1. Marginally Deformed Gauge Theories from Twistor String Theory Jun-Bao Wu (SISSA) based on work with Peng Gao hep-th/0611128v3 KITPC Beijing, October 18, 2007

2. Introduction In 2003, Witten found the relationship between perturbative N=4 SYM and twistor string theory. In 2003, Witten found the relationship between perturbative N=4 SYM and twistor string theory. Later, the twistor string theory corresponding to marginal deformations in SYM was studied by Kulaxizi and Zoubos. Later, the twistor string theory corresponding to marginal deformations in SYM was studied by Kulaxizi and Zoubos. They gave the prescriptions on how to compute the degree-one (MHV) amplitudes to the first order of the deformation parameters. They gave the prescriptions on how to compute the degree-one (MHV) amplitudes to the first order of the deformation parameters.

3. In our work, we generalized their prescription to the one for all of the amplitudes to the first order of the deformation parameters. In our work, we generalized their prescription to the one for all of the amplitudes to the first order of the deformation parameters. In the beta- and gamma-deformed theories, we gave the description to all orders of the deformation parameters and show that this description reproduces the right results. In the beta- and gamma-deformed theories, we gave the description to all orders of the deformation parameters and show that this description reproduces the right results.

4. Outline Tree-level amplitudes in gauge theory Tree-level amplitudes in gauge theory Twistor string theory for N=4 SYM Twistor string theory for N=4 SYM Twistor string theory for the SYM with margional deformations Twistor string theory for the SYM with margional deformations Beta-deformation Beta-deformation Gamma-deformation Gamma-deformation

5. Color decomposition The tree-level scattering amplitudes of gluons can be written as The tree-level scattering amplitudes of gluons can be written as Here the summation is over all of the non- cyclic permutations.

6. 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 matrices and is the identity matrix. We define, where is the Pauli matrices and is the identity matrix. If and only if, we can write as If and only if, we can write as We define and We define and

7. Chinese magic (II) We define the polarization vectors of the gauge bosons as We define the polarization vectors of the gauge bosons as We use the convention that all particles are outgoing. We use the convention that all particles are outgoing. Xu, Zhang and Chang, (1987). Xu, Zhang and Chang, (1987).

8. The MHV amplitudes Parke and Taylor (1986), Berends and Giele (1988). Parke and Taylor (1986), Berends and Giele (1988).

9. Why they are so simple? In 2003, Witten showed that to understand this we need to transform to twistor space introduced by Penrose. In 2003, Witten showed that to understand this we need to transform to twistor space introduced by Penrose. The simplest way to understand twistor space is the following: we consider a Minkowski space with signature (2, 2) instead (1, 3), and make a Fourier transformation with respect to The simplest way to understand twistor space is the following: we consider a Minkowski space with signature (2, 2) instead (1, 3), and make a Fourier transformation with respect to

10. The amplitudes in twistor space Instead of considering, we consider Instead of considering, we consider Consider l-loop amplitudes in N=4 with n gluons among whom q gluons are with negative helicity, the amplitude vanishes unless the n corresponding points in the twistor space are on an algebraic curve with genus g not larger than l and degree d=q-1+l. Consider l-loop amplitudes in N=4 with n gluons among whom q gluons are with negative helicity, the amplitude vanishes unless the n corresponding points in the twistor space are on an algebraic curve with genus g not larger than l and degree d=q-1+l.

11. Twistor string theory (I) Furthermore, Witten showed that the topological B-model in supertwistor space gives us the tree-level amplitudes. D1 branes needed to introduced here, they are a kind of instantons with respect to D5 branes and play an important role. Furthermore, Witten showed that the topological B-model in supertwistor space gives us the tree-level amplitudes. D1 branes needed to introduced here, they are a kind of instantons with respect to D5 branes and play an important role. The computations can use the connected instantons only (Roiban, Spradlin and Volovich 2004.) The computations can use the connected instantons only (Roiban, Spradlin and Volovich 2004.)

12. Twistor string theory (II) The prescription using the completely disconnected instantons leads to the CSW rules, using the MHV diagrams (Cachazo, Svrcek, Witten, 2004). The prescription using the completely disconnected instantons leads to the CSW rules, using the MHV diagrams (Cachazo, Svrcek, Witten, 2004). The prescriptions using the connected and completely disconnected instantons are shown to be equivalent by Gukov, Motl and Neitzke. A family of intermediate prescriptions are studied by Gukov et al and Bena, Bern and Kosower. The prescriptions using the connected and completely disconnected instantons are shown to be equivalent by Gukov, Motl and Neitzke. A family of intermediate prescriptions are studied by Gukov et al and Bena, Bern and Kosower.

13. Disconnected vs connected instanton

14. The amplitudes in N=4 SYM We denote the tree-level partial amplitudes of N=4 SYM by We denote the tree-level partial 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.

15. Computations using connected instatons These amplitudes can be computed in twistor string theory by using connected instantons These amplitudes can be computed in twistor string theory by using connected instantons are the wavefunctions of the external particles in supertwistor space, J ’ s are holomorphic currents in supertwistor space. are the wavefunctions of the external particles in supertwistor space, J ’ s are holomorphic currents in supertwistor space.

16. The measure Here C is a curve of genus 0 and degree d in supertwistor space. Here C is a curve of genus 0 and degree d in supertwistor space. is the measure on the moduli space of these curves. is the measure on the moduli space of these curves.

17. Twistor string theories for less supersymmetric gauge theories (I) It is quite interesting to find the twistor string theories for less supersymmetric gauge theories, some cases are studied: It is quite interesting to find the twistor string theories for less supersymmetric gauge theories, some cases are studied: The twistor string theory obtained by the orbifolding was studied by Giombi, Kulaxizi, Ricci, Robles-Llana, Trancanelli, Zoubos and Park, Ray. The twistor string theory obtained by the orbifolding was studied by Giombi, Kulaxizi, Ricci, Robles-Llana, Trancanelli, Zoubos and Park, Ray.

18. Twistor string theories for less supersymmetric gauge theories (II) The twistor string theory corresponding to the SYM obtained by marginal deformation was proposed by Kulaxizi and Zoubos. The twistor string theory corresponding to the SYM obtained by marginal deformation was proposed by Kulaxizi and Zoubos. Recently, Bedford, Papageogakis and Zoubos discussed the twitor string theories with flavors, these theories are corresponding to a class of UV finite N=2 SYM. Recently, Bedford, Papageogakis and Zoubos discussed the twitor string theories with flavors, these theories are corresponding to a class of UV finite N=2 SYM.

19. marginal deformations The superpotential of the supersymmetric marginally deformed theory can be written as The superpotential of the supersymmetric marginally deformed theory can be written as to the first order of the deformation parameters h ’ s which are totally symmetric. Here Here is the superpotential of the N=4 SYM.

20. Twistor string theory for deformed SYM (I) Kulaxizi and Zoubus proposed that in the corresponding twistor string theory, we need introduce a non-anti-commutative star product among the fermionic coordinates in supertwistor space. Kulaxizi and Zoubus proposed that in the corresponding twistor string theory, we need introduce a non-anti-commutative star product among the fermionic coordinates in supertwistor space. They further propose a nonanticommutative star product among the wavefunctions of the external particles in the supertwistor space. They further propose a nonanticommutative star product among the wavefunctions of the external particles in the supertwistor space.

21. The star product The definition of the nonanticommutative star products among the wavefunctions in the supertwistor space are The definition of the nonanticommutative star products among the wavefunctions in the supertwistor space are

22. The fermionic coordinates and are four fermionic coordinates in supertwistor space and are four fermionic coordinates in supertwistor space The V ’ s are defined as The V ’ s are defined as

23. Twistor string theory for deformed SYM (II) To the first order of the deformation parameters, the formula for the tree amplitudes are the following To the first order of the deformation parameters, the formula for the tree amplitudes are the following Kulaxizi and Zoubos (2004) for d=1, Kulaxizi and Zoubos (2004) for d=1, P. Gao and JW (2006) for the general d. P. Gao and JW (2006) for the general d.

24. Beta-deformation The superpotential of beta deformed theory are The superpotential of beta deformed theory are Lunin and Maldacena (2005) point out this deformation can be introduced via a star product among superfields in the N=4 theory Lunin and Maldacena (2005) point out this deformation can be introduced via a star product among superfields in the N=4 theory We expand the superpotential as We expand the superpotential as

25. Some simple calculations From this, we can read off From this, we can read off Then Then We can obtain We can obtainwhere

26. Two U(1) charges We consider the following U(1)*U(1) symmetries in the supertwistor space with the following charges: We consider the following U(1)*U(1) symmetries in the supertwistor space with the following charges: Then Then

27. The obtained amplitudes The obtained amplitudes are consistent with the ones in field theory side obtained by Lunin and Maldacena (2005) and Khoze (2005): The obtained amplitudes are consistent with the ones in field theory side obtained by Lunin and Maldacena (2005) and Khoze (2005):

28. All order prescription We propose the following all-order prescription of the star product We propose the following all-order prescription of the star product

29. The obtained amplitudes This star product is associative. This star product is associative. The obtained amplitudes are also correct: The obtained amplitudes are also correct: The descriptions using connected instantons and completely disconnected instantons are equivalent. The descriptions using connected instantons and completely disconnected instantons are equivalent.

30. The gamma deformation (I) The gamma deformed theory is non- supersymmetric and with three parameters. The gamma deformed theory is non- supersymmetric and with three parameters. Frolov, Roiban and Tseytlin (2005), Durnford, Georgiou and Khoze (2006). Frolov, Roiban and Tseytlin (2005), Durnford, Georgiou and Khoze (2006).

31. The gamma deformation (II) In the gamma deformed theory the deformations are introduced via the following star product among the superfields In the gamma deformed theory the deformations are introduced via the following star product among the superfields

32. The three charges Q ’ s are the following charges of three U(1) symmetries Q ’ s are the following charges of three U(1) symmetries

33. Star products in supertwistor space We consider the U(1)^3 symmetry in the supertwistor space with the following charges We consider the U(1)^3 symmetry in the supertwistor space with the following charges and the following all-order star product: and the following all-order star product:

34. The obtained amplitudes This star product is associative. The obtained amplitudes are the same as the ones in the field theory side: This star product is associative. The obtained amplitudes are the same as the ones in the field theory side:

35. The gluonic amplitudes In the supersymmetric marginally deformed theory, we consider the tree-level amplitudes with only the particles in the N=1 vector supermultiplet. These amplitudes are the same as the ones in the N=4 theory. It is a trivial results in field theory side. In the supersymmetric marginally deformed theory, we consider the tree-level amplitudes with only the particles in the N=1 vector supermultiplet. These amplitudes are the same as the ones in the N=4 theory. It is a trivial results in field theory side. We proved this from the twistor string theory to the first order of the deformation parameters. We proved this from the twistor string theory to the first order of the deformation parameters.

36. Summary We gave the prescription on computing all of the tree amplitudes in the theory with marginal deformations to the first order of the deformation parameters. We gave the prescription on computing all of the tree amplitudes in the theory with marginal deformations to the first order of the deformation parameters. In two special cases, we gave the prescription to all order of the deformation parameters and show that the amplitudes are reproduced correctly. In two special cases, we gave the prescription to all order of the deformation parameters and show that the amplitudes are reproduced correctly.

37. Open questions (I) One of the big questions is to give a prescription to all order of the deformation parameters for the generic marginal deformation. One of the big questions is to give a prescription to all order of the deformation parameters for the generic marginal deformation. How to prove the parity invariance in the theory with generic deformations? (The N=4 case was discussed by Roiban, Spradlin, Volovich and Witten) How to prove the parity invariance in the theory with generic deformations? (The N=4 case was discussed by Roiban, Spradlin, Volovich and Witten)

38. Open questions (II) Recently, Bedford, Papageogakis and Zoubos discussed the twitor string theories with flavors, these theories are corresponding to a class of UV finite N=2 SYM. It is also interesting to study the amplitudes corresponding to high degree curves in these twistor string theories. Recently, Bedford, Papageogakis and Zoubos discussed the twitor string theories with flavors, these theories are corresponding to a class of UV finite N=2 SYM. It is also interesting to study the amplitudes corresponding to high degree curves in these twistor string theories.