Pei-Ming Ho, EFSSIII on Particles and Fields, 7/5-7/10/2004

What is String Theory?

References:

Introduction to Superstring Theory, Schwarz, hep-ex/0008017.

A First Course in String Theory, Zwiebach (Cambridge) 2004

Lectures on String Theory, Lust, Theisen (Springer-Verlag) 1989.

What is String Theory? Polchinski, hep-th/9411028.

String Theory, I & II, Polchinski (Cambridge Monographs on Mathematical Physics) 1998.

Superstring Theory, I & II, Green, Schwarz, Witten (Cambridge Monographs on Mathematical Physics) 1988.

't Hooft's lecture on string theory: http://www.phys.uu.nl/~thooft/lectures/stringnotes.pdf.

Online Tutorial: http://www.sukidog.com/jpierre/strings/tutor.htm.

The Elegant Universe, TV program on PBS, http://www.pbs.org/wgbh/nova/elegant/.

The Elegant Universe, B. Greene, Random House (2000).

Lecture 1: introduction and bosonic string

Introduction:

Main features:

particles --> strings.

(0) most important feature = quantum gravity

(1) UV finite: Feynman diagrams = smooth manifolds

(2) supersymmetry: needed for the absence of closed string tachyon

(3) gauge symmetry: massless gauge fields appear in string spectrum

(4) no free parameter

(5) unifies kinematics and dynamics

(6) unifies all particles

String tension T = 1/(2p a^¢) is very large ® String is very short

oscillation of string = particle of certain mass and spin

caution: no experimental evidence

belief: consistency ®  strong constraint

-- String theory is at least a step toward unifying quantum mechanics, gravity and particle physics.

How to define a string theory?

(1) perturbative string + branes:

2d conformal field theory

(2) string field theory:

cubic, light-cone, boundary SFT

The correspondence between (1) and (2) is not totally clear.

What is the essence of string theory? (What is the source of magic?)

-- String theory is 21st century physics that accidentally fell into the 20th century.

Bosonic string in flat spacetime:

Action for a free string:

Nambu-Gotto action:

S = T ò d²s  (- h)^1/2,

where the induced metric h on the worldsheet is related to the spacetime metric g by

h_ab  = g_mn(x) ¶_a x^m ¶_b xⁿ.

x's are dynamic fields living on the worldsheet.

The worldsheet coordinates (s⁰ ,  s¹) = (t  , s) are just parametrizations with no physical meaning.

® worldsheet diffeomorphism (general coordinate transformation) is a gauge symmetry.

This is the natural generalization of the relativistic point partcle action

NG action is equivalent to the Polyakov action

S = (T/2) ò d²s  (- h)^1/2 h^ab g_mn(x) ¶_a x^m ¶_b xⁿ.

Here x and h are independent fields on the worldsheet.

But h is not dynamical since there is no time derivatives of h in S.

The equation of motion obtained from varying S with respect to h is a constraint on h.

It implies that h is proportional to the induced metric.

The proportional constant is undetermined because of the Weyl symmetry (scaling), which is a gauge symmetry.

Worldsheet gauge symmetries (diffeomorphism + Weyl) allow us to fix the gauge by choosing

h_ab  = h_ab.

The action becomes

S = (T/2) ò d²s  ¶_a x_m ¶^a x^m.

(We will only consider the case of Minkowski space g = h.)

One can also consider this action as the starting point of string theory.

There is no need to start with the NG action, since we don't know how to quantize it directly.

Equation of motion, boundary condition and solutions:

The EOM of x is the free wave equation.

S also tells us what are the boundary conditions available.

For closed strings, we impose periodic boundary conditions on x: x(t,s+2p) = x(t,s).

For open strings, one can take either the Neumann or the Dirichlet boundary condition.

The general solution of x for the open string with Neumann boundary condition is:

x = x₀ + pt/(pT) + i(pT)^-1/2 S_n¹0(a_n/n) e^-int cos(ns).

The general solution of x for the closed string is:

x = x₀ + pt/(2pT) + i(4pT)^-1/2 S_n¹0[(a_n/n) e^-in(t-s) + (a_n/n) e^-in(t+s)].

1st quantization:

Canonical quantization leads to

[a_m^m  , a_nⁿ  ] = m d_m+n h^mn.

The Hilbert space is defined by having creation operators acting on  the vacuum:

|0; k ñ :   a_m^m |0; k ñ = 0    for    m > 0.

Conformal symmetry:

The residual gauge symmetry is

z₊ ® f₊(z₊), z_- ® f_-(z_-),

where z_± = s ± t.

This symmetry is manifest if we express S in terms of  z_{± .}

For closed strings, f_± are independent periodic functions.

For open strings, they have to be the same.

Virasoro algebra is the Lie algebra of the symmetry group.

[L_m, L_n] = (m-n)L_m+n + (c/12) m (m² - 1) d_m+n.

Here c = central charge.

Classically it vanishes, and in the quantum theory of bosonic strings, c = d.

The last piece is an anomaly. One can try to cancel it by adding ghosts

S = (i/p) ò d²s  (c⁺ ¶_- b₊₊ + c^- ¶₊ b_-- ).

The anomaly is cancelled iff d = 26.

25+1 is the critical dimension for bosonic strings.

(For superstrings with N = 1 spacetime supersymmetry, the critical dimension is 9+1.

For superstrings with N = 2 spacetime susy, it is 2+2.

If we do not assume flat spacetime, the critical dimension is essentially arbitrary.)

The Virasoro generators (Fourier modes of the conserved charge) are

L_m = (1/2) S_nÎZ : a_m-n . a_n :

L₀ = (1/2) a₀ . a₀ + S_nÎZ+ : a_-n . a_n :

where a₀ = p(pT)^-1/2.

Exercise: Derive the Virasoro constraints (L₀ and L_m) in the classical theory.

Physical states are required to be annihilated by L_m with m > 0, and by (L₀ - 1).

A state for the open string with Neumann boundary condition looks like

f |0; k₀ ñ  + A_m a_-1^m  |0; k₁ ñ + (B_mn a_-1^m a_-1ⁿ + B_m a_-2^m ) |0; k₂ ñ + ....

To be physical, we need

k₀² = -1/a^¢ ,  k₁² = 0 ,  k₁.A = 0, etc.

There are zero norm states which are physical states and have vanishing inner product with all physical states.

They correspond to gauge transformations.

Physical states differ by zero norm states are physically the same.

An example is  k_1m a_-1^m  |0; k₁ ñ .

The spectrum of oscillations on a free string contains infinitely many modes of mass² = n / a^¢.

The coefficients ( f , A_m , B_mn , B_m ,...) correspond to spacetime particles/fields.

f = tachyon, A_m = massless U(1) gauge field,  B_mn = massive spin-2 "gauge" field.

We don't like bosonic strings because they have tachyons.

The spectrum contains tensor (gauge) fields of all ranks.

(String theory in flat spacetime may be a higher spin gauge theory with spontaneous symmetry breaking.)

For closed strings, the massless fields are  g_mn(x) , B_mn(x) , F(x).

They correspond to the closed string state

x_mn a_-1^m a_-1ⁿ  |0; k ñ ,

where k² = 0, k^m x_mn = 0, and there are zero modes.

Exercise: Find the Virasoro constraints and zero norm states for the massless level of the closed string.

Conclusion:

In general, string theory starts with a 2d CFT, viewed as the string worldsheet theory.

The target space of "matter" fields in the CFT is the spacetime.

Spectrum of the CFT is the field content of the spacetime field theory, which is called string field theory.

Question: How to define the string field theory from the 2d CFT?

Hint: Feynman defined perturbative QFT from QM using Feynman diagrams.

Lecture 2: perturbation theory and superstring

perturbative string theory

Feynman diagram

A perturbative QFT is defined by rules of Feynman diagrams.

A Feynman diagram of closed strings is a Riemann surface with handles and holes.

The Riemann surface is taken to be the base space of the 2d CFT.

A hole is viewed as an "external leg".

For each hole we specify an oscillation mode, like we specify quantum numbers for an external leg in QFT.

An operator is assigned to each oscillation mode.

The value of the Feynman diagram is given by the correlation function of these operators in the CFT for the given base space.

áV₁(k₁) V₂(k₂)...V_n(k_n) ñ = g_s^n-2 òDxDh e^iS V₁(k₁) V₂(k₂)...V_n(k_n).

(The path integral should be properly normalized by the symmetry group volume factor.)

For open strings, a Feynman diagram has boundaried and "half-holes".

One can also consider diagrams with both open and closed strings.

For example, the diagram of a sphere with n tachyon holes (tree level) gives

áV(k₁) V(k₂)...V(k_n) ñ  µ g_s^n-2 P_i òd²z_i P_i<j |z_i - z_j |^ki.kj/2 ,

where V(k) = òd²z  e^ik.x  is the operator for tachyon.

The loop expansion of QFT becomes now the handle expansion.

Feynman diagrams are UV finite.

Oscillation modes and background fields

Each oscillation mode of a free string corresponds to a spacetime field.

coherent state of a given mode interacts with a string = turning on a corresponding background

background ® worldsheet interaction

Bulk interactions ® closed string modes;    boundary interactions ® open string modes.

(Analogy: EM field background = coherent state of photons, and the background EM field turns on an interaction term in the Lagrangian of a charged matter field.)

Deformations of 2d CFT to a new CFT « physical states of  worldsheet theory

S = S₀ + (T/2) ò d²s  (-h)^1/2 (h^ab Dg_mn(x) ¶_a x^m ¶_b xⁿ + e^ab B_mn(x) ¶_a x^m ¶_b xⁿ + F(x) R₍₂₎ + ...).

Requiring that the quantum theory to be conformal (beta function = 0 for all couplings), one finds

R_mn + (1/4) H_mlr H_n^lr - 2D_mD_n F + ... = 0,

D_l H^l_mn - 2 (D_l F) H^l_mn + ... = 0,

4 (D_m F)² - 4  D_mD^m F + R + (1/12) H_mlr H^mlr + ... = 0,

where H = dB, and R is the spacetime scalar curvature.

Exercise: Derive the 1st eq. (Einstein eq.) to the leading order in a low energy, weak field expansion, assuming that B = F = 0.

The theory of gravity is obtained.

The B field is an antisymmetric tensor gauge potential.

The gauge transformation is B ® B + dL.

The field strength H is gauge invariant.

A string is a charge for the B field, as a generalization of point charges and vector gauge potentials.

The field F is called the dilaton and it plays a special role in string theory.

The Euler character is determined by the number of genus g

c = 2(1-g) = (1/4p) ò d²s  (-h)^1/2 R₍₂₎.

This means that the VEV of F effectively shifts the string coupling by

g_s ® g_s e^Fo.

The string coupling constant is determined by the VEV of dilaton, and is thus determined by the theory.

The physics of spacetime fields can be derived as a consequence of (worldsheet) conformal invariance,

or equivalently derived by Feynman diagram calculation.

The fact that conformal invariance determines string field interactions will be used to define SFT (nonperturbatively).

Superstrings:

Superstrings are strings with supersymmetry.

In addition to x, there must also be some fermions as superpartners.

One can either introduce worldsheet fermion/spacetime vectors  (RNS formulation),

or spacetime fermions/worldsheet scalars (GS formalism).

In either case, the spectrum will include spacetime fermions.

So far people have found 5 (consistent) superstring theories:

Type I, IIA, IIB, Heterotic SO(32), Heterotic E8 ´ E8.

Their low energy effective theories are 5 supergravities.

These superstrings are equivalent via dualities.

They are unified, together with the 11 dim. SUGRA, by the so-called M theory.

(We will talk about string dualities in Lecture 4.)

Up to some assumptions, we have a unique TOE.

How many string theories are there?

bosonic string, 5 superstrings, noncritical strings

Noncritical strings can be made critical by treating the Liouville field as a target space coordinate.

Then the Liouville potential is interpreted as a background.

® n dim. noncritical string » (n+1) dim. critical string.

(This is the case for 2d strings (1d noncritical strings), which are equivalent to the c = 1 Matrix model.)

RNS formalism for type II strings

The worldsheet action is

S = (T/2) ò d²s  (¶_a x_m ¶^a x^m - i y^m r^a ¶_a y_m),

where r^a are 1+1 dim. gamma matrices.

The two components of y are Majorana-Weyl spinors y_±.

The equations of motion for y_± are

¶_- y₊^m  = 0,    ¶₊ y_-^m = 0.

y₊^m = left movers,  y_-^m = right movers.

Worldsheet SUSY:

d x^m = ie y_-^m,    d y_-^m  = -2 ¶_- x^m  e.

There is an analogous symmetry for y₊^m.

® Virasoro constraints get superpartners.

® critical dim. = 10.

For the right moving modes they are:

(¶_- x)² + (i/2) y_-^m  ¶_- y_-m = 0.

y_-^m  ¶_- x_m = 0.

Exercise: Verify the equations of motion for y, the supersymmetry and the Virasoro constraint and its superpartner.

The action S demands that the boundary condition for open strings be

y₊^m (t, s) = ± y_-^m (t, s)   for   s = 0, p.

As a convention we take

y₊^m (t, 0) = y_-^m (t, 0),

and the theory has two sectors:

R: y₊^m (t, p) = y_-^m (t, p),

NS: y₊^m (t, p) = - y_-^m (t, p).

The solutions of y are

R:  y_-^m (t, s) = (1/Ö2) S_nÎZ  d^m_n  e^-in(t-s),

      y₊^m (t, 0) = (1/Ö2) S_nÎZ  d^m_n  e^-in(t+s),

NS:  y_-^m (t, s) = (1/Ö2) S_rÎZ+1/2  b^m_r  e^-ir(t-s),

         y₊^m (t, 0) = (1/Ö2) S_rÎZ+1/2  b^m_r  e^-ir(t+s).

Canonical quantization pairs up operators except

{ d^m₀ , dⁿ₀ } = h^mn d_{m+n ,}

which makes d^m₀ to act like spacetime 9+1 dim. gamma matrices on the Hilbert space.

® The vacuum states of the R-sector get spacetime spinor indices.

® There are spacetime fermions in the spectrum.

For closed strings there are R and NS sectors for both left and right movers.

Closed string states are like the tensor product of two copies of open string states.

We have R-R, R-NS, NS-R, NS-NS sectors. (This is not the end of the story.)

GSO (Gliozzi, Sherk, Olive) projection (for Type II):

GSO projection kills the tachyon in the NS sector, and results in spacetime SUSY.

GSO projection projects out states in the NS sector which has an odd number of b's,

and imposes a chirality condition in the R sector

For closed strings, one can choose the chirality condition for the left and right movers to be the same or opposite.

If same, we get Type IIB (chiral); if opposite, Type IIA (nonchiral).

Both have N  = 2 spacetime SUSY.

Type II low energy spectrum:

Let us count the low energy degrees of freedom.

NS-NS sector:

For both IIA and IIB, there are: the metric g_mn(x) , the NS-NS B-field B_mn(x) , and the dilaton F(x)

® (8*9/2!-1) + 8*7/2! + 1 = 35 + 28 + 1 = 64.

RR sector:

The massless gauge fields are different for IIA and IIB.

IIA: C1, C3 ® 8 + 8*7*6/3! = 8 + 56 = 64.

IIB: C0, C2, self-dual C4 ® 1 + 8*7/2! + (1/2)*(8*7*6*5/4!) = 1 + 28 + 35 = 64.

For both IIA and IIB there are 64 + 64 = 128 bosonic degrees of freedom.

NS-R/R-NS sector:

Y^m ® (8*16/2)*2 = 128.

(8 for the index m, 16 for MW spinor in 9+1 dim., (1/2) for fermion/boson comparison, 2 for NS-R and R-NS.)

Type I can be obtained from Type IIB by a projection.

Heterotic strings are obtained by putting together half of the bosonic string and half of the superstring.

Different "compactifications" of the extra 16 dim. of the bosonic half gives HO and HE.

Lecture 3: D-brane and phenomenology

Notes on D-branes, Polchinski, Chaudhuri, Johnson  [hep-th/9602052].

TASI Lectures on D-branes, Polchinski [hep-th/9611050].

String Solitons, Duff, Khuri, Lu: Phys. Rep. 259:213 (95) [hep-th/9412184].

D-branes as R-R charges:

Stable D-branes are:

IIA: D0, D2, D4, D6, D8.

IIB: D(-1), D1, D3, D5, D7, D9.

As R-R charges, Dp-branes are coupled to a rank-(p+1) gauge field C^(p+1),

with the interaction term in the action òC^(p+1),

which is invariant under that gauge transformation C^(p+1) ® C^(p+1) + dL^(p) .

This generalizes the usual electromagnetic field with point charges in 4 dim.

[Notation: By the superscript (p) I mean a differential p-form.

For example, B = (1/2) B_mn(x) dx^m dxⁿ   is a 2-form.

dx^m 's anticommute with each other, and is annihilated by d.

The exterior derivative d acts on a function like d = dx^m  ¶_m  and it satisfies Leibniz rule.]

The field strength of C^(p+1) is F^(p+2) = dC^(p+1), which is invariant.

The eom and Bianchi identity of F^(p+2) are:

d* F^(p+2) = 0     and     dF^(p+2) = 0.

® The dual  * F^(p+2) can also be viewed as a gauge field strength.

The prototype of this duality is the electromagnetic duality.

path-integral derivation:

S =  ò F*F/2 + dF*B.

B is the Lagrange multiplier used to impose the Bianchi identity.

Integrating out F ®

S =  ò dB*dB/2.

F ® dB = *F is the EM duality transformation.

Generalization of EM duality:

The "electro-magnetic" dual of C^(p+1) is a rank-(d-p-3) gauge field.

C^(p+1) ® F^(p+2) = dC^(p+1) ®  F^(d-p-2) = * F^(p+2) ® C^(d-p-3) such that F^(d-p-2) = dC^(d-p-3).

Thus Dp and D(d-p-4) are "electric" and "magnetic" charges for the same gauge potential C^(p+1).

Similarly, with respect to the NS-NS B-field, there is F1 and NS5.

D-branes as Dirichlet branes

We can understand open strings with Neumann boundary conditions as strings ending on space-filling branes.

If there is tachyon in the open string spectrum, the D-brane is unstable.

Oscillation modes on open strings correspond to fields living on D-branes.

Bosonic massless modes = F_a, A_i. (i = 0,1, ..., p; a = p+1, p+2, ..., d.) 

xⁱ = world-volume coordinates in the static gauge.

A_i = U(1) gauge potential, it couples to the string wordsheet via òA_i(x) (¶_t xⁱ ) dt.

F_a= Dirichlet boundary condition for x_a.

DBI action:

Low energy effective theory for fields living on D-branes can be calculated using

(1) scattering amplitudes from Feynman diagrams, or

(2) vanishing of beta functions.

The Dirac-Born-Infeld (DBI) action is

S = T_p ò d^p+1x  ( - det( G + B + 2p a^¢ F ) )^1/2  ,

where the D-brane tension is

T_p = 1/((2p)^p g_s l_s^p+1),

and l_s² = a^¢.   (Recall that  T = 1/(2p a^¢).)

G_ij  = g_mn(x) ¶_i x^m ¶_j xⁿ = the induced metric on the Dp-brane,

B = the induced B-field,

F = dA = field strength.

In addition to the gauge symmetry of A, there is the gauge transformation of B:

d B = d L,

d A = - L / 2p a^¢ ,

so that B + 2p a^¢ F is gauge invariant.

For N coincident branes, F_a, and A_i are promoted to N by N matrices

in the adjoint representation of U(N) gauge symmetry.

The low energy weak field effective theory is SYM_p+1 with U(N) gauge symmetry.

Start with the 9+1 dim. SYM action:

S = ò d⁹⁺¹x e^{- F}. [ (1/4) FF + (i/2) Y G^m (D_mY) ] .

Here Y is a Majorana-Weyl fermion in 9+1 dim.

(All fields are in the adjoint representation.

Then by dimensional reduction:

D_a ® i F_{a ,}

one obtains the SYM_p+1 action.

Exercise: Write down the action for 4 dim. SYM theory in terms of 4 dim. fields.

D-branes as solitonic solutions of SUGRA = (black) p-branes

ds² = Z^-1/2 h_mn dx^m dxⁿ  + Z^1/2 d_ab dx^a dx^b ,

e^2F  = Z^(3-p)/2 ,

where Z = 1 + c / r^7-p ,

r² = x^a x^a   and   c = l_s^7-p g_s Q (4p)^(5-p)/2 G((7-p)/2) .

This is the extremal solution with no finite horizon.

(Note that the string coupling is constant for D3-branes.)

It is a BPS state (i.e., it preserves partial SUSY and is thus stable).

Solutions with excitations and finite horizons are also known.

There are also stable BPS states corresponding to branes intersecting with one another.

How to kill 6 dimensions ?

Compactification:

There are 6 extra dimensions.

To get N = 1 spacetime SUSY, the extra dim. should be Calabi-Yau (CY) space.

To have chiral fermions in 3+1 dim., the extra dim. can not be a smooth manifold,

because a fermion in higher dimensions always reduces to fermions of both chiralities in 3+1 dim.

The number of families is determined by the topology (a betti number) of the CY3.

String scale l_s ?

l_p = G^1/2 = 1.6 ´10^-33 cm,

m_p = G^-1/2 = 1.2 ´10¹⁹ GeV.

The Planck scale is defined by 4d Newton constant.

If the size of extra dim. is ~ L, the 10d Newton constant is G₁₀ ~ G  L⁶.

®  m_p⁽¹⁰⁾  can be small if L is large.

In string theory, G₁₀  ~ g_s l_s⁸.  

®  If L ~  l_s , then  g_s l_s² ~ l_p².

® String scale is roughly the Planck scale if string coupling is of order 1.

(We hope that L is large so that strings can be found.)

High energy experiments ® the size of extra dimension is of scale 1/TeV or smaller.

Brane-world:

Arkani-Hamed et al. [hep-ph/9803315, hep-ph/9804398, hep-ph/9807344];

Randall, Sundrum [hep-ph/9905221, hep-th/9906064.]

If all SM particles come from open strings ending on D-branes,

gravity experiments ® the size of extra dim. is of scale 0.1 mm or smaller.

® The string scale can be close to TeV.

Solves one hierarchy problem but raises another.

Field theories resembling SM can be constructed by putting branes together.

Higgs mechanism has the geometric meaning of separating parallel branes.

If SM particles are open strings attached to the D-branes, the extra dimensions can be "large".

If extra dim. is not compact, it has to be warped for gravity localization.

Lecture 4: duality and holography

M theory and moduli space:

The term "M theory" is used to mean:

(1) the imaginary theory that unifies 5 superstrings and 11 dim. (quantum) SUGRA. (or we can still call it string theory.)

(2) the 11 dim. (quantum) SUGRA. (strings ® membranes.)

In 11 dim. SUGRA there are solutions corresponding to membranes and 5-branes.

T^M₂ = 1/((2p)² l_p³),

T^M₅ = (T^M₂)²/(2p).

T-duality:

IIA on S¹ of radius R_A with coupling g_A » IIB on S¹ of radius R_B = l_s² / R_A with coupling g_B  = g_A l_s / R_A

winding modes of F1 « momentum (KK) modes of F1

momentum modes of F1 « winding modes of F1

Dp-brane wound on S¹ « D(p-1)-brane

D(p-1)-brane  « Dp-brane wound on S¹

Mirror symmetry can be understood as a kind of T-duality.

Question: What is spacetime?

T-duality can be proven perturbatively.

Derivation of T-duality (flat spacetime):

For a bosonic closed string in flat space with x¹ compactified:

x = x_-(t-s) + x₊(t+s) ,

x¹ = x¹₀ + pt + ws + i S_n¹0[(a_n/n) e^-in(t-s) + (a_n/n) e^-in(t+s)],

p = m/R,   w = nR,

After the field redefinition  (x¹_- , x¹₊)  (x¹_- , - x¹₊),

then   p « w.

For an open string, the same field redefinition leads to the interchange of Neumann and Dirichlet boundary conditions.

Neumann:  ¶_sx¹ = - x_-^¢(t-s) + x₊^¢(t+s)  = 0,    Dirichlet:     ¶_tx¹ = x_-^¢(t-s) + x₊^¢(t+s) = 0.

Derivation of T duality (curved spacetime):

Assume that g is independent of x¹, but otherwise arbitrary.

S = (T/2) ò d²s  g_mn(x) ¶ x^m ¶ xⁿ .

It is equivalent to

S = (T/2) ò d²s  (g_mn(x) D x^m D xⁿ + q(¶A- ¶A)),

where       Dx¹ = ¶ x¹  + A ,     and      Dx^m = ¶ x^m       otherwise.

Integrating out A, A  ®

S = (T/2) ò d²s  (g^¢_mn(x)+B^¢_mn(x)) ¶ x^¢m ¶ x^¢n,

where

g^¢_mn = (1/g₁₁ ,  g_1i/g₁₁  ,  g_ij - g_1ig_1j/g₁₁),

B^¢_mn = (0,  g_1i/g₁₁  ,  0),

x^¢1 = q.

Exercise: Repeat with nonvanishing B-field.

IIA/M:

IIA with g_s , l_s » M theory compactified on S¹ of radius R = g_s l_s , l_p =  g_s^1/3 l_s

F1 « M2 wound on S¹

D0 « momentum modes

D2  « M2

D4 « M5 wound on S¹

NS5  « M5

Question: What is dimension?

HE/M:

HE » M theory compactified on S¹/Z₂

S-duality:

IIB is self-dual

F1 « D1

D3 is self-dual

D5 « NS5

(g_s, l_s) « (1/g_s, g_s^1/2 l_s)

Question: What is fundamental?

Type I » HO:

Another S-duality.

M/IIA/IIB triality:

M on T² » S^1´ S¹ with radius R₁, R₂  is equivalent to IIA on S¹ with radius R₁,

and is also equivalent to another IIA on S¹ with radius R₂ .

Both IIA are again equivalent to some IIB.

The two different IIB's must be equivalent.

Exercise: Go through the two lines of dualities to find out the duality between the two IIB theories.

AdS/CFT duality:

Review: [hep-th/9905111]

This is an example of the holographic principle of 't Hooft and Susskind for quantum gravity.

QG in higher dimensions  «  field theory without gravity in lower dimensions.

Best known example of AdS/CFT duality: IIB on AdS^{5 ´} S⁵ »  SYM₄

ds²_AdS = (R²/r²) dr²  + (r²/R²) (dx²) ,

where (dx²) represent the metric of Minkowski space, and

R⁴ = 4p g_s N l_s⁴  for N D3-branes.

The isometry SO(4,2) of AdS₅ « conformal group of N = 4 SYM.

The isometry SO(6) of S⁵ « R-symmetry.

The duality claims that:

[g_s , N]    «   [g_YM² = g_s , U(N)]  

Z_IIB[f₀] = Z_YM[f₀] .

The left hand side is the partition function of IIB theory on AdS5 ^´ S⁵.

It is approximated by SUGRA:  exp(-S_SUGRA[f₀]), where f₀ represents the boundary values of fields f in SUGRA.

The right hand side is the partition function of N = 4 U(N) SYM on Minkowski M₄ with the weight exp( òf₀ F ), where F is an operator in SYM.

For example, if f is the graviton, then F is the energy momentum tensor.

We can also talk about correspondences of states directly in simple cases.

For example: D(-1) in AdS « YM instanton.

The radial coordinate of D(-1) « the size of instanton.

Click here for Lecture 5.