Spectral Invariants

Abstract

In the previous chapter we have identified the gauge group canonically associated to any spectral triple and have derived the generalized gauge fields that carry an action of that gauge group.

7.A Divided Differences

We recall the definition of and some basic results on divided differences.

Definition 7.17

Let $f: \mathbb {R}\rightarrow \mathbb {R}$ and let $x_0, x_1, \ldots x_n$ be distinct points in $\mathbb {R}$. The divided difference of order $n$ is defined by the recursive relations

$$\begin{aligned} f[x_0]&= f(x_0), \\ f[x_0,x_1, \ldots x_n]&= \frac{ f[x_1, \ldots x_n] -f[x_0,x_1, \ldots x_{n-1}]}{x_{n} - x_0}. \end{aligned}$$

On coinciding points we extend this definition as the usual derivative:

$$ f[x_0, \ldots ,x \ldots , x \ldots x_n]:= \lim _{u \rightarrow 0} f[x_0, \ldots ,x+u \ldots , x \ldots x_n]. $$

Finally, as a shorthand notation, for an index set $I =\{i_1, \ldots , i_n \}$ we write

$$ f[x_{I}] = f[x_{i_1}, \ldots , x_{i_n}]. $$

Also note the following useful representation:

Proposition 7.18

For any $x_0, \ldots , x_n \in \mathbb {R}$,

$$ f[x_0, x_1, \ldots , x_n] = \int \limits _{\Delta _n} f^{(n)} \left( s_0 x_0 + s_1 x_1 + \cdots + s_n x_n\right) d^ns. $$

Proof

See Note 12 on page 133. $\Box$

Exercise 7.1

Prove Proposition 7.18 and show that it implies

$$ \sum _{i=0}^n f[x_0, \ldots , x_i,x_i, \ldots , x_n] = f'[x_0, x_1, \ldots , x_n]. $$

Proposition 7.19

For any $x_1, \ldots x_n \in \mathbb {R}$ for $f(x)= g(x^2)$ we have,

$$\begin{aligned} f[x_0,\cdots ,x_n] = \sum _I \left( \prod _{\{ i-1, i \} \subset I} (x_i+x_{i+1}) \right) g[x_I^2] , \end{aligned}$$

where the sum is over all ordered index sets $I= \{ 0 = i_0 < i_1<\cdots <i_k =n \}$ such that $i_{j} - i_{j-1} \le 2$ for all $1 \le j \le k$ (i.e. there are no gaps in $I$ of length greater than 1).

Proof

This follows from the chain rule for divided differences (see Note 13 on page 133): if $f = g \circ \phi$, then

$$ f[x_0, \ldots x_n] = \sum _{k=1}^n \sum _{ 0 = i_0 < i_1<\ldots <i_k =n } g[\phi (x_{i_0}), \ldots , \phi (x_{i_k})] \prod _{j=0}^{k-1} \phi [ x_{i_j}, \ldots , x_{i_{j+1}}]. $$

For $\phi (x) = x^2$ we have $\phi [x,y]= x+y$, $\phi [x,y,z]= 1$ and all higher divided differences are zero. Thus, if $i_{j+1} - i_j > 2$ then $\phi [ x_{i_j}, \ldots , x_{i_{j+1}}]=0$. In the remaining cases one has

$$ \phi [ x_{i_j}, \ldots , x_{i_{j+1}}]= \left\{ \begin{array}{ll} x_{i_j}+ x_{i_{j+1}} &{}\quad \text {if } \,i_{j+1} - i_j =1 \\ 1 &{}\quad \text {if } \,i_{j+1} - i_j =2 ,\\ \end{array}\right. $$

and in the above summation this selects precisely the index sets $I$. $\Box$

Example 7.20

For the first few terms, we have

$$\begin{aligned} f[x_0,x_1]&= (x_0+x_1) g[x_0^2, x_1^2],\\ f[x_0,x_1,x_2]&= (x_0+x_1)(x_1+x_2) g[x_0^2, x_1^2,x_2^2] + g[x_0^2,x_2^2] ,\\ f[x_0,x_1,x_2,x_3]&= (x_0+x_1)(x_1+x_2)(x_2+x_3) g[x_0^2,x_1^2, x_2^2,x_3^2] \\&\quad + (x_2+x_3) g[x_0^2,x_2^2,x_3^2] + (x_0+x_1) g[x_0^2,x_1^2, x_3^2]. \end{aligned}$$

Notes

Section 7.1 Spectral Action Functional

• 1. The spectral action principle was introduced by Chamseddine and Connes in [1, 2].

• 2. Note that we have put two restrictions on the fermions in the fermionic action $S_f$ of Definition 7.3. The first is that we restrict ourselves to even vectors in $\mathcal {H}^+$, instead of considering all vectors in $\mathcal {H}$. The second restriction is that we do not consider the inner product $\langle J\tilde{\psi }',D_\omega \tilde{\psi }\rangle$ for two independent vectors $\psi$ and $\psi '$, but instead use the same vector $\psi$ on both sides of the inner product. Each of these restrictions reduces the number of degrees of freedom in the fermionic action by a factor of $2$, yielding a factor of $4$ in total. It is precisely this approach that solves the problem of fermion doubling pointed out in [3] (see also the discussion in [4], Chap. 1, Sect. 16.3]). We shall discuss this in more detail in Chaps. 9 and 11, where we calculate the fermionic action for electrodynamics and the Standard Model, respectively.

Section 7.2 Expansions of the Spectral Action

• 3. For a complete treatment of the Laplace–Stieltjes transform, see [5].

• 4. Lemma 7.6 appeared as [4, Lemma 1.144].

• 5. Corollary 7.8 is [4], Theorem 1.145]. An analysis of the term ${{\mathrm{Tr}}}|D_\omega |^{-z} \big |_{z=0}$ therein, including a perturbative expansion in powers of $\omega$ has been obtained in [6].

• 6. Section 7.2.2 is based on [7].

• 7. The notation $\langle X_0, \ldots , X_n \rangle _{t,n}$ should not be confused with the zeta functions $\langle X_0,\ldots , X_n \rangle _z$ introduced in Chap. 5. However, they are related through the formula

  $$ \langle X_0, \ldots , X_n \rangle _{t,n} = \frac{(-1)^p}{2 \pi i } {{\mathrm{Tr}}}\int e^{-t \lambda } X_0 (\lambda -D^2)^{-1} X_1 \cdots A^n (\lambda -D^2)^{-1} d \lambda . $$

  Multiplying this expression by $t^{z-1}$ and integrating over $t$ eventually yields $\langle X_0,\ldots , X_n \rangle _z$. For details, we refer to [8], Appendix A].

• 8. For more details on Gâteaux derivatives, we refer to [9]. For instance, that the Gâteaux derivative of a linear map $F$ between Fréchet spaces is a linear map $F'(x)(\cdot )$ for any $x \in X$ is shown in [9], Theorem 3.2.5].

• 9. The expansion in Eq. 7.2.4 is asymptotic in the sense that the partial sums $\sum _{n=0}^N \frac{1}{n!} S_b^{(n)}(0)(\omega , \ldots , \omega )$ can be estimated to differ from $S_b[\omega ]$ by $\mathcal {O}(\Vert \omega \Vert ^{N+1})$. This is made precise in [7].

• 10. Theorem 7.14 was proved in [7]. A similar result was obtained in finite dimensions in [10] and in a different setting in [11]. Corollary 7.16 was obtained at first order for bounded operators [12].

• 11. There is a close connection between the spectral action, the Krein spectral shift function [13, 14], as well as the spectral flow of Atiyah and Lusztig [15–17]. One way to see this is from Theorem 7.11, where we can control the asymptotic expansion of the spectral action using the remainder terms $R_k$. In [11] these terms are analyzed and related to a spectral shift formula [13, 14] (see also the book [18] and the review [19], and references therein). In fact, under the assumption that $f$ has compact support, the first rest term $S_b[\omega ] - S_b^{(0)}(0)$ becomes

  $$ {{\mathrm{Tr}}}f(D+\omega ) - {{\mathrm{Tr}}}f(D) = \int _\mathbb {R}f(x) d ({{\mathrm{Tr}}}E_{D+\omega }(x)) - \int _\mathbb {R}f(x) d ({{\mathrm{Tr}}}E_{D}(x)), $$

  where $E_{D+\omega }$ and $E_D$ are the spectral projections of $D+\omega$ and $D$, respectively. After a partial integration, we then obtain [11][Theorem 3.9]

  $$\begin{aligned} (*)\qquad \qquad \qquad \qquad {{\mathrm{Tr}}}f(D+\omega ) - {{\mathrm{Tr}}}f(D) = \int _\mathbb {R}f'(x) \xi (x) dx , \end{aligned}$$

  where

  $$ \xi (x) = {{\mathrm{Tr}}}\left( E_{D+\omega }(x) - E_{D}(x) \right) $$

  is the so-called spectral shift function. Moreover, it turns out that the higher-order rest terms are related to higher-order spectral shift functions [20, 21].

  Let us also briefly describe the intriguing connection between the spectral shift function and the local index formula of Chap. 5. In fact, [22] (using a result from [23], Appendix B]) relates the index of $PuP$ which appears in the odd local index formula (Theorem 5.21) to the spectral flow $sf (\{ D_t\})$ of the family $D_t = (1-t)D+ t u Du^*= D + t u[D,u^*]$ for $0 \le t \le 1$. Roughly speaking, the spectral flow of such a family of operators is given by the net number of eigenvalues of $D_t$ that pass through $0$ in the positive direction when $t$ runs from $0$ to $1$. One then has

  $$ {{\mathrm{index}}}\,PuP = sf (\{ D_t\}_{t\in [0,1]}). $$

  The connection between spectral flow and the spectral shift function was first hinted at in [24] and has been worked out in [25, 26]. Essentially, these latter papers build on the observation that the spectral flow from $D_0 - x$ to $D_1-x$ for any real number $x$ is equal to the spectral shift function $\xi (x)$ defined above in terms of the spectral projections of $D_0$ and $D_1$. Note that for a path connecting $D$ and the unitarily equivalent operator $uDu^*$ the spectral shift function is a constant. In fact, since $D$ and $uDu^*$ have identical spectrum, the left-hand side of (${*}$) vanishes. Integration by parts on the right-hand side then ensures that $\xi$ is constant (and in fact equal to the above index).

  Eventually, a careful analysis of the spectral flow [27] (and [28] for the even case) allows one to prove the local index formula in the much more general setting of semi-finite spectral triples [22, 29–31].

  Another encounter of spectral shift and spectral flow is in the computation of the index of the operator $d/dt + A(t)$ with $A(t)$ a suitable family of perturbations $(t \in \mathbb {R}$). In fact, they were the operators studied by Atiyah, Patodi and Singer in [15–17]. The index of $d/dt + A(t)$ can be expressed in terms of the spectral flow of $A(t)$ under the assumptions that $A(\pm \infty )$ is boundedly invertible, and that $A(t)$ has discrete spectrum for all $t \in \mathbb {R}$. We refer to [32] for a careful historical account, and the extension of this result to relatively trace class perturbations $A(t)$.

Appendix 7.A Divided Differences

• 12. Proposition 7.18 is due to Hermite [33].

• 13. The chain rule for divided differences is proved in [34]. For Cauchy’s formula for divided differences, we refer to [35, Chap. I.1].