Pietro Frè University of Torino & Embassy of Italy in

the Russian Federation

Lectures on Group Theory for Supergravity and Brane Theorists

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Overview We illustrate Group Theory from the stand-point of a

Supergravity Theorist. These lectures deal only with mathematics and no supergravity

is actually discussed, yet the chosen topics are motivated by Supergravity/Brane Theory.

We deal with both finite and Lie groups in the perspective of their role in differential and complex algebraic geometry.

We emphasize geometrical/group theoretical structures and conceptions that have been motivated and uncovered by supergravity.

We aim at conveying the following message: geometry and groups are fundamental items in brane theories, AdS/CFT and supergravity. Not only: these physical theories have introduced new visions and conceptions in Geometry and this last aspect might turn out to be the most important and durable contribution of Supersymmetry to Science in general.

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Finite Group Theory Some elements of a theory which is quite old, usually not too much studied by high energy physicists, yet of growing relevance in the supergraivty/brane world

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Recalling some fundamental notions

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Order of elements and conjugacy classes

Ga∈ eah = h = order of a

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Still a few more general concepts

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Solvable Groups, Simple Groups

The definition requires that the group G should act on the group K as a transformation group.

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Linear Representations

In other words

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Reducible Representations

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Schurs’s Lemmas

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Characters

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Character orthogonality relations

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Example: the octahedral Group

There are 24 rotations of three dimensional space that map the octahedral into itself. They form a group O24 that consists of 5 conjugacy classes displayed above.

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Structure of the octahedral group

The group O24 is solvable

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Irreps of O24

χ2

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Irreps of O24 continued

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Character table of the octahedral group

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Crystallographic Groups

Lattice

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The ADE classification of SU(2) finite subgroups The problem of classifying finite rotation groups is Plato’s problem of regular solids. This problem is isomorphic to the classification of simple Lie algebras, of modular invariant CFTs, of Arnold simple singularities and of ALE manifolds, namely of gravitational instantons. It is one of the most profound items in the whole field of Mathematics and admits generalizations under the name of McKay correspondence. It plays an important role in supergravirty and AdS/CFT theories, also in modern algebraic geometry.

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SO(3) & SU(2) finite subgroups: preliminaries

Homomorphism SU(2) into SO(3)

where

Every rotation has an axis

We consider the preimage of the rotation in SU(2)

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Every SU(2) element has two poles

∃Poles of

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Formulating the argument Poles are equivalent if they are mapped one into the other by the group H

Equivalence classes

Stability subgroups of the classes

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The Diophantine inequality

Since

Only two cases

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The A solutions

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Structure of the DE solutions

There are in Γ elements of order at most of three different types

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D-solutions

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The E-solutions

1. Tetrahedral 2. Octahedral 3. Icosahedral

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The ADE classification

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The classification of simple Lie algebras The classification of semisimple Lie algebras is based on root spaces and Dynkin diagrams. Dynkin-Cartan theory, in all of its aspects, is of crucial relevance in many directions of Mathematical Physics and it is absolutely essential in Supergravity and Brane Theories.

Sophus Lie Felix Klein Wilhelm Killing Elie Cartan Eugenio Levi

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Cartan-Weyl basis of a complex Lie algebra

Cartan Killing metric

∆ is a finite collection of vectors in a r-dimensional Euclidian space

The elements α 2 ∆ are named the roots

The generators Hi span the Cartan subalgebra CSA (maximal abelian whose adjoint action is diagonalizable). The dimension r of the CSA is named the rank

Hermann Weyl

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Axiomatization of Root Systems

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Simply laced Lie algebras

Non simply laced Lie algebras

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Simple roots

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Weyl Group and Cartan matrix

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Coxeter-Dynkin diagrams Simple roots

# of lines joining αi with αj Coxeter graphs

Eugene Dynkin Harold Coxeter

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Dynkin diagrams Add an arrow

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Classification Theorem The possible Dynkin diagram and hence the simple complex Lie algebras are given by the following infinite series

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or by these 5 exceptional cases

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The ADE classification of simply laced Lie algebras

Elaborating the consequences of the axioms one reduces possible simply laced diagrams to the form on the right and finds the diophantine inequality

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Identification of the Classical Lie Algebras

Ar = SL(r+1,C) Br = SO(2r+1,C) Cr = Sp(2r,C) Dr = SO(2r,C)

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The weight lattice and linear representations

Fundamental weights

For any representation Γ of the Lie Algebra

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Weigths of a representations

Dirac notation (bra and kets)

One fundamental result of Weyl. The weights of a representation belong to the weight lattice.

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Another fundamental result of Weyl

Utilizing these two theorems the construction of all the weights of a representation can be easily encoded into an iterative computer algorithm starting from a highest weight

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Highest weight representations

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Example with A2

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The Weyl group

The complex Lie algebra SL(3,C) is provided by all 3£ 3 matrices that are traceless. The CSA is given by the diagonal set of traceless matrices.

The Weyl group of A2 is isomorphic to the permutation group of three objects S3

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The fundamental weights

The Weyl chamber

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Weights of the defining triplet representation

Matrices of the representation

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Weights of the adjoint representation, namely the roots

The root lattice is always sublattice of the weight lattice!

The highest root is the highest weight of the adjoint representation!

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Weights of the 15-dim rep.

This provides a more complicated, less trivial example

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Correspondence with Young tableaux and tensors

The reason why certain weights have multiplicity >1 is that there is more than one inequivalent way to fill the boxes of the corresponding Young tableau

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The Golden Splitting Supergravity has put into evidence some intrinsic properties of Lie Algebras that are purely mathematical yet obtain their proper interpretation only within the framework of those geometries that are well-adapted to supersymmetric field theories. One of these hidden jewels is the golden splitting that is the Lie algebra seed of the c-map

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Theory of Coset Manifolds A fundamental item in Supergravity and in many other branches of Physics. The classification of symmetric coset manifolds was the monumental work of Elie Cartan. Such a classification is also a classification of Real-Forms of the complex Lie algebras and contains the seeds of the Tits Satake projection, a very fundamental item in the applications of coset manifolds to Supergravity.

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Coset Manifolds

Decomposition of the Lie algebra

If The coset is reductive

Symmetric space

Basis of generators of the Lie algebra

All the geometric properties of the manifold are essentially encoded in the structure constants

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The structure of isometries

Killing vector fields on G/H

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An example: the hyperbolic hyperplane

For Standard parameterization of the coset

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Spheres or Hyperplanes

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The decomposition of the Lie algebra and the Killing vector field

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G is an H-bundle

Cartan Maurer forms

G is an H-bundle on G/H

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G/H as (pseudo)-Riemannian manifolds

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rank of the coset manifold

If rank=1

The spin connection uniquely determined

In any case

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The Riemann tensor

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Example (spheres or pseudo-spheres)

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Real sections and non-compact cosets

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Two extremal real sections

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Classification of real sections

STEP ONE

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From involution a new Lie algebra

The compact CSA is mapped into itself

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real sections and simple roots

Tits Satake diagrams

1

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All non compact cosets are normal 1

For maximally split real sections

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Kaehler Geometry and Hypergeometry Lie Group Theory is inextricably entangled with Differential Geometry. One of the most important and durable contributions of Supergravity to Science is encoded in the new geometrical structures that it has introduced or better clarified and developed. Kaehler, Hodge-Kaehler, Special Kaehler HyperKaehler and Quaternionic Kaehler manifolds are all advocated by and integrated into the fabrics of Supergravity

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Complex Manifolds

in a

Locally we can introduce complex coordinates

The question is whether we can do that globally

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Holomorphic transition functions

We can establish a global complex structure if J is integrable. This requires

where the following is the Nienjuis tensor that should vanish

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Connections on holomorphic bundles

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Connections are simpler on holomorphic bundles than in real geometry

Levi Civita connection

Curvature 2-form

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For the Levi Civita case we have

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Kaehler metrics

Hermitian metric

Kaehler 2-form

A hermitian metric is of the form:

A hermitian metric is Kaehler iff the Kaehler 2-form is closed

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Kaehler potential dK=0

gives rise to the same metric

N=1 supersymmetry requires that the scalar fields in the scalar multiplets should be the coordinates of a Kaehler manifold!

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Hypergeometry

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HyperKaehler and Quaternionic

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Triplet of Kaehler 2-forms

HyperKaehler

Quaternionic Kaehler

In the HyperKaehler case we can choose a frame where the three 2-forms are closed

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Holonomy restrictions

Formalism for hypergeometry

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Structure of the curvature 2-form

symplectic curvature

quaternionic algebra of the Kaehler forms

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Moment Maps Moment maps were born in hamiltonian mechanics but play a distinguished essential role in supersymmetric theories being essential building blocks of the Lagrangian and of the scalar potential. The use of moment-maps in supersymmetric field theories provided the framework of Kaehler and HyperKaehler quotient, one of the most prolific directions in modern differential and algebraic geometry. They are also essential items in the AdS/CFT correspondence.

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Moment Maps

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Moment maps of holomorphic Killing vector fields

Moment map

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In a more intrinsic language

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Poisson bracket

If

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Final form of the moment map

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Triholomorphic moment maps

HyperKaehler

Quaternionic Kaehler

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Triholomorphic Equivariance Equivariance

HyperKaehler

Quaternionic

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Special Geometry Special Kaehler Geometry is an entire new chapter of complex geometry created by supergravity with profound relations with the deformation theory of complex structures and Kaehler structures of Calabi-Yau three-folds. It also provides new visions on symmetric spaces and their structures.

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Hodge Kaehler manifolds

A

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Connection on the line bundle

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Special Kaehler manifolds

1

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Special Kaehler 2nd definition

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2nd definition continued

2

Kaehler potential

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Relation between the two definitions

The integrability conditions of the equations here on the left reproduces the statement on the Riemann tensor occurring in the first definition

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Symplectic transformations of the period matrix

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The transformation

The matrix N has a positive definite imaginary part and generalizes the notion of upper complex plane to what is known as the upper Siegel plane. The linear fractional transformations with symplectic matrices map the upper Siegel plane into itself just as SL(2,R) maps the Poincaré Lobachevsky plane into itself

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Quotient singularities and Kaehler quotient resolutions Finite Group Theory has an important bearing on the geometry of singular algebraic surfaces when considering orbifolds of the type Cn/¡. The resolutions of these singularities is done by means of Kaehler or HyperKaehler quotients. Gravitational instantons, the ALE-manifold are obtained in these way from C2/¡ where ¡ is a finite subgroup of SU(2). Hence we have a new incarnation of the ADE classification. These structures are very important for the study of the AdS4/CFT3 correspondence

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HyperKaehler quotients

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Complexified Killing fields & Kaehler potential

Solve for V

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ALE Manifolds

Except for the singular point

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Crepant resolutions

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Concept of aging

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Example with L168 and its subgroups

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C2/¡ singularities

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For ALE¡ manifolds

+ highest root

of the

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The McKay Dynkin Graphs

Coxeter numbers are the dimensions of irreps of ¡

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Kronheimer’s construction

Regular representation

:

¡ - invariant subspace

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Schur’s Lemma + McKay

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Solution of invariance constraints for Ak Decomposition into irreps (one)-dim.

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The gauge group

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Moment maps

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Eguchi Hanson space

It is far than obvious that this is a HyperKaehler metric ! Yet it is and it is a HyperKaehler quotient!

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The complex structure

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Relation with the Kronheimer approach

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The Kaehler potential retrieved

already constructed

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The c-map and the c*-map The c-map and the c*-map are purely mathematical structures put into evidence by supergravity. In alliance with the Tits Satake projection they provide very important tools in the study of the following two problems: 1. Gauging of supergravity theories and study of their vacua 2. Black Hole solutions of Supergravity Theories

In any case they provide new quality and new visions in geometry.

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Approaching the c-map

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above

XX

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The items of hypergeometry

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General formulae for su(2)

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c-map and triholomorphic moment maps

XX

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Types of isometries of the quaternionic manifold in the c-map image

General Form of the Killing Vector

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Killing vectors of Heis

The moment maps solving the equation are

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Moment map for the central charge

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Killing vectors of the special Kaehler and their moments maps

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Symmetric manifolds in the image of the c-map

XX

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Identifying UQ via its solvable Lie algebra

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Universal Heisenber algebras

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The Tits Satake Projection Although originally introduced in the mathematical literature, the Tits Satake projection reveals its profund meaning in the supergravity context, in particular in connection with the c-map and the c*-map. It allows to arrange manifolds into universality classes that provide a useful classification for supergravity models and the exploration of their general properties.

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Structure of the Tits Satake projection

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TS commutes also with the c*-map

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The concept of Paint Group We saw that each real form GR of a Lie algebra is in one-to-one correspondence with a symmetric space M=GR/H. This latter singles out a solvable Lie algebra as we have seen. The group of external Automorphisms of Solv(M) is the Paint Group.

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The sub Tits Satake algebra and the long roots

By definition GsubTS commutes with the Paint Group. The long roots are those whose projections are singlets under the Paint Group.

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The short roots

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A general structure above

We analyse an example where

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A primary example E(8,-24)

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The primary example continued

TS

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Decompositions

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Supergravity relevant symmetric spaces An analysis of supergravity models according to the classification of their scalar manifolds.

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General properties of these manifolds

GOLD

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Structure of the root systems according to the golden split and theTits Satake projection

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Classification of SUGRA symmetric spaces (non exotic)

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A most interesting TS class

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The exotic models

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Classification tab 1 non exotic

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Classification tab 2 non exotic

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Classification tab 3: non exotic

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N=8

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N=6

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Tits Satake of the N=6 theory F4

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N=5

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Detailed study of the F4 univ. class As an illustration we analyze one universality class

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The su(3,3) case

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The SO*(12) case

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The E(7,-25) case

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