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First-quantized N=4 Yang-Mills

arXiv:0812.4569[hep-th]

Machiko Hatsuda(KEK&Urawa),

Yu-tin Huang &Warren Siegel (YITP@Stony Brook)

I. Introduction

II. 1st-quantized “projective” superparticle

III. Projective superspace

IV. N=4 YM

V. Summary2009.2.10 @ Nagoya Univ.

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I. Introduction

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Motivation

Toward field theory with manifest symmetries

– by BRST:

– in superspace:

• AdS/CFT

Projective lightcone limit“A new AdS/CFT correspondence”

Nastase & Siegel, hep-th/0010106

“A new holographic limit of AdS5xS5”

Siegel & M.H., hep-th/0211184

“Superconformal spaces and implications for superstrings”

Siegel & M.H., arXiv:0709.4605

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translation Lorentz conformalboost

Conformal symmetry

by projective coordinate

• Conformal group SO(4,2)=SU(2,2)

• fractional linear transf.

• projective coordinate

ex. Infinitesimal tranf.

★Projective coordinates realize conformal sym. by fractional linear transf.

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[1] Supersymmetrization: SU(2,2) ⇒ SU(N|2,2)

[2] Minimization: SU(N|2,2)⇒OSp(N|4)

SU(N) Fermi

SU(2,2)Fermi

O(N) FermiT

Sp(4)Fermi

4N

4NN2-1

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N(N-1)/2

4N 10

OSp(n|2) metric Sp(2) metricOSp(N|4) metric

auxiliary coordinates

projective coordinates

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Superconformal symmetry

• Superconformal generators

• Super-twister on-shell generators

・・・fermi

・・・bose

SU(N|2,2) can be

made by only!

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II. 1st quantized “projective” superparticle

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BRST field theory

• Free scalar field action:

– BRST charge:

– Field:

– BRST gauge symmetry:

– Feynmann-Siegel gauge:

★How extend to super?

ghost for τ-diffeo

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BRST for super

• Spinning particle (NSR) :GSO & picture

• Brink-Schwarz superparticle (GS):• κsymmetry 1st class constrains ・・・1/4

• κgauge fixing ・・・・・・・・・・・・・・・・・・1/4

• 2nd class constraints・・・・・・・・・・・・・・1/4

⇒physical degrees of freedom・・・・・1/4

⇒ No need 2nd class constraints!

Can ¼ be obtained by cov. 1st class constraints only?

• Pure spinor: pure spinor condition on ghost

• Ghost pyramid: 1st class constraints only (∞reducible)

complicated

complicated

‟00 Berkovits

‟89 Siegel, ‟90 Mikovic, Rocek, Siegel, van Nieuwenhuizen, Yamron & van de Ven, ‟05 Lee & Siegel

‟81 Brink & Schwarz

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Regularize infinite number

of ghosts as

Then physical d.o.f.

becomes

⇒1/4

Spinor

ghost

ghost for ghost

ghost for ghost for ghost

Ghost pyramid Reducible constraints leads to ghost for ghost

★ Complicated YM coupling・・・

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Regularize infinite

number of ghosts as

Then physical d.o.f.

becomes

Separate projective

coordinate spinor

“Projective” superparticle

⇒1/2

⇒1/2

⇒ 1/2x1/2 = 1/4 !

“Projective” spinor

ghost

ghost for ghost

ghost for ghost for ghost

Auxiliary spinor

Ghost tower‟08 Huang, Siegel &M.H.

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• Supersymmetry generators:

• Action:

• Constraints:

– 2nd class:

– 1st class “κsymmetry”:

Brink-Schwarz superparticle

must be solved

can be gauged away

Covariant treatment is

difficult!

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“Projective” superparticle

• Covariant derivatives for projective coordinates

• 1st class constraint set

• Lightcone gauge

osp(N|4)

osp(N/2|2)2

¼ of 4N spinors !

( ref) PSU(4|2,2) case

‟01 Kamimura & M.H.

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: Fermi・・・internal space ghost

: Bose・・・κghost

: Fermi・・・τ-diffeo. ghost

BRST for “projective” superparticle

• BRST charge

– ghosts

★ Very simple expression

unifying κ-sym. & τ-diffeo. ！

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III. Projective superspace

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Simple (N=1) superspace

• Scalar multiplet action

– Superfield

– Susy transf.

★Susy transf. of fields is the one for coordinate.

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CP1

⊗

Extended(N＞1) superspace

• Hyper(N=2) superspace

– Automatically on-shell →no interaction

SU(N) internal symmetry requires coordinates !

• Harmonic(N=2,3,4) superspace

– 2-d. Sphere for SU(2)

• Projective(N=2) superspace

– CP1 for SU(2)

⊗

y plane

‟84 Galperin, Ivanov, Kalitzin, Ogievsky & Sokatchev

‟84 Karlhede, Lindstrom & Rocek‟88 Lindstrom & Rocek‟98 Gonzalez-Rey, Rocek, Wiles, Lindstrom & von Unge (Ref) N≧0 case ‟95 Hartwell & Howe, „00Heslop & Howe

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Projective(N=2) superspace

• Hypermultiplet action

– Real O(2) superfield

– Contour integral

Reduced to N=1 action

‟84 Karlhede, Lindstrom & Rocek

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Projective(N≧0) superspace

• Charge conjugation

– Inversion for N=2 case (CP1)

– Inversion for N case coordinates

– Superfield

• Real superconformal inv. action

0

Conjugate

is antipodal

map of

Riemann

sphere

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IV. N=4 YM

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YM-covariantized osp(N|4)

• YM-covariantization:

• Field strength:

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Field eq. for backgrond SYM

• Field strength relation from Bianchi

• Field eq. for N=4 (self-dual )

can be given by

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N=4 projective scalar field strength

• Projective gauge

• 4-point amplitude

is function only of projective coordinates

osp(2|2) osp(2|2)

(ref ‟07 Kallosh)

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V. Summary

• 1st quantized superparticle, described by

projective coordinates, is proposed.– 1st class constraints are bilinears of projective

coordinate derivatives with matrix indices.

– BRST has a simple form unifying κ-sym. &τ-diffeo.

• Projective superspace (N≧0) is proposed.– YM field strengths are introduced in it.

– A scalar superfield strength (N=4) is projective by whom 4-point amplitude is given.