Noncommutative Geometries in M-theory

1. Noncommutative Geometries in M-theory David Berman (Queen Mary, London) Neil Copland (DAMTP, Cambridge) Boris Pioline (LPTHE, Paris) Eric Bergshoeff (RUG, Groningen) TexPoint fonts used in EMF: AAAAAAAAAA

2. Introduction Noncommutative geometries have a natural realisation in string theory. M-theory is the nonperturbative description of string theory. How does noncommutative geometry arise in M-theory?

3. Outline • Review how noncommutative theories arise in string theory: a physical perspective. • M-theory as a theory of membranes and fivebranes. • The boundary term of membrane. • Its quantisation. • A physical perspective. • M2-M5 system and fuzzy three-spheres. • The degrees of freedom of the membrane.

4. Noncommutative geometry in string theory Simplest approach: the coupling of a string to a background two form B: For constant B field this is a boundary term: This is the action of the interaction of a charged particle in a magnetic field.

5. Noncommutative geometry in string theory We quantise this action (1st order) and we obtain: where Including the neglected kinetic terms

6. Noncommutative geometry in string theory Therefore, the open strings see a noncommutative space, in fact the Moyal plane. The field theory description of the low energy dynamics of open strings will then be modelled by field theory on a Moyal plane and hence the usual product will be replaced with the Moyal *product. Lets view this another way (usefull for later) .

7. Noncommutative geometry in string theory • Instead of quantising the boundary term of the open string consider the classical dynamics of an open string. • The boundary condition of the string in a background B field is: This can solved to give a zero mode solution:

8. Noncommutative geometry in string theory • The string is stretched into a length: • The canonical momentum is given by: The elogation of the string is proportional to the momentum:

9. Noncommutative geometry in string theory • The interactions will be via their end points thus in the effective field theory there will be a nonlocal interaction:

10. Noncommutative geometry in string theory • The effective metric arises from considering the Hamiltonian:

11. M-theory • For the purposes of this talk M-theory will be a theory of Membranes and Five-branes in eleven dimensional spacetime. • A membrane may end on a five-brane just as an open string may end on a D-brane. • The background fields of eleven dimensional supergravity are C3 , a three form potential and the metric. • What happens at the boundary of a membrane when there is a constant C field present? • What is the effective theory of the five-brane?

12. Boundary of a membrane • The membrane couples to the background three form via a pull back to the membrane world volume. • Constant C field, this becomes a boundary term: • 1st order action, quantise a la Dirac (This sort of action occurs in the effective theory of vortices; see eg. Regge, Lunde on He3 vortices).

13. Boundary of a membrane • Resulting bracket is for loops; the boundary of a membrane being a loop as opposed to the boundary of string being a point:

14. Strings to Ribbons • Look at the classical analysis of membranes in background fields. • The boundary condition of the membrane is This can be solved by:

15. Strings to ribbons • Where after calculating the canonical momentum • One can as before express the elongation of the boundary string as With

16. Strings to Ribbons • Thus the string opens up into a ribbon whose width is proportional to its momentum. • For thin ribbons one may model this at low energies as a string. • The membrane Hamiltonian in light cone formulation is given by: With g being the determinant of the spatial metric, for ribbon this becomes

17. Ribbons to strings • After expressing p0 in terms of P and integrating over rho the Hamiltonian becomes: • The Lagrangian density becomes • This is the Schild action of a string with tension C!

18. Strings to matrices • For those who are familiar with matrix regularisation of the membrane one may do the same here to obtain the matrix model with light cone Hamiltonian:

19. Interactions • The interactions would be nonlocal in that the membranes/ribbons would interact through their boundaries and so this would lead to a deformation from the point of view of closed string interactions. • Some loop space version of the Moyal product would be required.

20. Branes ending on branes • We have so far discussed the effective field theory on a brane in a background field. • Another interesting application of noncommutative geometry to string theory is in the description of how one brane may end on another.

21. Description of D-branes • When there are multiple D-branes, the low energy effective description is in terms of a non-abelian (susy) field theory. Branes ending on branes may be seen as solitonic configurations of the fields in the brane theory.

22. D3 brane perspective ½ BPS solution of the world volume theory N=1, BIon solution to nonlinear theory, good approximation in large k limit N>1, Monopole solution to the U(N) gauge theory Spike geometry D1 brane perspective ½ BPS solution of the world volume theory Require k>1, good approximation in large N limit. Fuzzy funnel geometry Branes ending on branes: k-D1, N-D3

23. D3 brane perspective Monopole equation D1 brane perspective Nahm Equation D1 ending on a D3

24. Nahm equation • Solution of the Nahm equation gives a fuzzy two sphere funnel: Where and

25. Fuzzy Funnel • The radius of the two sphere is given by With Which implies

26. BIon Spike • The BIon solution: • Agreement of the profile in the large N limit between BIon description and fuzzy funnel. • Also, agreement between spike energy per unit length; Chern Simons coupling; and fluctuations. • The Nahm Transform takes you between D1 and D3 brane descriptions of the system.

27. Trivial observation on fuzzy 2-spheres • Consider harmonics on a 2-sphere with cutoff, E. • Number of modes: • Where k is given by: • If the radius R is given by: • Then the number of modes in the large N limit scale as:

28. D1 ending on D3 branes BIon Spike Nahm Equation Fuzzy Funnel with a two sphere blowing up into the D3 M2 branes ending on M5 branes Self-dual string Basu-Harvey Equation Fuzzy Funnel with a three sphere blowing up in to M5 M2 branes ending on M5 branes

29. Self-dual string Solution to the ½ BPS equation on the M5 brane, BIon like spike gives the membrane

30. Basu-Harvey equation Where And G5 is a certain constant matrix Conjectured to be the equivalent to the Nahm equation for the M2-M5 system

31. Fuzzy funnel Solution Solution: Where Gi obeys the equation of a fuzzy 3-sphere

32. Properties of the solution • The physical radius is given by Which yields Agreeing with the self-dual string solution

33. From a Hamiltonian • Consider the energy functional • Bogmolnyi type construction yields

34. From a Hamiltonian • For more than 4 active scalars also require: • H must have the properties: • For four scalars one recovers B-H equation and H=G5

35. Properties of this solution • Just as for the D1 D3 system the fluctuation spectra matches and the tension matches. • There is no equivalent of the Nahm transform. • The membrane theory it is derived from is not understood.

36. Questions??? • Can the B-H equation be used to describe more than the M2 ending on a single M5? • How do the properties of fuzzy spheres relate to the properties of nonabelian membranes? • What is the relation between the B-H equation and the Nahm equation? • Supersymmetry??? • How many degrees of freedom are there on the membrane?

37. M-theory Calibrations • Configurations with less supersymmetry that correspond to intersecting M5 and M2 branes • Classified by the calibration that may be used to prove that they are minimal surfaces • Goal: Have the M2 branes blow up into generic M-theory calibrations

38. M-theory Calibrations Planar five-brane

39. M-theory Calibrations Intersecting five branes

40. M-theory Calibrations Intersecting five branes

41. M-theory Calibrations

42. M-theory Calibrations

43. M-theory Calibrations

44. The solutions • For example, two intersecting 5-branes This is a trivial superposition of the basic B-H solution. There are more solutions to these equations corresponding to nonflat solutions.

45. Calibrations • It is the calibration form g that goes into the generalised B-H equation. • Fuzzy funnels can successfully described all sorts of five-brane configurations. • Interesting to search for and understand the non-diagonal solutions.

46. Fuzzy Funnel description of membranes • We have seen a somewhat ad hoc description of membranes ending on five-brane configurations. Is there any further indication that this approach may have more merit?? • Back to the basic M2 ending on an M5. The basic equation is that of a fuzzy 3-sphere. • How many degrees of freedom are there on a fuzzy three sphere?

47. Fuzzy Three Sphere • Again consider the number of modes of a three sphere with a fixed UV cut-off • Number of modes scales as k^3 (large k limit) • k is given by • R is given by • Number of Modes

48. Non-Abelian Membranes • This recovers (surprisingly) the well known N dependence of the non-Abelian membrane theory (in the large N limit). • The matrices in the action were originally just any NxN matrices but the solutions yielded a representation of the fuzzy three sphere. • Other fuzzy three sphere properties: • The algebra of a fuzzy three sphere is nonassociative. • The associativity is recovered in the large N limit.

49. Relation to the Nahm Equation • To relate the Basu-Harvey equation to the Nahm equation we do this by introducing a projection. • Projection P should project out G4 and then the remaining projected matrices obey the Nahm equation. • Consider:

50. Projecting to Nahm • Properties: