Noncommutative CP N and CH N and their physics

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1 Journal of Physics: Conference Series OPEN ACCESS Noncommutative CP N and CH N and their physics To cite this article: Akifumi Sako et al 2013 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - Topical Review Richard J Szabo - A new approach to deformation equations of noncommutative KP hierarchies Aristophanes Dimakis and Folkert Müller- Hoissen - Deformation quantization quantization and the Klein--Gordon equation Philip Tillman Recent citations - Noncommutative deformations of locally symmetric Kähler manifolds Kentaro Hara and Akifumi Sako - Twisted Fock representations of noncommutative Kähler manifolds Akifumi Sako and Hiroshi Umetsu - Gauge theories on noncommutative PN and Bogomol nyi-prasad-sommerfield-like equations Akifumi Sako et al This content was downloaded from IP address on 28/12/2017 at 20:33

2 Noncommutative CP N and CH N and their physics Akifumi Sako Toshiya Suzuki and Hiroshi Umetsu Kushiro National College of Technology Otanoshike-Nishi Kushiro Japan Abstract. We study noncommutative deformation of manifolds by constructing star products. We start from a noncommutative R d and discuss more genaral noncommutative manifolds. In general star products can not be described in concrete expressions without some exceptions. In this article we introduce new examples of noncommutative manifolds with explicit star products. Karabegov s deformation quantization of CP N and CH N with separation of variables gives explicit calulable star products represented by gamma functions. Using the results of star products between inhomogeneous coordinates we find creation and anihilation operators and obtain the Fock representation of the noncommutative CP N and CH N. 1. Introduction Phyisical theories in noncommutative spaces are realized in several situations. For example in string theories low energy effective theories on D-branes in a constant background NS- NS B fields are given as noncommutative gauge theories. In the theories B field plays a role of an inverse of a noncommutative parameter matrix that represents noncommutativity of space coordinates. Another example in string theories is a series of matrix models which are constructed so that all dynamical variables are given by matricies. Spacetime coordinates themselves are also described as matrices and then the spacetime becomes a noncommutative space. By the way how can we obtain the noncommutative space? Let us consider a simple example the noncommutative R d using the Moyal product. The Moyal product in R d is defined as f g)x) = e i 2 θij i x y j fx)gy). y=x Here θ ij is an element of a skew symmetric constant matrix θ and is called a noncommutative parameter. The right hand side is denoted by fx)e i 2 θij i j gx) for simplicity. Note that the commutation relation of the coordinates are given by [x i x j ] = x i x j x j x i = iθ ij. In the usual commutative case the ordinary commutative product is employed to define a ring of functions. If we chose the Moyal product to define a ring of functions then the considering space become noncommutative because all products between arbitrary functions are replaced by the Moyal product. This is known as noncommutative R d. In the following we move to more general spaces. To make a notion of noncommutative deformation of space be mathematically rigor we introduce a definition of deformation quantization. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the authors) and the title of the work journal citation and DOI. Published under licence by Ltd 1

3 Definition 1. Deformation Quantization strong sense) Deformation Quantization is defined as follows. F is defined as a set of formal power series of : F := { f f = k f k k}. A star product of functions f and g is defined as f g = C k f g) k such that the product satisfies the following conditions. i) is associative product. ii) C k is a bidifferential operator. iii) C 0 and C 1 are defined as C 0 f g) = fg C 1 f g) = i {f g} 2 where {f g} is the Poisson bracket. iv) f 1 = 1 f. Deformation quantization of manifolds are given by this star product in a form of a formal power series in a deformation parameter. The power series is obtained as solutions of an infinite system of differential equations in general. The existence of the solution is proved for a wide class of manifolds however explicit expressions of deformation quantizations are constructed for few kinds of manifolds. For example Euclidean spaces are deformed by using the Moyal product as we saw above and on manifolds with spherically symmetric metrics explicit star products are given as concrete calcurable forms in the context of the Fedosov s deformation quantization [1]. The Fedosov s Deformation Quantization is constructed as the Moyal product on the Weyl algebra bundle in essentials. As mentioned above there are few noncommutative manifolds that have explicit expression of star products for example 2 dimensional manifolds with circular symmetric metrics. Following three examples are given by the methods of Fedosov s Deformation Quantization. 1). R 2 fr ϕ) gr ϕ) = f r 2 + 2y 1 r ϕ + y2 r ) exp i 2 g r 2 + 2y 1 r ϕ + y2 r ) y=0 y i ωij ) y j where r ϕ) is a polar coordinate and ω 12 = ω 21 = 1 ω 11 = ω 22 = 0. 2). CP 1 2y fr ϕ) gr ϕ) = f 1 r1 + r 2 ) + r 2 2y 1 r1 + r 2 ) + 1 ϕ + y2 r ) exp i ) 2y 2 y i ωij y j g 1 r1 + r 2 ) + r 2 2y 1 r1 + r 2 ) + 1 ϕ + y2 r ) y=0. 2

4 3). CH 1 fr ϕ) gr ϕ) =f g 2y 1 r1 r 2 ) + r 2 2y 1 r1 r 2 ) + 1 ϕ + y2 r ) exp i 2 2y 1 r1 r 2 ) + r 2 2y 1 r1 r 2 ) + 1 ϕ + y2 r ) y=0. y i ωij ) y j The purpose of this article is to introduce another way to have explicit expressions of star products in particular for Kähler manifolds and to construct star products for CP N and CH N by using special functions. 2. Deformation quantization of Kähler manifolds In this article noncommutative space is constructed by using deformation quantization. In the following we employ slight differenct deformation quantization from the one in the previous section. The deformation quantization is defined as follows. Definition 2. Let F be a set of formal power series of : { F := f f = f k k}. k A star product of f g F is defined as f g = C k f g) k such that the product satisfies the following conditions. i) is associative product. ii) C k is a bidifferential operator. iii) C 0 and C 1 is defined as C 0 f g) = fg C 1 f g) C 1 g f) = i{f g}. where {f g} is the Poisson bracket. iv) f 1 = 1 f. The difference from the definition of the deformation quantization in the previous section is in the condition for C 1. There is a star product called star product with separation of variables which is defined on Kähler manifolds. A Kähler manifold have a Kähler potential Φ and a Kähler 2-form ω ω = ig i jdz i d z j g i j = 2 Φ z i z j. is called a star product with separation of variables when a f = af 3

5 for holomorphic function a and f b = fb for anti-holomorphic function b. Karabegov showed that for arbitrary ω there exists a star product with separation variables [2] see also [3 4]). In this method of deformation qunatization a star product is constructed as a formal power series of differential operators. Let L f be a differential operator corresponding to a left multiplication by f: Then L f has the following form: L f g := f g. L f = n A n n=0 where formal power series of differential operators A n = a nα f) i Dī) α i Dī = gīj j ). It is required that L f satisfies L f 1 = f 1 = f L f L g h) = f g h) = f g) h = L Lf gh. L f which has the properties described above is determined by the following condition. [L f ī Φ + ī ] = 0 and A 0 = f. This condition is equivalent to the recursion relations [A n ī Φ] = [ ī A n 1 ]. If one obtains the operator L z i L z lf = z l f) L f is given by L f = α ) 1 α f L z z) α. α z Here α is a multi-index α = α 1 α m ). It is not easy to derive explicit expressions of star products in all order of by solving the recursion relation. In the next section this recursion relation is solved by using gamma functions for the case of CP N. 3. Noncommutative deformation of CP N From this section to the end of this article we studied noncommutative CP N and CH N. The detail derivations of the following results are found in [5]. Let z i i = 1 2 N) be inhomogeneous coordinates of CP N. Then the Kähler potential of CP N is given by Φ = ln 1 + z 2) z 2 = i z i z i ). 4

6 The complex metric g i j) is derived from the Kähler potential as Its inverse metric gīj ) is ds 2 = 2g i jdz i d z j g i j = i jφ = 1 + z 2 )δ ij z j z i 1 + z 2 ) 2 gīj = 1 + z 2 ) δ ij + z j z i). The following relations simplify our calculations of L f in the case of CP N ī1 ī2 īn Φ = 1) n 1 n 1)! ī1 Φ ī2 Φ īn Φ Riemann tensor: R i jk l = g i jg k l g i lg k j. Let us construct L z l L z lf = z l f). Recall that L z l is described as L z l = z l + D l + n A n where A n n 2) is a formal series of D k. Then we would like to have the concrete form of A n. We assume that A n has the following form A n = m=2 n=2 a n) m j 1 Φ j m 1 ΦD j1 D j m 1 D l where the coefficients a n) m do not depend on z i and z i. From [L z l ī Φ + ī ] = 0 A n are recursively determined by [A n ī Φ] = [ ī A n 1 ] n 2) where A 1 = D l. After some calculations we found the following recursion relation a n) m = a n 1) m 1 + m 1)an 1) m. and a n) 2 = a n 1) 2 = = a 2) 2 = 1. To solve these equations we introduce a generating function α m t) n=m From the recursion relation α m t) is determined as α 2 t) = n=2 t n a n) 2 = n=1 t n a n) m m 2). t n = n=2 t2 1 t m 1 α m t) = t m 1 1 nt = Γ1 m + 1 t ) Γ1 + 1 t ) m 2). 5

7 The coefficient a n) m is related to the Stirling number of the second kind Sn k) Summarizing the above calculations L z l L z l = z l + D l + = z l + n=2 a n) m = Sn 1 m 1). n becomes m=2 a n) m j 1 Φ j m 1 ΦD j1 D j m 1 D l α m ) j 1 Φ j ΦD j1 m 1 D j m 1 D l. m=1 Using these results star products among z i and z i are obtained as z i z j = z i z j z i z j = z i z j z i z j = z i z j 1) z i z j = z i z j + δ ij 1 + z 2 ) 2 F 1 1 1; 1 1/ ; z 2 ) + 1 zi z j 1 + z 2 ) 2 F 1 1 2; 2 1/ ; z 2 ). 2) Furthermore we can derive an explicite formula for L f for an arbitrary function f L f = n=0 α n ) g n! j1 k1 g jn kn D j1 D jn f ) D k1 D k n. This satisfies [L f ī Φ + ī ] = 0. This star product on CP N is characterized by a function of α n ). 4. Fock representation of CP N We have considered noncommutative CP N in the formalism of deformation quantization until the previous section. In the context of deformation quantization all objects in noncommutative CP N are described as formal power series. In physical theories phyisical quantities schould be expressed as convergent power seiries. Since much of finiteness or convergency in deformation quantization are not known we change our strategy here. To introduce a calcurable framework we consider a Fock representaion of CP N in this section. {z i j Φ i j = 1 2 N} and { z i jφ i j = 1 2 N} constitute 2N sets of the creation-annihilation operators under the star product [ i Φ z j] = δ [ ij z i z j] = 0 [ iφ j Φ] = 0 [ z i jφ ] = δ ij [ z i z j] = 0 [ ī Φ jφ ] = 0. So we can ascribe i Φ z j as annihilation operators and z i jφ as creation operators. Then e Φ/ = 1 + z 2 ) 1/ plays the role of the vacuume projection : i Φ e Φ/ = z j e Φ/ = 0 e Φ/ z i = e Φ/ jφ = 0 e Φ/ e Φ/ = e Φ/. Let us introduce a class of functions which is constructed by acting the creation-operators on the vacuume projection: := c mn z i 1 z im e Φ/ z j 1 z jn = c mn z i1 z im z j1 z jn e Φ/ 6

8 where we choose c mn = 1/ m!n!α m )α n ). These functions form a closed algebra: M k1 k r;l 1 l s = δ nr δ k 1 k n j 1 j n M i1 i m;l 1 l s where δ k 1 k n j 1 j n is defined as δ k 1 k n j 1 j n = 1 [ ] δ k 1 j n! 1 δ kn j n + permutations of j 1 j n ). We list the results of the operation of creation and anihilation operators to here. z k m + 1 = m + 1/ M ki 1 i m;j 1 j n kφ = m + 1) m + 1/ )M ki1 i m;j 1 j n m / m k Φ = δ kil M m i1 î l i m;j 1 j n z k 1 m = δ kil M m m / ) i1 î l i m;j 1 j n z k 1 = δ kjl M n n / ) i1 i m;j 1 ĵ l j n n / kφ = δ kjl M n i1 i m;j 1 ĵ l j n k Φ = n + 1) n + 1/ )k z k n + 1 = n + 1/ M i 1 i m;j 1 j nk. Using this we can construct field theories on noncommutative CP N by replacing all field elements of usual commutative field theories in CP N by matrices valued on these. For example we can define scalar field theories in this noncommutative CP N and it is possible to construct scalar soliton solutions like GMS soliton in this formulation. 5. Noncommutative deformation of CH N In this section we consider the noncommutative defromation of CH N by using Karabegov s deformation quantization with separation of varaibles. Any explicit expression of noncommutative CH N N 2) is not given until now. The Kähler potential of CH N is given by Φ = ln 1 z 2). The metric g i j and the inverse metric gīj are defined by g i j = i jφ = 1 z 2 )δ ij + z i z j 1 z 2 ) 2 gīj = 1 z 2 ) δ ij z i z j). 7

9 These are similar to the ones of CP N. The way to construct the star product is also as same as the one in section 3 so we skip the detail of the derivation. As similar to 1)-2) star products between inhomogeneous coordinates are given as z i z j =z i z j z i z j =z i z j z i z j = z i z j z i z j = z i z j + δ ij 1 z 2 ) 2 F 1 1 1; 1 + 1/ ; z 2 ) 1 + zi z j 1 z 2 ) 2 F 1 1 2; 2 + 1/ ; z 2 ). The explicit representation of the star product with separation of variables on CH N is given by L z l = z l + 1) m 1 β m ) j 1 Φ j ΦD j1 m 1 D j m 1 D l m=1 with β n t) = 1) n α n t) = Γ1/t) Γn + 1/t). A Fock representation of CH N is also given similary. As in the case of CP N sets {z i j Φ} and { z i jφ} satisfy the commutation relations for the creation-annihilation operators. Also e Φ/ is the vacuum projection operator and As in the case of CP N we consider a class of functions i Φ e Φ/ = 0 3) z i e Φ/ = 0 4) e Φ/ ī Φ = 0 5) e Φ/ z i = 0 6) e Φ/ e Φ/ = e Φ/. 7) = zi1 z im z j1 z jn m!n!βm )β n ) e Φ/ is totally symmetric under permutations of i s and j s respectively. Then we can show that these functions form a closed algebra N k1 k r;l 1 l s = δ nr δ k 1 k n j 1 j n N i1 i m;l 1 l s. 8) 8

10 Moreover the star products between and one of z k k Φ z k and kφ are calculated as follows z k m + 1 = m + 1/ N ki 1 i m;j 1 j n 9) m 1 + 1/ m k Φ = δ kil N m i1 î l i m;j 1 j n 10) z k 1 m = δ kil N mm 1 + 1/ ) i1 î l i m;j 1 j n 11) kφ = m + 1)m + 1/ )N ki1 i m;j 1 j n 12) z k 1 = δ kjl N nn 1 + 1/ ) i1 i m;j 1 ĵ l j n 13) k Φ = n + 1)n + 1/ )k 14) z k n + 1 = n + 1/ N i 1 i m;j 1 j nk 15) n 1 + 1/ kφ = δ kjl N n i1 i m;j 1 ĵ l j n. 16) 6. Summary In this article we studied noncommutative deformation of several manifolds by using deformation quantization. We started from a noncommutative R d by using the Moyal product and we moved into more genaral manifolds deformed with star products. For most manifolds star products can not be described in explicit expression but they are determined recursively. As exceptions R 2 CP 1 and CH 1 cases were reviewd and their star products are given by the method of Fedosov s deformation quantization. In this article we introduced new examples of noncommutative manifolds with explicit star products that is Karabegov s deformation quantization of CP N and CH N with separation of variables. Explicit calulable star products were given by using gamma functions. Using the results of star products between the inhomogeneous coordinates we found the creation and anihilation operators and obtained the Fock representations of the noncommutative CP N and CH N. Noncommutative CP N is also studied in [ ] and noncommutative CH 1 is constructed in [11]. This is the first time to construct an explicit noncommutative CH N for N 2. How can we construct physical theories on the noncommutative CP N and CH N? There are a lot of problems about finiteness convergency and so on in deformation quantization. Fock representations discussed in this article provide one of good ways to make physical field theories on the noncommutative manifolds. Indeed few soliton solutions are given on such noncommutative manifolds. It is important to construct frameworks of physics in the noncommutative manifolds defined by deformation quantization. Acknowledgments A.S. is supported by KAKENHI No Grant-in-Aid for Scientific Research C)). H.U. is supported by KAKENHI No Grant-in-Aid for Young Scientists B)). References [1] Fedosov B 1994 J. Differential Geom

11 [2] Karabegov A V 1996 Commun. Math. Phys [arxiv:hep-th/ ]. [3] Karabegov A V 1996 Funct. Anal. Appl [4] Karabegov A V 2011 An explicit formula for a star product with separation of variables [arxiv: [math.qa]]. [5] Sako A Suzuki T and Umetsu H 2012 J. Math. Phys [arxiv: [math-ph]]. [6] Bordemann M Brischle M Emmrich C Waldmann S 1996 Lett. Math. Phys [7] Balachandran A P Dolan B P Lee J H Martin X and O Connor D 2002 J. Geom. Phys [hepth/ ]. [8] Kitazawa Y 2002 Nucl. Phys. B [hep-th/ ]. [9] Karabali D Nair V P and Randjbar-Daemi S In Shifman M ed.) et al.: From fields to strings vol [hep-th/ ]. [10] Karabegov A 2008 Contemporary Math [11] Bieliavsky P Detournay S and Spindel P 2009 Commun. Math. Phys [arxiv: [math-ph]]. 10