Download On the road to $N=2$ supersymmetric Born-Infeld action - INFN-LNF

S. Belluccia
a
S. Krivonosb
A.Shcherbakova
A.Sutulinb
Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati , Italy
b
Bogoliubov Laboratory of Theoretical Physics, JINR
based on paper arXiv:1212.1902
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
Frontiers in Mathematical Physics Dubna 2012
2
Non-linear electrodynamics
M. Born, L. Infeld
Foundations of the new field theory
Proc.Roy.Soc.Lond. A144 (1934) 425-451
Introduced to remove the divergence of selfenergy of a charged point-like particle
S. Bellucci LNF INFN Italy
Frontiers in Mathematical Physics Dubna 2012
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The theory is duality invariant.
E. Schrodinger
Die gegenwartige Situation in
der Quantenmechanik
Naturwiss. 23 (1935) 807-812
This duality is related to the so-called electromagnetic duality in supergravity or T-duality in
string theory.
Duality constraint
S. Bellucci LNF INFN Italy
Frontiers in Mathematical Physics Dubna 2012
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N=1 SUSY:
Relies on PBGS from N=2 down to N=1
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
―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
Frontiers in Mathematical Physics Dubna 2012
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For a theory described by action
S[W,W] to be duality invariant, the
following must hold
S.Kuzenko, S. Theisen
Supersymmetric Duality Rotations
arXiv: hep-th/0001068
where Ma is an antichiral N=1 superfield, dual
to Wa
S. Bellucci
LNF INFN Italy
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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.
S. Bellucci
LNF INFN Italy
Frontiers in Mathematical Physics,
Dubna 2012
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The solution is then given in terms of
and has the following form
so that the theory is described by action
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LNF INFN Italy
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Different approaches:
Resulting actions are equivalent
—require the presence of another N=2 SUSY
which is spontaneously broken
—require
self-duality
along with non-linear
S. Bellucci,
E.Ivanov, S. Krivonos
andvector
N=4 superfield
supersymmetric Born-Infeld
shifts•N=2
of the
theories from nonlinear realizations
—try toS.find
an N=2S.analog
of N=1 equation
Kuzenko,
Theisen
•Towards the complete N=2 superfield BornSupersymmetric
Duality Rotations
Infeld
action with partially
broken N=4
S.supersymmetry
Ketov
and Super
Born-Infeld Theories
A •Superbranes
manifestly N=2
supersymmetric
Born-Infeld
from Nonlinear Realizations
action
S. Bellucci
LNF INFN Italy
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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
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How does A0 transform?
Again, how does A0 transform?
These fields turn out to be lower components of
infinite dimensional supermultiplet:
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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?
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LNF INFN Italy
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Making perturbation theory, one can find that
Therefore, up to this order, the action reads
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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
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LNF INFN Italy
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Inspired by lower terms in the series expansion,
it was suggested that the solution to Ketov
equation yields the following action
where
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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:
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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?
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LNF INFN Italy
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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.
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LNF INFN Italy
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This approximation is just a truncation
after which a little can be said about the hidden
N=2 SUSY.
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LNF INFN Italy
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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
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LNF INFN Italy
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Series expansion
Solution to Ketov equation, term by term:
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Some lower orders:
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new structures, not present in
Ketov solution, appear
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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
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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:
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The next term B14 introduces new structures:
This chain of appearance of new structures
seems to never end.
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Message learned from doing perturbative
expansion:
Higher orders in the perturbative expansion
contain terms of the following form:
etc.
written in terms of operators
and
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Introduction of the operators
and
is similar to the standard procedure in quantum
mechanics. By means of these operators, Ketov
equation
can be written in operational form
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LNF INFN Italy
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Once quantum mechanics is mentioned, one can
define its classical limit. In case under
consideration, it consists in replacing operators X
and
by functions:
and
In this limit, operational form of Ketov equation
transforms in an algebraic one
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LNF INFN Italy
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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.
S. Bellucci
LNF INFN Italy
Frontiers in Mathematical Physics,
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Inspired by the “classical” solution, one can try
to find the full solution using the ansatz
to emphasize the
quantum nature
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.
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LNF INFN Italy
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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.
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LNF INFN Italy
Frontiers in Mathematical Physics,
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With the help of operators X X and X3 one can
reproduce B2n+4 up to 18th order (included) by
means of the ansatz
the highest order that
we were able to check
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.
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LNF INFN Italy
Frontiers in Mathematical Physics,
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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
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LNF INFN Italy
Frontiers in Mathematical Physics,
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1.
2.
3.
4.
5.
S. Bellucci
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.
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