# Relate two different mod 2 indices: $\eta$ invariant and the number of zero modes of Dirac operator, associated to SU(2)

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.)

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:

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: $$$$Z^{\nu}_{SU(2)}[a] = \left(\frac{\det({D}_a-|m|)}{\det({D}_a+|m|)}\right)^\nu \stackrel{|m|\rightarrow \infty}{\longrightarrow} (-1)^{\nu N_0'}$$$$ where $$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.

• 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? – Sebastian Goette Mar 4 at 7:56