1. ARNOLD-BELTRAMI FLUX 2-BRANES Pietro Frè Torino University & Embassy of Italy in the Russian Federation VII Round Table Italy-Russia@Dubna November 27-th 2015

2. P. Fre, A.S. Sorin arXiv:1501.04604, arXiv:1504.06802 P. Fre, P. A. Grassi, A.S. Sorin arXiv:1509.09056 P.Fre, P. A. Grassi, L. Ravera, M. Trigiante arXiv:1511.06245 A research program developed in 2015

3. Mathematical Hydrodynamics Rewriting of Euler equations we obtain A flow is a smooth map from the time line to a Riemannian manifold The velocity field is a section of the tangent bundle lowering the indices we have a 1-form If H depends on x 2 M, the streamlines occur on level surfaces H=const and are two- dimensional. A necessary condition for chaos is Euler equation of classical hydrodynamics

4. Arnold Theorem There are only two possibilities Two-dimensional streamlines = no chaos Beltrami Equation On a 3-torus

5. Obtained by Fre & Sorin in 2014-2015 Classification of Beltrami fields on T 3 The cubic lattice is self –dual. The rotation subgroup of invariance is the octahedral group O 24 In the momentum lattice there are 4 types of orbits under O 24 The cubic latticeThe orbit of legth 6 The orbit of legth 8 The orbit of legth 12 The orbit of length 24 To each O 24 -orbit in momentum lattice of length r we associate a solution of Beltrami equation depending on r parameters F and in one case on 2r parameters F

6. The universal classifying group Translations are represented by linear transformations on parameters octahedral rotations are represented by linear transformations of parameters We obtain a direct product group Frobenius congruences We want to eliminate those roto-translations, that conjugated with translations can be reduced to pure rotations We are left with a discrete group (containing space groups of crystallography) Beltrami fields can be classified into irreducible representations of G 1536

7. A few info about G 1536 SOLVABLE GROUP 37 conjugacy classes 37 irreducible representations

8. Hydrodynamics to Supergravity A sentimental journey from… Could we use Beltrami Fields as Gauge Fields in some Supergravity Exact Solution interpreting them as Fluxes, rather than Flows? Yes we can! We need 7-dimensional supergravity and we have to look at 2-brane like solutions! This way of thinking eventually leads us to M-theory and to discover that Beltrami equation is just a subcase of Englert Equation

9. 2-branes in D=7 with fluxes The Action With fluxes, i.e.

10. Minimal D=7 Supergravity was constructed by P. Townsend and P.van Nieuwenhuizen and by E.Bergshoeff, I. Koh and E. Sezgin (up to 4-fermions) in 1984 We addressed the problem of clarifying its Free Differential Algebra (FDA) structure and construct it to all orders in the fermions. Embedding of Flux 2-branes into Supergravity The goal is eventually the holographic principle. The 3D-gauge theories on the brane world-sheet will inherit the discrete symmetries of the Arnold Beltrami fluxes in transverse space that produce the supergravity solution The FDA structure is the algebraic basis of all SUGRAS

11. The FDA of D=7 SUGRA  = dilaton  A = dilatino  A = gravitino 1-form  ab = spin connection V a = sieben-bein A  = triplet of gauge fields The complete FDA THE MINIMAL FDA The extension The superalgebra Cohomology classes of the superalgebra

12. The rheonomic parameterization These objects define a generalized connection in spinor space The determination of all these numerical coefficients is a several month long mission, both by hand and with the help of computers. It is boring but it is the essence of supergravity. From these formulae There follow both susy rules and field equations

13. The rheonomic parameterization continued All coefficients in all rheonomic parameterizations are completely fixed by the requirement that the rheonomic parameterization of the 3-form and 4-form curvature coexist. Their space-time components are dual to each other.

14. The structure of flux 2-branes solutions Beltrami equation

15. The Killing spinor equation Integrability condition Generalized curvature Generalized connection Must be a projector

16. The number of Killing spinors

17. The ABC 2-brane with N=0 susy

18. The N=0 solution with For the solution based on this vector field the rank of the generalized curvature in spinor space is always 16 and there are no Killing spinors

19. The 2-brane with N=1/4 susy

20. The N=1/4 solution For the solution based on this vector field the rank of the generalized curvature in spinor space is always 12 and there are 4 Killing spinors

21. The 2-brane with N=1/8 susy

22. The N=1/8 solution For the solution based on this vector field the rank of the generalized curvature in spinor space is always 14 and there are 2 Killing spinors

23. M2-branes in D=11 with Englert fluxes Englert Equation

24. Uplifting of Arnold-Beltrami flux branes

25. We have unveiled the relation between Beltrami equation and Englert equation and shown that crystallographic discrete symmetries can be transmitted to (supersymmetric) Gauge Theories in D=3 originating from M2-brane solutions. The game has just only started. We have now to analyse the very rich consequences of these constructions. CONCLUSIONS ??  Uplifting M2- branes Gauge Theories in D=3 with discrete symmetry 