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My major question in this post here is that:

How can we relate the following two mod 2 indices:

  1. $\eta$ invariant,

  2. the number of the zero modes of the Dirac operator $N_0'$ mod 2,

associated to the Stiefel-Whitney class $w_2(V_{SO(3)})$ of associated vector bundle of $SO(3)$,

relating to the local data:

  1. such as the 2nd Chern class $c_2$ of principal $SU(2)$ bundle, or the so called $F_a\wedge F_a$ term of $SU(2)$ field strength $F_a$? $$ \exp\big( { i \pi} \int\limits_{M_4} c_2 \big) = \exp\big( \frac{ i \pi}{8 \pi^2}\int\limits_{M_4} \text{Tr}\,F_a\wedge F_a \big) $$ (Note that by the local data, we mean such as the locally well-defined data, e.g. field strength/curvature measured by doing a local parallel transport of the principal SU(2) bundle.)

A short answer I already knew partially demystify my above question:

  1. $\eta$ invariant is obtained from a $\mathrm{Pin}^{+}\times_{\mathbb Z_2} SU(2)$-bundle, which the invariant is also its bordism invariant.

  2. the number of the zero modes of the Dirac operator $N_0'$ mod 2 is obtained from a $\mathrm{Pin}^{-}\times_{\mathbb Z_2} SU(2)$-bundle, which the invariant is also its bordism invariant.

  3. the 2nd Chern class $c_2$ or properly $\exp\big( { i \pi} \int\limits_{M_4} c_2 \big)$ is from a $\mathrm{O} \times SU(2)$-bundle, which the invariant is also its bordism invariant.

To relate Invariants of 1. and 2. to 3., we simply restrict the $\mathrm{Pin}^{+}\times_{\mathbb Z_2} SU(2)$-structure and $\mathrm{Pin}^{-}\times_{\mathbb Z_2} SU(2)$-structure to $\mathrm{O} \times SU(2)$-structure; then the two invariants (1. and 2.) will become the 3. invariant.

But this short reply does not resolve the local data definition problem of two different mod 2 indices on manifolds with the following structures:

$$\mathrm{Pin}^{+}\times_{\mathbb Z_2} SU(2), \text{ and } \mathrm{Pin}^{-}\times_{\mathbb Z_2} SU(2)$$


Detailed Background/Definitions of my questions:

Follow Annals of Physics 394, 244-293 (2018), DOI: 10.1016/j.aop.2018.04.025, by Guo, Putrov and Wang, I know that

  1. The $\eta$ invariant I refered to above is from a $\mathbb Z_4$ class in 4d ofa bordism group defined as $$\Omega_{\mathrm{Pin}^{+}\times_{\mathbb Z_2} SU(2)}^4\equiv \text{Hom}(\Omega^{\mathrm{Pin}^{+}\times_{\mathbb Z_2} SU(2)}_4,U(1))\cong \mathbb Z_4\times \mathbb Z_2$$ $$ Z^{\nu}[a]=\exp(2\pi i \nu \eta_{SU(2)}[a]), \qquad (\nu) \in \mathbb Z_4. $$

    • More details: We consider the case of $\nu$ massive Dirac fermions transforming under an orientation reversal map by $T'=CT$ matrix described below: Take $(CT)^2=1$ this requires a choice of $\mathrm{Pin}^{+}\times_{\mathbb Z_2} SU(2)$ structure on the manifold. One can consider the forgetful map $\text{Pin}^{+}\times_{\mathbb Z_2} SU(2)\rightarrow SU(2)/\mathbb Z_2\cong SO(3)$. So that any $\text{Pin}^{+}\times_{\mathbb Z_2} SU(2)$ structure on a 4-manifold $M_4$ gives an $SO(3)$ bundle $V_{SO(3)}$ on $M_4$. The $SO(3)$ bundle can be lifted to an $SU(2)$ bundle if $w_2(V_{SO(3)})=0$. This should be possible if the manifold admits $\text{Pin}^+$ structure. The obstruction to the existence of $\text{Pin}^+$ structure is also $w_2$ of the tangent bundle $TM_4$. If $w_2(TM_4)\neq 0$, then to define a $\text{Pin}^{+}\times_{\mathbb Z_2} SU(2)$ structure one can choose an $SO(3)$ bundle with $w_2(V_{SO(3)})=w_2(TM_4)$ (which is always possible), and lift it to $\text{Pin}^{+}\times_{\mathbb Z_2} SU(2)$.

    Consider partition function of such fermions with negative mass $m$ (normalized by partition function of fermions with positive mass) in presence of background $SU(2)$ connection $a$: $$ Z^{\nu}_{SU(2)}[a] = \left(\frac{\det({D}_a-|m|)}{\det({D}_a+|m|)}\right)^\nu \stackrel{|m|\rightarrow \infty}{\longrightarrow} \exp(2\pi i \nu \eta_{SU(2)}[a]) $$ where {${D}_a \equiv e^\mu_{\mu'} \gamma^{\mu'} (\partial_\mu+i \omega_\mu-i a_\mu)$} is the Dirac operator, its global definition was discussed in Ref above. Its $\eta$-invariant is defined, as usual, by the following formula $$ \eta_{SU(2)}=\frac{1}{2}(N_0+\lim_{s\rightarrow {0}+}\sum_{\lambda\neq 0} \text{sign}\,\lambda |\lambda|^{-s}) $$ where $\lambda$ are eigenvalues of ${D}_a$ and $N_0$ are the number of its zero modes. Since $\exp(2\pi i \nu \eta_{SU(2)})$ is cobordism invariant, the calculation of the bordism group tell us that $\eta_{SU(2)}\in \frac{1}{4}\mathbb{Z}$ and non-trivial fSPT classes generated by such massive Dirac fermions are effectively labelled by $\nu\in \mathbb Z_4$.

  2. The number of the zero modes of the Dirac operator I refered to above is from a $\mathbb Z_2$ class in 4d of $$\Omega_{\mathrm{Pin}^{-}\times_{\mathbb Z_2} SU(2)}^4\equiv \text{Hom}(\Omega^{\mathrm{Pin}^{-}\times_{\mathbb Z_2} SU(2)}_4,U(1))\cong (\mathbb Z_2)^3.$$ $$ Z^{\nu}=(-1)^{\nu N_0'},\qquad (\nu) \in \mathbb Z_2. $$

    • More details: In this case each eigenvalue of the Dirac operator is accompanied by an opposite one (that is by presenting an operator that anti-commutes with the Dirac operator and commutes with the transition functions). A similar calculation gives then: \begin{equation} Z^{\nu}_{SU(2)}[a] = \left(\frac{\det({D}_a-|m|)}{\det({D}_a+|m|)}\right)^\nu \stackrel{|m|\rightarrow \infty}{\longrightarrow} (-1)^{\nu N_0'} \end{equation} where $N_0'$ is the number of the zero modes of the Dirac operator. Its value mod 2 is a spin-topological invariant known as mod 2 index. The non-trivial fSPT classes generated by such massive Dirac fermions are effectively labelled by $\nu\in \mathbb Z_2$.

The notations of the bordism/cobordism group are fairly standard, where we also follow the Ref given above Annals of Physics 394, 244-293 (2018) and References therein.

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  • $\begingroup$ Please help me understand your question. Ad 1. $\eta$-invariant of which operator on which manifold? Ad 2. Which manifold is the Dirac operator $N_0'$ defined on? Ad 3. How does this relate to the manifold $M_4$ in 3? $\endgroup$ – Sebastian Goette Mar 4 at 7:56

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