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Dirac Matrix property suitable to finding sets of intersecting branes

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So, 11 dimensional supergravity has four oft-studied half-BPS states, the KK1 plane wave, the M2 brane, the M5 brane and the KK6 monopole. To figure out if we can find more solutions in the form of intersecting branes, it is useful to figure out the conditions under which

$$[\Gamma_{0,a_1, \ldots, a_M}, \Gamma_{0,b_1, \ldots, b_N}] = 0$$

or

$$\{\Gamma_{0,a_1, \ldots, a_M}, \Gamma_{0,b_2, \ldots, b_N}\}= 0$$

where $a_i, b_i \in \{1, 10\}$ are the usual spatial indices of 11 D supergravity. The former leads to intersecting branes if some $a_i = b_j$'s and the latter leads to rotated branes (or branes within branes) if some $a_i = b_j$'s.

This property is important because the half-BPS conditions of the form

$$\Gamma_{0,a_1, \ldots, a_M}\epsilon = \pm \epsilon$$

have a self-consistency requirement $(\Gamma_{0,a_1, \ldots, a_M})^2 = 1$ (which must be enforced whether you use the commutator or the anticommutator).

It should be possible to find all possible (M, N) tuples $(a_1, \ldots a_M)$ and $(b_1, \ldots b_N)$ satisfying either one of the two conditions above, using some techniques of the representation theory of Clifford algebras or simply some properties of Dirac matrices. I know a few examples:

(1) (0,1,2) and (0,3,4) commute (2) (0,1,2) and (0,1,3,4,5,6) commute (3) (0,1,2) and (0,1,3) anticommute (4) (0,1,2) and (0,1,2,3,4,5) anticommute

where the first ordered M-tuple defines the indices of $\Gamma$ from left to right, and the second N-tuple defines the indices of the other $\Gamma$. So, in this notation (1) means $[\Gamma_{012},\Gamma_{034}] = 0$.

Is there a general solution to this problem? Just writing down the gamma matrices and using the Clifford algebra to figure out which commutators/anticommutators are zero is obviously one way to do it, but I am wondering if there's a better way.


This post imported from StackExchange Physics at 2015-05-22 20:54 (UTC), posted by SE-user leastaction

asked May 21, 2015 in Theoretical Physics by leastaction (425 points) [ revision history ]
edited May 22, 2015 by Dilaton

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