Noncommutative Geometries in Mtheory

1
Noncommutative Geometries in M-theory

• David Berman (Queen Mary, London)
• Neil Copland (DAMTP, Cambridge)
• Boris Pioline (LPTHE, Paris)
• Eric Bergshoeff (RUG, Groningen)

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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?

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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.

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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.

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Noncommutative geometry in string theory

• We quantise this action (1st order) and we
  obtain
• where
• Including the neglected kinetic terms

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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)
  .

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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

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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

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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

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Noncommutative geometry in string theory

• The effective metric arises from considering the
  Hamiltonian

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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?

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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).

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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

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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

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Strings to ribbons

• Where after calculating the canonical momentum
• One can as before express the elongation of the
  boundary string as
• With

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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

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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!

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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

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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.

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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.

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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.

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Branes ending on branes k-D1, N-D3

• D3 brane perspective
• ½ BPS solution of the world volume theory
• N1, BIon solution to nonlinear theory, good
  approximation in large k limit
• Ngt1, Monopole solution to the U(N) gauge theory
• Spike geometry

• D1 brane perspective
• ½ BPS solution of the world volume theory
• Require kgt1, good approximation in large N limit.
• Fuzzy funnel geometry

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D1 ending on a D3

• D3 brane perspective
• Monopole equation

• D1 brane perspective
• Nahm Equation

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Nahm equation

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

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Fuzzy Funnel

• The radius of the two sphere is given by
• With
• Which implies

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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.

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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

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M2 branes ending on M5 branes

• 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

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Self-dual string

• Solution to the ½ BPS equation on the M5
  brane,
• BIon like spike gives the membrane

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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

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Fuzzy funnel Solution

• Solution
• Where Gi obeys the equation of a fuzzy
• 3-sphere

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Properties of the solution

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

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From a Hamiltonian

• Consider the energy functional
• Bogmolnyi type construction yields

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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
  HG5

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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.

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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?

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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

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M-theory Calibrations

• Planar five-brane

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M-theory Calibrations

• Intersecting five branes

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M-theory Calibrations

• Intersecting five branes

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M-theory Calibrations
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M-theory Calibrations
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M-theory Calibrations
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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.

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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.

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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?

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Fuzzy Three Sphere

• Again consider the number of modes of a three
  sphere with a fixed UV cut-off
• Number of modes scales as k3 (large k limit)
• k is given by
• R is given by
• Number of Modes

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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.

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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

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Projecting to Nahm

• Properties

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Apply to Basu-Harvey

• Project the Basu-Harvey equation
• Case i4, the equation vanishes
• Case i1,2,3 then one recovers the Nahm equation

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Projected Basu-Harvey equation

• Provided
• Giving (in the large N limit)

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Discussion

• Ad hoc attempts to generalise the Nahm equation
  have lead to interesting conjectures for the
  non-Abelian membrane theory.
• Successes include the incorporation of
  calibrations corresponding to various fivebrane
  intersections. The geometric profile,
  fluctuations and tensions match known results.
• The relation to the Nahm equation is through a
  projection (a bit different to the usual
  dimensional reduction.
• A key note of interest is the interpretation of
  the degrees of freedom of the membrane as
  coming from the fuzzy thee sphere.

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Conclusions

• Noncommutative geometry arises naturally in the
  effective theory of strings- Moyal plane, fuzzy
  2-sphere etc.
• M-theory is the nonperturbative version of string
  theory.
• It seems to require generalisations of these
  ideas to more exotic geometries.
• eg. Noncommutative loop spaces, deformed
  string interactions, fuzzy three spheres, the
  encoding nonabelian degrees of freedom.