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Presentation on theme: "1 “Y-formalism"— Presentation transcript:
Slide1
1
“Y-formalism & Curved Beta-Gamma Systems”
P.A. Grassi (Univ. of Piemonte Orientale)M. Tonin (Padova Univ.)I. O. (Univ. of the Ryukyus）
N.P.B (in press)
28 Jul.- 1 Aug.2008,
Yukawa Institute’s workshop
Slide2
Covariant quantization of Green-Schwarz superstring action (1984) Pure spinor formalism by N. Berkovitz (2000) = CFT on a cone SO(10)/U(5)A simple question:“What kind of conformal field theory can be constructed on a given hypersurface?”Sigma models on a constrained surface Difficult to compute the spectrum and correlation functions Chiral model of beta-gamma systems
Motivations of this study
2
Infinite radius limit
plus
holomorphy
Slide3
Chiral model of beta-gamma systems
3
An infinite tower of statesNon-trivial partition functionNeither operator nor functional formalismSome aspects are known: “Chiral de Rham Complex” by F. Malikov et al., math.AG/9803041 = N=2 superconformal field theory
The most interesting case Bosonic pure spinor formalism
One interesting approach:
Cech cohomology construction by Nekrasov, hep-th/0511008The procedure of gluing of free CFT on different patchesUnpractical (!) since it works only if the path structure is known
Slide4
Review of curved beta-gamma systems
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= World-sheet Riemann surface
= Target-space complex manifold surface
= Open covering of X
= Local coordinates in
= (1, 0)-form on
Action
of
Beta-gamma system (
Holomorphic
sector):
Slide5
5
Sigma model
Local coordinates on X
Hermitian
components
In conformal gauge, using first-order formalism
By construction, this action is a free,
conformal
field
theory.
Holomorphy
Infinite radius limit
Redefinition
Slide6
6
Basic OPE
Diffeomorphisms
Current
Anomaly term
Witten,
hep-th
/0504078
Nekrasov
,
hep-th
/0511008
Slide7
Y-formalism
7
M.
Tonin & I. O. , P.L.B520(2001)398; N.P.B639(2002)182;P.L.B606(2005)218; N.P.B727(2005)176; N.P.B779(2007)63
It relies on the existence of patches but it does not use itEasy to compute contact terms and anomalies in OPE’sEasy to construct b-ghost
We wish to use Y-formalism to study beta-gamma systems
Quantization of a system with constraints (on hypersurface)
Our strategy: A radically different way
Impose constraints at each step of computation without
s
olving the constraints!
Slide8
Y-formalism for beta-gamma models with quadratic constraint
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Target space manifold X = a hypersurface in n dimensions defined by constraints
= Homogeneous function of degree h
Gauge symmetry
Slide9
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Quadratic constraint
Pure
spinor
constraint
Conifold = singular CY space
Basic OPE
Slide10
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= Constant vector
Gauge symmetry
Slide11
Gauge-invariant currents
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Ghost number current
SO(N)
generators
Stress-energy tensor
Slide12
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Ghost number current
SO(N) generators
Stress-energy tensor
Cf.
Slide13
Current algebra
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Slide14
Adding other variables
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Purely bosonic beta-gamma systems
No BRST charge (needed for constructing physical states)No conformal field theory with zero central charge
Necessity for adding other variables!
Bosonic variables
Fermionic
variables
Slide15
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BRST charge
Stress-energy tensor
b-ghost
Slide16
Difficulty of treating constraints more than quadratic
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Slide17
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Conclusion
Construction of Y-formalism on a given hypersurfaceDerivation of algebra among currentsConstruction of quantum b-ghostCalculation of partition functionConstruction of Y-formalism on a given super-hypersurface
A remaining question:
How to treat systems with non-quadratic constraints?
