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N=4 superconformal mechanics and WDVV equations via superspace

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***

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N=4 superconformal mechanics and WDVV equations via superspace

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  1. 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*** * Laboratory of Mathematical Physics, Tomsk Polytechnic University ** Institut für Theoretische Physik, Leibniz Universität Hannover *** Bogoliubov Laboratory of Theoretical Physics, JINR

  2. 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 KirillPolovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09

  3. Conformal Mechanics Conformal Hamiltonian The dilatation and conformal boost generators (H, D, K) obey so(1,2) conformal algebra where Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09

  4. 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

  5. Hamiltonian formulation of N=4 superconformal mechanics and WDVV equations One introduces fermionic degrees of freedom Conformal algebra should be extended 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

  6. 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

  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. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09

  8. Superfieldapproach: N=4 supersymmetric action Let us define a set of N=4 superfields with one physical bosonic component restricted by the constraints these equations result in the conditions The most general N=4 supersymmetric action reads The bosonic part of the action has the very simple form with the notation Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09

  9. 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 We are interested in the subset of actions which features Superconformal invariance Flat kinetic term for bosons It is not clear how to find the solutions to this equation in full generality. Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09

  10. Superfieldapproach: 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 invariantonly when the transformation law is This demandrestricts the variable transformation by KirillPolovnikov et al. "N=4 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 KirillPolovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09

  12. 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. "N=4 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 superpotentialG(y), namely As expected, the superpotential G(y) determines both prepotentials U and F KirillPolovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09

  14. So, for the construction of N=4 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. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09

  15. Superfieldapproach: 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

  16. Superfieldapproach: 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. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09

  17. Superfieldapproach: 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. "N=4 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. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09

  19. Thanks for your attention! Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09

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