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

http://xxx.lanl.gov/abs/1212.1902

1. Born-Infeld theory and duality2. Supersymmetrization of Born-Infeld theorya) N=1b) Approaches to deal with N=23. Ketov equation and setup4. Description of the approach: perturbative

expansion5. “Quantum” and “classic” aspects6. Problems with the approach7. Conclusions

S. Bellucci LNF INFN Italy 2Frontiers in Mathematical Physics Dubna 2012

Non-linear electrodynamics

Introduced to remove the divergence of self-energy of a charged point-like particle


M. Born, L. InfeldFoundations of the new field theory

Proc.Roy.Soc.Lond. A144 (1934) 425-451

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


E. Schrodinger Die gegenwartige Situation in

der Quantenmechanik Naturwiss. 23 (1935) 807-812

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


J. Bagger, A. GalperinA new Goldstone multiplet for partially broken

supersymmetryPhys. Rev. D55 (1997) 1091-1098

M. Rocek, A. TseytlinPartial breaking of global D = 4 supersymmetry,

constrained superfields, and three-brane actionsPhys. Rev. D59 (1999) 106001

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

S. Bellucci LNF INFN Italy 6Frontiers in Mathematical Physics, Dubna 2012

S.Kuzenko, S. TheisenSupersymmetric Duality Rotations

arXiv: hep-th/0001068

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.


J. Bagger, A. GalperinA new Goldstone multiplet for partially

broken supersymmetryPhys. Rev. D55 (1997) 1091-1098

J. Bagger, A. GalperinA new Goldstone multiplet for partially

broken supersymmetryPhys. Rev. D55 (1997) 1091-1098

The solution is then given in terms of

and has the following form

so that the theory is described by action


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. TheisenSupersymmetric 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. KetovA manifestly N=2 supersymmetric Born-Infeld action

Resulting actions are equivalent

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 central charge trsf

parameters of broken SUSY trsf

How does A0 transform?

Again, how does A0 transform?

These fields turn out to be lower components of infinite dimensional supermultiplet:


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. Bellucci LNF INFN Italy 12Frontiers in Mathematical Physics, Dubna 2012

Making perturbation theory, one can find that

Therefore, up to this order, the action reads


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. KetovA manifestly N=2 supersymmetric Born-Infeld action

Mod.Phys.Lett. A14 (1999) 501-510

Inspired by lower terms in the series expansion, it was suggested that the solution to Ketov equation yields the following action

where


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:


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?


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.


This approximation is just a truncation

after which a little can be said about the hidden N=2 SUSY.


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


Series expansion

Solution to Ketov equation, term by term:


Some lower orders:


new structures, not present in Ketov solution, appear

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


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:


The next term B14 introduces new structures:

This chain of appearance of new structures seems to never end.



Message learned from doing perturbative expansion:

Higher orders in the perturbative expansion contain terms of the following form:

written in terms of operatorsetc.

and

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

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

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.

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.

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

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.


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

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


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 18th 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 X3.


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

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