A nonabelian M5 brane Lagrangian in a supergravity background
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Springer
Received: June 23, Revised: August 12, Accepted: August 30, Published: October 1,
2020 2020 2020 2020
Andreas Gustavsson Physics Department, University of Seoul, 13 Siripdae, Seoul 130-743, Korea
E-mail: [email protected] Abstract: We present a nonabelian Lagrangian that appears to have (2, 0) superconformal symmetry and that can be coupled to a supergravity background. But for our construction to work, we have to break this superconformal symmetry by imposing as a constraint on top of the Lagrangian that the fields have vanishing Lie derivatives along a Killing direction. Keywords: Chern-Simons Theories, Field Theories in Higher Dimensions, M-Theory, Supersymmetric Gauge Theory ArXiv ePrint: 2006.07557
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)001
JHEP10(2020)001
A nonabelian M5 brane Lagrangian in a supergravity background
Contents 1
2 The supersymmetric Lagrangian 2.1 Some comments
3 7
3 The supersymmetry variation of Lm
9
4 The supersymmetry variation of Lb
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5 Equations of motion 5.1 The on-shell Bianchi identity
19 20
6 The relation with the nonchiral Lagrangian
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7 Closure of supersymmetry variations
22
8 Deriving the fermionic equation of motion from selfduality
23
A Derivation of the Fierz identity
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B A consequence of the Killing spinor equation
24
1
Introduction
Finding the nonabelian M5 brane Lagrangian is a long-standing problem, but at the same time it has also been clear for a long time that a unique classial nonabelian Lagrangian for a selfdual tensor field with manifest (2, 0) superconformal symmetry can not exist [7, 11] and we will review the argument below. With the discovery of the M2 brane Lagrangians [1–3] a new hope was that also the M5 brane Lagrangian may be found if one relaxes some of the symmetries that should be present in the classical Lagrangian in the same spirit as one did for the ABJM Lagrangian [3] of multiple M2 branes that preserves only a subgroup of the SO(8) R-symmetry group. The worldvolume theory of flat M2’s has the bosonic symmetry group of AdS4 × S 7 . Since S 7 is a Hopf fiber bundle over CP2 there is a way of breaking its isometry group SO(8) down to SU(4) × U(1) corresponding to this Hopf fibration and it is only this latter R-symmetry that is manifest in the ABJM Lagrangian. For the M5’s on the other hand, we have the bosonic symmetry group of AdS7 × S 4 but here S 4 is not a Hopf circle-bundle so for the M5’s it may be better to consider an orbifolding of the AdS7 space which reduces the Lorentz symmetry rather than the R-symmetry. We will not attempt to orbifold AdS7 in this paper, but we will consider a nonabelian theory that breaks the Lorentz symmetry at the classical level of the Lagrangian. More generally we will present
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1 Introduction
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a candidate Lagrangian for M5’s on Lorentzian six-manifolds that has at least one Killing vector field that corresponds to an isometry direction. We break translational symmetry along this is
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