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Partial Breaking of N = 2 Supersymmetry and Decoupling Limit of Nambu-Goldstone Fermion in U(N) Gauge Model K.F. hep-th/0609039 Kazuhito Fujiwara(Osaka City Univ.) motivation Gauge-Gravity(Matrix) duality [Dijkgraaf-Vafa ’02] [Cachazo-Douglas-Seiberg-Witten ’02] N = 1 U(N) ・・・ N=2 The vector superfields The fermionic field ( index 0 refers to the overall U(1) part of U(N) ) is decoupled . in is a free fermion. low energy (SU(N) is confined ) fermionic shift symmetry free energy F (Matrix model) N=2? Find the relation between fermionic shift symmetry and spontaneous partial breaking of N = 2 supersymmetry. ( Derive the N = 1 action from spontaneous partial breaking of N = 2 supersymmetry. ) Table of Contents next N = 2 U(N) gauge model with spontaneous partial breaking of rigid SUSY(review) supercharge algebra resulting N = 1 U(N) action decoupling limit of Nambu-Goldstone fermion summary & future plans N = 2 U(N) gauge model [Antoniadis-Partouche-Taylor]; hep-th/9512006 Phys.Lett. B372 (1996) 83-87 N = 2 U(1) gauge model equipped with electric and magnetic FI terms. N= 2 N= 1 (on the vacuum) contribution from FI D-term and superpotential. [K.F-Itoyama-Sakaguchi]; hep-th/0409060 Prog.Theor.Phys. 113 (2005) 429-455 N = 2 U(N) gauge model described by N = 1 chiral superfields and N = 1 vector superfields cf. . [K.F-Itoyama-Sakaguchi]; hep-th/0503113 Nucl.Phys. B723 (2005) 33-52 (on vacua) [K.F-Itoyama-Sakaguchi]; hep-th/0510255 Nucl.Phys. B740 (2006) 58-78 Harmonic superspace formalism [Itoyama-Maruyoshi] hep-th/0603180 Local supersymmetry ingredients U(N) adjoint N = 1 chiral superfields overall U(1) U(N) adjoint N = 1 vector superfields Vector superfield strengths K hler potential where is an analytic function of (prepotential) , and K hler metric: Killing vector Killing potential : structure constant of U(N) gauge symmetry. . total action ( N =2 ) K hler term Counterterm for the U(N) gauging of Gauged kinetic term for the vector superfield Superpotential term condition for the supersymmetry FI D-term These parameters play a key role of partial breaking of N = 2 supersymmetry. forms the real part of an ``electric" FI term and forms the real part of a ``magnetic" FI term in [K.F.-Itoyama-Sakaguchi ’0510] We impose for later convenience. where analysis of vacua Let us examine the case with and gauge group U(N) is unbroken on vacua. scalar potential vacuum condition where denote reduces to ( evaluated at : index for non-Cartan generators) Nambu-Goldstone fermion :vacuum expectation value of . N=2→1! Table of Contents N = 2 U(N) gauge model with spontaneous partial breaking of rigid SUSY(review) next supercharge algebra resulting N = 1 U(N) action decoupling limit of Nambu-Goldstone fermion summary & future plans general supercurrent algebra central extension (constant) The N = 2 supersymmetric transformation rule is given as where We can obtain the 1st supercurrent where and from the action satisfy following relations, . After some algebra, we obtain The 2nd supercurrent of with . is given by the discrete transformation N = 2 Supercharge algebra is derived by It may be irrelevant to denote supercharges as because the supercharge corresponding to the broken supersymmetry is ill-defined. For convenience, we ignore this point here and write the divergent part explicitly. We obtain the central charge To get the resulting N = 1 supercharge algebra, we define . Anti-commutators of and and itself are given as ∞ central extension (divergent) is the unbroken generator is the broken generator. Table of Contents N = 2 U(N) gauge model with spontaneous partial breaking of rigid SUSY(review) supercharge algebra next resulting N = 1 U(N) action decoupling limit of Nambu-Goldstone fermion summary & future plans spinor fields mixing The spinor fields and are to be mixed as well as supercharges. Rewrite the N = 2 action in and . Following terms keep the same form as before with The F.I. term parameter so that we get the same form as before is absorbed into the superpotential terms shifted scalar fields The scalar fields The prepotential are to be sifted from its vacuum expectation value, is expanded in the shifted fields Similarly, derivatives of the prepotential are written as We can rewrite the action in the shifted scalar fields K hler metric K hler potential Killing potential with as Resulting N = 1 action where superpotential complex (×N = 2 supersymmetry.) We can write the off-shell N = 1 action by introducing auxillialy fields and form massive chiral multiplets form massless vector multilets The Nambu-Goldstone fermion is coupled with other fields. Table of Contents N = 2 U(N) gauge model with spontaneous partial breaking of rigid SUSY(review) supercharge algebra resulting N = 1 U(N) action next decoupling limit of Nambu-Goldstone fermion summary & future plans If the prepotential is a second order polynomial, there are no Yukawa couplings in the action and the Nambu-Goldstone fermion will be a free fermion. But derivatives of the superpotential become zero, This means that the superpotential does not contribute to the action and it recovers the N = 2 supersymmetry. (partial breaking does not occur!) This problem can be solved by a large limit of the parameters i.e. large limit of the electric and magnetic FI terms. reparametrization We reparametrize and The prepotential is written as Derivatives of the prepotential where and are and they are vanished at K hler metric and Killing potential structure constant Derivatives of the superpotential are cancellation ! The superpotential keeps non-trivial form at . decoupling limit Take a limit The superpotential , and the action is converted into is given as where binomial coefficient The Nambu-Goldstone fermion is decoupled from the other fields. We can rewrite the action where in superfield formalism as is the field strength of decoupling limit summary ・We obtain general N = 1 U(N) action, which include a free fermion, from spontaneously broken N = 2 supersymmetry. ・We conclude that the fermionic shift symmetry is identified with 2nd, spontaneously broken supersymmetry in the decoupling limit of the Nambu-Goldstone fermion in our model. future plans ・Relation between geometric transition and spontaneous partial breaking. ・Relation between domain wall solution and spontaneous partial breaking. ・Realization of our model in string theory.