Pure%20Spinor%20Superstrings%20from%20Supergravity

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Pure Spinor Superstrings fromSupergravity

• Pietro Fré
• _at_ Dubna September 2008

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Based on work with

• P.A.Grassi (UPO, Alessandria)
• M. Trigiante, R. DAuria (Politecnico di Torino)
• Hep-th 0803.1809, hep-th 0803.1703,
• Hep-th 0704.3413, hep-th 0808.1282
• Hep-th 0807.0044

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Where does this structure come from?

• What are the pure spinor fields?
• Where do the pure spinor constraints come from?
• How can we write the pure spinor superstrings on
  a generic curved background?
• How to solve the pure spinor superstring in Anti
  de Sitter backgrounds?

SUPERGRAVITY IS THE ANSWER
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The Rheonomic approach to SUGRA theories

• Neeman Regge 1978
• DAuria Fré 1979
• Castellani, DAuria, Fré,
• van Nieuwenhuizen, Pilch, Townsend 1980 - 1982

A simple idea, similar to analyticity, provided
by equations analogous to Cauchy Riemann
equations determines all possible supergravity
theories encoding their symmetries and their
dynamics into a single constructive principle
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Rheonomy of superspace
Vertical fermionic directions
tangent space to space-time
? x
PRINCIPLE OF RHEONOMY
space-time
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Flow chart to costruct a SUGRA
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Table of Supergravities in D10
Theories BOSE STATES BOSE STATES FERMI STATES FERMI STATES
  NS - NS R - R Left handed Right handed
Type II A
Type II B  
Heterotic SO(32)    
Heterotic E8 x E8  
Type I SO(32)  
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The Type II Lagrangians in D10
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String Origin of the NS sector
The bosonic part of the 2D ? - model
By conformal invariance leads to the effective
action of the NS target space fields. These
correspond to the zero modes in the NS spectrum
of string states
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The Ramond sector
In the NSR formulation of Superstrings there are
also world sheet fermions that are target space
vectors ?L? and ?R?
It is necessary to specify boundary conditions
for these fermions on the world sheet Riemann
surface. They can be periodic-periodic or
periodic antiperiodic or antiperiodic-antiperiodic
for ?L? and ?R?.
Depending on the various boundary conditions
(spin structures) we have the various sectors of
the string spectrum.
The periodic boundary conditions define the
Ramond sector. In the NR sector we find the
fermionic particles, in the RR we find the p-forms
There is no room in the sigma model for the RR
fields
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Let us illustrate the SUGRA construction à la
rheonomy FDA

• Starting from D10 type IIA supergravity
• Typically higher dimension SUGRAS involve p-form
  gauge fields
• The main question are
• What are the underlying algebras of which sugras
  are the gauge theories?
• How do we construct the gauge theories of such
  algebras?

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The answers are

• The underlying algebras are not Lie algebras nor
  super Lie algebras
• They are FDAs (free differential algebras)
• FDAs enlarge the category of (super) Lie algebras
  and are the proper structures to include p-forms
• FDAs differently from Lie algebras include their
  own gauging
• Merging Rheonomy with FDAs one constructs SUGRAS

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Free Differential Algebrasand Sullivan
structural Theorems
String Theory leads to p-form gauge fields that
are an essential part of all supergravity
multiplets in higher dimensions. The algebraic
structure underlying SUGRA in Dgt4 is that of an
FDA Free Differential Algebra
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Sullivans Structural Theorems
The analogue of Levis Theorem minimal versus
contractible algebras.
Sullivans first theorem
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Physical interpretation of Sullivans first
theorem
Differently from normal Lie algebras, FDAs
already contain their own gauging. The important
point is to identify the structure of the
underlying minimal algebra. The FDA is non
trivial only if the minimal algebra is non
trivial. The structure of this latter is obtained
by considering the zero curvature equations,
namely by modding out the contractible subalgebra
P. Fré Class. Quan. Grav. 1, (1984) L81
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Chevalley cohomology
The algebraic action of the coboundary ?
induces a sequence of maps
Cohomology groups
as usual...!
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Sullivans second Theorem
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Sullivans Theorems at work in D10 type IIA
The Maurer Cartan formulation of the super
Poincaré Lie algebra
For instance
FDA extension with B2 generator
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The complete FDA for type IIA SUGRA

• Black generators super Lie subalgebra
• Cyanide generators extension to FDA
• Red generators contractible generators
  curvatures

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The physical interpretation
LET US NOW SEE THE THEONOMIC PARAMETRIZATIONS
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The rheonomic parametrization of bosonic
curvatures

• Black forms outer directions
• Red curvature components inner components
• Blue forms inner directions
• Outer components linear expression in terms of
  inner ones
• THIS IS THE PRINCIPLE OF RHEONOMY
• IT ENCODES ALL THE DYNAMICS OF SUPERGRAVITY

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The rheonomic parametrization of fermionic
curvatures
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The field equations are justconsistency
conditions for.
THE RHEONOMIC PARAMETRIZATION
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Algebraic BRST quantization(Anselmi Frè (1994))
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In the case of type IIA Sugra
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Constrained BRST algebras

• Ghosts are associated with the quantum fixing of
  symmetries.
• Constraining ghosts liberating symmetries
• Let us consider
• Reinstalling diffeomorphism covariance deleting
  translation ghost ?a
• Reinstalling gauge covariance deleting a,c,b
• Is this consistent? Yes if additional constraints
  are imposed on the supersymmetry ghosts ?

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From the descent equations we obtain all the BRST
transformations
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BRST transformation of fields
BRST of physical fields
BRST of superghosts pure spinors
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Pure spinor constraints
When the superghosts are complexified these
constraints have a solution in terms of 22
independent parameters
This is very important in order to obtain a c0
conformal field theory!
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BRST invariant actions of type IIA superstring

• In the GS approach for all p-branes we have
• A ?supersymmetric action Aclass
• ?-supersymmetry is just the projection on the
  brane world-volume of bulk supersyemmtry
• We extend the bulk FDA with ghosts and we impose
  the pure spinor constraints we liberate bosonic
  symmetry
• The ?-supersymmetric action Aclass is not
  invariant under the constrained BRST
• We can always find a suitable gauge fermion such
  that

is invariant
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The antighost sector
The antighost sector is determined by the choice
of the gauge fixing terms.
We introduce a pair of spinor fields w with
ghost number g - 1 and a pair of spinor fields
d with ghost number g 0
The BRST invariant action
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Summarizing

• The algebraic structure underlying Supergravity
  and M theory is the category of FDAs.
• FDAs determine the structure of the Berkovits
  BRST operator and the appropriate scheme for
  string covariant quantization in any non trivial
  background.
• Let us apply this to a particular background

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Type IIA on AdS4 P3
This ansatz solves the supergravity field
equations and leads to a d4 theory with
preserved N6 susy on AdS4
The pure string action reduces to a ?-model on
the above super coset
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The Maurer Cartan forms ofOSp(64)
In terms of these forms we can cosntruct the
entire PS string action
?xy contains the vielbein Va and the spin
connection ?ab
AAB contains the vielbein V? and the spin
connection ???
?xA yields the gravitino 1-forms
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The structure of the final action

• The fields d appear quadratically and linear and
  can be eliminated by gaussian integration
• After this elimination we have
• a quadratic form in the MC of the supergroup plus
• Kinetic terms for the ?, w fermions plus
• Quartic terms in the fermions ?, w whose
  coefficients are provided by the Riemann tensor
  of the target space
• We have exactly the same structure as for the
  case of the string on AdS5 S5

Can we explicitly quantize the superstring on
these backgrounds constructing the corresponding
CFT ? Possibly!