Theories with a symmetry group not isomorphic to a matrix group

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Suppose we have a Lie group, $G$, and most importantly, there does not exist any continuous, injective homomorphism of $G$ into any $GL(n; \mathbb C)$. Are there any physical theories for which there is such a Lie group $G$ as a symmetry? Or does any such type of group appear in physics more generally?

One example I have been able to find is that the metaplectic group, that is, the double cover of $Sp_{2n}$, arises in the context of duality groups, and indeed it is not isomorphic to any matrix Lie group.

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Folland's book Harmonic analysis in phase space contains the application of the metaplectic group to geometric quantization. It is also relevant for the study of squeezed states in quantum optics.

Infinite-dimensional Lie groups are also relevant in physics; e.g. the Virasoro group in conformal field theory.
answered Jan 2, 2017 by (12,640 points)
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One of the most important examples is a stringy $\sigma$-model on $AdS_5 \times S^5$ background. The global symmetry group of this theory is $PSU(2,2 \vert 4)$ which is non-matrix, and $AdS_5 \times S^5$ itself is equal to $(SO(1,5) \times SO(6))/(SO(1,4) \times SO(5))$ which is a bosonic part of $PSU(2,2 \vert 4)/(SO(1,4) \times SO(5))$ supergroup.

answered Jan 7, 2017 by (670 points)
edited Jan 7, 2017

PSU(2,2|4) is a supergroup rather than a group. Do you mean that PSU(2,2|4) cannot be realized as a subsupergroup of some GL(m|n)? If so, why?

$AdS_5 \times S^5$ is more precisely the bosonic space underlying the superspace $PSU(2,2|4)/(SO(1,4) \times SO(5))$.

@40227 You are right that $AdS_5 \times S^5$ is a bosonic part of SUSY $\sigma$-model target superspace. I was somewhat imprecise.

$PSU(2,2 \vert 4)$ cannot be realized as a matrix supergroup due to the property of projectivity. It is not a specific feature of supergroups -- the  simplest example is $PSL(2, \mathbb{R})$, which is also cannot be realized in terms of matrices.

@Andrey Feldman  When one projectivizes, one divides by the center so one gets the image of the group under the adjoint representation, so one always gets a matrix group. In fact, $PSL(2,\mathbb{R})$ is the connected component of the identity in $SO(1,2)$, which is obviously a matrix group. Similarly, $PSU(2,2)$ is the connected component of the identity in $SO(2,4)$, which is obviously a matrix group.

@40227 Sorry, of course you are right. The fact that $PSU(2,2 \vert 4)$ is non-matrix is unrelated to projectivity. In order not to write a very long comment, let me refer you to section 3.2 in this review on $AdS/CFT$: https://arxiv.org/pdf/1104.2604v3.pdf, where this group is discussed.

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