Four-dimensional Noncommutative Gravity

George Manolakos

September 9, 2020

9th International Conference on New Frontiers in Physics (ICNFP 2020)

National Technical University of Athens

JHEP08(2020)001, e-Print: 1902.10922 [hep-th] Table of contents

1. First order formulation of General Relativity

2. Einstein gravity as a gauge theory in 4 dimensions

3. Conformal gravity as a gauge theory in 4 dimensions

4. Noncommutative framework - gauge theories

5. A four-dimensional noncommutative covariant space

6. A model for noncommutative four-dimensional gravity General Relativity

Second order formulation (Einstein gravity):

• independent field: metric tensor, gµν • curvature parametrized by Riemann tensor: ρ ρ ρ ρ λ ρ λ Rµνσ = ∂µΓνσ − ∂ν Γµσ + ΓµλΓνσ − ΓνλΓµσ λ λ λ • torsion: Tµν = Γµν − Γνµ √ • 1 R 4 action: S = 16πG d x −gR

First order formulation:

a ab • independent fields: vierbein and spin connection, eµ , ωµ • curvature parametrized by the curvature 2-form: c c Rµνab = ∂µωνab − ∂ν ωµab − ωµacων b − ωνacωµ b • a a a a c a c gr as gauge torsion: Tµν = ∂µeν − ∂ν eµ + ωµ beν − ων beµ • 1 R 1 a b cd action: S = 16πG 2 abcde ∧ e ∧ R (Palatini action) Einstein 4d gravity as a gauge theory

The algebra

• Employ the first order formulation of GR see for details: • Utiyama ’56, Kibble ’61, Gauge theory of Poincar´egroup ISO(1,3) McDowell-Mansuri ’77, • Ten generators (TranslationsP a & LTM ab) Kibble - Stelle ’85 Generators satisfy the commutation relations:

[Mab,Mcd] = ηacMdb − ηbcMda − ηadMcb + ηbdMca

[Pa,Mbc] = ηabPc − ηacPb , [Pa,Pb] = 0 where ηab = diag(−1, +1, +1, +1) and a, b, c, d = 1,..., 4. The gauging procedure

a ab • Intro of a gauge field for each generator: eµ , ωµ (transl, LT) - mix of gauge and spacetime • Define the covariant derivative → the iso(1, 3)-valued 1-form gauge connection is: a 1 ab Aµ = eµ (x)Pa + 2 ωµ (x)Mab • Transforms in the adjoint rep, according to the rule:

δAµ = ∂µ + [Aµ, ] • The gauge transformation parameter, (x) is expanded as:

a 1 ab (x) = ξ (x)Pa + 2 λ (x)Mab • Combining the above → transformations of the fields:

a a b a ab δeµ = ∂µξ − eµ λ b + ωµ ξb ab ab a cb b ca δωµ = ∂µλ − λ cωµ + λ cωµ • Gauge transf equivalent to diffeo transf (on-shell) → gauge invariance ←→ general covariance! Curvatures and action

• Curvatures of the fields are given by:

Rµν (A) = ∂µAν − ∂ν Aµ + [Aµ,Aν ]

• TensorR µν is also valued in iso(1, 3):

a ab Rµν (A) = Tµν Pa + Rµν Mab

• Combining the above → component tensor curvatures:

a a a b a b a Tµν = ∂µeν − ∂ν eµ + eµ ωνb − eν ωµb

ab ab ab cb a ac b 2nd order Rµν = ∂µων − ∂ν ωµ − ωµ ων c + ωµ ωνc • Dynamics of Einstein gravity cannot be obtained by an ISO(1,3) action of Yang-Mills type Stelle-West ’80, Kibble - Stelle ’85 • E-H action from de Sitter group, consider the Pontryagin index + use an auxiliary field, gauge fix it → break the symmetry to the Lorentz subgroup → scalar curvature action + torsionless condition Conformal 4d gravity as a gauge theory

• Group parametrizing the symmetry: SO(2,4)

• 15 generators:6 LTM ab, 4 translations,P a, 4 conformal boosts Ka and the dilatationD • Following the same procedure one calculates transf of the gauge fields and tensors after defining the gauge connection comm limit • Action is taken of Yang-Mills form • Initial symmetry breaks under certain constraints resulting to the Weyl action Kaku-Townsend-Van Nieu/zen ’77, Fradkin-Tseytlin ’85 • Alternative: Initial symmetry can break down to recover the Einstein action (two specific constraints are taken to hold simultaneously) Chamseddine ’02 The nc framework & gauge theories

The nc framework Szabo ’01

• Possible nc regime: Planck scale - coords may be considered non-commuting • Late 40’s: Nc structure of spacetime at small scales for an effective ultraviolet cutoff → control of divergences in qfts Snyder ’47 • Ignored → success of renormalization programme • 80’s: nc geometry revived after the generalization of diff structure Connes ’85, Woronowicz ’87 • Along with the definition of a generalized integration → Yang-Mills gauge theories on nc spaces Connes-Rieffel ’87 • Inspiration from qm: Operators instead of variables i • quantization of phase space of x , pj→ replace with Herm i i i operators:ˆx , pˆj satisfying:[ˆx , pˆj] = i~δj • Noncommutative space → quantization of space: xi → replace with operatorsˆxi (∈ A) satisfying: [ˆxi, xˆj] = iθij(ˆx) Connes ’94, Madore ’99 • Antisymmetric tensor θij(ˆx) - defines the nc of the space • Canonical case: θij (ˆx) = θij , i, j = 1,...,N For N = 2 → Moyal plane • ij ij k Lie-type case: θ (ˆx) = C kxˆ , i, j = 1,...,N For N = 3 → Noncommutative (fuzzy) sphere (SU(2)) • nc framework admits a matrix representation (operators)

• Derivation: ei(A) = [di,A], di ∈ A • Integration → Trace • Algebra of operators with normal multiplication ↔ algebra of functions with different product - star product • Weyl correspondence: 1-1 correspondence of operators and functions. Sternheimer ’98, Alvarez-Gaumez, Wadia ’01 The nc gauge theories • Natural intro of nc gauge theories through covariant nc coordinates: Xµ = Xµ + Aµ Madore-Schraml-Schupp-Wess ’00

• Obeys a covariant gauge transformation rule: δXµ = i[, Xµ]

• Aµ transforms in analogy with the gauge connection: δAµ = −i[Xµ, ] + i[, Aµ],(  - the gauge parameter) 4dncgravity • Definition of a nc covariant field strength tensor depends on the space:

• Canonical case: Fab = [Xa, Xb] − iθab • Lie-Type case: Fab = [Xa, Xb] − iCabcXc • Gauge theory could be abelian or non-abelian: • Abelian if  is a function in A • Non-abelian if  is matrix valued (Mat(A)) Non-Abelian case

B In nonabelian case, where are the gauge fields valued? • Let us consider the CR of two elements of an algebra: 1 1 [, A] = [AT A,ABT B] = {A,AB}[T A,T B]+ [A,AB]{T A,T B} 2 2 • Not possible to restrict to a matrix algebra: last term neither vanishes in nc nor is an algebra element • There are two options to overpass the difficulty: • Consider the universal enveloping algebra • Extend the generators and/or fix the rep so that the anticommutators close

B We employ the second option

(Note: replacing with the star-product → nc gauge theory with functions) The 4d covariant noncommutative space

Motivation for a 4d covariant nc space • Constructing gravitational models on nc spaces is non-trivial: nc deformations break Lorentz invariance • Special: covariant noncommutative spaces, i.e. coords transform as vectors under rotations • such an example is the fuzzy sphere (2d space) - coords are identified as rescaled SU(2) generators • Previous work on 3d nc gravity on the (already existing) 3 1,2 covariant spaces Rλ,Rλ L. Jonke, A. Chatz/dis, D. Jurman, GM, P. Manousselis, G. Zoupanos ’18 Hammou-Lagraa-Sheikh Jabbari ’02 Vitale-Wallet ’13, Vitale ’14 • Need of 4d covariant nc space to construct a gravity gauge theory...... so let us build one! Manousselis, G.M, Zoupanos ’20 Construction of the 4d covariant nc space Kimura ’02, Heckman-Verlinde ’15 Sperling-Steinacker ’17

• dS4 : homogeneous spacetime with constant curvature (positive) AB 2 1,4 • Described by the embedding η XAXB = R intoM

• We aim for a nc version of dS4

• Coords must satisfy[ Xa,Xb] = iθab, with θab to be determined • Analogy to the fuzzy sphere case: identification of the coordinates with generators of the SO(1,4) (isometry group of dS4)

• BUT: θab cannot be assigned to generators of the algebra → covariance breaks • Requiring covariance → use a group with larger symmetry → minimum extension: SO(1,5)(SO(6) in our language) • The SO(6) generators, JAB, A, B = 1,..., 6, satisfy the commutation relation:

[JAB,JCD] = i(δAC JBD + δBDJAC − δBC JAD − δADJBC )

• After decomposition in SO(4) notation, A = a, 5, 6, we identify the component generators as: 1 1 λ 1 Jab = Θab,Ja5 = Xa,Ja6 = Pa,J56 = h , ~ λ 2~ 2

where Xa, Pa andΘ ab are the coordinate, momentum and nc tensor, respectively • The above generators satisfy the following comm relations (among others):

λ2 [Xa,Xb] = i Θab, [Xm, Θnp] = i~(δmpXn − δmnXp) ~ • Therefore, coordinates are identified as SO(6) generators represented in a (large N) matrix representation • Covariance ensured - coords transform as vectors under rotations Noncommutative gauge theory of 4d gravity

• Formulation of gravity on the above space • Noncommutative gauge theory toolbox + the procedure described in the Einstein gravity case Kimura ’02, Heckman-Verlinde ’15 • Gauge the isometry group of the space, SO(5) as seen as a subgroup of the SO(6) we ended up • Anticommutators do not close → enlargement of the algebra + fix the representation Aschieri-Castellani ’09 L. Jonke, A. Chatz/dis, D. Jurman, GM, P. Manousselis, G. Zoupanos ’18 • Noncommutative gauge theory of SO(6)xU(1) (∼ U(4)) in the4 representation • The generators of the group are represented by combinations of the 4x4 gamma matrices in euclidean signature • Specifically, the 4x4 generators of the SO(6) group are: • six Lorentz rotation generators: i i M = − [Γ , Γ ] = − Γ Γ , a < b ab 4 a b 2 a b 1 • four generators for conformal boosts:K = Γ a 2 a i • four generators for translations:P = − Γ Γ a 2 a 5 • one generator for special conformal transformations: 1 D = − Γ 2 5 Also the following is included • one U(1) generator:1 • The above expressions of the generators allow the calculation of the algebra they satisfy:

[Mab,Mcd] = i(δacMbd + δbdMac − δbcMadδadMbc) , [Ka,Pb] = iδabD

[Ka,Kb] = iMab , [Pa,Pb] = iMab , [Pa,D] = iKa , [Ka,D] = −iPa

[Ka,Mbc] = i(δacKb − δabKc) , [Pa,Mbc] = i(δacPb − δabPc) , [D,Mab] = 0 • Gauging procedure → definition of the covariant coordinate: Xˆm = Xm ⊗ 1 + Am(X)

• Gauge connectionA m(X) taking values in the algebra: a ab a Am(X) = em (X)⊗Pa+ωm (X)⊗Mab(X)+bm (X)⊗Ka(X)+˜am(X)⊗D+am(X)⊗1 • Introduced a gauge field for each generator • Consider a gauge parameter (x): a ab ˜a  = 0(X) ⊗ 1 + ξ (X) ⊗ Ka + ˜0(X) ⊗ D + λ (X) ⊗ Mab + ξ (X) ⊗ Pa • Determine the field strength tensor: 2 iλ ˆ Rmn = [Xˆm, Xˆn] − Θmn ⊗ 1 , ~ ˆ where Θmn = Θmn ⊗ 1 + Bmn, where Bmn is a 2-form gauge field valued in U(4) transforming covariantly • The field strength tensor is also valued in the algebra → it is spanned on the generators

ab ˜ a a ˜ Rmn(X) = Rmn ⊗Mab+Rmn⊗Pa+Rmn⊗Ka+Rmn⊗D+Rmn⊗1 • Generators satisfy the following anticommutation relations: Smolin ’03 √ 1 2 {Mab,Mcd} = (δacδbd − δbcδad) 1l − abcdD 8 4 √ √ 2 {M ,K } = 2 P , {M ,P } = −  K ab c abcd d ab c 4 abcd d 1 1 {Ka,Kb} = δab1l , {Pa,Pb} = δab1l , {Ka,D} = {Pa,D} = 0 2 √ 8 2 {P ,K } = {M ,D} = −  M . a b ab 2 abcd cd • The necessary information for calculating the transformations of the fields and the component tensors is in these two slides! • Along with the general treatment of nc gauge theories ncgt The transformations of the fields:

ab ab ab ab a b 1 a bc δω = −i[Xm, λ ] − i[am, λ ] + i[0, ω ] − 2{ξ , b } − {λ , ω } m m m 2 c m 1 a b c d i cd i cd c d − {ξ˜ , e } + i[ξ , e ] + [˜0, ω ] + [λ , a˜m] − i[ξ˜ , b ] 2 m m abcd 2 m abcd 2 abcd m abcd a a a a a a 1 a b δe = −i[Xm, ξ˜ ] − i[am, ξ˜ ] + i[0, e ] − {ξ , a˜m} + {˜0, b } + {λ , e } m m m 4 b m 1 − {ξ˜ , ω ab} + i[ξc, ω bd] − i[λcd, b b ] 4 b m m abcd m abcd a a a a ab a 1 a b δb = −i[Xm, ξ ] − i[am, ξ ] + i[0, b ] − {ξ , ω } − 2{˜0, e } + {λ , b } m m b m m 2 b m ˜a bc d ˜b cd + {ξ , a˜m} + i[λ , em]abcd + i[ξ , ωm ]abcd

a a i ab i a δam = −i[Xm, 0] − i[am, 0] + i[ξ , b ] + i[˜0, a˜m] + [λ , ω ] + [ξ˜a, e ] m 2 ab m 2 m a a i ad bc δa˜m = −i[Xm, ˜0] − i[am, ˜0] + i[0, a˜m] + {ξa, e } − {ξ˜a, b } + [λ , ω ] m m 2 m abcd

(Transformations of the component of Bmn are calculated as well) The component curvatures:

a 1 ab Rmn = [Xm, an] − [Xn, am] + [am, an] + [b , bna] + [˜am, a˜n] + [ω , ω ] m 2 m nab a i~ + [ema, e ] − Bmn n λ2 ˜ a a Rmn = [Xm, a˜n] + [am, a˜n] − [Xn, a˜m] − [an, a˜m] − i{bma, en } + i{bna, em }

1 ab cd i~ +  [ω , ω ] − B˜mn 2 abcd m n λ2 a a a a a ab ab Rmn = [Xm, bn ] + [am, bn ] − [Xn, bm ] − [an, bm ] + i{bmb, ωm } − i{bnb, ωm }

a a b cd b cd i~ a + i{a˜m, e } − i{a˜n, e } +  ([e , ω ] − [e , ω ]) − B n m abcd m n n m λ2 mn ˜ a a a a a a a Rmn = [Xm, en ] + [am, en ] − [Xn, em ] − [an, em ] + i{bm , a˜n} − i{bn , a˜m} i − ([b b , ω cd] − [b b, ω cd]) − i{ω ab, e } + i{ω ab, e } − ~ B˜ a m n n m abcd m nb n mb λ2 mn ab ab ab ab ab a b c d Rmn = [Xm, ωn ] + [am, ωn ] − [Xn, ωm ] − [an, ωm ] + 2i{bm , bn } + ([bm, en ]

c d 1 cd cd ac b − [b , e ]) + ([˜am, ω ] − [˜an, ω ]) + 2i{ω , ω } n m abcd 2 n m abcd m n c i + 2i{e a, e b} − ~ B ab m n λ2 mn The commutative limit

• U(1) gauge field decouples, am = 0

• 2-form gauge field decouples, Bmn = 0 • commutators → partial derivatives

• anticommutators → multiplication

• up to some numerical coefficient

• Results of conformal gravity are recovered! conformal The constraints for the symmetry breaking

• We want to result with SO(4) symmetry out of SO(6) part • Employ the constraint of the torsionless condition: R˜(P ) = 0 a a • We adopt: em = bm and fix˜am = 0 ab ab • Solving the constraint we obtain an expression ωm = ωm (e, a): 3 ω ac = em abcd[D , e ] n 4 b m nd The action of the theory

• Choice of the action: SO(6)xU(1) invariant of Yang-Mills type:

mnrs S = TrtrΓ5RmnRrs

• (In the action we also include the kinetic term of the B field) • Applying the constraints, the expressions of the tensors we obtain the action with reduced gauge symmetry SO(4)xU(1) Conclusions & Future plans

• Successful construction of a 4d covariant noncommutative space • Description of gravity in a regime where coords can be considered nc • Study the Lorentz invariant action and try to relate it with the 4d Einstein-Hilbert - connect the large and low-energy regimes of the interaction • Include matter fields and examine a spontaneous symmetry breaking of the initial action • Obtain the equations of motion

Thank you for your attention!