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 Outline 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

3 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

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 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

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 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

8 Superfield approach: 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 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

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 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

11 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

12 Kirill 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

13 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 for the superpotential G(y), namely As expected, the superpotential G(y) determines both prepotentials U and F

14 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

15 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

16 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

17 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

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: ] S. Krivonos, O. Lechtenfeld, K. Polovnikov, N = 4 superconformal n-particle mechanics via superspace, Nucl. Phys. B 817 (2009) 265,[hep-th: ] Kirill Polovnikov et al. "N=4 SCM and WDVV equations via superspace" 29 July 2009, Dubna, SQS'09

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