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Affine Gauge Theory of Gravity and Its Reduction to the Riemann-Cartan

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Affine Gauge Theory of Gravity and Its Reduction to the Riemann-Cartan A±ne gauge gravity and its reduction to the Riemann-Cartan geometry Rodrigo F. Sobreiro Universidade Federal Fluminense Niter¶oi- Brazil NEB-14 : Recent Developments in Gravity Ioannina, Greece : 08-11 June 2010 1 Gravity as a (a±ne) gauge theory 2 Gravity as a (a±ne) gauge theory ² A±ne group: A(d; R) = GL(d; R) n Rd: [gl; gl] ⊆ gl ; [gl; r] ⊆ r ; [r; r] ⊆ 0 : 2 Gravity as a (a±ne) gauge theory ² A±ne group: A(d; R) = GL(d; R) n Rd: [gl; gl] ⊆ gl ; [gl; r] ⊆ r ; [r; r] ⊆ 0 : ² General linear group: GL(d; R) = O(d) ­ K(d): [o; o] ⊆ o ; [o; k] ⊆ k ; [k; k] ⊆ o : 2 Gravity as a (a±ne) gauge theory ² A±ne group: A(d; R) = GL(d; R) n Rd: [gl; gl] ⊆ gl ; [gl; r] ⊆ r ; [r; r] ⊆ 0 : ² General linear group: GL(d; R) = O(d) ­ K(d): [o; o] ⊆ o ; [o; k] ⊆ k ; [k; k] ⊆ o : ² Fundamental 1-form ¯elds (Utiyamma, Kibble, Zanelli, Hehl...): gauge-connection Y = ! + E, the vielbein e; and the 0=form premetric g. 2 Gravity as a (a±ne) gauge theory ² A±ne group: A(d; R) = GL(d; R) n Rd: [gl; gl] ⊆ gl ; [gl; r] ⊆ r ; [r; r] ⊆ 0 : ² General linear group: GL(d; R) = O(d) ­ K(d): [o; o] ⊆ o ; [o; k] ⊆ k ; [k; k] ⊆ o : ² Fundamental 1-form ¯elds (Utiyamma, Kibble, Zanelli, Hehl...): gauge-connection Y = ! + E, the vielbein e; and the 0=form premetric g. 2 ² Gauge transformations: a a a a b a ±! b = D® b ; ±E = D» + E ® b ; a a b c c ±e = ¡® be ; ±gab = ® agbc + ® bgac 3 ² Gauge transformations: a a a a b a ±! b = D® b ; ±E = D» + E ® b ; a a b c c ±e = ¡® be ; ±gab = ® agbc + ® bgac ² Relation with geometry: a b g = gabe ­ e ; ¡ = ¡(!; e) : 3 ² Gauge transformations: a a a a b a ±! b = D® b ; ±E = D» + E ® b ; a a b c c ±e = ¡® be ; ±gab = ® agbc + ® bgac ² Relation with geometry: a b g = gabe ­ e ; ¡ = ¡(!; e) : ² Gauge transformations are local and maintain spacetime ¯xed. 3 ² Gauge transformations: a a a a b a ±! b = D® b ; ±E = D» + E ® b ; a a b c c ±e = ¡® be ; ±gab = ® agbc + ® bgac ² Relation with geometry: a b g = gabe ­ e ; ¡ = ¡(!; e) : ² Gauge transformations are local and maintain spacetime ¯xed. ² e: gauge symmetry $ di®eomorphisms through A(d; R) $ ¤Td(M). 3 ² Gauge transformations: a a a a b a ±! b = D® b ; ±E = D» + E ® b ; a a b c c ±e = ¡® be ; ±gab = ® agbc + ® bgac ² Relation with geometry: a b g = gabe ­ e ; ¡ = ¡(!; e) : ² Gauge transformations are local and maintain spacetime ¯xed. ² e: gauge symmetry $ di®eomorphisms through A(d; R) $ ¤Td(M). 3 (Co)frame bundle 4 (Co)frame bundle 4 ² Figura 1: First order geometry. 5 ² Figura 1: First order geometry. 5 ² Figura 2: Gauge structure of gravity. 6 General gauge theories 6 General gauge theories 6 ² Figura 3: Basic principal bundle for gauge theories. 7 ² Gauge space and spacetime are independent structures. 7 ² Gauge space and spacetime are independent structures. ² The gauge bundle is static! So is the coframe bundle ! = !(e) ´ second order gravity. 7 ² Gauge space and spacetime are independent structures. ² The gauge bundle is static! So is the coframe bundle ! = !(e) ´ second order gravity. ² Dynamical gauge bundle: Base space = all independent gauge connections A called moduli space; Structure group G(N) = local group; Fibre: gauge orbits (group space copies): Ah = h¡1(d + A)h : 7 ² Gauge space and spacetime are independent structures. ² The gauge bundle is static! So is the coframe bundle ! = !(e) ´ second order gravity. ² Dynamical gauge bundle: Base space = all independent gauge connections A called moduli space; Structure group G(N) = local group; Fibre: gauge orbits (group space copies): Ah = h¡1(d + A)h : 7 ² Figura 4: 8 Gravity: QFT vs. Geometry 9 Gravity: QFT vs. Geometry a ² Problems with the coframe bundle: e¹ and gab are dimensionless; a e¹ induces a spacetime renormalization; EH action is a pure interacting term, there are no kinematical terms; Coframe bundle is static; ... 9 Gravity: QFT vs. Geometry a ² Problems with the coframe bundle: e¹ and gab are dimensionless; a e¹ induces a spacetime renormalization; EH action is a pure interacting term, there are no kinematical terms; Coframe bundle is static; ... ² Proposition for quantum gravity: start with independent gauge group A(d; R) and spacetime. Gauge space is NOT identi¯ed with the tangent bundle. 9 Gravity: QFT vs. Geometry a ² Problems with the coframe bundle: e¹ and gab are dimensionless; a e¹ induces a spacetime renormalization; EH action is a pure interacting term, there are no kinematical terms; Coframe bundle is static; ... ² Proposition for quantum gravity: start with independent gauge group A(d; R) and spacetime. Gauge space is NOT identi¯ed with the tangent bundle. ² At quantum regime: e and g are simple algebra-valued matter ¯elds. They are not M $ T mappings. Y is a gauge connection. 9 Gravity: QFT vs. Geometry a ² Problems with the coframe bundle: e¹ and gab are dimensionless; a e¹ induces a spacetime renormalization; EH action is a pure interacting term, there are no kinematical terms; Coframe bundle is static; ... ² Proposition for quantum gravity: start with independent gauge group A(d; R) and spacetime. Gauge space is NOT identi¯ed with the tangent bundle. ² At quantum regime: e and g are simple algebra-valued matter ¯elds. They are not M $ T mappings. Y is a gauge connection. a b ² Such id. must emerge naturally in some regime. g = gabe ­ e . 9 ² UV gauge theory $ IR geometry. 9 ² UV gauge theory $ IR geometry. ² ) Start with a MASSLESS dynamical gauge bundle for the a±ne group. 9 ² UV gauge theory $ IR geometry. ² ) Start with a MASSLESS dynamical gauge bundle for the a±ne group. ² Y : gauge ¯eld; e: matter ¯eld with dim=1; g: matter ¯eld with dim=1. Disables a possible gauge-geometry id. 9 ² UV gauge theory $ IR geometry. ² ) Start with a MASSLESS dynamical gauge bundle for the a±ne group. ² Y : gauge ¯eld; e: matter ¯eld with dim=1; g: matter ¯eld with dim=1. Disables a possible gauge-geometry id. ² Transition to geometry: dynamical e®ects induces mass parameters (dynamical masses, natural RG scales, Gribov pathologies...) e1 7¡! ¹e0 ; g1 7¡! ¹g0 : 9 ² UV gauge theory $ IR geometry. ² ) Start with a MASSLESS dynamical gauge bundle for the a±ne group. ² Y : gauge ¯eld; e: matter ¯eld with dim=1; g: matter ¯eld with dim=1. Disables a possible gauge-geometry id. ² Transition to geometry: dynamical e®ects induces mass parameters (dynamical masses, natural RG scales, Gribov pathologies...) e1 7¡! ¹e0 ; g1 7¡! ¹g0 : ² Thus, ¹ regulates UV-IR regimes. 9 ² UV gauge theory $ IR geometry. ² ) Start with a MASSLESS dynamical gauge bundle for the a±ne group. ² Y : gauge ¯eld; e: matter ¯eld with dim=1; g: matter ¯eld with dim=1. Disables a possible gauge-geometry id. ² Transition to geometry: dynamical e®ects induces mass parameters (dynamical masses, natural RG scales, Gribov pathologies...) e1 7¡! ¹e0 ; g1 7¡! ¹g0 : ² Thus, ¹ regulates UV-IR regimes. 9 Contraction 10 Contraction ² Principal bundles might be reduced to smaller bundles if there are contractible sectors. 10 Contraction ² Principal bundles might be reduced to smaller bundles if there are contractible sectors. ² The connection is as well reduced if, and only if, the contracted coset space is symmetric (the coset space is a tensor space with respect the stability group). 10 Contraction ² Principal bundles might be reduced to smaller bundles if there are contractible sectors. ² The connection is as well reduced if, and only if, the contracted coset space is symmetric (the coset space is a tensor space with respect the stability group). ² Known results (McInnes (1984); Sobreiro & Vasquez (2010)): A(d; R) 7¡! GL(d; R) 7¡! O(d) ; A(d; R) Ã IO(d) : 10 Contraction ² Principal bundles might be reduced to smaller bundles if there are contractible sectors. ² The connection is as well reduced if, and only if, the contracted coset space is symmetric (the coset space is a tensor space with respect the stability group). ² Known results (McInnes (1984); Sobreiro & Vasquez (2010)): A(d; R) 7¡! GL(d; R) 7¡! O(d) ; A(d; R) Ã IO(d) : ² So, all STATIC theories end up into the usual Riemann-Cartan orthogonal bundle. 10 Contraction ² Principal bundles might be reduced to smaller bundles if there are contractible sectors. ² The connection is as well reduced if, and only if, the contracted coset space is symmetric (the coset space is a tensor space with respect the stability group). ² Known results (McInnes (1984); Sobreiro & Vasquez (2010)): A(d; R) 7¡! GL(d; R) 7¡! O(d) ; A(d; R) Ã IO(d) : ² So, all STATIC theories end up into the usual Riemann-Cartan orthogonal bundle. ² Dynamical bundle: reduces as well, but... 10 Contraction ² Principal bundles might be reduced to smaller bundles if there are contractible sectors. ² The connection is as well reduced if, and only if, the contracted coset space is symmetric (the coset space is a tensor space with respect the stability group). ² Known results (McInnes (1984); Sobreiro & Vasquez (2010)): A(d; R) 7¡! GL(d; R) 7¡! O(d) ; A(d; R) Ã IO(d) : ² So, all STATIC theories end up into the usual Riemann-Cartan orthogonal bundle. ² Dynamical bundle: reduces as well, but... 10 ² All original ¯elds are dynamical: Y = w + q + E. The ¯bre (gauge orbit) decomposes under contraction: wL = L¡1(d + w)L; qL = L¡1qL ; EL = L¡1E; (1) 11 ² All original ¯elds are dynamical: Y = w + q + E.
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