The Emergence of 3+ 1D Einstein Gravity from Topological Gravity

Home , Nathan Seiberg, Peter van Nieuwenhuizen

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Instead of using super symmetry(SUSY) and super symmetry is not necessary. Second, it can handle ADS/CFT correspondence to define and understand general cases with or without cosmological constant. quantum gravity, here we attempt to consider the Surprisingly, recent development in condensed matter problem from a different angle. We would like to ask: physics indicates that T r[A∧A∧(dA+A∧A)]+ T r(B∧ Is there any TQFT in 3 + 1D that is closely related to F ) type actions[17–19]R do exist and might serveR as the the Einstein gravity, e.g., can we realize Einstein gravity most general 3 + 1D TQFT that describes nontrivial through a proper phase transition from a TQFT? There three-loop-braiding statistics[20, 21]. For discrete gauge are several advantages in TQFT approach to quantum group, they are known as Dijkgraaf-Witten theories[22]. gravity, as having already been demonstrated in the Now let us generalize the above action into Poincare 2 + 1D case. First, it is manifested renormalizable and group and define the following topological gauge theory:

1 a b cd ab a STop = ǫabcde ∧ e ∧ R + Bab ∧ R + B˜a ∧ T 2 Z Z Z 1 4 µνρσ a b cd 1 4 µνρσ ab 1 4 µνρσ a = d xǫ ǫabcdeµ eν Rρσ + d xǫ BµνabRρσ + d xǫ B˜µνaTρσ (1) 4 Z 4 Z 4 Z

Here B, B˜ are 2-form gauge fields which were first intro- Therefore the above action can be regarded as the duced in usual topological BF theory[23–25], and R,T 3 + 1D generalization of 2 + 1D topological gravity. Ap- are the usual curvature and torsion tensors: parently, the beta function vanishes for STop and it is renormalizable. The argument is exactly the same as the cd cd − cd ce d − ce d Rρσ = ∂ρωσ ∂σωρ + ωρ ωσe ωσ ωρe , 2+1D case, where the counterterms, if any, are integrals a a a ab ab Tµν = ∂µeν − ∂ν eµ + ωµ eνb − ων eµb (2) of local gauge invariant functional and can not renormal- ize the above action. Another straightforward argument The first term in the above action is the usual Einstein- is that for compact gauge groups, all the terms in the Cartan action. It is easy to verify that such a topological above actions are actually quantized and the beta func- action is invariant under local Lorentz symmetry trans- tion must vanish[17]. Similar to the 2+1D case, e,ω are formation. Interestingly, the total action is actually in- dimension one operators while B is dimension two opera- variant under the whole local Poincare symmetry trans- tor. Finally, the equation of motion implies the vanishing formation, if we properly define the gauge transformation of curvature and torsion tensors. of translational symmetry for 2-form gauge fields:

a a a a a ab ab a ˜ b cd eµ → eµ + Dµf ≡ eµ + ∂µf + ωµ fb R =0,T =0, ∇Ba = −ǫabcde ∧ R =0, ˜ → ˜ − b cd 1 c d Bµνa Bµνa ǫabcdf Rµν ∇Bab + (Bã ∧ eb − B˜b ∧ ea)= −ǫabcdT ∧ e =0 1 2 B → B − (B˜ f − B˜ f ) (3) µνab µνab 2 µνa b µνb a We note that the usual first order Einstein-Cartan action Quantization of topological gravity – Before discussing is not invariant under the above gauge transformation, the possible connection with 3 + 1D Einstein gravity, and that’s why it is not a well defined TQFT in 3 + let us proceed the standard canonical quantization for 1D. In addition, we can also define the following gauge the above topological gravity and explain its underlying physics. The Lagrangian density reads: transformation for 2-form gauge fields B˜µνa and Bµνab:

b b B˜µνa → B˜µνa + ∂µξ˜νa − ∂ν ξ˜µa + ωµa ξ˜νb − ωνa ξ˜µb i ab i a 1 ijk ab LTop = Π ∂0ω + π ∂0e + ǫ B0 R 1 ab i a i 2 iab jk Bµνab → Bµνab − [(ξ˜µaeνb − ξ˜νaeµb) − (ξ˜µbeνa − ξ˜νbeµa)] 2 1 ijk a a i 1 ijk b cd + ǫ B˜0iaTjk + e0 (∇iπ a + ǫ ǫabcdei Rjk ) and 2 2 ab i 1 i b i a +ω0 ∇iΠ ab + (π aei − π bei ) (5) Bµνab → Bµνab + Dµξνab − Dν ξµab 2 where the covariant derivative Dµ is deﬁned as: ab a where the canonical momentums of ωi and ei are ≡ c c i 1 ijk c d 1 ijk Dµξνab ∂µξνab + ωµa ξνcb + ωµb ξνac (4) deﬁned as Π ab = 2 ǫ ǫabcdej ek + 2 ǫ Bjkab and 3

i 1 ijk ˜ π a = 2 ǫ Bjka. Canonical quantization requires: vanishing of curvature and torsion at that scale. Mathe- matically, 3 + 1D TQFT can be described and classified cd i i cd − [ωj (y), Π ab(x)] = iδjδabδ(x y), by tensor 2-category theory and a possible way to gen- b i i b erate interesting dynamics is condensing loops(flux lines [ej (y), π a(x)] = iδ δ δ(x − y) j a in the context of topological gauge theory). If we further all other commutators = 0 (6) assume that the condensed loop carries a nontrivial linking Berry phase[28–30], a T r(B ∧ B) type term can be with the following flat-connection constraints: induced. Let us consider theR following term: 1 1 ǫijkR ab , ǫijkT a jk =0 jk =0 θ ab 2 2 Sθ = − Bab ∧ B (9) 1 2 Z ∇ πi = − ǫijkǫ e bR cd =0 i a 2 abcd i jk 1 This term breaks the 2-form gauge symmetry as well as ∇ Πi + (πi e b − πi e a) = 0 (7) the translational gauge symmetry explicitly. A micro- i ab 2 a i b i scopic derivation of the above term from loop condensa- Similar to the 2 + 1D topological gravity, the phase- tion is beyond the scope of this paper. Here we just in- space to be quantized is exactly the solutions of above troduce such a term phenomenologically to describe low constraints divided by the group of gauge transforma- energy dynamics and ignore all the microscopic details tions generated by the constraints. The quantum Hilbert of loop dynamics, which is the analog of using massive space is the flat connections of Poincare group modulo gauge boson to describe Abelian Higgs phase and consid- gauge transformations.(If we regard e and ω as coordi- ering the infinite massive limit for Higgs boson. (More nates while π and Π as momentums.)Of course, in order precisely, one can assume that the total action S is con- to define an ultraviolet(UV) complete theory, it is much sisting of two terms STop and SLoop at UV scale. SLoop better to use the algebraic framework of tensor 2-category describes the dynamics of closed loop and it can be ap- theory[26, 27](It is well known that the 2 + 1D Chern- proximated by Sθ in the loop condensed phase after tak- Simons theory can be described by the algebraic tensor ing the infinite massive limit for loop.) Remarkably, for category theory.) small θ, the total action S = STop + Sθ is still power- In fact, the above constraints are exactly the same as counting renormalizable since Sθ only contains dimension the usual BF theory of Poincare group, and the subtle four operators. A detailed calculation of beta functions difference only arises from the definition of physical ob- will be presented elsewhere. servable corresponding to loop like excitation, namely, The classical equation of motion for the total action S the Wilson surface operator. Let us rewrite the commu- reads: tation relations in terms of B, B,e,ω˜ : 1 Bab = Rab,T a =0, ǫ eb ∧ Rcd = −∇B˜ , ab ab − abcd a [ωi (x),Bjkcd(y)] = iǫijkδcdδ(x y), θ 1 [e a(x), B˜ (y)] = iǫ δaδ(x − y), ǫ T c ∧ ed + (B˜ ∧ e − B˜ ∧ e )= −∇B (10) i jkb ijk b abcd 2 a b b a ab d d [Bijab(x), B˜klc(y)] = iǫabcd(ei ǫjkl − ej ǫikl)δ(x − y), Insert the first two equations into the last equation, we all other commutators = 0. (8) have: In recent works, it has been pointed out that such mod- 1 ˜ ˜ 1 ified commutation relations actually imply the nontrivial (Ba ∧ eb − Bb ∧ ea)= − ∇Rab = 0 (11) 2 θ three-loop-braiding[17–19] statistics among flux lines of gauge fields, which makes it different from the usual BF The above equation can be rewritten in a compact form abcd a theory of Poincare group with trivial three-loop-braiding as ǫ Bã ∧ eb = 0, which further implies B˜ = 0. Thus, statistics. we eventually derive the vacuum Einstein-Cartan equa- Loop condensation and the emergence of Einstein grav- tion: ity – To this point, one may wonder why we are interested b cd in the 3 + 1D topological gravity theory which is some- ǫabcde ∧ R =0. (12) what trivial. Here we conjecture that quantum gravity is actually controlled by a topological gravity fixed point Einstein gravity as a non-commutative geometry – and the classical space-time vanishes at extremely high Now let us proceed the canonical quantization for the energy scale. Therefore it is quite natural to expect the total action S. The total Lagrangian density reads: 4

i ab i a 1 ijk ab ab L = Π ∂0ω + π ∂0e + ǫ B0 (R − θB ) ab i a i 2 iab jk jk 1 ijk a a i 1 ijk b cd ab i 1 i b i a + ǫ B˜0iaTjk + e0 (∇iπ a + ǫ ǫabcdei Rjk )+ ω0 ∇iΠ ab + (π aei − π bei ) (13) 2 2 2

i i where the canonical momentum Πab, πa have the same Similar to the case without cosmological constant term, definition as in 3+1D topological gravity. By integrating loop condensation will lead to Einstein-Cartan action out B0iab and B˜0ia, we derive the following constraints: with cosmological constant term, and the whole theory remains to be power-counting renormalizable. ab 1 ab a Bij = Rij ,Tij = 0 (14) Super symmetric generalization – Finally, let us discuss θ the SUSY generalization of 3 + 1D topological gravity. We note that the torsion free condition arises as a quan- Similar to the 2 + 1D topological gravity theory, we just tum constraint instead of equation of motion here. This need to introduce the gauge connection of super Poincare feature is very different from the usual Einstein-Cartan group and write the action as sT r[A ∧ A ∧ (dA + A ∧ theory. The canonical quantization conditions Eq.(8) im- A)] + sT r(B ∧ F ). For example,R for the N = 1 case, ply the following noncommutative geometry: we canR just express A, B and F as: ab ab − 1 [ωi (x), Rjkcd(y)] = iθǫijkδcdδ(x y), (15) ≡ ab a ¯ α Aµ ωµ Mab + eµPa + ψµαQ a ˜ a − 2 [ei (x), Bjkb(y)] = iǫijkδb δ(x y), ≡ 1 ab ã B α ˜ d − d − Bµν Bµν Mab + Bµν Pa + µναQ [Rijab(x), Bklc(y)] = iθǫabcd(ei ǫjkl ej ǫikl)δ(x y), 2 ≡ 1 ab a ¯ α In the semi-classical limit with, both R and e fields have Fµν Rµν Mab + Tµν Pa + RµναQ (18) 2 weak quantum fluctuations while ω and B˜ fields have strong quantum fluctuations. A very interesting obser- Here R¯µνα is the super curvature tensor defined as vation is that the small parameter θ enter the commuta- R¯µνα = Dµψ¯να − Dν ψ¯µα where Dµ is the covari- tion relation and it will be very interesting to understand ant derivative for spinon fields. However, as fermionic the relationship between θ and planck constant ~ in our loops(flux lines) can not be condensed, super symmetry future work. breaking already happens at very high energy scale when To this end, we see that the nature of quantum grav- bosonic loops condense and classical space-time emerges. ity is the emergence of non-commutative geometry via Thus, the super curvature R¯µνα always vanishes and the loop condensation from an underlying topological grav- semiclassical limit of 3 + 1D quantum gravity can still be ity theory. We stress that STop with vanishing R and T described by S at low energy. However, the SUSY gen- describes the absolute vacuum of our universe in the ab- eralization might provide us a natural way to extend the sence of classical space time, and it might provide a new our model to include fermionic matter fields. route towards resolving the black hole singularity as well Topological gravity in arbitrary dimensions and the as the big bang singularity. emergence of 3+1D space-time – Before conclusion, let Cosmological constant term – Our construction for us generalize topological gravity theory into arbitrary di- topological gravity action in 3 + 1D can be easily gen- mensions with the following gauge invariant action: eralized into the case with cosmological constant term: 1 ab ∧ a3 ∧···∧ an ′ Λ a b c d STop = ǫaba3...an R e e S = STop + ǫabcde ∧ e ∧ e ∧ e n − 2 Z Top 4! Z ∧ ab ˜ ∧ a Λ 4 µνρσ a b c d + Cab R + Ca T (19) = STop + d xǫ ǫabcde e e e (16) Z Z 4! Z µ ν ρ σ ˜ We only need to properly redefine the gauge transforma- where C and C are n − 2 forms. Interestingly, we see tion of translational symmetry: that it is only possible to introduce C ∧ C type term for four dimensional space-time. Thus,R we may start a a a a a ab eµ → eµ + Dµf ≡ eµ + ∂µf + ωµ fb with a model describing topological gravity in all di- Λ mensions(e.g. topological nonlinear sigma model of the B˜ → B˜ − ǫ f bR cd − ǫ (eb ec − eb ec )f d µνa µνa abcd µν 2 abcd µ ν ν µ Poincare group classifying space) and condense the loop, 1 only the four dimensional vielbein field admits a semi- B → B − (B˜ f − B˜ f ) (17) µνab µνab 2 µνa b µνb a classical limit that defines the classical space-time! 5

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