA = i✏AtAd+ d i✏AtA

• Lie derivative along a generic tangent vector ! where are a basis for tangent vectors on G (dual to ):

✏AtAtA

• its action on FDA forms:

�A

`✏BtB�A = d✏A � CA

BC ✏B�C

`✏BtBBi = �Ci

Bj ✏BBj � 1

p!Ci

BA1...Ap✏B�A1 ...�Ap+1

• Note: , are a cotangent basis for the “FDA manifold”.! Then

�A Bi

itB�A = �AB itBB

i = 0

Extended Lie derivatives

• computing the commutator of two Lie derivatives:

h`✏A1 tA , `✏B1 tB

i= `[✏A1 @A✏C2 �✏A2 @A✏C1 +✏A1 ✏B2 CC

AB]tC

✏i =1

(p� 1)!✏A1 ✏

B2 Ci

ABA1...Ap�1�A1 ...�Ap�1

where

is a composite parameter (p-1)-form

extended Lie derivative+ `✏iti

i✏iti• The extended Lie derivative is constructed via a !

generalized contraction operator

• on a generic form in the “FDA manifold” :!

! = !i1...inA1...AmBi1 ...Bin�A1 ...�Am

the generalized contraction acts as:

i✏jtj! = n ✏j!ji2...inA1...AmBi2 ...Bin�A1 ...�Am

where is a (p-1)-form. Still maps p-forms into (p-1)-forms ✏i

• Then the extended Lie derivative is defined by the Cartan ! formula:!

`✏iti = i✏itid+ d i✏iti

The extended Lie derivative

• commutes with d• satisfies the Leibniz rule• acts on the fundamental FDA fields as:

`✏jtj�A = 0

`✏jtjBi = d✏i + Ci

Aj�A✏j

• closes on the algebra:

+ `✏iti⇥`✏AtA , `✏jtj

⇤= `[`✏AtA

✏k+CkBj✏

B✏j ]tkh`✏i1ti , `✏j2tj

i= 0

Perotto, LC , LMP1996, hep-th/9509031

The FDA dual algebra

✏

[`tA , `tB ] = CCAB`tC +

1

(p� 1)!Ci

ABA1...Ap�1`�A1 ...�Ap�1ti

[`tA , `�B1 ...�Bp�1ti] = [Ck

Ai�B1...Bp�1

C1...Cp�1� (p� 1)C [B1

AC1�B2...Bp�1]C2...Cp�1

�ki ] `�C1 ...�Cp�1tk

[`�A1 ...�Ap�1ti, `�B1 ...�Bp�1tj

] = 0

• By taking constant parameters the algebra of Lie derivatives! becomes

• Extends the G Lie algebra of ordinary Lie derivatives

• In the following use simplified notations:

TA1...Ap�1

i ⌘ `�A1 ...�Ap�1tiTA ⌘ `tA ,

Nonassociativity

JABC ⌘ [[TA, TB ], TC ] + cyclic in ABC =

=1

(p� 1)!((p� 1)Ci

EA2...Ap�1[ABCEC]A1

� CjA1A2...Ap�1[ABC

iC]j)T

A1...Ap�1

i

• Jacobiator

in general nonvanishing

FDA symmetries close a nonassociative algebra

LC jhep 2014 , 1310.7185

Example 1: D=11 supergravity

• Jacobiator for two supersymmetry generators ! and a Lorentz generator

J↵�[ab] = ⌘c[a(C�b]d)↵�Tcd

Q↵, Q�Mab

with

T cd ⌘ `V cV dt

• when applied to the three-form A:

T cdA = `V cV dtA = d(V cV d)

Example 2: flux backrounds in closed string theory

[TA, TB ] = CCABTC + C•

ABCTC•

• Choose a 2-form B in the identity representation

• Dual FDA algebra:

[TA, TB• ] = �CB

ACTC• + C•

A•TB•

[TA• , TB

• ] = 0

• Jacobiator:

• Name generators as:

• Choose structure constants as: Cb

qa = �ba, C bqa = �ba, C•

abc = Rabc, C•q• = 3

TA = (xa, xa, q), T

B• = (pb, pb, q0)

yields the algebra of the R-flux model (Bakas,Lüst 2014)

[xa, xb] = l

3sRabcp

c, [xa, p

b] = i~�ba q

0, [pa, pb] = 0

[xa, xb] = 0, [xa, pb] = i~�ba q

0, [pa, pb] = 0

where the generators have been rescaled as xa ! lSx

a pb ! 1

lSpb q0 ! i~q

Jacobiator

[xa, xb, xb] = ~l3sRabc

Lüst, Bakas, Mylonas, Schupp, Szabo, Aschieri, Hohm, Zwiebach, Blumenhagen, Fuchs, Hassler, Sun, Hull, …

• FDA equations

d�a +Q�a = 0

d�a +Q�a = 0

dQ = 0

dB• + 3QB• +Rabc�a�b�c = 0

�

a $ xa �

a $ xa

: 2-form B•

pb pb TA•

allows identification of the momenta , with the “new” generators

Q $ q

Thank you