Constraint Structure of the Two-Dimensional Polynomial BF Theory

Background Field (BF ) Model Two-Dimensional BF Gravity MacDowell-Mansouri Gravity from BF Theory Quadratic Gravity from BF Theory

Constraint Structure of the two-dimensional Polynomial BF theory

C. E. Valcárcel

CMCC Universidade Federal do ABC

Natal-Brazil, July 2016

C. E. Valcárcel Constraint Structure of the two-dimensional Polynomial BF theory Background Field (BF ) Model Two-Dimensional BF Gravity MacDowell-Mansouri Gravity from BF Theory Quadratic Gravity from BF Theory Outline

1 Background Field (BF ) Model

2 Two-Dimensional BF Gravity

3 MacDowell-Mansouri Gravity from BF Theory

4 Quadratic Gravity from BF Theory

Topological Quantum Field Theories: Witten (Cohomological) Type. Schwarz Type.

Schwarz Type: Chern-Simons Topologically Massive Gauge Theories: S. Deser, R. Jackiw, S. Templeton, Annals Phys. 140 (1982) 372. (2 + 1)−Dimensional Gravity as an Exactly Soluble System: E. Witten, Nucl. Phys. B311 (1988) 46.

Schwarz Type: BF model Topological Field Theory, D. Birmingham, M. Blau, M. Rakowski, G. Thompson, Phys. Rep. 209 (1991) 129.

Ingredients: n−dimensional Manifold M . Gauge Group G. Gauge eld (or Connection) 1−form A. 2−form Field Strength (or Curvature) F = DA = dA + A ∧ A. (n − 2)−form B.

Action Z SBF = Tr (B ∧ F ). M

Equations of Motion:

Z h 1 i δS = Tr δB ∧ F + (−1)n− DB ∧ δA .

F = 0 : The connection A is at. DB = 0.

Gauge Transformation Gauge: δA = Dη. Shift: δB = Dχ. (Due to Bianchi Identity DF = 0)

BF in Two Dimensions

L2D = tr (B ∧ F ).

BF in Three Dimensions

L3D = tr (B ∧ F ) + α1tr (B ∧ B ∧ B).

BF in Four Dimensions

L4D = tr (B ∧ F ) + α2tr (B ∧ B) + α3tr (F ∧ F ).

Einstein-Hilbert Action: 1 Z √ S = dnx g (R − 2Λ). EH 16πG

Equation of motion: 1 0 Rµν − 2gµν R + gµν Λ = .

Considerations in two dimensions: For Λ = 0 the metric is undetermined. For Λ 6= 0 the metric is zero. Two-dim Gravity cannot be described by a Einstein-Hilbert Action!

Jackiw-Teitelboim

Z 2 √ SJT = d x gX (R − 2Λ).

R. Jackiw, C. Teitelboim, Quantum Theory of Gravity (1984). EoM: R − 2Λ = 0.

JT from BF Theory Gauge Theory of Two-Dimensional Quantum Gravity, K. Isler, C. A. Trugenberger, Phys. Rev. Lett. 63 (1989) 834. Gauge theory of Topological Gravity in 1 + 1 dimensions, A. H. Chamseddine, D. Wyler, Phys. Lett. B 228 (1989) 75. Quadratic gravity from BF theory in two and three dimensions, R. Paszko, R. da Rocha, Gen. Rel. Grav. 47, (2015) 94.

Ingredients: Group and Generators Group: SO (3).

Generators: MIJ = −MJI .

Algebra: [MIJ ,MKL] = ηILMJK − ηIK MJL + ηJK MIL − ηJLMIK .

Ingredients: Fields 1 1 IJ µ IJ A = 2Aµ MIJ dx , B = 2B MIJ .

Ingredients: Field Strength

IJ IJ IJ I KJ I KJ Fµν = ∂µ Aν − ∂ν Aµ + AµK Aν − AνK Aµ .

The Action:

1 Z 1 Z 2 µν IJ 2 µν IJ I KJ S = 4 d x ε BIJ Fµν = 2 d x ε BIJ ∂µ Aν + AµK Aν .

Characteristics of the BF action It is rst-order: Linear in the velocities IJ . ∂0Aν IJ There is no ∂0A0 (like electromagnetism). The system is constrained.

Dirac's Constraint Analysis P. A. M. Dirac, Lectures on Quantum Mechanics (New York: Yeshiva University) (1964). M. Henneaux, C. Teitelboim, Quantization of Gauge Systems, Princeton Univ. Press (1992).

Generators

Generators: MIJ = (Mab = ε¯abM, Ma2 = Pa). Algebra: b [M,Pa] = −ε¯a Pb, [Pa,Pb] = −ε¯abM.

Cartan variables a ab: Zweibein eµ ,ωµ

a b gµν = ηabeµ eν .

Torsion and Curvature

a a a b T ≡ de + ω b ∧ e . ab ab a cb R ≡ dω + ω c ∧ ω .

Gauge eld: 1 1 A = ωabM + ea P . µ 2 µ ab l µ a

Field Strength:

1 F ab = Rab − ea eb − eaeb , µν µν l2 µ ν ν µ 1 F a2 = T a . µν l µν

BF action

1 Z h i 2 µν ab 2 a2 S = 4 d x ε BabFµν + Ba2Fµν ,

BF Action

1 Z 2 1 2 1 S = d x B ε µν ε¯ Rab − e + ε µν B T a . 2 2 ab µν l2 l a µν

Equation of motion for the Ba eld:

0 µν a = ε Tµν .

Implication of the equations of motion Relation zweibein and spin-connection 0 µν a a b. = ε ∂µ eν +~ε bωµ eν Curvature form and Riemann tensor αβ ab α β . Rµν = Rµν ea eb Furthermore: µν ab 2 . ε ε¯abRµν = eR

BF and JT Gravity

1 Z √ 2 1 S = d2x gB R − , Λ = . 2 l2 l2

Modications Group SO (2,1) with metric ηIJ = diag (1,−1,1) or ηIJ = diag (1,−1,−1). Inonu-Wigner Contraction.

BF SUSY Orthosymplectic group OSP (1,1;1).

Noncommutative FJ Noncommutative Gravity in two dimensions, S. Cacciatori, et al. Class.Quant.Grav. 19 (2002) 4029.

Constraints Analysis Canonical Analysis of the Jackiw-Teitelboim Gravity in the Temporal Gauge, C. P. Constantinidis et. al. Class. Quant. Grav. 25 (2008), 125003. Quantization of the Jackiw-Teitelboim model, C. P. Constantinidis et. al. Phys. Rev. D 79 (2009) 084007.

Constraints from Hamilton-Jacobi Two-dimensional background eld gravity: A Hamilton-Jacobi analysis, M. C. Bertin, et. al. J. Math. Phys. 53 (2012), 102901. Three-dimensional background eld gravity: A Hamilton-Jacobi analysis, N. T. Maia, et. al. Class.Quant.Grav. 32 (2015), 185013. C. E. Valcárcel Constraint Structure of the two-dimensional Polynomial BF theory Background Field (BF ) Model Two-Dimensional BF Gravity MacDowell-Mansouri Gravity from BF Theory Quadratic Gravity from BF Theory Palatini formulation

Palatini formulation Z Λ 4 ¯abcd SPal = d x ε Rab ∧ ec ∧ ed − 6 ea ∧ eb ∧ ec ∧ ed .

MacDowell-Mansouri

Z Z SMM = Tr Fb∧ ?Fb = SPal + Tr (R ∧ ?R).

Unied Geometric theory o f gravity and supergravity, S.W. MacDowell, F. Mansouri. Phys.Rev.Lett. 38 (1977) 739.

Ingredients for the Riemannian case Group SO (5): Spherical space Λ > 0. Group SO (4,1): Hyperbolic space Λ < 0. Group ISO (4): Euclidean space Λ = 0.

L. Freidel, A. Starodubtsev (2005) Z κ2 IJ ¯IJKL4 S = BIJ ∧ F − 4 ε BIJ ∧ BKL .

Quantum gravity in terms of topological observables, e-Print: hep-th/0501191. General relativity with a topological phase: an action principle, L. Smolin, A. Starodubtsev, e-Print: hep-th/0311163.

BF Action Z 2 ab 2 a4 κ abcd S = Bab ∧ F + Ba4 ∧ F − 4 ε Bab ∧ Bcd ,

Equation of Motion 0 0 a4 1 a δBa4 = → = F = l T , 2 0 ab κ ¯abcd ab 1 a b δBab = → F = 2 ε Bcd = R − l2 e ∧ e .

MacDowell-Mansouri 1 Z S = ε¯abcd F ∧ F . 4κ2 ab cd

MacDowell-Mansouri

1 Z 1 S = − ε¯abcd R ∧ e ∧ e − e ∧ e ∧ e ∧ e 2κ2l2 ab c d 2l2 a b c d 1 Z + ε¯abcd [R ∧ R ]. 4κ2 ab cd

Terms: First line: Palatini Action with Cosmological Constant. Second line: Euler Class (Topological). Conserved charges for Gravity with locally AdS asymptotics, R. Aros et. al. Phys. Rev. Lett. 84 (2000),1964.

Adding more topological invariant terms: The presence of the term Tr[B ∧ B] adds more topological term: Euler, Pontryagin, Nieh-Yan. We also obtain the Holst term. The BF action reduce the number of parameters of this theory. The ratio of the cosmological term and the interaction is related to the Immirzi parameter.

Another formulations: Supergravity as constrained BF theory, R. Durka, J. Kowalski-Glikman, Phys. Rev. D 83 (2011) 124011. Gauged AdS-Maxwell algebra and Gravity, R. Durka et. al. Mod. Phys. Lett. A 26 (2011) 2689.

Back to two dimensions! Z 1 2 S = tr − B ∧ F + κ (PB)(PB)(QA) ∧ A . M 2 Quadratic gravity from BF theory in two and three dimensions, R. Paszko, R. Rocha, Gen. Rel. Grav. 47 (2015) 8, 94.

Projetors 1 1 ¯ ¯ ¯ ¯ ¯ PIJ,KL ≡ 2εIJ2εKL2, QIJ,KL ≡ 2εIJM εKLN εMN2.

Action of the projectors 0 (PM)ab = Mab, (PM)a2 = . b 0 (QM)a2 = ε¯a Mb2 (QM)ab = . C. E. Valcárcel Constraint Structure of the two-dimensional Polynomial BF theory Background Field (BF ) Model Two-Dimensional BF Gravity MacDowell-Mansouri Gravity from BF Theory Quadratic Gravity from BF Theory Polynomial BF (2)

Polynomial BF Action

2 Z 2 1 1 2 1 2κ 2 S = d x B ε µν ε¯ Rab − e + ε µν B T a + B e . 2 2 ab µν l2 2l a µν l2

Torsion free condition: µν a 0. ε Tµν = l2 2 B = − 8κ2 R − l2 .

Quadratic Gravity from Polynomial BF: 2Λ = 1/l2 1 Z √ 1 Z √ S = d2x g (R − 2Λ) − d2x gR2. 8κ2 64Λκ2

Some consideretions R2 is the lowest order higher derivative term introduced to get rid of the triviality of the Einstein-Hilbert action in two-dimensions. R2 is important in UV renormalization.

Foliation of space-time Z 1 1 2 a a S = d x B∂0ω + Ba∂0e1 + ω0G + e0 Ga . l l

Where: and 2κ2 b 2. G ≡ D1B Ga ≡ D1Ba + l ε¯abe1 B a We do not have ∂0ω0 and ∂0e0 . Canonical Momenta 0 0, 0 0. Π = πa = Primary Constraints: 0 0, 0 0. φ ≡ Π ≈ φa ≡ πa ≈

, a a . {ω (x),B (y)} = δ (x − y) {e1 (x),Bb} = lδb δ (x − y) Canonical Hamiltonian: 1 a . H0 = −ω0G − l e0 Ga

Consistency Condition: Primary Constraints must be conserved in time a Primary Hamiltonian: HP ≡ H0 + vφ + v φa.

Consistency: φ˙ = {φ,HP } = 0.

Secondary Constraints

a φ˙ = 0 → G φ˙ = 0 → Ga.

Algebra of Constraints

b {G (x),Ga (y)} = −ε¯a Gb (x). 2 {Ga (x),Gb (y)} = −ε¯ab 1 − 4κ B G (x).

Smeared constraints:

Z Z a a G (ζ) ≡ dx ζ (x)G (x), Ga (ζ ) ≡ dx ζ (x)Ga (x), Z Z a a Ge(λ) ≡ dx λ (x)φ (x), Gea (λ ) ≡ dx λ (x)φa (x).

Generators

n a bo a n o e0 ,Geb λ = λ , ω0,Ge(λ) = λ.

n a bo a e1 ,G (ζ) + Gb ζ = lD1ζ . 4 2 n bo κ a b ω,G (ζ) + G ζ = −D1ζ + ε¯ ζ e1 B. b l ab

Counting degrees of freedom (d.o.f) in Dirac's canonical formalism:

d.o.f = N − 2M − S.

N is the dimension of the phase-space. M is the number of rst-class constraints. S is the number of second-class constraints.

For Polynomial BF action we have 12: Due to a and their canonical momenta. N = eµ ,ωµ M = 6. Since we have (φ,φa,G ,Ga) S = 0. There is no second-class constraints

Batalin-Fradkin-Vilkovisky Formalism: E.S. Fradkin, G.A. Vilkovisky, Phys. Lett. B 55 (1975) 244. A. Batalin, G.A. Vilkovisky, Phys. Lett. B 69 (1977) 309.

Conditions for BFV The collection of primary and secondary

constraintsGA = (φ,φa,G ,Ga). Algebra C and B . {GA,GB } = UAB GC {H0,GA} = VA GB It is possible: C C UAB = UAB (q,p).

For the Polynomial BF action we have

4 b 3 1 4 2 U34 = −ε¯a , U44 = −ε¯ab − κ B .

3 4 1 4 1 V = −1, V = − δ a, V = − ε¯ bea. 1 2 l b 3 l a 0 1 3 b 2 4 b V = − ε¯ e 1 − 4κ B , V = ε¯ ω0. 4 l ab 0 4 a

Introduction of ghost A a a Vector of ghost η = (P,P ,c,c ) and PA = c,ca,P,Pa . Brackets A A η ,PB = −δB δ (x − y).

BRST Charge 1 A C A B Ω = η GA + 2PC UAB η η .

1 a a ¯ 1 4 2 a b ¯a b Ω = Pφ + P φa + cG + c Ga − 2εab − κ B Pc c + ε bPacc .

BRST Invariant Hamiltonian: {HB ,Ω} = 0

A B HB = H0 + η VA PB . HU = HB + {Ψ,Ω}.

A Gauge Fermion Ψ = PAχ A a a Gauge functions χ = (χ, χ ,0,0). Then Ψ = cχ + caχ . 1 a 1 a a Temporal Gauge 0 χ = γ ω , χ = γ (e0 − δ1 ).

1 1 1 1 0 a a 0 a {Ψ,Ω} = − ω0Π − (e0 − δ1 )π − Pc − P c . γ γ a γ γ a

Transition Amplitude:

Z Z h i A i Z = D[Fields]D[Ghosts] × expi dt q˙i p + η˙ PA − HU .

Transition amplitude:

Z Z 1 a a Z = De1 DωDBaDB expi dt e˙1 Ba + ω˙ B + G1 × Z , l gh

Ghost part

Z Z a a a Zgh = DcDcDc Dc expi dt [−cc˙ − lcac˙ ] Z 1 2 a a expi dt − ε¯ 1 1 − 4κ B cc + ε¯ c c . l a 1 a

Two dimensional dilaton theories Z 2 √ R 1 2 S (g,X ) = d x g 2 X − 2U (X )(∂X ) − V (X ) .

Dilaton gravity in two dimensions, D. Grumiller, W. Kummer, D. V. Vassilevich, Phys. Rep. 369,(2002) 327. Jackiw-Teitelboim Gravity: U (X ) = 0 and V (X ) = ΛX . β 2 2 KV Model: U (X ) = cte and V (X ) = 2 X − Λ.(R gravity with dynamical torsion). Almheiri-Polchinski model: Quadratic potential and RX 2 (2014).

Let us consider U (X ) = 0. Then, for the dilaton eld we have dV R = 2 . dX

We can choose V (X ) = 2ΛX 1 − 2κ2X 1 Z √ 1 Z √ S = d2x g (R − 2Λ) − d2x gR2. 8κ2 64Λκ2

Introducing Cartan variables:

Z 1 a a b ¯ a b S = Xdω + Xa de + ω b ∧ e − 2εabe ∧ e V (X ) .

Pros and Cons Dilaton theories englobe a large class of two-dimensional gravity models. BF theory is build as a gauge theory. Dilaton Gravity can be written as a Poisson-Sigma model. The rst-order Dilaton Gravity is valid on two dimensions. BF can be build in three and four dimensions.

In two dimensions Are valid the same extensions of Jackiw-Teitelboim gravity? Are valid the same extensions of MacDowell-Mansouri gravity? It is possible to introduce a dynamical torsions? Loop quantum gravity as in Jackiw-Teitelboim?

In three dimensions Is it useful to introduce a cosmological term in the three dimensional Polynomial BF action?.

C. E. Valcárcel Constraint Structure of the two-dimensional Polynomial BF theory