Presentations text content in 1 “Y-formalism
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
Slide2Covariant 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
Slide3Chiral 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
Slide4Review of curved beta-gamma systems
4
= 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):
Slide55
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
Slide66
Basic OPE
Diffeomorphisms
Current
Anomaly term
Witten,
hep-th
/0504078
Nekrasov
,
hep-th
/0511008
Slide7Y-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!
Slide8Y-formalism for beta-gamma models with quadratic constraint
8
Target space manifold X = a hypersurface in n dimensions defined by constraints
= Homogeneous function of degree h
Gauge symmetry
Slide99
Quadratic constraint
Pure
spinor
constraint
Conifold = singular CY space
Basic OPE
Slide1010
= Constant vector
Gauge symmetry
Slide11Gauge-invariant currents
11
Ghost number current
SO(N)
generators
Stress-energy tensor
Slide1212
Ghost number current
SO(N) generators
Stress-energy tensor
Cf.
Slide13Current algebra
13
Slide14Adding other variables
14
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
Slide1515
BRST charge
Stress-energy tensor
b-ghost
Slide16Difficulty of treating constraints more than quadratic
16
Slide1717
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?
1 “Y-formalism
Download Presentation - The PPT/PDF document "1 “Y-formalism" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.