Duality in String Theory

Home , Brane, Dilaton, E8 (mathematics), F4 (mathematics), String theory, Worldsheet

Yolanda Lozano (U. Oviedo)

2012 International Fall Workshop on Geometry and Physics, Burgos, August 2012

miércoles 29 de agosto de 2012 Duality: New insight into the non-perturbative regimes of both String Theory and QFT

In String Theory: T- and S-dualities

Witten’95

M-theory: Fully non-perturbative formulation of superstring theory. Uniﬁes gravity with particle physics interactions in a unique way.

miércoles 29 de agosto de 2012 Deﬁne M-theory as the quantum theory whose low energy limit is eleven dim supergravity

miércoles 29 de agosto de 2012 Deﬁne M-theory as the quantum theory whose low energy limit is eleven dim supergravity Theory of membranes →

miércoles 29 de agosto de 2012 Deﬁne M-theory as the quantum theory whose low energy limit is eleven dim supergravity Theory of membranes → A perturbative expansion cannot however be deﬁned from the SUGRA (no coupling constant) M-theory not known ⇒ (see M.P. García del Moral’s talk)

Gauge/gravity duality (Maldacena’97): Some gauge theories at strong (weak) coupling are dual to weakly (strongly) coupled string theories with one extra dimension

Gauge/gravity duality (Maldacena’97): Some gauge theories at strong (weak) coupling are dual to weakly (strongly) coupled string theories with one extra dimension Holographic duality

miércoles 29 de agosto de 2012 One of the most innovative ideas in Theoretical Physics in the recent past:

Possibility to apply string theory techniques to strongly coupled systems (quark-gluon plasma, condensed matter systems). Possibility to learn something about quantum gravity by studying low dimensional systems with a holographic description (concrete realization of QG and space-time as emergent phenomena).

miércoles 29 de agosto de 2012 1. Duality relations

miércoles 29 de agosto de 2012 1. Duality relations T-duality: Perturbative transformation in String Theory. Relates large and small distances (space-time duality)

In toroidal compactiﬁcations: x x +2πnR →

Momentum states: Winding states: m nR p = p = R α

2 α = l ; ls : string length ( ls 0 Field Theory limit) s → miércoles 29 de agosto de 2012 α R Symmetry of the bosonic string, maps Type IIA and → R : m n Type IIB and Het SO(32) and Het E8 E8 ↔ ×

miércoles 29 de agosto de 2012 α R Symmetry of the bosonic string, maps Type IIA and → R : m n Type IIB and Het SO(32) and Het E8 E8 ↔ ×

T T

miércoles 29 de agosto de 2012 α R Symmetry of the bosonic string, maps Type IIA and → R : m n Type IIB and Het SO(32) and Het E8 E8 ↔ ×

√α d g g st → R st g (a) (b) gst : String coupling constant

φ gst = e ; φ = dilaton (massless ﬁeld) g

(c)(d)

miércoles 29 de agosto de 2012 α R Symmetry of the bosonic string, maps Type IIA and → R : m n Type IIB and Het SO(32) and Het E8 E8 ↔ ×

√α d g g st → R st g (a) (b) gst : String coupling constant

φ gst = e ; φ = dilaton (massless ﬁeld) g

(c)(d) For more general compactiﬁcations: Abelian and non-Abelian T-duality (see later)

miércoles 29 de agosto de 2012 S-duality: Non-perturbative transformation in String and Field Theory. Relates the strong and weak coupling regimes. Interchanges electric (elementary states) and magnetic (solitons)

In a four dimensional Abelian Gauge Theory: ( g = coupling constant) 1 E B g + → Symmetry → g B E → →−

In a four dimensional Abelian Gauge Theory: ( g = coupling constant) 1 E B g + → Symmetry → g B E → →− In the presence of a θ term i θ F F : ∧ θ i 1 τ = + transforms as τ 2π g2 →−τ Together with τ τ +1 they generate the SL(2, Z) → S-duality group miércoles 29 de agosto de 2012 Also a symmetry of N=4 SYM in 4 dim (Montonen & Olive’s conjecture)

miércoles 29 de agosto de 2012 Also a symmetry of N=4 SYM in 4 dim (Montonen & Olive’s conjecture)

To test a non-perturbative duality relation:

miércoles 29 de agosto de 2012 Also a symmetry of N=4 SYM in 4 dim (Montonen & Olive’s conjecture)

To test a non-perturbative duality relation: Look at the BPS states (special class of supersymmetric states with the key property that their mass is completely determined by their charge) Stable and protected from quantum ⇒ corrections

miércoles 29 de agosto de 2012 Also a symmetry of N=4 SYM in 4 dim (Montonen & Olive’s conjecture)

i) Study their properties perturbatively at weak coupling in one theory ii) Extrapolate (safely) to strong coupling iii) Reinterpret in terms of non-perturbative conﬁgurations in the dual theory

miércoles 29 de agosto de 2012 In String Theory: S-duality transformation:

1 B2 C2 gst + → → gst C B 2 →− 2 ( B2 : NS-NS 2-form, C2 : RR 2-form) (generalized Maxwell ﬁelds with 2 antisym. indices, part of the massless spectrum)

miércoles 29 de agosto de 2012 In String Theory: S-duality transformation:

1 B2 C2 gst + → → gst C B 2 →− 2 ( B2 : NS-NS 2-form, C2 : RR 2-form) (generalized Maxwell ﬁelds with 2 antisym. indices, part of the massless spectrum)

In the presence of a C0 RR ﬁeld: i 1 λ = C0 + transforms as λ gst →−λ Together with λ λ +1 they generate the SL(2, Z) → S-duality group

miércoles 29 de agosto de 2012 Symmetry of Type IIB, maps Type I with Heterotic SO(32). Type IIA and Heterotic E E give different compactiﬁcations 8 × 8 of M-theory in the strong coupling limit

1 S1 S /Z2

T T S

miércoles 29 de agosto de 2012 Symmetry of Type IIB, maps Type I with Heterotic SO(32). Type IIA and Heterotic E E give different compactiﬁcations 8 × 8 of M-theory in the strong coupling limit

1 S1 S /Z2

T T S Ω

miércoles 29 de agosto de 2012 Symmetry of Type IIB, maps Type I with Heterotic SO(32). Type IIA and Heterotic E E give different compactiﬁcations 8 × 8 of M-theory in the strong coupling limit

1 S1 S /Z2

T T S Ω

To get deeper into these dualities in String Theory:

miércoles 29 de agosto de 2012 2. Some basic facts about String Theory

miércoles 29 de agosto de 2012 2. Some basic facts about String Theory Type II theories:

B2 , Cp ( p odd IIA, p even IIB ) NS-NS, RR

miércoles 29 de agosto de 2012 2. Some basic facts about String Theory Type II theories:

B2 , Cp ( p odd IIA, p even IIB ) NS-NS, RR

The (fundamental) string is an electric source for B2 . The object that couples magnetically is called and NS5-brane: dB = dB ∗ 2 6

miércoles 29 de agosto de 2012 2. Some basic facts about String Theory Type II theories:

B2 , Cp ( p odd IIA, p even IIB ) NS-NS, RR

miércoles 29 de agosto de 2012 2. Some basic facts about String Theory Type II theories:

B2 , Cp ( p odd IIA, p even IIB ) NS-NS, RR

The (fundamental) string is an electric source for B2 . The object that couples magnetically is called and NS5-brane: dB = dB ∗ 2 6 A class of p-branes, called Dirichlet branes (Dp-branes) are electric sources for Cp+1 All these branes are BPS objects, satisfying q = T : 1 1 1 1 1 TF 1 = 2 ; TDp = p+1 ; TNS5 = 6 2 ls ls gst ls gst (non-perturbative) miércoles 29 de agosto de 2012 They all occur as classical solitonic solutions of the low energy ( ls 0 ) effective actions (SUGRAs) →

miércoles 29 de agosto de 2012 They all occur as classical solitonic solutions of the low energy ( ls 0 ) effective actions (SUGRAs) → Crucial discovery (Polchinski’95): RR charged p-branes = Dirichlet p-branes: (p+1) dimensional hypersurfaces on which open strings can end (previously predicted by T-duality):

D0D1 D2

The important implication is that their dynamics can now be determined at weak coupling using open string perturbation theory!

miércoles 29 de agosto de 2012 For ex. conformal invariance of the worldsheet theory ↔ Equations of motion of Seﬀ = Sbulk + SD brane with − SD brane a U(1) gauge theory (U(N) for a system of N − D-branes)

Type I: C2 ,C6 (D1, D5 branes)

Heterotic: B2 ,B6 (F1, NS5 branes)

Type I: C2 ,C6 (D1, D5 branes)

Heterotic: B2 ,B6 (F1, NS5 branes)

Type I and Heterotic SO(32) contain as well an SO(32) N=1 SYM vector multiplet ( E8 E8 for Heterotic E8 E8 ) × ×

miércoles 29 de agosto de 2012 Now we can go back to S-duality:

1 B2 C2 gst + → → gst C B 2 →− 2

miércoles 29 de agosto de 2012 Now we can go back to S-duality:

1 B2 C2 gst + → → gst C B 2 →− 2 Is a symmetry of Type IIB (F 1 D1; NS5 D5) ↔ ↔

miércoles 29 de agosto de 2012 Now we can go back to S-duality:

1 B2 C2 gst + → → gst C B 2 →− 2 Is a symmetry of Type IIB (F 1 D1; NS5 D5) ↔ ↔ It relates Type I and Heterotic SO(32):

C2 B2 D1 F1 → D5 NS5

miércoles 29 de agosto de 2012 Now we can go back to S-duality:

1 B2 C2 gst + → → gst C B 2 →− 2 Is a symmetry of Type IIB (F 1 D1; NS5 D5) ↔ ↔ It relates Type I and Heterotic SO(32):

C2 B2 D1 F1 → D5 NS5

The strong coupling limit of Type IIA and Heterotic E E 8 × 8 is M-theory (more difﬁcult to see..).

miércoles 29 de agosto de 2012 3. The gauge/gravity duality

Maldacena’97:

Equivalence between Type IIB string theory on AdS S5 5 × and 4 dim =4 supersymmetric SU(N) Yang-Mills theory N

AdS5 : anti-de Sitter space in 5 dim (maximally symmetric solution of the Einstein eqs. with a negative cosmological constant)

miércoles 29 de agosto de 2012 3. The gauge/gravity duality

Maldacena’97:

Equivalence between Type IIB string theory on AdS S5 5 × and 4 dim =4 supersymmetric SU(N) Yang-Mills theory N

AdS5 : anti-de Sitter space in 5 dim (maximally symmetric solution of the Einstein eqs. with a negative cosmological constant) First piece of evidence: The symmetry group of 5 dim anti-de Sitter space matches precisely with the group of conformal symmetries of =4 SYM AdS/CFT N ⇒

miércoles 29 de agosto de 2012 Starting point: Study of N coincident D3-branes in Type IIB in the large N limit, based on two dual descriptions: - As solution of the classical eqs. of motion of ST/SUGRA - As a D-brane system

5 AdS5 S emerges as the near horizon geometry of the × 4 4 D3-brane solution. The radius is given by R =4πgsNls

5 AdS5 S emerges as the near horizon geometry of the × 4 4 D3-brane solution. The radius is given by R =4πgsNls The U(N) gauge theory living on the branes becomes 4 dim 2 =4SYM, with gYM = gs : N Conformally invariant ∞ 2 2g Topological large N expansion N − fg(λ) with ‘t Hooft 2 g=0 parameter λ = gYMN

4 4 Through the correspondence: R =4πλ ls

miércoles 29 de agosto de 2012 4 4 λ = gsNR=4πλ ls imply

N , ﬁnite λ gs 0 Planar limit Classical →∞ ⇔ → ⇒ ↔ limit of IIB ST

miércoles 29 de agosto de 2012 4 4 λ = gsNR=4πλ ls imply

N , ﬁnite λ gs 0 Planar limit Classical →∞ ⇔ → ⇒ ↔ limit of IIB ST

λ ls 0 Strong ‘t Hooft coupling limit →∞ ⇔ → ⇒ ↔ SUGRA limit

miércoles 29 de agosto de 2012 4 4 λ = gsNR=4πλ ls imply

N , ﬁnite λ gs 0 Planar limit Classical →∞ ⇔ → ⇒ ↔ limit of IIB ST

λ ls 0 Strong ‘t Hooft coupling limit →∞ ⇔ → ⇒ ↔ SUGRA limit Strong-weak coupling duality ⇒

miércoles 29 de agosto de 2012 4 4 λ = gsNR=4πλ ls imply

N , ﬁnite λ gs 0 Planar limit Classical →∞ ⇔ → ⇒ ↔ limit of IIB ST

λ ls 0 Strong ‘t Hooft coupling limit →∞ ⇔ → ⇒ ↔ SUGRA limit Strong-weak coupling duality ⇒

Many tests: For each ﬁeld in the 5 dim bulk Operator in the dual CFT ↔ D-branes Baryons, domain walls, etc.. ↔ ...

miércoles 29 de agosto de 2012 Less supersymmetric extensions Type IIB on AdS Y with Y an Einstein manifold bearing 5 × 5 5 5-form ﬂux:

2 2 2 2 Y5 C6 with C Ricci-ﬂat such that ds = dr + r ds ↔ 6 C6 Y5 SUSY preserved Y5 is Sasaki-Einstein ( C6 is Ricci-ﬂat ⇔ ⇔ and Kähler (Calabi-Yau 3-fold))

miércoles 29 de agosto de 2012 Less supersymmetric extensions Type IIB on AdS Y with Y an Einstein manifold bearing 5 × 5 5 5-form ﬂux:

2 2 2 2 Y5 C6 with C Ricci-ﬂat such that ds = dr + r ds ↔ 6 C6 Y5 SUSY preserved Y5 is Sasaki-Einstein ( C6 is Ricci-ﬂat ⇔ ⇔ and Kähler (Calabi-Yau 3-fold))

5 6 Examples: Y5 = S , C6 = C

1,1 SU(2) SU(2) Y = T = × , C6 is the conifold 5 U(1) p,q Y5 = Y ,p,q coprime integers (includes quasi- regular and irregular SE manifolds)

miércoles 29 de agosto de 2012 Less supersymmetric extensions Type IIB on AdS Y with Y an Einstein manifold bearing 5 × 5 5 5-form ﬂux:

2 2 2 2 Y5 C6 with C Ricci-ﬂat such that ds = dr + r ds ↔ 6 C6 Y5 SUSY preserved Y5 is Sasaki-Einstein ( C6 is Ricci-ﬂat ⇔ ⇔ and Kähler (Calabi-Yau 3-fold))

5 6 Examples: Y5 = S , C6 = C

1,1 SU(2) SU(2) Y = T = × , C6 is the conifold 5 U(1) p,q Y5 = Y ,p,q coprime integers (includes quasi- regular and irregular SE manifolds) Dual to N=1 SCFTs arising from a stack of D3-branes ↔ sitting at the tip of the CY cone

miércoles 29 de agosto de 2012 4. More on T-duality

4.1. D-branes from open strings 4.2. T-duality for more general backgrounds 4.3. Non-Abelian T-duality

miércoles 29 de agosto de 2012 4. More on T-duality

4.1. D-branes from open strings 4.2. T-duality for more general backgrounds 4.3. Non-Abelian T-duality

4.1. D-branes from open strings Closed strings in a toroidal compactiﬁcation:

i Solution of the equations of motion: ∂+∂ X =0 + − boundary condition Xi(τ, σ +2π)=Xi(τ, σ)+2πnR :

i i m X (τ, σ)=X + α τ + nRσ + .. 0 R

miércoles 29 de agosto de 2012 α i i T-duality transformation: R ∂+X ∂+X → R → i i m n ⇔ ∂ X ∂ X ↔ − →− − (d d : Hodge duality) →∗

miércoles 29 de agosto de 2012 α i i T-duality transformation: R ∂+X ∂+X → R → i i m n ⇔ ∂ X ∂ X ↔ − →− − (d d : Hodge duality) For open strings: σ [0, π] →∗ ∈

Boundary: m2 ∂Σ = σ = constant { }

/ 0 m1

µ Neumann boundary conditions: ∂σX ∂Σ =0 | (zero momentum ﬂux at the end points)

miércoles 29 de agosto de 2012 α i i T-duality transformation: R ∂+X ∂+X → R → i i m n ⇔ ∂ X ∂ X ↔ − →− − (d d : Hodge duality) For open strings: σ [0, π] →∗ ∈

Boundary: m2 ∂Σ = σ = constant { }

/ 0 m1

µ Neumann boundary conditions: ∂σX ∂Σ =0 | (zero momentum ﬂux at the end points)

Dirichlet boundary conditions: X˙ i =0; i = p +1,...,9 |∂Σ

miércoles 29 de agosto de 2012 Xµ = Xm,Xi m =0,...,p non-compact { } The end-points of the strings are ﬁxed on a (p+1) dimensional hypersurface Dp-brane ≡

D0D1 D2

miércoles 29 de agosto de 2012 Xµ = Xm,Xi m =0,...,p non-compact { } The end-points of the strings are ﬁxed on a (p+1) dimensional hypersurface Dp-brane ≡

D0D1 D2

Open strings T-duality Dp-branes

miércoles 29 de agosto de 2012 Xµ = Xm,Xi m =0,...,p non-compact { } The end-points of the strings are ﬁxed on a (p+1) dimensional hypersurface Dp-brane ≡

D0D1 D2

Open strings T-duality Dp-branes

Dynamical due to the open strings attached

miércoles 29 de agosto de 2012 4.2. T-duality for more general backgrounds

(Buscher’88; Rocek, Verlinde’92)

The fundamental string couples to the metric, NS-NS 2-form and dilaton through a non-linear sigma-model:

1 µ ν µ ν 1 (2) S = gµν dX dX + Bµν dX dX + R φ 4πα ∧∗ ∧ 4π

miércoles 29 de agosto de 2012 4.2. T-duality for more general backgrounds

(Buscher’88; Rocek, Verlinde’92)

The fundamental string couples to the metric, NS-NS 2-form and dilaton through a non-linear sigma-model:

1 µ ν µ ν 1 (2) S = gµν dX dX + Bµν dX dX + R φ 4πα ∧∗ ∧ 4π

In the presence of an Abelian isometry: δXµ = kµ / g =0, B = dω ,i dφ =0 Lk Lk k

miércoles 29 de agosto de 2012 4.2. T-duality for more general backgrounds

(Buscher’88; Rocek, Verlinde’92)

The fundamental string couples to the metric, NS-NS 2-form and dilaton through a non-linear sigma-model:

1 µ ν µ ν 1 (2) S = gµν dX dX + Bµν dX dX + R φ 4πα ∧∗ ∧ 4π

In the presence of an Abelian isometry: δXµ = kµ / g =0, B = dω ,i dφ =0 Lk Lk k i) Go to adapted coordinates: Xµ = θ,Xα such that { } θ θ + and ∂ (backgrounds) = 0 → θ

miércoles 29 de agosto de 2012 ii) Gauge the isometry: dθ Dθ = dθ + A → A non-dynamical gauge ﬁeld / δA = d −

iii) Add a Lagrange multiplier term: θ˜dA , such that

θ˜ dA =0 A exact D → ⇒ (in a topologically trivial worldsheet) + ﬁx the gauge: A =0 Original theory →

miércoles 29 de agosto de 2012 ii) Gauge the isometry: dθ Dθ = dθ + A → A non-dynamical gauge ﬁeld / δA = d −

iii) Add a Lagrange multiplier term: θ˜dA , such that

θ˜ dA =0 A exact D → ⇒ (in a topologically trivial worldsheet) + ﬁx the gauge: A =0 Original theory →

iv) Integrate the gauge ﬁeld + ﬁx the gauge: θ =0 Dual sigma model: → θ,Xα θ˜,Xα and { } → { }

miércoles 29 de agosto de 2012 1 B0α g0αg0β B0αB0β g˜00 = ;˜g0α = ;˜gαβ = gαβ − g00 g00 − g00

g0α g0αB0β g0βB0α B˜0α = ; B˜αβ = Bαβ − g00 − g00

φ˜ = φ log g Buscher’s formulae − 00

miércoles 29 de agosto de 2012 1 B0α g0αg0β B0αB0β g˜00 = ;˜g0α = ;˜gαβ = gαβ − g00 g00 − g00

g0α g0αB0β g0βB0α B˜0α = ; B˜αβ = Bαβ − g00 − g00

φ˜ = φ log g Buscher’s formulae − 00

2 - Conformally invariant (to ﬁrst order in ls )

miércoles 29 de agosto de 2012 1 B0α g0αg0β B0αB0β g˜00 = ;˜g0α = ;˜gαβ = gαβ − g00 g00 − g00

g0α g0αB0β g0βB0α B˜0α = ; B˜αβ = Bαβ − g00 − g00

φ˜ = φ log g Buscher’s formulae − 00

2 - Conformally invariant (to ﬁrst order in ls ) - Involutive transformation: S˜ S θ˜ θ˜ + →

miércoles 29 de agosto de 2012 1 B0α g0αg0β B0αB0β g˜00 = ;˜g0α = ;˜gαβ = gαβ − g00 g00 − g00

g0α g0αB0β g0βB0α B˜0α = ; B˜αβ = Bαβ − g00 − g00

φ˜ = φ log g Buscher’s formulae − 00

2 - Conformally invariant (to ﬁrst order in ls ) - Involutive transformation: S˜ S θ˜ θ˜ + → - (M,M˜ ) different geometries and topologies For ex. M = S3 ; M˜ = S2 S1 × (Alvarez, Alvarez-Gaumé, Barbón, Y.L.’93)

miércoles 29 de agosto de 2012 - Arbitrary worldsheets? (symmetry of string perturbation theory):

(a)(b) Non-trivial topologies + compact isometry orbits ⇒

miércoles 29 de agosto de 2012 - Arbitrary worldsheets? (symmetry of string perturbation theory):

(a)(b) Non-trivial topologies + compact isometry orbits ⇒

Large gauge transformations: d =2πn ; n Z ∈ γ

miércoles 29 de agosto de 2012 - Arbitrary worldsheets? (symmetry of string perturbation theory):

(a)(b) Non-trivial topologies + compact isometry orbits ⇒

Large gauge transformations: d =2πn ; n Z γ ∈ To ﬁx them:

miércoles 29 de agosto de 2012 - Arbitrary worldsheets? (symmetry of string perturbation theory):

(a)(b) Non-trivial topologies + compact isometry orbits ⇒

Large gauge transformations: d =2πn ; n Z γ ∈ To ﬁx them: Multivalued Lagrange multiplier: dθ˜ =2πm ; m Z ∈ such that γ

[exact] dA =0 + [harmonic] A exact → ⇒

miércoles 29 de agosto de 2012 - Arbitrary worldsheets? (symmetry of string perturbation theory):

(a)(b) Non-trivial topologies + compact isometry orbits ⇒

Large gauge transformations: d =2πn ; n Z γ ∈ To ﬁx them: Multivalued Lagrange multiplier: dθ˜ =2πm ; m Z ∈ such that γ

[exact] dA =0 + [harmonic] A exact → ⇒ The gauging procedure works for all genera ⇒ (Alvarez, Alvarez-Gaumé, Barbón, Y.L.’93)

miércoles 29 de agosto de 2012 In fact, going to phase space: L(θ, θ˙,Xα) H(θ,P ,Xα) → θ

miércoles 29 de agosto de 2012 In fact, going to phase space: L(θ, θ˙,Xα) H(θ,P ,Xα) → θ ˜ The canonical transformation from θ,Pθ θ,P˜ : { } → { θ} P = θ˜ θ − generates the dual background! P˜ = θ θ − (Alvarez, Alvarez-Gaumé, Y.L.´94)

`Minimal´approach to Abelian T-duality ⇒

`Minimal´approach to Abelian T-duality ⇒ In this set-up: ∂ Xi ∂ Xi + → + is generalized to ∂ Xi ∂ Xi − →− − ∂ θ˜ = g ∂ θ +(g B ) ∂ Xα + 00 + 0α − 0α + α ∂ θ˜ = (g00 ∂ θ +(g0α + B0α) ∂ X ) − − − −

miércoles 29 de agosto de 2012

For the open strings: Neumann boundary conditions are generalized to

α α g θ + g X B X˙ =0 00 0α − 0α β β g θ + g X + B θ˙ B X˙ =0 0α αβ 0α − αβ

miércoles 29 de agosto de 2012

For the open strings: Neumann boundary conditions are generalized to

α α g θ + g X B X˙ =0 00 0α − 0α β β g θ + g X + B θ˙ B X˙ =0 0α αβ 0α − αβ and they are mapped to ˙ θ˜ =0 β ˙ β g˜ θ˜ +˜g X + B˜ θ˜ B˜ X˙ =0 0α αβ 0α − αβ

miércoles 29 de agosto de 2012

For the open strings: Neumann boundary conditions are generalized to

θ˜ : Dirichlet ⇒ D-branes Xα : Neumann ⇒ (Borlaf, Y.L.’96)

miércoles 29 de agosto de 2012

For the open strings: Neumann boundary conditions are generalized to

θ˜ : Dirichlet ⇒ D-branes Xα : Neumann ⇒ (Borlaf, Y.L.’96) Suitable for the study of boundary conditions (also fermions)

miércoles 29 de agosto de 2012 i i Quantum mechanically: HeF = He˜ F implies:

i [θ,θ˜] ψ [θ˜]=N(k) θ(σ) e F φ [θ(σ)] k D k In our case: 1 = dσ(θθ˜ θθ˜) F 2 1 − S Derive global properties Arbitrary genus Riemann surfaces

miércoles 29 de agosto de 2012 4.3. Non-Abelian T-duality

(De la Ossa, Quevedo’93; Alvarez, Alvarez-Gaume, Y.L.’95)

miércoles 29 de agosto de 2012 4.3. Non-Abelian T-duality

(De la Ossa, Quevedo’93; Alvarez, Alvarez-Gaume, Y.L.’95)

Non-Abelian continuous isometry: Xm gmXn ,g G → n ∈

miércoles 29 de agosto de 2012 4.3. Non-Abelian T-duality

(De la Ossa, Quevedo’93; Alvarez, Alvarez-Gaume, Y.L.’95)

Non-Abelian continuous isometry: Xm gmXn ,g G → n ∈ i) Gauge it: dXm DXm = dXm + AmXn → n 1 A Lie algebra of G A g(A + d)g− ∈ →

miércoles 29 de agosto de 2012 4.3. Non-Abelian T-duality

(De la Ossa, Quevedo’93; Alvarez, Alvarez-Gaume, Y.L.’95)

Non-Abelian continuous isometry: Xm gmXn ,g G → n ∈ i) Gauge it: dXm DXm = dXm + AmXn → n 1 A Lie algebra of G A g(A + d)g− ∈ → ii) Add a Lagrange multiplier term: Tr(χF ) F = dA A A − ∧ 1 χ Lie Algebra of G χ gχg− , such that ∈ , → χ F =0 A exact D → ⇒ (in a topologically trivial worldsheet) + ﬁx the gauge: A =0 Original theory ⇒

miércoles 29 de agosto de 2012 iii) Integrate the gauge ﬁeld + ﬁx the gauge Dual theory →

miércoles 29 de agosto de 2012 iii) Integrate the gauge ﬁeld + ﬁx the gauge Dual theory → General properties: - Loss of symmetries in the dual Non-involutive ⇒ - Original and dual very different geometrically

Example: Principal chiral model with group SU(2):

Geometrically: S3

1 1 L = Tr(g− dg g− dg); g SU(2) ∧∗ ∈ Invariant under: g h gh ; h ,h SU(2) → 1 2 1 2 ∈ Choose: g hg ; h SU(2) → ∈

miércoles 29 de agosto de 2012 1 L˜ = δ χk + χ χ dχi dχj 1+χ2 ij − ijk i j ∧∗ 1 Invariant under χ hχh− ; h SU(2) → ∈

A - Higher genus generalization? Set to zero Wγ = Pe γ

miércoles 29 de agosto de 2012 1 L˜ = δ χk + χ χ dχi dχj 1+χ2 ij − ijk i j ∧∗ 1 Invariant under χ hχh− ; h SU(2) → ∈

A - Higher genus generalization? Set to zero Wγ = Pe γ - Global properties?

miércoles 29 de agosto de 2012 1 L˜ = δ χk + χ χ dχi dχj 1+χ2 ij − ijk i j ∧∗ 1 Invariant under χ hχh− ; h SU(2) → ∈

A - Higher genus generalization? Set to zero Wγ = Pe γ - Global properties? - Conformal invariance not proved in general

miércoles 29 de agosto de 2012 1 L˜ = δ χk + χ χ dχi dχj 1+χ2 ij − ijk i j ∧∗ 1 Invariant under χ hχh− ; h SU(2) → ∈

A - Higher genus generalization? Set to zero Wγ = Pe γ - Global properties? - Conformal invariance not proved in general - Non-involutive

miércoles 29 de agosto de 2012 1 L˜ = δ χk + χ χ dχi dχj 1+χ2 ij − ijk i j ∧∗ 1 Invariant under χ hχh− ; h SU(2) → ∈

A - Higher genus generalization? Set to zero Wγ = Pe γ - Global properties? - Conformal invariance not proved in general - Non-involutive

True symmetry in String Theory?

miércoles 29 de agosto de 2012 Partial results about its realization as a canonical transformation: (Y.L.’95)

miércoles 29 de agosto de 2012 Partial results about its realization as a canonical transformation: (Y.L.’95) A system of coordinates adapted to the non-commuting isometries does not exist

miércoles 29 de agosto de 2012 Partial results about its realization as a canonical transformation: (Y.L.’95) A system of coordinates adapted to the non-commuting isometries does not exist For principal chiral models: 1 1 L = Tr(g− dg g− dg); g G ∧∗ ∈ 1 a Use Maurer-Cartan forms: g− dg Ωa(θ)dθ ≡ Then: 1 ak bk k k a b H = ω ω Π Π + Ω Ω θ θ 4 a b a b

k k ak i ki Ωa = ΩaT , ω (θ)Ωa(θ)=δ

miércoles 29 de agosto de 2012 The generating functional

a [χ, θ]= dσ Tr (χ Ωa(θ)∂σθ ) F 1 S generates the canonical transformation that gives the dual: δ Π = F a −δθa δ Π˜ = F i δχi

miércoles 29 de agosto de 2012 The generating functional

a [χ, θ]= dσ Tr (χ Ωa(θ)∂σθ ) F 1 S generates the canonical transformation that gives the dual: δ Π = F a −δθa δ Π˜ = F i δχi

In this approach: - Involutive - Generalize to arbitrary genus Riemann surfaces - Global properties

miércoles 29 de agosto de 2012 Interesting as a solution generating technique (Sfetsos, Thompson’10; Y.L., O Colgain, Sfetsos, Thompson’11; Itsios, Y.L., O Colgain, Sfetsos’12):

miércoles 29 de agosto de 2012 Interesting as a solution generating technique (Sfetsos, Thompson’10; Y.L., O Colgain, Sfetsos, Thompson’11; Itsios, Y.L., O Colgain, Sfetsos’12): In the context of gauge/gravity duality: Apply it to the AdS S5 Type IIB background: 5 ×

5 F5 = 2Vol(AdS5) 2Vol(S ) F = dC − 5 4

5 F5 = 2Vol(AdS5) 2Vol(S ) F = dC − 5 4 Dual Type IIA background: dr2 r2 cos2 θ ds˜2 = ds2(AdS ) + 4(dθ2 +sin2 θdφ2)+ + ds2(S2) 5 cos2 θ cos4 θ + r2 F ,F ,F ,F =0 2 4 6 8

miércoles 29 de agosto de 2012 Less supersymmetric background related to some N=2 SCFT’s

miércoles 29 de agosto de 2012 Less supersymmetric background related to some N=2 SCFT’s In general, less SUSY backgrounds with no clear dual CFT.

Progress about the non-invertibility:

miércoles 29 de agosto de 2012 Less supersymmetric background related to some N=2 SCFT’s In general, less SUSY backgrounds with no clear dual CFT.

Progress about the non-invertibility:

T Type IIA SUGRA Type IIB SUGRA Abelian (Bergshoeff, Hull, S1 S1 Ortín’95) 9 dim SUGRA

miércoles 29 de agosto de 2012 Less supersymmetric background related to some N=2 SCFT’s In general, less SUSY backgrounds with no clear dual CFT.

Progress about the non-invertibility:

T Type IIA SUGRA Type IIB SUGRA Abelian (Bergshoeff, Hull, S1 S1 Ortín’95) 9 dim SUGRA

T Type IIA SUGRA Type IIB SUGRA

Non-Abelian 2 3 S R S × 7 dim SUGRA

miércoles 29 de agosto de 2012 5. Conclusions