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

Aﬃne 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 (aﬃne) gauge theory

2 Gravity as a (aﬃne) gauge theory

• Aﬃne group: A(d, R) = GL(d, R) n Rd:

[gl, gl] ⊆ gl , [gl, r] ⊆ r , [r, r] ⊆ 0 .

2 Gravity as a (aﬃne) gauge theory

• Aﬃne group: A(d, R) = GL(d, R) n Rd:

[gl, gl] ⊆ gl , [gl, r] ⊆ r , [r, r] ⊆ 0 .

• General linear group: GL(d, R) = O(d) ⊗ K(d):

[o, o] ⊆ o , [o, k] ⊆ k , [k, k] ⊆ o .

2 Gravity as a (aﬃne) gauge theory

• Aﬃne group: A(d, R) = GL(d, R) n Rd:

[gl, gl] ⊆ gl , [gl, r] ⊆ r , [r, r] ⊆ 0 .

• General linear group: GL(d, R) = O(d) ⊗ K(d):

[o, o] ⊆ o , [o, k] ⊆ k , [k, k] ⊆ o .

• Fundamental 1-form ﬁelds (Utiyamma, Kibble, Zanelli, Hehl...): gauge-connection Y = ω + E, the vielbein e; and the 0=form premetric g.

2 Gravity as a (aﬃne) gauge theory

• Aﬃne group: A(d, R) = GL(d, R) n Rd:

[gl, gl] ⊆ gl , [gl, r] ⊆ r , [r, r] ⊆ 0 .

• General linear group: GL(d, R) = O(d) ⊗ K(d):

[o, o] ⊆ o , [o, k] ⊆ k , [k, k] ⊆ o .

• Fundamental 1-form ﬁelds (Utiyamma, Kibble, Zanelli, Hehl...): gauge-connection Y = ω + E, the vielbein e; and the 0=form premetric g.

2 • Gauge transformations:

a a a a b a δω b = Dα b , δE = Dξ + E α b , a a b c c δe = −α be , δgab = α agbc + α bgac

3 • Gauge transformations:

a a a a b a δω b = Dα b , δE = Dξ + E α b , a a b c c δe = −α be , δgab = α agbc + α bgac

• Relation with geometry:

a b g = gabe ⊗ e , Γ = Γ(ω, e) .

3 • Gauge transformations:

a a a a b a δω b = Dα b , δE = Dξ + E α b , a a b c c δe = −α be , δgab = α agbc + α bgac

• Relation with geometry:

a b g = gabe ⊗ e , Γ = Γ(ω, e) .

• Gauge transformations are local and maintain spacetime ﬁxed.

3 • Gauge transformations:

a a a a b a δω b = Dα b , δE = Dξ + E α b , a a b c c δe = −α be , δgab = α agbc + α bgac

• Relation with geometry:

a b g = gabe ⊗ e , Γ = Γ(ω, e) .

• Gauge transformations are local and maintain spacetime ﬁxed.

• e: gauge symmetry ↔ diﬀeomorphisms through A(d, R) ↔ ∗Td(M).

3 • Gauge transformations:

a a a a b a δω b = Dα b , δE = Dξ + E α b , a a b c c δe = −α be , δgab = α agbc + α bgac

• Relation with geometry:

a b g = gabe ⊗ e , Γ = Γ(ω, e) .

• Gauge transformations are local and maintain spacetime ﬁxed.

• e: gauge symmetry ↔ diﬀeomorphisms through A(d, R) ↔ ∗Td(M).

3 (Co)frame bundle

4 (Co)frame bundle

4 •

Figura 1: First order geometry.

5 •

Figura 1: First order geometry.

5 •

Figura 2: Gauge structure of gravity.

6 General gauge theories

6 •

Figura 3: Basic principal bundle for gauge theories.

7 • Gauge space and spacetime are independent structures.

• The gauge bundle is static! So is the coframe bundle ω = ω(e) ≡ second order gravity.

7 • Gauge space and spacetime are independent structures.

• The gauge bundle is static! So is the coframe bundle ω = ω(e) ≡ second order gravity.

• 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 • Gauge space and spacetime are independent structures.

• The gauge bundle is static! So is the coframe bundle ω = ω(e) ≡ second order gravity.

• 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 •

Figura 4: 8 Gravity: QFT vs. Geometry

9 Gravity: QFT vs. Geometry

a • Problems with the coframe bundle: eµ and gab are dimensionless; a eµ 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

• Proposition for quantum gravity: start with independent gauge group A(d, R) and spacetime. Gauge space is NOT identiﬁed with the tangent bundle.

9 Gravity: QFT vs. Geometry

• Proposition for quantum gravity: start with independent gauge group A(d, R) and spacetime. Gauge space is NOT identiﬁed with the tangent bundle.

• At quantum regime: e and g are simple algebra-valued matter ﬁelds. They are not M ↔ T mappings. Y is a gauge connection.

9 Gravity: QFT vs. Geometry

• Proposition for quantum gravity: start with independent gauge group A(d, R) and spacetime. Gauge space is NOT identiﬁed with the tangent bundle.

• At quantum regime: e and g are simple algebra-valued matter ﬁelds. They are not M ↔ T mappings. Y is a gauge connection.

a b • Such id. must emerge naturally in some regime. g = gabe ⊗ e .

9 • UV gauge theory ↔ IR geometry.

• ⇒ Start with a MASSLESS dynamical gauge bundle for the aﬃne group.

9 • UV gauge theory ↔ IR geometry.

• ⇒ Start with a MASSLESS dynamical gauge bundle for the aﬃne group.

• Y : gauge field; e: matter field with dim=1; g: matter field with dim=1. Disables a possible gauge-geometry id.

9 • UV gauge theory ↔ IR geometry.

• ⇒ Start with a MASSLESS dynamical gauge bundle for the aﬃne group.

• Y : gauge field; e: matter field with dim=1; g: matter field with dim=1. Disables a possible gauge-geometry id.

• Transition to geometry: dynamical eﬀects induces mass parameters (dynamical masses, natural RG scales, Gribov pathologies...)

e1 7−→ µe0 , g1 7−→ µg0 .

9 • UV gauge theory ↔ IR geometry.

• ⇒ Start with a MASSLESS dynamical gauge bundle for the aﬃne group.

• Y : gauge field; e: matter field with dim=1; g: matter field with dim=1. Disables a possible gauge-geometry id.

• Transition to geometry: dynamical eﬀects induces mass parameters (dynamical masses, natural RG scales, Gribov pathologies...)

e1 7−→ µe0 , g1 7−→ µg0 .

• Thus, µ regulates UV-IR regimes.

9 • UV gauge theory ↔ IR geometry.

• ⇒ Start with a MASSLESS dynamical gauge bundle for the aﬃne group.

• Y : gauge field; e: matter field with dim=1; g: matter field with dim=1. Disables a possible gauge-geometry id.

• Transition to geometry: dynamical eﬀects induces mass parameters (dynamical masses, natural RG scales, Gribov pathologies...)

e1 7−→ µe0 , g1 7−→ µg0 .

• Thus, µ regulates UV-IR regimes.

9 Contraction

10 Contraction

• Principal bundles might be reduced to smaller bundles if there are contractible sectors.

10 Contraction

• Principal bundles might be reduced to smaller bundles if there are contractible sectors.

• 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

• Principal bundles might be reduced to smaller bundles if there are contractible sectors.

• 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).

• 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

• Principal bundles might be reduced to smaller bundles if there are contractible sectors.

• 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).

• Known results (McInnes (1984); Sobreiro & Vasquez (2010)):

A(d, R) 7−→ GL(d, R) 7−→ O(d) , A(d, R) Ã IO(d) .

• So, all STATIC theories end up into the usual Riemann-Cartan orthogonal bundle.

10 Contraction

• Principal bundles might be reduced to smaller bundles if there are contractible sectors.

• 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).

• Known results (McInnes (1984); Sobreiro & Vasquez (2010)):

A(d, R) 7−→ GL(d, R) 7−→ O(d) , A(d, R) Ã IO(d) .

• So, all STATIC theories end up into the usual Riemann-Cartan orthogonal bundle.

• Dynamical bundle: reduces as well, but...

10 Contraction

• Principal bundles might be reduced to smaller bundles if there are contractible sectors.

• 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).

• Known results (McInnes (1984); Sobreiro & Vasquez (2010)):

A(d, R) 7−→ GL(d, R) 7−→ O(d) , A(d, R) Ã IO(d) .

• So, all STATIC theories end up into the usual Riemann-Cartan orthogonal bundle.

• Dynamical bundle: reduces as well, but...

10 • All original ﬁelds are dynamical: Y = w + q + E. The ﬁbre (gauge orbit) decomposes under contraction:

wL = L−1(d + w)L, qL = L−1qL , EL = L−1E, (1)

11 • All original ﬁelds are dynamical: Y = w + q + E. The ﬁbre (gauge orbit) decomposes under contraction:

wL = L−1(d + w)L, qL = L−1qL , EL = L−1E, (1)

• The original matter ﬁelds:

e = L−1e , gL = L−1g(L−1)T , (2)

11 • All original ﬁelds are dynamical: Y = w + q + E. The ﬁbre (gauge orbit) decomposes under contraction:

wL = L−1(d + w)L, qL = L−1qL , EL = L−1E, (1)

• The original matter ﬁelds:

e = L−1e , gL = L−1g(L−1)T , (2)

• So q and E are now additional matter ﬁelds. Also, e and E obey the same transformation rule!

11 • All original ﬁelds are dynamical: Y = w + q + E. The ﬁbre (gauge orbit) decomposes under contraction:

wL = L−1(d + w)L, qL = L−1qL , EL = L−1E, (1)

• The original matter ﬁelds:

e = L−1e , gL = L−1g(L−1)T , (2)

• So q and E are now additional matter ﬁelds. Also, e and E obey the same transformation rule!

11 • Finally (Sobreiro & Vasquez (2010)): A 7−→ O ⊕ E ⊕ Q. An Aﬃne gauge theory reduces to the usual orthogonal one with additional matter ﬁelds.

12 • Finally (Sobreiro & Vasquez (2010)): A 7−→ O ⊕ E ⊕ Q. An Aﬃne gauge theory reduces to the usual orthogonal one with additional matter ﬁelds.

Identifying geometry

• First option: identify the aﬃne gauge theory with the cotangent bundle of spacetime ≡ Metric-Aﬃne Gravity (Hehl (1976))

12 • Finally (Sobreiro & Vasquez (2010)): A 7−→ O ⊕ E ⊕ Q. An Aﬃne gauge theory reduces to the usual orthogonal one with additional matter ﬁelds.

Identifying geometry

• First option: identify the aﬃne gauge theory with the cotangent bundle of spacetime ≡ Metric-Aﬃne Gravity (Hehl (1976))

• Second option: Reduce to the Riemann-Cartan bundle and then identify it with the cotangent bundle. Careful to keep all degrees of freedom.

12 • Finally (Sobreiro & Vasquez (2010)): A 7−→ O ⊕ E ⊕ Q. An Aﬃne gauge theory reduces to the usual orthogonal one with additional matter ﬁelds.

Identifying geometry

• First option: identify the aﬃne gauge theory with the cotangent bundle of spacetime ≡ Metric-Aﬃne Gravity (Hehl (1976))

• Second option: Reduce to the Riemann-Cartan bundle and then identify it with the cotangent bundle. Careful to keep all degrees of freedom.

• So, contracting the aﬃne curvature:

a a a Ω b = R b + V b , a a a b Ξ = DE − q bE ,

12 where

a a a c R b = dw b + w cw b , a a a c V b = Dq b − q cq b ,

And the minimal couplings:

a a a b T = De − q be , c c Qab = Dgab + q agcb + q bgca ,

13 where

a a a c R b = dw b + w cw b , a a a c V b = Dq b − q cq b ,

And the minimal couplings:

a a a b T = De − q be , c c Qab = Dgab + q agcb + q bgca ,

• Then, identifying with geometry (e 7−→ µ coframes):

gab = δab + γab , a b g = δabe ⊗ e .

13 Thus

T a = Dea , c c Qab = Dγab + 2qab + q aγbc + q bγac .

14 Thus

T a = Dea , c c Qab = Dγab + 2qab + q aγbc + q bγac .

• Conclusion: START- gauge conn.Y and matter.(e, g). END- gauge conn.w, vielbein.e, matter.(E, q, γ) ≡ Riemann-Cartan geometry plus matter.

Conclusions and perspectives

14 Thus

T a = Dea , c c Qab = Dγab + 2qab + q aγbc + q bγac .

• Conclusion: START- gauge conn.Y and matter.(e, g). END- gauge conn.w, vielbein.e, matter.(E, q, γ) ≡ Riemann-Cartan geometry plus matter.

Conclusions and perspectives

• Independently of the starting ”action”: any aﬃne gauge theory reduces to the Riemann-Cartan one with additional matter ﬁelds.

14 Thus

T a = Dea , c c Qab = Dγab + 2qab + q aγbc + q bγac .

• Conclusion: START- gauge conn.Y and matter.(e, g). END- gauge conn.w, vielbein.e, matter.(E, q, γ) ≡ Riemann-Cartan geometry plus matter.

Conclusions and perspectives

• Independently of the starting ”action”: any aﬃne gauge theory reduces to the Riemann-Cartan one with additional matter ﬁelds.

• Gauge-spacetime independence ⇒ Euclidean spacetime at quantum level. So, we know what to do!

14 • Are those ﬁelds related with dark issues?

• Which is the correct mechanism of mass generation?

14 • Are those ﬁelds related with dark issues?

• Which is the correct mechanism of mass generation?

• Quantum sector (Spin-1 theory) vs. Geometric sector (Spin-2?)

14 • Are those ﬁelds related with dark issues?

• Which is the correct mechanism of mass generation?

• Quantum sector (Spin-1 theory) vs. Geometric sector (Spin-2?)

• Inclusion of (fermionic and gauge) matter.

14 • Are those ﬁelds related with dark issues?

• Which is the correct mechanism of mass generation?

• Quantum sector (Spin-1 theory) vs. Geometric sector (Spin-2?)

• Inclusion of (fermionic and gauge) matter.

• Formalization of a theory!

14 • Are those ﬁelds related with dark issues?

• Which is the correct mechanism of mass generation?

• Quantum sector (Spin-1 theory) vs. Geometric sector (Spin-2?)

• Inclusion of (fermionic and gauge) matter.

• Formalization of a theory!

• To be continued...