Franco PEZZELLA INFN - Naples Division

Based on: L. De Angelis, G. Gionti S.J., R. Marotta and F. P., Comparing Double String Theories, arXiv: 1312.7367, JHEP 04 (2014) 171

Naples – May 15, 2014

PLAN OF THE TALK

Main References

A.  A. Tseytlin, Phys. Lett. B242 (1990) 163 ; Nucl. Phys. B350 (1991) 39. A. Giveon, M. Porrati and E. Rabinovici, Phys. Rep. 244 (1994) 77. C. Hull, JHEP 0909 (2009) 099. B. Zwiebach, arXiv: 1109.1782 [hep-th] 2011.

!  About Duality in Field Theory.

!  Reminding some aspects of String Theory.

!  T-duality in Bosonic Closed String.

!  Looking for a T-duality symmetric formulation of String Theory. Toy model: Two-dimensional free scalar field. Floreanini-Jackiw Lagrangian and quantization.

!  T-duality symmetric string action (Tseytlin). Quantization.

!  Conclusion and perspectives.

The term duality has been used time and again in Physics, always for very noble causes!

The wave-particle duality was the precursor of quantum theory.

Further example: Kramers-Wienner duality

This property and its generalizations have been found in many more statistical mechanical systems, some of them (such as lattice gauge theory) are of interest to particle physicists.

Two-dimensional Ising model for magnets with spins living on a square lattice and interacting according to a nearest-neighbour interaction with a strength J

partition function Z(K) function of the temperature through K=J/(kBT)

Z(K) Z*(K*) Partition function of the Ising model formulated on the dual lattice constructed from the original one by selecting the central points of each square of the lattice.

= if

€

sinh2K =1

sinh2K*

The formulation on the original lattice provides a good description of the system at high T (weak coupling), while the one on the dual lattice gives a description of the same system at low temperature (strong coupling).

ABOUT DUALITY IN FIELD THEORY

Duality indicates the existence of two equivalent descriptions of the same model using different fields.

Ex: Duality between the massive Thirring model and the Sine-Gordon model in 2D

L = Ψ iγ µ ∂µ +m( )Ψ−g2Ψγ µΨΨγ µΨ

€

Φ≈ Ψ Ψbound state from the point of view of the Thirring model.

The coupling costants are related by

The weak coupling regime in one model is the strong coupling regime in the other and vice versa!

Knowing the explicit relation allows perturbative calculations in the variables of the original theory both in the strong and in the weak coupling regimes.

ELECTRIC-MAGNETIC DUALITY IN 4D

in the absence of sources.

€

E →E cosφ − BsinφB→ Bcosφ + E sinφ€

∇⋅ E = 0 , ∇∧ B =∂E∂t

, ∇⋅ B = 0 , ∇∧ E = −∂B∂t

The free Maxwell equations are invariant under the exchange of electric and magnetic fields.

This nice duality is destroyed when electric sources are introduced, unless also magnetic ones are included.

€

E → BB → − E

€

φ = −π2

discrete duality

€

∇⋅ (E + iB) = ρe + iρm

∇∧ (E + iB) = i ∂∂t(E + iB) + i( je + ijm )

€

E + iB → eiϕ (E + iB)ρe + iρm → eiϕ (ρe + iρm )je + ijm → eiϕ ( je + ijm )

For particles with both electric and magnetic charges qe and qm respectively:

€

qe +iqm →eiϕ (qe +iqm )

But a thory with both electric and magnetic charges can be consistently quantized only if the Dirac quantization condition is satisfied:

€

qei qm

j = 2πhηij invariant for

€

φ = −π2

Both the electric and magnetic charges are quantized in terms of an elementary electric q0

e and magnetic charges q0m satisfying the Dirac condition:

€

qe0 qm

0 = 2πhn0

A theory in which the fundamental electric charge q0e is small,

corresponding to a perturbative electric theory, is necessarily a theory in which the magnetic charges are large, corresponding to a strongly interacting magnetic theory and viceversa.

One can have a perturbative theory either in the electric or in the magnetic charge, but not in both of them.

In field theory, magnetic charges can arise as solitons which are non-trivial solutions of the field equations with localized energy density.

While massless monopoles and dyons are best described as solitons in the original description of the theory (the description used to analyze the theory in the UV), the theory in the infrared can be essentially understood in terms of these distinguished particles.

In particular, there is a distinguished subclass of supersymmetric gauge theories, namely those with extended supersymmetry (N=2 theories) in which it is almost always possible to find a dual weakly coupled description (electric-magnetic duality), allowing thus to determine the structure of the low-energy effective action and, consequently, the infrared dynamics of the theory (Seiberg-Witten).

These models are just distant relatives of QCD.

If QCD had a dual description and the explicit transformations were known, then one could have perturbative control over both the asymptotically free and the confined phase!

Recent years have seen a remarkable development in field theory along these lines.

There exists a semi-realistic theory with some of the desired properties!

N. Seiberg and E. Witten have solved the N=2 SYM theory by using a sort of electric-magnetic duality showing that breaking N=2 down to N=1 gives an electric confinement.

General dualization between a p-form and a D-(p+2)-form in D dimensions.

In particular, in D=2, there is a duality between:

€

L = −12∂αφ∂αφ

and

€

˜ L = − 12∂α ˜ φ ∂α ˜ φ

with

€

∂αφ = εαβ∂β ˜ φ

€

∂0φ = ∂1˜ φ

∂1φ = ∂0˜ φ

It will shown soon that this duality plays an important role soon in T-duality. Let’s see why…

€

*dφ = d ˜ φ

Hodge duality

The so-called 2nd String Revolution has allowed to give a positive answer! String dualities relate various weakly coupled string theories:

The known string models are connected by a net of dualities: S-duality, T-duality and U-duality.

Dualities have revealed the rich structure of ST allowing to discover new structures and new phenomena such as D-branes and mirror symmetry.

S-duality is a transformation that relates a theory with coupling constants gs to a (possibly) different theory with coupling constant 1/gs.

DO SIMILAR RELATIONS EXIST IN STRING THEORY?

Field theory being the low-energy limit of string, it is clear that string duality implies field theory duality, but not the other way around.

REMINDING SOME ASPECTS OF STRING THEORY

String theory is the most promising candidate for providing a consistent quantum theory of gravity unified with the other fundamental

interactions.

The theory is based on extended objects - strings - rather than point particles.

GAUGE THEORIES

GRAVITY

As a string evolves in time, it sweeps out a two-dimensional surface in space-time: world-sheet,

parametrized by two “coordinates”:

€

S = −T2

d2ξ −h∫ hαβ γαβ

auxiliary metric

€

γαβ = ∂αXµ ∂βX

ν Gµν

induced metric The string coordinates on the space-time are given by a mapping

€

X µ : M 2 →R1,D−1 non-linear sigma-model

Polyakov action

€

Sbos = −12πα '

dA∫ τ

σ

€

X µ (τ,σ )€

ξ0 = τ , ξ1 =σ .

target space

string background (allowed space-time for string propagation)

two-dim scalar field

String Theory is an extension of Field Theory

General Relativity reduces to Special Relativity in the limit c → ∞

Quantum Mechanics reduces to Classical Mechanics in the limit ħ → 0

String Theory reduces to (Quantum) Field Theory in the limit α’ → 0

String Theory is able to predict the dimensionality of the space-time where it lives!

Bosonic string theory (unrealistic) Superstring theory

To make contact with our four-dimensional world, extra dimensions have to be compactified.

€

D = 26D =10

Toroidal compactification (simplest case)

The theory is consistently quantized only if:

Usual formulation: the Polyakov action

Symmetries of the action:

• Poincaré transformations in the space-time (global)

• Weyl invariance (local)

• Reparametrizations of the world-sheet coordinates (local)

€

S = −T2

dτ−∞

+∞

∫ dσ0

π

∫ −hhαβ∂αXµ∂βX

νηµν

€

σα → ʹ′ σ α = ʹ′ σ α (τ,σ) (σ0 = τ ,σ1 =σ)

€

α,β = 0,1

These invariances can be used for fixing:

€

hαβ =ηαβ =−1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

conformal gauge

€

γαβ

Constraints:

€

δSδhαβ

≈ Tαβ = 0 Virasoro algebra:

€

Lm ,Ln{ }PB

= i(m − n)Lm+n

€

aµν = −aνµ

In light-cone coordinates

€

T ++ = T −− = 0

€

(σ+,σ−) = (τ +σ,τ −σ)

€

S = −T2

dσ dτηαβ−∞

+∞

∫0

π

∫ ∂αXµ∂βX

νηµν

€

S =T2

dσ dτ ( ˙ X 2 − X '2 )−∞

+∞

∫0

π

∫ ˙ X = ∂τ X ; X '= ∂σ X

Massless wave equation in D=2

closed string solution:

€

X(τ,σ +π ) = X(τ,σ )

€

XL (τ +σ) =12x +α ' p(τ +σ ) +

i22α ' 1

nn≠0∑ α ne

−2in(τ+σ )

XR (τ −σ) =12x +α ' p(τ −σ ) +

i22α ' 1

nn≠0∑ αne

−2in(τ−σ )

periodicity boundary conditions

center-of-mass position string momentum

€

X(τ,σ ) = x + 2α ' pτ +i22α ' 1

nn≠0∑ α ne

−2in(τ+σ ) +i22α ' 1

nn≠0∑ αne

−2in(τ−σ )

Left and right moving modes

€

αnµ ,αm

ν{ }PB

= α nµ,α m

ν{ }PB

= imδm+n,0ηµν

αnµ ,α m

ν{ }PB

= 0

€

. , .{ }PB→−i . , .[ ]

These are interpreted as harmonic oscillators raising and lowering operators for negative and positive subscript respectively.

SPECTRUM MASSLESS OF THE BOSONIC CLOSED STRINGS:

€

|Ωµν >= α−1µ α −1

ν | 0,k >

spin-2 particle GRAVITON, scalar particle DILATON, antisymmetric second-rank tensor KALB-RAMOND (potential for the torsion)

€

N = N =1€

Lm ,Ln[ ] = (n −m)Lm+n in the critical dimensions D=26, D=10

4 GRAVITON AMPLITUDES

€

A ≈ <V (k1)V (k2)V (k3)V (k4 ) >Vertex operator associated to the graviton state

€

V (X)

€

X µ (τ,σ), Xν (τ,σ '){ }PB

= ˙ X µ (τ,σ ), ˙ X ν (τ,σ '){ }PB

= 0 ˙ X µ (τ,σ ),Xν (τ,σ '){ }PB

= T −1δ(σ −σ ')ηµν

€

αnµ ,αm

ν[ ] = α nµ,α m

ν[ ] = mδm+n,0ηµν

αnµ ,α m

ν{ }PB

= 0

Through computing scattering amplitudes of these massless states, one can get the effective gravitational action involving the fields associated with string massless states Gµν, Bµν and ϕ:

€

Seff = −12k 2

d10∫ X −G e−2φ R(10) + 13!FµνρF

µνρ − 4DµφDµφ

⎧ ⎨ ⎩

⎫ ⎬ ⎭

gravitational coupling Ricci tensor field strength of the Kalb-Ramond

covariant derivatives on the target space

dimensionally reduced to 4D Einstein-Hilbert action

Alternatively obtained starting from the action of the strings in background fields:

€

S = −14πα '

d2ξ ∂αXµ∂αXνGµν (X) +εαβ∂αX

µ∂αXνBµν (X) −α 'R(2){ }∫

and, to the lowest order in the string parameter α’ , by imposing the vanishing of the β-functions relative to each coupling.

This encodes the requirement of scale invariance of the quantum theory.

In the presence of dimensions compactified on a circle of radius R extra modes appear!

€

X a (τ,σ +π ) = X a (τ,σ ) + 2πRwa ; wa ∈Z winding modes

representing the number of times the string winds around the compact dimension

momentum modes

€

pa =kaR

, ka ∈Zare quantized along the compact dimension.

while

€

X(τ,σ ) = x + 2α ' pτ + 2Rwσ +i22α ' 1

nn≠0∑ α ne

−2in(τ+σ ) +i22α ' 1

nn≠0∑ αne

−2in(τ−σ )

€

XL (τ +σ) =12

(x + ˜ x ) + α ' kR

+ wR⎛

⎝ ⎜

⎞

⎠ ⎟ (τ +σ) +

i2

2α ' 1nn≠0

∑ α ne−2in(τ +σ ) ;

XR (τ −σ) =12

(x − ˜ x ) + α ' kR− wR

⎛

⎝ ⎜

⎞

⎠ ⎟ (τ −σ) +

i2

2α ' 1nn≠0

∑ αne−2in(τ−σ ) .

€

α ' pL

€

α ' pR

T-DUALITY IN BOSONIC CLOSED STRING THEORY

€

H = 2(N + N − 2) + ʹ′ α kR⎛

⎝ ⎜

⎞

⎠ ⎟ 2

+Rwʹ′ α

⎛

⎝ ⎜

⎞

⎠ ⎟ 2⎡

⎣ ⎢

⎤

⎦ ⎥

€

α ' pL ,

€

α ' pR contribute to the Hamiltonian

T-duality transformation

€

w↔ kA closed string theory compactified on a circle of radius R is dual to a closed string

theory compactified on a circle of inverse radius. This is also an exact symmetry of the Hamiltonian but it is not manifest in the action.

left invariant by a

T-duality implies that in many cases two different geometries for the extra-dimensions are physically equivalent.

T-duality symmetry is a clear indication that ordinary geometric concepts can break down in string theory at the string scale.

The interchange of w and k means that the momentum excitations in one description correspond to winding mode excitations in the dual description and vice versa.

Hence….

The presence of compact dimensions implies the existence of the following modes for bosonic closed strings:

momentum modes

€

pa =kaR

, ka ∈Z

winding modes

€

€

X a (τ,σ +π ) = X a (τ,σ ) + 2πRwa ; wa ∈Z

dual coordinates

€

X a , pa

˜ X a , wa

quantized along the compact direction.

representing the number of times the string winds around the compact dimension

T-DUALITY PROVIDES THIS “NEW” COORDINATE!

imply also a transformation on the string coordinate X along the compact dimension:

€

R↔ α 'R

; k↔ w

€

X(τ,σ ) = XL (τ +σ) + XR (τ −σ )↔ ˜ X (τ,σ ) = XL (τ +σ) − XR (τ −σ )

associated with the winding mode!

IT WOULD BE INTERESTING TO FIND A MANIFESTLY T-DUAL INVARIANT FORMULATION OF THE BOSONIC CLOSED STRING THEORY!

The T-duality transformations

An advantage of the “extended” formulation is that the duality symmetry becomes an off-shell symmetry of the world-sheet action. This makes the duality invariance of the scattering amplitudes and of the effective action manifest.

All of this implies that, if interested in writing down the complete effective field theory of a compactified bosonic closed string, one has to include both momentum excitations and winding excitations or, equivalently

The fields associated with the string states will depend on

€

X i = (X a , ˜ X a ) .

The effective closed string field theory would look like:

€

S = dX a∫ d ˜ X a L X a, ˜ X a( ).Hence, the DOUBLE STRING effective field theory is a DOUBLE FIELD THEORY.

€

X a and ˜ X a .

THE T-DUALITY SYMMETRIC FORMULATION MAY BE CONSIDERED AS A NATURAL GENERALIZATION OF THE STANDARD ONE AT THE STRING SCALE.

This has to be true, in particular, for the effective gravitational action involving the fields associated with string massless states gµν, Bµν and ϕ

€

Seff = −12k 2

d10∫ X −G e−2φ R(10) + 13!FµνρF

µνρ − 4DµφDµφ

⎧ ⎨ ⎩

⎫ ⎬ ⎭

dimensionally reduced to 4D Einstein-Hilbert action

QUESTIONS

!  WHAT DOES THIS ACTION BECOME IN THE LIGHT THAT ALL THE FIELDS ARE “DOUBLED”?

!  WHAT SYMMETRIES AND WHAT PROPERTIES WOULD IT HAVE?

HOW WOULD THE CLOSED STRING LOOK LIKE WHEN T- DUALITY IS MADE MANIFEST?

…hopefully shedding light on aspects of string gravity so far unexplored.

TOROIDAL COMPACTIFICATIONS WITH BACKGROUNDS

€

X µ (σ +π,τ ) = X µ (σ,τ )X a (σ +π,τ ) = X a (σ,τ ) + 2πRawa wa ∈Z

€

S = −T2

d2∫ σ (Gabηαβ + Babε

αβ )∂αXa∂βX

b

€

π a =δSδ ˙ X a

= T(Gab˙ X b − Bab X 'b )

€

pa = dσπ a0

π

∫ =kaRa

€

(a =1,...,d)

INVARIANCE UNDER

€

O(d,d;Ζ)

€

ka ↔ wa ; M↔M-1

€

M =G − BG−1B −G−1BBG−1 G−1

⎛

⎝ ⎜

⎞

⎠ ⎟

In this case the Hamiltonian reads as:

€

H =12

V t M V + N + N

€

V =wa

ka

⎛

⎝ ⎜

⎞

⎠ ⎟

generalized metric 2dX2d

To have a more fundamental action of the (closed) bosonic string theory in which T-duality is manifest. MAIN GOAL

€

˜ S = d2∫ σ12

˜ G abηαβ∂α ˜ X a∂β ˜ X b +εαβ ˜ B ab∂α ˜ X a∂β ˜ X b( ) +εαβ∂α ˜ X a∂β ˜ X b

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

€

˜ G = G − BG−1B

€

˜ B = B −GB−1G

Duality conditions:

€

∗dX = d ˜ X

€

∗dXL = dXL , ∗ dXR = −dXR

€

S = −T2

dσ dτ−∞

+∞

∫0

π

∫ ∂α ˜ X ∂α ˜ X

€

∂σ X = ∂τ ˜ X ; ∂τ X = ∂σ ˜ X

SCALAR-SCALAR DUALITY IN D=2

In the simplest case of a circle compactification:

One can get an explicit dual formulation under the exchange

€

X ↔ ˜ X

first considering a the simplest case of a free scalar field in D=2.

A general free scalar field in 2D Minkowski space described by the usual Lagrangian density

€

L = −12∂αφ∂

αφ =12

( ˙ φ 2 −φ'2 )

admits a Hodge dual defined by:

with

€

∂σφ = ∂τ ˜ φ , ∂τφ = ∂σ ˜ φ

€

∗dφ = d ˜ φ

It is possible to rewrite L in such a way that the two fields appear on an equal footing.

€ €

€

∂α ˜ φ = −εαβ ∂βφ ε 01 =1

€

€

∂α ∂αφ = ∂α ∂

α ˜ φ = 0

€

φ↔ ˜ φ Invariance under

Looking for a T-duality symmetric formulation of String Theory: Two-dimensional free scalar field. Floreanini-Jackiw Lagrangian and quantization.

Procedure

- First step Rewrite L in “first-order” form:

€

L p,φ[ ] = p ˙ φ −12p2 −

12

˙ φ 2

€

p = ˙ φ

auxiliary field

- Second step

Introduce a Lagrange multiplier b

€

L p,φ[ ] = p ˙ φ −12p2 −

12

˙ φ 2 + b(p − ˜ φ ')

€

L[φ, ˜ φ ] = ˙ φ ̃ φ '− 12

˜ φ '2 − 12

˙ φ 2

€

p = ˜ φ '

equations of motion:

€

They are dual to each other on the mass-shell

€

∂σ φ = ∂τ ˜ φ , ∂τφ = ∂σ ˜ φ

invariant under

€

φ↔ ˜ φ

€ €

φ →∂σ ∂σφ −∂τ ˜ φ [ ] = 0 →∂σφ −∂τ ˜ φ = f (τ )˜ φ →∂σ ∂σ ˜ φ −∂τφ[ ] = 0 → ∂σ ˜ φ −∂τφ = ˜ f (τ )

The invariances under

€

φ →φ + g(τ)˜ φ → ˜ φ + ˜ g (τ)

allow to gauge-fix

€

f (τ) = ˜ f (τ) = 0

€

Lsym =12

˙ φ ˜ φ '+ 12φ' ˜ ˙ φ −

12φ '2 − 1

2˜ φ '2

Furthermore Lsym can be diagonalized by introducing a pair of “new” scalar fields:

€

φ =12φ+ + φ−( )

˜ φ =12φ+ −φ−( )

Floreanini-Jackiw Lagrangians for chiral and antichiral fields

Equations of motion

€

φ+ →∂σ ˙ φ + −φ'+[ ] = 0

φ_ →∂σ ˙ φ + + φ'−[ ] = 0

€

˙ φ + −φ'+ = f (τ )˙ φ + φ'− = ˜ f (τ )€

€

˙ φ + = φ'+˙ φ − = −φ '−

on-shell they become respectively functions of

€

σ ± τ

chiral, anti-chiral

No Lorentz local invariance.

€

Lsym (φ, ˜ φ ) = L+(φ+) + L−(φ−)

L±(φ±) = ±12

˙ φ ±φ '± −12φ'2±

SYMMETRIES

€

Lsym , L± invariant under space-time translations

€

L± invariant under

€

Lsym invariant under

€

δLφ = τφ '+σ ˜ φ ' , δL ˜ φ = τ ˜ φ '+σφ '

€

δφ = τ φ'+σ ˙ φ on-shell

usual 2-dim Lorentz rotation

Lorentz invariance is recovered on-shell.

€

δLφ± = (τ ±σ)φ'± ,δφ± = f (τ ±σ)φ'± ,δφ± = τ ˙ φ ± +σφ± '

“conformal”

“scale”

€

The FJ Lagrangians belong to a general class of first-order Lagrangians:

€

L =12

qi cij ˙ q j −V (q) i, j =1,...,N

canonically conjugate momentum to qj .

€

Qjk ≡ Tj , Tk{ }PB

= c jk ≠ 0 All constraints are second class.

characterized by N primary constraints

€

Tj ≡ p j −12qi cij ≈ 0

€

detcij ≠ 0

Dirac bracket for any two functions of the phase-space variables:

€

f ,g{ }D≡ f ,g{ }

P− f ,Tj{ }

P(Q−1) jk Tk,g{ }

P

The Dirac formalism allows for a transition to the quantum theory:

€

qi , q j[ ] = i(c −1)ij

qi , p j[ ] =12iδ ij

pi, p j[ ] = −14icij

€

L =12

qi cij ˙ q j −V (q) i, j =1,...,N

€

≡14

dσdσ'φ '(σ)ε(σ −σ') ˙ φ '(σ')∫ −12

dσ∫ φ'2 (σ)

€

φ , ˜ φ behave like “non-commuting” phase space type coordinates.

€

i f ,g{ }D→ ˆ f , ˆ g [ ]

€

L =14

dσdσ'χ(σ)ε(σ −σ') ˙ χ (σ')∫ −12

dσ∫ χ2(σ)

€

χ σ( ) = ±φ'± (σ) qi →χ(σ)

V (q)→12

dσ χ2 σ( )∫ cij →12ε (σ −σ ')

€

φ±(σ,τ), φ±(σ ',τ ')[ ] = ±i ε2(σ −σ ')

φ±(σ,τ) , π ±(σ ',τ ')[ ] =i2δ '(σ −σ ')

π ±(σ,τ) , π ±(σ ',τ ')[ ] = −i8ε (σ −σ ')

€

π(σ,τ ) =14

d∫ σ 'χ(σ ',τ)ε(σ '−σ)

φ(σ,τ ) ≡ π(σ,τ ) − 14

d∫ σ 'χ(σ ',τ)ε(σ '−σ) ≈ 0

with

primary constraints

∇a ≡ eaα ∂α ; e ≡dete α

a

Cij = Cji ; Mij = M jiInvariant

under diffeomorphisms ξα →ξ 'α (ξ ) acting as χ 'i (ξ 'α )= χ i (ξ ) ; e α'a = e β

a ∂ξβ

∂ξ 'α

under Weyl transformations e αa → e α

'a = λ(ξ )e αa

e α'a →Λ b

a (ξ )e αb

δe αa =α (ξ )ε b

a eαb

SO(1,1) matrix

δ eS =α (ξ )δSδe α

a ε ba eα

b = 0 if ε ab tab = 0

with

tab = −

2eδSδe α

a eαb

condition to be imposed

€

S eαa,χi[ ] = −

12

d2∫ ξe Cij∇0χi∇1χ

j + Mij∇1χi∇1χ

j[ ]

€

i, j =1,...,N

Invariance under local Lorentz transformations has to hold since physical observables are independent on the choice of the vielbein.

Requirement of on-shell local Lorentz invariance has to be made:

T-duality symmetric action (Tseytlin). Quantization.

€

hαβ = eaα ηab ebβ

€

tab = −δa

b Cij∇0χi∇1χ

j + Mij∇1χi∇1χ

j[ ] +δ0bCij∇aχ

i∇1χj +δ1

bCij∇0χi∇aχ

j

+ 2δ1bMij∇aχ

i∇1χj

Weyl invariance

€

0 =δSδeα

a λeαa =

λ2eta

a

eaα

tab = 0

analogously to what happens in the ordinary formulation Tαβ = −2

T gδSδgαβ

= 0

on the solutions of the equations of motion for the zweibein is satifisfied and the local Lorentz invariance holds.

Reparametrization + Weyl + Local Lorentz inv. gauge choice for e αa

In particular, e αa = δ α

a flat gauge

∇1 Cij∇0χj +Mij∇1χ

j( ) = 0

dτ Cij−∞

+∞

∫ δχ i ∂0χj⎡⎣ ⎤⎦ σ =0

σ =π= 0 δχ i = 0 or, alternatively, ∂0χ

i = 0 at σ =0,π

Equations of motion for €

taa = Tr ta

b[ ] = 0

Equations of motion (with C and M costant) for χ i

€

∂1 Cij∂0χj + Mij∂1χ

j[ ] = 0

€

− dτ δ−∞

+∞

∫ χ i(Cij∂0χj + Mij∂1χ

j )σ =0

σ =π

+12

dτCij−∞

+∞

∫ ∂0χjδχ i[ ]

σ =0

σ =π

surface integral

€

Cij∂0χj + Mij∂1χ

j = gi(τ)arbitrary function

Further local gauge symmetry of the action:

€

χi →χ' i = χi + f i(τ,σ ) with

€

∇1 fi = 0

€

C∂0 f = gfix

€

Cij∂0χj + Mij∂1χ

j = 0

€

12

dτCij−∞

+∞

∫ ∂0χjδχ i[ ]

σ =0

σ =π

= 0

open strings

closed strings €

∂0χi = 0 σ = 0,π

€

δχ i = 0

€

ε ab tab = 0 = − ∇0χi Cij +∇1 χ

i Mij[ ](C−1) jk Ckl ∇0 χl + Mkl∇1χ

l[ ] +

−∇1χi(C −MC−1M)ij∇1χ

j

€

C = MC−1M

the matrix C can be always put in the following form:

€

C = diag (1,..., 1,-1,...,-1)

p q

€

⇒ C = C−1

€

C = MCMdefines the indefinite orthogonal group O(p,q) of NxN matrices M with N=p+q.

€

S = −12

d2∫ ξe ∇0χ−µ ∇1χ−

µ − ∇0χ+ν ∇1χ+

ν + Mij∇1χi∇1χ

j

ν =1

q

∑µ =1

p

∑⎡

⎣ ⎢

⎤

⎦ ⎥

p two-dimensional chiral scalar fields q two-dimensional antichiral scalar fields

In the flat gauge and along the solutions of the equations of motion for

€

χ i

C becomes the O(D,D) metric in the 2D-dimensional target space with coordinates χ i

ds2 = dχ i Cij dχj

S describes a mixture of D chiral scalars and D antichiral scalars which are the components of

The absence of a quantum Lorentz anomaly implies p=q=D with 2D=N.

€

χi =χ−

µ

χ+ν

⎛

⎝ ⎜

⎞

⎠ ⎟ i =1, ..., 2D µ, ν =1,...,D

€

Cij∇0 χj + Mij∇1 χ

j = 0

C = MC−1M

€

−εab Cij ∇b χ j + Mij∇aχ

j = 0

can be put in a covariant form: The two equations

constraints imposed in the covariant formulation (Hull).

Cij = −Ωij ≡ −0µν Ιµ

ν

Ιµν 0µν

⎛

⎝⎜⎜

⎞

⎠⎟⎟

M parametrized by D2 =D(D +1)

2+D(D −1)

2independent elements

symmetric D-dim matrix G antisymmetric D-dim matrix B

M = G − BG−1B BG−1

−G−1B G−1

⎛

⎝⎜

⎞

⎠⎟ generalized 2Dx2D metric (defined up to a sign)

invariant under ℜ∈O(D,D)

background transformation

€

χi =X µ

˜ X µ

⎛

⎝ ⎜

⎞

⎠ ⎟ In the “non-chiral” basis

€

C = MC−1M⇒ M −1 =Ω−1MΩ−1

€

X µ =12χ+

µ + χ−µ( )

˜ X µ =12δµν χ+

ν − χ−ν( )

€

S =12

dξe Ωij ∇0χi∇1χ

j −Mij∇1χi∇1χ

j[ ]∫

€

€

χ'=ℜχ M' = ℜ-tMℜ ℜtΩℜ =Ω

In particular, for

€

ℜ =Ω

€

S = −12

d2∫ ξ e[ ∇0X µ∇1˜ X µ +∇0

˜ X µ∇1Xµ − (G − BG−1B)µν ∇1Xµ∇1X

ν

−(BG−1)µν ∇1X

µ∇1˜ X ν + (G−1B)µ

ν ∇1˜ X µ∇1

˜ X ν − (G−1)µν ∇1˜ X µ∇1

˜ X ν ]

one gets the familiar T-duality invariance under

€

X ⇔ ˜ X

€

M ⇔ M −1

S = − T2

d 2ξ e∫ Cij ∇0χi∇1χ

j +Mij∇1χi∇1χ

j⎡⎣ ⎤⎦

is candidate to describe a bosonic string in a constant background, made of G and B, and compactified on a torus TD exhibiting a manifest T-duality invariance O(D,D).

€

χi

€

X µ (τ,σ +π ) = X µ (τ,σ ) + 2πlω µ

˜ X µ (τ,σ +π ) = ˜ X µ (τ,σ ) + 2πl2 pµ

and

€

ω µ

lpµ

⎛

⎝ ⎜

⎞

⎠ ⎟

being a vector spanning a Lorentzian lattice

€

ΛD,D

€

T =12πl2

€

O(D,D;Z)

€

O(D,D)

interpreted as string coordinates on a double torus T2D defined by

It is possible to block-diagonalize simultaneously C and M through the matrix

€

(T −1)ij =12

(G−1)µν (G−1)µν

(−E tG−1)µν (EG−1)µ

ν

⎛

⎝ ⎜

⎞

⎠ ⎟

introducing new coordinates

€

Φi = Tijχj ≡ (XRµ,XLµ )

€

E =G + B

€

C→C -1 ≡G−1 00 −G−1

⎛

⎝ ⎜

⎞

⎠ ⎟ = T − tCT −1

€

M →G -1 ≡G−1 00 G−1

⎛

⎝ ⎜

⎞

⎠ ⎟ = T −tMT −1

generalized metric

in which the R and L sectors are decoupled even in the presence of B!.

€

T −1 is not an element of the group O(d,d).

The transformations leaving the two metrics invariant belong to the subgroup O(d)xO(d) of the original group O(d,d).

An O(d,d) element transforms the generalized metric.

€

−εab Cij ∇b χ j + Mij∇aχ

j = 0

€

*dXR = −dXR ; * dXL = dXL

€

XR τ − (σ +π )[ ] = XR (τ −σ ) − 2πl2pR

XL τ + (σ +π )[ ] = XL (τ +σ ) + 2πl2pL

€

−lpRlpL

⎛

⎝ ⎜

⎞

⎠ ⎟ = T

wlp⎛

⎝ ⎜

⎞

⎠ ⎟

€

XL (τ +σ) = xL + 2l2pL (τ +σ) + il α nne−2in(τ+σ )

n≠0∑

XR (τ −σ) = xR + 2l2pR (τ −σ) + il αn

ne−2in(τ−σ )

n≠0∑

formally identical to the usual expansion of the right and left coordinates.

€

S =T2

d2∫ ξ LR + LL[ ]

LL,R = ±12∂0XL ,R

t G−1∂1XL ,R −12∂1XL,R

t G−1∂1XL,R

Floreanini-Jackiw Lagrangians with a non-vanishing Kalb-Ramond field as a background.

flat gauge

The relation between (XR, XL) and

€

X(τ,σ ) = x + 2l2G−1 p − B wl

⎡

⎣ ⎢ ⎤

⎦ ⎥ τ + 2lwσ +

il2

e−2inτ

nn≠0∑ G−1 αne

2inσ +α ne−2inσ[ ]

˜ X (τ,σ ) = ˜ x + 2l2 BG−1p + (G − BG−1B) wl

⎡

⎣ ⎢ ⎤

⎦ ⎥ τ + 2l2 pσ

+il2

e−2inτ

nn≠0∑ −E tG−1αne

2inσ + EG−1α ne−2inσ[ ]

€

x =12

G−1(xL + xR ) ; ˜ x = 12

(EG−1xL − E tG−1xR )

€

pR =12

p − E wl

⎡

⎣ ⎢ ⎤

⎦ ⎥ ; pL =

12

p + E t wl

⎡

⎣ ⎢ ⎤

⎦ ⎥

These relations reproduce the same ones in the usual closed string compactification on a torus in the presence of a B-field.

€

(X, ˜ X )

Hamiltonian

€

H = d∫ σ ∂1Φt G −1∂1Φ

generalized metric in the chiral coordinates

The analysis of the constraints and quantization lead to the following results:

€

X(τ,σ) , ˜ X t (τ,σ '){ }DB

= −1Tε(σ −σ ')I

P(τ,σ) , X t (τ,σ '){ }DB

= ˜ P (τ,σ) , ˜ X (τ,σ '){ }DB

=12δ(σ −σ ')I

P(τ,σ) , ˜ P t (τ,σ '){ }DB

= −T4

Gδ '(σ −σ ')I

conjugate momentum with respect to

€

X, ˜ X

non-commuting tori

€

pR ,L , x tR ,L[ ] = −iG ; αn , α tm[ ] = ˜ α m , ˜ α n

t[ ] = mGδm+n,0 ; αm , ˜ α n[ ] = 0

The string coordinates on T2d are non-commuting phase space but generate the usual commutation relations for the Fourier modes.

€

L n =T4

dσe2in(τ +σ )

0

π

∫ ∂+XLt G−1∂+XL =

12

α mt

m∈Z∑ G−1α n−m − aδn,0

Ln =T4

dσe2in(τ −σ )

0

π

∫ ∂−XRt G−1∂−XR =

12

αmt

m∈Z∑ G−1αn−m − aδn,0

formally identical to the usual Virasoro generators.

€

D = 26is again the critical dimension.

WHEN THE USUAL FORMULATION IS OBTAINED?

Assuming that the compactification radius

€

R >> α ' , then the winding modes

are very massive so the dual coordinate

At intermediate scales

€

R ≈ α ' the duality symmetry

At small scales

€

R << α '

THE T-DUALITY SYMMETRIC FORMULATION MAY BE CONSIDERED AS A NATURAL GENERALIZATION OF THE STANDARD ONE AT THE STRING SCALE.

€

˜ X decouples.

€

X ↔ ˜ X is restored.

it is the coordinate

€

X that decouples.

CONCLUSION AND PERSPECTIVES

The non-covariant sigma-model, defined on a double torus and in presence of non-vanishing metric and Kalb-Ramond backgrounds, differs for the non-commutative behavior of the string coordinates with respect the usual formulation but, in spite of that, the commutation relations of the modes are the same!

IMPLICATIONS ON THE GRAVITATIONAL EFFECTIVE THEORY

As in the standard manifestly Lorentz invariant formulation, vertex operators are associated with the physical states, now expressed in terms of X and its dual.

NEXT STEP Using Graviton Vertex Operators for computing scattering amplitudes in order to get gravitational low-energy effective action

This could shed light on string gravity and its “corrections” to the Einstein-Hilbert action due to the T-duality.

WORTH TO CONTINUE!