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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.

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  • $\begingroup$ Perhaps: according to 1611.07874 (below eq. 2.1), the symplectic theories are not spin, for any $N,K$, unless I misunderstood something. $\endgroup$ – AccidentalFourierTransform Jun 21 '18 at 16:27
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One way to see whether it is spin is to consider the Chern-Simons theory with gauge group given by the universal covering. Then the original Chern-Simons theory is obtained by gauging a center one-form symmetry, and the theory is spin if and only if the generators of this one-form symmetry contain a line of half-integer spin.

For instance, in SO(N)K = Spin(N)K/Z2 you only need to compute the spin of the line in Spin(N)K that generates this Z2 one-form symmetry (which is K/2 mod 1, so SO(N)K is spin iff K is odd).

In general, the spin can be computed by the Casimir, and the generator of the center one-form symmetry can be obtained from the outer automorphism of affine Lie algebra acting on the trivial representation. Both of these have closed form formulas for any Lie algebra e.g. in the big yellow book https://books.google.com/books/about/Conformal_Field_Theory.html?id=mcMbswEACAAJ Equivalently, the theory is spin iff the instanton contribution on closed 4-manifold is an integer multiple of 2\pi for spin manifold and a half-integer multiple of 2\pi on non-spin manifold.

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  • $\begingroup$ This assumes that the theory with simply-connected group is never spin. Is this always true? How can we prove that? Is it possible to obtain spin TQFTs with simply-connected group by taking the level to be half-integral? $\endgroup$ – AccidentalFourierTransform Jul 8 at 20:53
  • $\begingroup$ for the theory to be spin you need it to have a bulk dependence of the form pi int w2(TM) cup x2 = pi int x2 cup x2 where x2 is some characteristic class of the gauge bundle and w2(TM) is the second Stiefel-Whitney class of the tangent bundle (whose trivialization is the spin structure). x2 measures the monopole number in 3d. So if the gauge group is simply connected I think it is always non-spin. $\endgroup$ – user236418 Jul 9 at 19:59

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