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International Workshop “Supersymmetries & Quantum Symmetries - SQS'09” July 29 – August 3, 2009, Dubna. N=4 superconformal mechanics and WDVV equations via superspace. Kirill Polovnikov* Anton Galajinsky* Olaf Lechtenfeld** Sergey Krivonos*** - PowerPoint PPT Presentation
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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
International Workshop“Supersymmetries & Quantum Symmetries - SQS'09”July 29 – August 3, 2009, Dubna
Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09 2
Outline1. Introduction: Conformal Mechanics
2. Hamiltonian formulation of N = 4 superconformal mechanics and WDVV equations
3. Superfield approach• N = 4 supersymmetric action• Superconformal symmetry• Inertial co-ordinates• Examples
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Conformal MechanicsConformal Hamiltonian
where
(H, D, K) obey so(1,2) conformal algebra
The dilatation and conformal boost generators
Kirill Polovnikov et al. "N=4 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. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
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Hamiltonian formulation of N=4 superconformal mechanics and WDVV
equations
Conformal algebra should be extended
One introduces fermionic degrees of freedom
A minimal ansatz to close the su(1,1|2) algebra reads
where
Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
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N=4 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. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
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1. Introduction: Conformal Mechanics
2. Hamiltonian formulation of N = 4 superconformal mechanics and WDVV equations
3. Superfield approach• N = 4 supersymmetric action• Superconformal symmetry• Inertial co-ordinates• Examples
Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
Outline
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Superfield approach: N=4 supersymmetric action
Let us define a set of N=4 superfields with one physical bosonic component
restricted by the constraintsthese equations result
in the conditions
The most general N=4 supersymmetric action reads
The bosonic part of the action has the very
simple formwith the notation
Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
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Imposing N=4 superconformal symmetry: here we restrict our consideration to the special case of SU(1,1|2) superconformal symmetry. Its natural realization is
where the superfunction E collects all SU(1,1|2) parameters
One may check that the constraints are invariant under the N=4 superconformalgroup if the superfields transform like
1. Superconformal invariance
2. Flat kinetic term for bosons
It is not clear how to find the solutions to this equation in full generality.
We are interested in the subset of actions which features
Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
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Superfield approach: Inertial co-ordinatesWe 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
Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
This demand restricts the variable transformation by
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Rewriting the constraints in the y-frame one can find
where
One can show that
The consistency condition is
Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
12Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
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
Hence, there exists a prepotential F obeying the WDVV equation.
Playing a little bit with obtained equations one can find
Thus
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Furthermore, some contractions simplify
Thus, all the ‘structure equations of the Hamiltonian approach are fulfilled precisely by
The bosonic potential
where
Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
With the help of the ‘dual superfields’ w, one can give a simple expression forthe superpotential G(y), namely
As expected, the superpotential G(y) determines both prepotentials U and F
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So, for the construction of N=4 superconformal mechanical models, in principle one needs to solve only two equations, namely
All other relations and conditions (including WDVV) follow from these!
or
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. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
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Superfield approach: Examples
1. Two dimensional systems: all equations can be resolved in a general case
with
Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
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2. Three dimensional systems: some particular solutionsFor B_3 solution of WDVV equation without radial term
we found the inertial coordinates
which yield the dual coordinates
Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
Superfield approach: Examples
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2. Three dimensional systems: some particular solutionsFor B_3 solution of WDVV equation with radial term
we found the inertial coordinates
which yield the dual coordinates
Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
Superfield approach: Examples
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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. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09
19Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09