T-DUALITY FOR MASSIVE STATES IN STRING THEORY Jnan Maharana March - - PowerPoint PPT Presentation

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T-DUALITY FOR MASSIVE STATES IN STRING THEORY

Jnan Maharana March 8, 2013

In Memory of Sumitra bigskip arXiv:1302.1719

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INTRODUCTION

• String theory is very rich in its symmetry contents. The duality

symmetries - T-duality and S-duality - have played very important roles in our understandings of string dynamics in diverse dimensions.

• T-duality is well understood from perturbative perspective.

It has played many useful roles. In some sense this symmetry pre-dates string theory and it appeared in the study of strong interactions in the guise of Igi-Matsuda, Dolen-Horn-Schmid (FESR) Duality. This led to the Veneziano Model, Dual Models and eventually to String Theory.

2 / 44

INTRODUCTION

• String theory is very rich in its symmetry contents. The duality

symmetries - T-duality and S-duality - have played very important roles in our understandings of string dynamics in diverse dimensions.

• T-duality is well understood from perturbative perspective.

It has played many useful roles. In some sense this symmetry pre-dates string theory and it appeared in the study of strong interactions in the guise of Igi-Matsuda, Dolen-Horn-Schmid (FESR) Duality. This led to the Veneziano Model, Dual Models and eventually to String Theory.

• Consider two body scattering amplitude

a + b → c + d

2 / 44

• OLD DUALITY:

Σ s − channel Res = Σ t − channel Regge

3 / 44

• OLD DUALITY:

Σ s − channel Res = Σ t − channel Regge

• In the String Theoritic description a string is exchanged - it looks like

the same for both the channels (see later)

3 / 44

• OLD DUALITY:

Σ s − channel Res = Σ t − channel Regge

• In the String Theoritic description a string is exchanged - it looks like

the same for both the channels (see later)

• Let us consider a closed bosonic string in the Minkowski target space.

The action is S = 1 2

• dτdσˆ

ηˆ

µˆ ν∂aX ˆ µ∂aX ˆ ν

where ˆ µ, ˆ ν = 0, 1, 2...ˆ D − 1, ˆ ηˆ

µˆ ν = diag(1, −1, −1..)

3 / 44

• OLD DUALITY:

Σ s − channel Res = Σ t − channel Regge

• In the String Theoritic description a string is exchanged - it looks like

the same for both the channels (see later)

• Let us consider a closed bosonic string in the Minkowski target space.

The action is S = 1 2

• dτdσˆ

ηˆ

µˆ ν∂aX ˆ µ∂aX ˆ ν

where ˆ µ, ˆ ν = 0, 1, 2...ˆ D − 1, ˆ ηˆ

µˆ ν = diag(1, −1, −1..)

• The canonical Hamiltonian density is

Hc = 1 2

• P2 + X ′2
• 3 / 44

• Pˆ

µ = ∂L ˙ X ˆ

µ , are the canonical momenta, over-dot and prime stand for

τ and σ derivatives.

4 / 44

• Pˆ

µ = ∂L ˙ X ˆ

µ , are the canonical momenta, over-dot and prime stand for

τ and σ derivatives.

• Consider evolution of a closed string in the presence of constant

backgrounds, G (0)

ˆ µˆ ν and B(0) ˆ µˆ ν

S = 1 2

• dσdτ
• G (0)

ˆ µˆ ν ∂aX ˆ µ∂aX ˆ ν + ǫabB(0) ˆ µˆ ν ∂aX ˆ µ∂bX ˆ ν

• The corresponding Hamiltonian density is
• Hc = 1

2Z TM0(G (0), B(0))Z

4 / 44

• where

Z = P X ′

• M0 =
• G (0)−1

−G (0)−1B(0) B(0)G (0)−1 G (0) − B(0)G (0)−1B(0)

• 5 / 44

• where

Z = P X ′

• M0 =
• G (0)−1

−G (0)−1B(0) B(0)G (0)−1 G (0) − B(0)G (0)−1B(0)

• Under the interchange P ↔ X ′, the Hamiltonian density remains

invariant M0 ↔ M−1

5 / 44

• where

Z = P X ′

• M0 =
• G (0)−1

−G (0)−1B(0) B(0)G (0)−1 G (0) − B(0)G (0)−1B(0)

• Under the interchange P ↔ X ′, the Hamiltonian density remains

invariant M0 ↔ M−1

• The Hamiltonian density is also invariant under the global O(ˆ

D, ˆ D) transformation:

5 / 44

• The Z-vector and M0-matrix transform as

Z → Ω0Z, M0 → Ω0M0ΩT

0 , η0 → η0,

Ω0 ∈ O(ˆ D, ˆ D)

• where η0 is the O(ˆ

D, ˆ D) metric. η0 = 1 1

• 1 is ˆ

D × ˆ D unit matrix. Z is 2ˆ D-dimensional vector, and M0 is 2ˆ D × 2ˆ D symmetric matrix.

6 / 44

• The Z-vector and M0-matrix transform as

Z → Ω0Z, M0 → Ω0M0ΩT

0 , η0 → η0,

Ω0 ∈ O(ˆ D, ˆ D)

• where η0 is the O(ˆ

D, ˆ D) metric. η0 = 1 1

• 1 is ˆ

D × ˆ D unit matrix. Z is 2ˆ D-dimensional vector, and M0 is 2ˆ D × 2ˆ D symmetric matrix.

• In general the backgrounds may depend on the string coordinates X ˆ

µ.

The worldsheet action is a σ-model action. These X-dependent backgrounds, in such a case, satisfy the β-function equations - the equations of motion.

6 / 44

• Consider a scenario where the backgrounds depend only on some of

the coordinates, X µ and are independent of the rest Y α i.e. X ˆ

µ =

• X µ, Y α
• ,

µ = 0, 1..D − 1, α = D, ..ˆ D − 1 with ˆ D = D + d.

7 / 44

• Consider a scenario where the backgrounds depend only on some of

the coordinates, X µ and are independent of the rest Y α i.e. X ˆ

µ =

• X µ, Y α
• ,

µ = 0, 1..D − 1, α = D, ..ˆ D − 1 with ˆ D = D + d.

• Let us decompose the backgrounds as follows when they depend only
• n the set of string coordinates X µ (Hassan-Sen)

G (0)

ˆ µˆ ν =

gµν(X) Gαβ(X)

• B(0)

ˆ µˆ ν =

bµν(X) Bαβ(X)

• 7 / 44

• Consider a scenario where the backgrounds depend only on some of

the coordinates, X µ and are independent of the rest Y α i.e. X ˆ

µ =

• X µ, Y α
• ,

µ = 0, 1..D − 1, α = D, ..ˆ D − 1 with ˆ D = D + d.

• Let us decompose the backgrounds as follows when they depend only
• n the set of string coordinates X µ (Hassan-Sen)

G (0)

ˆ µˆ ν =

gµν(X) Gαβ(X)

• B(0)

ˆ µˆ ν =

bµν(X) Bαβ(X)

• Introduce a pair of vectors V and W of dimensions 2D and 2d

respectively V = Pµ X ′µ

• ,

W = Pα Y ′α

• 7 / 44

• The canonical Hamiltonian density is expressed as

Hc = 1 2

• VT ˜

MV + WTMW

• Whereas ˜

M(X) is a 2D × 2D matrix, M(X) is another 2d × 2d

• matrix. These two matrices are

˜ M = gµν −gµρbρν bµρgρν gµν − bµρgρλbλν

• M =

G αβ −G αγBγβ BαγG γβ Gαβ − BαγG γδBδβ

• Let us focus on the second term and define

H2 = 1 2WTMW Note that H2 is invariant under the global O(d, d) transformations given below M → ΩMΩT, W → ΩW, ΩTηΩ = η, Ω ∈ O(d, d), η = 1 1

• 8 / 44

• Now 1 being d × d unit matrix and W is the O(d, d) vector.
• Since ˜

M and V are inert under this duality transformation, Hc is O(d, d) invariant.

9 / 44

• Now 1 being d × d unit matrix and W is the O(d, d) vector.
• Since ˜

M and V are inert under this duality transformation, Hc is O(d, d) invariant.

• REMARKS:

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• Now 1 being d × d unit matrix and W is the O(d, d) vector.
• Since ˜

M and V are inert under this duality transformation, Hc is O(d, d) invariant.

• REMARKS:
• If we consider toroidal compactification, on T d, and Y α are the

compact coordinates, then the duality group is O(d, d).

9 / 44

• Now 1 being d × d unit matrix and W is the O(d, d) vector.
• Since ˜

M and V are inert under this duality transformation, Hc is O(d, d) invariant.

• REMARKS:
• If we consider toroidal compactification, on T d, and Y α are the

compact coordinates, then the duality group is O(d, d).

• The moduli G and B parametrize the coset

O(d,d) O(d)×O(d).

9 / 44

• Now 1 being d × d unit matrix and W is the O(d, d) vector.
• Since ˜

M and V are inert under this duality transformation, Hc is O(d, d) invariant.

• REMARKS:
• If we consider toroidal compactification, on T d, and Y α are the

compact coordinates, then the duality group is O(d, d).

• The moduli G and B parametrize the coset

O(d,d) O(d)×O(d).

• In general for compactification on T d, the S-S prescription is to

express ˆ eA

M =

• er

µ(X)

Aβ

µ(X)E a β(X)

E a

α(X)

• The D-dimensional spacetime metric is: gµν(X) = er

µes νg0 rs, g0 rs being

D-dimensional Lorentzian signature flat metric.

• Gαβ = E a

αE b β δab is the metric along compact directions. Aβ µ are the

abelian gauge fields due to the isometries associated with d-dimensional torus.

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• The antisymmetric tensor gets decomposed as

ˆ BMN(ˆ X) = bµν(X) Bµα(X) Bνβ(X) Bαβ(X)

• Note that there are d abelian gauge fields Bµα(X) appearing due to

compactification of ˆ B-field.

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• The antisymmetric tensor gets decomposed as

ˆ BMN(ˆ X) = bµν(X) Bµα(X) Bνβ(X) Bαβ(X)

• Note that there are d abelian gauge fields Bµα(X) appearing due to

compactification of ˆ B-field.

• The worldsheet action for closed bosonic string in ˆ

D-dimensions takes the following form in the presence of backgrounds ˆ G(ˆ X) and ˆ B(ˆ X) (depending in string coordinates) ˆ S = 1 2

• dσdτ
• ˆ

Gˆ

µˆ ν∂a ˆ

X ˆ

µ∂b ˆ

X ˆ

ν + ǫab ˆ

Bˆ

µˆ ν∂a ˆ

X ˆ

µ∂b ˆ

X ˆ

ν

• 10 / 44

• The antisymmetric tensor gets decomposed as

ˆ BMN(ˆ X) = bµν(X) Bµα(X) Bνβ(X) Bαβ(X)

• Note that there are d abelian gauge fields Bµα(X) appearing due to

compactification of ˆ B-field.

• The worldsheet action for closed bosonic string in ˆ

D-dimensions takes the following form in the presence of backgrounds ˆ G(ˆ X) and ˆ B(ˆ X) (depending in string coordinates) ˆ S = 1 2

• dσdτ
• ˆ

Gˆ

µˆ ν∂a ˆ

X ˆ

µ∂b ˆ

X ˆ

ν + ǫab ˆ

Bˆ

µˆ ν∂a ˆ

X ˆ

µ∂b ˆ

X ˆ

ν

• REMARKS:

10 / 44

• The antisymmetric tensor gets decomposed as

ˆ BMN(ˆ X) = bµν(X) Bµα(X) Bνβ(X) Bαβ(X)

• Note that there are d abelian gauge fields Bµα(X) appearing due to

compactification of ˆ B-field.

• The worldsheet action for closed bosonic string in ˆ

D-dimensions takes the following form in the presence of backgrounds ˆ G(ˆ X) and ˆ B(ˆ X) (depending in string coordinates) ˆ S = 1 2

• dσdτ
• ˆ

Gˆ

µˆ ν∂a ˆ

X ˆ

µ∂b ˆ

X ˆ

ν + ǫab ˆ

Bˆ

µˆ ν∂a ˆ

X ˆ

µ∂b ˆ

X ˆ

ν

• REMARKS:
• When we compactify the on T d and backgrounds ˆ

G and ˆ B are independent of compact coordinates Y α then the action can be expressed in terms of string coordinates X µ and Y α and these couple to backgrounds g, G, Aα

µ, b, B and Bµα which depend on X µ(σ, τ)

• nly.

10 / 44

• In this scenario, we introduce a set of dual coordinates ˜

Y corresponding to each compact coordinate Y and dual backgrounds ˜ G and ˜ B and define a dual action ˆ ˜

• S. The two sets of equations of

motion derived from ˆ S and ˆ ˜ S can be suitably combined to express O(d, d) covariant equations of motion (see JM and JHS for details).

11 / 44

• In this scenario, we introduce a set of dual coordinates ˜

Y corresponding to each compact coordinate Y and dual backgrounds ˜ G and ˜ B and define a dual action ˆ ˜

• S. The two sets of equations of

motion derived from ˆ S and ˆ ˜ S can be suitably combined to express O(d, d) covariant equations of motion (see JM and JHS for details).

• We may ask whether it is possible to construct vertex operators for

massive excited states compactified on T d in an O(d, d) invariant manner?

11 / 44

• In this scenario, we introduce a set of dual coordinates ˜

Y corresponding to each compact coordinate Y and dual backgrounds ˜ G and ˜ B and define a dual action ˆ ˜

• S. The two sets of equations of

motion derived from ˆ S and ˆ ˜ S can be suitably combined to express O(d, d) covariant equations of motion (see JM and JHS for details).

• We may ask whether it is possible to construct vertex operators for

massive excited states compactified on T d in an O(d, d) invariant manner?

• Is is possible to generalize the results to superstring?

11 / 44

MASSIVE EXCITED STATES

• Excited massive stringy states are interesting.
• At very high energy (Planckian energy) scatterings stringy states are

important (Gross, Mende, Amati, Ciafoloni, Veneziano..). It is conjectured that there might be symmetry enhancements in the limit α′ → ∞.

• There are evidences for existence of gauge symmetry associated with

massive states: Evans, Ovrut, Kubota, Veneziano...

12 / 44

MASSIVE EXCITED STATES

• Excited massive stringy states are interesting.
• At very high energy (Planckian energy) scatterings stringy states are

important (Gross, Mende, Amati, Ciafoloni, Veneziano..). It is conjectured that there might be symmetry enhancements in the limit α′ → ∞.

• There are evidences for existence of gauge symmetry associated with

massive states: Evans, Ovrut, Kubota, Veneziano...

• For closed strings in background of massless states when one

computes β-function to higher loops and sums up it is necessary to include effects of massive excited states. (Das, Sathiapalan, Itoi, Watabiki).

12 / 44

MASSIVE EXCITED STATES

• Excited massive stringy states are interesting.
• At very high energy (Planckian energy) scatterings stringy states are

important (Gross, Mende, Amati, Ciafoloni, Veneziano..). It is conjectured that there might be symmetry enhancements in the limit α′ → ∞.

• There are evidences for existence of gauge symmetry associated with

massive states: Evans, Ovrut, Kubota, Veneziano...

• For closed strings in background of massless states when one

computes β-function to higher loops and sums up it is necessary to include effects of massive excited states. (Das, Sathiapalan, Itoi, Watabiki).

• Interactions in higher spin massless field theory gets insight from

higher spin stringy state vertex operators in appropriate limit. Sagnotti, Tarona...

12 / 44

• T-duality Properties of Excited Massive States
• We consider a simple scenario. Let us envisage evolution of the string

in constant background G (0), independent of coordinates. Under the interchange P ↔ X ′ the Hamiltonian remains invariant if G (0) ↔ G (o)−1. The vertex operator can be cast in a T-duality invariant form in this case.

• Recall that this G (0)-tensor can carry coordinate dependence.
• Let us recall some of the essential properties of the vertex operators.

We work in the weak field approximation through out.

• where

T++ = 1 2(ˆ G (0)

ˆ µˆ ν ∂X ˆ µ∂X ˆ ν), T−− = 1

2(ˆ G (0)

ˆ µˆ ν ¯

∂X ˆ

µ ¯

∂X ˆ

ν)

• Here ˆ

G (0)

ˆ µˆ ν = (1, −1, −1...). The stress energy momentum tensors are

defined with flat target space metric.

• Moreover, ∂X ˆ

µ = ˙

X ˆ

µ + X ′ˆ µ, ¯

∂X ˆ

µ = ˙

X ˆ

µ − X ′ˆ µ

13 / 44

• T-duality Properties of Excited Massive States
• We consider a simple scenario. Let us envisage evolution of the string

in constant background G (0), independent of coordinates. Under the interchange P ↔ X ′ the Hamiltonian remains invariant if G (0) ↔ G (o)−1. The vertex operator can be cast in a T-duality invariant form in this case.

• Recall that this G (0)-tensor can carry coordinate dependence.
• Let us recall some of the essential properties of the vertex operators.

We work in the weak field approximation through out.

• The vertex operators ˆ

Φn(Xˆ

µ), n, referring to the level of excited state

are required to be (1, 0) and (0, 1) primaries with respect to T++ and T−− respectively.

• where

T++ = 1 2(ˆ G (0)

ˆ µˆ ν ∂X ˆ µ∂X ˆ ν), T−− = 1

2(ˆ G (0)

ˆ µˆ ν ¯

∂X ˆ

µ ¯

∂X ˆ

ν)

• Here ˆ

G (0)

ˆ µˆ ν = (1, −1, −1...). The stress energy momentum tensors are

defined with flat target space metric.

• Moreover, ∂X ˆ

µ = ˙

X ˆ

µ + X ′ˆ µ, ¯

∂X ˆ

µ = ˙

X ˆ

µ − X ′ˆ µ

13 / 44

• These vertex operators satisfy ’Equations of Motion’ and certain

’Transversality (gauge) Conditions’ due to above requirements.

• Example:

Consider the graviton background in the weak field approximation:

• Gˆ

µˆ ν(X ˆ µ) = G (0) ˆ µˆ ν + hˆ µˆ ν(X ˆ µ). The corresponding constrains are:

∇2hˆ

µˆ ν = 0, and ∂ ˆ µhˆ µˆ ν = 0

14 / 44

• These vertex operators satisfy ’Equations of Motion’ and certain

’Transversality (gauge) Conditions’ due to above requirements.

• Example:

Consider the graviton background in the weak field approximation:

• Gˆ

µˆ ν(X ˆ µ) = G (0) ˆ µˆ ν + hˆ µˆ ν(X ˆ µ). The corresponding constrains are:

∇2hˆ

µˆ ν = 0, and ∂ ˆ µhˆ µˆ ν = 0

• First Excited Massive Level:

The vertex operator is ˆ Φ1 = ˆ V(1)

1

+ ˆ V(2)

1

+ ˆ V(3)

1

+ ˆ V(4)

1

• We define ˆ

V (i)

1 , i = 1, 2, 3, 4 as vertex functions. Thus a vertex

• perators is sum of vertex functions as above.

14 / 44

• The four vertex functions are

ˆ V (1)

1

= A(1)

ˆ µˆ ν,ˆ µ′ˆ ν′(X)∂X ˆ µ∂X ˆ ν ¯

∂X ˆ

µ′ ¯

∂X ˆ

ν′

ˆ V (2)

1

= A(2)

ˆ µˆ ν,ˆ µ′(X)∂X ˆ µ∂X ˆ ν ¯

∂2X ˆ

µ′

ˆ V (3)

1

= A(3)

ˆ µ,ˆ µ′ˆ ν′(X)∂2X ˆ µ ¯

∂X ˆ

µ′ ¯

∂X ˆ

ν′

ˆ V (4)

1

= A(4)

ˆ µ,ˆ µ′(X)∂2X ˆ µ ¯

∂2X ˆ

µ′

15 / 44

• The four vertex functions are

ˆ V (1)

1

= A(1)

ˆ µˆ ν,ˆ µ′ˆ ν′(X)∂X ˆ µ∂X ˆ ν ¯

∂X ˆ

µ′ ¯

∂X ˆ

ν′

ˆ V (2)

1

= A(2)

ˆ µˆ ν,ˆ µ′(X)∂X ˆ µ∂X ˆ ν ¯

∂2X ˆ

µ′

ˆ V (3)

1

= A(3)

ˆ µ,ˆ µ′ˆ ν′(X)∂2X ˆ µ ¯

∂X ˆ

µ′ ¯

∂X ˆ

ν′

ˆ V (4)

1

= A(4)

ˆ µ,ˆ µ′(X)∂2X ˆ µ ¯

∂2X ˆ

µ′

• Unprimed and primed indices correspond to right moving and left

moving sectors respectively (∂X ˆ

µ and ∂X ˆ µ′). When we demand ˆ

Φ1 to be (1, 1) with respect to T±±, then ˆ V (i)

1 (i.e.A(i)) are constrained -

they are not all independent. (i) Only ˆ V (1)

1

is (1, 1) on its own; however the other three vertex functions: ˆ V (2)

1

− ˆ V (4)

1

are related to ˆ V (1)

1

.

15 / 44

• The (1, 1) constraint on ˆ

Φ1 implies (a) Each A(i) satisfies a mass shell condition. ( ˆ ∇2 − 2)A(1)

ˆ µˆ ν,ˆ µ′ˆ ν′(X) = 0,

( ˆ ∇2 − 2)A(2)

ˆ µˆ ν,ˆ µ′(X) = 0

( ˆ ∇2 − 2)A(3)

ˆ µ,ˆ µ′ˆ ν′(X) = 0,

( ˆ ∇2 − 2)A(4)

ˆ µ,ˆ µ′(X) = 0

where ˆ ∇2 is ˆ D-dimensional Laplacian defined in flat space. (b) Following relations show how A(i) are related: Transversality (gauge) conditions are A(2)

ˆ µˆ ν,ˆ µ′ = ∂ˆ ν′A(1) ˆ µˆ ν,ˆ µ′ˆ ν′, A(3) ˆ µ,ˆ µ′ˆ ν′ = ∂ˆ νA(1) ˆ µˆ ν,ˆ µ′ˆ ν′, A(4) ˆ µ,ˆ µ′ = ∂ˆ ν′∂ˆ νA(1) ˆ µˆ ν,ˆ µ′ˆ ν′

The other set i.e. Transversality (gauge) conditions are A(1)ˆ

µ ˆ µ

,ˆ

µ′ˆ ν′ +2∂ ˆ µ∂ˆ νA(1) ˆ µˆ ν,ˆ µ′ˆ ν′ = 0

and A(1)

ˆ µˆ ν,ˆ µ′ ˆ µ′

+ 2∂ ˆ

µ′∂ˆ ν′A(1) ˆ µˆ ν,ˆ µ′ˆ ν′ = 0

16 / 44

• Simple Example: Graviton Vertex Consider the Graviton vertex in

ˆ D-dimension and compactify on T d. Look at the part involving compact coordinates Y α. In weak field approximation: Gαβ(X) = δαβ + hαβ(X), α, β = 1, 2..d Vh = hαβ′∂Y α ¯ ∂Y β′ use Pα = δαβ ˙ Y α to express the vertex operator as Vh = hαβ′PαPβ − hαβ′Y ′αY ′β′ − hα

β′PαY ′β′ + hα′ β Pα′Y ′β

We have retained the memory where P and Y ′ came from through appearance of unprimed and primed indices.

17 / 44

• Simple Example: Graviton Vertex Consider the Graviton vertex in

ˆ D-dimension and compactify on T d. Look at the part involving compact coordinates Y α. In weak field approximation: Gαβ(X) = δαβ + hαβ(X), α, β = 1, 2..d Vh = hαβ′∂Y α ¯ ∂Y β′ use Pα = δαβ ˙ Y α to express the vertex operator as Vh = hαβ′PαPβ − hαβ′Y ′αY ′β′ − hα

β′PαY ′β′ + hα′ β Pα′Y ′β

We have retained the memory where P and Y ′ came from through appearance of unprimed and primed indices.

• The vertex operator can be expressed as

Vh = HmnWmWn − K n

mWmWn

Note that W is the O(d, d) vector defined earlier. Vh be O(d, d) invariant if Hmn → Ωm′

n Ωn′ n Hm′n′, Wm → Ωm m′Wm′, K n m → Ωm′ m Ωn n′K n′ m′

17 / 44

• The vertex operator along noncompact directions (setting Aα

µ = 0 for

simplicity) is Vh = hµν′(X)∂X µ ¯ ∂X ν′ in the weak field approximation.

• The spacetime coordinates and the tensors are inert under O(d, d)

transformations. Therefore, the full vertex operator is T-duality invariant.

• For excited massive levels, we follow a similar approach. However, we

shall encounter complications since number of vertex functions keep increasing as we go to higher and higher excited level. Therefore, we have to adopt a suitable procedure.

18 / 44

• The vertex operator along noncompact directions (setting Aα

µ = 0 for

simplicity) is Vh = hµν′(X)∂X µ ¯ ∂X ν′ in the weak field approximation.

• The spacetime coordinates and the tensors are inert under O(d, d)

transformations. Therefore, the full vertex operator is T-duality invariant.

• For excited massive levels, we follow a similar approach. However, we

shall encounter complications since number of vertex functions keep increasing as we go to higher and higher excited level. Therefore, we have to adopt a suitable procedure.

• As we noticed, when graviton and antisymmetric tensor backgrounds

are considered in lower dimensions, we see appearance of scalars and gauge fields in addition to graviton and antisymmetric tensors, g(X) and b(X). We consider the case analogous to Hassan-Sen prescription in what follows.

18 / 44

T-DUALITY FOR EXCITED STATES

• Let us consider the scenario where tensors ˆ

V (i)

1

(i.e. A(i)) are coordinate independent - this is analog of constant G and B. Focus

• n ˆ

V (1)

1

ˆ V (1)

1

= A(1)

ˆ µˆ ν,ˆ µ′ˆ ν′∂X ˆ µ∂X ˆ ν ¯

∂X ˆ

µ′ ¯

∂X ˆ

ν′

∂X ˆ

µ = P ˆ µ + X ′ˆ µ. We can express the above equation in terms of

P ˆ

µ and X ′ˆ µ. It has following form in the expanded version.

(I). G (1)

ˆ µˆ ν,ˆ ρˆ λP ˆ µP ˆ νP ˆ ρP ˆ λ - Products of momenta only.

(II). G (2)

ˆ µˆ ν,ˆ ρˆ λX ′ˆ µX ′ˆ νX ′ˆ ρX ′ˆ λ - Product of X ′ only.

(III). G (3)

ˆ µˆ ν,ˆ ρˆ λP ˆ µP ˆ νP ˆ ρX ′ˆ λ - Product of three momenta and one X ′.

There are four such terms altogether. (IV ). G (4)

ˆ µˆ ν,ˆ ρˆ λX ′ˆ µX ′ˆ νX ′ˆ ρP ˆ λ - Product of three X ′ and one P. There

are also four such terms.

19 / 44

• (V ). G (5)

ˆ µˆ ν,ˆ ρˆ λP ˆ µP ˆ νX ′ˆ ρX ′ˆ λ - There are six terms of this type: Product

• f pair of P’s and a pair of X ′’s.

20 / 44

• (V ). G (5)

ˆ µˆ ν,ˆ ρˆ λP ˆ µP ˆ νX ′ˆ ρX ′ˆ λ - There are six terms of this type: Product

• f pair of P’s and a pair of X ′’s.
• There are total 16 terms in (I) − (V ). A careful inspection shows that

the vertex function A(1) remains invariant under P ↔ X ′ if following relations hold: G (1) ↔ G (2), G (3) ↔ G (4)

20 / 44

• (V ). G (5)

ˆ µˆ ν,ˆ ρˆ λP ˆ µP ˆ νX ′ˆ ρX ′ˆ λ - There are six terms of this type: Product

• f pair of P’s and a pair of X ′’s.
• There are total 16 terms in (I) − (V ). A careful inspection shows that

the vertex function A(1) remains invariant under P ↔ X ′ if following relations hold: G (1) ↔ G (2), G (3) ↔ G (4)

• The six terms in (V ) rearrange themselves appropriately.

20 / 44

• (V ). G (5)

ˆ µˆ ν,ˆ ρˆ λP ˆ µP ˆ νX ′ˆ ρX ′ˆ λ - There are six terms of this type: Product

• f pair of P’s and a pair of X ′’s.
• There are total 16 terms in (I) − (V ). A careful inspection shows that

the vertex function A(1) remains invariant under P ↔ X ′ if following relations hold: G (1) ↔ G (2), G (3) ↔ G (4)

• The six terms in (V ) rearrange themselves appropriately.
• This duality in analog of G ↔ G −1 under τ ↔ σ - A(1) are constant

as was G.

20 / 44

• (V ). G (5)

ˆ µˆ ν,ˆ ρˆ λP ˆ µP ˆ νX ′ˆ ρX ′ˆ λ - There are six terms of this type: Product

• f pair of P’s and a pair of X ′’s.
• There are total 16 terms in (I) − (V ). A careful inspection shows that

the vertex function A(1) remains invariant under P ↔ X ′ if following relations hold: G (1) ↔ G (2), G (3) ↔ G (4)

• The six terms in (V ) rearrange themselves appropriately.
• This duality in analog of G ↔ G −1 under τ ↔ σ - A(1) are constant

as was G.

• Consider the case where A(1)

ˆ µˆ ν,ˆ µ′ˆ ν′(X)∂X ˆ µ∂X ˆ ν ¯

∂X ˆ

µ′ depends on

X µ, µ = 0, 1, ..D − 1 and is independent of internal coordinates Y α.

20 / 44

• The vertex function will be decomposed into following classes.

21 / 44

• The vertex function will be decomposed into following classes.
• (i) A tensor A(1)

µν,µ′ν′ which has spacetime Lorentz indices.

• (ii) Another: Three Lorentz indices and one internal index.
• (iii) A tensor with two Lorentz indices and two internal indices.
• (iv) A tensor with one Lorentz index and three internal indices.
• (v) A tensor with all internal indices: A(1)

αβ,α′β′ which contracts with

∂Y α∂Y β ¯ ∂Y α′ ¯ ∂Y β′. If we express this vertex function in terms of Pα and Y ′α etc. we note that there will be 16 terms - for discrete T-duality transformations along internal directions the tensor A(1)

αβ,α′β′

behaves like a constant tensor.

21 / 44

• The vertex function will be decomposed into following classes.
• (i) A tensor A(1)

µν,µ′ν′ which has spacetime Lorentz indices.

• (ii) Another: Three Lorentz indices and one internal index.
• (iii) A tensor with two Lorentz indices and two internal indices.
• (iv) A tensor with one Lorentz index and three internal indices.
• (v) A tensor with all internal indices: A(1)

αβ,α′β′ which contracts with

∂Y α∂Y β ¯ ∂Y α′ ¯ ∂Y β′. If we express this vertex function in terms of Pα and Y ′α etc. we note that there will be 16 terms - for discrete T-duality transformations along internal directions the tensor A(1)

αβ,α′β′

behaves like a constant tensor.

• our goal in to construct O(d, d) invariant vertex operators and we

take clue from construction of graviton vertex operator. Recall W = P Y ′

• 21 / 44

• Let us closely look at the types of terms we have

(I) PαPβPα′Pβ′ (II) Y ′αY ′βY ′α′Y ′β′. (III) PαPβPα′Y ′β′ and three more terms. (IV) Y ′αY ′βPα′Y ′β′ Also three more terms. (V) PαPβY ′α′Y ′β′. Altogether six terms.

• These 16 terms originate from products of ∂Y α ¯

∂Y α′.... There are terms with positive and negative signs in (III), (IV ) and (V ). When we inspect (I) and (II) we note that they can be combined to form product of W-vectors which will contract with a suitable O(d, d) tensor.

22 / 44

• Let us closely look at the types of terms we have

(I) PαPβPα′Pβ′ (II) Y ′αY ′βY ′α′Y ′β′. (III) PαPβPα′Y ′β′ and three more terms. (IV) Y ′αY ′βPα′Y ′β′ Also three more terms. (V) PαPβY ′α′Y ′β′. Altogether six terms.

• These 16 terms originate from products of ∂Y α ¯

∂Y α′.... There are terms with positive and negative signs in (III), (IV ) and (V ). When we inspect (I) and (II) we note that they can be combined to form product of W-vectors which will contract with a suitable O(d, d) tensor.

• The terms appearing in (III) and (IV ) can be appropriately

rearranged to obtain products of W vectors.

• The six terms in (V ) have pair of momenta and Y ′. They nicely

arrange to be products of W-vectors.

22 / 44

• Let us closely look at the types of terms we have

(I) PαPβPα′Pβ′ (II) Y ′αY ′βY ′α′Y ′β′. (III) PαPβPα′Y ′β′ and three more terms. (IV) Y ′αY ′βPα′Y ′β′ Also three more terms. (V) PαPβY ′α′Y ′β′. Altogether six terms.

• These 16 terms originate from products of ∂Y α ¯

∂Y α′.... There are terms with positive and negative signs in (III), (IV ) and (V ). When we inspect (I) and (II) we note that they can be combined to form product of W-vectors which will contract with a suitable O(d, d) tensor.

• The terms appearing in (III) and (IV ) can be appropriately

rearranged to obtain products of W vectors.

• The six terms in (V ) have pair of momenta and Y ′. They nicely

arrange to be products of W-vectors.

• Therefore, we have products W’s which we contract with appropriate

O(d, d) tensors such that resulting vertex functions are O(d, d) invariant.

22 / 44

• If we have a doublet

Y ′ P

• we can express it as ηW which is also an O(d, d) vector. Therefore,

there are products of W and ηW.

23 / 44

• If we have a doublet

Y ′ P

• we can express it as ηW which is also an O(d, d) vector. Therefore,

there are products of W and ηW.

• The other vertex functions (look at only internal directions)

V (2)

1

= A(2)

αβ,α′(X)∂Y α∂Y β ¯

∂2Y α′ V (3)

1

= A(3)

α,α′β′(X)∂2Y α∂Y α′∂Y β′

V (4)

1

= A(4)

α,α′(X)∂2Y α ¯

∂2Y α′

• Explicit calculations show that these can be also cast in duality

invariant forms; however it is not so straight forward since higher derivatives of σ and τ appear. Moreover, as we consider higher and higher excited levels more vertex functions appear with higher derivatives.

23 / 44

• We encounter, generically, terms of two different types:

(a) Products like: ∂Y ∂Y ...¯ ∂Y ¯ ∂Y .. which has only ∂ or ¯ ∂ acting on string coordinates. (b) Higher derivatives and their products, typically of the form: ∂mY ∂Y ...¯ ∂nY ¯ ∂Y .. The structure of a vertex function, at a given level, is constrained by level matching condition and by requirement that it should be (1, 1).

24 / 44

• We encounter, generically, terms of two different types:

(a) Products like: ∂Y ∂Y ...¯ ∂Y ¯ ∂Y .. which has only ∂ or ¯ ∂ acting on string coordinates. (b) Higher derivatives and their products, typically of the form: ∂mY ∂Y ...¯ ∂nY ¯ ∂Y .. The structure of a vertex function, at a given level, is constrained by level matching condition and by requirement that it should be (1, 1).

• Consider a vertex function (leading trajectory) of second massive level:

V (1)

2

= C (1)

αβγ,α′β′γ′(X)∂Y α∂Y β∂Y γ ¯

∂Y α′ ¯ ∂Y β′ ¯ ∂Y γ′

• We can express V (1)

2

in terms of Pα, Y ′α as before. Rearrange the terms ( here we 64 terms!) and cast in O(d, d) invariant form.

• For second level the vertex operator has nine vertex functions and up

to ∂3Y and ¯ ∂3Y . Thus present method becomes unmanageable soon.

24 / 44

HIGHER EXCITED LEVELS

• Observations:

(a) The basic building blocks of any vertex functions are: ∂Y α = Pα + Y ′α and ¯ ∂Y α = Pα − Y ′α (b) Each vertex function at every mass level is EITHER string of products of these basic blocks OR these blocks are acted upon by ∂ and ¯ ∂ respectively so that the vertex function has desired

• dimensions. Note terms like ∂ ¯

∂Y do not appear since they vanish by worldsheet equations of motion. (c) It is not convenient to deal with P ± Y ′ for our purpose - we project them out from O(d, d) vector W.

25 / 44

HIGHER EXCITED LEVELS

• Observations:

(a) The basic building blocks of any vertex functions are: ∂Y α = Pα + Y ′α and ¯ ∂Y α = Pα − Y ′α (b) Each vertex function at every mass level is EITHER string of products of these basic blocks OR these blocks are acted upon by ∂ and ¯ ∂ respectively so that the vertex function has desired

• dimensions. Note terms like ∂ ¯

∂Y do not appear since they vanish by worldsheet equations of motion. (c) It is not convenient to deal with P ± Y ′ for our purpose - we project them out from O(d, d) vector W.

• Introduce projection operators

P± = 1 2(1 ± ˜ σ3), ˜ σ3 = 1 −1

• Here 1 is 2d × 2d matrix and the diagonal elements of ˜

σ3 are d × d unit matrix.

25 / 44

• The canonical momentum Pα and Y ′α are projected out as follows.

P = P+W, Y ′ = P−W Consequently, P + Y ′ = 1 2

• P+W + ηP−W
• , P − Y ′ = 1

2

• P+W − ηP−W
• 26 / 44

• The canonical momentum Pα and Y ′α are projected out as follows.

P = P+W, Y ′ = P−W Consequently, P + Y ′ = 1 2

• P+W + ηP−W
• , P − Y ′ = 1

2

• P+W − ηP−W
• Note: η flips lower component Y ′ of W to upper component.

26 / 44

• The canonical momentum Pα and Y ′α are projected out as follows.

P = P+W, Y ′ = P−W Consequently, P + Y ′ = 1 2

• P+W + ηP−W
• , P − Y ′ = 1

2

• P+W − ηP−W
• Note: η flips lower component Y ′ of W to upper component.
• When we have only products of P + Y ′ and P − Y ′,we first express

them as products of O(d,d) vectors and contract these vector indices with suitably constructed O(d, d) tensors.

26 / 44

• The canonical momentum Pα and Y ′α are projected out as follows.

P = P+W, Y ′ = P−W Consequently, P + Y ′ = 1 2

• P+W + ηP−W
• , P − Y ′ = 1

2

• P+W − ηP−W
• Note: η flips lower component Y ′ of W to upper component.
• When we have only products of P + Y ′ and P − Y ′,we first express

them as products of O(d,d) vectors and contract these vector indices with suitably constructed O(d, d) tensors.

• Some of the vertex functions have structures where ∂ and ¯

∂ act on P + Y ′ and P − Y ′ respectively.

26 / 44

• In order to cast these vertex functions in desired form, we define

∆τ = P+∂τ, ∆σ = P+∂σ, ∆±(τ, σ) = 1 2(∆τ ± ∆σ)

27 / 44

• In order to cast these vertex functions in desired form, we define

∆τ = P+∂τ, ∆σ = P+∂σ, ∆±(τ, σ) = 1 2(∆τ ± ∆σ)

• There are two useful relations

∂(P + Y ′) = ∆+(τ, σ)

• P+W + ηP−W
• ¯

∂(P − Y ′) = ∆−(τ, σ)

• P+W − ηP−W
• 27 / 44

• In order to cast these vertex functions in desired form, we define

∆τ = P+∂τ, ∆σ = P+∂σ, ∆±(τ, σ) = 1 2(∆τ ± ∆σ)

• There are two useful relations

∂(P + Y ′) = ∆+(τ, σ)

• P+W + ηP−W
• ¯

∂(P − Y ′) = ∆−(τ, σ)

• P+W − ηP−W
• We utilize above two relations to express the basic buliding blocks

and their worldsheet derivatives appearing in any vertex function as products of O(d, d) vectors

27 / 44

• In order to cast these vertex functions in desired form, we define

∆τ = P+∂τ, ∆σ = P+∂σ, ∆±(τ, σ) = 1 2(∆τ ± ∆σ)

• There are two useful relations

∂(P + Y ′) = ∆+(τ, σ)

• P+W + ηP−W
• ¯

∂(P − Y ′) = ∆−(τ, σ)

• P+W − ηP−W
• We utilize above two relations to express the basic buliding blocks

and their worldsheet derivatives appearing in any vertex function as products of O(d, d) vectors

• The product of such vectors are to be contracted with appropriate

O(d, d) tensors to construct T-duality invariant vertex functions. Recall that when we construct O(d, d) invariant vertex operators for Gαβ and Bαβ we introduce M-matrix and contract W-vector with it.

27 / 44

• With these prescriptions, we shall construct duality invariant vertex

functions for excited massive levels. Let us consider nth excited massive level.

28 / 44

• With these prescriptions, we shall construct duality invariant vertex

functions for excited massive levels. Let us consider nth excited massive level.

• The dimension of all right movers constructed from ∂Y and powers of

∂ acting on ∂Y should be n + 1. Same holds for left moving sector.

28 / 44

• With these prescriptions, we shall construct duality invariant vertex

functions for excited massive levels. Let us consider nth excited massive level.

• The dimension of all right movers constructed from ∂Y and powers of

∂ acting on ∂Y should be n + 1. Same holds for left moving sector.

• Consider right moving sector of the type Πn+1

1

∂Y αi and left moving sector of the same type i.e. Πn+1

1

¯ ∂Y αi

28 / 44

• With these prescriptions, we shall construct duality invariant vertex

functions for excited massive levels. Let us consider nth excited massive level.

• The dimension of all right movers constructed from ∂Y and powers of

∂ acting on ∂Y should be n + 1. Same holds for left moving sector.

• Consider right moving sector of the type Πn+1

1

∂Y αi and left moving sector of the same type i.e. Πn+1

1

¯ ∂Y αi

• A vertex function at this level takes the form

Vα1,α2...αn+1,α′

1α′ 2...α′ n+1(X)Πn+1

1

∂YαiΠn+1

1

¯ ∂Yα′

i 28 / 44

• With these prescriptions, we shall construct duality invariant vertex

functions for excited massive levels. Let us consider nth excited massive level.

• The dimension of all right movers constructed from ∂Y and powers of

∂ acting on ∂Y should be n + 1. Same holds for left moving sector.

• Consider right moving sector of the type Πn+1

1

∂Y αi and left moving sector of the same type i.e. Πn+1

1

¯ ∂Y αi

• A vertex function at this level takes the form

Vα1,α2...αn+1,α′

1α′ 2...α′ n+1(X)Πn+1

1

∂YαiΠn+1

1

¯ ∂Yα′

i

• We know how to covert Πn+1

1

∂Y α′

i and Πn+1

1

¯ ∂Y α′

i to products

projected O(d, d) vectors W.

28 / 44

• With these prescriptions, we shall construct duality invariant vertex

functions for excited massive levels. Let us consider nth excited massive level.

• The dimension of all right movers constructed from ∂Y and powers of

∂ acting on ∂Y should be n + 1. Same holds for left moving sector.

• Consider right moving sector of the type Πn+1

1

∂Y αi and left moving sector of the same type i.e. Πn+1

1

¯ ∂Y αi

• A vertex function at this level takes the form

Vα1,α2...αn+1,α′

1α′ 2...α′ n+1(X)Πn+1

1

∂YαiΠn+1

1

¯ ∂Yα′

i

• We know how to covert Πn+1

1

∂Y α′

i and Πn+1

1

¯ ∂Y α′

i to products

projected O(d, d) vectors W.

• A generic vertex function is of the form

∂pY αi∂qY αj∂rY αk...¯ ∂p′Y α′

i ¯

∂q′Y α′

j ¯

∂r′Y α′

k...,

With constraints: p + q + r = n + 1, p′ + q′ + r′ = n + 1

28 / 44

• APPENDIX
• Look at second excited level:

Φ2 = V (1)

2

+ V (2)

2

+ V (3)

2

+ V (4)

2

+ V (5)

2

+ V (6)

2

+ V (7)

2

+ V (6)

2

+ V (8)

2

where V (2)

2

= C (2)

αβ,α′β′γ′∂2Y α∂Y β ¯

∂Y α′ ¯ ∂Y β′ ¯ ∂Y γ′ V (3)

2

= C (3)

αβγ,α′β′∂Y α∂Y β∂Y γ ¯

∂2Y α′ ¯ ∂Y β′ V (4)

2

= C (4)

α,α′β′γ′∂3Y αY α′ ¯

∂Y β′ ¯ ∂Y γ′ V (5)

2

= C (5)

αβγ,α′∂Y α∂Y β∂Y γ ¯

∂3Y α′ V (6)

2

= C (6)

αβ,α′β′∂2Y α∂Y β ¯

∂2Y α′ ¯ ∂Y β′ V (7)

2

= C (7)

α α′β′∂3Y α ¯

∂2Y α′ ¯ ∂Y β′ + C (8)

αβ,α′∂2Y α∂Y β ¯

∂3Y α′ V (8)

2

= C (9)

α,α′∂3Y α ¯

∂3Y α′ This illustrates how the number of vertex functions grow with levels.

29 / 44

• REMARKS:
• To achieve this objective, we go through following steps:

STEP I: Rewrite ∂pY = ∂p−1(P + Y′), ¯ ∂p′(P − Y′) = ¯ ∂p′−1(P − Y′) STEP II: Using the projection operators ∂p−1(P + Y′) = ∆+p−1(P + Y′), ¯ ∂p′−1(P − Y′) = ∆−p′−1(P − Y′)

30 / 44

• REMARKS:
• The above expression has to be converted into products of W’s and

derivatives to be an O(d, d) tensor. The constraints satisfied by p, q, r.. determine the rank of the tensor. This tensor is contracted with a tensor with appropriate O(d, d) transformation properties in

• rder to obtain the duality invariant vertex operator.
• To achieve this objective, we go through following steps:

STEP I: Rewrite ∂pY = ∂p−1(P + Y′), ¯ ∂p′(P − Y′) = ¯ ∂p′−1(P − Y′) STEP II: Using the projection operators ∂p−1(P + Y′) = ∆+p−1(P + Y′), ¯ ∂p′−1(P − Y′) = ∆−p′−1(P − Y′)

30 / 44

• STEP III:

∆+p−1(P + Y ′) = ∆+p−1

• P+W + ηP−W
• ,

∆p′−1(P − Y ′) = ∆−p′−1

• P+W − ηP−W
• With these relations a generic vertex function an expressed as product
• f W vectors its ∆± derivatives which are contracted with suitable

tensors. Vn+1 = Aklm..,k′l′m′..(X)∆+p−1Wk

+∆q−1 +

Wl

+∆+r−1Wm + ..

∆−p′−1Wk′

− ∆−p′−1Wl′ −∆−p′−1Wm′ −

(1) where W± = (P+W ± ηP−W). p + q + r = n + 1 and p′ + q′ + r′ = n + 1. Indices {k, l, m; k′, l′, m′} of W± correspond to O(d, d) components. Aklm..,k′l′m′..(X) is O(d, d) tensor.

31 / 44

• Vertex function Vn+1 will be O(d, d) invariant if the tensor

Aklm..,k′l′m′..(X) transforms as Aklm..,k′l′m′.. → Ωp

kΩq l Ωr m...Ωp′ k′Ωq′ l′ Ωr′ m′Apqr..,p′q′r′..

32 / 44

• Vertex function Vn+1 will be O(d, d) invariant if the tensor

Aklm..,k′l′m′..(X) transforms as Aklm..,k′l′m′.. → Ωp

kΩq l Ωr m...Ωp′ k′Ωq′ l′ Ωr′ m′Apqr..,p′q′r′..

• So far we have discussed only those vertex functions which have

internal indices contracting with derivatives of compact coordinates. There are two other possibilities: (a) Set of vertex functions which have only have spacetime coordinates (like ∂X, ¯ ∂X and higher derivatives) contracting with spacetime tensors. Such vertex functions are O(d, d) invariant since T-duality transformations do not affect them. (b) There are set of vertex functions with mixed indices (Lorentz and internal). It the terms like ∂Y , ¯ ∂Y and higher derivatives which need to be converted to W’s and the derivatives to obtain T-duality invariant vertex functions.

32 / 44

• Vertex function Vn+1 will be O(d, d) invariant if the tensor

Aklm..,k′l′m′..(X) transforms as Aklm..,k′l′m′.. → Ωp

kΩq l Ωr m...Ωp′ k′Ωq′ l′ Ωr′ m′Apqr..,p′q′r′..

• So far we have discussed only those vertex functions which have

internal indices contracting with derivatives of compact coordinates. There are two other possibilities: (a) Set of vertex functions which have only have spacetime coordinates (like ∂X, ¯ ∂X and higher derivatives) contracting with spacetime tensors. Such vertex functions are O(d, d) invariant since T-duality transformations do not affect them. (b) There are set of vertex functions with mixed indices (Lorentz and internal). It the terms like ∂Y , ¯ ∂Y and higher derivatives which need to be converted to W’s and the derivatives to obtain T-duality invariant vertex functions.

• A vertex function of the form

Tµαiαjµ′α′

iα′ j∂mX µ∂pY αi∂qY αj...¯

∂m′X µ′ ¯ ∂p′Y α′

i ¯

∂q′Y α′

j..

can expressed in T-duality invariant manner (note that spacetime vectors.. are T-duality-inert).

32 / 44

T-DUALITY FOR NSR STRING

• We encounter some difficulties when we investigate T-duality for NSR
• string. Consider free NSR string. The σ ↔ τ duality (analog of

P ↔ X ′ for closed bosonic string) is not exhibited since the worldsheet spinor part is not endowed with this symmetry.

33 / 44

T-DUALITY FOR NSR STRING

• We encounter some difficulties when we investigate T-duality for NSR
• string. Consider free NSR string. The σ ↔ τ duality (analog of

P ↔ X ′ for closed bosonic string) is not exhibited since the worldsheet spinor part is not endowed with this symmetry.

• Issues related to T-duality for NSR strings have been addressed in the

past: (Das+JM, Siegel, E. Alvarez, L. Alvarez-Gaume, Lozano, Hassan, Curtright, Uematsu, Zachos, ....).

33 / 44

T-DUALITY FOR NSR STRING

• We encounter some difficulties when we investigate T-duality for NSR
• string. Consider free NSR string. The σ ↔ τ duality (analog of

P ↔ X ′ for closed bosonic string) is not exhibited since the worldsheet spinor part is not endowed with this symmetry.

• Issues related to T-duality for NSR strings have been addressed in the

past: (Das+JM, Siegel, E. Alvarez, L. Alvarez-Gaume, Lozano, Hassan, Curtright, Uematsu, Zachos, ....).

• Basically the gaol is two fold: (i) Consider NSR string in background

ˆ G and ˆ B, compactify on T d and explore the T-duality symmetry. (ii) Whether vertex functions of excited massive levels can be cast in T-duality invariant form.

33 / 44

T-DUALITY FOR NSR STRING

• We encounter some difficulties when we investigate T-duality for NSR
• string. Consider free NSR string. The σ ↔ τ duality (analog of

P ↔ X ′ for closed bosonic string) is not exhibited since the worldsheet spinor part is not endowed with this symmetry.

• Issues related to T-duality for NSR strings have been addressed in the

past: (Das+JM, Siegel, E. Alvarez, L. Alvarez-Gaume, Lozano, Hassan, Curtright, Uematsu, Zachos, ....).

• Basically the gaol is two fold: (i) Consider NSR string in background

ˆ G and ˆ B, compactify on T d and explore the T-duality symmetry. (ii) Whether vertex functions of excited massive levels can be cast in T-duality invariant form.

• The convenient starting point is to consider two dimensional

worldsheet action in superspace in the presence of backgrounds and then compactify on T d.

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• The action in the NS-NS massless background is

S = −1 2

• dσdτd2θD ˆ

Φˆ

µ

• ˆ

Gˆ

µˆ ν(ˆ

Φ) − γ5 ˆ Bˆ

µˆ ν(ˆ

Φ)

• D ˆ

Φˆ

ν

Here ˆ G ˆ

µν(ˆ

Φ) and ˆ B ˆ

µν(ˆ

Φ) are the graviton and and 2-form backgrounds which depend on the superfield ˆ Φ. It has expansion in component fields as ˆ Φˆ

µ = X ˆ µ + ¯

θψˆ

µ + ¯

ψˆ

µθ + 1

2 ¯ θθF ˆ

µ

where X ˆ

µ, ψˆ µ and F ˆ µ are the bosonic, fermionic and auxiliary fields

respectively.

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• The action in the NS-NS massless background is

S = −1 2

• dσdτd2θD ˆ

Φˆ

µ

• ˆ

Gˆ

µˆ ν(ˆ

Φ) − γ5 ˆ Bˆ

µˆ ν(ˆ

Φ)

• D ˆ

Φˆ

ν

Here ˆ G ˆ

µν(ˆ

Φ) and ˆ B ˆ

µν(ˆ

Φ) are the graviton and and 2-form backgrounds which depend on the superfield ˆ Φ. It has expansion in component fields as ˆ Φˆ

µ = X ˆ µ + ¯

θψˆ

µ + ¯

ψˆ

µθ + 1

2 ¯ θθF ˆ

µ

where X ˆ

µ, ψˆ µ and F ˆ µ are the bosonic, fermionic and auxiliary fields

respectively.

• The covariant derivatives in superspace are defined to be

Dα = ∂ ∂¯ θα − i(γaθ)α∂a, Dα = − ∂ ∂θα + i(¯ θγa)α∂a where ∂a stands for worldsheet derivatives (σ and τ) and the convention for γ matrices are γ0 = 1 1

• , γ1 =

−1 1

• , γ5 = γ0γ1 =

1 −1

• 34 / 44

• The equations of motion are

D

• ˆ

Gˆ

µˆ ν(ˆ

Φ) − γ5 ˆ Gˆ

µˆ ν(ˆ

Φ)

• D ˆ

Φˆ

ν = 0

Let us adopt Hassan-Sen compactification scheme ˆ Gˆ

µˆ ν =

gµν(φ) Gij(φ)

• , ˆ

Bˆ

µˆ ν =

Bµν(φ) Bij(φ)

• 35 / 44

• The equations of motion are

D

• ˆ

Gˆ

µˆ ν(ˆ

Φ) − γ5 ˆ Gˆ

µˆ ν(ˆ

Φ)

• D ˆ

Φˆ

ν = 0

Let us adopt Hassan-Sen compactification scheme ˆ Gˆ

µˆ ν =

gµν(φ) Gij(φ)

• , ˆ

Bˆ

µˆ ν =

Bµν(φ) Bij(φ)

• Note: backgrounds depend only on spacetime superfields, φµ. We

decompose the superfields: ˆ Φˆ

µ = (φµ, W i) where

µ, ν = 0, 1, 2..D − 1 and i, j = 1, 2, ..d with ˆ D = D + d. The two superfields can be expanded as φµ = X µ + ¯ θψµ + ¯ ψµθ + 1 2 ¯ θθF µ and W i = Y i + ¯ θχi + ¯ χiθ + 1 2 ¯ θθF i

35 / 44

• Consider equation of motion of superfields along compact directions.

The action S = −1 2

• dσdτd2θDW i
• Gij(φ) − γ5Bij(φ)
• DW j

The equations of motion are D

• Gij(φ) − γ5Bij(φ)
• DW j
• = 0

This is just a conservation law since backgrounds depend on φµ. We may introduce a free dual superfield, Wi, satisfying following equation locally

• Gij(φ) − γ5Bij(φ)
• DW j = D

Wi satisfying the constraints: DD Wi = 0.

36 / 44

• Consider equation of motion of superfields along compact directions.

The action S = −1 2

• dσdτd2θDW i
• Gij(φ) − γ5Bij(φ)
• DW j

The equations of motion are D

• Gij(φ) − γ5Bij(φ)
• DW j
• = 0

This is just a conservation law since backgrounds depend on φµ. We may introduce a free dual superfield, Wi, satisfying following equation locally

• Gij(φ) − γ5Bij(φ)
• DW j = D

Wi satisfying the constraints: DD Wi = 0.

• This is analog of dual coordinate introduced in case of closed

compactified bosonic string. Introduce a Lagrangian density in first

• rder formalism
• L = 1

2 ¯ Σi

• Gij(φ) − γ5Bij(φ)
• Σj − ¯

ΣiD Wi

36 / 44

• The ¯

Σi variation leads to

• Gij(φ) − γ5Bij(φ)
• Σj = D

Wi and Wi variation implies D ¯ Σi = 0. Therefore, when Σi = DW i we recover original equation.

37 / 44

• The ¯

Σi variation leads to

• Gij(φ) − γ5Bij(φ)
• Σj = D

Wi and Wi variation implies D ¯ Σi = 0. Therefore, when Σi = DW i we recover original equation.

• Introduce a dual Lagrangian density in terms of the dual superfields,
• Wi and a set of dual backgrounds Gij(φ) and Bij(φ); whereas the

former of the two backgrounds is symmetric in its indices the latter is antisymmetric. Lf

W = −1

2D Wi

• Gij(φ) − γ5Bij(φ)
• D

Wj

37 / 44

• The ¯

Σi variation leads to

• Gij(φ) − γ5Bij(φ)
• Σj = D

Wi and Wi variation implies D ¯ Σi = 0. Therefore, when Σi = DW i we recover original equation.

• Introduce a dual Lagrangian density in terms of the dual superfields,
• Wi and a set of dual backgrounds Gij(φ) and Bij(φ); whereas the

former of the two backgrounds is symmetric in its indices the latter is antisymmetric. Lf

W = −1

2D Wi

• Gij(φ) − γ5Bij(φ)
• D

Wj

• The two dual backgrounds, (G, B), are related to the original

background fields, (G, B) as follows: G =

• G − BG −1B

−1 and B = −

• G − BG −1B

−1 BG −1

37 / 44

• Note that G is symmetric and B is antisymmetric (the combination

(G − BG −1B) is symmetric). The equation of motion resulting from dual Lagrangian is D

• G(φ) − γ5B(φ)
• D

W

• = 0

Note: We may identify W as the dual superfield of W from here.

38 / 44

• Note that G is symmetric and B is antisymmetric (the combination

(G − BG −1B) is symmetric). The equation of motion resulting from dual Lagrangian is D

• G(φ) − γ5B(φ)
• D

W

• = 0

Note: We may identify W as the dual superfield of W from here.

• Strategy: combine the equations of motion from original Lagrangian

and the dual Lagrangian in such a way that combined equation is in O(d, d) covariant form (both the equations motion are conservation laws). The two equations are DW i = γ5(G −1B)i

jDW j + G −1ijD

Wj D Wi = γ5(G−1B)j

iD

Wj + G−1

ij DW j

38 / 44

• Note that G is symmetric and B is antisymmetric (the combination

(G − BG −1B) is symmetric). The equation of motion resulting from dual Lagrangian is D

• G(φ) − γ5B(φ)
• D

W

• = 0

Note: We may identify W as the dual superfield of W from here.

• Strategy: combine the equations of motion from original Lagrangian

and the dual Lagrangian in such a way that combined equation is in O(d, d) covariant form (both the equations motion are conservation laws). The two equations are DW i = γ5(G −1B)i

jDW j + G −1ijD

Wj D Wi = γ5(G−1B)j

iD

Wj + G−1

ij DW j

• Define an O(d, d) vector

U = W i

• Wi
• 38 / 44

• Define M-matrix similar to one we know of

M =

• 1G −1

γ5G −1B −γ5BG −1 1G − 1BG −1B

• 39 / 44

• Define M-matrix similar to one we know of

M =

• 1G −1

γ5G −1B −γ5BG −1 1G − 1BG −1B

• Here 1 is the 2 × 2 unit matrix and γ5 is two dimensional diagonal

matrix defined earlier. The M matrix has properties of the familiar M-matrix introduced in dimensional reduction of closed bosonic string: M ∈ O(d, d) and corresponding metric is η. The dimensions are further doubled due to the presence of two component Majorana fermions and is reflected by the appearance of 1 and γ5 in the M-matrix.

39 / 44

• The two equations can be combined to a form

DU = MηU It follows from the definition of the O(d, d) vector U that DDU = 0. It holds by virtue of the fact that the two components of U satisfy DDW i = 0 and DD Wi = 0 from our original equations (they are dual superfields of each other).

40 / 44

• The two equations can be combined to a form

DU = MηU It follows from the definition of the O(d, d) vector U that DDU = 0. It holds by virtue of the fact that the two components of U satisfy DDW i = 0 and DD Wi = 0 from our original equations (they are dual superfields of each other).

• Therefore, we arrive at an O(d, d) covariant equations of motion for

coordinates along compact directions D

• MηU
• = 0

40 / 44

• The two equations can be combined to a form

DU = MηU It follows from the definition of the O(d, d) vector U that DDU = 0. It holds by virtue of the fact that the two components of U satisfy DDW i = 0 and DD Wi = 0 from our original equations (they are dual superfields of each other).

• Therefore, we arrive at an O(d, d) covariant equations of motion for

coordinates along compact directions D

• MηU
• = 0
• This generalizes the closed string O(d, d) covariant equations of

motion to NSR superstring.

• The equations of motion associated with noncompact coordinates are

T-duality invariant since the coordinates and the backgrounds are inert under the duality transformations.

40 / 44

• With our construction of T-duality invariant vertex functions for

closed strings, we can study the case of NSR string.

41 / 44

• With our construction of T-duality invariant vertex functions for

closed strings, we can study the case of NSR string.

• There are two types of generic vertex functions:

(i) DW i1DW i2...DW imDW j1DW j2...DW jm.These correspond to leading Regge trajectories. (ii) D

pW i1D qW i2....D rW imDp′W j1Dq′W j2...Dr′W jm and we require

p + q + r = p′ + q′ + r′.

41 / 44

• With our construction of T-duality invariant vertex functions for

closed strings, we can study the case of NSR string.

• There are two types of generic vertex functions:

(i) DW i1DW i2...DW imDW j1DW j2...DW jm.These correspond to leading Regge trajectories. (ii) D

pW i1D qW i2....D rW imDp′W j1Dq′W j2...Dr′W jm and we require

p + q + r = p′ + q′ + r′.

• In analogy with bosonic string for case (i) U is an O(d, d) vector with

W i and Wi as upper and lower components. As before introduce two projection operators

• P+ =

1

• P− =

1

• 41 / 44

• We can write DW i1, DW j1... as projections of U . Moreover, as in

the bosonic case, define a doublet operators D = D D

• Introduce projectors
• ∆+ =

1

• ∆− =

1

• 42 / 44

• We can write DW i1, DW j1... as projections of U . Moreover, as in

the bosonic case, define a doublet operators D = D D

• Introduce projectors
• ∆+ =

1

• ∆− =

1

• Thus vertex functions having form (i) can expressed as
• ∆−D

P+Uα1.... ∆−D P+Uαm ∆+D P+Uβ1.... ∆+D P+Uβm

42 / 44

• We can write DW i1, DW j1... as projections of U . Moreover, as in

the bosonic case, define a doublet operators D = D D

• Introduce projectors
• ∆+ =

1

• ∆− =

1

• Thus vertex functions having form (i) can expressed as
• ∆−D

P+Uα1.... ∆−D P+Uαm ∆+D P+Uβ1.... ∆+D P+Uβm

• The O(d, d) invariant vertex function for nth massive level is

Vn+1 = Tα1..αmβ1..βm ∆−D P+Uα1.... ∆−D P+Uαm ∆+D P+Uβ1.... ∆+D P+U

42 / 44

• Invariance is assured if the O(d, d) vector transforms as:

Uα1 → Ωα1

α′

1Uα′ 1 and

Tα1,..αmβ1..βm → Ωα′

1

α1..Ωα′

m

αmΩβ′

1

β1...Ωβ′

m

βmTα′

1..α′ mβ′ 1...β′ m

• Consider the case (ii) above where higher order super-derivatives like

(D)p and (D)p′ appear. We can convert these type of terms to desired forms using the projection operators already at our disposal

• ∆p

−

P+Uα1 ∆q

−

P+Uα2.. ∆r

−

P+Uαm ∆p′

+

P+Uβ1 ∆p′

+

P+Uβ2.. ∆p′

+

P+Uβm

• Note that this is product of O(d, d) vectors and is a tensor of rank

p + q + r satisfying the level matching condition. The vertex function can be obtained by contracting with an appropriate tensor.

• The case vertex functions having mixed indices can be dealt with

analogous to bosonic string. Those with only spacetime indices are inert under T-duality.

43 / 44

CONCLUSIONS

• We considered closed bosonic string compactified on T d and

spacetime metric is flat.

44 / 44

CONCLUSIONS

• We considered closed bosonic string compactified on T d and

spacetime metric is flat.

• We constructed vertex operators for excited massive levels. A simple

example, for first massive level, exhibited how Z2 type T-duality appears.

44 / 44

CONCLUSIONS

• We considered closed bosonic string compactified on T d and

spacetime metric is flat.

• We constructed vertex operators for excited massive levels. A simple

example, for first massive level, exhibited how Z2 type T-duality appears.

• A method is proposed to construct O(d, d) invariant vertex functions

for excited massive levels of closed string.

44 / 44

CONCLUSIONS

• We considered closed bosonic string compactified on T d and

spacetime metric is flat.

• We constructed vertex operators for excited massive levels. A simple

example, for first massive level, exhibited how Z2 type T-duality appears.

• A method is proposed to construct O(d, d) invariant vertex functions

for excited massive levels of closed string.

• T-duality properties of NSR string are investigated in the presence of

its NS-NS massless excitations. It was shown, when it is compactified

• n T d, the equations of motion can be expressed in an O(d, d)

covariant form when one introduces dual coordinates (superfields) along compact directions.

44 / 44

CONCLUSIONS

• We considered closed bosonic string compactified on T d and

spacetime metric is flat.

• We constructed vertex operators for excited massive levels. A simple

example, for first massive level, exhibited how Z2 type T-duality appears.

• A method is proposed to construct O(d, d) invariant vertex functions

for excited massive levels of closed string.

• T-duality properties of NSR string are investigated in the presence of

its NS-NS massless excitations. It was shown, when it is compactified

• n T d, the equations of motion can be expressed in an O(d, d)

covariant form when one introduces dual coordinates (superfields) along compact directions.

• The excited massive level vertex operators for compactified string in

the NS-NS sector can be expressed in O(d, d) invariant form.

44 / 44

CONCLUSIONS

• We considered closed bosonic string compactified on T d and

spacetime metric is flat.

• We constructed vertex operators for excited massive levels. A simple

example, for first massive level, exhibited how Z2 type T-duality appears.

• A method is proposed to construct O(d, d) invariant vertex functions

for excited massive levels of closed string.

• T-duality properties of NSR string are investigated in the presence of

its NS-NS massless excitations. It was shown, when it is compactified

• n T d, the equations of motion can be expressed in an O(d, d)

covariant form when one introduces dual coordinates (superfields) along compact directions.

• The excited massive level vertex operators for compactified string in

the NS-NS sector can be expressed in O(d, d) invariant form.

• Explicit examples can be studied for type IIB compactified on

M4 ⊗ S3 ⊗ T 4.

44 / 44