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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?