Courant algebroids and string low energy effective actions

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Thesis

Reference

Courant algebroids and string low energy effective actions

VALACH, Fridrich

Abstract

Cette thèse est consacrée à l'application des méthodes de la géométrie de Poisson dans le contexte de la théorie des cordes, notamment pour étudier la limite d'énergie basse et les dualités de cette dernière. Les résultats contenus dans la thèse sont les suivants: - Reformulation des notions de tenseur de Ricci et de courbure scalaire généralisées aux algébroïdes de Courant. - Dérivation de ces tenseurs de courbure à partir de la variation d'une action naturelle. - Preuve de la compatibilité de la T-dualité de Poisson-Lie avec les transformations infinitésimales du groupe de renormalisation à une boucle, dans la configuration générale. - Preuve de la compatibilité de la T-dualité de Poisson-Lie avec la partie bosonique des équations d'arrière-plan pour les 5 théories des supercordes. - Découverte des nouvelles classes de solutions d'équations de supergravité modifiées sur des espaces symétriques.

VALACH, Fridrich. Courant algebroids and string low energy effective actions. Thèse de doctorat : Univ. Genève, 2019, no. Sc. 5374

DOI : 10.13097/archive-ouverte/unige:122409 URN : urn:nbn:ch:unige-1224093

Available at:

http://archive-ouverte.unige.ch/unige:122409

Disclaimer: layout of this document may differ from the published version.

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UNIVERSITÉ DE GENÈVE Section de Mathématiques

FACULT´E DE SCIENCES Docteur P. ˇSevera

Courant Algebroids And String Low Energy Effective Actions

TH` ESE

Présentée à la Faculté des sciences de l’Université de Genève Pour obtenir le grade de Docteur ès sciences, mention mathématiques

Par

Fridrich VALACH

de

Kom´arno (Slovaquie)

Th` ese No. 5374

GEN`EVE

Centre d’impression de l’UNIGE 2019

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R´ esum´ e

Cette thèse est consacrée à l’application des méthodes de la géométrie de Poisson dans le contexte de la théorie des cordes, notamment pour étudier la limite d’énergie basse et les dualités de cette dernière. Les résultats contenus dans la thèse sont les suivants:

• Reformulation des notions de tenseur de Ricci et de courbure scalaire généralisées aux algébro¨ıdes de Courant, en utilisant des objets naturels par rapport aux pullbacks et réductions.

• D´erivation de ces tenseurs de courbure `a partir de la variation d’une action naturelle.

• Preuve de la compatibilité de la T-dualité de Poisson-Lie avec les transformations infinitésimales du groupe de renormalisation à une boucle, dans la configuration générale avec les “dressing cosets” et les specta- teurs.

• Preuve de la compatibilité de la T-dualité de Poisson-Lie avec la partie bosonique des équations d’arrière-plan pour les 5 théories des supercordes.

• Découverte des nouvelles classes de solutions d’équations de super- gravité modifiées sur des espaces symétriques.

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Abstract

This thesis is devoted to the application of methods of Poisson geometry in the context of string theory, namely to study its low energy limit and its dualities. The results contained in the thesis are the following ones:

• Reformulation of the notions of the generalized Ricci tensor and scalar curvature on Courant algebroids, using objects natural w.r.t. pull-backs and reductions.

• Derivation of these curvature tensors from the variation of a natural action functional.

• Proof of the compatibility of the Poisson-Lie T-duality with the one- loop renormalization group flow, in the general setup with dressing cosets and spectators.

• Proof of the compatibility of the Poisson-Lie T-duality with the bosonic part of the superstring background equations for the 5 superstring theories.

• Discovery of new classes of solutions of modified supergravity equations on symmetric spaces.

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Acknowledgment

First, I would like to thank my supervisor Pavol ˇSevera, for his patient guid- ance, continuous support, jokes, and all the time he spent sharing his knowl- edge with me. Furthermore, I would like to express my gratitude to my colleagues for all the constructive fun we had in the office. Special thanks also go to my brother and my family for the all their help and support. Fi- nally, I thank my friends from the dancing ensemble FS Kamz´ık, for making my life merrier.

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Chapter 1 Introduction

In recent decades, we have witnessed many interesting developments and important discoveries in theoretical physics, originating from the search for a theory of quantum gravity. In particular, a great deal of new mathematical concepts and methods were discovered in the study of string theory, with a deep impact on the field of mathematical physics.

In this thesis we focus on one of these discoveries, namely the connection between Courant algebroids, NQ manifolds, Poisson-Lie T-duality, string sigma models, and supergravities (or low energy limits of string theory). In two recent papers by the author and his supervisor, which are incorporated in this thesis, some of these links have been elucidated. In particular, we have shown the compatibility of the Poisson-Lie T-duality with the one-loop renormalization group flow, and with the string background equations. Let us comment briefly on the importance of these results.

The Poisson-Lie T-duality is a vast non-abelian generalization of the T- duality of string theory. The latter provides a crucial strand in the dense web of dualities between various superstring theories and M-theory. In turn, these dualities offer an invaluable tool for the investigation of the (often non- perturbative) aspects of the theory. In other words, since string theory is a notoriously complex subject, dualities provide some very useful “bridges”, connecting two seemingly different setups.

Nevertheless, the Poisson-Lie T-duality, while reasonably well understood on the classical level, poses some problems in the quantum setup. One of the questions raised by the quantization is the question of the compatibility of the renormalization group flow with the duality. This has been answered positively in the general case, in the above papers.

In fact, the author believes that the theory developed in these two works provides a solid ground for a general investigation of the areas of string low energy limits and dualities. In addition to the compatibility proofs, it leads to

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a new method of discovering solutions to (modified) supergravity equations, via purely algebraic methods.

Although the original motivation for these studies comes from string theory, the results described in this thesis can be interpreted in a purely mathematical way. We define and study the Ricci flow for generalized metrics on arbitrary Courant algebroids. In a particular case this corresponds to the Ricci flow for metric connections with torsion. Our proof of its compatibility with Courant algebroid pullbacks and reductions then provides new insights into the nature of the Ricci flow.

The text is stuctured as follows: First, in chapters 2 and 3 we provide some background material. More specifically, in Chapter 2 we start by dis- cussing Poisson-Lie groups and Lie algebroids. This naturally paves a way towards Courant algebroids, to which we turn next. After explaining basic constructions we reformulate the theory in the language of NQ manifolds.

We then outline the AKSZ construction and we finish the chapter with the description of the Poisson-Lie T-duality, supplemented by examples.

Chapter 3 is more physical - we introduce some basic concepts from string theory, first in the purely bosonic case and then briefly in the case of super- strings.

In Chapter 4 we summarize the results of the two articles, which we then include in the remaining two chapters. In the first one (Chapter 5) we provide a proof of the compatibility of the Poisson-Lie T-duality with the one-loop renormalization group flow, in the general case of spectators.

This is done by reformulating the notion of the generalized Ricci tensor for Courant algebroids.

The rest is done in the subsequent article in Chapter 6. In particular, we extend the proof to the equivariant supergravity setup, reformulate the notion of scalar curvature, and find new classes of solutions to the modified type II supergravity equations on symmetric spaces.

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Chapter 2 Courant algebroids and Poisson-Lie T-duality

We assume that the reader is familiar with the basics of Poisson geometry and Lie theory. The objects appearing in this chapter are finite dimensional and the summation convention is assumed.

2.1 Poisson-Lie groups

This section describes several notions and results from [11, 12, 13]. We begin with

Definition 2.1.1. A Poisson-Lie group is a Lie group equipped with a Pois- son structure such that the multiplication map G×G→G is Poisson.

The last condition can be written in terms of the Poisson bivector π as π_gh =g·π_h+π_g·h,

for g, h ∈G. Hereg· and ·h denote the push-forward by the left translation by g and the right translation by h, respectively.

Remark 2.1.2. Notice that, in particular, π vanishes at the identity, meaning that Poisson-Lie groups are never symplectic. Similarly, if i: G → G is the inverse map, we have i∗π =−π. Thus, requiring that the inverse map is Poisson would neccessarily lead to π= 0.

An “infinitesimal” object associated to a Poisson-Lie group is the Lie bialgebra.

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Definition 2.1.3. A Lie bialgebra is a Lie algebra g together with a map δ: g → g∧g whose dual induces a Lie algebra structure on g^∗ and which satisfies the cocycle condition

ad⁽²⁾_x δ(y)−ad⁽²⁾_y δ(x)−δ([x, y]) = 0, ∀x, y ∈g, where ad⁽²⁾_x stands for ad_x⊗1 + 1⊗ad_x.

For a given Poisson-Lie group, we obtain the map δ (on its Lie algebra) by linearizing the Poisson bivector field at the identity of G, i.e.

δ(x) := (L_x_˜π)₁,

where ˜xis any vector field whose value at the identity isx∈g. This construction can be shown to give a one-to-one correspondence between Lie bialgebras and 1-connected Poisson-Lie groups.

To continue, recall the following definition.

Definition 2.1.4. Let V be a vector space equipped with a symmetric or skew-symmetric non-degenerate bilinear form. We then say that a subspace W ⊂V is

- isotropic if W ⊂W^⊥, - coisotropic if W ⊃W^⊥, - Lagrangian if W =W^⊥.

Remark 2.1.5. In particular, isotropic subspaces cannot be more than half- dimensional, coisotropic subspaces cannot be less then half-dimensional, and Lagrangian subspaces are exactly half-dimensional. Furthemore, in the case of the symmetric bilinear form, Lagrangian subspaces exist only if the bilinear form has the symmetric signature, i.e. (n/2, n/2) for n = dimV.

Now, there exists another type of structure equivalent to Lie bialgebras, which is often more convenient to work with.

Definition 2.1.6. AManin tripleis the triple(d,a,b), wheredis a Lie algebra with an invariant inner product and a, b are complementary Lagrangian Lie subalgebras of d. The Lie algebra d is called the Drinfeld double.

For (g, δ) a Lie bialgebra, we define a Manin triple as follows: We set a := g, b := g^∗, and we extend the Lie algebra structure on g and g^∗ to d :=g⊕g^∗ in a unique way such that it leaves the canonical inner product

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(given by the pairing) on g⊕g^∗ invariant. Explicitly, for x ∈ g, ξ ∈ g^∗ we have

[x, ξ] = ad^∗_xξ−ad^∗_ξx.

Again, this can be shown to give a one-to-one correspondence between Lie bialgebras and Manin triples. It also proves that the dual of a Lie bialgebra is also a Lie bialgebra.

Definition 2.1.7. Poisson-Lie groups G, G^∗, corresponding to a Lie bialgebra and its dual, are called dual Poisson-Lie groups.

Example 2.1.8 (Semi-abelian Drinfeld doubles). As our Lie bialgebra we take any gwithδ = 0. This givesd=gng^∗, whereg acts on its dual by the coadjoint action. The dual pair of Poisson-Lie groups is then (G,g^∗). Here G is equipped with the zero Poisson structure, and g^∗ (taken as an abelian group) has the KKS Poisson structure.

Example 2.1.9. Let g be a real semisimple split Lie algebra (e.g. sl_n). We setd=g⊕¯g(here bar denotes the opposite inner product),ais the diagonal embedding g → d, and b consists of those elements of b₊⊕b− (direct sum of the Borel subalgebra and its opposite Borel subalgebra) that coincide on the Cartan subalgebra.

To finish, let us define an “easier” analogue of the Manin triples.

Definition 2.1.10. A Manin pair is a pair(g,h) of a Lie algebra g with an invariant inner product and a Lagrangian subalgebra h⊂g.

Example 2.1.11. We take d = gl₂ with the inner product hx, xi = detx.

We can take h to be the Lie subalgebra of matrices with a vanishing bottom (resp. top) row, or alternatively, the subalgebra consisting of matrices with vanishing left (resp. right) column.

2.2 Lie algebroids

Although not directly relevant for our study of dualities, we shall now briefly discuss Lie algebroids, introduced in [20]. This will provide a natural bridge towards Courant algebroids.

Definition 2.2.1. A Lie algebroid is a triple (A,[·,·], ρ) where E → M is a vector bundle together with an R-linear Lie bracket [·,·] on the space of sections Γ(E) and a vector bundle map ρ: A → T M, called the anchor, satisfying

∞

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Remark 2.2.2. It is easy to show that as a consequence of the definition, the anchor map induces a Lie algebra map between Γ(E) and Γ(T M).

To get a better intuition for the definition, we now list a few illustrative examples.

Example 2.2.3. For M a manifold, the triple (T M,[·,·],id), with [·,·] the commutator of vector fields, is a Lie algebroid.

Example 2.2.4. Forg a Lie algebra, we have the Lie algebroid over a point (g,[·,·],0).

Example 2.2.5. Generalizing the previous example, we can take, forga Lie algebra and M a manifold, the bundle of Lie algebras (M ×g,[·,·],0).

Example 2.2.6. Generalizing yet again, suppose we have an action of a Lie algebra g on a manifold M (i.e. a Lie algebra homomorphism g→Γ(T M)).

Then we have a unique Lie algebroid structure on the trivial bundleM×g→ M, with the anchor given by the Lie algebra action and the bracket on constant sections corresponding to the bracket on g. This is the action Lie algebroid.

Example 2.2.7. Let P → M be a principal G-bundle. The Atiyah Lie algebroid consists of the vector bundleT P/G →M with the bracket inherited from the Lie bracket of vector fields on P and with ρ being the obvious projection to T M.

An important connection between Poisson manifolds and Lie algebroids is provided by the following construction.

Example 2.2.8. Let (M, π) be a Poisson manifold. Letπ^]: T^∗M →T M be the map (π^](α))(β) = π(α, β). We then have a Lie algebroid (T^∗M,[·,·]_π, π^]), where

[α, β]_π :=L_π^]_(α)β−L_π^]_(β)α−dπ(α, β).

A more conceptual viewpoint of Lie algebroids will be provided in section 2.4

2.3 Courant algebroids

We now come to the main tool for our study of dualities.

Definition 2.3.1 ([19]). A Courant algebroid (from now on we shall often refer to it as CA) is a quadruple (E,[·,·],h·,·i, ρ) consisting of

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- a vector bundle E →M, - an R-linear bracket

[·,·] : Γ(E)×Γ(E)→Γ(E),

- a non-degenerate symmetric bilinear form h·,·i on the fibres of E, - a vector bundle map (the anchor) ρ:E →T M,

satisfying for every u, v, w ∈Γ(E) and f ∈C^∞(M) [u,[v, w]] = [[u, v], w] + [v,[u, w]]

[u, f v] =f[u, v] + (ρ(u)f)v ρ(u)hv, wi=h[u, v], wi+hv,[u, w]i [u, v] + [v, u] =ρ^tdhu, vi,

where ρ^t: T^∗M →E^{∗ h·}−−→^,·i E is the transpose of the anchor.

Remark 2.3.2. Notice that the bracket is not required to be skew-symmetric, but we still ask the Jacobi identity to hold. (There is an equivalent definition, which uses the skew-symmetrized version of the bracket.) Note also that if ρ has a constant rank then E/kerρ^t is a Lie algebroid.

Courant algebroids originated in the study of generalizations of Drinfeld doubles/Manin triples in the Lie algebroid context (c.f. the standard CA below). For a review of their history see [17].

Again, it follows from the axioms that for x, y ∈Γ(E), ρ([x, y]) = [ρ(x), ρ(y)].

Combining this with the last condition in the definition, we get that 0→T^∗M ^ρ

t

−→E −→^ρ T M →0 (2.1)

is a chain complex.

Let us now look at different classes of Courant algebroids.

Definition 2.3.3. We say a Courant algebroid E is - regular if ρ has a constant rank,

- transitive if ρ is surjective,

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- exact if the sequence (2.1) is exact.

In particular, exact =⇒ transitive =⇒ regular. Also, notice that E is exact iff it is transitive and rkE = 2 dimM.

Standard two classes of examples of CAs are

Example 2.3.4. Forg a Lie algebra with and invariant inner producth·,·i, we set (g,[·,·],h·,·i,0), where we treatgas a vector bundle over a point. This is a transitive CA.

Example 2.3.5. Let M be a manifold and H a closed 3-form on M. We then take E =T M⊕T^∗M with the standard pairing,ρ the projection onto T M, and the bracket given by

[x+α, y+β] = [x, y] +L_xα−di_yβ+H(x, y,·), (2.2) for x, y ∈ Γ(T M), α, β ∈ Γ(T^∗M). Notice that E is exact. In the special case H = 0 we call E the standard Courant algebroid[6].

The standard Courant algebroid provides an interesting unifying picture for Poisson and (pre-)symplectic geometry, via Dirac structures, investigated in [6, 7]:

Definition 2.3.6. LetE be a Courant algebroid. We say a subbundleL⊂E is a Dirac structure if it is Lagrangian and involutive, i.e. L^⊥ = L and [Γ(L),Γ(L)]⊂Γ(L).

It is straightforward to check the following.

Proposition 2.3.7. LetE be the standard CA andL⊂E a Dirac structure.

- If L∩T M = 0 then L is a graph of Poisson tensor (seen as a map T^∗M →T M). Any Poisson tensor arises in this way.

- If L∩T^∗M = 0 then L is a graph of a closed 2-form (seen as a map T M →T^∗M). Any closed 2-form arises in this way.

Our last class of examples comes from actions of Lie algebras on manifolds.

Example 2.3.8. [18] Let gbe a Lie algebra with an invariant inner product h·,·i. Suppose g acts on M with coisotropic stabilizers. Then there is a unique CA structure on the trivial bundle M ×g → M for which the inner product is given by h·,·i, the bracket of constant sections coincide with the bracket of g, and for which the anchor is given by the action. Such a CA is called an action Courant algebroid. If g acts transitively with Lagrangian stabilizers, the resulting CA is exact.

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Exact CAs can be classified [23]. The idea is to use a Lagrangian splitting of the short exact sequence (2.1) (such a splitting always exists). We then have E ∼=T M ⊕T^∗M as vector bundles, and a quick calculation shows that the Dorfman bracket has the form (2.2) for some closed H ∈ H³(M). If we change the splitting, we obtain a different H⁰ which differs from H only by an exact 3-form. This leads to

Theorem 2.3.9. Any exact CA has the form from ex. 2.3.5. For a given base M the exact Courant algebroids are classified by H³(M).

Classification of transitive CAs is more complicated. It is, however, not too difficult to prove the following [23].

Theorem 2.3.10. Any transitive Courant algebroid over M is locally of the form T M ⊕T^∗M ⊕g for some Lie algebra g with an invariant inner product. The pairing on T M ⊕T^∗M is then the standard one, the anchor is the projection to T M and the bracket has the form

[x+α+a, y+β+b] = [x, y] +L_xα−di_yβ+L_xb−L_ya+ [a, b]_g+ρ^t(hda, bi), where x, y ∈Γ(T M), α, β ∈Γ(T^∗M), and a, b∈Γ(M ×g).

There are two important notions that are relevant for the Poisson-Lie T-duality - CA pullbacks [18] and reductions of equivariant CAs [23, 3].

Definition 2.3.11. Suppose E → M is a CA and f: M⁰ → M a smooth map. We say that a CA structure on E⁰ :=f^∗E is a CA-pullback ofE if we have the following compatibility (prime now indicates the corresponding CA operations on E⁰):

hf^∗x, f^∗yi⁰ =f^∗hx, yi, [f^∗x, f^∗y]⁰ =f^∗[x, y],

f_∗ρ⁰(f^∗x) =ρ(x), for every x, y ∈Γ(E).

Notice that only the anchor ρ⁰ is not determined by f. We shall discuss examples in section 2.6.

Definition 2.3.12. Let g be a Lie algebra with an invariant bilinear symmetric form (possibly degenerate). A g-equivariant Courant algebroid is a CA E →M with a linear map χ: g→Γ(E), injective at every point on M, satisfying

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The Lie algebra g thus acts on E by x7→[χ(x),·], and therefore also on M, via x 7→ ρ(χ(x)). If the action on E integrates to an action of a connected Lie group G, we say that E is G-equivariant.

Definition 2.3.13. SupposeE →M isG-equivariant, with a free and proper action on M. We then define, for any x∈M,

(E_/G)x := (χx(g))^⊥/χx(g⁰),

where g⁰ is the kernel of the bilinear form on g. After taking the quotient by G, we get a vector bundle E_/G → M/G, which has a CA structure induced from E.

Notice that ifE is exact and g has the trivial pairing, then E_/G is exact as well (for a nontrivial pairing it is transitive).

A classification result from [24] is

Theorem 2.3.14. Suppose G acts freely and transitively on M. Then there exists aG-equivariant exact CA over M (which induces the given action ofG on M) iff the first Pontryagin class [hF, Fi_g]of M →M/Gvanishes. In that case, there is a free and transitive action of H³(M/G) on the isomorphism classes of such G-equivariant exact CAs.

2.4 NQ manifolds

We now turn to a more conceptual viewpoint of both Lie and Courant algebroids, namely the one offered by NQ manifolds. Those formalize the idea of “spaces” for which the algebra of functions forms a non-negatively graded commutative algebra (hence the ‘N’ from NQ), instead of the commutative algebra of functions on usual manifolds. They will be furthermore equipped with some geometric structure, namely a cohomological vector field (usually denoted Q, hence the ‘Q’ from NQ), and sometimes with a symplectic form.

This section is mostly based on the works [29, 1, 23, 26].

Definition 2.4.1. Let W =L∞

i=0W_i be an N-graded vector space.¹ We define the N-graded symmetric algebraSW as the quotient of the tensor algebra of W by the ideal generated by x⊗y−(−1)^|x||y|y⊗x, for x, y homogeneous elements of W of degrees |x| and |y|, respectively.

Definition 2.4.2. An N-manifold M is an ordinary manifold M together with a sheaf of N-graded algebras, which is locally isomorphic toC^∞(U)×SV, for V =L∞

i=1Vi a finite-dimenional N-graded vector space with trivial degree 0 part (U is an open subset of M). We say M is the base or body of M.

1Here we understandNas the additive semigroup{0,1,2, . . .}.

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One should think of homogeneous basis elements ofV as coordinates (in addition to the coordinates on M), and the sections of the structure sheaf as functions on M.

Definition 2.4.3. A map M → M⁰ between two N-manifolds M, M⁰ is a homomorphism C^∞(M⁰)→C^∞(M) of graded commutative algebras.

We now state an equivalent but somehow more elegant (and concise) definition of the N-manifold from [23].

Definition 2.4.4. An N-manifold is a supermanifold with an action of the semigroup (R,×) such that −1 acts as the parity operator (i.e. it flips the signs of odd functions). A map between two N-manifolds is a map of supermanifolds preserving the (R,×)-action.

Let us now make a few comments and compare the two definitions. First of all, the (R,×)-action here corresponds to the N-grading. More precisely, if a function f satisfies f(λ · x) = λ^mf(x) for every λ ∈ R, we call m the degree of f. The fact that the degree is always a non-negative integer follows from the role of the degrees as exponents in a Taylor expansion in the semigroup R.² Finally, the condition concerning the action of −1 expresses the compatibility between the Nand Z² grading. Notice also that the action of 0∈Ris a projection onto the base manifold (i.e. it kills all the coordinates of non-zero degree).

Example 2.4.5. If E → M is a vector bundle (in the ordinary manifold setup), we define E[n] to be the graded manifold obtained by shifting the degree of the fiber coordinates by n. The most important example of this construction is the shifted tangent bundle T[1]M, having the property that C^∞(T[1]M) ∼= Ω(M). The definition of (shifted) vector bundles extends naturally to the case when M and E are N-manifolds.

We shall also extend to the N-manifold setup all the standard operations and structures from differential geometry, following the viewpoint of functions as elements of an algebra. For instance, a vector field is defined as a derivation of the algebra of functions. As an example, notice that every N-manifold M possesses a canonical vector field related to the grading:

Definition 2.4.6. Let M be an N-manifold. We define the Euler vector field E by:

Ef = (degf)f,

2The action of (R,×) is given by a mapC^∞(M)→C^∞(M×R). We can thus Taylor

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for f a function of a definite degree. Thus, denoting the coordinates by x^a, E =X

a

(degxâ)xâ∂_xâ. Alternatively, E generates the action of (R,×).

Similarly, differential forms on an N-manifoldMare understood as functions on T[1]M. In particular, this means that as a graded commutative algebra, the degree in Ω(M) (we shall call it the total degree deg_tot) is the sum of the form degree and the degree on M (i.e. the one measured by the Euler vector field on M; we shall simply refer to it as the degree). For example, if f, g₁, . . . , g_k are functions of a definite degree,

deg_tot(f dg₁. . . dg_k) = degf+ degg₁+· · ·+ degg_k+k = deg(f dg₁. . . dg_k) +k We shall now consider N-manifolds equipped with an additional structure.

Definition 2.4.7. An NQ-manifold is an N-manifold together with a degree 1 vector field Q satisfying Q² = 0.

Example 2.4.8. If Mis an N-manifold then T[1]Mhas a canonical structure of an NQ-manifold, with Q being the de Rham differential (understood as a derivation of the algebra C^∞(T[1]M) ∼= Ω(M)). To be more explicit, let us choose coordinatesxâ onMand denote the corresponding coordinates on the tangent fibers by ξâ. ThenQ=ξâ∂_xâ.

Notice also that if M is an ordinary manifold then T[1]M is equipped with a canonical measure, with R

T[1]Mf = R

Mf^top, where on the RHS we understand f as a differential form onM and then take its top part.

We are now ready to see an equivalent (and more conceptual) definition of the Lie algebroid.

Theorem 2.4.9. [31] There is a one-to-one correspondence between Lie algebroids and NQ-manifolds with coordinates only in degrees 0 and 1, given as follows:

Suppose E is a Lie algebroid. Let us choose a local trivialization given by a choice of sections e_a spanning the fibers at every point. We will denote the corresponding fiber coordinates by e^a and the coordinates on the base by xⁱ. Then the corresponding NQ-manifold is M=E[1], with the differential given by

Q=−¹₂câ_bc(x)e^be^c∂eâ+ρⁱ_a(x)eâ∂_xⁱ,

where ρ is the anchor, and c are the structure constants relative to the trivialization, i.e. [e_a, e_b] =c^c_ab(x)e_c.

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Remark 2.4.10. In the case when the base of the Lie algebroid is a point, we get the realization of Lie algebras as NQ-manifolds, namely

M=g[1], Q_CE =−¹₂c^a_bce^be^c∂_e^a. In other words, C^∞(g[1]) = V

g^∗, equipped with QCE, is the Chevalley- Eilenberg algebra corresponding to g.

Remark 2.4.11. Similarly, T[1]g[1] has a well-defined differential, given by the sum of the de Rham differential and the lift of the differential on g[1]

to the tangent bundle (i.e. taking Q to be the Lie derivative w.r.t. Q_CE of Ω(g[1]) ∼= C^∞(T[1]g[1])). The differential graded algebra C^∞(T[1]g[1]) is then the Weil algebra of g.

Definition 2.4.12. An NQ symplectic manifold of degree n is an NQ- manifold together with a symplectic form (i.e. a closed, non-degenerate two- form) ω of degree n, satisfying L_Qω = 0.

Notice that the non-degeneracy of ω implies that only coordinates of degrees 0, . . . , ncan be present on an NQ symplectic manifold of degree n.

Example 2.4.13. If M is an NQ manifold with no coordinates of degree larger than n, then T^∗[n]M has a canonical structure of an NQ symplectic manifold of degreen, with the usual cotangent symplectic form and with the differential lifted from the one on M.

One difference between the non-graded and graded setup comes from the simple observation:

Ifαis a closed p-form of degree n, withp, n >0, then (using the Cartan’s formula) we have

α= _n¹L_Eα= ¹_ndi_Eα, implying α is exact.

For example,i_Qωis a closed 1-form of degreen+ 1>0. Thus there exists a function H of degree n+ 1 satisfying i_Qω = dH, i.e. H is a Hamiltonian for the vector field Q. The condition Q² = 0 then translates to {H, H}= 0, where {·,·} is the Poisson bracket on Minduced by ω. This is the classical master equation. Conversely, any H satisfying {H, H} = 0 gives rise to a vector field Q satisfying Q² = 0 andL_Qω = 0.

Let us now look at the NQ symplectic manifolds in the lowest degrees.

First, an NQ symplectic manifold of degree 0 is an ordinary manifold with Q= 0 and ω an ordinary symplectic form. Thus, NQ symplectic manifolds of degree 0 are in one-to-one correspondence with (ordinary) symplectic manifolds.

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Theorem 2.4.14. There is a one-to-one correspondence between NQ symplectic manifolds of degree 1 and Poisson manifolds via (M, π) 7→ T^∗[1]M, where the latter corresponds to the Lie algebroid of(M, π). Explicitly,H_T^∗_[1]M is the Poisson bivector, seen as a function on T^∗[1]M.

Finally, we have

Theorem 2.4.15. [21, 23] There is a one-to-one correspondence between NQ symplectic manifolds of degree 2 and Courant algebroids, described as follows:

Suppose E is a Courant algebroid. We choose a local trivialization (over U ⊂ M) given by a choice of sections ea spanning the fibers at every point.

Let xⁱ, p_i be the coordinates on T^∗U. Then the corresponding NQ symplectic manifold is given locally as T^∗[2]U×V[1], with

ω =dpidxⁱ+gabdeâde^b, H =−¹₆câbc(x)eâe^be^c+ρⁱ_a(x)pieâ,

where ρ is the anchor, g is the inner product on E and c_abc(x) = he_a[e_b, e_c]i.

Example 2.4.16. An NQ symplectic manifold corresponding to an exact CA with the three-form η (here we reserve the letterH to the Hamiltonian) is

T^∗[2]T[1]M, H =d−η.

Here we see das vector field onT[1]M, hence a function onT^∗[2]T[1]M, and η as a function on T[1]M pulled back to T^∗[2]T[1]M.

Let us pick (degree 0) coordinates xⁱ on M, ξⁱ the induced (degree 1) coordinates on the tangent fibers, and takep_i andπ_i to be the corresponding dual coordinates for xⁱ andξⁱ, respectively (degπ_i = 1, degp_i = 2). We then have

H =p_iξⁱ− ¹₆η_ijk(x)ξⁱξ^jξ^k. Remark 2.4.17. We thus get a natural sequence

symplectic manifolds→Poisson manifolds→Courant algebroids→ · · · In passing we remark that higher terms in this sequence also play an important role in mathematical physics [2].

Finally, let us mention that the process of equivariant reduction of CAs corresponds, in the NQ language, to a Hamiltonian reduction, for details see [25].

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2.5 AKSZ construction

We now turn to a beautiful construction by Alexandrov-Kontsevich-Schwarz- Zaboronski [1]. One can use it to produce various classical field theories governed by an action satisfying (automatically) the classical master equation.

We shall, however, only look at the degree zero part of the story (in the BV sense), as this is sufficient for the our (non-BV) purposes.

A starting data for this construction consists simply of - an oriented compact (n+ 1)-dimensional manifoldN - an NQ symplectic manifoldM of degreen

- a suitable boundary condition, if∂N 6=∅.

The fields are maps

f: T[1]N → M.

Their dynamics is governed by the action S(f) =

Z

T[1]N

i_d(f^∗θ)−f^∗H,

with θ = i_Eω/n satisfying dθ =ω (we restrict ourselves to the case n > 0).

By construction, the extrema of this functional are the maps T[1]N → M which preserve the differential (i.e. f∗d =Q). Importantly, if ∂N =∅ then the resulting field theory is topological.

We now briefly discuss the situation in the two lowest degrees.

Example 2.5.1. [10] For n = 1, M corresponds to a Poisson manifold and dimN = 2. The resulting field theory is called the Poisson sigma model and it plays a central role in several fields of mathematical physics, such as the deformation quantization [16, 4],³ symplectic groupoids [5],⁴ and 2D gravity theories [10, 28].⁵

Example 2.5.2. [9, 22] For n= 2, Mis given by a CA and dimN = 3. We call the resulting model the Courant sigma model. In particular, for E =g we recover the Chern-Simons theory. In general, using the notation from Theorem 2.4.15 we get the action

S = Z

N

p_idxⁱ+¹₂g_abeâde^b−ρⁱ_a(x)p_ieâ+¹₆c_abc(x)eâe^be^c. (2.3)

3The Kontsevich star product can be seen as a 3-point function of the Poisson sigma model on a disk.

4The phase space of the Poisson sigma model on a strip (with vanishing boundary conditions) is the symplectic groupoid corresponding to the Poisson manifold.

5Gravity theories can be obtained by identifying the 1-form part of the fieldf with the

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2.6 Poisson-Lie T-duality

This section discusses the concept of Poisson-Lie T-duality, coming from the article [14] of Klimˇc´ık and ˇSevera. We shall follow the exposition from the works [23, 24, 25], which provide a reformulation of the phenomenon using the language of Courant algebroids. We start by introducing the last remaining structure we need for our description of the duality.

Definition 2.6.1. [8] LetE be a CA. Ageneralized metric V₊ is a half-rank subbundle of E on which the induced inner product is positive definite.

Remark 2.6.2. This is not the most general definition - one can relax both the signature and rank condition. However, for the purpose of this section, the above definition is sufficient.

Furthermore, different equivalent definitions of the generalized metric can be found in the literature. Namely, one often considers, instead of V₊, the reflection operator R w.r.t. the subbundle, or the inner producth·, R(·)i.

Example 2.6.3. Consider a generalized metric V+ ⊂ E in an exact CA.

Recall that, choosing a Lagrangian splitting of the sequence 2.1, we have an identification E ∼= T M ⊕T^∗M with both T M and T^∗M Lagrangian subbundles. Since the inner product on the generalized metric is positive definite, V₊ ∩ T^∗M = 0. In particular, one can see V₊ as a graph of a C^∞(M)-linear map T M → T^∗M, i.e. as a section e of T^∗M ⊗T^∗M, with positive definite symmetric part.

Now, it is easy to see that a change in the choice of the Lagrangian splitting only affects the skew-symmetric part of e. Moreover, there exists exactly one Lagrangian splitting for which the skew-symmetric part of e vanishes. Denoting the 3-form corresponding to this splitting by H and the symmetric part ofe byg, we see that the generalized metric on an exact CA is the same as the pair (g, H) of a metric and a closed 3-form [8].

This data can then be used to construct a sigma model, describing the classical dynamics of the field ϕ: Σ→M, for Σ ∼=R×S¹ a cylinder with a fixed pseudo-conformal structure.⁶ The action is given by

S(ϕ) = Z

Σ

g(∂ϕ,∂ϕ) +¯ Z

Y

ϕ^∗H, (2.4)

6It is possible to formulate the theory on other 2D manifolds, but to get a Hamiltonian picture (which we need for the proof of the Poisson-Lie T-duality below) we need to single out a time coordinate, leaving us with a cylinder. This corresponds to a classical description of the closed string. (For an open string, one needs to consider an infinite strip instead. We will not develop the corresponding theory here.)

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where ∂ and ¯∂ are the components of the de Rham differential on Σ corresponding to the two respective null directions, and Y is the solid cylinder (here we extend the map f to Y; it is easy to see that for the purposes of variation, the value of f inside of the cylinder is irrelevant).

We are now ready to describe the duality.

Definition 2.6.4. Let E →M be a transitive CA with a generalized metric V₊, and f⁰: M⁰ → M, f⁰⁰: M⁰⁰ → M be surjective submersions. Suppose we have exact CA pullback structures on E⁰ := f^0∗E and E⁰⁰ := f^00∗E. We then say that the pairs (E⁰, f^0∗V₊) and (E⁰⁰, f^00∗V₊) of CAs with generalized metrics are Poisson-Lie T-dual.

The reason for this name is the following result.

As discussed above, the pairs (E⁰, f^0∗V₊) and (E⁰⁰, f^00∗V₊) can be trans- lated into pairs (g⁰, H⁰) and (g⁰⁰, H⁰⁰) on M⁰ and M⁰⁰, respectively, which in turn yield two sigma models of the type 2.4. The claim is that these two sigma models are isomorphic up to finitely many degrees of freedom. The isomorphism here is to be understood as that of Hamiltonian systems, i.e. as a symplectomorphism between the two corresponding (infinite-dimensional) phase spaces, which furthermore preserves the Hamiltonians.

Remark 2.6.5. The above “finite-dimensional discrepancy” is something present already in the original T-duality of string theory. It is the quantization condition that resolves the issue in string theory, imposing a momentum constraint which is lacking on the classical level.

Remark 2.6.6. In fact a plurality, instead of a duality, seems to be a more fitting term (see [30]), as a given transitive CA might admit multiple different CA pullbacks.

Before sketching the proof of this fact, let us describe some examples.

Example 2.6.7. Suppose (g,h) is a Manin pair, with corresponding Lie groups G, H. We take E = g, M⁰ = G/H, and E⁰ = G/H ×g the action Courant algebroid, coming from the action of G on G/H. Since h is La- grangian, E⁰ is exact. Thus any choice of a generalized metric on g leads to a sigma model 2.4 with the target space G/H. The duality now comes from the choice of various Manin pairs with the same g(i.e. from the choice of various Lagrangian subalgebras h of g). This is the Poisson-Lie T-duality without spectators.

An important special case is the one of the Manin triple (g,h,h⁰).⁷ It is actually this case that was studied first and whence the Poisson-Lie T-duality derives its name.

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An even more important and more special case is given by the abelian Drinfeld doubleg=R². Denoting the basis elements byeandf (s.t.he, fi= 1), the generalized metric is always of the form V₊ = span(e+rf) with r a positive constant. The manifoldsM⁰ andM⁰⁰can be both taken to be circles, with vanishing 3-forms, but with inverse radii.⁸ This is the original T-duality of string theory.

Example 2.6.8. Let us consider a more general case. We start by fixing a Lie group G with an invariant inner product on its Lie algebra g. We then take a principalG-bundleP →P/Gand an exactG-equivariant CA ˜E →P (it exists iff the first Pontryagin class of P vanishes, see Theorem 2.3.14).

As our transitive CA we setE := ˜E_/G. SupposeH ⊂Gis a subgroup with h ⊂ g Lagrangian. We can then take the exact CA E⁰ :=E_/H. Notice that E⁰ is a CA pullback of E, mediated by the map P/H → P/G. The duality here again corresponds to different choices of the Lagrangian subalgebra h (with a fixed generalized metric on E). This is the Poisson-Lie T-duality with spectators (the manifold P/G corresponds to the spectators). In the special case of P =G we recover the previous example.

Let us now sketch the proof of the duality assertion [25].

Suppose E is a CA with a generalized metric V+. Let us denote the reflection w.r.t. V₊ by R. We consider the Courant sigma model given by E with the source N being a solid cylinder. Since ∂N 6=∅, we need to add a boundary condition. This will be given by V+, namely we fix a pseudo- conformal structure on ∂N and require that on this boundary

∗e^a=R^a_be^b, p_i = 1

4g_abR^a_c∂R^b_d

∂xⁱe^ce^d. (2.5) In this expression we borrowed the notation from Theorem 2.4.15 and denoted by ∗ the Hodge operator for the conformal structure.⁹

Let now E be an exact CA with a generalized metric V+, inducing a splitting E ∼= T M ⊕T^∗M and a pair (g, H). The action functional 2.3 is then

S= Z

N

p_idxⁱ+ ¹₂π_idξⁱ +¹₂ξⁱdπ_i−p_iξⁱ+ ¹₆H_ijk(x)ξⁱξ^jξ^k

= Z

N

p_i(dxⁱ−ξⁱ) +π_idξⁱ+¹₆H_ijk(x)ξⁱξ^jξ^k+ ¹₂ Z

∂N

ξⁱπ_i.

8The metricsM⁰ andM⁰⁰are homogeneous and thus only given by their radius.

9Recall thate^a are 1-forms on a two-dimensional boundary, implying that∗e^a indeed depends only on the conformal class of the metric, i.e. on the conformal structure.

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Interpreting p_i as a Lagrange multiplier, we obtain the condition ξⁱ = dxⁱ. Notice that the first part of the boundary condition 2.5 can be written, in this case, as

(π_i)₊=g_ij(ξ^j)₊, (π_i)−=−g_ij(ξ^j)−,

where the subscripts ±denote the components in the null directions. Using the boundary condition and the identification ξⁱ =dxⁱ, we get

S = Z

∂N

g_ij(ξⁱ)₊(ξ^j)−+ Z

N

x^∗H= Z

∂N

g(∂x,∂x) +¯ Z

N

x^∗H,

recovering exactly the action (2.4).

The crucial observation is now the following: IfE⁰ is an exact CA which is a CA pullback of E → M via a surjective submersion M⁰ → M, then the NQ symplectic manifold given by E is a coisotropic reduction of the one given by E⁰. Suppose we pick a generalized metric V₊ onE and pull it back to a generalized metricV₊⁰ onE⁰. It is then easy to convince oneself that the corresponding Hamiltonian systems (i.e. the phase spaces with Hamiltonians) are also related by a coisotropic reduction. The last remaining step is to check that the latter coistropic submanifold has a finite codimension in the phase space given by E⁰ and V₊⁰. From this the duality follows, as now any given pair of Poisson-Lie T-dual CAs produces a pair of Hamiltonian systems which are both “almost-isomorphic” to the one produced by E.

Remark 2.6.9. For the string theory purposes one requires that the metric on the target space has an indefinite (Minkowski) signature. This would correspond to the change of signature in the definition of the generalized metric. All the conclusions of this section would then go through unchanged, except that in order to get a well defined two-tensor on M one would need to impose the condition V₊∩T^∗M = 0 by hand.

2.7 Equivariant Poisson-Lie T-duality

We will end this chapter by mentioning a generalized version of the Poisson- Lie T-duality, the so called equivariant Poisson-Lie T-duality, introduced originally under the name ofdressing cosets [15]. It works with gauged sigma models and extends significantly the number of examples of the duality. We here follow the description from [27].¹⁰

10Although, for simplicity, we again put a restriction on the rank and signature of the generalized metric.

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Definition 2.7.1. Let s be a Lie algebra and E an s-equivariant CA (here we take s with the zero bilinear form). An admissible generalized metric V₊⊂E is an s-invariant subbundle satisfying

rankV+ = ¹₂rankE−dims, hχ(s), ui= 0, ∀s∈s, u∈Γ(V₊), and on which the induced inner product is positive definite.

In particular, if the action of s on M integrates to a free and proper action of a connected Lie groupS, then the admissible generalized metricV+

descends to a generalized metric (V₊)_red on the reduced CA E_/S.

Definition 2.7.2. Let E → M be an s-equivariant CA with an admissible generalized metric V+. Suppose f⁰: M⁰ → M, f⁰⁰: M⁰⁰ → M are surjective submersions and we have exact CA pullback structures on E⁰ := f^0∗E and E⁰⁰ :=f^00∗E. Then bothE⁰ andE⁰⁰ares-equivariant, with admissible generalized metrics f^0∗V+ andf^00∗V+. If the actions ofS onM⁰ andM⁰⁰ are free and proper, we say that the reduced pairs (E_/S⁰ ,(f^0∗V₊)_red) and (E_/S⁰⁰ , ,(f^00∗V₊)_red) are (equivariantly) Poisson-Lie T-dual.

Again, for this setup, it can be shown that the corresponding sigma models are “almost isomorphic”. The proof is similar to the one in the previous section, but one now has to perform an additional Hamiltonian reduction w.r.t. to the action of the loop group LS on the phase space given byE.

Example 2.7.3. The simplest case consists of E = g for a Lie algebra g with an invariant inner product, and s⊂g an isotropic Lie subalgebra. The admissible generalized metric is now given by an s-invariant subspace of s^⊥. As before, dual models would arise from various choices of Lagrangian Lie subalgebras of g, which now in addition have to be transverse to s.

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Modern Phys. A 12 (1997), pp. 1405–1430.

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[4] A. Cattaneo, G. Felder,A path integral approach to the Kontsevich quantization formula, Commun. Math. Phys. 212, 591–611 (2000).

[5] A. Cattaneo, G. Felder,Poisson sigma models and symplectic groupoids, (ed. Klaas Landsman, M. Pflaum, M. Schlichenmeier), Progress in Math- ematics 198, 61–93 (Birkh¨auser, 2001).

[6] T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), 631–661.

[7] T. Courant, A. Weinstein,Beyond Poisson structures, in Actions hamil- toniennes de groupes. Troisième théorème de Lie (Lyon, 1986), Travaux en Cours, Vol. 27, Hermann, Paris, 1988, 39–49.

[8] M. Gualtieri, Generalized K¨ahler geometry, Commun.Math.Phys. 331 (2014) no.1, 297–331.

[9] N. Ikeda, Chern-Simons Gauge Theory coupled with BF Theory, Int. J.

Mod. Phys. A18 (2003) 2689–2702.

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Phys. 235 (1994) 435-464.

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[13] V.G. Drinfel’d, Quasi-Hopf algebras, Leningrad Math. J., 1, 1990, 1419–1457.

[14] C. Klimˇc´ık, P. ˇSevera,Dual non-Abelian T-duality and the Drinfeld double. Phys.Lett. B 351 (1995), 455–462.

[15] C. Klimˇc´ık, P. ˇSevera, Dressing cosets, Physics Letters B, Vol. 381, No.

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Chapter 3 String Theory and Supergravity

The goal of this chapter is to introduce the reader to the most neccessary basics of string and superstring theory. For a more thorough treatment, we advice the reader to consult the corresponding literature. In particular, the author recommends the lecture notes by David Tong [6], which we follow to a large extent in this chapter, and the books by Polchinski [3, 4]. Furthermore, there is a briefer but more modern introduction by Becker-Becker-Schwarz [5]. To find more details one can also consider the classical exposition by Green-Schwarz-Witten [1, 2].

We shall start our discussion with the bosonic string and we then pass briefly to the superstring setup. Since the presented material has by now become standard, we shall omit the detailed references, directing the inter- ested reader instead to the above books and lecture notes (and the references therein).

3.1 Rudiments of the bosonic string theory

Consider a freely moving particle in a spacetime M, given by the action S= length,

yielding geodesics as the extremal trajectories. The idea is now to look at what happens if one considers extended objects, i.e. objects whose spacetime trajectory cuts out a d+ 1-dimensional submanifold ofM. Ford = 1 we talk about strings, the d >1 case corresponds to branes. The natural action

S_{N G} = volume

is called theNambu-Goto action. We understand this as a field theory, where the fields are maps Σ →M, with Σ a d+ 1-dimensional manifold. In what

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follows, we shall focus on the string case.¹

For the purposes of quantization this action turns out to be not so convenient, which is why it is standard to pass to a classically equivalentPolyakov action

SP(X, h) = 1 4πα⁰

Z

Σ

g(∂αX, ∂βX)h^αβvolh,

where we now supplemented the map X: Σ → M with an additional dy- namical field h, understood as an (auxiliary) metric on the two-dimensional source Σ.² Similarly, we introduced a standard (classically irrelevant) factor involving a constant α⁰.³

The Polyakov action possesses, in addition to the reparametrization invariance, the so called Weyl symmetry, namely S_P is invariant under the changeh_αβ →Λ²h_αβ, for Λ any function.⁴ Since these are gauge symmetries, we can now do a gauge fixing procedure, imposing (suppose for now that Σ =R²) a special form of h, which we choose for convenience to be the one of Minkowski metric η. This leads to

S(X) = 1 4πα⁰

Z

Σ

g(∂X,∂X),¯ (3.1)

with ∂, ¯∂ are projections of the de Rham differential into the null directions.

In addition, we have to supplement the action with a constraint, namely the equation of motion of h,⁵

g(∂_αX, ∂_βX)− ¹₂η_αβη^γδg(∂_γX, ∂_δX) = 0.

A crucial observation is that after the gauge fixing, there is still the conformal symmetry (on Σ) left. To see why this is the case, suppose that we take a conformal diffeomorphism F of Σ. Using both the diffeomorphism invariance and the Weyl symmetry of the Polyakov action, we get

S(F^∗X) =S_P(F^∗X, η) = S_P(X,(F⁻¹)^∗η) =S_P(X, η) =S(X).

1This case turns out to be very special, in particular it allows a convenient quantization leading to a conformal field theory.

2Classical correspondence between these theories is established by using the equations of motion forhto eliminate the auxiliary metric from the action, recovering the Nambu- Goto action.

3This notation comes from the original attempt to interpret string theory as the theory of strong interactions,α⁰ corresponding to the Regge slope.

4The Polyakov action can be in principle written for Σ of any dimension, but the Weyl symmetry is only present in the 2D case.

5This, by definition, corresponds to the vanishing of the stress-energy tensor.

Courant algebroids and string low energy effective actions

Thesis

Reference

Courant algebroids and string low energy effective actions

Courant Algebroids And String Low Energy Effective Actions

TH` ESE

Fridrich VALACH

Th` ese No. 5374

R´ esum´ e

Abstract

Acknowledgment

Contents

Chapter 1 Introduction

Chapter 2

Courant algebroids and Poisson-Lie T-duality

2.1 Poisson-Lie groups

2.2 Lie algebroids

2.3 Courant algebroids

2.4 NQ manifolds

2.5 AKSZ construction

2.6 Poisson-Lie T-duality

2.7 Equivariant Poisson-Lie T-duality

Bibliography

Chapter 3

String Theory and Supergravity

3.1 Rudiments of the bosonic string theory