1 “y-formalism & curved beta-gamma systems” p.a. grassi (univ. of piemonte orientale) m....
DESCRIPTION
Chiral model of beta-gamma systems 3 An infinite tower of states Non-trivial partition function Neither operator nor functional formalism Some aspects are known: “Chiral de Rham Complex” by F. Malikov et al., math.AG/ = N=2 superconformal field theory The most interesting case Bosonic pure spinor formalism One interesting approach: Cech cohomology construction by Nekrasov, hep-th/ The procedure of gluing of free CFT on different patches Unpractical (!) since it works only if the path structure is knownTRANSCRIPT
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“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
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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
Infinite radius limit plusholomorphy
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Chiral model of beta-gamma systems
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
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Review of curved beta-gamma systems
= 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):
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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
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Basic OPE
Diffeomorphisms
Current
Anomaly term Witten, hep-th/0504078Nekrasov, hep-th/0511008
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Y-formalism
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 withoutsolving the constraints!
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Y-formalism for beta-gamma models with quadratic constraint
Target space manifold X = a hypersurface in n dimensions defined by constraints
= Homogeneous function of degree h
Gauge symmetry
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Quadratic constraint
Pure spinor constraint
Conifold = singular CY space
Basic OPE
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= Constant vector
Gauge symmetry
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Gauge-invariant currents
Ghost number current
SO(N) generators
Stress-energy tensor
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Ghost number current
SO(N) generators
Stress-energy tensor
Cf.
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Current algebra
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Adding other variables
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
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BRST charge
Stress-energy tensor
b-ghost
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Difficulty of treating constraints more than quadratic
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Conclusion
1. Construction of Y-formalism on a given hypersurface
2. Derivation of algebra among currents3. Construction of quantum b-ghost4. Calculation of partition function5. Construction of Y-formalism on a given super-hypersurface
A remaining question:How to treat systems with non-quadratic constraints?