Mass spectrum of chiral ten-dimensional <i>N=2</i> supergravity on <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>S</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></span>

H. J. Kim; L. J. Romans; P. van Nieuwenhuizen

doi:10.1103/PhysRevD.32.389

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Mass spectrum of chiral ten-dimensional N=2 supergravity on $S^{5}$

H. J. Kim

L. J. Romans

P. van Nieuwenhuizen

Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11794

California Institute of Technology, Pasadena, California 91125

Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11794

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Phys. Rev. D 32, 389 – Published 15 July, 1985

DOI: https://doi.org/10.1103/PhysRevD.32.389

Abstract

We discuss the spontaneous compactification of chiral N=2 ten-dimensional supergravity from ten to five dimensions on $S^{5}$ . Harmonic analysis on $S^{5}$ is used to compute the complete mass spectrum. Our results indicate that scalars and spinors in different SO(6) multiplets have different masses, even within the ‘‘massless’’ supermultiplet. We show that the conformal diffeomorphisms, which remain after imposing certain covariant gauge conditions for the general coordinate invariance, can be used to gauge away twice as many modes as there are gauge parameters. A doubleton multiplet of pure gauge modes is identified, and all modes in the massless supermultiplet lie at the beginning of infinite towers of modes.

References (22)

J. H. Schwarz, Nucl. Phys. B 226, 269 (1983); M. B. Green and J. H. Schwarz, Phys. Lett. 122 B, 143 (1983); J. H. Schwarz and P. C. West, ibid. 126 B, 301 (1983).
P. S. Howe and P. C. West, Nucl. Phys. B 238, 181 (1984).
For reviews, see J. H. Schwarz, Phys. Rep. 89, 223 (1982); M. B. Green, Surv. High Energy Phys. 3, 127 (1983).
L. Alvarez-Gaumé and E. Witten, Nucl. Phys. B 234, 269 (1983).
E. Witten, Princeton report, 1983 (unpublished).
S. R. Das, Phys. Lett. 148 B, 39 (1984).
Curved indices are denoted by μ hat , ν hat=0,...,9, μ , ν = 0 ,..., 4, and α , β = 5 ,..., 9 while corresponding flat indices are denoted by m hat , n hat (m hat,n hat = 0 ,..., 9), m,n (m,n=0 ,..., 4), and a,b (a,b=5,..., 9). Our metric is η ${}_{m hat n hat}$ =diag( - , + ,..., + ). To go to the η ${}_{m hat n hat}$ =diag(+ ,..., +) convention, one replaces ε ${}_{μ ν ρ σ τ}$ by i ε ${}_{μ ν ρ σ τ}$ where epsilon is a true tensor in d=5 spacetime, and ε ${}_{μ hat ν hat ρ hat σ hat τ hat α hat β hat γ hat δ hat ε hat}$ by i ε ${}_{μ hat ν hat ρ hat σ hat τ hat α hat β hat γ hat σ hat ε hat}$ where the 10-index epsilon symbol is a true tensor in d=10.
P. G. O. Freund and M. A. Rubin, Phys. Lett. 97 B, 233 (1980).
N. P. Warner and P. van Nieuwenhuizen, Commun. Math. Phys. (to be published); L. J. Romans, Phys. Lett. 153B, 392 (1985).
E. Cremmer, J. Scherk and J. H. Schwarz, Phys. Lett. 84 B, 83 (1979); E. Cremmer, in Superspace and Supergravity, edited by S. W. Hawking and M. Rocek (Cambridge University Press, Cambridge, 1983).
K. Pilch, P. K. Townsend and P. van Nieuwenhuizen, Phys. Lett. 136 B, 38 (1984).
H. Nicolai and E. Sezgin, Phys. Lett. 143 B, 389 (1984).
M. Günaydin, P. van Nieuwenhuizen, and N. P. Warner, Caltech Report No. CALT-68-1184, (unpublished).
M. Günaydin and N. Marcus, Class. Quan. Grav. 2, L11 (1985).
P. K. Townsend and P. van Nieuwenhuizen, Phys. Lett. 125 B, 41 (1984).
M. Pernici, K. Pilch and P. van Nieuwenhuizen, Phys. Lett. 143 B, 103 (1984).
P. van Nieuwenhuizen, Class. Quan. Grav. 2, 1 (1985).
M. Pernici, K. Pilch, and P. van Nieuwenhuizen, Nucl. Phys. B (to be published).
M. Günaydin, L. J. Romans, and N. P. Warner, Phys. Lett. 154B, 268 (1985).
L. Castellani, R. D'Auria, P. Fré, K. Pilch and P. van Nieuwenhuizen, Class. Quan. Grav. 1, 339 (1984).
The last result δ (eb) = - lambda follows from δ $A_{α β γ δ}$ = $ξ^{epsilon}$ $F_{ε α β γ δ}$ - 4 partial[ ${}_{{}$ α }( $ξ^{epsilon}$ $A_{β γ δ] ε}$ ) by taking the lowest-order terms and marking a compensating gauge transformation. Under the sum of the conformal diffeomorphism and the compensating gauge transformation with parameters Λ ${}_{β γ δ}$ = - 4 $ξ^{epsilon}$ $A_{ε β γ δ}$ , the other components of $A_{μ hat ν hat ρ hat σ hat}$ are all inert, except, as required by self-duality, $A_{ν ρ σ τ}$ . Under conformal diffeomorphism, $A_{ν ρ σ τ}$ transforms as δ $A_{ν ρ σ τ}$ = $ξ^{mu}$ $F_{μ ν ρ σ τ}$ - 4 partial[ ${}_{{}$ ν }( $ξ^{mu}$ $A_{ρ σ τ] μ}$ ) so that δ $b_{ν ρ σ τ}$ = - ( ${partial}^{μ}$ λ )ε ${}_{μ ν ρ σ τ}$ if one adds a compensating gauge transformation with parameters Λ ${}_{ν ρ σ}$ = - 4 $ξ^{mu}$ $A_{μ ν ρ σ}$ . One may verify that all field equations are invariant under these combined conformal diffeomorphisms and gauge transformations.
L. Mezincescu and P. K. Townsend, Ann. Phys. (N.Y.) (to be published).

[c1] J. H. Schwarz, Nucl. Phys. B 226, 269 (1983); M. B. Green and J. H. Schwarz, Phys. Lett. 122 B, 143 (1983); J. H. Schwarz and P. C. West, ibid. 126 B, 301 (1983).

[c2] P. S. Howe and P. C. West, Nucl. Phys. B 238, 181 (1984).

[c3] For reviews, see J. H. Schwarz, Phys. Rep. 89, 223 (1982); M. B. Green, Surv. High Energy Phys. 3, 127 (1983).

[c4] L. Alvarez-Gaumé and E. Witten, Nucl. Phys. B 234, 269 (1983).

[c5] E. Witten, Princeton report, 1983 (unpublished).

[c6] S. R. Das, Phys. Lett. 148 B, 39 (1984).

[c7] Curved indices are denoted by μ hat , ν hat=0,...,9, μ , ν = 0 ,..., 4, and α , β = 5 ,..., 9 while corresponding flat indices are denoted by m hat , n hat (m hat,n hat = 0 ,..., 9), m,n (m,n=0 ,..., 4), and a,b (a,b=5,..., 9). Our metric is η ${}_{m hat n hat}$ =diag( - , + ,..., + ). To go to the η ${}_{m hat n hat}$ =diag(+ ,..., +) convention, one replaces ε ${}_{μ ν ρ σ τ}$ by i ε ${}_{μ ν ρ σ τ}$ where epsilon is a true tensor in d=5 spacetime, and ε ${}_{μ hat ν hat ρ hat σ hat τ hat α hat β hat γ hat δ hat ε hat}$ by i ε ${}_{μ hat ν hat ρ hat σ hat τ hat α hat β hat γ hat σ hat ε hat}$ where the 10-index epsilon symbol is a true tensor in d=10.

[c8] P. G. O. Freund and M. A. Rubin, Phys. Lett. 97 B, 233 (1980).

[c9] N. P. Warner and P. van Nieuwenhuizen, Commun. Math. Phys. (to be published); L. J. Romans, Phys. Lett. 153B, 392 (1985).

[c10] E. Cremmer, J. Scherk and J. H. Schwarz, Phys. Lett. 84 B, 83 (1979); E. Cremmer, in Superspace and Supergravity, edited by S. W. Hawking and M. Rocek (Cambridge University Press, Cambridge, 1983).

[c11] K. Pilch, P. K. Townsend and P. van Nieuwenhuizen, Phys. Lett. 136 B, 38 (1984).

[c12] H. Nicolai and E. Sezgin, Phys. Lett. 143 B, 389 (1984).

[c13] M. Günaydin, P. van Nieuwenhuizen, and N. P. Warner, Caltech Report No. CALT-68-1184, (unpublished).

[c14] M. Günaydin and N. Marcus, Class. Quan. Grav. 2, L11 (1985).

[c15] P. K. Townsend and P. van Nieuwenhuizen, Phys. Lett. 125 B, 41 (1984).

[c16] M. Pernici, K. Pilch and P. van Nieuwenhuizen, Phys. Lett. 143 B, 103 (1984).

[c17] P. van Nieuwenhuizen, Class. Quan. Grav. 2, 1 (1985).

[c18] M. Pernici, K. Pilch, and P. van Nieuwenhuizen, Nucl. Phys. B (to be published).

[c19] M. Günaydin, L. J. Romans, and N. P. Warner, Phys. Lett. 154B, 268 (1985).

[c20] L. Castellani, R. D'Auria, P. Fré, K. Pilch and P. van Nieuwenhuizen, Class. Quan. Grav. 1, 339 (1984).

[c21] The last result δ (eb) = - lambda follows from δ $A_{α β γ δ}$ = $ξ^{epsilon}$ $F_{ε α β γ δ}$ - 4 partial[ ${}_{{}$ α }( $ξ^{epsilon}$ $A_{β γ δ] ε}$ ) by taking the lowest-order terms and marking a compensating gauge transformation. Under the sum of the conformal diffeomorphism and the compensating gauge transformation with parameters Λ ${}_{β γ δ}$ = - 4 $ξ^{epsilon}$ $A_{ε β γ δ}$ , the other components of $A_{μ hat ν hat ρ hat σ hat}$ are all inert, except, as required by self-duality, $A_{ν ρ σ τ}$ . Under conformal diffeomorphism, $A_{ν ρ σ τ}$ transforms as δ $A_{ν ρ σ τ}$ = $ξ^{mu}$ $F_{μ ν ρ σ τ}$ - 4 partial[ ${}_{{}$ ν }( $ξ^{mu}$ $A_{ρ σ τ] μ}$ ) so that δ $b_{ν ρ σ τ}$ = - ( ${partial}^{μ}$ λ )ε ${}_{μ ν ρ σ τ}$ if one adds a compensating gauge transformation with parameters Λ ${}_{ν ρ σ}$ = - 4 $ξ^{mu}$ $A_{μ ν ρ σ}$ . One may verify that all field equations are invariant under these combined conformal diffeomorphisms and gauge transformations.

[c22] L. Mezincescu and P. K. Townsend, Ann. Phys. (N.Y.) (to be published).

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Mass spectrum of chiral ten-dimensional N=2 supergravity on $S^{5}$

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Mass spectrum of chiral ten-dimensional N=2 supergravity on S5

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References (22)

Mass spectrum of chiral ten-dimensional N=2 supergravity on $S^{5}$