Which Chern-Simons TQFTs are spin?
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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}{1,f}, with ff 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′U(N)K,K+NK′ is spin iff K+K′K+K′ is odd; and that SU(N)KSU(N)K is never spin. What about other CS theories, such as: the classical Lie groups
Spin(N)KSpin(N)K,
Sp(2N)KSp(2N)K,
and the standard quotients,
SU(N)K/ZMSU(N)K/ZM (with M|NM|N),
SO(N)K:=Spin(N)K/Z2SO(N)K:=Spin(N)K/Z2,
PSO(2N)K:=SO(2N)K/Z2PSO(2N)K:=SO(2N)K/Z2,
PSp(2N)K:=Sp(2N)K/Z2PSp(2N)K:=Sp(2N)K/Z2
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 KK is odd. If this is correct, then only PSUPSU 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).
hope that help you...
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′U(N)K,K+NK′ is spin iff K+K′K+K′ is odd; and that SU(N)KSU(N)K is never spin. What about other CS theories, such as: the classical Lie groups
Spin(N)KSpin(N)K,
Sp(2N)KSp(2N)K,
and the standard quotients,
SU(N)K/ZMSU(N)K/ZM (with M|NM|N),
SO(N)K:=Spin(N)K/Z2SO(N)K:=Spin(N)K/Z2,
PSO(2N)K:=SO(2N)K/Z2PSO(2N)K:=SO(2N)K/Z2,
PSp(2N)K:=Sp(2N)K/Z2PSp(2N)K:=Sp(2N)K/Z2
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 KK is odd. If this is correct, then only PSUPSU 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).
hope that help you...
Answered by
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Hey mate ^_^
Spin Chern-Simons and Spin TQFTs are Chern-Simons TQFTs are spin as gauge theory whose action depends on the spin structure of the 3-manifold.
#Be Brainly❤️
Spin Chern-Simons and Spin TQFTs are Chern-Simons TQFTs are spin as gauge theory whose action depends on the spin structure of the 3-manifold.
#Be Brainly❤️
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