Geometry dg.differential – Links two different mod 2 indexes: $ eta $ invariant and the number of null modes of the Dirac operator, associated with SU (2)

My main question in this post here is that:

How can we relate the two following indices mod 2:

  1. $ eta $ invariant,

  2. the number of zero modes of the Dirac operator,

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

regarding local data like the 2nd class of Chern $ c_2 $ of principal $ US (2) $ package, or the so-called $ F_a wedge F_a $ term of $ US (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 local datawe mean for example locally well defined data, eg. field strength / curvature measured by performing local parallel transport of the main packet SU (2).)



Detailed context of the questions:

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

  1. the $ eta $ invariant I referred to above is from one $ mathbb Z_4 $ class
    in 4d ofa a bordering group defined as
    $$ Omega _ { mathrm {Pin} ^ {+} times _ { mathbb Z_2} SU (2)} ^ 4 equiv text {Hom} ( Omega ^ { mathrm {Pin} ^ {+ } times _ { mathbb Z_2} SU (2)} _ 4, U (1)) Z_4 times mathbb Z_2 $$
    $$
    Z ^ { nu}[a]= exp (2 pi i nu eta_ {SU (2)}[a])
    qquad ( nu) in mathbb Z_4.
    $$

  2. The number of zero modes of the above mentioned Dirac operator comes from a $ mathbb Z_2 $ class
    in 4d's
    $$ Omega _ { mathrm {Pin} ^ {-} times _ { mathbb Z_2} SU (2)} ^ 4 equiv text {Hom} ( Omega ^ { mathrm {Pin} ^ {- } times _ { mathbb Z_2} SU (2)} _ 4, U (1)) cong ( mathbb Z_2) ^ 3. $$
    $$
    Z ^ { nu} = (- 1) ^ { nu N_0}, qquad ( nu) in mathbb Z_2.
    $$

    • More details: in this case, each eigenvalue of the Dirac operator is accompanied by an opposite value (ie by presenting an operator who does not commute with the l & # 39; 39; operator Dirac and who commutes with the transition functions). A similar calculation then gives:
      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 & # 39;
      end {equation}

      or $ N_0 $ is the number of zero modes of the operator Dirac. Its mod 2 value is a topological spin invariant known as the mod 2 index. Non-trivial fSPT classes generated by such Dirac fermions are effectively labeled by $ nu in mathbb Z_2 $.

The notations of the bordism / cobordism group are quite classic, where we also follow the above reference Annals of Physics 394, 244-293 (2018) and references therein.