3-Sasakian geometry from M2 branes

1
3-Sasakian geometry from M2 branes

• Daniel L. Jafferis
• Rutgers University

Based on work with A. Tomasiello X. Yin D.
Gaiotto. and work in progress
Kähler and Sasakian Geometry in Rome 19 June, 2009
2
Outline

• Introduction
• N3 Chern-Simons-matter and 3-Sasakian
  7-manifolds
• Quantum corrections to N3 CSM moduli spaces and
  duals to AdS4 with D6 branes
• N2 speculations

3
Motivation

• Extend AdS/CFT correspondence between 4d gauge
  theory and SE 5-manifolds to relation between 3d
  Chern-Simons theories and 3-Sasakian and SE
  7-manifolds.
• Requires extra data of a U(1) action that
  commutes with Reeb vector. What happens when this
  action has fixed loci?
• New mathematical objects associated to 3-cycles
  in 8-manifolds with flux?

4
M2 branes

• These are 21 dimensional objects in 11d
  M-theory. Their explicit description in physics
  was mysterious since their discovery over a
  decade ago.
• Now we have found that they carry
  (super)conformally invariant theory of
  Chern-Simons coupled to charged scalars.

5
AdS/CFT

• Equivalence between 4(3)d theory on the boundary
  and AdS5(4) x SE5(7)
• Focus on BPS sector (roughly topological sector)
• Classical limit
• Moduli space of stable representations of a
  quiver
• Moduli space of stable objects in the derived
  category of coherent sheaves on the CY cone

6
Simplest version

• The moduli space of the abelian quiver, thought
  of as a quotient of the set of solutions to some
  equations is the CY cone itself.
• Example
• A1, A2, B1, B2. Equations dW 0 for W Tr(A1 B1
  A2 B2 A1 B2 A2 B1).
• Results in C4//U(1) acting by 1, 1, -1, -1.

This is the cone over T1,1
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• The correspondence also says that stable
  representations of the path algebra
• correspond to stable objects in the derived
  category of compactly supported sheaves in the
  conifold.
• At the level of equations, Kings theorem says
  that solving µ? and quotienting by U(N) is
  equivalent to imposing an algebraic stability
  condition and quotienting by GL(N).

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/U(1)_R
SE7 with U(1)B action
CY4 cone
KE6
/U(1)B
//U(1)B
M6
//U(1)B
/U(1)R
KE4
SE5
CY3 cone
M2 theory for SE7 is related to D3 theory for SE5
M6 is an S2 bundle over KE4
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3-S with U(1)B action
hK2 cone
/U(1)B
M6
///U(1)B
S3/G
hK1 cone
M2 theory for 3-S is related to hyperKähler
quiver for hK1
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Toric hyperKähler 8-manifold
Bielawski, Dancer Gauntlett, Gibbons,
Papadopulos, Townsend

• We know the associated M2 field theory in the
  case all p 1 this last summer.
• New development in March p0 as well.

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Hyperkahler singularity

• In particular, the pair of U(1) isometries of the
  T2 fiber are compatible with the hyperkahler
  structure, and one obtains the hypertoric
  manifold where N is the
  kernel of the map
• In the case of D6 branes in CP3, this reduces to

Bielawski Dancer
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D-term equation for M2

• These are now cubic equations, and the analog of
  Kings theorem is not known. Here a,b are Lie
  algebra indices, and i,j are representation
  indices.
• One branch of solutions has all
• Here k is diagonal on each U(N) factor, indexed
  by m.

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D6 branes in AdS_4

• We now know a large class of quiver
  CSM theories describing a stack of M2 branes at a
  hypertoric singularity. In the t Hooft limit,
  the dual geometry is a warped product
• Introducing D6 branes wrapping an internal
  3-cycle ( in the case) adds
  fundamental hypermultiplets to the quiver.
• Interestingly, conformality is preserved.

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Dual CFT for .

• One of the most symmetric 3-Sasakians, its cone
  is , and it can be written as a quotient
  
• Classical moduli space is , but quantum
  corrected to
• Attempts in the 90s where close

k
-k
Billo Fabbri Fre Merlatti Zaffaroni
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Quantum correction

• The cone is modified to
• where U(1) acts as U(1) B on M and with the
  natural charge m action on C2
• Applied to C4 this results in the cone over N010

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ADHM quiver for D2 in D6

• In addition to the branch of moduli space where
  M2 branes probe the geometry including the lift
  of the D6 branes (which is always via quantum
  correction), the D2 branes may dissolve into the
  D6 branes (M2 branes fractionate).
• Fundamentals now get a VEV, quiver is exactly
  ADHM quiver.

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Fantasy

• The quiver theory describing D3 branes at a
  singular Calabi-Yau 3-fold can be determined by
  resolving the singularity, thus blowing up the
  fractional branes into wrapped D5 and D7 branes.
  Mathematically, the arrows in the quiver are the
  Ext groups between a primitive objects in the
  derived category of coherent sheaves.
• It would be extremely interesting to find the
  analog for M2 brane theories.

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• The physically natural picture would be the
  resolution of fractional M2 branes into wrapped
  M5 branes. There are two differences, however
• The fractional branes are typically pure torsion
• There are no supersymmetric 3-cycles in
  hyperkahler, CY4, Spin(7) 8-manifolds.
• The resolution is probably that M-theory 4-form
  flux plays a crucial role. Conjecture that there
  are supersymmetric resolutions of CY4
  singularities with flux. Fuzzy M5 branes would
  naturally have s-rule.

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Quantum correction to hypermultiplet moduli space

• In Yang-Mills theories with eight supercharges,
  the moduli space of hypermultiplets is normally
  not corrected, since one can promote the coupling
  to a vector superfield, which decouples from the
  hypers.
• In CSM theories, no such argument exists for the
  CS level. However, corrections to the metric must
  respect the hyperkahler structure.
• We will find a correction of this type.

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Branches of CSM moduli space

• In Chern-Simons theories, there is no Coulomb
  branch, as the vector multiplets are all
  effectively massive.
• N3 supersymmetry protects the dimensions of
  ordinary chiral operators formed from the matter
  fields the quantum correction depends on the
  existence of monopole operators.
• Rich structure of branches distinguished by the
  spectrum of allowed monopoles.

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Chern-Simons-matter theory

• We first consider the case with N2 susy. It
  consists of a vector multiplet in the adjoint of
  the gauge group, and chiral multiplets in
  representations
• The kinetic term for the chiral multiplets
  includes couplings
• There is the usual D term

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We integrate out D, , and
Note that this action has classically marginal
couplings. It is has been argued that it does not
renormalize, up to shift of k, and so is a CFT.
Gaiotto Yin
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N3 CS-matter

• To obtain a more supersymmetric theory, begin
  with N4 YM-matter. Then add the CS term,
  breaking to N3.
• Thus we add a chiral multiplet, ,with no
  kintetic term in the adjoint, and the matter
  chiral multiplets, must come in pairs.
• There is a superpotential,
  from the CS term.

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• Integrating out one obtains the same action
  as before, but with a superpotential
• These N3 theories are completely rigid, and
  hence superconformal. It is impossible to have
  more supersymmetry in a YM-CS-matter theory, but
  we shall see that for particular choices of gauge
  groups and matter representations, the pure CSM
  can have enhanced supersymmetry.
• Schwarz Gaiotto Yin

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Simple example

• Consider a U(1) x U(1) CSM theory, with a BF-like
  Chern-Simons coupling
• Take a pair of matter hypers,
  in the fundamental of the first and second
  U(1).
• In this theory the supersymmetry is enhanced to
  N4 one can check that the boson-fermion
  coupling is invariant under a separate SU(2)
  acting of each fundamental hyper.

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Classical moduli space

• The superpotential is dictated by N3
  supersymmetry to be
• Thus there are two branches, and
  , on which the respective U(1) is unbroken.
• Naively, one would quotient by the nontrivially
  acting U(1), but would leave 3d, so cant be.
• is only invariant under a
  Z_k .

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Extra massless fields at origin

• The two branches intersect at the origin, where
  there are extra massless fields. In particular,
  on which is parameterized by
  the X fields have a mass
• We will see that integrating out these fields
  changes the singularity at the origin.

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Mukhi-Papageorgakis effect

• Forget about the mutliplet for a
  moment. Going onto the moduli space by turning on
  gives a mass to the broken gauge field,
  b.
• Integrating out b gives Yang-Mills kinetics to
  the unbroken gauge field, a! It can then be
  dualized to a scalar, which transforms
  under the U(1) in the same way as the phase of
  the hyper Y, but with charge k. The Z_k arises by
  gauge fixing.

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Correction to the hyperkahler metric

• As familiar from the Coulomb branch of N4 21
  gauge theories, integrating out a charged massive
  hypermultiplet at 1 loop gives rise to a term
• Note that this already introduces a Yang-Mills
  term for the gauge field a.

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• Before integrating out the broken gauge field b,
  we dualize a, treating F_a as the fundamental
  variable .
• Integrating out F_a leads to
• The U(1)_b acts on the space of
  . The metric is nontrivial due to the quantum
  correction as seen.

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Monopoles in the chiral ring

• There are monopole operators in YM-CS-matter
  theories, which we follow to the IR CSM.
• In radial quantization, it is a classical
  background with magnetic flux
  , and constant scalar, . Of course, in
  the CSM limit,
• It is crucial that Y is not charged under a.
• Call this monopole operator T.

Borokhov Kapustin Wu
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CS induced charge of T

• The Chern-Simons term induces a charge for the
  operator T we have just defined. Writing
  in the
  monopole background, it is a particle of charge n
  k under U(1)_b
• Equivalently, in radial quantization, the Gauss
  law constraint is modified, and some matter field
  zero modes must be turned on.

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Anomalous dimension

• The dimension of the monopole operator will be
  the sum of the two contributions
• and the dimension of the scalar fields used in
  the dressing.
• This was calculated in Borokhov-Kapustin-Wu by
  quantizing the matter fields in the monopole
  background with constant

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• The result was that the spectrum of fermions from
  the hypermultiplets became asymmetric,
• The spectrum of scalars was found to be
  symmetric. Thus only the fermions contributed to
  the vacuum energy, which is exactly the anomalous
  dimension of the operator.
• We will include the CS terms simply noting this
  operator is charged under the gauge group.
• This is sensible since the matter fields needed
  to dress the operator are neutral under the
  magnetic U(1). Needed for it to be in chiral ring

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Our example

• We have monopoles , the
  first two on the branch , and the latter
  pair on
• Each has one hypermultiplet charged under the
  associated U(1), so it gets a dimension ½
• The CS induced charge of T is (0,k) under the
  U(1) x U(1) gauge group.
• The chiral operators on are
• exactly as expected for

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D6 branes in AdS_4 x CP3

• We consider introducing D6 branes wrapping the
  AdS. This should be similar to adding D7 branes
  in AdS5.
• They wrap an cycle in the internal
  manifold. Thus there is a Z_2 Wilson line,
  distinguishing two types of D6 branes.
• One can also add D6 branes to more general N3
  AdS4 backgrounds, where they wrap

DJ Tomasiello CSM quiver with n nodes
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IIB engineering

• Consider N D3 branes wrapping a circle and
  intersecting an NS5 and (1,k) 5 brane. This
  engineers the ABJM theory.
• Add some D5 branes, some intersecting each half
  of the stack of D3. Breaks supersymmetry to N3,
  and adds fundamentals to the quiver.

D5
NS5
(1,k) 5
D5
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M-theory lift

• T-dualize the circle NS5 branes turn into
  Taub-NUT, D5 charge become D6, D3 becomes D2.
• Near the D2 horizon, lift to M-theory
• Gibbons-Gauntless-Papadopolus-Townsend showed
  this is purely geometry

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Hyperkahler singularity

• In particular, the pair of U(1) isometries of the
  T2 fiber are compatible with the hyperkahler
  structure, and one obtains the hypertoric
  manifold where N is the
  kernel of the map
• In the case of D6 branes in CP3, this reduces to

Bielawski Dancer
40
Quantum corrected geometric branch

• There are ordinary chiral operators of the form
• On the moduli space of diagonal matrices, the
  diagonal U(1)N is unbroken, and there are
  monopoles operators with such magnetic fluxes.
• They have CS induced charge k, and anomalous
  dimension m/2.
• For m1, k1, at dimension 1, one has 8 gauge
  invariant operators as expected for

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Completely Higgsed branch

• If the number of fundamentals is at least twice
  the rank of the gauge groups, there is a branch
  in which the entire gauge symmetry is Higgsed.
• This branch must have all moments set to zero,
  resulting exactly the ordinary Kahler quotient
  for the ADHM quiver of N instantons of rank m on
  C2/Z_n.
• FI parameters resolve the singularity, each node
  is a fractional brane that blows up into a D4.

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Supergravity limit

• The volume of these 3-Sasakians is known
• This implies that the radius of curvature in
  M-theory is given by
• It is a warped compactification, but using the
  inverse of the lightest D0 mass, a typical value
  of

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Stuffing fundamentals with dof

• It is simplest to determine the number of degrees
  of freedom at high temperature from the M-theory
  supergravity limit. It is dominated by the large
  AdS4 black hole, and the internal manifold only
  enters via the four dimension Planck scale.
• Note the enhancement of N m!