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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
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
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
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
KLT relations inString Theory
SupergravityMultiplets fromYang-MillsMultiplets
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Table of Contents1 Clues for an unexpected relationship
KLT 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 pyramid7 Conformal Pyramid8 Conclusions and future work
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
KLT relations inString Theory
SupergravityMultiplets fromYang-MillsMultiplets
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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)
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
KLT relations inString Theory
SupergravityMultiplets fromYang-MillsMultiplets
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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)
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
KLT relations inString Theory
SupergravityMultiplets fromYang-MillsMultiplets
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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)
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
KLT relations inString Theory
SupergravityMultiplets fromYang-MillsMultiplets
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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)
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
KLT relations inString Theory
SupergravityMultiplets fromYang-MillsMultiplets
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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)
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
What is aDivisionAlgebra?
R, C, H, O fromCayley-DicksonDoubling
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Table of Contents1 Clues for an unexpected relationship
KLT 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 pyramid7 Conformal Pyramid8 Conclusions and future work
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
What is aDivisionAlgebra?
R, C, H, O fromCayley-DicksonDoubling
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
What is aDivisionAlgebra?
R, C, H, O fromCayley-DicksonDoubling
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
What is aDivisionAlgebra?
R, C, H, O fromCayley-DicksonDoubling
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
What is aDivisionAlgebra?
R, C, H, O fromCayley-DicksonDoubling
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
What is aDivisionAlgebra?
R, C, H, O fromCayley-DicksonDoubling
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
What is aDivisionAlgebra?
R, C, H, O fromCayley-DicksonDoubling
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
What is aDivisionAlgebra?
R, C, H, O fromCayley-DicksonDoubling
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
What is aDivisionAlgebra?
R, C, H, O fromCayley-DicksonDoubling
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
What is aDivisionAlgebra?
R, C, H, O fromCayley-DicksonDoubling
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
What is aDivisionAlgebra?
R, C, H, O fromCayley-DicksonDoubling
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
What is aDivisionAlgebra?
R, C, H, O fromCayley-DicksonDoubling
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Table of Contents1 Clues for an unexpected relationship
KLT 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 pyramid7 Conformal Pyramid8 Conclusions and future work
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Supergravity
• Low energy limit of string theory• General relativity + supersymmetry (local susy parameter).• Field content- supergravity multiplets.• Characterised by scalar coset groups.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Supergravity
• Low energy limit of string theory
• General relativity + supersymmetry (local susy parameter).• Field content- supergravity multiplets.• Characterised by scalar coset groups.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Supergravity
• Low energy limit of string theory• General relativity + supersymmetry (local susy parameter).
• Field content- supergravity multiplets.• Characterised by scalar coset groups.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Supergravity
• Low energy limit of string theory• General relativity + supersymmetry (local susy parameter).• Field content- supergravity multiplets.
• Characterised by scalar coset groups.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Supergravity
• Low energy limit of string theory• General relativity + supersymmetry (local susy parameter).• Field content- supergravity multiplets.• Characterised by scalar coset groups.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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!
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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!
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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!
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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!
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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!
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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!
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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!
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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, whichdescribes transformations of different supercharges intoeach other.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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.
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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)
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The Scalarcosets ofSupergravity
N=1,2,4,8 SuperYang Mills overthe divisionalgebras
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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)
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Table of Contents1 Clues for an unexpected relationship
KLT 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 pyramid7 Conformal Pyramid8 Conclusions and future work
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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)
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
The real projective plane
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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)
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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)
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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)
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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)
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Isometries of projective planes
• isom(RP2) ∼= so(3)• isom(CP2) ∼= su(3)• isom(HP2) ∼= sp(3)• isom(OP2) ∼= f4
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Isometries of projective planes
• isom(RP2) ∼= so(3)
• isom(CP2) ∼= su(3)• isom(HP2) ∼= sp(3)• isom(OP2) ∼= f4
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Isometries of projective planes
• isom(RP2) ∼= so(3)• isom(CP2) ∼= su(3)
• isom(HP2) ∼= sp(3)• isom(OP2) ∼= f4
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Isometries of projective planes
• isom(RP2) ∼= so(3)• isom(CP2) ∼= su(3)• isom(HP2) ∼= sp(3)
• isom(OP2) ∼= f4
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Isometries of projective planes
• isom(RP2) ∼= so(3)• isom(CP2) ∼= su(3)• isom(HP2) ∼= sp(3)• isom(OP2) ∼= f4
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
What about the exceptionalgroups?
• isom((C⊗O)P2) ∼= e6• isom((H⊗O)P2) ∼= e7• isom((O⊗O)P2) ∼= e8
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
What about the exceptionalgroups?
• isom((C⊗O)P2) ∼= e6
• isom((H⊗O)P2) ∼= e7• isom((O⊗O)P2) ∼= e8
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
What about the exceptionalgroups?
• isom((C⊗O)P2) ∼= e6• isom((H⊗O)P2) ∼= e7
• isom((O⊗O)P2) ∼= e8
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
What about the exceptionalgroups?
• isom((C⊗O)P2) ∼= e6• isom((H⊗O)P2) ∼= e7• isom((O⊗O)P2) ∼= e8
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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
Supergravityas the square
of SuperYang-Mills
Silvia Nagy
Outline
Clues for anunexpectedrelationship
The divisionalgebras
A(super)quickintro toSupergravityand SuperYang-Mills
The MagicSquare
Projective planes
Isometries of theprojective planes
The MagicSquare
A MagicSquare ofSupergravities
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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
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
Tensoring theMultiplets
MAGIC!
Magic pyramid
ConformalPyramid
Conclusionsand futurework
Table of Contents1 Clues for an unexpected relationship
KLT 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 pyramid7 Conformal Pyramid8 Conclusions and future work
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
Tensoring theMultiplets
MAGIC!
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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 !
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
Tensoring theMultiplets
MAGIC!
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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 !
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
Tensoring theMultiplets
MAGIC!
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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 !
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
Tensoring theMultiplets
MAGIC!
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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 !
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
Tensoring theMultiplets
MAGIC!
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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 !
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
Tensoring theMultiplets
MAGIC!
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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 !
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
Tensoring theMultiplets
MAGIC!
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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 !
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
Tensoring theMultiplets
MAGIC!
Magic pyramid
ConformalPyramid
Conclusionsand futurework
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
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
Table of Contents1 Clues for an unexpected relationship
KLT 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 pyramid7 Conformal Pyramid8 Conclusions and future work
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
SYM in various dimensions
• Remember the lagragian:
L = −14Tr(Fµν ,F
µν)− i2Tr(λ̄, γµDµλ) (19)
• 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!
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
SYM in various dimensions
• Remember the lagragian:
L = −14Tr(Fµν ,F
µν)− i2Tr(λ̄, γµDµλ) (19)
• 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!
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
SYM in various dimensions
• Remember the lagragian:
L = −14Tr(Fµν ,F
µν)− i2Tr(λ̄, γµDµλ) (19)
• 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!
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
SYM in various dimensions
• Remember the lagragian:
L = −14Tr(Fµν ,F
µν)− i2Tr(λ̄, γµDµλ) (19)
• 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!
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
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!
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
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!
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
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!
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
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!
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
The Magic Pyramid
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
4 dimensions
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
Table of Contents1 Clues for an unexpected relationship
KLT 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 pyramid7 Conformal Pyramid8 Conclusions and future work
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
Conformal Pyramid
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
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
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
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
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
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
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
Table of Contents1 Clues for an unexpected relationship
KLT 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 pyramid7 Conformal Pyramid8 Conclusions and future work
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
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].
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
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].
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
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].
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
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].
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
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].
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