From D-Branes to M-
Branes:
Neil Lambert
CERN
University of Nis
22 Oct 2010
Up from String Theory
Plan
• Introduction
• What is String Theory?
• D-branes
• M-Theory
• M-branes
• Conclusions
Standard
Model
GUT
scale
LHC
Quantum Gravity
The World (as seen from CERN)
• The Standard Model of particle physics is
incredibly successful
– Describes structure and interactions of all matter*
fr...
• String Theory seems capable of
describing all that we expect in one
consistent framework:
– Quantum Mechanics and Genera...
What is String Theory?
Well in fact we know an awful lot (although
not what string theory really is)
• (perturbative) quantum field theory
assumes that the basic states are point-
like particles
– Interactions occur when tw...
• Point particles are replaced by 1-
dimensional strings
– Multitude of particles correspond to the lowest
harmonics of an...
• Feynman diagrams merge and become
smooth surfaces
• Only one coupling constant: gs
- Vacuum expectation value of a scala...
• A remarkable feature is that gravity comes
out of the quantum theory, unified with
gauge forces
• The dimension of space...
The World (as seen from the
Multiverse)
D-Branes
• In addition to strings, String Theory
contains D-branes:
– p-dimensional surfaces in spacetime
• 0-brane = poin...
• These open strings give dynamics to the D-
brane
• At lowest order the dynamics are those of U(n)
Super-Yang-Mills
– gYM is determined from gs
– Light modes on the worldvolume arise from
the open strings (Higg’s mechanism)
• Mass = lengt...
• At low energy D-branes appear as (extremal)
charged black hole solutions
– Singularity is extended along p-dimensions
• ...
What is M-Theory?
• But not all is perfect in String Theory
– Are there really 10500
vacua?
– Can one make any observable ...
• Now all 5 are all thought to be related as
different aspects of single theory:
M-theory
• How?
Duality
• Two theories ar...
• The classic example of duality occurs in
Maxwell’s equations without sources:
– ‘electric’ variables:
– ‘magnetic’ varia...
• M-theory moduli space:
• M-theory moduli space:
at strong coupling
10D
• M-theory moduli space in 3D:
X11
• An 11D metric tensor becomes a 10D
metric tensor plus a vector and a scalar
Scalar that controls the
size of the 11th
di...
• Thus the String Theory dilaton has a
geometric interpretation as the size of the
11th
dimension
– But the vev of is gs
–...
Type IIA String Theory M-Theory
0-Branes gravitational wave along X11
Strings
2-branes
4-branes
5-branes
6-Branes Kaluza-K...
Type IIA String Theory M-Theory
0-Branes gravitational wave along X11
Strings
2-branes
4-branes
5-branes
6-Branes Kaluza-K...
Type IIA String Theory M-Theory
0-Branes gravitational wave along X11
Strings
2-branes
4-branes
5-branes
6-Branes Kaluza-K...
• So there are no strings in M-theory
– 2-branes and 5-branes
• In particular no open strings and no gs
– No perturbative ...
• In string theory you can derive the
dynamics of multiple D-branes from
symmetries:
– Effective theory has 16 supersymmet...
• Can we derive the dynamics of M2-branes
from symmetries?
– Conformal field theory
• Strong coupling (IR) fixed point of ...
• Can we derive the dynamics of M2-branes
from symmetries?
– Conformal field theory
• Strong coupling (IR) fixed point of ...
• The Yang-Mills theories living on D-branes are
determined by the susy variation
• Here we find a Lie-algebra with a bi-l...
• What is required for M2-branes?
– Now and so we require
– Thus we need a triple product: 3-algebra
– Closure implies a g...
• The fundamental identity implies the
gauge symmetry acts as a
(non simple) Lie algebra acting on
• 3-algebra data is equ...
• This gives a maximally supersymmetric
Lagrangian with SO(8) R-symmetry
[Bagger,NL]
• ‘twisted’ Chern-Simons gauge theory...
• But it turns out to only have one example:
– a,b,c,d = 1,2,3,4
• SU(2)xSU(2) Chern-Simons at level (k,-k)
and matter in ...
• Need to generalize:
– Weak coupling arises from orbifold
– Consider C4
/Zk
0
B
B
@
Z1
Z2
Z3
Z4
1
C
C
A »
0
B
B
@
!
!
!¡ ...
• From the 3-algebra this is achieved if the triple
product is no longer totally anti-symmetric:
• Consistency requires a ...
• These theories were was first proposed by
[Aharony, Bergman, Jafferis and
Maldacena]
• They gave a brane diagram derivat...
• In terms of the D3-brane SYM
worldvolume theory:
– Integrating out D5/D3-strings and flowing to IR
gives a U(n)xU(n) CS ...
IIB
D3 : 1 2 3
NS5 : 1 2 4 5 6
(1; k)5 : 1 2 4µ 5µ 6µ 7µ 8µ 9µ
+ T ¡ duality along x3
IIA
D2 : 1 2
KK : ^3 7 8 9
KK=D6 : ^...
• The final configuration is just n M2s in a
curved background preserving 3/16 susys.
– Metric can be written explicitly
–...
• One success of these models is an
understanding of the mysterious n3/2
growth of
the degrees of freedom
– Free energy = ...
• How does one recover D2-branes from
this [Mukhi, Papageorgakis]
– Give a vev to a scalar field
• breaks U(n)xU(n) U(n) a...
• What can we learn about M-theory?
– Hints at microscopic dynamics of M-branes
• e.g. in the N=8 theory one finds mass = ...
• Mass deformations give fuzzy vacua:
– M2-branes blow up into fuzzy M5-branes
– Can we learn about M5-branes
• Also M2s c...
• There are also infinite dimensional totally
antisymmetric 3-algebras: Nambu bracket
– Related to M5-branes?
• Infinitely...
Conclusions
• M-Theory and M-branes are poorly understood
but there has been much recent progress:
– Complete proposal for...
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Neil Lambert - From D-branes to M-branes

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Lecture by prof. dr Neil Lambert (Department of Mathematics, King’s College, London, UK & CERN, Geneva, Switzerland) on October 22, 2010 at the Faculty of Science and Mathematics, Nis, Serbia.

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Neil Lambert - From D-branes to M-branes

  1. 1. From D-Branes to M- Branes: Neil Lambert CERN University of Nis 22 Oct 2010 Up from String Theory
  2. 2. Plan • Introduction • What is String Theory? • D-branes • M-Theory • M-branes • Conclusions
  3. 3. Standard Model GUT scale LHC Quantum Gravity The World (as seen from CERN)
  4. 4. • The Standard Model of particle physics is incredibly successful – Describes structure and interactions of all matter* from deep inside nucleons upwards • General Relativity is also very successful – Describes physics on large to cosmologically large scales • But they are famously hard to reconcile – GR is classical – Standard Model is an effective low-energy theory * Well maybe 20% of it
  5. 5. • String Theory seems capable of describing all that we expect in one consistent framework: – Quantum Mechanics and General Covariance – Standard Model-like gauge theory – General Relativity – Cosmology (inflation)?
  6. 6. What is String Theory? Well in fact we know an awful lot (although not what string theory really is)
  7. 7. • (perturbative) quantum field theory assumes that the basic states are point- like particles – Interactions occur when two particles meet:
  8. 8. • Point particles are replaced by 1- dimensional strings – Multitude of particles correspond to the lowest harmonics of an infinite tower of modes
  9. 9. • Feynman diagrams merge and become smooth surfaces • Only one coupling constant: gs - Vacuum expectation value of a scalar field – the dilaton Á
  10. 10. • A remarkable feature is that gravity comes out of the quantum theory, unified with gauge forces • The dimension of spacetime is 10 • Must compactify to 4D • There appear to be a plethora of models with Standard Model-like behaviour – Estimated 10500 4D vacua Landscape
  11. 11. The World (as seen from the Multiverse)
  12. 12. D-Branes • In addition to strings, String Theory contains D-branes: – p-dimensional surfaces in spacetime • 0-brane = point particle • 1-brane = string • 2-brane = membrane • etc…. – Non-perturbative states: Mass ~ 1/gs – End point of open strings
  13. 13. • These open strings give dynamics to the D- brane • At lowest order the dynamics are those of U(n) Super-Yang-Mills
  14. 14. – gYM is determined from gs – Light modes on the worldvolume arise from the open strings (Higg’s mechanism) • Mass = length of a stretched string between the branes – Vast applications to model building m
  15. 15. • At low energy D-branes appear as (extremal) charged black hole solutions – Singularity is extended along p-dimensions • Thus D-branes have both a Yang-Mills description as well as a gravitational one – Exact counting of black hole microstates – AdS/CFT
  16. 16. What is M-Theory? • But not all is perfect in String Theory – Are there really 10500 vacua? – Can one make any observable predictions? • What is String Theory really? – The construction of vibrating interacting strings is just a perturbative device, not a definition of the theory • What are strongly coupled strings? • Furthermore why 5 perturbative string theories – Type I – Type II A & B – Heterotic E8xE8 & SO(32)
  17. 17. • Now all 5 are all thought to be related as different aspects of single theory: M-theory • How? Duality • Two theories are dual if they describe the same physics but with different variables. e.g. S-duality gs ↔ 1/gs
  18. 18. • The classic example of duality occurs in Maxwell’s equations without sources: – ‘electric’ variables: – ‘magnetic’ variables: Self-dual dF = 0 d ? F = 0 F = dA d ? dA = 0 F = ?dAD d ? dAD = 0
  19. 19. • M-theory moduli space:
  20. 20. • M-theory moduli space: at strong coupling 10D
  21. 21. • M-theory moduli space in 3D: X11
  22. 22. • An 11D metric tensor becomes a 10D metric tensor plus a vector and a scalar Scalar that controls the size of the 11th dimensionU(1) gauge field 10D metric g¹ º = µ e¡ 2Á=3g¹ º e4Á=3Aº e4Á=3A¹ e4Á=3 ¶
  23. 23. • Thus the String Theory dilaton has a geometric interpretation as the size of the 11th dimension – But the vev of is gs – String perturbation theory is an expansion about a degenerate 11th dimension – As gs ∞ an extra dimension opens up • 11D theory in the infinite coupling limit. • Predicts a complete quantum theory in eleven dimensions: M-Theory – Effective action is 11D supergravity – Little else is known Á Á
  24. 24. Type IIA String Theory M-Theory 0-Branes gravitational wave along X11 Strings 2-branes 4-branes 5-branes 6-Branes Kaluza-Klein monopoles 2-branes 5-branes M-Branes
  25. 25. Type IIA String Theory M-Theory 0-Branes gravitational wave along X11 Strings 2-branes 4-branes 5-branes 6-Branes Kaluza-Klein monopoles 2-branes 5-branes M-Branes
  26. 26. Type IIA String Theory M-Theory 0-Branes gravitational wave along X11 Strings 2-branes 4-branes 5-branes 6-Branes Kaluza-Klein monopoles 2-branes 5-branes purely gravitational excitations The branes of M-theory M-Branes
  27. 27. • So there are no strings in M-theory – 2-branes and 5-branes • In particular no open strings and no gs – No perturbative expansion – No microscopic understanding • The dynamics of a single M-branes act to minimize their worldvolumes – With other fields related by supersymmetry • M2 [Bergshoeff, Sezgin, Townsend] • M5 [Howe, Sezgin, West] • What about multiple M-branes?
  28. 28. • In string theory you can derive the dynamics of multiple D-branes from symmetries: – Effective theory has 16 supersymmetries and breaks SO(1,9) → SO(1,p) x SO(9-p) – This is in agreement with maximally supersymmetric Yang-Mills gauge theory L = ¡ 1 4 tr(F2 ) ¡ 1 2 tr(DXi )2 + i 2 tr(¹ª¡¹ D¹ ª) + itr(¹ª¡i [Xi ; ª]) + 1 4 tr([Xi ; Xj ])2
  29. 29. • Can we derive the dynamics of M2-branes from symmetries? – Conformal field theory • Strong coupling (IR) fixed point of 3D SYM – No perturbation expansion – The only maximally supersymmetric Lagrangians are Yang-Mills theories • Wrong symmetries for M-Theory • need SO(1,2) x SO(8) not SO(1,2) x SO(7)
  30. 30. • Can we derive the dynamics of M2-branes from symmetries? – Conformal field theory • Strong coupling (IR) fixed point of 3D SYM – No perturbation expansion – The only maximally supersymmetric Lagrangians are Yang-Mills theories • Wrong symmetries for M-Theory • need SO(1,2) x SO(8) not SO(1,2) x SO(7) • Well that turns out not to be true
  31. 31. • The Yang-Mills theories living on D-branes are determined by the susy variation • Here we find a Lie-algebra with a bi-linear anti- symmetric product: • Closure of the susy algebra leads to gauge symmetry: • Consistency of this implies the Jacobi identity: ±ª = ¡¹ ¡i D¹ Xi ² + [Xi ; Xj ]¡i j ¡10² + : : : [¢; ¢] : A ­ A ! A ±Xi = [¤; Xi ] [¤; [X; Y ]] = [[¤; X]; Y ] + [X; [¤; Y ]]
  32. 32. • What is required for M2-branes? – Now and so we require – Thus we need a triple product: 3-algebra – Closure implies a gauge symmetry: – Consistency requires a generalization of the Jacobi identity (fundamental identity) ¡012² = ² ¡012ª = ¡ª [¢; ¢; ¢] : A ­ A ­ A ! A ±ª = ¡¹ ¡i D¹ Xi ² + [XI ; XJ ; XK ]¡I J K ² ±X = [X; A; B] [X; Y; Z; [A; B]] = [[X; A; B]; Y; Z] + [A; [Y; A; B]; Z] + [X; Y; [Z; A; B]]
  33. 33. • The fundamental identity implies the gauge symmetry acts as a (non simple) Lie algebra acting on • 3-algebra data is equivalent to specifying a Lie-algebra with a (split) metric and a representation acting on vector space space (with an invariant metric). ±X = [X; A; B] A A g g
  34. 34. • This gives a maximally supersymmetric Lagrangian with SO(8) R-symmetry [Bagger,NL] • ‘twisted’ Chern-Simons gauge theory • Conformal, parity invariant L = ¡ 1 2 tr(D¹ XI ; D¹ XI ) + i 2 tr(¹ª; ¡¹ D¹ ª) + i 4 tr(¹ª; ¡I J [XI ; XJ ; ª]) + 1 12 tr([XI ; XJ ; XK ])2 +LC S LC S = X k 4¼ tr( ~A ^ d ~A + 2i 3 ~A ^ ~A ^ ~A)
  35. 35. • But it turns out to only have one example: – a,b,c,d = 1,2,3,4 • SU(2)xSU(2) Chern-Simons at level (k,-k) and matter in the bi-fundamental • Vacuum moduli space: • Two M2-branes in R8 /Z2 – agrees with M-theory when k=2 [Ta ; Tb ; Tc ] = 2¼ k "abcd Td integer Mk = (R8 £ R8 )=D2k M2 = (R8 =Z2 £ R8=Z2)=Z2
  36. 36. • Need to generalize: – Weak coupling arises from orbifold – Consider C4 /Zk 0 B B @ Z1 Z2 Z3 Z4 1 C C A » 0 B B @ ! ! !¡ 1 !¡ 1 1 C C A 0 B B @ Z1 Z2 Z3 Z4 1 C C A ! = e2¼i =k
  37. 37. • From the 3-algebra this is achieved if the triple product is no longer totally anti-symmetric: • Consistency requires a related fundamental identity • For example we can take (for nxm matrices): • Resulting action is similar to the N=8 case but: – U(n)xU(m) Chern-Simons theory at level (k,-k) with matter in the bifundamental [X; Y ; ¹Z] = ¡[Y; X; ¹Z] X,Y,Z are Complex Scalar Fields [X; Y ; ¹Z] = 2¼ k (XZy Y ¡ Y Zy X) Mk;n = Symn (R8 =Zk )
  38. 38. • These theories were was first proposed by [Aharony, Bergman, Jafferis and Maldacena] • They gave a brane diagram derivation – Consider the following Hannay-Witten picture
  39. 39. • In terms of the D3-brane SYM worldvolume theory: – Integrating out D5/D3-strings and flowing to IR gives a U(n)xU(n) CS theory with level (k,-k) coupled to bi-fundamental matter – N=3 is enhanced to N=6
  40. 40. IIB D3 : 1 2 3 NS5 : 1 2 4 5 6 (1; k)5 : 1 2 4µ 5µ 6µ 7µ 8µ 9µ + T ¡ duality along x3 IIA D2 : 1 2 KK : ^3 7 8 9 KK=D6 : ^3 4µ 5µ 6µ 7µ 8µ 9µ + lift to M ¡ theory M ¡ theory M2 : 1 2 KK : ^3 7 8 9 ^10 KK : ^3 4µ 5µ 6µ 7µ 8µ 9µ ^10
  41. 41. • The final configuration is just n M2s in a curved background preserving 3/16 susys. – Metric can be written explicitly – smooth except where the centre's intersect – near horizon limit gives n M2's in R8 /Zk. – Preserved susy's are enhanced to 6/16. • Note that this works for all n and all k – even k=1,2 where we expect N=8 susy • Two supersymmetries are not realized in the Lagrangian (carry U(1) charge) • For k=1 even the centre of mass mode is obscured
  42. 42. • One success of these models is an understanding of the mysterious n3/2 growth of the degrees of freedom – Free energy = f(λ)n2 • λ= n/k 1 λ <<1 • f(λ) = λ-1/2 λ>>1 • This has recently been confirmed in Chern- Simons Theory for all λ [Drukker,Marino,Putrov]
  43. 43. • How does one recover D2-branes from this [Mukhi, Papageorgakis] – Give a vev to a scalar field • breaks U(n)xU(n) U(n) and SO(8) SO(7) – becomes a dynamical U(n) gauge field • Similar to a Higg’s effect where a non-dynamical vector eats a scalar to become dynamical – g2 Y M = v2=k v = hX8i X8 L = k v2 LU (n) SY M (XI 6= 8) + O(kv¡ 3)
  44. 44. • What can we learn about M-theory? – Hints at microscopic dynamics of M-branes • e.g. in the N=8 theory one finds mass = area of a triangle with vertices on an M2
  45. 45. • Mass deformations give fuzzy vacua: – M2-branes blow up into fuzzy M5-branes – Can we learn about M5-branes • Also M2s can end on M5’s: Chern-Simons gauge fields become dynamical [ZA ; ZB ; ¹ZB ] = mB A ZB
  46. 46. • There are also infinite dimensional totally antisymmetric 3-algebras: Nambu bracket – Related to M5-branes? • Infinitely many totally anti-symmetric 3- algebras with a Lorentzian metric – Seem to be equivalent to 3D N=8 SYM but with manifest SO(8) and conformal symmetry [X; Y; Z] = ?(dX ^ dY ^ dZ) Functions on a 3-manifold
  47. 47. Conclusions • M-Theory and M-branes are poorly understood but there has been much recent progress: – Complete proposal for the effective Lagrangian of n M2’s in R8 /Zk – Novel highly supersymmetric Chern-Simons gauge theories based on a 3-algebra. – Gives a Lagrangian description of strongly coupled 3D super Yang-Mills • M5-branes remain very challenging as does M- Theory itself but hopefully progress will be made – M2-brane CFT’s ‘define’ M-theory in AdS4xX7

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