M. Dimitrijević, Noncommutative models of gauge and gravity theories

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Balkan Workshop BW2013
Beyond the Standard Models
25 – 29 April, 2013, Vrnjačka Banja, Serbia

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M. Dimitrijević, Noncommutative models of gauge and gravity theories

  1. 1. Balkan Workshop BW2013-Beyond the Standard Models-,Vrnjaˇcka Banja, April 25th-29th, 2013Noncommutative gauge and gravity theoriesMarija Dimitrijevi´cUniversity of Belgrade, Faculty of Physics,Belgrade, SerbiaM. Dimitrijevi´c, V. Radovanovi´c and H. ˇStefanˇci´c, arXiv: 1207.4675[hep-th]P. Aschieri, L. Castellani and M. Dimitrijevi´c, arXiv: 1207.4346[hep-th]
  2. 2. OverviewNC geometryMotivationMain results-product approachNC gauge theoryNC gravityGravity as a gauge theoryAdS inspired NC actionConclusions & Outlook
  3. 3. NC geometry: motivationWhy Noncommutativity? Physics at small distances (highenergies) not very well understood.Original motivation: to solve the problem of divergences in QFT.Recent motivation: appears in string theory, playground forQuantum theory of gravity, new effects in QFT, . . .[Douglas, Nekrasov ’01; Szabo ’01, ’06,; Castellani ’00;. . . ]Why deformation of gravity?: No renormalizable gravity theory(yet), modified gravity theory could explain the problems such asDark matter and Dark energy,. . .Why deformation of gauge theories?: Standard model is not thefinal theory, renormalizability, SUSY, cosmological constantproblem, DM particles,. . .Different approaches: Connes, Wess (Munich group), Grosse,. . .
  4. 4. NC geometry: main resultsMathematics: C∗-algebras, deformation quantizaton, Hopfalgebras and quantum groups,. . .Physics:-renormalizability of NC field theories: UV/IR mixing,renormalizable models [Minwalla, Raamsdonk ’99; Grosse, Wulkenhaar ’05;Buri´c, Radovanovi´c ’04]-NC Standard Model: predictions of new processes (new vertices)[Wess et al. ’01; Trampeti´c et al. ’04,’05]. New vertices:γγγ, γγZ, γZZ, ZZZ, . . .ff γγ, ff γZ, ffZZ, udγW , . . .
  5. 5. -NC SUSY theories: models, renormalization [Seiberg ’03; Britto, Feng,Rey ’03; Dimitrijevi´c, Radovanovic´c ’09].-NC gravity: twist approach [Wess et al. ’05, ’06; Ohl, Schenckel ’09],gauge theory of Lorentz/Poincar´e group [Chamseddine ’01,’04, Cardela,Zanon ’03, Aschieri, Castellani ’09,’12]: new terms in the action appear(Rµνρσ)n, . . . .
  6. 6. NC geometry: -product approachNoncommutative space ˆAˆx , generated by ˆxµ coordinatesµ = 0, 1, . . . n such that:[ˆxµ, ˆxν] = Θµν(ˆx). (1)It is an associative free algebra generated by ˆxµ and divided by theideal generated by relations (1). Differential calculus, integral,symmetries [Wess et al. ∼1990s] can be discussed, but. . .-product geometry: represent ˆAˆx on the space of commutingcoordinates, but keep track of the deformationˆAˆx → Axˆf (ˆx) → f (x) and ˆf (ˆx)ˆg(ˆx) → f g(x).
  7. 7. Example: θ-constant (canonical) deformation leads to Moyal-Weyl-product:f g (x) =∞n=0i2n 1n!θρ1σ1. . . θρnσn∂ρ1 . . . ∂ρn f (x) ∂σ1 . . . ∂σn g(x)= f · g +i2θρσ(∂ρf ) · (∂σg) + O(θ2). (2)Associative, noncommutative; c. conjugation: (f g)∗ = g∗ f ∗.Derivatives, integral are well definedd4x f g = d4x g f = d4x fg. (3)The -product (2) enabled: construction of quantum field theoriesand analysis of their renormalizability properties, construction ofNC Standard Model and the analysis of its phenomenologicalconsequences,. . .
  8. 8. NC gauge theoryWe work with θ-constant NC space:f · g → f g = f · g + i2 θαβ(∂αf )(∂βg)−18 θαβθκλ(∂α∂κf )(∂β∂λg) + . . .α, Φ, Aµ, Fµν → ˆα, ˆΦ, ˆAµ,ˆFµν = ∂µˆAν − ∂νˆAµ − i[ˆAµ , ˆAν]δαΨ = iαΨ → δαˆΨ = i ˆα ˆΨδαΦ = i[α, Φ] → δαˆΦ = i[ˆα , ˆΦ]δαAµ = ∂µ + i[α, Aµ] → δαˆAµ = ∂µ ˆα + i[ˆα , ˆAµ]δαFµν = i[α, Fµν] → δαˆFµν = i[ˆα , ˆFµν]
  9. 9. Idea of the Seiberg-Witten map: NC gauge transformations areinduced by the commutative gauge transformations, δα → δα.Thenˆα = ˆα(α, Aµ), ˆAµ = ˆAµ(Aµ), ˆΦ = ˆΦ(Φ, Aµ). (4)NC gauge transformations have to close(δαδβ − δβδα)ˆΦ = δ−i[α,β]ˆΦ.This conditions enables to solve for ˆα(α, Aµ), [Ulker, Yapiskan ’08,Aschieri, Castellani ’11]:ˆα(n+1)= −14(n + 1)θκλ{ˆAκ , ∂λ ˆα}(n), (5)with(A B)(n)= A(n)B(0)+ A(n−1)B(1)+ . . .+A(0) (1)B(n−1)+ A(1) (1)B(n−2)+ . . .
  10. 10. The SW-map solution for the NC gauge field ˆAµ follows fromδαˆAµ = ∂µ ˆα + i[ˆα , ˆAµ]:A(n+1)µ = −14(n + 1)θκλ{ˆAκ , ∂λˆAµ + ˆFλµ}(n), (6)with the NC field-strength tensor ˆFλµ.Similarly, one calculates solutions of the SW-map for matter fields,field-strength tensor,. . .NC U(1) action, expanded up to first order in the NC parameterθαβS = −14d4x ˆFµνˆFµν= d4x −14FµνFµν+ θαβ(14FαβFµνFµν− FαµFβνFµν) .(7)The commutative abelian gauge theory becomes a NC nonabeliangauge theory ⇒ Photon self-interactions appear!
  11. 11. The main result of the SW-map: No new degrees of freedom, NCgauge theory and the corresponding commutative gauge theoryhave the same number of degrees of freedom! This enablesconstrustion and analysis of NC gauge theories.NC Standard Model [Wess et al. ’01,’02]:-phenomenology: [Trampeti´c et al. ’03,’05, . . . ],-renormalization: [Buri´c, Radovanovi´c ’03,. . . , 11; Martin, Tamarit ’05,’07,. . . ]
  12. 12. NC gravity: gravity as a gauge theoryLocalization of space-time symmetry ⇒ gravity.Start from the MacDowell-Mansouri action [MacDowell, Mansouri ’77]:S =116πGNd4xl216µνρσabcd R abµν R cdρσ +√−gR − 2√−gΛ (8)and Λ = −3/l2,√−g = det e aµ . Comments:-variables in (8) are spin connection ωµ and vielbeins eµ. They areindependent, 1st order formalsim.-(8) is invariant under SO(1, 3) gauge symmetry, while thediffeomorphism symmetry appears as a consequence of SSB, see[Stelle, West ’80].-varying (8) with respect to ωµ and vielbeins eµ gives equations ofmotion for these fields.-(8) written in the 2nd order formalism has three terms:Gauss-Bonnet topological term, Einstein-Hilbert term and thecosmological constant term.
  13. 13. NC gravity: AdS inspired NC actionThe NC generalization of the action (8) isS = −il264πGNd4x µνρσTr (ˆRµνˆRρσγ5) (9)−2il2Tr (ˆRµνˆEρˆEσγ5) +1l4Tr (ˆEµˆEνˆEρˆEσγ5) ,with:-NC SO(1, 3) gauge potential: ˆωµ = ωµ + ˆω(1)µ + ˆω(2)µ + . . .-NC vielbeins: ˆEµ = eµ + ˆE(1)µ + ˆE(2)µ + . . .-NC curvature tensor:ˆRµν = ∂µ ˆων − ∂ν ˆωµ − i[ˆωµ , ˆων]= Rµν + ˆR(1)µν + ˆR(2)µν + . . . .
  14. 14. Solutions of the SW map we insert into the action (9) andcalculate corrections. We find:S(0)= commutative action (8)S(1)= 0 unfortunately!!!S(2)= very complicated, not manifestly gauge invariant,no explicit results existed until June 2012.Shortcut: SW map for composite fields [Aschieri, Castellani, Dimitrijevi´c,’12].An example:(ˆEµˆEν)(1)= ˆE(1)µ eν + eµˆE(1)ν +i2θαβ∂αeµ∂βeν (10)= −14θαβ{ωα, ∂β(eµeν) + Dβ(eµeν)} +i2θαβ(Dαeµ)(Dβeν).
  15. 15. Inserting these expressions and calculating traces explicitly, weobtainS(2)GB = −l21024πGNθκλθρσ µναβabcd d4x R cdαβ R abµκ R mnνρ Rλσmn−12R cdαβ R abµκ R mnνλ Rρσmn + R mnαβ RµκmnR abνρ R cdλσ+R mnµρ RνσmnR abακ R cdβλ −12R abακ R cdβλ R mnρσ Rµνmn , (11)S(2)CC = −1512πGN l2θκλθαβd4xe 6R abκα Rλβab − 3R abαβ Rκλab+4R γδαβ (Dκeµ)a(Dλeγ)b(eµa eδb + eµb eδa) (12)−4R γδαβ (Dκeγ)a(Dλeδ)a + 4(DκDαeµ)a(DλDβeν )b(eµa eνb − eµb eνa )−8R γδκα (Dβeγ)a(Dλeδ)a + 8R γδκα (Dβeγ)a(Dλeµ)b(eµa eδb + eµb eδa) ,
  16. 16. S(2)EH = −1512πGNθαβθκλd4xe12R µνκλ R γδαβ Rµνγδ+Rκλρσ(12R µναβ R ρσµν − 2R µσαβ R ρµ +12RR ρσαβ+4R ρσβν R να + 4R ρα R σβ − 4R νραµ R µσβν )−2R µνκλ R γδαµ Rβνγδ − RR γδκα Rλβγδ−Rµνρσ(2R µνκα R ρσλβ + 4R µσκα R νρλβ )−4R να R γδκβ Rλνγδ + 2Rαµρσ(2R µνκβ R ρσλν−4R ρλ R µσκβ + 4R νρκβ R µσλν + 2R ρσκβ R µλ )+4eµb eνc ((DκRαβ)mb(DλRµν )cm − 2(DκRαµ)mb(DλRβν )cm)+(Dκeρ)m(Dλeσ)m(2R µναβ R ρσµν + 8R µραβ R σµ + 2RR ρσαβ−8R ρσαµ R µβ − 8R ρβ R σα − 8R νραµ R µσβν )−2(DκDαeρ)c(DλDβeσ)d(R(eρc eσd − eσc eρd ) − 3R ρν eνc eσd+3R σν eνc eρd + R ρν eσc eνd − R σν eρc eνd + 2R ρσµν eµc eνd )+4(Dλeσ)aR bmµν (Dαeρ)m(R µνκβ (eρa eσb − eσa eρb ) (13)+2R µσκβ (eνa eρb − eρa eνb ) − 2R µρκβ (eνa eσb − eσa eνb ) + 2R ρσκβ eµa eνb ) .
  17. 17. Conclusions & OutlookNC gauge theory-NCSM: no new fields, but new interactions appear;renormalizability and phenomenology studied,. . .-NC gravity: MacDowell-Mansouri action studied, expansionup to second order in the NC parameter written in amanifestly gauge covariant way; phenomenologicalconsequences remain to be studied,. . .future investigation-phenomenological consequences of NC gravity-NC SO(2, 3) symmetry and SSB in NC theory-SW map for composite fields and renormalization of gaugetheories

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