All spectral action calculations below, except SU(3), use the Poisson summation formula. To calculate the spectral action of SU(3) we apply a multivariate version of the Euler-Maclaurin formula.
Introduction
The quaternionic cosmology and the spectral action
The Dirac spectra for SU(2)/Q8
Trivial spin structure: nonperturbative spectral action
Consider a test function for the Poisson summation formula that has the form h(u) =g(4u+s . 2) for some s∈Z. This gives an expression for the spectral action on S=SU(2)/Q8 with a trivial spin structure and with a sphere S3 =SU(2) of radius one, which is of the form
Nontrivial spin structures: nonperturbative spectral action
The case with the 3-sphere SU(2) = S3 with radius a is then analogous, where the spectrum is scaled by a factor a−1, which is the same as changing Λ to Λa in the above expressions, so that one ( 2.7) is obtained. It is sufficient to note that the sum of the two polynomials (2.14) interpolating the spectral multiplicities,.
Poincar´ e homology sphere
- Generating functions for spectral multiplicities
- The Dirac spectrum of the Poincar´ e sphere
- The double cover Spin(4) → SO(4)
- The spectral multiplicities
- The spectral action for the Poincar´ e sphere
It is shown in [4] that the generating functions of the spectral multiplicities have the form These are calculated directly from the Taylor coefficients of the generating functions of the spectral multiplicities (4.100) and (4.101).
Flat tori
The spectral action on the flat tori
We note that the nonperturbative spectral performance is independent of the choice of spin structure. However, as in the case of spherical manifolds, we show here that the nonperturbative spectral behavior is independent of the spin structure.
The structure of Dirac spectra of Bieberbach manifold
Recalling the torus case
The spectral action for G2
The case of G2(a)
The decomposition of Z3 used to calculate this contribution to the spectral action is shown in Figure 3.2. When we include the contribution (3.1) due to the asymmetric component, we see that the spectral action of the space G2-(a) is equal to.
The case of G2(b) and G2(d)
The case of G2(c)
The spectral action for G3
The case of G3(a) and G3(b)
Combining this with the asymmetric contribution (3.1), we see that the spectral effect of spaces G3(a) and G3(b) is equal to. The dashed lines indicate one of the boundary lines which define the region ˜I together with its images under the symmetries of λklm.
The spectral action for G4
The case of G4(a)
Then we see that the contribution of the symmetric component of the spectrum to the spectral action is. Combining this with the asymmetric component, we find that the spectral action is given by (3.13).
The case of G4(b)
It is reasonable to expect that it will also give a multiple of the spectral function of the torus with the proportionality factor HL2/(4√ .3).
The spectral action for G6
The method used to calculate the spectral action is a very slight change to the one used in [16]. Then one obtains a non-perturbative expression for the spectral action using the Poisson summation formula. In section 4.11 we calculate the Dirac spectrum and spectral action in the case where Γ is the binary icosahedral group.
The expression for the spectral action derived here is the same as that found in [50].
Spin structures on homogeneous spaces
Dirac operator on homogeneous spaces
4.9) The notation p denotes the projection of ∈onto the kernel h, and the angle brackets denote the pairing of tangent vectors via the Riemannian metric.
Dirac spectra for lens spaces with Berger metric
Bk+1 is not equal to 0, and with respect to these two vectors D0non again gives the matrix expression given by Equation 4.15, with eigenvalues given by Equation 4.16. In the case N ≡2 mod 4, the analysis proceeds exactly as when N ≡0 mod 4, except for some small changes that do not change the spectrum. Note that the second row of the table corresponds to the case n =mN −1, in which case,n+ 1 =mN, which accounts for the factor of mN in the multiple.
Unlike the case where N is even, HomZN(Vn,Σ3) can be non-trivial regardless of whether n is even or odd.
Spectral action of round lens spaces
Round metric, T = 1
We now have the following form of the Dirac spectrum, which is still not quite the final form. The final form of the spectrum of the lens space LN equipped with the round metric, with N equal, is obtained when one realizes that the first row of Table 4.21 already completely describes the spectrum as soon as one lets the parameter take values throughout Z as opposed to just in N. Hence we have the following expression for the Dirac spectrum in the round, odd case.
The final form of the spectrum in the odd case is given by the following corollary.
Computing the spectral action
The last row can be split into two parts and combined with the third and fifth rows, changing the multiplicity in each case for b = N2−1. Now to calculate the spectral action, we proceed as in [16], and use the Poisson summation formula, except here we sum over N−22 arithmetic progressions instead of just one. Since zif ∈ S(R), so are the functions fb(j) and so we have the estimate as Λ approaches plus infinity,.
In the case where N is odd, the interpolating polynomials are collected in the following table.
Dirac spectra for dicyclic spaces with Berger metric
Unlike the case of lens space, the expression for the spectrum is the same whether N is even or odd. As in the case where N is even, k is an integer only when n is odd, which means that HomΓ(Vn,Σ3) is trivial unless n is odd. These eigenvectors form the basis for HomΓ(Vn,Σ3), and we see that the spectrum has the same expression as when N is even.
The Dirac operator on the dicyclic space S3/Γ equipped with the Berger metric corresponding to parameter T > 0, and the trivial spin structure has the following spectrum:.
Spectral action of round dicyclic space
Round metric, T =1
In this case, the third and fourth rows of 4.53 can be decomposed into N/2 parts and written as. By writing 2 and 2t+ 1 alternately, we obtain the following definitive form of the spectrum in the even case. By alternately writing as 2a, and 2a+ 1, and also by alternately writing as 2 and 2t+ 1, the first two rows of 4.53 can be written as respectively.
Next, the third and fourth lines of 4.53 are analyzed into N arithmetic progressions, and then each set of progressions is separated into two groups.
Computing the spectral action
Once this is done, it is easy to check that the lines coming from the positive spectrum combine perfectly with the lines coming from the negative spectrum, just like in the case of lens spaces.
Generating function method
The double cover Spin(4) → SO(4)
The complex semirotational representation ρ+ is just the projection onto SL3, where we identify SL3 with SU(2) via. The next complex semirotational representation ρ− is the projection onto SR3, where we identify SR3 with SU(2) via. In this paper, when Γ is the binary tetrahedral group, the binary octahedral group, or the binary icosahedral group, we choose the corresponding rotation structure.
It is clear that it raises the identity map, and therefore it corresponds to a rotation structure.
Dirac spectrum of round binary tetrahedral coset space
We don't need to worry about the smaller values of k since we can just check them by hand. When you have also verified that the polynomials interpolate the spectrum for small eigenvalues, you have shown by induction (12 inductions in parallel) that the polynomials interpolate that part of the spectrum in the positive reals. Proposition 4.9.1 For the round binary tetrahedral space with the trivial spin structure, the spectrum of the canonical Dirac operator Dis is contained in the set {±(3/2 +k)|k∈N}.
Therefore, by Lemma 4.5.3 we have calculated the spectral action of the binary tetrahedral cost space.
Dirac spectrum of round binary octahedral coset space
Proposition 4.10.1 For the round binary octahedral space with trivial spin structure, the spectrum of the canonical Dirac operator Dis lies in the set {±(3/2 +k)|k∈N}.
Dirac spectrum of round Poincar´ e homology sphere
Following the method developed in [16], and we calculate the spectral action of the quotient spaces S3/Γ equipped with the twisted Dirac operator corresponding to a finite-dimensional representation α of Γ as follows. Since the polynomial on the right-hand side is a multiple of the polynomial of the spectral multiplicities of the Dirac spectrum of the sphereS3 (see [16]), from this we will obtain the relation between the non-perturbative spectral action of the twisted Dirac operator DαΓ on S3/ Γ and the spectral effect on the sphere, see theorem 5.1.1 below. In all cases, we explicitly calculate the polynomials of the spectral multiplicities and check that (5.2) is satisfied.
Our calculations are based on a result of Cisneros-Molina, [19], on the explicit form of the Dirac spectra of twisted Dirac operators DαΓ, which we recall here below.
Twisted Dirac spectra of spherical space forms
In the sequel, we describe how to obtain equation (5.2) by explicitly analyzing the cases of the different spherical space forms: lens spaces, dicyclic group and binary tetrahedral, octahedral and icosahedral groups.
Lens spaces, odd order
Let us denote the irreducible representations of Zn by χt that send the generator to exp(2πitN. Proposition 5.3.1 The irreducible characters χEk of the irreducible representations of SU(2) restricted to Zn, odd, decompose in χ [t] of Zn .For the argument to go through, one must also check the special case Pc+Γ(1/2) = 0.
Note that in the statement of Theorem 5.2.1 the first rule holds even if we take k=−1, since the multiplicity for this value yields zero.
Lens spaces, even order
Since the coefficients of the polynomial are additive with respect to the direct sum, it is sufficient to consider only irreducible representations. To complete the calculation of the spectral function, the condition (5.3) still needs to be verified. We have the following lemma, which is obtained by equating the coefficients Pm+ and Pm−0 and also covers the cases of binary tetrahedral, octahedral and icosahedral groups.
Dicyclic group
A sign in the subscript indicates the sign of the traces of the elements in the conjugation class as elements of SU(2). One can decompose the characters χEk into the irreducible characters by inspection, and with a few counts obtain the following theorems.
Binary tetrahedral group
Then, since each character decomposes uniquely into irreducibles, we have a unique expression for χEk as a linear combination. A is necessarily invertible due to the uniqueness of the column of coefficients c, so c is given by z.
Binary octahedral group
Binary icosahedral group
Sums of polynomials
One-parameter family of Dirac operators D t
Since π is injective, the action of D2t is determined by the action of Tt, which in turn is written as a combination of 1⊗Cas, Cas⊗1 and ∆Cas. Since U(g) acts as left invariant differential operators on L2(G), it acts as the identity of the dual componentsVλ∗. But to know how Tt, i.e. D2t, acts on Vρ⊗Vλ, we need to know the effect of ∆ Cas, which can be obtained if one knows the direct sum decomposition of Vρ⊗Vλ into irreducible components, the so-called Clebsch-Gordan decomposition.
In fact, we reduce the study of the spectrum of Dt2 to the Clebsch-Gordan decomposition of Vρ⊗Vλ.
Spectral action for SU (2)
Spectrum of Dirac Laplacian of SU (3)
Spectrum for t = 1/3
In the case of t = 1/3, the expression of the spectrum becomes much simpler, since we no longer need to take into account the Clebsch-Gordan decomposition. We will later apply the Poisson summation formula to the result of Theorem 6.5.2, and we will make use of the neat property that the multiplicity of (p,0) and (0, q) is zero for p, q∈N.
Derivation of the spectrum
To derive the Dirac Laplacian spectrum, the pairing of weights must first be analyzed. We have listed the irreducible representations of SU(3) as well as the action of the Casimir operator on them. To write the spectrum of the Dirac Laplacian, the only obstacle now is to understand the term ∆ Cas in Theorem 6.3.6; that is
Each summand in the left column appears once if (p, q) lies in the set of parameter values listed in the right column.
Spectral action for SU (3)
- t = 1/3
- General t and the Euler-Maclaurin formula
- Analysis of remainders
- Analysis of main terms
Using Theorem 6.5.1, one can write the spectral action in terms of eight summations of the form. The terms in the Taylor expansions of the integrals yield the asymptotic expansion of the spectral action. In this way, one can obtain the large-O behavior of the spectral action with respect to Λ to any desired order.
When the asymptotic expansion of the spectral action is calculated with the Euler-Maclaurin formula, due to the chain rule, the negative powers, Λ−j appear only with derivativesf(k)(0), k≥j.
Details of the Calculations
- The Identity Term
- The terms (2i)! b 2i ∂h ∂ 2i RR
- The terms (2i)! b 2i (2j)! b 2j
- Boundary Term R
- The Boundary Terms (2i)! b 2i ∂h ∂ 2i R
- Corner Term
Here y5 is O(Λ−10), which is enough to suppress the positive powers of Λ in the remaining part of the expression if you work up to constant order. For large values of i and j, the powers of Λ1 suppress the powers of Λ that appear in the rest of the expression. Vittorio, "A flat universe from high-resolution maps of the cosmic microwave background radiation." Nature no.
Marcolli, "Boundary conditions of the RGE flow in the non-commutative geometry approach to particle physics and cosmology." Phys.
Lattice decomposition for the I 2 contribution to the spectral action of G2(a)
Lattice decomposition for G2(b), (d) computation. Two regions
Lattice decomposition for G2(c) computation. Two regions
Lattice decomposition for G3 computation. Six regions and the set l = m
Lattice decomposition for G4(a) computation. Four regions
Lattice decomposition for G4(b) computation. Four regions
Lattice decomposition for G6 computation. Four regions
