Lorentz spinor in Lorentz $\rm Spin(3,1)$ signature and the real structure?

Question

In this paper:

• J. Wang, X. Wen and E. Witten, "A new ${\rm SU}(2)$ anomaly", J. Math. Phys. 60 (2019) 052301,

it says the following in p.2,

It says for $3+1$ dimensional spacetime, the Weyl spinor with $SU(2)$ isospin $1/2$ , "Lorentz signature always carry a real structure; if a fermion field appears in the Lagrangian, so does its hermitian adjoint"

What does it mean to be real ? If :

• Weyl spinor is complex in $\text {Spin}(3,1)$, and
• its $SU(2)$ isospin $1/2$ is pseudoreal in $SU(2) = \text {Spin} (3)$,

why do we get a Lorentz signature always carry a real structure (instead of just complex or pseudoreal)? Does it mean the whole Weyl spinor is in a real representation (4 component) of $\text {Spin} (3,1)$ and $SU(2)$ together?

What is the emphasis to say "In the Euclidean signature, nothing like that is true in general; what happens depends on the spacetime dimension"? Isnt that the Lorentz signature : real, pseudoreal, or complex also depends on the spacetime dimension ?

This post imported from StackExchange Physics at 2020-12-03 13:06 (UTC), posted by SE-user annie marie heart

Answer 1

The highlighted statement is quite gnomic. There are many like it the paper. The claim that an an odd number of zero modes makes the path integral measure change sign under a gauge transformation by the central element $-1\in {\rm SU}(2)$ is likewise a mystery. The measure has both a $d\bar \psi_0$ and $d\psi_0$ for each zero mode, and so is invariant under the simultaneous transformation $\psi\to -\psi$ and $\bar\psi \to -\bar \psi$. Further, while it is clear that the mod 2 index of the five dimensional Dirac operator connects to the five dimensional operator $(-1)^F$, (which comes from the mapping torus periodic boundary condition in the $S^1$ direction) and is given by the ordinary four dimensional index reduced mod 2, I do not see how they can claim that this relates to the four dimensional $(-1)^F$ which is not well defined on $S^4$. I look forward to some interesting answers to your question.

This post imported from StackExchange Physics at 2020-12-03 13:06 (UTC), posted by SE-user mike stone