Supersymmetric CP N Sigma Model on Noncommutative Superspace

Transcription

1 Supersymmetric CP N Sigma Model on Noncommutative Superspace Takeo Inami and Hiroaki Nakajima arxiv:hep-th/ v2 1 Mar 2004 Department of Physics, Chuo University Kasuga, Bunkyo-ku, Tokyo , Japan. Abstract We construct a closed form of the action of the supersymmetric CP N sigma model on noncommutative superspace in four dimensions. We show that this model has N = 1 2 supersymmetry and that the transformation law is not modified. The supersymmetric CP N sigma model on noncommutative superspace in two dimensions is obtained by dimensionally reducing the model in four dimensions. nakajima@phys.chuo-u.ac.jp

2 1 Introduction Noncommutative geometry [1] appears in M-theory, string theory and condensed matter physics. Noncommutative field theories are known to describe the effective theory of string in a constant NS-NS B field [2]. (2+1)- dimensional noncommutative field theories have been applied to the quantum Hall effect. In supersymmetric field theories, there are a few alternatives in introducing non(anti)commutativity of the supercoordinates (x µ, θ α, θ α ) 1. In particular, supersymmetric Yang-Mills theory on noncommutative superspace [4, 5, 6] describes the effective field theory of string in a constant selfdual graviphoton background [6, 7, 8]. In the field theoritical view point, these theories keep N = 1 2 supersymmetry and have some interesting properties. In this letter, we construct the supersymmetric nonlinear sigma model whose target space is CP N (CP N SNLSM) on noncommutative superspace in four and two dimensions. Low-dimensional SNLSMs on ordinary superspace have interesting properties. In two dimensions, the CP N SNLSM is integrable, i.e., it has infinitely many conservation laws. It shares important properties with four-dimensional supersymmetric gauge theories, such as asymptotic freedom and dynamical mass gap. In three dimensions, the CP N SNLSM has been investigated using the large-n expansion [9]. As we will see below, since the Kähler potential of SNLSM is generally non-polynomial, the action of SNLSM on noncommutative superspace has infinitely many terms [10]. It is difficult to study the properties of this model either perturbatively or non-perturbatively. We introduce an auxiliary vector superfield to linearize the CP N SNLSM, mimicking the commutative case [11]. Once introducing the vector superfield, we can eliminate all auxiliary fields and obtain a closed form of the action. 1 We follow the notation of [3]. 1

3 2 Noncommutative Superspace 2.1 Noncommutative Superspace We recapitulate noncommutative supersupace, closely following Seiberg [6]. We consider four-dimensional N = 1 supersymmetric field theories on the noncommutative superspace. The non(anti)commutativity is introduced by where y µ is the chiral coordinate {θ α, θ β } = C αβ, {θ α, θ α } = { θ α, θ β} = 0, [y µ, y ν ] = [y µ, θ α ] = [y µ, θ α ] = 0, (1) y µ = x µ + iθσ µ θ. (2) The product of functions of θ is Weyl ordered by using the Moyal product, which is defined by f(θ) g(θ) = f(θ) exp ( 12 ) Cαβ g(θ) θ α θ β = f(θ) [1 12 ] Cαβ θ α θ det C g(θ). β (θθ) (θθ) The supercovariant derivatives are defined by (3) D α = θ α + 2iσµ α α θ α y µ, D α = θ α. (4) Since D α and D α do not contain θ, their anticommutation relations are same as those on the commutative superspace. {D α, D β } = 0, { D α, D β} = 0, {D α, D α } = 2iσ µ α α y µ. (5) The supercharges are defined by Q α = θ α, Q α = θ + α 2iθα σ µ α α y µ. (6) 2

4 Since Q α contains θ, the anticommutation relations are modified as follows. {Q α, Q β } = 0, {Q α, Q α } = +2iσ µ α α y µ, (7) { Q α, Q β} = 4C αβ σ µ α α σν β β y µ y. (8) ν Furthermore, Q α does not act as derivations on the Moyal product of fields Q α (f g) ( Q α f) g + f ( Q α g). (9) 2 Then Q α is not a symmetry of the theory in general, hence we have N = 1 2 supersymmetry. 2.2 Superfields The chiral superfield is defined by D α Φ = 0, and hence, Φ = Φ(y, θ). In terms of the component fields, it is given by Φ(y, θ) = φ(y) + 2θψ(y) + θθf(y), (10) where θθ = θ 1 θ 2 + θ 2 θ 1 and is Weyl ordered. The antichiral superfield is defined by D α Φ = 0, and hence, Φ = Φ(ȳ, θ), where ȳ µ is given by ȳ µ = y µ 2iθσ µ θ, (11) [ȳ µ, ȳ ν ] = 4 θ θc µν, C µν = C αβ ǫ βγ (σ µν ) γ α. (12) In the component fields, it is convenient to express the antichiral superfield in terms of y and θ and to Weyl order the θs Φ(y 2iθσ θ, θ) = φ(y) + 2 θ ψ(y) 2iθσ µ θ µ φ(y) + θ θ [ ] F(y) + 2iθσ µ µ ψ(y) + θθ 2 φ(y). (13) 3

5 We also need the U(1) vector superfield in constructing the CP N SNLSM later. The vector superfield is written in the Wess-Zumino gauge as ] V (y, θ, θ) = θσ µ θaµ (y) + iθθ θ λ(y) i θ θθ [λ α α (y) ǫ αβc βγ σ µ { λ γ γ γ, A µ } θθ θ θ [ D(y) i µ A µ (y) ]. (14) The C-deformed part in the θ θθ term is introduced in order that the component fields transform canonically under the gauge transformation. The powers of V are obtained by [ V 2 = θ θ 1 2 θθa µa µ 1 2 Cµν A µ A ν ] + i 2 θ αc αβ σ µ β α [A µ, λ α ] 1 8 C 2 λ λ, (15) V 3 = 0, (16) where C 2 = C µν C µν = 4 det C. 3 Supersymmetric CP N Sigma Model on Noncommutative Superspace The Lagrangian of four-dimensional N = 1 supersymmetric nonlinear sigma model (SNLSM) is written using the Kähler potential K as L = d 2 θd 2 θ K(Φ, Φ). (17) The same expression can be used for N = 2 SNLSM in two dimensions. The results derived below for the four-dimensional case hold true for the twodimensional case after slight modification. See the argument of dimensional reduction to two dimensions in the end of this section. 4

6 In the case of a single pair of chiral and antichiral superfields, the Berezin integration in eq.(17) with the noncommutativity (1) was calculated in [10]. It is given by L = L(C = 0) + (det C) n [A n F 2n 1 + B n F 2n ], (18) n=1 where A n and B n are functions of the component fields. Eq.(18) contains infinitely many terms since generally K is not a polynomial and the powers of θ are nonzero. We have not found a good way to analyze this model as it stands. Some SNLSMs are expressed as supersymmetric gauge theories. Such SNLSMs contain the model whose target space is a Hermitian symmetric space [12] (e.g. CP N and Grassmannian G N,M ), T CP N 2 and T G N,M. We construct the CP N SNLSM on noncommutative superspace as the noncommutative extension of [11] using the result of [13]. We start from the following Lagrangian L = d 2 θd 2 θ [ Φi e V Φ i V ], (19) where i = 1, 2,..., N + 1. V is the U(1) vector superfield. It is written modulo total derivatives in terms of component fields as L = F i F i i ψ i σ µ D µ ψ i D µ φi D µ φ i φ i Dφ i + i 2 ( φ i λψ i ψ i λφ i ) + i 2 Cµν φi F µν F i 1 16 C 2 φi λ λf i 1 2 C αβ (D µ φi )σ µ β α λ α ψ i α (20) 1 2 D. Here D µ is the gauge covariant derivative defined by D µ φ i = µ φ i + i 2 A µφ i, D µ ψ i = µ ψ i + i 2 A µψ i. (21) 2 T CP N denotes the cotangent bundle of CP N. T G N,M is similar. 5

7 Following [13], we redefine the antichiral superfields Φ i in the Lagrangian (20) as Φ i (ȳ, θ) = φ i (ȳ) + 2 θ ψ i (ȳ) θ θ[ + F i (ȳ) + ic µν µ ( φ i A ν )(ȳ) 1 ] 4 Cµν φi A µ A ν (ȳ), (22) so that the component fields transform canonically under the gauge transformation. Eq.(20) contains the auxiliary fields F i, F i, D, λ, λ and A µ. They have the role of imposing constraints on the fields as follows. F i : F i = 0, (23) F i : F i + i 2 Cµν φi F µν 1 16 C 2 φi λ λ = 0, (24) D : φi φ i = 1, (25) λ α : φi ψα i = 0, (26) λ α i : ψi α φ i C 2 φi λ α F i 1 C αβ (D φi µ )σ µ β α 2 ψi α = 0, (27) A µ : 1 2 ψ i σ µ ψ i + i 2 ( φ i µ φ i µ φi φ i ) 1 2 ( φ i φ i )A µ +ic µν ν ( φ i F i ) i 2 2 Cαβ σ µ β α λ α φi ψα i = 0. (28) After eliminating F i and F i, the Lagrangian (20) takes a simple form L = D µ φi D µ φ i i ψ i σ µ D µ ψ i, (29) with the constraints φ i φ i = 1, (30) φ i ψα i = 0, (31) ψ i αφ i + ic αβ σ µ β α (D φ µ i )ψα i = 0, (32) A µ = i( φ i µ φ i µ φi φ i ) + ψ i σ µ ψ i. (33) 6

8 The constraints (30-32) are solved as follows ( ) ) φ i 1 ϕ a ā 1 ( ϕ =, 1 + ϕϕ 1 φi =, (34) 1 + ϕϕ 1 ( ) ψα i = 1 χ P ij χ j a 1 + ϕϕ α, χi α = α, (35) 0 ψ i α = 1 [ χ j α P ji ic αβ σ µ φ ] i 1 + ϕϕ β α ( φj µ )P jk χ k α, χ i α = ) ( χ ā α, (36) 0 where a, ā = 1, 2,..., N. P ij = δ ij φ i φj is a projection operator which satisfies P 2 = P, φi P ij = P ij φ j = 0. (37) Substituting eqs.(33-36) into eq.(29), the Lagrangian becomes L = L 0 + L C, (38) L 0 = g a b µ ϕ a µ ϕ b ig a b χ b σ µ D µ χ a 1 4 R a bc d(χ a χ c )( χ b χ d), (39) L C = 2g a bg c d C αβ (σ µν ) γ β χa α χc γ ( µ ϕ b)( ν ϕ d), (40) where g a b, D µ χ a and R a bc d are given by g a b = (1 + ϕϕ)δ ab ϕāϕ b (1 + ϕϕ) 2, D µ χ a = µ χ a + Γ a bc ( µϕ b )χ c, (41) Γ a bc ga d b g c d, R a bc d g aē c (g fē dg f b) = g a bg c d + g a dg c b. (42) g a b is the Fubini-Study metric of CP N. Γ a bc and R a bc d are the Christoffel symbol and the Riemann curvature tensor respectively. In the CP 1 case, the C-deformed part L C vanishes. L (CP1 ) C = 2(1 + ϕϕ) 4 C αβ (σ µν ) γ β χ αχ γ ( µ ϕ)( ν ϕ) = 0. (43) We study supersymmetry of the Lagrangian (38). In the C = 0 case, the N = 1 supersymmetry transformation is generated by Q α and Q α. Q α 7

9 generates the transformation δϕ a = 2ξχ a, δ ϕā = 0, (44) δχ a α = 2Γ a bc(ξχ b )χ c α, δ χā α = 2i( σ µ ξ) α µ ϕā. (45) In the present case with C 0, the same transformation (44) and (45) give δg a b = c g a b δϕ c + c g δϕ c a b = 2g b bγ ac b (ξχc ), (46) [ δ(g a bχ a α µ ϕ b) = 2gb bγ ac b (ξχc )χ a α ] 2Γbc a (ξχb )χ c α g a b µ ϕ b = 0. (47) We then obtain [ δl C = δ 2C αβ (σ µν ) γ β (g a bχ a α µ ϕ b)(g c dχ c γ ν ϕ d) ] = 0. (48) We have shown that the Lagrangian (38) is invariant under the N = 1 2 supersymmetry transformation (44) and (45). Using dimensional reduction, we obtain the Lagrangian of CP N SNLSM on noncommutative superspace in two dimensions. L 2D = 1 ( ) 2 g AB z ϕ A z ϕ B + ig a b χ b L D z χ a L + χ b R D z χ a R + R a bc dχ a L χ b L χ c R χ d R + 2g a bg c d (C 11 χ a L χc L C22 χ a R χc R )ǫµν ( µ ϕ b)( ν ϕ d), (49) where ϕ A = (ϕ a, ϕā), g AB = 0 g a b, χ a α = g bā 0 4 Discussion χa L χ a R, χā α = χāl. (50) In this letter, we have studied the supersymmetric CP N sigma model on noncommutative superspace. We have constructed a closed form of the Lagrangian of the model (38). We have found that the N = 1 supersymmetry 2 transformation law of the model is not modified. In two dimensions ordinary NLSMs with extended supersymmetry have a few remarkable properties. 8 χār

10 i) Models are integrable (at least at classical level). ii) They have good UV divergence properties, i.e., finite to certain loops for N = 2 and finite to all loops for N = 4. iii) They possess instantons. It is interesting to see whether these nice properties hold true for twodimensional SNLSM on noncommutative superspace. Acknowledgements We thank Chong-Sun Chu, Ko Furuta, Takeo Araki, Katsushi Ito and Muneto Nitta for useful conversations and comments. This work is supported partially by Chuo University grant for special research and research grants of Japanese ministry of Education and Science, Kiban C and Kiban B. References [1] A. Connes, M. R. Douglas and A. Schwarz, Noncommutative geometry and matrix theory: Compactification on tori, JHEP 9802 (1998) 003 [arxiv:hep-th/ ]; M. R. Douglas and N. A. Nekrasov, Noncommutative field theory, Rev. Mod. Phys. 73 (2001) 977 [arxiv:hep-th/ ]. [2] N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 9909 (1999) 032 [arxiv:hep-th/ ]. [3] J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton University Press, [4] D. Klemm, S. Penati and L. Tamassia, Non(anti)commutative superspace, Class. Quant. Grav. 20 (2003) 2905 [arxiv:hep-th/ ]. 9

11 [5] J. de Boer, P. A. Grassi and P. van Nieuwenhuizen, Non-commutative superspace from string theory, Phys. Lett. B 574 (2003) 98 [arxiv:hep-th/ ]. [6] N. Seiberg, Noncommutative superspace, N = 1 2 supersymmetry, field theory and string theory, JHEP 0306 (2003) 010 [arxiv:hep-th/ ]. [7] H. Ooguri and C. Vafa, The C-deformation of gluino and non-planar diagrams, Adv. Theor. Math. Phys. 7 (2003) 53 [arxiv:hep-th/ ]; Gravity induced C-deformation, arxiv:hep-th/ [8] N. Berkovits and N. Seiberg, Superstrings in graviphoton background and N = supersymmetry, JHEP 0307 (2003) 010 [arxiv:hep-th/ ]. [9] T. Inami, Y. Saito and M. Yamamoto, Vanishing next-to-leading corrections to the β-function of the SUSY CP N 1 model in three dimensions, Prog. Theor. Phys. 103 (2000) 1283 [arxiv:hep-th/ ]. [10] B. Chandrasekhar and A. Kumar, D = 2, N = 2, supersymmetric theories on non(anti)commutative superspace, arxiv:hep-th/ [11] E. Cremmer and J. Scherk, The supersymmetric nonlinear sigma model in four-dimensions and its coupling to supergravity, Phys. Lett. B 74 (1978) 341; A. D Adda, P. Di Vecchia and M. Luscher, Confinement and chiral symmetry breaking in CP n 1 models with quarks, Nucl. Phys. B 152 (1979) 125. [12] K. Higashijima and M. Nitta, Supersymmetric nonlinear sigma models as gauge theories, Prog. Theor. Phys. 103 (2000) 635 [arxiv:hep-th/ ]. 10

12 [13] T. Araki, K. Ito and A. Ohtsuka, Supersymmetric gauge theories on noncommutative superspace, Phys. Lett. B 573 (2003) 209 [arxiv:hep-th/ ]. 11