first-quantized n=4 yang-millsmoriyama/slides/slide_hatsuda.pdf · 2010. 4. 1. · 1...
TRANSCRIPT
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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
15
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.