Supergravity as the square of Super Yang-Millsunexpected relationship The division algebras A (super)quick intro to Supergravity and Super Yang-Mills The Magic Square A Magic Square

Supergravityas the square

of SuperYang-Mills

Silvia Nagy

Outline

Clues for anunexpectedrelationship

The divisionalgebras

A(super)quickintro toSupergravityand SuperYang-Mills

The MagicSquare

A MagicSquare ofSupergravities

Magic pyramid

ConformalPyramid

Conclusionsand futurework

Supergravity as the square of Super Yang-Mills

Silvia Nagy

Imperial College London,based on work done in collaboration with:

A.Anastasiou, L. Borsten, M. J. Duff and L. J. HughesarXiv:1301.4176

SCGSC, November 7, 2013


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Magic pyramid

ConformalPyramid


1 Clues for an unexpected relationshipKLT relations in String TheorySupergravity Multiplets from Yang-Mills Multiplets

2 The division algebrasWhat is a Division Algebra?R,C,H,O from Cayley-Dickson Doubling

3 A (super)quick intro to Supergravity and Super Yang-MillsThe Scalar cosets of SupergravityN=1,2,4,8 Super Yang Mills over the division algebras

4 The Magic SquareProjective planesIsometries of the projective planesThe Magic Square

5 A Magic Square of SupergravitiesTensoring the MultipletsMAGIC!

6 Magic pyramid

7 Conformal Pyramid

8 Conclusions and future work


of SuperYang-Mills

Silvia Nagy

Outline


KLT relations inString Theory

SupergravityMultiplets fromYang-MillsMultiplets



The MagicSquare


Magic pyramid

ConformalPyramid


Table of Contents1 Clues for an unexpected relationship

KLT relations in String TheorySupergravity Multiplets from Yang-Mills Multiplets





6 Magic pyramid7 Conformal Pyramid8 Conclusions and future work


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


Scattering Amplitudes

• Simplest example: Parke-Taylor formula for MHVscattering of n gluons:

AMHV (1, 2, ..., n) = 1< 12 >< 23 > ... < n1 >

(1)

• 3-graviton scattering amplitude

MMHV (1, 2, 3) = 1(< 12 >< 13 >< 23 >)2

(2)


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


KLT relations in String Theory

Any closed string vertex operator for the emission of a closedstring state (e.g. a graviton) is a product of open string states:

Vclosed = Vopenleft × V̄

openright (3)


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


Supergravity Multiplets fromYang-Mills Multiplets

• Tensor 2 YM multiplets of opposite chirality to get TypeIIA SuGra:

(8V + 8C)⊗ (8V + 8S) = (1 + 28 + 35V + 8V + 56V)B+ (8S + 8C + 56S + 56C)F (4)

• Tensor 2 YM multiplets of the same chirality to get TypeIIB SuGra:

(8V + 8C)⊗ (8V + 8C) = (1 + 28 + 35V + 1 + 28 + 35C)B+ (8S + 8S + 56S + 56S)F (5)


of SuperYang-Mills

Silvia Nagy

Outline



What is aDivisionAlgebra?

R, C, H, O fromCayley-DicksonDoubling


The MagicSquare


Magic pyramid

ConformalPyramid










of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


What is a Division Algebra?

• A division algebra is a ring in which every nonzero elementhas a multiplicative inverse, but multiplication is notnecessarily commutative.

• A normed division algebra K comes equipped with a norm:

|ab| = |a||b| (6)

• Division algebras have no zero divisors.


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


Cayley-Dickson construction• Start with real numbers R. A complex number a + bi ,

with a, b ∈ R can be thought of as a pair (a, b) with thefollowing rules:

• multiplication:

(a, b)(c , d) = (ac − db, ad + cb) (7)

• conjugation:(a, b)∗ = (a,−b) (8)

• Similarly, a quaternion can be defined as a pair ofcomplexes (x , y) with:

• multiplication

(x , y)(z ,w) = (xz − wy∗, x∗w + zy) (9)

• conjugation(x , y)∗ = (x∗,−y) (10)

• Octonions are constructed in exactly the same way from 2quaternions.


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


The four division algebras

• R-real, commutative,associative normed division algebra

• C-commutative, associative normed division algebra• H-associative normed division algebra• O-normed division algebra


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid



• R-real, commutative,associative normed division algebra• C-commutative, associative normed division algebra

• H-associative normed division algebra• O-normed division algebra


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid



• R-real, commutative,associative normed division algebra• C-commutative, associative normed division algebra• H-associative normed division algebra

• O-normed division algebra


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid



• R-real, commutative,associative normed division algebra• C-commutative, associative normed division algebra• H-associative normed division algebra• O-normed division algebra


of SuperYang-Mills

Silvia Nagy

Outline




The Scalarcosets ofSupergravity

N=1,2,4,8 SuperYang Mills overthe divisionalgebras

The MagicSquare


Magic pyramid

ConformalPyramid










of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


Supergravity

• Low energy limit of string theory• General relativity + supersymmetry (local susy parameter).• Field content- supergravity multiplets.• Characterised by scalar coset groups.


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


Supergravity

• Low energy limit of string theory

• General relativity + supersymmetry (local susy parameter).• Field content- supergravity multiplets.• Characterised by scalar coset groups.


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


Supergravity

• Low energy limit of string theory• General relativity + supersymmetry (local susy parameter).

• Field content- supergravity multiplets.• Characterised by scalar coset groups.


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


Supergravity

• Low energy limit of string theory• General relativity + supersymmetry (local susy parameter).• Field content- supergravity multiplets.

• Characterised by scalar coset groups.


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


Supergravity

• Low energy limit of string theory• General relativity + supersymmetry (local susy parameter).• Field content- supergravity multiplets.• Characterised by scalar coset groups.


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


What are the scalar cosets?

• Symmetries of theories obtained by reduction on variousmanifolds

• Study symmetries of scalars• General form of transformation:

V ′ = OVΛ (11)

• V obtained by exponentiating scalars with CartanGenerator+positive root generators

• Λ is the global symmetry transformation (G group)• O is the compensating transformation• Example: 2 torus reduction gives the scalar coset SL(2)SO(2)• Minimal variety corresponds to maximal ideal• Scalars determine symmetries of all fields!


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid




• Study symmetries of scalars

• General form of transformation:

V ′ = OVΛ (11)




of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid





V ′ = OVΛ (11)


• Λ is the global symmetry transformation (G group)• O is the compensating transformation

• Example: 2 torus reduction gives the scalar coset SL(2)SO(2)• Minimal variety corresponds to maximal ideal• Scalars determine symmetries of all fields!


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid





V ′ = OVΛ (11)


• Λ is the global symmetry transformation (G group)• O is the compensating transformation• Example: 2 torus reduction gives the scalar coset SL(2)SO(2)

• Minimal variety corresponds to maximal ideal• Scalars determine symmetries of all fields!


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid





V ′ = OVΛ (11)


• Λ is the global symmetry transformation (G group)• O is the compensating transformation• Example: 2 torus reduction gives the scalar coset SL(2)SO(2)• Minimal variety corresponds to maximal ideal

• Scalars determine symmetries of all fields!


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid





V ′ = OVΛ (11)




of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


Super Yang-Mills

• N = 1 Super YM Lagrangian:

L = −14Tr(Fµν ,F

µν)− i2Tr(λ̄, γµDµλ) (12)

• SYM multiplets• SYM theories are characterised by the R-symmetry, which

describes transformations of different supercharges intoeach other.


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


Super Yang-Mills




• SYM multiplets

• SYM theories are characterised by the R-symmetry, whichdescribes transformations of different supercharges intoeach other.


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


Super Yang-Mills




• SYM multiplets• SYM theories are characterised by the R-symmetry, which

describes transformations of different supercharges intoeach other.


of SuperYang-Mills

Silvia Nagy

Outline






The MagicSquare


Magic pyramid

ConformalPyramid


N = 1, 2, 4, 8 Super Yang Millsover the division algebras

In 3 dimensions, we can write the Lagrangian:

L =− 14FAµνF

Aµν − 12Dµφ

∗ADµφA + i λ̄AγµDµλA

− 14g2fBC

AfDEA〈φB |φD〉〈φC |φE 〉

+i

2gfBC

A(

(λ̄AφB)λC − λ̄A(φ∗BλC ))

(13)


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes

Isometries of theprojective planes

The MagicSquare


Magic pyramid

ConformalPyramid










of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid


The projective plane

• A set of points and lines together with a relation betweenthem, satisfying the following axioms:

• For any two distinct points, there is a unique line on whichthey both lie.

• For any two distinct lines, there is a unique point whichlies on both of them.

• There exist four points, no three of which lie on the sameline.

• The terms point and line are interchangeable in the abovedefinition.

• Important: compactness, any two lines intersect


of SuperYang-Mills

Silvia Nagy

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The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid


A more intuitive definition

For any field F , the projective plane FP2 is the set ofequivalence classes of non-zero points in F 3, where theequivalence relation is given by:

(x , y , z) ≡ (rx , ry , rz) (14)


of SuperYang-Mills

Silvia Nagy

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The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid


The real projective plane


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid


Some simple Lie Algebras

• so(n) = {x ∈ R[n] : x† = −x , tr(x) = 0}

• su(n) = {x ∈ C[n] : x† = −x , tr(x) = 0}• sp(n) = {x ∈ H[n] : x† = −x}• Describe collectively by sa(n,A)


of SuperYang-Mills

Silvia Nagy

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The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid



• so(n) = {x ∈ R[n] : x† = −x , tr(x) = 0}• su(n) = {x ∈ C[n] : x† = −x , tr(x) = 0}

• sp(n) = {x ∈ H[n] : x† = −x}• Describe collectively by sa(n,A)


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid



• so(n) = {x ∈ R[n] : x† = −x , tr(x) = 0}• su(n) = {x ∈ C[n] : x† = −x , tr(x) = 0}• sp(n) = {x ∈ H[n] : x† = −x}

• Describe collectively by sa(n,A)


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid



• so(n) = {x ∈ R[n] : x† = −x , tr(x) = 0}• su(n) = {x ∈ C[n] : x† = −x , tr(x) = 0}• sp(n) = {x ∈ H[n] : x† = −x}• Describe collectively by sa(n,A)


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid


Isometries of projective planes

• isom(RP2) ∼= so(3)• isom(CP2) ∼= su(3)• isom(HP2) ∼= sp(3)• isom(OP2) ∼= f4


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid



• isom(RP2) ∼= so(3)

• isom(CP2) ∼= su(3)• isom(HP2) ∼= sp(3)• isom(OP2) ∼= f4


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid



• isom(RP2) ∼= so(3)• isom(CP2) ∼= su(3)

• isom(HP2) ∼= sp(3)• isom(OP2) ∼= f4


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid



• isom(RP2) ∼= so(3)• isom(CP2) ∼= su(3)• isom(HP2) ∼= sp(3)

• isom(OP2) ∼= f4


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid



• isom(RP2) ∼= so(3)• isom(CP2) ∼= su(3)• isom(HP2) ∼= sp(3)• isom(OP2) ∼= f4


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid


What about the exceptionalgroups?

• isom((C⊗O)P2) ∼= e6• isom((H⊗O)P2) ∼= e7• isom((O⊗O)P2) ∼= e8


of SuperYang-Mills

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Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid



• isom((C⊗O)P2) ∼= e6

• isom((H⊗O)P2) ∼= e7• isom((O⊗O)P2) ∼= e8


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid



• isom((C⊗O)P2) ∼= e6• isom((H⊗O)P2) ∼= e7

• isom((O⊗O)P2) ∼= e8


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid



• isom((C⊗O)P2) ∼= e6• isom((H⊗O)P2) ∼= e7• isom((O⊗O)P2) ∼= e8


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare

Projective planes


The MagicSquare


Magic pyramid

ConformalPyramid


The magic square

• Define the magic square by:

M(AL,AR) = isom((AL ⊗ AR)P2) (15)

•AL/AR R C H OR SL(2,R) SU(2, 1) USp(4, 2) F4(−20)C SU(2, 1) SU(2, 1)× SU(2, 1) SU(4, 2) E6(−14)H USp(4, 2) SU(4, 2) SO(8, 4) E7(−5)O F4(−20) E6(−14) E7(−5) E8(8)

Table : Magic square


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Tensoring theMultiplets

MAGIC!

Magic pyramid

ConformalPyramid










of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare



MAGIC!

Magic pyramid

ConformalPyramid


Tensoring the Multiplets• In 3 dimensions, we tensor together left and right

multiplets of Super YM, for N = 1, 2, 4, 8

•NL(SYM) +NR(SYM) = NSuGra (16)

• Equivalently:|AL|+ |AR | = NSuGra (17)

• Remenber the fields are: ψA and φA, with A = 1, ...|AL,R |• The scalars of supergravity will be:

φSuGra =

(ψA ⊗ ψB

φA ⊗ φB

)(18)

• They are points in (AL ⊗ AR)2• This is affine space, need to take projective closure − >

(AL ⊗ AR)P2 !


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare



MAGIC!

Magic pyramid

ConformalPyramid



multiplets of Super YM, for N = 1, 2, 4, 8•

NL(SYM) +NR(SYM) = NSuGra (16)



φSuGra =

(ψA ⊗ ψB

φA ⊗ φB

)(18)


(AL ⊗ AR)P2 !


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare



MAGIC!

Magic pyramid

ConformalPyramid






• Remenber the fields are: ψA and φA, with A = 1, ...|AL,R |

• The scalars of supergravity will be:

φSuGra =

(ψA ⊗ ψB

φA ⊗ φB

)(18)


(AL ⊗ AR)P2 !


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare



MAGIC!

Magic pyramid

ConformalPyramid







φSuGra =

(ψA ⊗ ψB

φA ⊗ φB

)(18)


(AL ⊗ AR)P2 !


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare



MAGIC!

Magic pyramid

ConformalPyramid







φSuGra =

(ψA ⊗ ψB

φA ⊗ φB

)(18)

• They are points in (AL ⊗ AR)2

• This is affine space, need to take projective closure − >(AL ⊗ AR)P2 !


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare



MAGIC!

Magic pyramid

ConformalPyramid







φSuGra =

(ψA ⊗ ψB

φA ⊗ φB

)(18)


(AL ⊗ AR)P2 !


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare



MAGIC!

Magic pyramid

ConformalPyramid


Magic Square of Supergravities

AL/AR R C H O

R SL(2,R) SU(2, 1) USp(4, 2) F4(−20)C SU(2, 1) SU(2, 1)× SU(2, 1) SU(4, 2) E6(−14)H USp(4, 2) SU(4, 2) SO(8, 4) E7(−5)O F4(−20) E6(−14) E7(−5) E8(8)

Table : Magic square


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Magic pyramid

ConformalPyramid










of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Magic pyramid

ConformalPyramid


SYM in various dimensions

• Remember the lagragian:



• We want its SUSY variation to vanish• We get a term of the form:

Tr(λ, γµ[(�γµλ), λ]) (20)

• Only vanishes in 3,4,6 and 10 dimensions!


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Magic pyramid

ConformalPyramid






• We want its SUSY variation to vanish

• We get a term of the form:

Tr(λ, γµ[(�γµλ), λ]) (20)



of SuperYang-Mills

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Outline




The MagicSquare


Magic pyramid

ConformalPyramid






• We want its SUSY variation to vanish• We get a term of the form:

Tr(λ, γµ[(�γµλ), λ]) (20)



of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Magic pyramid

ConformalPyramid


What have we learnt so far?

• Extended Super YM theories characterised by divisionalgebras in 3D.

• Tensor them to get supergravities, whose global symmetrygroups are given by the magic square construction.

• Pure super- Yang-Mills theories only exist in 3,4,6 and 10dimensions!


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Magic pyramid

ConformalPyramid


The Magic Pyramid


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Magic pyramid

ConformalPyramid


4 dimensions


of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Magic pyramid

ConformalPyramid










of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Magic pyramid

ConformalPyramid


Conformal Pyramid


of SuperYang-Mills

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Outline




The MagicSquare


Magic pyramid

ConformalPyramid


Conformal Pyramid formula

• The U-duality groups given by the formula:

H(AL,AR ,A) = R(AL,A)⊕ R(AR ,A) + |A| · AL × AR+ St(AP2) (21)

• WhereR(AL,R ,A) ∼ sa(|AL,R |,A) (22)

• Prediction for D=10


of SuperYang-Mills

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Outline




The MagicSquare


Magic pyramid

ConformalPyramid










of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Magic pyramid

ConformalPyramid


Conclusions and future work

• Division algebras provide further evidence for the idea thatSupergravity is, in a sense, the square of a gauge theory.

• Supergravity pyramid, Lagrangian.• Local symmetries of SuGra (diffeomorphisms, gauge +

local SUSY) from gauge symmetries of SYM.

• L. Borsten, M. J. Duff, L. J. Hughes and S. Nagy, “Amagic square from Yang-Mills squared,” arXiv:1301.4176[hep-th].

• A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes andS. Nagy, “Super Yang-Mills, division algebras and triality,”arXiv:1309.0546 [hep-th].


of SuperYang-Mills

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Outline




The MagicSquare


Magic pyramid

ConformalPyramid




• Supergravity pyramid, Lagrangian.

• Local symmetries of SuGra (diffeomorphisms, gauge +local SUSY) from gauge symmetries of SYM.




of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Magic pyramid

ConformalPyramid









of SuperYang-Mills

Silvia Nagy

Outline




The MagicSquare


Magic pyramid

ConformalPyramid








Clues for an unexpected relationshipKLT relations in String TheorySupergravity Multiplets from Yang-Mills Multiplets

The division algebrasWhat is a Division Algebra?R,C,H,O from Cayley-Dickson Doubling

A (super)quick intro to Supergravity and Super Yang-MillsThe Scalar cosets of SupergravityN=1,2,4,8 Super Yang Mills over the division algebras

The Magic SquareProjective planesIsometries of the projective planesThe Magic Square

A Magic Square of SupergravitiesTensoring the MultipletsMAGIC!

Magic pyramidConformal PyramidConclusions and future work

Documents

Supergravity as the square of Super Yang-Millsunexpected relationship The division algebras A (super)quick intro to Supergravity and Super Yang-Mills The Magic Square A Magic Square