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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

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

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Infinite radius limit

plus

holomorphy

Chiral model of beta-gamma systems

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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

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):

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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

/0504078

Nekrasov

,

hep-th

/0511008

Y-formalism

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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!

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

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Quadratic constraint

Pure

spinor

constraint

Conifold = singular CY space

Basic OPE

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= Constant vector

Gauge symmetry

Gauge-invariant currents

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Ghost number current

SO(N)

generators

Stress-energy tensor

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Ghost number current

SO(N) generators

Stress-energy tensor

Cf.

Current algebra

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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

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BRST charge

Stress-energy tensor

b-ghost

Difficulty of treating constraints more than quadratic

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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?