# projective representation of supergroup

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In fact, I am not very clear about what I am asking, but I am looking for a concrete example of supergroup which has non-trivial projective representation(some supergroup similar to usual Lie group SO(3), which has $H^2(SO(3),U(1))=Z_2$ as group cohomology.)

This post imported from StackExchange MathOverflow at 2015-11-20 14:55 (UTC), posted by SE-user Yingfei Gu
asked Apr 16, 2015
retagged Nov 21, 2015

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You are asking for interesting 2-cocycles on super Lie groups which are not just 2-cocycles on the underlying bosonic Lie groups. Here is one example:

Exceptional and fermionic (p+2)-cocycles for $p \in \mathbb{N}$ appear on super-Poincaré groups $\mathrm{Spin}(d-1,1) \rtimes \mathbb{R}^{d-1,1\vert \mathbf{N}}$ (where $\mathbf{N}$ denotes a choice real spinor representation) for a finite number of triples $(d,\mathbf{N},p)$. In string theory, the table of these nontrivial triples is called the brane scan since there is one such for every spacetime dimension $d$ for supergravity with $\mathbf{N}$-supersymmetries in which super p-branes may propagate.

Hence the 2-cocycles correspond to 0-branes. In particular $\mathbb{R}^{9,1\vert 16 + \overline{16}}$ (the type IIA supersymmetry super Lie group) carries such a 2-cocycle, corresponding to the D0-brane in typeII A string theory.

The corresponding projective representations are equivalently ordinary representations of the corresponding central extension. The central extension classified by the D0-brane 2-cocycle on the 10d supertranslation group is curious: it's 11d super-translation group. (This is a super Lie theoretic incarnation of the physics lore that type IIA string theory grows an 11th dimension via D0-brane condensation.)

There is loads of further interesting super Lie theoretic structure hidden in the super Lie cocycles of the brane scan. See The brane bouquet for more.

answered Nov 20, 2015 by (6,095 points)

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