Pure Spinor Helicity Methods

Rutger Boels

Niels Bohr International Academy, Copenhagen

R.B., arXiv:0908.0738 [hep-th]

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 1 / 25 in books?

Why you should pay attention, an experiment

The greatest common denominator of the audience?

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25 Why you should pay attention, an experiment

The greatest common denominator of the audience in books?

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25 Why you should pay attention, an experiment

The greatest common denominator of the audience in books:

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25 Why you should pay attention, an experiment

The greatest common denominator of the audience in science books?

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 2 / 25 for all legs simultaneously

pure spinor helicity methods: precise control over Poincaré and Susy quantum numbers

Why you should pay attention

subject: calculation of scattering amplitudes in D > 4 with many legs

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 3 / 25 for all legs simultaneously

Why you should pay attention

subject: calculation of scattering amplitudes in D > 4 with many legs

pure spinor helicity methods: precise control over Poincaré and Susy quantum numbers

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 3 / 25 Why you should pay attention

subject: calculation of scattering amplitudes in D > 4 with many legs

pure spinor helicity methods: precise control over Poincaré and Susy quantum numbers for all legs simultaneously

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 3 / 25 Outline

1 Motivation

2 Covariant representation theory of Poincaré algebra Spin algebra Susy algebra

3 Outlook

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 4 / 25 <=== Zürich

Every talk on amplitudes should mention ...

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 5 / 25 Every talk on amplitudes should mention ...

<=== Zürich

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 5 / 25 one-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08]

I uses 6D trees high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others]

I uses 10D, 11D trees recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known

is there a D > 4 analogue of: MHV amplitudes? (any recent buzzword in D = 4?)

pure spinor spaces are higher dimensional twistor spaces

Why study amplitudes in higher dimensions?

loops in four dimensions ← dimensional regularization

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25 recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known

is there a D > 4 analogue of: MHV amplitudes? (any recent buzzword in D = 4?)

pure spinor spaces are higher dimensional twistor spaces

Why study amplitudes in higher dimensions?

loops in four dimensions ← dimensional regularization one-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08]

I uses 6D trees high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others]

I uses 10D, 11D trees

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25 little is known

is there a D > 4 analogue of: MHV amplitudes? (any recent buzzword in D = 4?)

pure spinor spaces are higher dimensional twistor spaces

Why study amplitudes in higher dimensions?

loops in four dimensions ← dimensional regularization one-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08]

I uses 6D trees high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others]

I uses 10D, 11D trees recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25 is there a D > 4 analogue of: MHV amplitudes? (any recent buzzword in D = 4?)

pure spinor spaces are higher dimensional twistor spaces

Why study amplitudes in higher dimensions?

loops in four dimensions ← dimensional regularization one-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08]

I uses 6D trees high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others]

I uses 10D, 11D trees recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25 pure spinor spaces are higher dimensional twistor spaces

Why study amplitudes in higher dimensions?

loops in four dimensions ← dimensional regularization one-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08]

I uses 6D trees high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others]

I uses 10D, 11D trees recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known

is there a D > 4 analogue of: MHV amplitudes? (any recent buzzword in D = 4?)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25 Why study amplitudes in higher dimensions?

loops in four dimensions ← dimensional regularization one-loop: pure Yang-Mills [Giele, Kunszt, and Melnikov, 08]

I uses 6D trees high loops: N = 4, N = 8 [Bern, Dixon, Kosower et. al.], [others]

I uses 10D, 11D trees recent quantum leaps in four: what is special about four? string theory (analytic S-matrix type techniques) little is known

is there a D > 4 analogue of: MHV amplitudes? (any recent buzzword in D = 4?)

pure spinor spaces are higher dimensional twistor spaces

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 6 / 25 Outline

1 Motivation

2 Covariant representation theory of Poincaré algebra Spin algebra Susy algebra

3 Outlook

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 7 / 25 maximal set of commuting operators: kµ, Wµ µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k

1,3 ∃ vectors n1 and n2 such that q, qˆ, n1, n2 span R , q · ni = 0

massive polarization vectors

µ µ µ 1 µ µ µ ((n1 ± in2) · k) µ k mq e± = √ n ± in − q e = − 2 1 2 q · k 0 m q · k

broken gauge theory in D = 4 → [R.B., Christian Schwinn, to appear]

Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98]

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k

1,3 ∃ vectors n1 and n2 such that q, qˆ, n1, n2 span R , q · ni = 0

massive polarization vectors

µ µ µ 1 µ µ µ ((n1 ± in2) · k) µ k mq e± = √ n ± in − q e = − 2 1 2 q · k 0 m q · k

broken gauge theory in D = 4 → [R.B., Christian Schwinn, to appear]

Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98]

maximal set of commuting operators: kµ, Wµ

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 1,3 ∃ vectors n1 and n2 such that q, qˆ, n1, n2 span R , q · ni = 0

massive polarization vectors

µ µ µ 1 µ µ µ ((n1 ± in2) · k) µ k mq e± = √ n ± in − q e = − 2 1 2 q · k 0 m q · k

broken gauge theory in D = 4 → [R.B., Christian Schwinn, to appear]

Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98]

maximal set of commuting operators: kµ, Wµ µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 massive polarization vectors

µ µ µ 1 µ µ µ ((n1 ± in2) · k) µ k mq e± = √ n ± in − q e = − 2 1 2 q · k 0 m q · k

broken gauge theory in D = 4 → [R.B., Christian Schwinn, to appear]

Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98]

maximal set of commuting operators: kµ, Wµ µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k

1,3 ∃ vectors n1 and n2 such that q, qˆ, n1, n2 span R , q · ni = 0

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 broken gauge theory in D = 4 → [R.B., Christian Schwinn, to appear]

Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98]

maximal set of commuting operators: kµ, Wµ µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k

1,3 ∃ vectors n1 and n2 such that q, qˆ, n1, n2 span R , q · ni = 0

massive polarization vectors

µ µ µ 1 µ µ µ ((n1 ± in2) · k) µ k mq e± = √ n ± in − q e = − 2 1 2 q · k 0 m q · k

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 broken gauge theory in D = 4 → [R.B., Christian Schwinn, to appear]

Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98]

maximal set of commuting operators: kµ, Wµ µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k

1,3 ∃ vectors n1 and n2 such that q, qˆ, n1, n2 span R , q · ni = 0

massive polarization vectors

µ µ µ 1 µ µ µ ((n1 ± in2) · k) µ k mq e± = √ n ± in − q e = − 2 1 2 q · k 0 m q · k

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 broken gauge theory in D = 4 → [R.B., Christian Schwinn, to appear]

Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98]

maximal set of commuting operators: kµ, Wµ µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k

1,3 ∃ vectors n1 and n2 such that q, qˆ, n1, n2 span R , q · ni = 0

massive polarization vectors

µ µ µ 1 µ µ µ k mq e± = √ n˜ ± in˜ (k) e = − 2 1 2 0 m q · k

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 Warmup: massive vectors in four dimensions higher D massless vectors decompose into massive D = 4 massive spinor helicity: e.g. [Kleiss, Stirling, 85], [Dittmaier, 98]

maximal set of commuting operators: kµ, Wµ µ 2 kµk = m , pick spin axis through light-like vector q qµW qµk νΣρσ R = R1 = µ = µνρσ z q 2q · k 2q · k

1,3 ∃ vectors n1 and n2 such that q, qˆ, n1, n2 span R , q · ni = 0

massive polarization vectors

µ µ µ 1 µ µ µ k mq e± = √ n˜ ± in˜ (k) e = − 2 1 2 0 m q · k

broken gauge theory in D = 4 → [R.B., Christian Schwinn, to appear]

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 8 / 25 Pauli-Lubanski vector tensors k[µΣνρ] from previous use: 2 I choose q such that q = 0 1,D−1 I choose q, qˆ, ni to form an ortho-normal basis of R ˜ (ni ·k) I construct ni = ni − q (q·k) for representation theory choose Cartan generators as 1 Rj ≡ n˜ n˜ Σ = qn n (k Σ ) q 2 2j−1 2j 1 2 [µ νρ] ~ j j eigenvalues form a weight vector h: Rqe = h e, ~h = (0,..., ±1,...) polarization vectors (in q lightcone gauge)

µ 1 µ µ e (0,..., ±j ,...) = √ n˜ ± in˜ (k) 2 2j−1 2j

requires choice of complex structure

Higher dimensional massless vectors

given kµ, little group is ISO(D − 2)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 9 / 25 Pauli-Lubanski vector tensors k[µΣνρ] from previous use: 2 I choose q such that q = 0 1,D−1 I choose q, qˆ, ni to form an ortho-normal basis of R ˜ (ni ·k) I construct ni = ni − q (q·k) for representation theory choose Cartan generators as 1 Rj ≡ n˜ n˜ Σ = qn n (k Σ ) q 2 2j−1 2j 1 2 [µ νρ] ~ j j eigenvalues form a weight vector h: Rqe = h e, ~h = (0,..., ±1,...) polarization vectors (in q lightcone gauge)

µ 1 µ µ e (0,..., ±j ,...) = √ n˜ ± in˜ (k) 2 2j−1 2j

requires choice of complex structure

Higher dimensional massless vectors

given kµ, little group is SO(D − 2)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 9 / 25 from previous use: 2 I choose q such that q = 0 1,D−1 I choose q, qˆ, ni to form an ortho-normal basis of R ˜ (ni ·k) I construct ni = ni − q (q·k) for representation theory choose Cartan generators as 1 Rj ≡ n˜ n˜ Σ = qn n (k Σ ) q 2 2j−1 2j 1 2 [µ νρ] ~ j j eigenvalues form a weight vector h: Rqe = h e, ~h = (0,..., ±1,...) polarization vectors (in q lightcone gauge)

µ 1 µ µ e (0,..., ±j ,...) = √ n˜ ± in˜ (k) 2 2j−1 2j

requires choice of complex structure

Higher dimensional massless vectors

given kµ, little group is SO(D − 2) Pauli-Lubanski vector tensors k[µΣνρ]

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 9 / 25 for representation theory choose Cartan generators as 1 Rj ≡ n˜ n˜ Σ = qn n (k Σ ) q 2 2j−1 2j 1 2 [µ νρ] ~ j j eigenvalues form a weight vector h: Rqe = h e, ~h = (0,..., ±1,...) polarization vectors (in q lightcone gauge)

µ 1 µ µ e (0,..., ±j ,...) = √ n˜ ± in˜ (k) 2 2j−1 2j

requires choice of complex structure

Higher dimensional massless vectors

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 9 / 25 polarization vectors (in q lightcone gauge)

µ 1 µ µ e (0,..., ±j ,...) = √ n˜ ± in˜ (k) 2 2j−1 2j

requires choice of complex structure

Higher dimensional massless vectors

given kµ, little group is SO(D − 2) Pauli-Lubanski vector tensors k[µΣνρ] from previous use: 2 I choose q such that q = 0 1,D−1 I choose q, qˆ, ni to form an ortho-normal basis of R ˜ (ni ·k) I construct ni = ni − q (q·k) for representation theory choose Cartan generators as 1 Rj ≡ n˜ n˜ Σ = qn n (k Σ ) q 2 2j−1 2j 1 2 [µ νρ] ~ j j eigenvalues form a weight vector h: Rqe = h e, ~h = (0,..., ±1,...)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 9 / 25 Higher dimensional massless vectors

µ 1 µ µ e (0,..., ±j ,...) = √ n˜ ± in˜ (k) 2 2j−1 2j

requires choice of complex structure Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 9 / 25 q: choice of Cartan generators and choice of gauge D = 4: little group is Abelian can compare different legs: ~ µ ~ ~ ~ eµ ki , hi · e kj , hj = −δ hi + hj

Application: higher dimensional YM amplitudes at tree level

h(±1, 0,...)i1 (0, ±1,...)i2 ... (0,..., ±1)iD/2 i = 0

‘helicity equal’ in D = 4 ‘one helicity unequal’ does not vanish in D > 4 class of Einstein gravity amplitudes through KLT six gluon open string amplitude [Oprisa, Stieberger, 2005]

Remarks and a quick application massive vectors: SO(D − 1) ∃ generalization to all integer spins (e.g. gravity)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 10 / 25 can compare different legs: ~ µ ~ ~ ~ eµ ki , hi · e kj , hj = −δ hi + hj

Application: higher dimensional YM amplitudes at tree level

h(±1, 0,...)i1 (0, ±1,...)i2 ... (0,..., ±1)iD/2 i = 0

‘helicity equal’ in D = 4 ‘one helicity unequal’ does not vanish in D > 4 class of Einstein gravity amplitudes through KLT six gluon open string amplitude [Oprisa, Stieberger, 2005]

Remarks and a quick application massive vectors: SO(D − 1) ∃ generalization to all integer spins (e.g. gravity) q: choice of Cartan generators and choice of gauge D = 4: little group is Abelian

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 10 / 25 Application: higher dimensional YM amplitudes at tree level

h(±1, 0,...)i1 (0, ±1,...)i2 ... (0,..., ±1)iD/2 i = 0

‘helicity equal’ in D = 4 ‘one helicity unequal’ does not vanish in D > 4 class of Einstein gravity amplitudes through KLT six gluon open string amplitude [Oprisa, Stieberger, 2005]

Remarks and a quick application massive vectors: SO(D − 1) ∃ generalization to all integer spins (e.g. gravity) q: choice of Cartan generators and choice of gauge D = 4: little group is Abelian can compare different legs: ~ µ ~ ~ ~ eµ ki , hi · e kj , hj = −δ hi + hj

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 10 / 25 ‘helicity equal’ in D = 4 ‘one helicity unequal’ does not vanish in D > 4 class of Einstein gravity amplitudes through KLT six gluon open string amplitude [Oprisa, Stieberger, 2005]

Application: higher dimensional YM amplitudes at tree level

h(±1, 0,...)i1 (0, ±1,...)i2 ... (0,..., ±1)iD/2 i = 0

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 10 / 25 Remarks and a quick application massive vectors: SO(D − 1) ∃ generalization to all integer spins (e.g. gravity) q: choice of Cartan generators and choice of gauge D = 4: little group is Abelian can compare different legs: ~ µ ~ ~ ~ eµ ki , hi · e kj , hj = −δ hi + hj

Application: higher dimensional YM amplitudes at tree level

h(±1, 0,...)i1 (0, ±1,...)i2 ... (0,..., ±1)iD/2 i = 0

‘helicity equal’ in D = 4 ‘one helicity unequal’ does not vanish in D > 4 class of Einstein gravity amplitudes through KLT six gluon open string amplitude [Oprisa, Stieberger, 2005]

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 10 / 25 Outline

1 Motivation

2 Covariant representation theory of Poincaré algebra Spin algebra Susy algebra

3 Outlook

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 11 / 25 → deﬁnition of pure spinors

guess: eµ(~h) ∼ ξγµψ

want: µ µ qµe = kµe = ... = 0 therefore: µ µ µ µ ξ (qµγ ) = (kµγ ) ψ = 0 or ξ (kµγ ) = (qµγ ) ψ = 0

D other inner products → total 2 annihilation conditions

D i µ need spinors annihilated by 2 generators vµγ such that hv i , v j i = 0 ∀i, j

Towards spinor representations

, − polarization vectors and q and k span R1 D 1 needed: similar basis of spinors would like: vectors in terms of spinors

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 12 / 25 → deﬁnition of pure spinors

D i µ need spinors annihilated by 2 generators vµγ such that hv i , v j i = 0 ∀i, j

Towards spinor representations

, − polarization vectors and q and k span R1 D 1 needed: similar basis of spinors would like: vectors in terms of spinors

guess: eµ(~h) ∼ ξγµψ

want: µ µ qµe = kµe = ... = 0 therefore: µ µ µ µ ξ (qµγ ) = (kµγ ) ψ = 0 or ξ (kµγ ) = (qµγ ) ψ = 0

D other inner products → total 2 annihilation conditions

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 12 / 25 → deﬁnition of pure spinors

Towards spinor representations

, − polarization vectors and q and k span R1 D 1 needed: similar basis of spinors would like: vectors in terms of spinors

guess: eµ(~h) ∼ ξγµψ

want: µ µ qµe = kµe = ... = 0 therefore: µ µ µ µ ξ (qµγ ) = (kµγ ) ψ = 0 or ξ (kµγ ) = (qµγ ) ψ = 0

D other inner products → total 2 annihilation conditions

D i µ need spinors annihilated by 2 generators vµγ such that hv i , v j i = 0 ∀i, j

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 12 / 25 Towards spinor representations

, − polarization vectors and q and k span R1 D 1 needed: similar basis of spinors would like: vectors in terms of spinors

guess: eµ(~h) ∼ ξγµψ

want: µ µ qµe = kµe = ... = 0 therefore: µ µ µ µ ξ (qµγ ) = (kµγ ) ψ = 0 or ξ (kµγ ) = (qµγ ) ψ = 0

D other inner products → total 2 annihilation conditions

D i µ need spinors annihilated by 2 generators vµγ such that hv i , v j i = 0 ∀i, j → deﬁnition of pure spinors

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 12 / 25 deﬁne

+ 1 µ − 1 qµ µ γ ≡ √ kµγ γ ≡ √ γ 0 2 0 2 q · k + i +,i µ − i −,i µ γ ≡ √ eµ γ γ ≡ √ eµ γ i 2 i 2 D Clifford algebra: 2 copies of fermionic harmonic oscillator a b {γi , γj } = δij δa,−b

j 1 + − + − 1 quantum numbers Rq = 2 [γj , γj ] = γj γj − 2 quantum numbers ↔ annihilation conditions, 2hi ~ 2hi ~ γi ψ(h) = 0 no sum , γi ξ(h) = 0 no sum µ ~ µ ~ γµk ψ(h) = 0 γµq ξ(h) = 0 . phases, schmases

Spinor representations spinors transform under Clifford algebra {γµ, γν} = 2ηµν

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 13 / 25 D Clifford algebra: 2 copies of fermionic harmonic oscillator a b {γi , γj } = δij δa,−b

Spinor representations spinors transform under Clifford algebra {γµ, γν} = 2ηµν deﬁne

+ 1 µ − 1 qµ µ γ ≡ √ kµγ γ ≡ √ γ 0 2 0 2 q · k + i +,i µ − i −,i µ γ ≡ √ eµ γ γ ≡ √ eµ γ i 2 i 2

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 13 / 25 j 1 + − + − 1 quantum numbers Rq = 2 [γj , γj ] = γj γj − 2 quantum numbers ↔ annihilation conditions, 2hi ~ 2hi ~ γi ψ(h) = 0 no sum , γi ξ(h) = 0 no sum µ ~ µ ~ γµk ψ(h) = 0 γµq ξ(h) = 0 . phases, schmases

Spinor representations spinors transform under Clifford algebra {γµ, γν} = 2ηµν deﬁne

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 13 / 25 phases, schmases

Spinor representations spinors transform under Clifford algebra {γµ, γν} = 2ηµν deﬁne

j 1 + − + − 1 quantum numbers Rq = 2 [γj , γj ] = γj γj − 2 quantum numbers ↔ annihilation conditions, 2hi ~ 2hi ~ γi ψ(h) = 0 no sum , γi ξ(h) = 0 no sum µ ~ µ ~ γµk ψ(h) = 0 γµq ξ(h) = 0 .

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 13 / 25 Spinor representations spinors transform under Clifford algebra {γµ, γν} = 2ηµν deﬁne

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 13 / 25 for real momenta, ﬁxing one spinor product ﬁxes all by algebra, dependent on phase conventions

spinor helicity There is a phase convention for which

µ X µ X kµγ = ψ~hψ~h qµγ = ξ~hξ~h ~h ~h 1 1 q = ξ γ ξ k = ψ γ ψ µ 2 ~h µ ~h µ 2 ~h µ ~h

for any ~h. Similar formulae for polarization vectors.

representation is redundant (D = 4, 1), (D = 6, 2), (D = 10, 4)

Spinor helicity there is a natural spinor inner product ~ 0 0 ~ 0 0 ~ ~ 0 0 0† ψ h0, h ψ h0, h ∼ δ h0 + h0 δ h − h where ψ ψ ≡ ψ γ0ψ

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 14 / 25 spinor helicity There is a phase convention for which

µ X µ X kµγ = ψ~hψ~h qµγ = ξ~hξ~h ~h ~h 1 1 q = ξ γ ξ k = ψ γ ψ µ 2 ~h µ ~h µ 2 ~h µ ~h

for any ~h. Similar formulae for polarization vectors.

representation is redundant (D = 4, 1), (D = 6, 2), (D = 10, 4)

Spinor helicity there is a natural spinor inner product ~ 0 0 ~ 0 0 ~ ~ 0 0 0† ψ h0, h ψ h0, h ∼ δ h0 + h0 δ h − h where ψ ψ ≡ ψ γ0ψ for real momenta, ﬁxing one spinor product ﬁxes all by algebra, dependent on phase conventions

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 14 / 25 representation is redundant (D = 4, 1), (D = 6, 2), (D = 10, 4)

spinor helicity There is a phase convention for which

µ X µ X kµγ = ψ~hψ~h qµγ = ξ~hξ~h ~h ~h 1 1 q = ξ γ ξ k = ψ γ ψ µ 2 ~h µ ~h µ 2 ~h µ ~h

for any ~h. Similar formulae for polarization vectors.

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 14 / 25 Spinor helicity there is a natural spinor inner product ~ 0 0 ~ 0 0 ~ ~ 0 0 0† ψ h0, h ψ h0, h ∼ δ h0 + h0 δ h − h where ψ ψ ≡ ψ γ0ψ for real momenta, ﬁxing one spinor product ﬁxes all by algebra, dependent on phase conventions

spinor helicity There is a phase convention for which

µ X µ X kµγ = ψ~hψ~h qµγ = ξ~hξ~h ~h ~h 1 1 q = ξ γ ξ k = ψ γ ψ µ 2 ~h µ ~h µ 2 ~h µ ~h

for any ~h. Similar formulae for polarization vectors.

representation is redundant (D = 4, 1), (D = 6, 2), (D = 10, 4)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 14 / 25 pick a frame in which q = (1, 0,..., 0, 1), n˜i = (0,..., 1i , 0,...). pick a gamma matrix representation. µ ﬁnd eigenvectors of qµγ ξ = 0 other quantum numbers are then easy ← momentum independent µ all solutions to the massless Dirac equation by applying γµk

this ﬁxes the action of all generators on all states obtained states form a basis of all spinors

numerics & lightcone analysis...

Calculability 1 1 choose highest weight ξ( 2 ,... 2 ) (qξ = 0) deﬁne (phases of) other states using (ordered) lowering operators, e.g. 1 1 1 ψ(− ) ≡ γ−γ−ξ( ) = −γ−γ−ξ( ) 2 0 1 2 1 0 2

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 15 / 25 pick a frame in which q = (1, 0,..., 0, 1), n˜i = (0,..., 1i , 0,...). pick a gamma matrix representation. µ ﬁnd eigenvectors of qµγ ξ = 0 other quantum numbers are then easy ← momentum independent µ all solutions to the massless Dirac equation by applying γµk

numerics & lightcone analysis...

Calculability 1 1 choose highest weight ξ( 2 ,... 2 ) (qξ = 0) deﬁne (phases of) other states using (ordered) lowering operators, e.g. 1 1 1 ψ(− ) ≡ γ−γ−ξ( ) = −γ−γ−ξ( ) 2 0 1 2 1 0 2 this ﬁxes the action of all generators on all states obtained states form a basis of all spinors

numerics & lightcone analysis...

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 15 / 25 Calculability 1 1 choose highest weight ξ( 2 ,... 2 ) (qξ = 0) deﬁne (phases of) other states using (ordered) lowering operators, e.g. 1 1 1 ψ(− ) ≡ γ−γ−ξ( ) = −γ−γ−ξ( ) 2 0 1 2 1 0 2 this ﬁxes the action of all generators on all states obtained states form a basis of all spinors

numerics & lightcone analysis...

pick a frame in which q = (1, 0,..., 0, 1), n˜i = (0,..., 1i , 0,...). pick a gamma matrix representation. µ ﬁnd eigenvectors of qµγ ξ = 0 other quantum numbers are then easy ← momentum independent µ all solutions to the massless Dirac equation by applying γµk

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 15 / 25 4D spinor helicity proposal for 6D [Cheung, O’ Donnal, 09] lightcone frame

generalization to all spinor representations (e.g. gravitini) solutions to the massive Dirac equation, use

k 2 k − q = k [ 2q · k

choice of complex structure? → pure spinor spaces

study supersymmetry ﬁnd more amplitudes (on-shell recursion!)

Some remarks Checks

Extensions

Intriguing further structure

To do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 16 / 25 generalization to all spinor representations (e.g. gravitini) solutions to the massive Dirac equation, use

k 2 k − q = k [ 2q · k

choice of complex structure? → pure spinor spaces

study supersymmetry ﬁnd more amplitudes (on-shell recursion!)

Some remarks Checks 4D spinor helicity proposal for 6D [Cheung, O’ Donnal, 09] lightcone frame Extensions

Intriguing further structure

To do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 16 / 25 choice of complex structure? → pure spinor spaces

study supersymmetry ﬁnd more amplitudes (on-shell recursion!)

Some remarks Checks 4D spinor helicity proposal for 6D [Cheung, O’ Donnal, 09] lightcone frame Extensions generalization to all spinor representations (e.g. gravitini) solutions to the massive Dirac equation, use

k 2 k − q = k [ 2q · k Intriguing further structure

To do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 16 / 25 study supersymmetry ﬁnd more amplitudes (on-shell recursion!)

k 2 k − q = k [ 2q · k Intriguing further structure choice of complex structure? → pure spinor spaces To do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 16 / 25 Some remarks Checks 4D spinor helicity proposal for 6D [Cheung, O’ Donnal, 09] lightcone frame Extensions generalization to all spinor representations (e.g. gravitini) solutions to the massive Dirac equation, use

k 2 k − q = k [ 2q · k Intriguing further structure choice of complex structure? → pure spinor spaces To do study supersymmetry ﬁnd more amplitudes (on-shell recursion!)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 16 / 25 Outline

1 Motivation

2 Covariant representation theory of Poincaré algebra Spin algebra Susy algebra

3 Outlook

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 17 / 25 4D: derived from 1 Lagrangian [Grisaru, Pendleton, Van Nieuwenhuizen, 76] 2 on-shell susy algebra [Grisaru, Pendleton, 77] further only 4D fundamental massive multiplet [Schwinn, Weinzierl, 06]: derivation 1 should not require off-shell information

In any susy S-matrix theory

0 = h0|S Φin Q |0i = h0|S [Q, Φin]|0i = h0|SQ |ini

need: knowledge of action of supersymmetry on free in-state need: covariant supersymmetry representation theory

Susy Ward identities

susy relates scattering amplitudes with different particles independent of coupling constants

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 18 / 25 should not require off-shell information

In any susy S-matrix theory

0 = h0|S Φin Q |0i = h0|S [Q, Φin]|0i = h0|SQ |ini

need: knowledge of action of supersymmetry on free in-state need: covariant supersymmetry representation theory

Susy Ward identities

susy relates scattering amplitudes with different particles independent of coupling constants 4D: derived from 1 Lagrangian [Grisaru, Pendleton, Van Nieuwenhuizen, 76] 2 on-shell susy algebra [Grisaru, Pendleton, 77] further only 4D fundamental massive multiplet [Schwinn, Weinzierl, 06]: derivation 1

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 18 / 25 In any susy S-matrix theory

0 = h0|S Φin Q |0i = h0|S [Q, Φin]|0i = h0|SQ |ini

need: knowledge of action of supersymmetry on free in-state need: covariant supersymmetry representation theory

Susy Ward identities

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 18 / 25 need: covariant supersymmetry representation theory

Susy Ward identities

In any susy S-matrix theory

0 = h0|S Φin Q |0i = h0|S [Q, Φin]|0i = h0|SQ |ini

need: knowledge of action of supersymmetry on free in-state

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 18 / 25 Susy Ward identities

In any susy S-matrix theory

0 = h0|S Φin Q |0i = h0|S [Q, Φin]|0i = h0|SQ |ini

need: knowledge of action of supersymmetry on free in-state need: covariant supersymmetry representation theory

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 18 / 25 and expand everything into the pure spinor basis, e.g. X Q ≡ Q ψ(k,~h) + Q0 ξ(q,~h) ~h ~h ~h

from which we get →

{Q , Q } = 2 Q0 = 0 ~h ~h ~h

in terms of Lorentz invariant generators

~ ~ ~ Q~h|k, gi = |k, g + hi → action of any Q known covariantly

Covariant action of supersymmetry generators take µ {Q, Q} = 2kµγ

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 19 / 25 from which we get →

{Q , Q } = 2 Q0 = 0 ~h ~h ~h

in terms of Lorentz invariant generators

~ ~ ~ Q~h|k, gi = |k, g + hi → action of any Q known covariantly

Covariant action of supersymmetry generators take µ {Q, Q} = 2kµγ and expand everything into the pure spinor basis, e.g. X Q ≡ Q ψ(k,~h) + Q0 ξ(q,~h) ~h ~h ~h

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 19 / 25 in terms of Lorentz invariant generators

~ ~ ~ Q~h|k, gi = |k, g + hi → action of any Q known covariantly

Covariant action of supersymmetry generators take µ {Q, Q} = 2kµγ and expand everything into the pure spinor basis, e.g. X Q ≡ Q ψ(k,~h) + Q0 ξ(q,~h) ~h ~h ~h

from which we get →

{Q , Q } = 2 Q0 = 0 ~h ~h ~h

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 19 / 25 Covariant action of supersymmetry generators take µ {Q, Q} = 2kµγ and expand everything into the pure spinor basis, e.g. X Q ≡ Q ψ(k,~h) + Q0 ξ(q,~h) ~h ~h ~h

from which we get →

{Q , Q } = 2 Q0 = 0 ~h ~h ~h

in terms of Lorentz invariant generators

~ ~ ~ Q~h|k, gi = |k, g + hi → action of any Q known covariantly

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 19 / 25 Example: N = 1, D = 10 bosonic states are e.g. ~h = (±1, 0, 0, 0) (# = 8) ~ 1 1 1 fermionic ones are h = (± 2 , ± 2 , ± 2 , a), last ‘bit’ ﬁxed by chirality drop last bit

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 20 / 25 Example: N = 1, D = 10 bosonic states are e.g. ~h = (±1, 0, 0, 0) (# = 8) ~ 1 1 1 fermionic ones are h = (± 2 , ± 2 , ± 2 , a), last ‘bit’ ﬁxed by chirality drop last bit

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 20 / 25 choice of highest weight state component → split algebra into creation and annihilation operators Susy now acts naturally on the coherent state

i P (Qc θi ) i ηi θi 0 e |k, ηi i = e |k, ηi + ηi i

i (Qaθi ) e |k, ηi i = |k, ηi + θi i compare [Arkani-Hamed,Cachazo,Kaplan, 08] for N = (4, 8), D = 4 (8 + 6 additional states) coherent state scattering amplitudes naturally supersymmetric same conclusions? → next slide

Susy amplitudes: coherent states

every fermionic harmonic oscillator has a coherent state representation, i (Q ηi ) ~ |kηi i = e |k, htopi

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 21 / 25 Susy now acts naturally on the coherent state

i P (Qc θi ) i ηi θi 0 e |k, ηi i = e |k, ηi + ηi i

Susy amplitudes: coherent states

every fermionic harmonic oscillator has a coherent state representation, i (Q ηi ) ~ |kηi i = e |k, htopi choice of highest weight state component → split algebra into creation and annihilation operators

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 21 / 25 compare [Arkani-Hamed,Cachazo,Kaplan, 08] for N = (4, 8), D = 4 (8 + 6 additional states) coherent state scattering amplitudes naturally supersymmetric same conclusions? → next slide

Susy amplitudes: coherent states

i P (Qc θi ) i ηi θi 0 e |k, ηi i = e |k, ηi + ηi i

i (Qaθi ) e |k, ηi i = |k, ηi + θi i

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 21 / 25 same conclusions? → next slide

Susy amplitudes: coherent states

i P (Qc θi ) i ηi θi 0 e |k, ηi i = e |k, ηi + ηi i

i (Qaθi ) e |k, ηi i = |k, ηi + θi i compare [Arkani-Hamed,Cachazo,Kaplan, 08] for N = (4, 8), D = 4 (8 + 6 additional states) coherent state scattering amplitudes naturally supersymmetric

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 21 / 25 Susy amplitudes: coherent states

i P (Qc θi ) i ηi θi 0 e |k, ηi i = e |k, ηi + ηi i

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 21 / 25 I any chiral/antichiral susy transformation (allowed if U(1)R is unbroken)

does the susy transformation exist? D = 4: free parameters (‘one helicity unequal’ = 0)

D > 4: ﬁxes susy transformation (‘one helicity unequal’ 6= 0)

can use susy to shift one ηi coherent state parameter to zero, without phase factors 0 amplitudes vanish to all orders in ~, α , gs,... applies to all top states which share a coherent state parameter Yang-Mills, Einstein gravity, string theory ... an important subtlety

Vanishing amplitudes consider n particle ‘all plus’ amplitude, Z n ! ~ Y 4 A(ki , h) = dηi A(ki , ηi ) i=1

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25 I any chiral/antichiral susy transformation (allowed if U(1)R is unbroken)

does the susy transformation exist? D = 4: free parameters (‘one helicity unequal’ = 0)

D > 4: ﬁxes susy transformation (‘one helicity unequal’ 6= 0)

0 amplitudes vanish to all orders in ~, α , gs,... applies to all top states which share a coherent state parameter Yang-Mills, Einstein gravity, string theory ... an important subtlety

Vanishing amplitudes consider n particle ‘all plus’ amplitude, Z n ! ~ Y 4 A(ki , h) = dηi A(ki , ηi ) i=1

can use susy to shift one ηi coherent state parameter to zero, without phase factors

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25 I any chiral/antichiral susy transformation (allowed if U(1)R is unbroken)

does the susy transformation exist? D = 4: free parameters (‘one helicity unequal’ = 0)

D > 4: ﬁxes susy transformation (‘one helicity unequal’ 6= 0)

applies to all top states which share a coherent state parameter Yang-Mills, Einstein gravity, string theory ... an important subtlety

Vanishing amplitudes consider n particle ‘all plus’ amplitude, Z n ! ~ Y 4 A(ki , h) = dηi A(ki , ηi ) i=1

can use susy to shift one ηi coherent state parameter to zero, without phase factors 0 amplitudes vanish to all orders in ~, α , gs,...

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25 I any chiral/antichiral susy transformation (allowed if U(1)R is unbroken)

does the susy transformation exist? D = 4: free parameters (‘one helicity unequal’ = 0)

D > 4: ﬁxes susy transformation (‘one helicity unequal’ 6= 0)

an important subtlety

Vanishing amplitudes consider n particle ‘all plus’ amplitude, Z n ! ~ Y 4 A(ki , h) = dηi A(ki , ηi ) i=1

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25 I any chiral/antichiral susy transformation (allowed if U(1)R is unbroken)

does the susy transformation exist? D = 4: free parameters (‘one helicity unequal’ = 0)

D > 4: ﬁxes susy transformation (‘one helicity unequal’ 6= 0)

Vanishing amplitudes consider n particle ‘all plus’ amplitude, Z n ! ~ Y 4 A(ki , h) = dηi A(ki , ηi ) i=1

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25 I any chiral/antichiral susy transformation (allowed if U(1)R is unbroken)

D = 4: free parameters (‘one helicity unequal’ = 0)

D > 4: ﬁxes susy transformation (‘one helicity unequal’ 6= 0)

Vanishing amplitudes consider n particle ‘all plus’ amplitude, Z n ! ~ Y 4 A(ki , h) = dηi A(ki , ηi ) i=1

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25 D > 4: ﬁxes susy transformation (‘one helicity unequal’ 6= 0)

Vanishing amplitudes consider n particle ‘all plus’ amplitude, Z n ! ~ Y 4 A(ki , h) = dηi A(ki , ηi ) i=1

I any chiral/antichiral susy transformation (allowed if U(1)R is unbroken)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25 Vanishing amplitudes consider n particle ‘all plus’ amplitude, Z n ! ~ Y 4 A(ki , h) = dηi A(ki , ηi ) i=1

I any chiral/antichiral susy transformation (allowed if U(1)R is unbroken) D > 4: ﬁxes susy transformation (‘one helicity unequal’ 6= 0)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 22 / 25 ∃ ﬁeld theory derivation for N = 1 D = 10

∃ extension to massive reps (reproduces D = 4 fundamental massive) BPS representations by decomposition [Fayet, 78]

generic central charges (’technicality’)

Remarks

crosscheck

extensions

to do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 23 / 25 ∃ extension to massive reps (reproduces D = 4 fundamental massive) BPS representations by decomposition [Fayet, 78]

generic central charges (’technicality’)

Remarks

crosscheck ∃ ﬁeld theory derivation for N = 1 D = 10

extensions

to do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 23 / 25 generic central charges (’technicality’)

Remarks

crosscheck ∃ ﬁeld theory derivation for N = 1 D = 10

extensions ∃ extension to massive reps (reproduces D = 4 fundamental massive) BPS representations by decomposition [Fayet, 78]

to do

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 23 / 25 Remarks

crosscheck ∃ ﬁeld theory derivation for N = 1 D = 10

extensions ∃ extension to massive reps (reproduces D = 4 fundamental massive) BPS representations by decomposition [Fayet, 78]

to do generic central charges (’technicality’)

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 23 / 25 join the fun!

I non-zero amplitudes? → on-shell recursion (work in progress) I less nuts and bolts? I connection to Berkovits’ pure spinors? I spontaneous symmetry breaking?

Conclusions and outlook

complete spinor helicity construction from covariant representation theory quantum numbers under control

I class of vanishing amplitudes

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 24 / 25 Conclusions and outlook

complete spinor helicity construction from covariant representation theory quantum numbers under control

I class of vanishing amplitudes

join the fun!

I non-zero amplitudes? → on-shell recursion (work in progress) I less nuts and bolts? I connection to Berkovits’ pure spinors? I spontaneous symmetry breaking?

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 24 / 25 Example: N = 1, D = 11

Rutger Boels (NBIA) Pure Spinor Helicity Methods 15th String Workshop, Zürich 25 / 25