Towards modified bimetric theories within non-product spectral geometry

We discuss class of doubled geometry models with diagonal metrics. Based on the analysis of known examples we formulate a hypothesis that supports treating them as modified bimetric gravity theories. Certain steps towards the generic case are then performed.


II. THE GENERIC DIAGONAL MODEL
A framework of spectral geometry, allowing for an equivalent description of geometric objects in terms of algebraic data, originates from the observation that the geometry of a compact spin Riemannian manifold M can be encoded in the collection of data (C ∞ (M), L 2 (M), D M ) [2], where L 2 (M) is the Hilbert space of square-integrable spinors, and D M is the Dirac operator, which can be written locally (with the use of the spin connection ω) as iγ µ (∂ µ + ω µ ).
This system of data is a prerequisite for the notion of a spectral triple, a set (A, H, D) consisting of a unital * -algebra A represented in a faithful way on a Hilbert space H, on which the (possibly unbounded) densely defined self-adjoint operator D acts. In the generic case it is assumed that the commutators [D, a], a ∈ A, are well-defined and can be (uniquely) extended to an element from B(H), bounded operators on H. Furthermore, the resolvent of D has to be compact. Several further comptability conditions are imposed for certain applications [6,9].
In addition to the aforementioned canonical spectral triple associated to a manifold M, the finite dimensional ones are well-understood [11,12]. In this case both an algebra A F and a Hilbert space H F are finite dimensional, and D F is just a matrix. One can go one step further and consider products of spectral triples considered so far. The almost-commutative geometry is a result of such a construction, where the first spectral triple in the product is the canonical one for a manifold M, and the other one is finite. In the case with dim M = 4, the Dirac operator for the resulting triple is (pointwisely) D M ⊗ 1 + γ 5 ⊗ D F , where γ 5 is the usual grading in the Clifford algebra associated to the manifold M.
However, even for the product space with finite part being just the two points set, this is not the most general Dirac operator one can consider. Indeed, the operator with a field Φ, which for our purposes is taken to be a constant, is an example of another candidate [14]. Here D 1 , D 2 are two Dirac operators for M, but considered with two different Riemannian metrics g 1 , g 2 . The operator γ is a straightforward generalization of γ 5 : γ * = γ, {γ, γ a } = 0 for all anti-Hermitian γ a generating the Clifford algebra, {γ a , γ b } = −2δ ab 1, but now γ 2 = κ = ±1 (instead of requiring κ = 1). These models are refered to as the doubled geometries [13]. We consider geometries of this type with the metric on each sheet chosen to be of the form where a j , for j = 0, 1, 2, 3, are constants. The spin connection ω is identically zero since for the coframe {θ a } we have dθ a = 0 for every a = 0, . . . , 3, and the resulting Dirac operator therefore reads, D = 3 j=0 1 a j γ j ∂ j . The corresponding doubled geometry constructed out of these two sheets is therefore described by a Dirac operator of the form The associated Laplace operator is hence given by and one can then easily read the decomposition of its symbol into the homogeneous parts, Since our first goal is to determine the leading terms of the spectral action, we have to find the symbol of the inverse of the Laplace operator, what can be achieved by using the standard methods of pseudodifferential calculus [15]. (In the above equation Tr Cl denotes the trace performed over the Clifford algebra and Tr is the usual matrix trace over two-by-two matrices.) In our case we get so that The only nonzero elements of the matrix b 0 are on its diagonal and they are equal to

and as a result of a straightforward computation we get
The resulting spectral action is therefore of the form and where we have already introduced effective parametrization, and ommited the irrelevant global multiplicative constant. Therefore, the problem of finding the potential term describing the interaction between the two diagonal metrics reduces to compute linear combination of the integrals of the form Moreover, from the Eqn. (II.13) it immediately follows that V (g 1 , g 2 ) = V (g 2 , g 1 ), that is, V is symmetric under the interchange g 1 ↔ g 2 . We further conjecture that the potential term can be written as for some function V. By the symmetry of V , to prove this claim it is enough to show that the function V ′ (g 1 , g 2 ) := V (g 1 ,g 2 ) 2π 2 √ det g 2 depends only on the eigenvalues of g −1 2 g 1 . We illustrate this hypothesis on a simple nontrivial example -the Hopf model -discussed in the forthcoming section.

III. THE HOPF MODEL
We consider here models with diagonal metrics given by g 00 = g 11 = b 2 and g 22 = g 33 = a 2 , for which then have ,

(III.4)
Let us introduce the following notation (III.5) and µ sin 2 θ dS a 2 1 cos 2 ϕ(a 2 1 cos 2 θ + b 2 1 sin 2 θ) sin 2 ϕ 2 a 2 2 cos 2 ϕ(a 2 2 cos 2 θ + b 2 2 sin 2 θ) sin 2 ϕ 2 , (III.6) and notice that I 0,c = I 1,c = I 2,c = I 3,c , I 0,s = I 1,s , I 2,s = I 3,s , (III.7) so that In order to find the final form of the potential it remains to compute the integrals I 0,c , I 0,s and I 2,s . The result reads, F (a 1 , a 2 , b 1 , b 2 ) + G(a 1 , a 2 , b 1 , b 2 )) , (III.9) where F (a 1 , a 2 , b 1 (III.11) We observe that for a 1 = a 2 the potential reduces to V (g 1 , g 2 ) = 2π 2 a 2 Since the logarithm vanishes if and only if a 1 a 2 = b 1 b 2 it would be, in principle, interesting to consider the limit of V (g 1 , g 2 ) when b 1 tends to b 2 a 1 a 2 . The value of the function V is indeterminated in this case, but the limit may still exists. Indeed, as a result we get Introducing the new variable x = b 1 b 2 and y = a 1 a 2 we can write (III.14) We observe that what illustrates the hypothesis.

IV. COMMENT ON GENERIC METRICS
The so far examined examples of doubled geometries suggest that these models can be thought of as certain modifications of bimetric theories as the potential term possesses features characteristic to this type of modified gravity theories. Despite the fact that series of non-trivial examples are already analysed, the derivation of the action in the generic case is still an open problem. In the approach we are using the main chalenge is related with the computation of certain integrals of rational functions defined over higher spheres: with smooth A µν , which can be further written as A µν = Ω(δ µν + ǫ µν ) with Ω ∈ R and ǫ µν being symmetric and traceless. In the formula above ξ α is the αth coordinate of vector ξ.
We make here some comments on the analysis of doubled geometry models in case when the tensors A µν does not differ sufficiently from the diagonal ones. This is not identical to the situation where the metrics are small perturbation of the Euclidean ones -for the discussion of the latter we refer to [19].
Assuming that max µ,ν ǫ µν ∞ is sufficiently small we can expand in these parameters and are polynomial integrals over higher spheres which can be evaluated generalizing the methods from [18] -see also [19] for further discussion. Denoting
Then N 2m is a number of such fillings for which there is no j such that T j is of the form 2l − 1|2l , for some l = 1, ..., m.
Since, by symmetry of ǫ, any nonzero term in ǫ γ 1 γ 2 ...ǫ γ 2m−1 γ 2m I γ 1 ...γ 2m produces tr(ǫ m ), we have ǫ γ 1 γ 2 ...ǫ γ 2m−1 γ 2m I γ 1 ...γ 2m = N 2m c m tr(ǫ m ), (IV.7) and the problem reduces to finding coefficients c m . Since area(S 3 ) = 2π 2 we get c 1 = π 2 2 . Moreover, by using the generalization of [18,Prop. 2] (see also [19,Prop As a result, In order to apply this result to a specific term of the action one has to first solve the combinatorial problem of finding the coefficients N 2m , up to required order in m. We postpone for the future research the problem of determining set of metrics for which the rate of convergence of the above series is satisfactory for all the terms that appear in the action functional.

V. CONCLUSIONS AND OUTLOOK
The discussed doubled geometry model is an interesting possibility of going beyond the General Relativity. The explicit functional form of its action is derivable in the same way as the Hilbert-Einstein's one but with the use of a different geometry instead of the classical manifold. Here we extended the existing family of known examples for which the features characteristic to bimetric gravity models are present. We also made further steps towards the analysis of models that are beyond the class of such whose action is analytically computable. We remark that yet another approach based on a different type of noncommutativity can produce bimetric type of models [20]. It will be interesting to find some deeper relations between these two formulations -we postpone this for a future research.