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 Mechanics

Conformal 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 superconformal

group 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-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

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 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. "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 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. "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