Principal Chiral Model: T-Dualities and Doubling Principal Chiral Model: Franco Pezzella T-Dualities and Doubling Introduction and Motivation

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Principal Chiral Model: T-dualities and Doubling Principal Chiral Model: Franco Pezzella T-dualities and Doubling Introduction and Motivation

Principal Chiral Model Franco Pezzella A Drinfel’d Double INFN - Naples Section - Italy

Born Geometry with V. E. Marotta and P. Vitale

Dual Principal [arXiv:1903.01243 [hep-th]] Chiral Models JHEP 1808(2018)185 [arXiv:1804.00744 [hep-th]] The Doubled Principal and F. Bascone Chiral Model [arXiv:1904.03727 [hep-th]] + work in progress Conclusion and Outlook Workshop on Recent Developments in Strings and Gravity - Corfu September 15, 2019

1 / 39 September 15, 2019 2 Principal Chiral Model

3 A Drinfel’d Double

4 Born Geometry

5 Dual Principal Chiral Models

6 The Doubled Principal Chiral Model

7 Conclusion and Outlook

Plan of the talk

Principal Chiral Model: T-dualities and Doubling 1 Introduction and Motivation Franco Pezzella

Introduction and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

2 / 39 3 A Drinfel’d Double

4 Born Geometry

5 Dual Principal Chiral Models

6 The Doubled Principal Chiral Model

7 Conclusion and Outlook

Plan of the talk

Principal Chiral Model: T-dualities and Doubling 1 Introduction and Motivation Franco Pezzella

Introduction 2 Principal Chiral Model and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

2 / 39 4 Born Geometry

5 Dual Principal Chiral Models

6 The Doubled Principal Chiral Model

7 Conclusion and Outlook

Plan of the talk

Principal Chiral Model: T-dualities and Doubling 1 Introduction and Motivation Franco Pezzella

Introduction 2 Principal Chiral Model and Motivation

Principal Chiral Model 3 A Drinfel’d Double A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

2 / 39 5 Dual Principal Chiral Models

6 The Doubled Principal Chiral Model

7 Conclusion and Outlook

Plan of the talk

Principal Chiral Model: T-dualities and Doubling 1 Introduction and Motivation Franco Pezzella

Introduction 2 Principal Chiral Model and Motivation

Principal Chiral Model 3 A Drinfel’d Double A Drinfel’d Double

Born 4 Born Geometry Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

2 / 39 6 The Doubled Principal Chiral Model

7 Conclusion and Outlook

Plan of the talk

Principal Chiral Model: T-dualities and Doubling 1 Introduction and Motivation Franco Pezzella

Introduction 2 Principal Chiral Model and Motivation

Principal Chiral Model 3 A Drinfel’d Double A Drinfel’d Double

Born 4 Born Geometry Geometry

Dual Principal Chiral Models 5 Dual Principal Chiral Models The Doubled Principal Chiral Model

Conclusion and Outlook

2 / 39 7 Conclusion and Outlook

Plan of the talk

Principal Chiral Model: T-dualities and Doubling 1 Introduction and Motivation Franco Pezzella

Introduction 2 Principal Chiral Model and Motivation

Principal Chiral Model 3 A Drinfel’d Double A Drinfel’d Double

Born 4 Born Geometry Geometry

Dual Principal Chiral Models 5 Dual Principal Chiral Models The Doubled Principal Chiral Model 6 The Doubled Principal Chiral Model Conclusion and Outlook

2 / 39 Plan of the talk

Principal Chiral Model: T-dualities and Doubling 1 Introduction and Motivation Franco Pezzella

Introduction 2 Principal Chiral Model and Motivation

Principal Chiral Model 3 A Drinfel’d Double A Drinfel’d Double

Born 4 Born Geometry Geometry

Dual Principal Chiral Models 5 Dual Principal Chiral Models The Doubled Principal Chiral Model 6 The Doubled Principal Chiral Model Conclusion and Outlook 7 Conclusion and Outlook

2 / 39 d On a d-torus T , with constant backgrounds Gab and Bab, (Abelian) T-duality is described by O(d, d; Z) transformations. By exchanging momentum and winding modes, it implies that the short distance behavior is governed by long distance behavior in the dual torus T˜d . The indeﬁnite orthogonal group O(d, d) naturally appears in the Hamiltonian description of the bosonic string in the target space M with two peculiar structures, the generalized metric H and the O(d, d) invariant metric η.

T-Duality

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

3 / 39 The indeﬁnite orthogonal group O(d, d) naturally appears in the Hamiltonian description of the bosonic string in the target space M with two peculiar structures, the generalized metric H and the O(d, d) invariant metric η.

T-Duality

Principal Chiral Model: T-dualities and Doubling T-duality provides a powerful tool for investigating the structure Franco Pezzella of the spacetime from the string point of view by relating, in the usual σ-model approach, backgrounds which otherwise would be Introduction and Motivation considered diﬀerent. Principal Chiral Model d On a d-torus T , with constant backgrounds Gab and Bab, A Drinfel’d Double (Abelian) T-duality is described by O(d, d; Z) transformations. Born By exchanging momentum and winding modes, it implies that Geometry

Dual Principal the short distance behavior is governed by long distance behavior Chiral Models in the dual torus T˜d . The Doubled Principal Chiral Model

Conclusion and Outlook

3 / 39 T-Duality

Dual Principal the short distance behavior is governed by long distance behavior Chiral Models in the dual torus T˜d . The Doubled Principal The indeﬁnite orthogonal group O(d, d) naturally appears in the Chiral Model Hamiltonian description of the bosonic string in the target space Conclusion and Outlook M with two peculiar structures, the generalized metric H and the O(d, d) invariant metric η.

3 / 39 1 ˙ a ˙ b 0a 0b Hamiltonian density H = 4πα0 Gab(X X + X X ) rewritten as: t 1 ∂ X ∂ X H = σ H(G, B) σ 4πα0 2πα0P 2πα0P where the generalized metric is introduced: G − BG −1B BG −1 H(G, B) = −G −1BG −1 H results to be proportional to the squared length of the generalized vector AP as measured by the generalized metric H.

Generalized Metric and O(d, d) Invariant Metric

Principal L ∗ Chiral Model: Generalized vector AP in TM T M T-dualities and Doubling

Franco a ∂ 0 a ∂L Pezzella AP (X ) = ∂σX a + 2πα Padx Pa = a ∂x ∂(∂τ X ) Introduction and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

4 / 39 H results to be proportional to the squared length of the generalized vector AP as measured by the generalized metric H.

Generalized Metric and O(d, d) Invariant Metric

Principal L ∗ Chiral Model: Generalized vector AP in TM T M T-dualities and Doubling

Franco a ∂ 0 a ∂L Pezzella AP (X ) = ∂σX a + 2πα Padx Pa = a ∂x ∂(∂τ X ) Introduction and Motivation

Principal Chiral Model 1 ˙ a ˙ b 0a 0b A Drinfel’d Hamiltonian density H = 4πα0 Gab(X X + X X ) rewritten as: Double

4 / 39 Generalized Metric and O(d, d) Invariant Metric

Principal L ∗ Chiral Model: Generalized vector AP in TM T M T-dualities and Doubling

Franco a ∂ 0 a ∂L Pezzella AP (X ) = ∂σX a + 2πα Padx Pa = a ∂x ∂(∂τ X ) Introduction and Motivation

Principal Chiral Model 1 ˙ a ˙ b 0a 0b A Drinfel’d Hamiltonian density H = 4πα0 Gab(X X + X X ) rewritten as: Double

Born t Geometry 1 ∂σX ∂σX H = 0 0 H(G, B) 0 Dual Principal 4πα 2πα P 2πα P Chiral Models The Doubled where the generalized metric is introduced: Principal Chiral Model G − BG −1B BG −1 Conclusion H(G, B) = and Outlook −G −1BG −1 H results to be proportional to the squared length of the generalized vector AP as measured by the generalized metric H.

4 / 39 Generalized Metric and O(d, d) Invariant Metric

Principal L ∗ Chiral Model: Generalized vector AP in TM T M T-dualities and Doubling

Franco a ∂ 0 a ∂L Pezzella AP (X ) = ∂σX a + 2πα Padx Pa = a ∂x ∂(∂τ X ) Introduction and Motivation

Principal Chiral Model 1 ˙ a ˙ b 0a 0b A Drinfel’d Hamiltonian density H = 4πα0 Gab(X X + X X ) rewritten as: Double

4 / 39 The generalized metric is an element of O(d, d): Ht ηH = η t All the admissible generalized vectors satisfying AP ηAP = 0 are 0 related by an O(d, d) transformation via AP = T AP . 0 For AP to solve the ﬁrst constraint, a compensating transformation T −1 has to be applied to H. , i.e. H0 = (T −1)t H(T −1).

Constraints and O(d, d) invariant Metric

Principal Chiral Model: T-dualities In terms of AP the constraints and Doubling Franco a b 0a 0b a 0b Pezzella Gab(X˙ X˙ + X X ) = 0 ; GabX˙ X = 0

The Doubled Principal Chiral Model

Conclusion and Outlook

5 / 39 t All the admissible generalized vectors satisfying AP ηAP = 0 are 0 related by an O(d, d) transformation via AP = T AP . 0 For AP to solve the ﬁrst constraint, a compensating transformation T −1 has to be applied to H. , i.e. H0 = (T −1)t H(T −1).

Constraints and O(d, d) invariant Metric

Principal Chiral Model: T-dualities In terms of AP the constraints and Doubling Franco a b 0a 0b a 0b Pezzella Gab(X˙ X˙ + X X ) = 0 ; GabX˙ X = 0

Conclusion and Outlook

5 / 39 0 For AP to solve the ﬁrst constraint, a compensating transformation T −1 has to be applied to H. , i.e. H0 = (T −1)t H(T −1).

Constraints and O(d, d) invariant Metric

Principal Chiral Model: T-dualities In terms of AP the constraints and Doubling Franco a b 0a 0b a 0b Pezzella Gab(X˙ X˙ + X X ) = 0 ; GabX˙ X = 0

Introduction and Motivation t t become AP HAP = 0 AP ηAP = 0. Principal Chiral Model The ﬁrst sets H to zero ; the second completely determines the A Drinfel’d dynamics, rewritten in terms of the O(d, d) invariant metric: Double 0 1 Born η = . Geometry 1 0 Dual Principal t Chiral Models The generalized metric is an element of O(d, d): H ηH = η The Doubled t Principal All the admissible generalized vectors satisfying AP ηAP = 0 are Chiral Model 0 related by an O(d, d) transformation via AP = T AP . Conclusion and Outlook

5 / 39 Constraints and O(d, d) invariant Metric

Principal Chiral Model: T-dualities In terms of AP the constraints and Doubling Franco a b 0a 0b a 0b Pezzella Gab(X˙ X˙ + X X ) = 0 ; GabX˙ X = 0

5 / 39 This indeed shows that the background B can be created from the G-background through a B-transformation.

Principal Chiral Model: T-dualities and Doubling

Franco Pezzella H and its inverse H−1 can be rewritten in products:

Introduction and Motivation 1 B G 0 1 0 Principal H(G, B) = −1 Chiral Model 0 1 0 G −B 1

A Drinfel’d Double Born 1 0 G −1 0 1 −B Geometry H−1(G, B) = Dual Principal B 1 0 G 0 1 Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

6 / 39 Principal Chiral Model: T-dualities and Doubling

Franco Pezzella H and its inverse H−1 can be rewritten in products:

Introduction and Motivation 1 B G 0 1 0 Principal H(G, B) = −1 Chiral Model 0 1 0 G −B 1

A Drinfel’d Double Born 1 0 G −1 0 1 −B Geometry H−1(G, B) = Dual Principal B 1 0 G 0 1 Chiral Models

The Doubled Principal This indeed shows that the background B can be created from Chiral Model the G-background through a B-transformation. Conclusion and Outlook

6 / 39 Through the use of auxiliary fields, one gets the dual Polyakov d action S˜[X˜; G˜, B˜] on T˜ with string coordinates Xã and a connected to S[X ; G, B] by X → Xã and suitable transformations of (G, B) through the Büscherrules. Abelian T-duality: based on the presence of global Abelian isometries in the target spaces of both the paired sigma models.

O(D, D) and constant backgrounds

Principal Chiral Model: T-dualities and Doubling

Franco d d Pezzella On T (Gab and Bab constant) with its isometries U(1) , the e.o.m.’s for the string coordinates are a set of conservation laws Introduction and Motivation on the world-sheet: Principal Chiral Model α α αβ b αβ b αβ ˜ ∂αJa = 0 Ja = h Gab∂βX + Bab∂βX h ≡ ∂βXa A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

7 / 39 Abelian T-duality: based on the presence of global Abelian isometries in the target spaces of both the paired sigma models.

O(D, D) and constant backgrounds

Principal Chiral Model: T-dualities and Doubling

Conclusion and Outlook

7 / 39 O(D, D) and constant backgrounds

Principal Chiral Model: T-dualities and Doubling

Born Through the use of auxiliary fields, one gets the dual Polyakov Geometry d action S˜[X˜; G˜, B˜] on T˜ with string coordinates Xã and Dual Principal a Chiral Models connected to S[X ; G, B] by X → Xã and suitable The Doubled transformations of (G, B) through the Büscherrules. Principal Chiral Model Abelian T-duality: based on the presence of global Abelian Conclusion and Outlook isometries in the target spaces of both the paired sigma models.

7 / 39 The manifestly T-duality O(d, d) symmetric formulation may be considered as a natural generalization of the standard one at the string scale. At compactiﬁcation radius R >> α0 it reproduces the usual Polyakov action, at R << α0 its dual. First example of Born geometry.

T-dual invariant bosonic string formulation

Principal Chiral Model: a T-dualities Doubling the coordinates, i.e. putting both the coordinates X and Doubling ˜ A a ˜ and the dual ones Xa in a generalized vector X = (X , Xa) Franco Pezzella (A = 1,... 2d), (a = 1,... d) it is natural to replace the standard notation in string theory based on G and B by η and H. Introduction and Motivation Tseytlin action for constant backgrounds (A. A. Tseytlin, 1990 Principal Chiral Model and 1991) highlights the role of the generalized vector X and of the

A Drinfel’d two metrics: Double Z T h A B A B i Born S = − dτdσ ∂τ X ∂σX ηAB − ∂σX ∂σX HAB Geometry 2 Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

8 / 39 At compactiﬁcation radius R >> α0 it reproduces the usual Polyakov action, at R << α0 its dual. First example of Born geometry.

T-dual invariant bosonic string formulation

A Drinfel’d two metrics: Double Z T h A B A B i Born S = − dτdσ ∂τ X ∂σX ηAB − ∂σX ∂σX HAB Geometry 2 Dual Principal Chiral Models The Doubled The manifestly T-duality O(d, d) symmetric formulation may be Principal Chiral Model considered as a natural generalization of the standard one at the Conclusion string scale. and Outlook

8 / 39 First example of Born geometry.

T-dual invariant bosonic string formulation

8 / 39 T-dual invariant bosonic string formulation

8 / 39 Non-Abelian T-duality refers to the existence of a global Abelian isometry on the target space of one of the two σ-models and of a global non-Abelian isometry on the other [de la Ossa and Quevedo, 1992, Alvarez, Alvarez-Gaum´eand Lozano, 1993 - M. Rocek and E. Verlinde, 1993] The notion of non-abelian T-duality is still lacking some of the key features of its Abelian counterpart. A canonical procedure is missing that would yield the original theory if one is given its non-Abelian dual. The Poisson-Lie T-duality [Klimˇcikand Severa, 1996] generalizes the previous deﬁnitions to all the other cases.

What about more general settings?

Principal Chiral Model: T-dualities and Doubling Are there similar aspects in more general settings, for instance, Franco Pezzella when target space is curved or non-compact and/or, in

Introduction particular, for other kinds of T-dualities? and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

9 / 39 The notion of non-abelian T-duality is still lacking some of the key features of its Abelian counterpart. A canonical procedure is missing that would yield the original theory if one is given its non-Abelian dual. The Poisson-Lie T-duality [Klimˇcikand Severa, 1996] generalizes the previous deﬁnitions to all the other cases.

What about more general settings?

Principal Chiral Model: T-dualities and Doubling Are there similar aspects in more general settings, for instance, Franco Pezzella when target space is curved or non-compact and/or, in

Introduction particular, for other kinds of T-dualities? and Motivation Non-Abelian T-duality refers to the existence of a global Abelian Principal Chiral Model isometry on the target space of one of the two σ-models and of

A Drinfel’d a global non-Abelian isometry on the other [de la Ossa and Double Quevedo, 1992, Alvarez, Alvarez-Gaum´eand Lozano, 1993 - M. Born Geometry Rocek and E. Verlinde, 1993]

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

9 / 39 The Poisson-Lie T-duality [Klimˇcikand Severa, 1996] generalizes the previous deﬁnitions to all the other cases.

What about more general settings?

Principal Chiral Model: T-dualities and Doubling Are there similar aspects in more general settings, for instance, Franco Pezzella when target space is curved or non-compact and/or, in

A Drinfel’d a global non-Abelian isometry on the other [de la Ossa and Double Quevedo, 1992, Alvarez, Alvarez-Gaum´eand Lozano, 1993 - M. Born Geometry Rocek and E. Verlinde, 1993] Dual Principal The notion of non-abelian T-duality is still lacking some of the key Chiral Models features of its Abelian counterpart. A canonical procedure is missing The Doubled Principal that would yield the original theory if one is given its non-Abelian Chiral Model dual. Conclusion and Outlook

9 / 39 What about more general settings?

Principal Chiral Model: T-dualities and Doubling Are there similar aspects in more general settings, for instance, Franco Pezzella when target space is curved or non-compact and/or, in

9 / 39 The relevant structure for the existence of dual counterparts: the Drinfel’d double of G together with the notion of Poisson-Lie symmetries. Also it allows to establish more connections with Generalized Geometry since tangent and cotangent vector ﬁelds of G may be respectively related to the span of the Lie algebra ganditsdualg˜. The Doubled Geometry may play a role in describing the generalized dynamics on the tangent bundle TD ' D × d used to describe within a single action both dually related models.

Why a Principal Chiral Model?

Principal Chiral Model: T-dualities and Doubling

Franco Principal Chiral Models: σ-models whose target space is a Lie Pezzella group G are very helpful in understanding Abelian, Non-Abelian Introduction and Poisson-Lie T-dualities. and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

10 / 39 Also it allows to establish more connections with Generalized Geometry since tangent and cotangent vector ﬁelds of G may be respectively related to the span of the Lie algebra ganditsdualg˜. The Doubled Geometry may play a role in describing the generalized dynamics on the tangent bundle TD ' D × d used to describe within a single action both dually related models.

Why a Principal Chiral Model?

Principal Chiral Model: T-dualities and Doubling

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

10 / 39 The Doubled Geometry may play a role in describing the generalized dynamics on the tangent bundle TD ' D × d used to describe within a single action both dually related models.

Why a Principal Chiral Model?

Principal Chiral Model: T-dualities and Doubling

Franco Principal Chiral Models: σ-models whose target space is a Lie Pezzella group G are very helpful in understanding Abelian, Non-Abelian Introduction and Poisson-Lie T-dualities. and Motivation Principal The relevant structure for the existence of dual counterparts: Chiral Model the Drinfel’d double of G together with the notion of A Drinfel’d Double Poisson-Lie symmetries. Born Geometry Also it allows to establish more connections with Generalized Dual Principal Geometry since tangent and cotangent vector ﬁelds of G may be Chiral Models respectively related to the span of the Lie algebra ganditsdualg˜. The Doubled Principal Chiral Model

Conclusion and Outlook

10 / 39 Why a Principal Chiral Model?

Principal Chiral Model: T-dualities and Doubling

Conclusion generalized dynamics on the tangent bundle TD ' D × d used and Outlook to describe within a single action both dually related models.

10 / 39 For every decomposition of the Drinfel’d double D into dually related subgroups G, G˜, it is possible to deﬁne a couple of PCM’s having as target conﬁguration space either of the two subgroups. Every PCM has its dual counterpart for which the role of G and its dual G˜ is interchanged : G ↔ G˜. The set of all decompositions (d, g, g˜), plays the role of the modular space of sigma models mutually connected by an O(d, d) transformation. In particular, for the manifest Abelian T-duality of the string model on the d-torus, the Drinfel’d double is D = U(1)2d and its modular space, is in one-to-one correspondence with O(d, d; Z).

Drinfel’d double

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

11 / 39 Every PCM has its dual counterpart for which the role of G and its dual G˜ is interchanged : G ↔ G˜. The set of all decompositions (d, g, g˜), plays the role of the modular space of sigma models mutually connected by an O(d, d) transformation. In particular, for the manifest Abelian T-duality of the string model on the d-torus, the Drinfel’d double is D = U(1)2d and its modular space, is in one-to-one correspondence with O(d, d; Z).

Drinfel’d double

Principal Chiral Model: The Drinfel’d double of G: Lie group D, with dimension twice T-dualities and Doubling the one of G. Its Lie algebra d can be decomposed into a pair of Franco Pezzella maximally isotropic sub-algebras, g, g˜ with respect to a non-degenerate invariant bilinear form on d. Introduction and Motivation For every decomposition of the Drinfel’d double D into dually Principal ˜ Chiral Model related subgroups G, G, it is possible to deﬁne a couple of

A Drinfel’d PCM’s having as target conﬁguration space either of the two Double subgroups. Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

11 / 39 The set of all decompositions (d, g, g˜), plays the role of the modular space of sigma models mutually connected by an O(d, d) transformation. In particular, for the manifest Abelian T-duality of the string model on the d-torus, the Drinfel’d double is D = U(1)2d and its modular space, is in one-to-one correspondence with O(d, d; Z).

Drinfel’d double

The Doubled Principal Chiral Model

Conclusion and Outlook

11 / 39 In particular, for the manifest Abelian T-duality of the string model on the d-torus, the Drinfel’d double is D = U(1)2d and its modular space, is in one-to-one correspondence with O(d, d; Z).

Drinfel’d double

11 / 39 Drinfel’d double

A Drinfel’d PCM’s having as target conﬁguration space either of the two Double subgroups. Born Geometry Every PCM has its dual counterpart for which the role of G and Dual Principal ˜ ˜ Chiral Models its dual G is interchanged : G ↔ G. The Doubled The set of all decompositions (d, g, g˜), plays the role of the Principal Chiral Model modular space of sigma models mutually connected by an Conclusion O(d, d) transformation. and Outlook In particular, for the manifest Abelian T-duality of the string model on the d-torus, the Drinfel’d double is D = U(1)2d and its modular space, is in one-to-one correspondence with O(d, d; Z).

11 / 39 Action written in terms ofg −1dg, the Lie algebra valued left-invariant one-forms with g ∈ SU(2): Z 1 −1 2 −1 2 S = dtdσ Tr (g ∂t g) − (g ∂σg) 4 2 R trace → scalar product on the Lie algebra su(2), non-degenerate and invariant. Z 1 i j i j S = dtdσ (A δij A − J δij J ) 2 2 R i i −1 i −1 i currents A and J : g ∂t g = 2A ei , g ∂σg = 2J ei , ei = σi /2.

i −1 i −1 A = Tr (g ∂t g)ei J = Tr (g ∂σg)ei

SU(2) Principal Chiral Model

Principal Chiral Model: SU(2) Principal Chiral Model: target space SU(2) and source T-dualities 1,1 and Doubling space R endowed with the metric hαβ = diag(−1, 1). Franco Pezzella

Introduction and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

12 / 39 SU(2) Principal Chiral Model

Principal Chiral Model: SU(2) Principal Chiral Model: target space SU(2) and source T-dualities 1,1 and Doubling space R endowed with the metric hαβ = diag(−1, 1). Franco −1 Pezzella Action written in terms ofg dg, the Lie algebra valued

Introduction left-invariant one-forms with g ∈ SU(2): and Motivation Z Principal 1 −1 2 −1 2 Chiral Model S = dtdσ Tr (g ∂t g) − (g ∂σg) 4 2 A Drinfel’d R Double Born trace → scalar product on the Lie algebra su(2), Geometry

Dual Principal non-degenerate and invariant. Chiral Models Z The Doubled 1 i j i j Principal S = dtdσ (A δij A − J δij J ) Chiral Model 2 2 R Conclusion and Outlook i i −1 i −1 i currents A and J : g ∂t g = 2A ei , g ∂σg = 2J ei , ei = σi /2.

i −1 i −1 A = Tr (g ∂t g)ei J = Tr (g ∂σg)ei

12 / 39 At fixed t, all g’s constant at infinity form an infinite dimensional Lie group SU(2)(R) ≡ Map(R, SU(2)), given by smooth maps g : σ ∈ R → g(σ) ∈ SU(2). At fixed time, the currents J and A take values in the Lie algebra su(2)(R) of the group SU(2)(R).

Principal Chiral Model: T-dualities and Doubling

Franco Equations of motion: Pezzella

Introduction ∂t A =∂σJ, and Motivation

Principal ∂t J =∂σA − [A, J] Chiral Model

A Drinfel’d Double → integrability condition for the the existence of a g ∈ SU(2) i i Born that allows the expression of the two currents A and J . Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

13 / 39 Principal Chiral Model: T-dualities and Doubling

Franco Equations of motion: Pezzella

Introduction ∂t A =∂σJ, and Motivation

Principal ∂t J =∂σA − [A, J] Chiral Model

A Drinfel’d Double → integrability condition for the the existence of a g ∈ SU(2) i i Born that allows the expression of the two currents A and J . Geometry At fixed t, all g’s constant at infinity form an infinite Dual Principal Chiral Models dimensional Lie group SU(2)(R) ≡ Map(R, SU(2)), given by The Doubled Principal smooth maps g : σ ∈ R → g(σ) ∈ SU(2). Chiral Model At fixed time, the currents J and A take values in the Lie Conclusion and Outlook algebra su(2)(R) of the group SU(2)(R).

13 / 39 Inﬁnitesimal generators of the Lie algebra su(2)(R): δ X (σ) = X a(σ) , i i δg a(σ)

with Lie brackets:

0 k 0 [Xi (σ), Xj (σ )] = cij Xk (σ)δ(σ − σ )

deﬁning the current algebra su(2)(R) ' su(2) ⊗ C ∞(R).

Lagrangian formalism

Principal Chiral Model: T-dualities and Doubling Franco The tangent bundle description of the dynamics is given in terms Pezzella of( J, A): A left generalized velocities and J left conﬁguration Introduction and Motivation space coordinates.

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

14 / 39 Lagrangian formalism

A Drinfel’d Double a δ Xi (σ) = Xi (σ) , Born δg a(σ) Geometry

Dual Principal Chiral Models with Lie brackets:

The Doubled Principal 0 k 0 Chiral Model [Xi (σ), Xj (σ )] = cij Xk (σ)δ(σ − σ ) Conclusion and Outlook deﬁning the current algebra su(2)(R) ' su(2) ⊗ C ∞(R).

14 / 39 Topologically is the manifold S 3 × R3. As a group, T ∗SU(2) ' SU(2) n R3. As a Poisson manifold it is symplectomorphic to the group SL(2, C) (same topology). T ∗SU(2) and SL(2, C) are both Drinfel’d doubles of the group SU(2). The former is said classical double. i (J , Ii ) conjugate variables with J conﬁguration space coordinates and I left generalized momenta:

δL −1 j j Ii = −1 i = δij (g ∂t g) = δij A . δ (g ∂t g)

Hamiltonian Formalism

Principal Chiral Model: T-dualities ∗ and Doubling The target phase space is naturally given by T SU(2) (Drinfel’d Franco double) Pezzella .

Introduction and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

15 / 39 As a group, T ∗SU(2) ' SU(2) n R3. As a Poisson manifold it is symplectomorphic to the group SL(2, C) (same topology). T ∗SU(2) and SL(2, C) are both Drinfel’d doubles of the group SU(2). The former is said classical double. i (J , Ii ) conjugate variables with J conﬁguration space coordinates and I left generalized momenta:

δL −1 j j Ii = −1 i = δij (g ∂t g) = δij A . δ (g ∂t g)

Hamiltonian Formalism

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

15 / 39 As a Poisson manifold it is symplectomorphic to the group SL(2, C) (same topology). T ∗SU(2) and SL(2, C) are both Drinfel’d doubles of the group SU(2). The former is said classical double. i (J , Ii ) conjugate variables with J conﬁguration space coordinates and I left generalized momenta:

δL −1 j j Ii = −1 i = δij (g ∂t g) = δij A . δ (g ∂t g)

Hamiltonian Formalism

Principal Chiral Model: T-dualities ∗ and Doubling The target phase space is naturally given by T SU(2) (Drinfel’d Franco double) Pezzella . 3 3 Introduction Topologically is the manifold S × R . and Motivation As a group, T ∗SU(2) ' SU(2) 3. Principal n R Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

15 / 39 T ∗SU(2) and SL(2, C) are both Drinfel’d doubles of the group SU(2). The former is said classical double. i (J , Ii ) conjugate variables with J conﬁguration space coordinates and I left generalized momenta:

δL −1 j j Ii = −1 i = δij (g ∂t g) = δij A . δ (g ∂t g)

Hamiltonian Formalism

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

15 / 39 i (J , Ii ) conjugate variables with J conﬁguration space coordinates and I left generalized momenta:

δL −1 j j Ii = −1 i = δij (g ∂t g) = δij A . δ (g ∂t g)

Hamiltonian Formalism

Born ∗ Geometry T SU(2) and SL(2, C) are both Drinfel’d doubles of the group

Dual Principal SU(2). The former is said classical double. Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

15 / 39 Hamiltonian Formalism

Born ∗ Geometry T SU(2) and SL(2, C) are both Drinfel’d doubles of the group

Dual Principal SU(2). The former is said classical double. Chiral Models (Ji , I ) conjugate variables with J conﬁguration space The Doubled i Principal coordinates and I left generalized momenta: Chiral Model

Conclusion and Outlook δL −1 j j Ii = −1 i = δij (g ∂t g) = δij A . δ (g ∂t g)

15 / 39 Principal Chiral Model: T-dualities and Doubling Hamiltonian: Franco Pezzella Z Z 1 ij i j 1 −1 IJ Introduction H = dσ(Ii Ij δ + J J δij ) = dσ II (H0 ) IJ and Motivation 2 2 R R Principal Chiral Model i ∗ II = (Ii , J ) components of the current 1-form on T SU(2) and A Drinfel’d Double ij Born −1 IJ δ 0 Geometry (H0 ) = 0 δij Dual Principal Chiral Models ∗ The Doubled is a Riemannian metric on T SU(2). Principal Chiral Model The Hamiltonian description of the Principal Chiral Model on −1 Conclusion SU(2) naturally involves the Riemannian generalized metric H and Outlook 0 on the cotangent bundle.

16 / 39 The I ’s generators of the Lie algebra su(2)(R). The J’s span an Abelian algebra a. I and J span the inﬁnite-dimensional current algebra c1 = su(2)(R) n a.

Principal Chiral Model: T-dualities and Doubling The ﬁrst-order Lagrangian, together with the canonical one-form Franco and the symplectic form, allows to determine the equal-time Pezzella Poisson brackets : Introduction and Motivation 0 k 0 Principal {Ii (σ), Ij (σ )} =ij Ik (σ)δ(σ − σ ) Chiral Model j 0 j k 0 j 0 0 A Drinfel’d {Ii (σ), J (σ )} =ki J (σ)δ(σ − σ ) − δi δ (σ − σ ) Double i j 0 Born {J (σ), J (σ )} =0 Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

17 / 39 Principal Chiral Model: T-dualities and Doubling The ﬁrst-order Lagrangian, together with the canonical one-form Franco and the symplectic form, allows to determine the equal-time Pezzella Poisson brackets : Introduction and Motivation 0 k 0 Principal {Ii (σ), Ij (σ )} =ij Ik (σ)δ(σ − σ ) Chiral Model j 0 j k 0 j 0 0 A Drinfel’d {Ii (σ), J (σ )} =ki J (σ)δ(σ − σ ) − δi δ (σ − σ ) Double i j 0 Born {J (σ), J (σ )} =0 Geometry

Dual Principal Chiral Models The Doubled The I ’s generators of the Lie algebra su(2)( ). Principal R Chiral Model The J’s span an Abelian algebra a. Conclusion and Outlook I and J span the inﬁnite-dimensional current algebra c1 = su(2)(R) n a.

17 / 39 Deformed brackets by parameter τ (imaginary): 0 2 k 0 {Ii (σ), Ij (σ )} =(1 − τ )ij Ik (σ)δ(σ − σ ) j 0 2 k j 0 2 2 j 0 0 {Ii (σ), J (σ )} =(1 − τ ) J (σ)ki δ(σ − σ ) − (1 − τ ) δi δ (σ − σ )) i j 0 2 2 ij 0 {J (σ), J (σ )} =(1 − τ )τ k Ik (σ)δ(σ − σ ). The new brackets correspond to the inﬁnite-dimensional Lie algebra c2 ' sl(2, C)(R) isomorphic to the current algebra modelled on the Lorentz algebra sl(2, C).

Principal Equations of motion: Chiral Model: T-dualities k and Doubling ∂t Ij (σ) = {H, Ij (σ)} = ∂σJ δkj (σ), Franco Pezzella and Introduction j j kj jl k and Motivation ∂t J (σ) = {H, J (σ)} = ∂σIk δ (σ) − k Il J (σ). Principal Chiral Model It is possible to give an equivalent description of the dynamics in

A Drinfel’d terms of a new one-parameter family of Poisson algebras [Rajeev, Double 1989] and modiﬁed Hamiltonians, with the currents playing a Born Geometry symmetric role.

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

18 / 39 Principal Equations of motion: Chiral Model: T-dualities k and Doubling ∂t Ij (σ) = {H, Ij (σ)} = ∂σJ δkj (σ), Franco Pezzella and Introduction j j kj jl k and Motivation ∂t J (σ) = {H, J (σ)} = ∂σIk δ (σ) − k Il J (σ). Principal Chiral Model It is possible to give an equivalent description of the dynamics in

A Drinfel’d terms of a new one-parameter family of Poisson algebras [Rajeev, Double 1989] and modiﬁed Hamiltonians, with the currents playing a Born Geometry symmetric role.

Dual Principal Deformed brackets by parameter τ (imaginary): Chiral Models 0 2 k 0 The Doubled {Ii (σ), Ij (σ )} =(1 − τ )ij Ik (σ)δ(σ − σ ) Principal j 0 2 k j 0 2 2 j 0 0 Chiral Model {Ii (σ), J (σ )} =(1 − τ ) J (σ)ki δ(σ − σ ) − (1 − τ ) δi δ (σ − σ )) Conclusion i j 0 2 2 ij 0 and Outlook {J (σ), J (σ )} =(1 − τ )τ k Ik (σ)δ(σ − σ ). The new brackets correspond to the infinite-dimensional Lie algebra c2 ' sl(2, C)(R) isomorphic to the current algebra modelled on the Lorentz algebra sl(2, C). 18 / 39 The previous equations of motion remain unmodified. Alternative description of one and the same dynamics even considering deformed algebras of SL(2, C). Rescale the fields according to: I Ji i → I → Ji . 1 − τ 2 i 1 − τ 2

The rescaled Hamiltonian Hτ becomes identical to the undeformed one H, while the Poisson algebra acquires the form: 0 k 0 {Ii (σ), Ij (σ )} = ij Ik (σ)δ(σ − σ ), j 0 k j 0 j 0 0 {Ii (σ), J (σ )} = J (σ)ki δ(σ − σ ) − δi δ (σ − σ ), i j 0 2 ij 0 {J (σ), J (σ )} = τ k Ik (σ)δ(σ − σ ).

Principal Modiﬁed Hamiltonian: Chiral Model: T-dualities Z and Doubling 1 ij i j Hτ = dσ (Ii Ij δ + J J δij ). Franco 2(1 − τ 2)2 Pezzella R

Introduction and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

19 / 39 Rescale the ﬁelds according to: I Ji i → I → Ji . 1 − τ 2 i 1 − τ 2

Principal Modiﬁed Hamiltonian: Chiral Model: T-dualities Z and Doubling 1 ij i j Hτ = dσ (Ii Ij δ + J J δij ). Franco 2(1 − τ 2)2 Pezzella R

Introduction and Motivation The previous equations of motion remain unmodiﬁed.

Principal Alternative description of one and the same dynamics even Chiral Model considering deformed algebras of SL(2, C). A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

19 / 39 Principal Modiﬁed Hamiltonian: Chiral Model: T-dualities Z and Doubling 1 ij i j Hτ = dσ (Ii Ij δ + J J δij ). Franco 2(1 − τ 2)2 Pezzella R

Introduction and Motivation The previous equations of motion remain unmodiﬁed.

Principal Alternative description of one and the same dynamics even Chiral Model considering deformed algebras of SL(2, C). A Drinfel’d Double Rescale the ﬁelds according to: Born i Geometry I J i → I → Ji . Dual Principal 2 i 2 Chiral Models 1 − τ 1 − τ

The Doubled The rescaled Hamiltonian Hτ becomes identical to the Principal Chiral Model undeformed one H, while the Poisson algebra acquires the form: Conclusion 0 k 0 and Outlook {Ii (σ), Ij (σ )} = ij Ik (σ)δ(σ − σ ), j 0 k j 0 j 0 0 {Ii (σ), J (σ )} = J (σ)ki δ(σ − σ ) − δi δ (σ − σ ), i j 0 2 ij 0 {J (σ), J (σ )} = τ k Ik (σ)δ(σ − σ ).

19 / 39 From the deformed Poisson brackets it is possible to derive the Poisson brackets of the new generators: i j 0 ij k 0 0 {K (σ), K (σ )} = iτf k K (σ ) δ(σ − σ ) together with 0 k 0 {Ii (σ), Ij (σ )} = ij Ik (σ)δ(σ − σ ) j 0 k 0 j jk 0 0 j 0 0 {Ii (σ), K (σ )} = K (σ )ki + iτf i Ik (σ ) δ(σ − σ ) − δi δ (σ − σ )

The K’s span the sb(2, C)(R) Lie algebra with structure constants ij ijl f k = l3k . The Lie algebra c2 ≡ sl(2, C)(R) has been expressed as c2 = su(2)(R) on sb(2, C)(R).

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

20 / 39 The K’s span the sb(2, C)(R) Lie algebra with structure constants ij ijl f k = l3k . The Lie algebra c2 ≡ sl(2, C)(R) has been expressed as c2 = su(2)(R) on sb(2, C)(R).

Principal Introduce new generators showing the bi-algebra structure of Chiral Model: T-dualities sl(2, )( ). and Doubling C R Franco Keeping the generators of su(2)(R) unmodiﬁed, consider the Pezzella linear combination: Introduction i i li3 and Motivation K (σ) = J (σ) − iτ Il (σ). Principal Chiral Model A Drinfel’d From the deformed Poisson brackets it is possible to derive the Double Poisson brackets of the new generators: Born Geometry i j 0 ij k 0 0 {K (σ), K (σ )} = iτf k K (σ ) δ(σ − σ ) Dual Principal Chiral Models together with The Doubled Principal 0 k 0 Chiral Model {Ii (σ), Ij (σ )} = ij Ik (σ)δ(σ − σ )

Conclusion j 0 k 0 j jk 0 0 j 0 0 and Outlook {Ii (σ), K (σ )} = K (σ )ki + iτf i Ik (σ ) δ(σ − σ ) − δi δ (σ − σ )

20 / 39 The Lie algebra c2 ≡ sl(2, C)(R) has been expressed as c2 = su(2)(R) on sb(2, C)(R).

Conclusion j 0 k 0 j jk 0 0 j 0 0 and Outlook {Ii (σ), K (σ )} = K (σ )ki + iτf i Ik (σ ) δ(σ − σ ) − δi δ (σ − σ )

The K’s span the sb(2, C)(R) Lie algebra with structure constants ij ijl f k = l3k .

20 / 39 Principal Introduce new generators showing the bi-algebra structure of Chiral Model: T-dualities sl(2, )( ). and Doubling C R Franco Keeping the generators of su(2)(R) unmodiﬁed, consider the Pezzella linear combination: Introduction i i li3 and Motivation K (σ) = J (σ) − iτ Il (σ). Principal Chiral Model A Drinfel’d From the deformed Poisson brackets it is possible to derive the Double Poisson brackets of the new generators: Born Geometry i j 0 ij k 0 0 {K (σ), K (σ )} = iτf k K (σ ) δ(σ − σ ) Dual Principal Chiral Models together with The Doubled Principal 0 k 0 Chiral Model {Ii (σ), Ij (σ )} = ij Ik (σ)δ(σ − σ )

Conclusion j 0 k 0 j jk 0 0 j 0 0 and Outlook {Ii (σ), K (σ )} = K (σ )ki + iτf i Ik (σ ) δ(σ − σ ) − δi δ (σ − σ )

The K’s span the sb(2, C)(R) Lie algebra with structure constants ij ijl f k = l3k . The Lie algebra c2 ≡ sl(2, )( ) has been expressed as C R 20 / 39 c2 = su(2)(R) on sb(2, C)(R). It is equipped with two non-degenerate invariant scalar products: hu, vi = 2Im(Tr(uv)) ∀u, v ∈ sl(2, C) (u, v) = 2Re(Tr(uv)) ∀u, v ∈ sl(2, C).

hu, vi deﬁnes two maximally isotropic subalgebras k i i k ki i j ij k [ei , ej ] = iij ek , [˜e , ej ] = ijk e˜ +iek f j , [˜e , e˜ ] = if k e˜ i spanned by {ei } and the linear combinatione ˜ = bi − ij3ej . su(2) and sb(2, C) maximally isotropic with respect to <, >: i j j j hei , ej i = 0, he˜ , e˜ i = 0, hei , e˜ i = δi

SL(2, C) as a Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

21 / 39 hu, vi deﬁnes two maximally isotropic subalgebras k i i k ki i j ij k [ei , ej ] = iij ek , [˜e , ej ] = ijk e˜ +iek f j , [˜e , e˜ ] = if k e˜ i spanned by {ei } and the linear combinatione ˜ = bi − ij3ej . su(2) and sb(2, C) maximally isotropic with respect to <, >: i j j j hei , ej i = 0, he˜ , e˜ i = 0, hei , e˜ i = δi

SL(2, C) as a Drinfel’d Double

Principal Chiral Model: The Lie algebra sl(2, C) is deﬁned by the Lie brackets: T-dualities and Doubling k k k [ei , ej ] = iij ek [ei , bj ] = iij bk [bi , bj ] = −iij ek . Franco Pezzella with Introduction σ1 σ2 σ3 and Motivation e = , e = , e = generators of su(2) 1 2 2 2 3 2 Principal Chiral Model bi = iei , i = 1, 2, 3 A Drinfel’d Double Born It is equipped with two non-degenerate invariant scalar products: Geometry Dual Principal hu, vi = 2Im(Tr(uv)) ∀u, v ∈ sl(2, ) Chiral Models C The Doubled (u, v) = 2Re(Tr(uv)) ∀u, v ∈ sl(2, ). Principal C Chiral Model

Conclusion and Outlook

21 / 39 su(2) and sb(2, C) maximally isotropic with respect to <, >: i j j j hei , ej i = 0, he˜ , e˜ i = 0, hei , e˜ i = δi

SL(2, C) as a Drinfel’d Double

21 / 39 SL(2, C) as a Drinfel’d Double

Conclusion and Outlook hu, vi deﬁnes two maximally isotropic subalgebras k i i k ki i j ij k [ei , ej ] = iij ek , [˜e , ej ] = ijk e˜ +iek f j , [˜e , e˜ ] = if k e˜ i spanned by {ei } and the linear combinatione ˜ = bi − ij3ej . su(2) and sb(2, ) maximally isotropic with respect to <, >: C 21 / 39 i j j j hei , ej i = 0, he˜ , e˜ i = 0, hei , e˜ i = δi SL(2, C) is Drinfel’d double of SU(2) and SB(2, C) with polarization sl(2, C) = su(2) ./ sb(2, C) and (sl(2, C), su(2), sb(2, C)) is a Manin triple. SU(2) and SB(2, C) are then dual groups.

Manin Triple

Principal Chiral Model: T-dualities and Doubling

Franco Pezzella Each subalgebra acts on the other one non-trivially, by Introduction co-adjoint action: and Motivation

Principal i i k ki Chiral Model [˜e , ej ] = i jk e˜ + if j ek A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

22 / 39 Manin Triple

Principal Chiral Model: T-dualities and Doubling

Franco Pezzella Each subalgebra acts on the other one non-trivially, by Introduction co-adjoint action: and Motivation

Principal i i k ki Chiral Model [˜e , ej ] = i jk e˜ + if j ek A Drinfel’d Double

Born Geometry SL(2, C) is Drinfel’d double of SU(2) and SB(2, C) with Dual Principal Chiral Models polarization sl(2, C) = su(2) ./ sb(2, C) and The Doubled (sl(2, ), su(2), sb(2, )) is a Manin triple. Principal C C Chiral Model SU(2) and SB(2, C) are then dual groups. Conclusion and Outlook

22 / 39 Principal Chiral Model: T-dualities and Doubling Doubled notation: Franco Pezzella ei i eI = i , ei ∈ su(2), e˜ ∈ sb(2, C) . Introduction e˜ and Motivation

Principal Chiral Model The ﬁrst scalar product then becomes:

A Drinfel’d Double j 0 δi Born (eI , eJ ) = ηIJ = i Geometry δj 0 Dual Principal Chiral Models which is O(3, 3) invariant by construction. The second scalar The Doubled Principal product yields: Chiral Model Conclusion pj and Outlook δij ip3δ (eI , eJ ) = ip ij ik3 jl3 . δ jp3 δ − δkl

23 / 39 Principal Chiral Model: T-dualities and Doubling

Franco Pezzella

Introduction The splitting d = C+ ⊕ C− with C+, C− spanned by {ei }, {bi } and Motivation respectively, deﬁnes a positive deﬁnite metric H on d via: Principal Chiral Model

A Drinfel’d H = ( , )C+ − ( , )C− Double

Born Geometry satisfying T Dual Principal H ηH = η Chiral Models

The Doubled namely H is a pseudo-orthogonal O(3, 3) metric. Principal Chiral Model

Conclusion and Outlook

24 / 39 They are related (and indeed equivalent) to the standard SU(2) chiral model by the O(3, 3) transformation i i li3 K (σ) = J (σ) − iτ Il (σ), symmetry of the dynamics because it maps solutions into solutions .

Principal Chiral Model: A whole family of models exists, described by the Hamiltonians T-dualities j and Doubling labelled by the parameter τ in terms of IJ = (Ij , K ): Franco Z Pezzella 1 −1 LM Hτ = dσ IL(Hτ ) IM Introduction 2 and Motivation R Principal with H−1 being the Riemannian generalized metric Chiral Model τ A Drinfel’d ij ip3 Double −1 h (τ) iτ δpj Hτ = jp3 Born iτδip δij Geometry

Dual Principal Chiral Models where: ij ij 2 ia3 jb3 The Doubled h (τ) = δ − τ δab . Principal Chiral Model

Conclusion and Outlook

25 / 39 Principal Chiral Model: A whole family of models exists, described by the Hamiltonians T-dualities j and Doubling labelled by the parameter τ in terms of IJ = (Ij , K ): Franco Z Pezzella 1 −1 LM Hτ = dσ IL(Hτ ) IM Introduction 2 and Motivation R Principal with H−1 being the Riemannian generalized metric Chiral Model τ A Drinfel’d ij ip3 Double −1 h (τ) iτ δpj Hτ = jp3 Born iτδip δij Geometry

Dual Principal Chiral Models where: ij ij 2 ia3 jb3 The Doubled h (τ) = δ − τ δab . Principal Chiral Model

Conclusion and Outlook They are related (and indeed equivalent) to the standard SU(2) chiral model by the O(3, 3) transformation i i li3 K (σ) = J (σ) − iτ Il (σ), symmetry of the dynamics because it maps solutions into solutions .

25 / 39 The Hamiltonian description naturally involves the Riemannian metric: ij −1 IJ δ 0 (H0 ) = 0 δij one of the structures defining a Born geometry on T ∗SU(2) T ∗SU(2) is a Born manifold: a phase space equipped with a 3 2 structure (η, κ, H0) with κ ∈ End(su(2) n R ) such that κ = 1 with su(2) eigenspace of κ associated with the eigenvalue +1 and R3 eigenspace with the eigenvalue −1. The structures < · , · > and κ satisfy a compatibility condition 3 < κ(ξ), ψ >= − < κ(ψ), ξ >, ∀ξ, ψ ∈ su(2) n R , which defines a two-form ω on su(2) n R3. Defining relations: −1 −1 −1 −1 η H0 = H0 η ω H0 = −H0 ω.

Born Geometry

Principal −1 Chiral Model: What is the geometrical meaning of Hτ emerging in the T-dualities and Doubling deﬁnition of the alternative Hamiltonian Hτ ?

Franco Pezzella

Introduction and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

26 / 39 T ∗SU(2) is a Born manifold: a phase space equipped with a 3 2 structure (η, κ, H0) with κ ∈ End(su(2) n R ) such that κ = 1 with su(2) eigenspace of κ associated with the eigenvalue +1 and R3 eigenspace with the eigenvalue −1. The structures < · , · > and κ satisfy a compatibility condition 3 < κ(ξ), ψ >= − < κ(ψ), ξ >, ∀ξ, ψ ∈ su(2) n R , which deﬁnes a two-form ω on su(2) n R3. Deﬁning relations: −1 −1 −1 −1 η H0 = H0 η ω H0 = −H0 ω.

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

26 / 39 Deﬁning relations: −1 −1 −1 −1 η H0 = H0 η ω H0 = −H0 ω.

Born Geometry

Principal −1 Chiral Model: What is the geometrical meaning of Hτ emerging in the T-dualities and Doubling definition of the alternative Hamiltonian Hτ ? Franco The Hamiltonian description naturally involves the Riemannian Pezzella metric: ij Introduction −1 IJ δ 0 and Motivation (H0 ) = 0 δij Principal Chiral Model one of the structures defining a Born geometry on T ∗SU(2) A Drinfel’d ∗ Double T SU(2) is a Born manifold: a phase space equipped with a 3 2 Born structure (η, κ, H0) with κ ∈ End(su(2) n R ) such that κ = 1 Geometry with su(2) eigenspace of κ associated with the eigenvalue +1 Dual Principal 3 Chiral Models and R eigenspace with the eigenvalue −1. The structures The Doubled Principal < · , · > and κ satisfy a compatibility condition Chiral Model 3 Conclusion < κ(ξ), ψ >= − < κ(ψ), ξ >, ∀ξ, ψ ∈ su(2) n R , and Outlook which defines a two-form ω on su(2) n R3.

26 / 39 Born Geometry

26 / 39 τ-dependent B-transformation

1 iτB eB(τ) = ∈ O(3, 3) 0 1

with components of the tensor B given by Bij = ij3

Hτ is obtained by the B-transformation acting on H0 :

−B(τ)t B(τ) Hτ = e H0e .

The family of equivalent Hamiltonian descriptions of the SU(2) PCM can be understood in terms of a one-parameter family of Born geometries for T ∗SU(2), corresponding, for each choice of the parameter τ, to a speciﬁc splitting of the phase space, with the value τ = 0 the canonical splitting.

B-transformations

Principal Chiral Model: The deformed Hamiltonian Hτ also gives a Riemannian metric T-dualities ∗ and Doubling on T SU(2) and is a B-transformation of the metric H0. Franco Pezzella

Introduction and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

27 / 39 Hτ is obtained by the B-transformation acting on H0 :

−B(τ)t B(τ) Hτ = e H0e .

B-transformations

Principal Chiral Model: The deformed Hamiltonian Hτ also gives a Riemannian metric T-dualities ∗ and Doubling on T SU(2) and is a B-transformation of the metric H0. Franco Pezzella τ-dependent B-transformation

Introduction and Motivation 1 iτB eB(τ) = ∈ O(3, 3) Principal 0 1 Chiral Model

A Drinfel’d Double with components of the tensor B given by Bij = ij3 Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

27 / 39 The family of equivalent Hamiltonian descriptions of the SU(2) PCM can be understood in terms of a one-parameter family of Born geometries for T ∗SU(2), corresponding, for each choice of the parameter τ, to a speciﬁc splitting of the phase space, with the value τ = 0 the canonical splitting.

B-transformations

Introduction and Motivation 1 iτB eB(τ) = ∈ O(3, 3) Principal 0 1 Chiral Model

Conclusion and Outlook

27 / 39 B-transformations

Introduction and Motivation 1 iτB eB(τ) = ∈ O(3, 3) Principal 0 1 Chiral Model

A Drinfel’d Double with components of the tensor B given by Bij = ij3 Born Geometry Hτ is obtained by the B-transformation acting on H0 : Dual Principal Chiral Models −B(τ)t B(τ) Hτ = e H0e . The Doubled Principal Chiral Model The family of equivalent Hamiltonian descriptions of the SU(2) Conclusion and Outlook PCM can be understood in terms of a one-parameter family of Born geometries for T ∗SU(2), corresponding, for each choice of the parameter τ, to a speciﬁc splitting of the phase space, with the value τ = 0 the canonical splitting.

27 / 39 i Hamiltonian vector ﬁelds XK i · := {·, K } associated to the coordinates functions K i for T ∗SU(2), close a non-Abelian algebra according to the following:

ij [XK i , XK j ] = X{K i ,K j } = iτfk XK k because of the non-trivial Poisson bracket i j 0 ij k 0 0 {K (σ), K (σ )} = iτf k K (σ )δ(σ − σ ) → the constant structures of the dual Lie algebra sb(2, C) appear. A dual formulation of this property can be given in terms of the Hamiltonian vector ﬁelds associated with the currents Ii that close the Lie algebra su(2).

Poisson-Lie structure

Principal Chiral Model: T-dualities and Doubling The PCM, in the Hamiltonian formulation given by any Hτ is a Franco Pezzella Poisson-Lie sigma-model.

Introduction and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

28 / 39 Poisson-Lie structure

Principal Chiral Model: T-dualities and Doubling The PCM, in the Hamiltonian formulation given by any Hτ is a Franco Pezzella Poisson-Lie sigma-model. i Introduction Hamiltonian vector ﬁelds XK i · := {·, K } associated to the and Motivation coordinates functions K i for T ∗SU(2), close a non-Abelian Principal Chiral Model algebra according to the following:

A Drinfel’d Double ij [XK i , XK j ] = X{K i ,K j } = iτfk XK k Born Geometry

Dual Principal because of the non-trivial Poisson bracket Chiral Models i j 0 ij k 0 0 {K (σ), K (σ )} = iτf k K (σ )δ(σ − σ ) → the constant The Doubled Principal structures of the dual Lie algebra sb(2, C) appear. Chiral Model

Conclusion A dual formulation of this property can be given in terms of the and Outlook Hamiltonian vector ﬁelds associated with the currents Ii that close the Lie algebra su(2).

28 / 39 In the limit iτ → 0, it reproduces the semi-direct sum su(2)(R) n a, while the limit iα → 0 yields sb(2, C)(R) n a. For all non zero values of the two parameters, the algebra is isomorphic to sl(2, C), and, by suitably rescaling the ﬁelds, one gets a two-parameter family of models, all equivalent to the PCM.

Poisson-Lie dual models

Principal Chiral Model: T-dualities and Doubling Two-parameter generalization in the Poisson algebra generated i Franco by Ii and K by adding another imaginary parameter α making Pezzella the role of the subalgebras su(2)(R) and sb(2, C)(R) symmetric: Introduction and Motivation

Principal 0 k 0 Chiral Model {Ii (σ), Ij (σ )} = iα ij Ik (σ)δ(σ − σ ) i j 0 ij k 0 0 A Drinfel’d {K (σ), K (σ )} = iτf k K (σ )δ(σ − σ ) Double

Born j 0 k 0 j jk 0 0 j 0 0 Geometry {Ii (σ), K (σ )} = iαK (σ )ki + iτf i Ik (σ ) δ(σ−σ )−δi δ (σ−σ ) Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

29 / 39 For all non zero values of the two parameters, the algebra is isomorphic to sl(2, C), and, by suitably rescaling the ﬁelds, one gets a two-parameter family of models, all equivalent to the PCM.

Poisson-Lie dual models

Principal 0 k 0 Chiral Model {Ii (σ), Ij (σ )} = iα ij Ik (σ)δ(σ − σ ) i j 0 ij k 0 0 A Drinfel’d {K (σ), K (σ )} = iτf k K (σ )δ(σ − σ ) Double

29 / 39 Poisson-Lie dual models

Principal 0 k 0 Chiral Model {Ii (σ), Ij (σ )} = iα ij Ik (σ)δ(σ − σ ) i j 0 ij k 0 0 A Drinfel’d {K (σ), K (σ )} = iτf k K (σ )δ(σ − σ ) Double

Born j 0 k 0 j jk 0 0 j 0 0 Geometry {Ii (σ), K (σ )} = iαK (σ )ki + iτf i Ik (σ ) δ(σ−σ )−δi δ (σ−σ ) Dual Principal Chiral Models The Doubled In the limit iτ → 0, it reproduces the semi-direct sum su(2)( ) a, Principal R n Chiral Model while the limit iα → 0 yields sb(2, C)(R) n a. Conclusion and Outlook For all non zero values of the two parameters, the algebra is isomorphic to sl(2, C), and, by suitably rescaling the ﬁelds, one gets a two-parameter family of models, all equivalent to the PCM.

29 / 39 Poisson-Lie dual models

Principal 0 k 0 Chiral Model {Ii (σ), Ij (σ )} = iα ij Ik (σ)δ(σ − σ ) i j 0 ij k 0 0 A Drinfel’d {K (σ), K (σ )} = iτf k K (σ )δ(σ − σ ) Double

29 / 39 Since the role of I and K is now symmetric, one can perform an O(3, 3) transformation which exchanges the momenta Ii with the ﬁelds K i , thus obtaining a new two-parameter family of models, dual to the PCM.

Principal Chiral Model: T-dualities and Doubling The dynamics derived from this algebra is equivalent to the

Franco dynamics following from the original underfomed algebra on Pezzella T ∗SU(2) and the undeformed Hamiltonian, if the new Introduction Hamiltonian is considered: and Motivation

Conclusion and Outlook

30 / 39 Principal Chiral Model: T-dualities and Doubling The dynamics derived from this algebra is equivalent to the

Franco dynamics following from the original underfomed algebra on Pezzella T ∗SU(2) and the undeformed Hamiltonian, if the new Introduction Hamiltonian is considered: and Motivation

Principal Z Chiral Model 1 −1 LM Hτα = dσ IL(Hτ,α) IM . A Drinfel’d 2 Double R Born with Geometry ij ! h (τ) iτip3δ Dual Principal H −1 = (iα)2 pj Chiral Models τ,α jp3 2 iτδip (iα) δij The Doubled Principal Chiral Model Since the role of I and K is now symmetric, one can perform an Conclusion and Outlook O(3, 3) transformation which exchanges the momenta Ii with the ﬁelds K i , thus obtaining a new two-parameter family of models, dual to the PCM.

30 / 39 The Dual Principal Chiral Models are Poisson-Lie sigma models.

Principal Chiral Model: T-dualities and Doubling

Franco Pezzella The O(3, 3) transformation Introduction and Motivation K˜(σ) = I (σ), I˜(σ) = K(σ) Principal Chiral Model A Drinfel’d when applied to Hτα and to the corresponding Poisson algebra Double leads to a new family of models having target space Born Geometry conﬁguration the group manifold of SB(2, C) spanned by the Dual Principal ﬁelds K˜ and momenta I˜i . These are the Dual Principal Chiral Chiral Models i

The Doubled Models. Principal Chiral Model

Conclusion and Outlook

31 / 39 Principal Chiral Model: T-dualities and Doubling

The Doubled Models. Principal Chiral Model The Dual Principal Chiral Models are Poisson-Lie sigma models. Conclusion and Outlook

31 / 39 Principal Chiral Model SB(2, C) - Lagrangian approach

Principal Chiral Model: Lagrangian for the PCM SB(2, C) involving T-dualities and Doubling g˜ :(t, σ) → SB(2, C), one-forms valued in the Lie algebra Franco sb(2, C) and T r non-degenerate product only invariant under Pezzella SB(2, C) action: Introduction and Motivation 1 Z S˜ = T r φ∗(˜g −1dg˜) ∧ φ∗(˜g −1dg˜) Principal 2 1,1 Chiral Model R A Drinfel’d ˜∗ −1 −1 i −1 i Double with φ (˜g dg˜) = (˜g ∂t g˜)i e˜ dt + (˜g ∂σg˜)i e˜ dσ Born ˜ −1 ˜ −1 Geometry Ai = (˜g ∂t g˜)i , Ji = (˜g ∂σg˜)i

Dual Principal Chiral Models The Lagrangian becomes then: The Doubled 1 Z Principal L˜ = dσ(A˜ hij A˜ − J˜hij J˜ ) Chiral Model 2 i j i k Conclusion R and Outlook At fixed t, all elementsg ˜ constant at the infinity form the infinite-dimensional Lie group SB(2, C)(R) ≡ Map(R, SB(2, C)), given by smooth maps g˜ : σ ∈ R → g˜(σ) ∈ SB(2, C) which are constant at infinity. 32 / 39 Principal Chiral Model SB(2, C) - Hamiltonian approach

Principal i ij Chiral Model: By introducing left momenta I˜ = A˜j h and inverting for the T-dualities and Doubling generalized velocities, one obtains the Hamiltonian: Franco Pezzella Z ˜ 1 ˜ ˜−1 IJ ˜ H = dσII (K0 ) IJ Introduction 2 and Motivation R Principal with Chiral Model h ij 0 A Drinfel’d K˜0 = Double 0 hij Born Geometry j and I˜J = (I˜ , J˜j ). Dual Principal Chiral Models Equal-time Poisson brackets The Doubled Principal ˜i ˜j 0 ij ˜k 0 Chiral Model {I (σ), I (σ )} =f k I (σ)δ(σ − σ ), Conclusion ˜i ˜ 0 ˜ ki 0 i 0 0 and Outlook {I (σ), Jj (σ )} =Jk (σ)f j δ(σ − σ ) − δj δ (σ − σ ), 0 {J˜i (σ), J˜j (σ )} =0 Dual Born geometry.

33 / 39 i By using the Lie algebra basis eI = (ei , e˜ ) one has:

−1 I −1 0I γ ∂t γ = Q˙ eI , ; γ ∂σγ = Q eI .

Q˙ I , Q0I , left generalized coordinates, respectively given by:

I −1 0I −1 Q˙ = Tr γ ∂t γeI , Q = Tr γ ∂σγeI

with Tr the Cartan-Killing metric of sl(2, C).

The Doubled Principal Chiral Model

Principal Chiral Model: T-dualities and Doubling Group valued ﬁeld Φ : R1,1 → γ ∈ SL(2, C) and the Franco −1 Pezzella left-invariant Maurer-Cartan one-form γ dγ ∈ sl(2, C):

Introduction ∗ −1 −1 −1 and Motivation Φ (γ dγ) = γ ∂t γdt + γ ∂σγdσ

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

34 / 39 The Doubled Principal Chiral Model

Principal Chiral Model: T-dualities and Doubling Group valued ﬁeld Φ : R1,1 → γ ∈ SL(2, C) and the Franco −1 Pezzella left-invariant Maurer-Cartan one-form γ dγ ∈ sl(2, C):

Introduction ∗ −1 −1 −1 and Motivation Φ (γ dγ) = γ ∂t γdt + γ ∂σγdσ

Principal Chiral Model

A Drinfel’d i Double By using the Lie algebra basis eI = (ei , e˜ ) one has: Born Geometry γ−1∂ γ = Q˙ I e , ; γ−1∂ γ = Q0I e . Dual Principal t I σ I Chiral Models I 0I The Doubled Q˙ , Q , left generalized coordinates, respectively given by: Principal Chiral Model I −1 0I −1 Conclusion Q˙ = Tr γ ∂t γeI , Q = Tr γ ∂σγeI and Outlook with Tr the Cartan-Killing metric of sl(2, C).

34 / 39 η (Lorentzian) and H (Riemannian) are the left-invariant metrics on SL(2, C) induced, respectively, by the pairings 2ImTr() and 2ReTr() on sl(2, C). They are two of the structures deﬁning a Born geometry on SL(2, C). The degrees of freedom are doubled. Performing a gauging of its global symmetries both the Lagrangian models, with SU(2) and SB(2, C) target conﬁguration spaces, can be retrieved.

Lagrangian

Principal Chiral Model: The Lagrangian density can be rewritten in terms of the left T-dualities I and Doubling generalized coordinates Q˙ as follows: Franco Pezzella 1 L = (k η + H) Q˙ I Q˙ J − Q0I Q0J Introduction IJ and Motivation 2

Principal Chiral Model with j j3 ! A Drinfel’d δij kδi + i Double (kη + H) = IJ kδi − i (δij + i j δkl ) Born j j3 k3 l3 Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

35 / 39 Lagrangian

Principal Chiral Model with j j3 ! A Drinfel’d δij kδi + i Double (kη + H) = IJ kδi − i (δij + i j δkl ) Born j j3 k3 l3 Geometry

Dual Principal Chiral Models η (Lorentzian) and H (Riemannian) are the left-invariant metrics The Doubled on SL(2, ) induced, respectively, by the pairings 2ImTr() and Principal C Chiral Model 2ReTr() on sl(2, C). They are two of the structures deﬁning a Conclusion Born geometry on SL(2, ). and Outlook C The degrees of freedom are doubled. Performing a gauging of its global symmetries both the Lagrangian models, with SU(2) and SB(2, C) target conﬁguration spaces, can be retrieved.

35 / 39 The Legendre transform gives: 1 Z H = dσ [(k η + H)−1]IJ I I + (k η + H) JI JJ . 2 I J IJ R

Poisson brackets:

0 00 K 0 00 {II (σ ), IJ (σ )} = CIJ IK δ(σ − σ ) 0 J 00 J K 0 00 J 0 0 00 {II (σ ), J (σ )} = CKI J δ(σ − σ ) − δI δ (σ − σ ) {JI (σ0), JJ (σ00)} = 0

with Q0I → JI

Hamiltonian Formalism

Principal Chiral Model: Canonical momentum: T-dualities and Doubling δL Franco ˜i ˙ J Pezzella II = (Ii , I ) = = (k η + H)IJ Q . δQ˙ I Introduction and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

36 / 39 Poisson brackets:

0 00 K 0 00 {II (σ ), IJ (σ )} = CIJ IK δ(σ − σ ) 0 J 00 J K 0 00 J 0 0 00 {II (σ ), J (σ )} = CKI J δ(σ − σ ) − δI δ (σ − σ ) {JI (σ0), JJ (σ00)} = 0

with Q0I → JI

Hamiltonian Formalism

Principal Chiral Model: Canonical momentum: T-dualities and Doubling δL Franco ˜i ˙ J Pezzella II = (Ii , I ) = = (k η + H)IJ Q . δQ˙ I Introduction and Motivation The Legendre transform gives: Principal Chiral Model 1 Z A Drinfel’d H = dσ [(k η + H)−1]IJ I I + (k η + H) JI JJ . Double 2 I J IJ Born R Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

36 / 39 Hamiltonian Formalism

Dual Principal Chiral Models Poisson brackets: The Doubled Principal 0 00 K 0 00 Chiral Model {II (σ ), IJ (σ )} = CIJ IK δ(σ − σ ) Conclusion 0 J 00 J K 0 00 J 0 0 00 and Outlook {II (σ ), J (σ )} = CKI J δ(σ − σ ) − δI δ (σ − σ ) {JI (σ0), JJ (σ00)} = 0

with Q0I → JI

36 / 39 Derivation of a whole family of equivalent PCM models described in terms of current algebra of the group SL(2, C). They can be interpreted in terms of Born geometries related by B-transformations. O(3, 3) transformations allow to ﬁnd a parametric family of T-dual PCM models, with target conﬁguration space the group SB(2, C, the Poisson-Lie dual of SU(2) in the Iwasawa decomposition of the Drinfel’d double SL(2, C). They exhibit Poisson-Lie symmetries. A further double PCM with the group manifold of SL(2, C) has been constructed.

Conclusion

Principal Chiral Model: T-dualities and Doubling

Franco Recovering many aspects of the Abelian T-duality. Pezzella

Introduction and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

37 / 39 They can be interpreted in terms of Born geometries related by B-transformations. O(3, 3) transformations allow to ﬁnd a parametric family of T-dual PCM models, with target conﬁguration space the group SB(2, C, the Poisson-Lie dual of SU(2) in the Iwasawa decomposition of the Drinfel’d double SL(2, C). They exhibit Poisson-Lie symmetries. A further double PCM with the group manifold of SL(2, C) has been constructed.

Conclusion

Principal Chiral Model: T-dualities and Doubling

Franco Recovering many aspects of the Abelian T-duality. Pezzella Derivation of a whole family of equivalent PCM models Introduction and Motivation described in terms of current algebra of the group SL(2, C). Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

37 / 39 O(3, 3) transformations allow to ﬁnd a parametric family of T-dual PCM models, with target conﬁguration space the group SB(2, C, the Poisson-Lie dual of SU(2) in the Iwasawa decomposition of the Drinfel’d double SL(2, C). They exhibit Poisson-Lie symmetries. A further double PCM with the group manifold of SL(2, C) has been constructed.

Conclusion

Principal Chiral Model: T-dualities and Doubling

A Drinfel’d B-transformations. Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

37 / 39 A further double PCM with the group manifold of SL(2, C) has been constructed.

Conclusion

Principal Chiral Model: T-dualities and Doubling

A Drinfel’d B-transformations. Double O(3, 3) transformations allow to ﬁnd a parametric family of Born Geometry T-dual PCM models, with target conﬁguration space the group Dual Principal Chiral Models SB(2, C, the Poisson-Lie dual of SU(2) in the Iwasawa The Doubled decomposition of the Drinfel’d double SL(2, C). They exhibit Principal Chiral Model Poisson-Lie symmetries.

Conclusion and Outlook

37 / 39 Conclusion

Principal Chiral Model: T-dualities and Doubling

37 / 39 Extension to String Theory on Group Manifolds. To apply this scheme of construction to the world-sheet action. In this case, a manifestly O(d, d)-invariant action may be written, considering that the conﬁguration space is a diﬀerentiable manifold. It would be interesting to A doubled world-sheet string action, as discussed for Principal Chiral Models, could be written and then perform the low energy limit. This limit result should reproduce all the results so far obtained in Double Field Theory.

Outlook

Principal Chiral Model: T-dualities and Doubling

Franco Pezzella Extension with a Wess-Zumino term: this could provide a deeper insight on the geometric structures of String Theory on AdS . Introduction 3 and Motivation

Principal Chiral Model

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

38 / 39 To apply this scheme of construction to the world-sheet action. In this case, a manifestly O(d, d)-invariant action may be written, considering that the conﬁguration space is a diﬀerentiable manifold. It would be interesting to A doubled world-sheet string action, as discussed for Principal Chiral Models, could be written and then perform the low energy limit. This limit result should reproduce all the results so far obtained in Double Field Theory.

Outlook

Principal Chiral Model: T-dualities and Doubling

A Drinfel’d Double

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

38 / 39 Outlook

Principal Chiral Model: T-dualities and Doubling

Franco Pezzella Extension with a Wess-Zumino term: this could provide a deeper insight on the geometric structures of String Theory on AdS . Introduction 3 and Motivation Extension to String Theory on Group Manifolds. Principal Chiral Model To apply this scheme of construction to the world-sheet action. A Drinfel’d Double In this case, a manifestly O(d, d)-invariant action may be

Born written, considering that the conﬁguration space is a Geometry diﬀerentiable manifold. It would be interesting to A doubled Dual Principal Chiral Models world-sheet string action, as discussed for Principal Chiral The Doubled Models, could be written and then perform the low energy limit. Principal Chiral Model This limit result should reproduce all the results so far obtained Conclusion in Double Field Theory. and Outlook

38 / 39 Principal Chiral Model: T-dualities and Doubling

Franco Pezzella

Introduction and Motivation

Principal Chiral Model

A Drinfel’d Double Thank you for your attention.

Born Geometry

Dual Principal Chiral Models

The Doubled Principal Chiral Model

Conclusion and Outlook

39 / 39