12-dimensional supergravity





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There is a sensible theory of supergravity in a total of 12 spacetime dimensions. Even though this requires an exotic non-Lorentzian signature of (10,2)(10,2) (hence with a “2-dimensional time”) it has been argued that this is a better starting point for obtaining low-dimensional supergravity theory by KK-compactification, since it yields some lower-dimensional theories that are missed when starting with 11-dimensional supergravity, notably type IIB supergravity in 10 dimensions, hence relates to F-theory as 11-dimensional supergravity relates to M-theory (e.g. Nishino 97b, Hewson 97). (A theory in (9,3)(9,3) signature has also been proposed in (Kriz 05).)

It is an oft-repeated folklore that the highest number of spacetime dimensions for supergravity to make sense is 11, realized by 11-dimensional supergravity. However, there are some assumptions that go into this conclusion. First of all, the argument goes that after KK-compactification to 4-dimensions there must not appear supermultiplets with mass-less fields of spin >2\gt 2, since another folklore argument states that quantum field theory in 3+13+1 dimensions with fields of spin larger than 2 is inconsistent.

(This in turn needs further qualification: Consistent quantum field theory with an infinite tower of higher spin fields is consistent, this is called higher spin gauge theory arising as the vanishing string tension-limit of string field theory. Ever since this discovery, the modified folklore is that field theories with a finite number of higher spin fields is inconsistent.)

Since acting with a supersymmetry generator on elements of a supermultiplet increases spin by 1/2, this argument requires that there are at most (2(2))×2=8(2 - (-2)) \times 2 = 8 super charges in (3+1)d, hence corresponding to N=8 d=4 supergravity.

This, in turn, requires, by the rules of KK-compactification, that

  1. there be only a single supercharge in dimension 10+110+1, since the irreducible real spin representation of Spin(10,1)Spin(10,1) has real dimension 32, which branches as 3284\mathbf{32} \mapsto 8 \cdot \mathbf{4} under Spin(3,1)Spin(10,1)Spin(3,1) \hookrightarrow Spin(10,1);

  2. there cannot be any supercharge in dimension 11+111+1, since the irreducible real spin representation of Spin(11,1)Spin(11,1) has real dimension 64, which branches as 64164\mathbf{64} \mapsto 16 \cdot \mathbf{4} under Spin(3,1)Spin(11,1)Spin(3,1) \hookrightarrow Spin(11,1).

However, the second conclusion here is evaded by a change of spacetime signature: The irreducible real spin representation of Spin(10,2)Spin(10,2) still happens to be of dimension 32 and still branches as 3284\mathbf{32} \mapsto 8 \cdot \mathbf{4}.


The 2+12+1-brane in 10+210+2 dimensions

There is supposed to be a consistent fundamental super p-brane on 10+210+2-dimensional supergravity backgrounds, whose double dimensional reduction yields the M2-brane in 11-dimensional supergravity and further the superstrings not just of type IIA supergravity but also (?) of type IIB supergravity. The worldvolume of this p-brane has 4 spacetime dimensions with signature (2,2)(2,2). Therefore some authors refer to this as a “2+2”-brane, even though this does not mesh well with the naming convention of pp-branes in Lorentzian signature. Since Lorentzian pp-branes have (p+1)(p+1)-dimensional worldvolume, the systematic naming here would be “2+1”-brane.

See (Blencowe-Duff 88, section 7, Hewson-Perry 96, Nishino 97b)



On the 2+12+1-brane in 10+210+2 dimensions

On supergravity in 9+39 + 3 dimensions

Last revised on August 29, 2018 at 03:05:49. See the history of this page for a list of all contributions to it.