Spectral Action for a One–Parameter Family of Dirac–Type Operators on SU(2) and its Inflation Model

We analyze the Dirac Laplacian of a one-parameter family of Dirac operators on a compact Lie group, which includes the Levi-Civita, cubic, and trivial Dirac operators. More specifically, we describe the Dirac Laplacian action on any Clifford module in terms of the action of the Lie algebra’s Casimir element on finitedimensional irreducible representations of the Lie group. Using this description of the Dirac Laplacian, we explicitly compute spectrum for the one-parameter family of Dirac Laplacians on SU(2), and then using the Poisson summation formula, the full asymptotic expansion of the spectral action. The technique used to explicitly compute the spectrum applies more generally to any Lie group where one can concretely describe the weights and corresponding irreducible representations, as well as decompose tensor products of an irreducible representation with the Weyl representation into irreducible components. Using the full asymptotic expansion of the spectral action, we generate the inflation potential and slow-roll parameters for the corresponding pure gravity inflationary theory. 1 One-parameter family of Dirac-type operators Dt In this section, we follow Agricola [2] to define a family of Dirac operators, which interpolates the Levi-Civita, cubic, and trivial Dirac operators. Let the Lie algebra g of G be identified as the space of left invariant tangent vector fields on G, then one obtains a family of connections ∇X := ∇ 0 X + t[X, ·], where ∇ is the trivial connection with respect to the left trivialization, and ∇ is the trivial connection with respect to the right trivialization. Let 〈·, ·〉 denote a positive definite invariant metric on g. One checks that ∇ is a metric so(g) connection, and the torsion, T (X,Y ) = (2t− 1)[X,Y ], vanishes when t = 1/2. Hence one can think of the Levi-Civita connection, ∇, as the middle connection in the linear interpolation between the left trivial connection and the right trivial connection. The so(g) connection ∇ lifts to a metric spin(g) connection ∇̂t [11] given by the formula ∇̂X = ∇ 0 X + t 1 4 ∑ k,l 〈X, [Xk, Xl]〉XkXl. Let C l(g) denote the Clifford algebra generated by g with the relation XY + Y X = −2〈X,Y 〉 for X,Y ∈ g. Let {Xi} denote the set of orthonormal basis of g with respect to the metric 〈·, ·〉. Let us define the Dirac operator in C l(g)⊗ U(g) given by ∇̂t to be Dt := ∑ i Xi ⊗Xi + tH ∈ C l(g)⊗ U(g), (1) ∗Subject classification: AMS 58B34,22E46