An Algebraic Topological Construct of Classical Loop Gravity and the prospect of Higher Dimensions
aa r X i v : . [ g r- q c ] M a y An Algebraic Topological Construct of ClassicalLoop Gravity and the prospect of HigherDimensions
Madhavan VenkateshCentre For Fundamental Research and Creative EducationBangalore, IndiaMay 2, 2013
Abstract
In this paper, classical gravity is reformulated in terms of loops, via analgebraic topological approach. The main component is the loop group,whose elements consist of pairs of cobordant loops. A Chas-Sullivan prod-uct is described on the cobordism, and three other products, namely the’vertical’, ’horizontal’ and ’total’ products are re-introduced. (They havealready been defined in an earlier paper by the author). A loop calculusis introduced on the space of loops, consisting of the loop variation func-tional,the loop derivative, Mandelstam derivative, and what the authorwishes to call, the Gambini-Pullin contact functional. The loop derivativehappens to be a generator of the group of loops, and the Gambini-Pullinfunctional is an infinitesimal generator of diffeomorphisms. A toy model ofgravity is formulated in terms of the above, and it is proven that the totalproduct provides for the Maurer-Cartan structure of the space. Further,new quantities labeled as ’momenta’, ’velocity’ and ’energy’ are introducedin terms of the loop products and loop derivatives. The prospect of re-constructing general relativity in higher dimensions is explored, with the’momenta’ and ’velocity’ being the basic loop variables. In the process, itis proven that the ’momenta’ and ’velocity’ behave as cobordant loops inhigher dimensions.
This paper initiates the new loop program that will encompass the subjectstalked of, below. Forthcoming papers will be based on the higher dimensionalloop representation of gravity ,the possible quantization approaches and thequantum theory itself The aim of this formalism is, at its ends, to construct a1uantum theory of gravity and provide for a step towards unification. First, a5+1 split of gravity is done, and the constraints are linearized in terms of thenew loop variables. Then, the Master Constraint is introduced and the princi-pal constraint that is, Q, which corresponds to the ’energy’ in ordinary dimen-sions;(arises due to more degrees of freedom in higher dimensions) is packedinto it. With the classical theory completely reformulated (in the Ashtekarapproach), prospects of quantization will be explored including loop quantiza-tion, twistor, deformation and Berezin-Toeplitz quantization. In a pre-quantumset-up; it is found that the Grassmannian of the Hilbert Space is actually aCalabi-Yau manifold, with a U ( ) compactification, analogous to Kaluza-Kleintheory. This is particularly interesting as the theory will incorporate super-symmetry into it. In order to recover the physics of the U ( ), a detour intosurgery theory will be made, involving a handle decomposition, using the factthat the compactified piece is K¨ahler with an exact curvature two-form. As forthe quantum states, polynomials in knots are used; as they are both Cobordismand diffeomorphism invariant and prove to be ideal for the scenario.So, these developments in prospect are the motivation for the new loop ap-proach. We begin with a compact Lie Group G and the Lie algebra associated with it g . We have the commutation relation [ g , g ] = ˆ g . The loop group LG is thegroup of maps from the circle S into G, with ˆ g being the loop algebra endowedwith it, and ΩG the loop space. We restrict the elements of LG to be orderedpairs of disjoint cobordant loops. We endow with ΩG a loop homology H anda Chas-Sullivan type product, ◦ that, along with the action of the ’loop bracket’ { (cid:5) , (cid:5) } makes H into a Gerstenhaber algebra. It can easily be shown that thebrackets follow the Jacobi identity. We now restrict the Killing form of g to itsCartan sub-algebra, thus, inducing a metric on the Lie group manifold M . Wehave, the Maurer-Cartan structure equation dφ = −
12 [ φ, φ ] , where φ is the Maurer-Cartan 1-form; that also behaves as the connectionform,or, in the language of Reiris-Spallanzani [4], it is the universal one-form.(Note: This one-form is the one whose curvature happens to be the loop deriva-tive introduced by Gambini-Pullin. This is an extremely useful tool to constructa (loop) polynomial action, which can be varied, using the loop derivative andthe connection derivative to give the equations of motion, or, more precisely,the vacuum Einstein equations;in another form). in this section, we give a description of the three products defined in , the ”verti-cal”,”horizontal” and ”total” products. We deal with two free, analytic loops α β to express the products between them. First, we define a quantity knownas the holonomy average,(which will be used to define the products themselves) γ = 12 (cid:26)(cid:18)I dx a dx b φ ab (cid:19) + (cid:18)I dy a dy b φ ab (cid:19)(cid:27) . Definition 1
We define the ’vertical product’ as the following composition: α ⊕ β = ( α ◦ β ) ◦ γ, where the ◦ is the Chas-Sullivan loop product, that forms a graded, commutativeassociative algebra. Definition 2
The ”horizontal product” is written as : α ⊖ β = ( α ◦ γ ) + ( β ◦ γ ) . Definition 3
Further, we write the ’total product’ as a composition of the two; α ⊛ β = { ( α ⊕ β ) ◦ ( α ⊖ β ) } . Proposition 4
The integral over all space of the total product ⊛ between twoloops, α and β can be represented as the curvature of the universal connectionform, dφ . We leave the reader to ponder over this, while a formal and rather indirect proofwill be presented after introduction to some loop calculus, that is, the threederivatives; the loop derivative, the Mandelstam derivative and the Gambini-Pullin variation (or connection functional).
Definition 5
We define the loop variation functional α with path ̺ x ◦ ̺ x bybasing its point of conjoiment to another point x + ǫu , with the composition ofan infinetisimal vector at that point, δu , which is also tangential to the originalpath of the loop. That is, δα = ̺ x ◦ δu ◦ ̺ x + ǫu (1) Definition 6
We now define the loop derivative ∆ ab acting on the loop α asfollows: ∆ ab ( α x ) = ∂ a δ b ( x ) − ∂ b δ a ( x ) + [ δ a ( x ) , δ b ( x )] (2) Definition 7
The connection contact functional, or the Gambini-Pullin varia-tion, is defined for a loop, with respect to a connection φ : δαδφ a ( x ) = I α dx b δ ( y − x ) ∆ ab ( α x ) φ ( x ) . (3) Definition 8
We introduce what is called the Mandelstam derivative. Here, itacts like how a covariant derivative does for Gauge theories:D a α ( ̺ x ) = ∂ a α ( x ) + i φ a ( x ) α ( x ) (4)3 .3 Proof of Proposition 4, and further notes Proof:
Proposition 4 states: Z ΩG α ⊛ β = dφ First, we consider the space of smooth maps C ∞ (cid:0) S , G (cid:1) , on a Lie group Gwith respect to a specific inner product. Naturally, the loop algebra ˆ g is thetangent space and the space of sections of the pullback bundle P ( T G ).We consider the connection to be constructed compatible with the quanti-zation condition: φ ab = φ a φ b − φ b φ a + φ [ a,b ] We define the curvature operator, depending on the Sobolev space parameter s , as: Θ s ( α, β ) = φ sα φ sβ − φ sβ φ sα + φ s [ α,β ] . Here, the parameter s is 1 /
2, as we are dealing with a flat connection. Finally,we define the inner product on LG , for two loops α and β as follows: < α, β > = { (1 + ∆) s ( α ⊛ β ) } , where ∆ is the Chas-Sullivan operator, that, along with the composition ◦ and the ”loop brackets ” {· , ·} make the homology H into a Gerstenhaber quaBatalin-Vilkovisky algebra. It appears, as if the inner-product defined, is a littlevague, but; the product ⊛ appears in the term, giving the complete structureof the group LG to the inner product. So, the definition of the inner product interms of the ”total product ” is justified. Now, we exploit the K¨ahler structureof the loop group to further the proof. Let LG be the group of loops andΩG be the loop space. The curvature operator and the Chas-Sullivan productare defined with respect to the cobordism ∂W = α ⊔ β . In this notation, thesymplectic form on LG is given by ω ( α, β ) = Z ΩG < α, β >, (5)when an arbitrary loop ζ ∈ ˆ g can be written as the following Fourier series, ζ (cid:0) e πit (cid:1) = X n> (cid:0) ζ n e πint − iζ n e − πint (cid:1) . Here, the symplectic form is just the curvature 2-form, which; can as well bewritten as dφ . Expanding the inner product, we have, ω ( α, β ) = dφ = Z ΩG (1 + ∆) s ( α ⊛ β ) (6)The (1 + ∆) s becomes trivial for real exponents(as s = 1 /
2, which indicatesa flat connection). So, neglecting that term, the proof is complete. S ( α, β ) = Z { ( α ⊕ β ) + ( α ⊖ β ) } √ gd x, where g is the metric induced by restricting the Killing form κ to a Cartan sub-algebra of g . The equations of motion for such a set-up can easily be derived byvarying the action with respect to the loops(i.e. acting the loop functional δ ,defined in Definition 5, on it). Quite clearly, this is equivalent to the Einstein-Hilbert formalism. We vary the action (with respect to the loops) to obtainthe Ricci-flat equations of motion, (by acting the loop variation functional) asfollows: δS = Z (cid:8)(cid:2)(cid:0) ̺ x ◦ δu ◦ ̺ x + ǫ u (cid:1) ◦ β ◦ γ (cid:3) + (cid:2) α ◦ (cid:0) ̺ y ◦ δv ◦ ̺ y + ǫ v (cid:1) ◦ γ (cid:3)(cid:9) (7)+ Z (cid:2)(cid:8)(cid:0) ̺ x ◦ δu ◦ ̺ x + ǫ u (cid:1) ◦ γ (cid:9) + (cid:8)(cid:0) ̺ y ◦ δv ◦ ̺ y + ǫ v (cid:1) ◦ γ (cid:9)(cid:3) − Z [ { ( α ◦ β ◦ γ ) } + { ( α ◦ γ ) + ( β ◦ γ ) } ] (cid:0) g ab δg ab (cid:1) = 0 . where the loops α and β have a variation in their paths from points x to x + ǫ u and y to y + ǫ v respectively. We denote, by ̺ , the path; and by δu and δv , infinitesimal vectors at points u and v respectively.Let us go a little bit deeper, into this interesting set-up. We derive thecanonical momenta, ˜ π , by varying the action with respect to the connection asfollows: ˜ π = δδφ (cid:26)(cid:18)Z α ⊕ β (cid:19) + (cid:18)Z α ⊖ β (cid:19)(cid:27) = (cid:20)(cid:26)Z (cid:18) δαδφ ◦ β ◦ γ (cid:19) + (cid:18) α ◦ δβδφ ◦ γ (cid:19)(cid:27) + (cid:26)Z (cid:18) δαδφ ◦ γ (cid:19) + Z (cid:18) δβδφ ◦ γ (cid:19)(cid:27)(cid:21) = (cid:26)Z (cid:18)(cid:18)I α dx b δ ( y − x ) ∆ ab ( α x ) φ a (cid:19) ◦ β ◦ γ (cid:19) + (cid:18) α ◦ (cid:18)I β dy b δ ( x − y ) ∆ ab ( β y ) φ a (cid:19) ◦ γ (cid:19)(cid:27) (8)+ (cid:26)Z (cid:18)(cid:18)I α dx b δ ( y − x ) ∆ ab ( α x ) φ a (cid:19) ◦ γ (cid:19) + Z (cid:18)(cid:18)I β dy b δ ( x − y ) ∆ ab ( β y ) φ a (cid:19) ◦ γ (cid:19)(cid:27) We now define a quantity, which we shall call ’velocity’ for brevity, as: ̟ = Z i X (cid:26)Z ( α ⊕ β ) + Z ( α ⊖ β ) (cid:27) , (9)5here the i X is the interior derivative, with respect to a Hamiltonian vectorfield, that satisfies the symplectic equation i X ω = d H. Further, we write an-other quantity, called ’Energy’(Q); as a total product between the momenta and’velocity’ as follows: Q = Z (˜ π ⊛ ̟ ) . (10)Note: Equipped with this definition, we can now write the structure equa-tion for another space in terms of the above total product, which means, the’energy’ Q for this space corresponds to the structure of another, wherein; ourfundamental loops would be ˜ π and ̟ , and not α and β . In order to do so, welook for a space wherein ˜ π and ̟ behave as fundamental loops. The answerto this, is that we move to higher dimensions. In ordinary dimensions (that is,4) ˜ π and ̟ are the momenta and ’velocity’ respectively. When we move to ahigher dimension ( D + 2 = 6), we choose our loops to be ˜ π and ̟ ; and havethem behave the same way as the ordinary loop variables, here, do. In this section, we attempt to validate the claim that the variables ˜ π and ̟ can be used, or, behave as loops in higher dimensions (in particular, 6). Theproducts introduced in the previous section hold for our new loop variables ˜ π and ̟ , generated by the loop algebra [ g , g ] = Ω g . We denote by LG the groupof analytic loops whose elements consist of ordered pairs of cobordant loops,two of which, are ˜ π and ̟ . We choose higher dimensions, or, in particular, 6dimensions for our momenta ˜ π and the velocity ̟ to behave as loops. We needthem as observables for a higher dimensional theory, and can be sure about thefact that it would reproduce the same theory, in lower dimensions, as, it caneasily be seen, that; the total product, not only provides for the structure of atheory, but also the dynamics of another, in a different dimension!. Lemma 9
The loops ˜ π and ̟ are cobordant in 6 dimensions.Proof: Let LG be the group of loops whose elements are ordered pairs of cobor-dant loops. Let ˜ π ∈ LG and ̟ ∈ LG . We need to prove that ˜ π and ̟ arecobordant. We denote the loop algebra, generating the loops as Ω g , arisingfrom the Lie algeebra g with Lie Group G and Group manifold M , which is anoriented Riemannian six-fold. Cobordisms are defined with respect to loops onΩG, the loop space. It is known that cobordisms are equivalence relations. Itis enough if we show that cobordisms of ˜ π and ̟ ’s cobordism class forms anequivalence class on LG . We observe the construct of the variables ˜ π and ̟ .We see that, the polynomial in loops in the two, are the same; but the actualterm differs by the operator acting on them. In the case of the former it is theGambini-Pullin functional , taken with respect to a flat connection φ ; whereas6n the latter, it is just the interior derivative taken with respect to a Hamiltonianvector field X. Now, if we prove that the loops ˜ π and ̟ are diffeomorphic, weget a result that is a direct consequence of the above lemma. Also, we get another interesting theorem stating that: ( Note: we put a direct proof for this onhold, and this theorem shall be proved as a direct result of following theorem) Theorem 10
To every loop ζ ∈ LG , there exists one and only one unique non-trivial diffeomorphic loop, and they form an ordered pair in LG , ie. they arecobordant. It is easier to prove this in our case as ˜ π contains the Gambini-Pullin functional,which is an infinitesimal generator of diffeomorphisms. Also, if we prove that thetwo loops are isotopic, then,it would mean they are diffeomorphic; which in turn,implies that they are cobordant. Let us examine the terms in ˜ π and ̟ . ˜ π hasa Gambini-Pullin functional that gets rid of terms unrelated to the connection.The initial expression (cid:0)R α ⊕ β + R α ⊖ β (cid:1) behaves as a 2-form (due to the totalproduct, being a two-form), resulting in a 1-form after the operation. In the caseof ̟ , we have the interior product acting on the same intitial expression; andit is quite clear that the resultant expression here too, is a 1-form (as i X mapsp-forms to (p-1) forms). We write the two loops as ˜ π (Σ , M ) and ̟ (Σ , M ) ,where the Σs are two surfaces and M and M are two submanifolds of M . Weneed to show that if there is an embedding Σ ∐ ( − Σ ) ֒ → ∂W , then; under theinduced embedding: (Σ × R ) ` (Σ × R ) ֒ → ∂W × R , M ` ( −M ), boundsan oriented annulus in M × R . We can prove this via the Whitney embeddingtheorem. The above embeddings result in two points with opposite intersectionnumbers, say x and y . Now, by the embedding theorem; there exists an isotopyon M , that is constant in a neighbourhood of M ∩ M − { x, y } ; f : M → M ∩ M . It is also apparent, that there is an induced structureon M × R , that is an S × (0 ,
1) embedding; that is topologically equivalent tothe annulus. This proves, that the two loops are isotopic, therefore, cobordantand diffeomorphic. In this paper, we have initiated some theorems and supplemented them withproofs; that will be helpful to construct a theory of quantum general relativity,prospects of which have been put forward in the ’Introduction’ section. Thesetopological constructions go on to be extremely useful; when building a physicaltheory. This paper initiates the foundations of the program that will eventuallylead to an algebraic topological formalism of quantum gravity. It is intendedto use the usual techniques of ’Loop Quantum Gravity’; to the classical theoryof general relativity (ie. the Ashtekar variables) as well as exploring prospectsof quantization; that include loop quantization, twistor; and Berezin-Toeplitzquantization in forthcoming papers. 7 cknowledgements
I would like to thank Martin Reiris and Jorge Pullin for a clarification. This workwas carried out at the Center For Fundamental Research And Creative Educa-tion (CFRCE), Bangalore, India, under the guidance of Dr B S Ramachandrawhom I wish to acknowledge. I would like to acknowledge the Director Ms.Pratiti B R for creating the highly charged research atmosphere at CFRCE.I would also like to thank my fellow researchers Magnona H Shastry,VasudevShyam, Karthik T Vasu and Arvind Dudi.
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