Wednesday, 7 October 2020

Does $GL(N,mathbb{R})$ own spinor representation? Which group is its covering group? (Kaku's QFT textbook)



In Kaku's QFT textbook page 54, there is a saying:



$GL(N)$ does not have any finite-dimensional spinorial representation.



This implicates that $GL(N)$ owns infinite-dimensional spinorial representation. While in my opinion, a group's spinorial representation is the representation of its universal covering group. And the connnected compenent of $GL(n,\mathbb{R})$ ($n>2$) group is not simply connected and its fundamental group is $\mathbb{Z}_2$. So what group is its covering group?


My question:




  1. Since the connnected compenent of $GL(n,\mathbb{R})$ ($n>2$) group is not simply connected and according to Lie's theorem, there exist a simple connnected Lie group whose Lie algebra is $gl(n,\mathbb{R})$ , then what's this covering group of $GL(n,\mathbb{R})$? While I cannot imagine which group can cover the $GL(n,\mathbb{R})$.





  2. Now that $GL(N)$ owns infinite-dimensional spinorial representation, can show me explicitly, or give me some reference which have solved this problem.





Answer



I) Recall that since the Lie group $SO(N)\subset GL(N)$ is a proper subgroup of $GL(N)$, then functorially speaking, an irreducible representation of $GL(N)$ is also a (possible reducible) representation of $SO(N)$, but not necessarily the other way around.


When Ref. 1 states



There are no finite-dimensional spinorial representations of $GL(N)$,




it means in this context that the finite-dimensional spinor representation of $SO(N)$ does not arise from a finite-dimensional representation of $GL(N)$.


II) For$^1$ $N>2$, the (double) covering group of the general linear group $$GL(N,\mathbb{R})~\cong~ \mathbb{R}_{>0}\times SL(N,\mathbb{R})$$ is the metalinear group $ML(N,\mathbb{R})$. The metalinear group is a subgroup of the metaplectic group $Mp(2N,\mathbb{R})$ in twice the dimension. The reason that the metaplectic group $Mp(2N,\mathbb{R})$ has no non-trivial finite-dimensional representations is closely related to a similar fact for the Heisenberg Lie algebra.


References:




  1. M. Kaku, QFT, 1993; p. 54. and p. 640.




  2. M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, Vol. 2, 1986; p. 272.





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$^1$ $N=2$ is a special case, since $\pi_1(SL(2,\mathbb{R}),*)=\mathbb{Z}$, cf. e.g. this Phys.SE post and this Wikipedia page.


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