N=4 superconformal mechanics and WDVV equations via superspace

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N4 superconformal mechanics and WDVV equations
via superspace
International Workshop Supersymmetries Quantum
Symmetries - SQS'09 July 29 August 3, 2009,
Dubna

• Kirill Polovnikov
• Anton Galajinsky Olaf Lechtenfeld Sergey
  Krivonos
• Laboratory of Mathematical Physics, Tomsk
  Polytechnic University
• Institut für Theoretische Physik, Leibniz
  Universität Hannover
• Bogoliubov Laboratory of Theoretical Physics,
  JINR

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Outline

• Introduction Conformal Mechanics
• Hamiltonian formulation of N 4 superconformal
  mechanics and WDVV equations
• Superfield approach
• N 4 supersymmetric action
• Superconformal symmetry
• Inertial co-ordinates
• Examples

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Conformal Mechanics
Conformal Hamiltonian
The dilatation and conformal boost generators
(H, D, K) obey so(1,2) conformal algebra
where
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
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Example Calogero model
Hamiltonian of the n-particles Calogero model

• Calogero model features
• integrable many-particles system in one
  dimension
• exactly solvable quantum mechanical system

• Applications
• Condensed matter physics
• Supergravity and Superstring theory (AdS/CFT
  correspondence)
• Black holes physics
• Interacting supermultiplets

Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
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Hamiltonian formulation of N4 superconformal
mechanics and WDVV equations
One introduces fermionic degrees of freedom
Conformal algebra should be extended
A minimal ansatz to close the su(1,12) algebra
reads
where
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
6
N4 superconformal Hamiltonian can be written as
where the bosonic potential takes form
and two prepotentials F and U obbey the following
system of PDE
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
7
Outline

• Introduction Conformal Mechanics
• Hamiltonian formulation of N 4 superconformal
  mechanics and WDVV equations
• Superfield approach
• N 4 supersymmetric action
• Superconformal symmetry
• Inertial co-ordinates
• Examples

Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
8
Superfield approach N4 supersymmetric action
Let us define a set of N4 superfields with one
physical bosonic component restricted by the
constraints
these equations result in the conditions
The most general N4 supersymmetric action reads
The bosonic part of the action has the very
simple form
with the notation
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
9
Imposing N4 superconformal symmetry here we
restrict our consideration to the special case of
SU(1,12) superconformal symmetry. Its natural
realization is
where the superfunction E collects all SU(1,12)
parameters
One may check that the constraints are invariant
under the N4 superconformal group if the
superfields transform like
We are interested in the subset of actions which
features

1. Superconformal invariance
2. Flat kinetic term for bosons

It is not clear how to find the solutions to this
equation in full generality.
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
10
Superfield approach Inertial co-ordinates
We are looking for inertial coordinates, in which
the bosonic action takes the form
After transforming to the y-frame, the
superconformal transformations become nonlinear
However, the action is invariant only when the
transformation law is
This demand restricts the variable transformation
by
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
11
Rewriting the constraints in the y-frame one can
find
where
The consistency condition is
One can show that
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
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So, our flat connection is symmetric in all three
indices. It can be in case if and only if the
inverse Jacobian is integrable
In these notations one can rewrite
Thus
Playing a little bit with obtained equations one
can find
Hence, there exists a prepotential F obeying the
WDVV equation.
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
13
Furthermore, some contractions simplify
The bosonic potential
Thus, all the structure equations of the
Hamiltonian approach are fulfilled precisely by
where
With the help of the dual superfields w, one
can give a simple expression for the
superpotential G(y), namely
As expected, the superpotential G(y) determines
both prepotentials U and F
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
14
So, for the construction of N4 superconformal
mechanical models, in principle one needs to
solve only two equations, namely
or
All other relations and conditions (including
WDVV) follow from these!
There also possible one more way to solve
obtained equations If prepotential F is known
otherwise, e.g. from solving the WDVV equation,
it is easier to reconstruct superfeilds u or w
from
Their advantage is the linearity, which allows
superposition.
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
15
Superfield approach Examples
1. Two dimensional systems all equations can be
resolved in a general case
with
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
16
Superfield approach Examples
2. Three dimensional systems some particular
solutions
For B_3 solution of WDVV equation without radial
term
we found the inertial coordinates
which yield the dual coordinates
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
17
Superfield approach Examples
2. Three dimensional systems some particular
solutions
For B_3 solution of WDVV equation with radial term
we found the inertial coordinates
which yield the dual coordinates
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
18
The talk is based on joint works A. Galajinsky,
O. Lechtenfeld, K. Polovnikov, N 4 mechanics,
WDVV equations and roots, JHEP 03 (2009) 113,
hep-th 0802.4386 S. Krivonos, O.
Lechtenfeld, K. Polovnikov, N 4
superconformal n-particle mechanics via
superspace, Nucl. Phys. B 817 (2009)
265, hep-th 0812.5062
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09
19
Thanks for your attention!
Kirill Polovnikov et al. "N4 SCM and WDVV
equations via superspace" 29
July 2009, Dubna, SQS'09