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On the road to N=2 supersymmetric Born- Infeld action

On the road to N=2 supersymmetric Born- Infeld action. S. Bellucci a S. Krivonos b A.Shcherbakov a A.Sutulin b a Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati , Italy b Bogoliubov Laboratory of Theoretical Physics, JINR. based on paper arXiv:1212.1902.

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On the road to N=2 supersymmetric Born- Infeld action

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  1. On the road to N=2 supersymmetric Born-Infeld action S. Belluccia S. Krivonosb A.Shcherbakova A.Sutulinb a Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati , Italy b Bogoliubov Laboratory of Theoretical Physics, JINR based on paper arXiv:1212.1902

  2. Brief summary Born-Infeld theory and duality Supersymmetrization of Born-Infeld theory N=1 Approaches to deal with N=2 Ketov equation and setup Description of the approach: perturbative expansion “Quantum” and “classic” aspects Problems with the approach Conclusions Frontiers in Mathematical Physics Dubna 2012

  3. Born-Infeld theory M. Born, L. Infeld Foundations of the new field theoryProc.Roy.Soc.Lond. A144 (1934) 425-451 Non-linear electrodynamics Introduced to remove the divergence of self-energy of a charged point-like particle Frontiers in Mathematical Physics Dubna 2012

  4. Born-Infeld theory E. Schrodinger Die gegenwartige Situation in der Quantenmechanik Naturwiss. 23 (1935) 807-812 The theory is duality invariant. This duality is related to the so-called electro-magnetic duality in supergravity or T-duality in string theory. Duality constraint Frontiers in Mathematical Physics Dubna 2012

  5. Supersymmetrization of Born-Infeld J. Bagger, A. Galperin A new Goldstone multiplet for partially broken supersymmetry Phys. Rev. D55 (1997) 1091-1098 N=1 SUSY: Relies on PBGS from N=2 down to N=1 supersymmetry is spontaneously broken, so that only ½ of them is manifest Goldstone fields belong to a vector (i.e. Maxwell) supermultiplet where V is an unconstraint N=1 superfield M. Rocek, A. Tseytlin Partial breaking of global D = 4 supersymmetry, constrained superfields, and three-brane actions Phys. Rev. D59 (1999) 106001 Frontiers in Mathematical Physics Dubna 2012

  6. N=1 SUSY BI and duality S.Kuzenko, S. Theisen Supersymmetric Duality Rotations arXiv: hep-th/0001068 For a theory described by action S[W,W] to be duality invariant, the following must hold where Ma is an antichiral N=1 superfield, dual to Wa Frontiers in Mathematical Physics, Dubna 2012

  7. Solution to the duality constraint J. Bagger, A. Galperin A new Goldstone multiplet for partially broken supersymmetry Phys. Rev. D55 (1997) 1091-1098 A non-trivial solution to the duality constraint has a form where N=1 chiral superfield Lagrangian is a solution to equation Due to the anticommutativity of Wa, this equation can be solved. Frontiers in Mathematical Physics, Dubna 2012

  8. Solution to the equation The solution is then given in terms of and has the following form so that the theory is described by action Frontiers in Mathematical Physics, Dubna 2012

  9. N=2 supersymmetrization of BI Different approaches: require the presence of another N=2 SUSY which is spontaneously broken require self-duality along with non-linear shifts of the vector superfield try to find an N=2 analog of N=1 equation Resulting actions are equivalent • S. Bellucci, E.Ivanov, S. Krivonos • N=2 and N=4 supersymmetric Born-Infeld theories from nonlinear realizations • Towards the complete N=2 superfield Born-Infeld action with partially broken N=4 supersymmetry • Superbranes and Super Born-Infeld Theories from Nonlinear Realizations S. Kuzenko, S. Theisen Supersymmetric Duality Rotations S. Ketov A manifestly N=2 supersymmetric Born-Infeld action Frontiers in Mathematical Physics, Dubna 2012

  10. N=2 BI with another hidden N=2 The basic object is a chiral complex scalar N=2 off-shell superfield strength W subjected to Bianchi identity The hidden SUSY (along with central charge transformations) is realized as where parameters of broken SUSY trsf parameters of central charge trsf Frontiers in Mathematical Physics, Dubna 2012

  11. N=2 BI with another hidden N=2 How does A0 transform? Again, how does A0 transform? These fields turn out to be lower components of infinite dimensional supermultiplet: Frontiers in Mathematical Physics, Dubna 2012

  12. Infinitely many constraints A0 is good candidate to be the chiral superfield Lagrangian. To get an interaction theory, the chiral superfields An should be covariantly constrained: What is the solution? Frontiers in Mathematical Physics, Dubna 2012

  13. Finding the solution Making perturbation theory, one can find that Therefore, up to this order, the action reads Frontiers in Mathematical Physics, Dubna 2012

  14. N=2 analog of S. Ketov A manifestly N=2 supersymmetric Born-Infeld action Mod.Phys.Lett. A14 (1999) 501-510 It was claimed that in N=2 case the theory is described by the action where A is chiral superfield obeying N=2 equation Frontiers in Mathematical Physics, Dubna 2012

  15. Ketov solution to eq. Inspired by lower terms in the series expansion, it was suggested that the solution to Ketov equation yields the following action where Frontiers in Mathematical Physics, Dubna 2012

  16. Properties of the action Reproduces correct N=1 limit. Contains only W, D4W and their conjugate. Being defined as follows the action is duality invariant. The exact expression is wrong: Frontiers in Mathematical Physics, Dubna 2012

  17. Set up So, if there exists another hidden N=2 SUSY, the chiral superfield Lagrangian is constrained as follows Corresponding N=2 Born-Infeld action How to find A0? Frontiers in Mathematical Physics, Dubna 2012

  18. Set up Observe that the basic equation is a generalization of Ketov equation: Remind that this equation corresponds to duality invariant action. So let us consider this equation as an approximation. Frontiers in Mathematical Physics, Dubna 2012

  19. Set up This approximation is just a truncation after which a little can be said about the hidden N=2 SUSY. Frontiers in Mathematical Physics, Dubna 2012

  20. Perturbative solution to Ketov eq. Equivalent form of Ketov equation: The full action acquires the form Total derivative terms in B are unessential, since they do not contribute to the action Frontiers in Mathematical Physics, Dubna 2012

  21. Perturbative solution to Ketov eq. Series expansion Solution to Ketov equation, term by term: Frontiers in Mathematical Physics, Dubna 2012

  22. Perturbative solution to Ketov eq. Some lower orders: new structures, not present in Ketov solution, appear Frontiers in Mathematical Physics, Dubna 2012

  23. Perturbative solution to Ketov eq. Due to the irrelevance of total derivative terms in B , expression for B8 may be written in form that does not contain new structures For B10 such a trick does not succeed, it can only be simplified to Frontiers in Mathematical Physics, Dubna 2012

  24. Perturbative solution to Ketov eq. One can guess that to have a complete set of variables, one should add new objects to those in terms of which Ketov’s solution is written: Indeed, B12 contains only these four structures: Frontiers in Mathematical Physics, Dubna 2012

  25. Perturbative solution to Ketov eq. The next term B14 introduces new structures: This chain of appearance of new structures seems to never end. Frontiers in Mathematical Physics, Dubna 2012

  26. Perturbative solution to Ketov eq. Message learned from doing perturbative expansion: Higher orders in the perturbative expansion contain terms of the following form: written in terms of operators the full solution can not be represented as some function depending on finite numberof its arguments Unfortunately, this type of terms is not the only one that appears in the higher orders etc. and Frontiers in Mathematical Physics, Dubna 2012

  27. “Quantum” aspects of the pert. sol. Introduction of the operators is similar to the standard procedure in quantum mechanics. By means of these operators, Ketov equation can be written in operational form and Frontiers in Mathematical Physics, Dubna 2012

  28. “Classical” limit Once quantum mechanics is mentioned, one can define its classical limit. In case under consideration, it consists in replacing operators X by functions: In this limit, operational form of Ketov equation transforms in an algebraic one and and Frontiers in Mathematical Physics, Dubna 2012

  29. “Classical” limit This equation can immediately be solved as Curiously enough, this is exactly the expression proposed by Ketov as a solution to Ketov equation! Clearly, this is not the exact solution to the equation, but a solution to its “classical” limit, obtained by unjustified replacement of the operators by their “classical” expressions. Frontiers in Mathematical Physics, Dubna 2012

  30. Operational perturbative expansion Inspired by the “classical” solution, one can try to find the full solution using the ansatz Up to tenth order, operators X and X are enough to reproduce correctly the solution. The twelfth order, however, can not be reproduced by this ansatz: so that new ingredients must be introduced. to emphasize the quantum nature Frontiers in Mathematical Physics, Dubna 2012

  31. Operational perturbative expansion The difference btw. “quantum” and the exact solution in 12th order is equal to where the new operator is introduced as Obviously, since it vanishes the classical limit. Frontiers in Mathematical Physics, Dubna 2012

  32. Operational perturbative expansion With the help of operators X X and X3 one can reproduce B2n+4 up to 18th order (included) by means of the ansatz Unfortunately, in the 20th order a new “quantum” structure is needed. It is not an operator but a function: which, obviously, disappears in the classical limit. the highest order that we were able to check Frontiers in Mathematical Physics, Dubna 2012

  33. Operational perturbative expansion The necessity of this new variable makes all analysis quite cumbersome and unpredictable, because we cannot forbid the appearance of this variable in the lower orders to produce the structures already generated by means of operators X, bX and Frontiers in Mathematical Physics, Dubna 2012

  34. Conclusions We investigated the structure of the exact solution of Ketov equation which contains important information about N=2 SUSY BI theory. Perturbative analysis reveals that at each order new structures arise. Thus, it seems impossible to write the exact solution as a function depending on finite number of its arguments. We proposed to introduce differential operators which could, in principle, generate new structures for the Lagrangian density. With the help of these operators, we reproduced the corresponding Lagrangian density up to the 18th order. The highest order that we managed to deal with (the 20-th order) asks for new structures which cannot be generated by action of generators X and X3. Frontiers in Mathematical Physics, Dubna 2012

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