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Supersymmetry algebra cohomology. IV. Primitive elements in all dimensions from D = 4 to D = 11
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10.1063/1.4804953
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1 Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, D-30167 Hannover, Germany
J. Math. Phys. 54, 052302 (2013)
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### References

• By Friedemann Brandt
• Source: J. Math. Phys. 54, 052302 ( 2013 );
1.
1. F. Brandt, “Supersymmetry algebra cohomology. I. Definition and general structure,” J. Math. Phys. 51, 122302 (2010);
http://dx.doi.org/10.1063/1.3515844
1.e-print arXiv:0911.2118v5 [hep-th].
2.
2. A. Van Proeyen, “Tools for supersymmetry,” e-print arXiv:hep-th/9910030v6.
3.
3. F. Brandt, “Supersymmetry algebra cohomology. II. Primitive elements in 2 and 3 dimensions,” J. Math. Phys. 51, 112303 (2010);
http://dx.doi.org/10.1063/1.3515845
3.e-print arXiv:1004.2978v3 [hep-th].
4.
4. F. Brandt, “Supersymmetry algebra cohomology. III. Primitive elements in four and five dimensions,” J. Math. Phys. 52, 052301 (2011);
http://dx.doi.org/10.1063/1.3583554
4.e-print arXiv:1005.2102v2 [hep-th].
5.
5. M. V. Movshev, A. Schwarz and R. Xu, “Homology of Lie algebra of supersymmetries,” e-print arXiv:1011.4731v1 [hep-th].
6.
6. M. V. Movshev, A. Schwarz and R. Xu, “Homology of Lie algebra of supersymmetries and of super Poincare Lie algebra,” Nucl. Phys. B 854, 483 (2012);
http://dx.doi.org/10.1016/j.nuclphysb.2011.08.023
6.e-print arXiv:1106.0335v3 [hep-th].
7.
7. F. Brandt, “Aspects of supersymmetric BRST cohomology,” Strings, Gauge Fields, and the Geometry Behind, edited by A. Rebhan et al. (World Scientific, 2012), p. 87;
7.e-print arXiv:1201.3638v1 [hep-th].
8.
8.In D = 4, 8, …, there is an alternative choice of C for which the matrices ΓaC−1 are antisymmetric1,2 and which therefore requires N ∈ {2, 4, …}, excluding the case N = 1 (more precisely: a case N = 1 with nonvanishing anticommutators in (1.1)).
9.
9.This notation deviates somewhat from the notation used in Ref. 1 where Hg(sgh) denotes H(sgh) in the sector of ghost number g and Hp,*(sgh) denotes H(sgh) in the sector of c-degree p.
10.
10.The first part of Eq. (5.9) and Lemma 5.1 imply for some polynomials pija(ξ) and some . does not depend on ξ2, does not depend on ξ1, and is symmetric under ξ1↔ξ2 whereas all terms in depend on both ξ1 and ξ2 and is antisymmetric under ξ1↔ξ2. As a consequence, the second part of Eq. (5.9) implies that p11 a(ξ), p12 a(ξ), p22 a(ξ) are such that is -exact (or vanishes).
11.
11.In D = 6 one has for (N+, N) = (2, 0) and (N+, N) = (0, 2):where we used that vanishes for (N+, N) = (2, 0) and (N+, N) = (0, 2),12 and that also vanishes because ΓcC−1 is antisymmetric in D = 6, see (3.18).
12.
12.Equations (2.12) and (6.1) imply for (N+, N) = (2, 0), and for (N+, N) = (0, 2). follows from the “completeness relation” of the Γ-matrices in D = 5 which reads and implieswhere we used that (ΓaC−1)(5) and are antisymmetric, whereas (ΓabC−1)(5) is symmetric, see (3.18).
13.
13.For instance:where we used that and vanish because ΓaC−1 is antisymmetric in D = 6, see (3.18).
14.
14.Using the structure of sghc7 (which consists of bilinears and ) and sghc1, …, sghc6 (which only involve bilinears and ), one infers from (2.6) for p = 1 that is at least linear in both ξ1(6) and ξ2(6). This gives with constant 8 × 8-matrices Kr which can be chosen according to Kr ∈ {(C−1)(6),(ΓaC−1)(6),(ΓabC−1)(6),(ΓabcC−1)(6), since these matrices form a basis for 8 × 8-matrices. Equation (2.6) then leads to as in (7.9), up to a coboundary in which can be neglected.
15.
15.For N = 1 all cocycles with p > 0 depend at least linearly on both ξ1(7) and ξ2(7), see first comment at the end of Sec. VIII. In the case p = 1, this implies with constant 8 × 8-matrices Kr which can be chosen according to Kr ∈ {(C−1)(7),(ΓaC−1)(7),(ΓabC−1)(7),(ΓabcC−1)(7)} since these matrices form a basis for 8 × 8-matrices. Equation (2.6) then leads to as in (8.13), up to a coboundary in which can be neglected.
16.
16.The structures of sghc9 (which consists of bilinears and ) and sghc1, …, sghc8 (which only involve bilinears ) imply that with constant 16 × 16-matrices Kr which can be chosen according to Kr ∈ {(C−1)(8),(ΓaC−1)(8), ..., (ΓabcdC−1)(8), , ..., since these matrices form a basis for 16 × 16-matrices. Equation (2.6) then leads to as in (9.11), up to a coboundary in which can be neglected.
17.
17.In the case, p = 1 we use that every cocycle ω1 must be at least linear in the supersymmetry ghosts since no nonvanishing linear combination of the ca with constant coefficients is sgh-closed. This yields as in (10.13). Lemma 9.1 yields as in (10.13). Equation (10.18) follows from the facts that every monomial corresponds to a cocycle in proportional to one of the , and that , analogously to the situation in D = 6.12,13
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2013-05-20
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