Marginally Deformed GaugMarginally Deformed Gauge Theories from Twistor Stre Theories from Twistor Str

ing Theorying TheoryJun-Bao WuJun-Bao Wu

(SISSA)(SISSA)based on work with Peng Gaobased on work with Peng Gao

hep-th/0611128hep-th/0611128v3v3KITPC Beijing, October 18, 2007KITPC Beijing, October 18, 2007

IntroductionIntroduction

• In 2003, Witten found the relationship betweeIn 2003, Witten found the relationship between perturbative N=4 SYM and twistor string theon perturbative N=4 SYM and twistor string theory.ry.

• Later, the twistor string theory corresponding Later, the twistor string theory corresponding to marginal deformations in SYM was studied to marginal deformations in SYM was studied by Kulaxizi and Zoubos.by Kulaxizi and Zoubos.

• They gave the prescriptions on how to computThey gave the prescriptions on how to compute the e the degree-one (MHV)degree-one (MHV) amplitudes to the first amplitudes to the first order of the deformation parameters.order of the deformation parameters.

• In our work, we generalized their prescription In our work, we generalized their prescription to the one for to the one for allall of the amplitudes to the first of the amplitudes to the first order of the deformation parameters. order of the deformation parameters.

• In the beta- and gamma-deformed theories, In the beta- and gamma-deformed theories, we gave the description to we gave the description to all ordersall orders of the of the deformation parameters and show that this deformation parameters and show that this description reproduces the right results.description reproduces the right results.

OutlineOutline

• Tree-level amplitudes in gauge theory Tree-level amplitudes in gauge theory

• Twistor string theory for N=4 SYMTwistor string theory for N=4 SYM

• Twistor string theory for the SYM with margional deformationsTwistor string theory for the SYM with margional deformations

• Beta-deformationBeta-deformation

• Gamma-deformationGamma-deformation

Color decompositionColor decomposition

• The tree-level scattering amplitudes The tree-level scattering amplitudes of gluons can be written asof gluons can be written as

Here the summation is over all of the Here the summation is over all of the non-cyclic permutations.non-cyclic permutations.

Chinese magic (I)Chinese magic (I)

• The use of two-component spinors:The use of two-component spinors:• For four-moment ,For four-moment , We define , where is the Pauli We define , where is the Pauli

matrices and is the identity matrix. matrices and is the identity matrix. If and only if , we can write asIf and only if , we can write as• We define and We define and

Chinese magic (II)Chinese magic (II)

• We define the polarization vectors of the gauge We define the polarization vectors of the gauge bosons as bosons as

• We use the convention that all particles are outWe use the convention that all particles are outgoing.going.

• Xu, Zhang and Chang, (1987).Xu, Zhang and Chang, (1987).

The MHV amplitudesThe MHV amplitudes

• Parke and Taylor (1986), Berends and GielParke and Taylor (1986), Berends and Giele (1988).e (1988).

Why they are so Why they are so simplesimple? ?

• In 2003, Witten showed that to understand thiIn 2003, Witten showed that to understand this we need to transform to s we need to transform to twistortwistor space introd space introduced by Penrose.uced by Penrose.

• The simplest way to understand twistor space The simplest way to understand twistor space is the following: we consider a Minkowski spacis the following: we consider a Minkowski space with signature (2, 2) instead (1, 3), and make e with signature (2, 2) instead (1, 3), and make a Fourier transformation with respect to a Fourier transformation with respect to

The amplitudes in twistor spacThe amplitudes in twistor spacee• Instead of considering , we consider Instead of considering , we consider

• ConsiderConsider l l-loop amplitudes in N=4 with -loop amplitudes in N=4 with n n gluogluons among whom ns among whom qq gluons are with negative he gluons are with negative helicity, the amplitude vanishes unless the licity, the amplitude vanishes unless the nn corr corresponding points in the twistor space are on aesponding points in the twistor space are on an algebraic curve with genusn algebraic curve with genus g not larger than lg not larger than l and degree and degree d=q-1+ld=q-1+l..

Twistor string theory (I)Twistor string theory (I)

• Furthermore, Witten showed that the topologiFurthermore, Witten showed that the topological B-model in supertwistor space cal B-model in supertwistor space gives us the tree-level amplitudes. D1 branes ngives us the tree-level amplitudes. D1 branes needed to introduced here, they are a kind of ineeded to introduced here, they are a kind of instantons with respect to D5 branes and play astantons with respect to D5 branes and play an important role.n important role.

• The computations can use the The computations can use the connectedconnected inst instantons only (Roiban, Spradlin and Volovich 20antons only (Roiban, Spradlin and Volovich 2004.)04.)

Twistor string theory (II)Twistor string theory (II)

• The prescription using the The prescription using the completely disconnectcompletely disconnected ed instantons leads to the CSW rules, using the Minstantons leads to the CSW rules, using the MHV diagrams (Cachazo, Svrcek, Witten, 2004).HV diagrams (Cachazo, Svrcek, Witten, 2004).

• The prescriptions using the connected and complThe prescriptions using the connected and completely disconnected instantons are shown to be etely disconnected instantons are shown to be eqequivalentuivalent by Gukov, Motl and Neitzke. A family of by Gukov, Motl and Neitzke. A family of inintermediate termediate prescriptions are studied by Gukov et prescriptions are studied by Gukov et al and Bena, Bern and Kosower.al and Bena, Bern and Kosower.

Disconnected vs connected instaDisconnected vs connected instantonnton

The amplitudes in N=4 SYMThe amplitudes in N=4 SYM

• We denote the tree-level partial We denote the tree-level partial amplitudes of N=4 SYM by amplitudes of N=4 SYM by

• They are coefficients of They are coefficients of

• These are the center objects of this talk.These are the center objects of this talk.

Computations using connected inComputations using connected instatonsstatons• These amplitudes can be computed in twistoThese amplitudes can be computed in twisto

r string theory by using connected instantons r string theory by using connected instantons

• are the wavefunctions are the wavefunctions of the external of the external particlesparticles in supertwistor space, J’s are hol in supertwistor space, J’s are holomorphic currents in supertwistor space.omorphic currents in supertwistor space.

The measure The measure

• Here C is a curve of genus 0 and degree Here C is a curve of genus 0 and degree d in supertwistor space.d in supertwistor space.

• is the measure on the moduli space is the measure on the moduli space of these curves.of these curves.

Twistor string theories for less suTwistor string theories for less supersymmetric gauge theories (I)persymmetric gauge theories (I)• It is quite interesting to find the twistor string It is quite interesting to find the twistor string

theories for theories for less supersymmetricless supersymmetric gauge theor gauge theories, some cases are studied:ies, some cases are studied:

• The twistor string theory obtained by the The twistor string theory obtained by the orbiorbifolding folding was studied by Giombi, Kulaxizi, Ricci, was studied by Giombi, Kulaxizi, Ricci, Robles-Llana, Trancanelli, Zoubos and Park, Robles-Llana, Trancanelli, Zoubos and Park, Ray.Ray.

Twistor string theories for less suTwistor string theories for less supersymmetric gauge theories (II)persymmetric gauge theories (II)• The twistor string theory corresponding to the The twistor string theory corresponding to the

SYM obtained SYM obtained by marginal deformationby marginal deformation was pr was proposed by Kulaxizi and Zoubos.oposed by Kulaxizi and Zoubos.

• Recently, Bedford, Papageogakis and Zoubos Recently, Bedford, Papageogakis and Zoubos discussed the twitor string theories with discussed the twitor string theories with flavorflavorss, these theories are corresponding to a class o, these theories are corresponding to a class of UV finite N=2 SYM.f UV finite N=2 SYM.

marginal deformationsmarginal deformations

• The superpotential of the supersymmetric marThe superpotential of the supersymmetric marginally deformed theory can be written asginally deformed theory can be written as

to the first order of the deformation parameters to the first order of the deformation parameters

h’s which are totally symmetric.h’s which are totally symmetric.• Here Here

is the superpotential of the N=4 SYM.is the superpotential of the N=4 SYM.

Twistor string theory for deformeTwistor string theory for deformed SYM (I)d SYM (I)• Kulaxizi and Zoubus proposed that in the corKulaxizi and Zoubus proposed that in the cor

responding twistor string theory, we need inresponding twistor string theory, we need introduce a troduce a non-anti-commutative star producnon-anti-commutative star productt among the among the fermionicfermionic coordinates in supert coordinates in supertwistor space.wistor space.

• They further propose a They further propose a nonanticommutative nonanticommutative star productstar product among the among the wavefunctions wavefunctions of th of the external particles in the supertwistor space.e external particles in the supertwistor space.

The star productThe star product

• The definition of the nonanticommutatiThe definition of the nonanticommutative star products among the wavefunctiove star products among the wavefunctions in the supertwistor space arens in the supertwistor space are

The fermionic coordinatesThe fermionic coordinates

• and are four fermionic coordinaand are four fermionic coordinates in supertwistor spacetes in supertwistor space

• The V’s are defined asThe V’s are defined as

Twistor string theory for deformed Twistor string theory for deformed SYM (II)SYM (II)

• To theTo the first first order of the deformation parorder of the deformation parameters, the formula for the tree amplitameters, the formula for the tree amplitudes are the followingudes are the following

• Kulaxizi and Zoubos (2004) for d=1, Kulaxizi and Zoubos (2004) for d=1, • P. Gao and JW (2006) for the P. Gao and JW (2006) for the general dgeneral d..

Beta-deformationBeta-deformation

• The superpotential of beta deformed theThe superpotential of beta deformed theory are ory are

• Lunin and Maldacena (2005) point out this Lunin and Maldacena (2005) point out this deformation can be introduced via a star deformation can be introduced via a star product among superfields in the N=4 theproduct among superfields in the N=4 theoryory

• We expand the superpotential as We expand the superpotential as

Some simple calculationsSome simple calculations

• From this, we can read off From this, we can read off • Then Then

• We can obtainWe can obtain

where where

Two U(1) chargesTwo U(1) charges

• We consider the following U(1)*U(1) symWe consider the following U(1)*U(1) symmetries in the supertwistor space with tmetries in the supertwistor space with the following charges:he following charges:

• Then Then

The obtained amplitudesThe obtained amplitudes

• The obtained amplitudes are consistent The obtained amplitudes are consistent with the ones in field theory side obtainewith the ones in field theory side obtained by Lunin and Maldacena (2005) and Khd by Lunin and Maldacena (2005) and Khoze (2005):oze (2005):

All order prescriptionAll order prescription

• We propose the following We propose the following all-orderall-order prescription of the star productprescription of the star product

The obtained amplitudes The obtained amplitudes

• This star product is This star product is associativeassociative..• The obtained amplitudes are also correcThe obtained amplitudes are also correc

t:t:

• The descriptions using connected instanThe descriptions using connected instantons and completely disconnected instatons and completely disconnected instantons are equivalent.ntons are equivalent.

The gamma deformation (I)The gamma deformation (I)

• The gamma deformed theory is The gamma deformed theory is non-supnon-supersymmetricersymmetric and and with three parameterswith three parameters..

• Frolov, Roiban and Tseytlin (2005), DurnFrolov, Roiban and Tseytlin (2005), Durnford, Georgiou and Khoze (2006).ford, Georgiou and Khoze (2006).

The gamma deformation (II)The gamma deformation (II)

• In the gamma deformed theory the defoIn the gamma deformed theory the deformations are introduced via the followinrmations are introduced via the following star product among the superfields g star product among the superfields

The three chargesThe three charges

• Q’s are the following charges of three Q’s are the following charges of three U(1) symmetries U(1) symmetries

Star products in supertwistor spaStar products in supertwistor spacece• We consider the U(1)^3 symmetry in the We consider the U(1)^3 symmetry in the

supertwistor space with the following chsupertwistor space with the following chargesarges

• and the following and the following all-orderall-order star product: star product:

The obtained amplitudesThe obtained amplitudes

• This star product is This star product is associativeassociative. The obt. The obtained amplitudes are the same as the onained amplitudes are the same as the ones in the field theory side:es in the field theory side:

The gluonic amplitudesThe gluonic amplitudes

• In the supersymmetric marginally deformed thIn the supersymmetric marginally deformed theory, we consider the tree-level amplitudes witeory, we consider the tree-level amplitudes with only the particles in the N=1 vector supermulth only the particles in the N=1 vector supermultiplet. These amplitudes are the same as the oniplet. These amplitudes are the same as the ones in the N=4 theory. It is a trivial results in field es in the N=4 theory. It is a trivial results in field theory side.theory side.

• We proved this from the twistor string theory to We proved this from the twistor string theory to the first order of the deformation parameters.the first order of the deformation parameters.

SummarySummary

• We gave the prescription on computing We gave the prescription on computing allall of the tree amplitudes in the theory with of the tree amplitudes in the theory with marginal deformations to the first order of marginal deformations to the first order of the deformation parameters. the deformation parameters.

• In two special cases, we gave the In two special cases, we gave the prescription to prescription to all orderall order of the deformation of the deformation parameters and show that the amplitudes parameters and show that the amplitudes are reproduced correctly. are reproduced correctly.

Open questionOpen questions (I)s (I)

• One of the big questions is to give a One of the big questions is to give a prescription to prescription to all orderall order of the of the deformation parameters for thedeformation parameters for the genergenericic marginal deformation. marginal deformation.

• How to prove the How to prove the parity invarianceparity invariance in in the theory with generthe theory with genericic deformation deformations?s? (The N=4 case was (The N=4 case was discusseddiscussed by by RoibRoiban, Spradlin, Volovich and Wittenan, Spradlin, Volovich and Witten))

Open questions (II)Open questions (II)

• Recently, Bedford, Papageogakis and ZoRecently, Bedford, Papageogakis and Zoubos discussed the twitor string theories ubos discussed the twitor string theories with with flavorsflavors, these theories are correspo, these theories are corresponding to a class of UV finite N=2 SYM. It is nding to a class of UV finite N=2 SYM. It is also interesting to study the amplitudes also interesting to study the amplitudes corresponding to corresponding to high degree high degree curves in tcurves in these twistor string theories.hese twistor string theories.