# Which Chern-Simons TQFTs are spin?

Refs.1&2 prove several level/rank dualities among different 3d Chern-Simons theories. An important point is that some dualities involve, on one side, a theory that depends on the spin structure, and on the other side, one that does not. To have a meaningful duality, the authors insist that the non-spin theories must be regarded as spin, by tensoring them with the (almost) trivial sTQFT $\{1,f\}$, with $f$ a fermion. They most often leave this factor implicit.

Now my problem is that it is rather non-trivial for me to tell whether a certain CS theory is spin or not, i.e., whether this factor is required. The only general method I can think of is to find the spectrum of lines, and check whether there is a fermion that braids trivially to the rest of lines, a computation that becomes prohibitively complicated as we increase the rank of the group.

Ref.1 claims that $U(N)_{K,K+NK'}$ is spin iff $K+K'$ is odd; and that $\mathrm{SU}(N)_K$ is never spin. What about other CS theories, such as: the classical Lie groups

• $\mathrm{Spin}(N)_K$,

• $\mathrm{Sp}(2N)_K$,

and the standard quotients,

• $\mathrm{SU}(N)_K/\mathbb Z_M$ (with $M|N$),

• $\mathrm{SO}(N)_K:=\mathrm{Spin}(N)_K/\mathbb Z_2$,

• $\mathrm{PSO}(2N)_K:=\mathrm{SO}(2N)_K/\mathbb Z_2$,

• $\mathrm{PSp}(2N)_K:=\mathrm{Sp}(2N)_K/\mathbb Z_2$

When are these theories spin? Is there any "easy" way to tell, or is computing the spectrum the only general approach?

Note: While ref.1 focuses on the unitary groups, ref.2's main concern is the orthogonal ones. If I am not mistaken (cf. the discussion around eq. 2.6), the authors claim that all the orthogonal groups are spin iff $K$ is odd. If this is correct, then only $\mathrm{PSU}$ and the symplectic groups remain to be determined.

(For completeness, it would also be nice to know about the exceptional groups too, and their quotients, but this is not a priority here).

References.

1. Po-Shen Hsin, Nathan Seiberg - Level/rank Duality and Chern-Simons-Matter Theories.

2. Clay Cordova, Po-Shen Hsin, Nathan Seiberg - Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups.

• Perhaps: according to 1611.07874 (below eq. 2.1), the symplectic theories are not spin, for any $N,K$, unless I misunderstood something. – AccidentalFourierTransform Jun 21 '18 at 16:27