1. 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:1212.1902arXiv:1212.1902

2. 1.Born-Infeld theory and duality 2.Supersymmetrization of Born-Infeld theory a)N=1 b)Approaches to deal with N=2 3.Ketov equation and setup 4.Description of the approach: perturbative expansion 5.“Quantum” and “classic” aspects 6.Problems with the approach 7.Conclusions S. Bellucci LNF INFN Italy 2Frontiers in Mathematical Physics Dubna 2012

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

4. 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 S. Bellucci LNF INFN Italy 4Frontiers in Mathematical Physics Dubna 2012 E. Schrodinger Die gegenwartige Situation in der Quantenmechanik Naturwiss. 23 (1935) 807-812

5. 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 S. Bellucci LNF INFN Italy 5Frontiers in Mathematical Physics Dubna 2012 J. Bagger, A. Galperin A new Goldstone multiplet for partially broken supersymmetry Phys. Rev. D55 (1997) 1091-1098 M. Rocek, A. Tseytlin Partial breaking of global D = 4 supersymmetry, constrained superfields, and three-brane actions Phys. Rev. D59 (1999) 106001

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

7. 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 W a, this equation can be solved. S. BellucciLNF INFN Italy 7Frontiers in Mathematical Physics, Dubna 2012 J. Bagger, A. Galperin A new Goldstone multiplet for partially broken supersymmetry Phys. Rev. D55 (1997) 1091-1098 J. Bagger, A. Galperin A new Goldstone multiplet for partially broken supersymmetry Phys. Rev. D55 (1997) 1091-1098

8. The solution is then given in terms of and has the following form so that the theory is described by action S. BellucciLNF INFN Italy 8Frontiers in Mathematical Physics, Dubna 2012

9. 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 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 S. BellucciLNF INFN Italy 9Frontiers in Mathematical Physics, Dubna 2012 S. Ketov A manifestly N=2 supersymmetric Born-Infeld action Resulting actions are equivalent

10. 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 S. BellucciLNF INFN Italy 10Frontiers in Mathematical Physics, Dubna 2012 parameters of central charge trsf parameters of broken SUSY trsf

11. How does A 0 transform? Again, how does A 0 transform? These fields turn out to be lower components of infinite dimensional supermultiplet: S. BellucciLNF INFN Italy 11Frontiers in Mathematical Physics, Dubna 2012

12. 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? S. BellucciLNF INFN Italy 12Frontiers in Mathematical Physics, Dubna 2012

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

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

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

16. 1.Reproduces correct N=1 limit. 2.Contains only W, D4W and their conjugate. 3.Being defined as follows the action is duality invariant. 4. The exact expression is wrong: S. BellucciLNF INFN Italy 16Frontiers in Mathematical Physics, Dubna 2012

17. 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? S. BellucciLNF INFN Italy 17Frontiers in Mathematical Physics, Dubna 2012

18. 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. S. BellucciLNF INFN Italy 18Frontiers in Mathematical Physics, Dubna 2012

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

20. 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 S. BellucciLNF INFN Italy 20Frontiers in Mathematical Physics, Dubna 2012

21. Series expansion Solution to Ketov equation, term by term: S. BellucciLNF INFN Italy 21Frontiers in Mathematical Physics, Dubna 2012

22. Some lower orders: S. BellucciLNF INFN Italy 22Frontiers in Mathematical Physics, Dubna 2012 new structures, not present in Ketov solution, appear

23. 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 S. BellucciLNF INFN Italy 23Frontiers in Mathematical Physics, Dubna 2012

24. 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: S. BellucciLNF INFN Italy 24Frontiers in Mathematical Physics, Dubna 2012

25. The next term B14 introduces new structures: This chain of appearance of new structures seems to never end. S. BellucciLNF INFN Italy 25Frontiers in Mathematical Physics, Dubna 2012

26. S. BellucciLNF INFN Italy 26Frontiers in Mathematical Physics, Dubna 2012 Message learned from doing perturbative expansion: Higher orders in the perturbative expansion contain terms of the following form: written in terms of operators etc. and

27. 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 S. BellucciLNF INFN Italy 27Frontiers in Mathematical Physics, Dubna 2012 and

28. 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 S. BellucciLNF INFN Italy 28Frontiers in Mathematical Physics, Dubna 2012 and

29. This equation can immediately be solved as Curiously enough, this is exactly the expression proposed by Ketov as a solution to Ketov equation! S. BellucciLNF INFN Italy 29Frontiers in Mathematical Physics, Dubna 2012 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.

30. 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. S. BellucciLNF INFN Italy 30Frontiers in Mathematical Physics, Dubna 2012 to emphasize the quantum nature

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

32. With the help of operators X X and X3 one can reproduce B 2n+4 up to 18 th order (included) by means of the ansatz Unfortunately, in the 20 th order a new “quantum” structure is needed. It is not an operator but a function: which, obviously, disappears in the classical limit. S. BellucciLNF INFN Italy 32Frontiers in Mathematical Physics, Dubna 2012 the highest order that we were able to check

33. 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 S. BellucciLNF INFN Italy 33Frontiers in Mathematical Physics, Dubna 2012

34. 1.We investigated the structure of the exact solution of Ketov equation which contains important information about N=2 SUSY BI theory. 2.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. 3.We proposed to introduce differential operators which could, in principle, generate new structures for the Lagrangian density. 4.With the help of these operators, we reproduced the corresponding Lagrangian density up to the 18 th order. 5.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 X 3. S. BellucciLNF INFN Italy 34Frontiers in Mathematical Physics, Dubna 2012