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It is said that writing down an action in presence of a self-dual field strength is subtle and not known till date (February 2016). The familiar example people give is that of type IIB super-gravity which has a self-dual 5-form field strength. Can someone elaborate on what exactly the subtlety is?

Qmechanic
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Physics Moron
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2 Answers2

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Suppose you have a self-dual five-form field strength $F_5=*F_5$. The kinetic term of this field strength is proportional to $$ \int F_5\wedge*F_5=\int F_5\wedge F_5=-\int F_5\wedge F_5 $$ where I used $A\wedge B=(-1)^{pq}\, B\wedge A$ for $p$-form $A$ and $q$-form $B$ in the second equality. So you can conclude that $$ \int F_5\wedge*F_5=0\,. $$ This is the subtlety you mentioned. People usually impose self-dual condition on $F_5$ after obtaining equations of motion. But this is not a satisfactory resolution, because we want to obtain self-dual condition as a part of equations of motion.

(Edit) The above answer was my naive understanding. For more detailed explanation, see page 313 of Becker, Becker and Schwarz.

Frame
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  1. In the meantime, Ashok Sen has developed a covariant variational principle for self-dual (or anti-self-dual) forms, suitable for quantization, cf. Refs. 1-4.

  2. It is compatible with the traditional ad hoc pseudoaction approach, where the (anti)self-dual condition is imposed after the variation of the pseudoaction at the level of the EOMs.

  3. The caveat is that it introduces an (otherwise harmless) decoupled/un-physical/non-interacting extra DOF with a negative definite kinetic term, i.e. a ghost.

References:

  1. A. Sen, Covariant Action for Type IIB SUGRA, JHEP 07 (2016) 01, arXiv:1511.08220.

  2. A. Sen, Self-dual forms: Action, Hamiltonian and Compactification, J. Phys. A: Math. Theor. 53 084002, arXiv:1903.12196.

  3. E. Andriolo, N. Lambert & C. Papageorgakis, Geometrical Aspects of An Abelian (2,0) Action, JHEP 04 (2020) 200, arXiv:2003.10567.

  4. C.M. Hull, Covariant Action for Self-Dual $p$-Form Gauge Fields in General Spacetimes, arXiv:2307.04748.

  5. P. Vanichchapongjaroen, Covariant M5-brane action with self-dual 3-form, JHEP 05 (2021) 039, arXiv:2011.14384.

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