aa r X i v : . [ h e p - t h ] N ov November, 2013 ‘Analytic Continuation’ of N = 2 Minimal Model
Yuji Sugawara ∗ Department of Physical Sciences, College of Science and Engineering,Ritsumeikan University, Shiga 525-8577, Japan
Abstract
In this paper we discuss what theory should be identified as the ‘analytic continuation’with N → − N of the N = 2 minimal model with the central charge ˆ c = 1 − N . Weclarify how the elliptic genus of the expected model is written in terms of holomorphic linear combinations of the ‘modular completions’ introduced in [1] in the SL (2) N +2 /U (1)-supercoset theory. We further discuss how this model could be interpreted as a kindof ‘compactified’ model of the SL (2) N +2 /U (1)-supercoset in the ( e R , e R)-sector, in whichonly the discrete spectrum appears in the torus partition function and the potential IR-divergence due to the non-compactness of target space is removed. We also briefly argueon possible definitions of the sectors with other spin structures. ∗ [email protected] Introduction
The N = 2 minimal model is one of the most familiar rational superconformal field theories intwo-dimension [2]. This is defined by the supercoset theory of SU (2) k /U (1) with level k = N − N = 2 Landau-Ginzburg (LG) model withthe superpotential W ( X ) = X N + [lower powers] [4]. One of good features of N = 2 minimalmodels is a very simple formula of elliptic genus [5] ; Z ( N − ( τ, z ) = θ (cid:0) τ, N − N z (cid:1) θ (cid:0) τ, N z (cid:1) , (1.1)which nicely behaves under modular transformations as well as the spectral flows. Namely, thefunction Z ( N − ( τ, N z ) is a weak Jacobi form [6] of weight 0 and index N ( N − .It is an old question what theory should be identified as the ‘analytic continuation’ of the N =2 minimal model under N → − N . A naive guess of the answer would be the SL (2) N +2 /U (1)-supercoset theory that has the expected central charge ˆ c (cid:0) ≡ c (cid:1) = 1 + N . However, it is not completely correct due to the following reasons; • The SL (2) /U (1)-supercoset theory contains both discrete and continuous spectra of pri-mary fields, while the N = 2 minimal model only has discrete spectra. It is not likely tobe the case that such two theories are directly connected by an analytic continuation ofthe parameter of theory. • It has been known that the elliptic genera of the SL (2) /U (1)-supercoset theories showsnon-holomorphicity with respect to the modulus τ of the world-sheet torus [7, 1], whereasthe elliptic genera of the minimal model (1.1) are manifestly holomorphic.In this paper we shall try to give a precise answer to this question. In other words, we willfocus on the problems; (i) What is the superconformal model with ˆ c = 1 + N which has Z ( τ, z )( ∝ ” Z ( − N − ( τ, z )”) = θ (cid:0) τ, N +1 N z (cid:1) θ (cid:0) τ, N z (cid:1) , (1.2)as its elliptic genus? (ii) Then, does this model have any relationship with the SL (2) N +2 /U (1)-supercoset?1his paper is organized as follows:In section 2 we shall demonstrate our mathematical results. We make a detailed analysison the holomorphic function (1.2), and prove the main theorem which addresses the preciserelation between (1.2) and the ‘modular completions’ introduced in [1] in the SL (2) N +2 /U (1)-supercoset. We will further present a physical interpretation of this mathematical result andsome discussions in section 3. In section 4 we present the summary and some comments. In this section we address some mathematical results. The main claims will be expressed in(2.24) and (2.31).
We first prepare relevant notations. To begin with, we introduce the symbol of ‘IR-part’ justfor convenience; [ f ( τ, z )] := lim τ → i ∞ f ( τ, z ) , (2.1)where f ( τ, z ) is assumed to be holomorphic around the cusp τ = i ∞ . We consider a holomorphic function defined byΦ ( N ) ( τ, z ) := θ (cid:0) τ, N +1 N z (cid:1) θ (cid:0) τ, N z (cid:1) . (2.2)This is obtained by the formal replacement N → − N in the elliptic genus of N = 2 minimalmodel (1.1). The function (2.2) possesses the next modular and spectral flow properties withˆ c ≡ N ; Φ ( N ) ( τ + 1 , z ) = Φ ( N ) ( τ, z ) , Φ ( N ) (cid:18) − τ , zτ (cid:19) = e iπ ˆ cτ z Φ ( N ) ( τ, z ) . (2.3)Φ ( N ) ( τ, z + mτ + n ) = ( − m + n q − ˆ c m y − ˆ cm Φ ( N ) ( τ, z ) , ( m, n ∈ N Z ) . (2.4)2n other words, Φ ( N ) ( τ, N z ) is a weak Jacobi form of weight 0 and index N ˆ c ≡ N ( N +2)2 ;Φ ( N ) ( τ, N z ) ∈ J [0 , N ( N + 2)2 ] , where we set J [ w, d ] := { weak Jacobi forms of weight w and index d } . (2.5)The IR-part of (2.2) is evaluated as (cid:2) Φ ( N ) ( τ, z ) (cid:3) = y − N X v =0 y vN . (2.6)We next introduce the ‘spectral flow operator’ s ( a,b ) ( a, b ∈ Z ) defined by s ( a,b ) · f ( τ, z ) := ( − a + b q ˆ c a y ˆ ca e πi abN f ( τ, z + aτ + b ) , (2.7)and set Φ ( N )( a,b ) ( τ, z ) := s ( a,b ) · Φ ( N ) ( τ, z ) ( a, b ∈ Z ) (2.8)Since having a periodicityΦ ( N )( a + Nm,b + Nn ) ( τ, z ) = Φ ( N )( a,b ) ( τ, z ) , ( ∀ m, n ∈ Z ) , we may assume a, b ∈ Z N . Its modular property is written asΦ ( N )( a,b ) ( τ + 1 , z ) = Φ ( N )( a,a + b ) ( τ, z ) , Φ ( N )( a,b ) (cid:18) − τ , zτ (cid:19) = e iπ ˆ cτ z Φ ( N )( b, − a ) ( τ, z ) , (2.9)The IR-part of Φ ( N )( a,b ) ( τ, z ) is computed as h Φ ( N )( a,b ) ( τ, z ) i = δ ( N ) a, N X v =1 e πi bN v y − + vN + y − + [ a ] N , (2.10)where we introduced the notation [ a ], defined by [ a ] ≡ a ( mod N ), 0 ≤ [ a ] ≤ N −
1, and δ ( N ) r,s denotes the ‘mod N Kronecker delta’. In fact, we findΦ ( N )(0 ,b ) ( τ, z ) = y − N +12 N e πi b N (cid:16) − y N +1 N e πi bN (cid:17) y − N e πi b N (cid:16) − y N e πi bN (cid:17) + O ( q ) = N X v =0 e πi bN v y − + vN + O ( q ) , and also (for a = 0), Φ ( N )( a,b ) ( τ, z ) = y − + [ a ] N + O ( q N ) , which proves (2.10). 3 .1.2 Modular Completions Let us introduce the ‘modular completion’ of the extended discrete characters (of the e R-sector)[8, 9, 10] in the SL (2) /U (1)-supercoset according to [1]. For the case of ˆ c = 1 + N , ( ∀ N ∈ Z > ),this function is defined as b χ ( N, dis ( v, a ; τ, z ) := θ ( τ, z )2 πη ( τ ) X n ∈ a + N Z r ∈ v + N Z (cid:26)Z R + i ( N − dp − Z R − i dp ( yq n ) (cid:27) e − πτ p r N ( yq n ) rN p − ir y nN q n N − yq n ≡ χ ( N, dis ( v, a ; τ, z ) + θ ( τ, z )2 πη ( τ ) X n ∈ a + N Z r ∈ v + N Z Z R − i dp e − πτ p r N p − ir ( yq n ) rN y nN q n N . ( v, a ∈ Z N ) , (2.11)where χ ( N, dis denotes the extended discrete character introduced in [8, 9, 10] (written in theconvention of [1]); χ ( N, dis ( v, a ; τ, z ) ≡ X n ∈ a + N Z ( yq n ) vN − yq n y nN q n N θ ( τ, z ) iη ( τ ) , ( v = 0 , , . . . , N, a ∈ Z N ) . (2.12)The first line (2.11) naturally appears through the analysis of partition function of SL (2) /U (1)-supercoset [1], and the second line just comes from the contour deformation.The modular completion of Appell function (or the ‘Appell-Lerch sum) K (2 N ) ( τ, z ) [11, 12]is given as [13] ; b K (2 N ) ( τ, z ) := K (2 N ) ( τ, z ) − X m ∈ Z N R m,N ( τ ) Θ m,N ( τ, z ) , (2.13)where we set R m,N ( τ ) := 1 iπ X r ∈ m +2 N Z Z R − i dp e − πτ p r N p − ir q − r N , (2.14)which is generically non-holomorphic due to explicit τ -dependence.The next ‘Fourier expansion relation’ [1] will be useful for our analysis; y aN q a N b K (2 N ) (cid:18) τ, z + aτ + bN (cid:19) θ ( τ, z ) iη ( τ ) = X v ∈ Z N e πi vbN b χ ( N, dis ( v, a ; τ, z ) , (2.15) The relation to the notation given in Chapter 3 of [13] are as follows; K (2 k ) ( τ, z ) ≡ f ( k ) u =0 ( z ; τ ) , b K (2 k ) ( τ, z ) ≡ ˜ f ( k ) u =0 ( z ; τ ) . K (2 k ) and χ dis ( v, a ) given in [10].The modular transformation formulas for the modular completions (2.11), (2.13) are writtenas [1, 13] b χ ( N, dis (cid:18) v, a ; − τ , zτ (cid:19) = e iπ ˆ cτ z N − X v ′ =0 X a ′ ∈ Z N N e πi vv ′− ( v +2 a )( v ′ +2 a ′ )2 N b χ ( N, dis ( v ′ , a ′ ; τ, z ) , (2.16) b χ ( N, dis ( v, a ; τ + 1 , z ) = e πi aN ( v + a ) b χ ( N, dis ( v, a ; τ, z ) . (2.17) b K (2 N ) (cid:18) − τ , zτ (cid:19) = τ e iπ Nτ z b K (2 N ) ( τ, z ) , (2.18) b K (2 N ) ( τ + 1 , z ) = b K (2 N ) ( τ, z ) . (2.19)The spectral flow property is summarized as follows; b χ ( N, dis ( v, a ; τ, z + rτ + s ) = ( − r + s e πi v +2 aN s q − ˆ c r y − ˆ cr b χ ( N, dis ( v, a + r ; τ, z ) , ( ∀ r, s ∈ Z ) . (2.20) b K (2 N ) ( τ, z + rτ + s ) = q − Nr y − Nr b K (2 N ) ( τ, z ) , ( ∀ r, s ∈ Z ) . (2.21)More detailed formulas in general cases of b χ ( N,K ) dis ( v, a ) with ˆ c = 1 + KN , ( N, K ∈ Z > ) aresummarized in Appendix B. Let us start our main analysis. We begin with introducing the holomorphic functions X ( N ) ( v, a ; τ, z )as the ‘Fourier transforms’ of Φ ( N )( a,b ) ( τ, z ); X ( N ) ( v, a ; τ, z ) := 1 N X b ∈ Z N e − πi bN ( v + a ) Φ ( N )( a,b ) ( τ, z ) , (2.22)in other words, Φ ( N )( a,b ) ( τ, z ) = N − X v =0 e πi bN ( v + a ) X ( N ) ( v, a ; τ, z ) . (2.23)The main formula that we would like to prove in this section is X ( N ) ( v, a ; τ, z ) = b χ ( N, dis ( v, a ; τ, z ) + b χ ( N, dis ( N − v, a + v ; τ, z ) . (2.24)5n order to achieve this formula we first consider the elliptic genera of SL (2) N +2 /U (1)-supercoset with ˆ c = 1 + N [7, 1]. We set Z ( τ, z ) := b K (2 N ) (cid:16) τ, zN (cid:17) θ ( τ, z ) iη ( τ ) , (2.25)and e Z ( τ, z ) := 1 N X a,b ∈ Z N s ( a,b ) · Z ( τ, z ) ≡ N X a,b ∈ Z N q a N y aN e πi abN b K (2 N ) (cid:18) τ, z + aτ + bN (cid:19) θ ( τ, z ) iη ( τ ) . (2.26)Here, e Z ( τ, z ) is identified with the elliptic genus of axial supercoset of SL (2) N +2 /U (1) (‘cigar’[14]), while Z ( τ, z ) is associated with the vector supercoset;[vector SL (2) /U (1)] ∼ = [ Z N -orbifold of axial SL (2) /U (1)] ∼ = [ N -fold cover of ‘trumpet’] , as is shown in [15]. The following identities play crucial roles; Z ( a,b ) ( τ, z ) := s ( a,b ) · Z ( τ, z ) = N − X v =0 e πi bN ( v + a ) b χ ( N, dis ( v, a ; τ, z ) , (2.27) e Z ( a,b ) ( τ, z ) := s ( a,b ) · e Z ( τ, z ) ≡ N X α,β ∈ Z N e − πi N ( aβ − bα ) Z ( α,β ) ( τ, z ) ! = N − X v =0 e πi bN ( v + a ) b χ ( N, dis ( N − v, a + v ; τ, z ) , (2.28)which are proven in [1, 15].We also note that the IR-parts of Z ( a,b ) and e Z ( a,b ) are given as[ Z ( a,b ) ( τ, z )] = δ ( N ) a, ( N − X v =1 e πi bN v y − + vN + 12 (cid:16) y − + y (cid:17)) (2.29) h e Z ( a,b ) ( τ, z ) i = y − + [ a ] N + δ ( N ) a, (cid:16) y − y − (cid:17) . (2.30)With these preparations we shall prove the next identity; Z ( τ, z ) + e Z ( τ, z ) = Φ ( N ) ( τ, z ) , (2.31)6rom which the identity (2.24) is readily derived by using (2.27), (2.28) as well as the definition X ( N ) ( v, a ) (2.22). [proof of (2.31)] We set Φ ( N ) ′ ( τ, z ) := Z ( τ, z ) + e Z ( τ, z ), and will prove Φ ( N ) ′ ( τ, z ) = Φ ( N ) ( τ, z ).We first enumerate relevant properties of Φ ( N ) ′ ( τ, z ); (i) modular and spectral flow properties : We first note that Φ ( N ) ′ ( τ, z ) possesses the expected modular and spectral flow properties;Φ ( N ) ′ ( τ + 1 , z ) = Φ ( N ) ′ ( τ, z ) , Φ ( N ) ′ (cid:18) − τ , zτ (cid:19) = e iπ ˆ cτ z Φ ( N ) ′ ( τ, z ) , (2.32) s ( Na,Nb ) · Φ ( N ) ′ ( τ, z ) = Φ ( N ) ′ ( τ, z ) , ( ∀ a, b ∈ Z ) . (2.33)They are shown from the same properties of Z ( τ, z ) as well as the fact that s ( a,b ) actsmodular covariantly. (ii) holomorphicity : Recall the fact that Z ( τ, z ) is written as Z ( τ, z ) = θ ( τ, z ) iη ( τ ) " K (2 N ) (cid:16) τ, zN (cid:17) − X m ∈ Z N R m,N ( τ )Θ m,N (cid:18) τ, zN (cid:19) , (2.34)and the second term is non-holomorphic. Let us consider the ‘ Z N -orbifold action’, that is1 N X a,b ∈ Z N s ( a,b ) · [ ∗ ] , to this non-holomorphic correction term. Because of the simple identity1 N X a,b ∈ Z N q a N y aN e πi abN Θ m,N (cid:18) τ, N ( z + aτ + b ) (cid:19) = Θ − m,N (cid:18) τ, zN (cid:19) , (2.35)we obtain[non-hol. corr. term] Z N -orbifolding = ⇒ − X m ∈ Z N R m,N ( τ )Θ − m,N (cid:18) τ, zN (cid:19) θ ( τ, z ) iη ( τ ) = − X m ∈ Z N R − m,N ( τ )Θ m,N (cid:18) τ, zN (cid:19) θ ( τ, z ) iη ( τ ) = 12 X m ∈ Z N n R m,N ( τ ) − δ ( N ) m, o Θ m,N (cid:18) τ, zN (cid:19) θ ( τ, z ) iη ( τ ) ( ∵ R − m,N ( τ ) = − R m,N ( τ ) + 2 δ ( N ) m, ) . ( N ) ′ ( τ, z ) are strictly canceled out, andwe can rewrite it asΦ ( N ) ′ ( τ, z ) = θ ( τ, z ) iη ( τ ) " K (2 N ) (cid:16) τ, zN (cid:17) + 1 N X a,b ∈ Z N q a N y aN e πi abN K (2 N ) (cid:18) τ, z + aτ + bN (cid:19) − Θ ,N (cid:18) τ, zN (cid:19)(cid:21) . (2.36)This is manifestly holomorphic. (iii) IR-part : Recall[ Z ( τ, z )] = 12 (cid:16) y + y − (cid:17) + N − X v =1 y − + vN , [ e Z ( τ, z )] = 12 (cid:16) y + y − (cid:17) , (2.37)and thus h Φ ( N ) ′ ( τ, z ) i = N X v =0 y − + vN (cid:0) ≡ (cid:2) Φ ( N ) ( τ, z ) (cid:3) (cid:1) . (2.38)Moreover, we can show h Φ ( N ) ′ ( a,b ) ( τ, z ) i = δ ( N ) a, N X v =1 e πi bN v y − + vN + y − + [ a ] N (cid:16) ≡ h Φ ( N )( a,b ) ( τ, z ) i (cid:17) , (2.39)due to (2.29), (2.30) and (2.10).In this way, we can conclude that both of Φ ( N ) ( τ, N z ) and Φ ( N ) ′ ( τ, N z ) are holomorphicJacobi forms of weight 0 and index N ˆ c ≡ N ( N +2)2 which share the IR-part, namely,Φ ( N ) ( τ, N z ) , Φ ( N ) ′ ( τ, N z ) ∈ J (cid:20) , N ( N + 2)2 (cid:21) , (cid:2) Φ ( N ) ( τ, N z ) (cid:3) = h Φ ( N ) ′ ( τ, N z ) i = N X v =0 y − N + v . Consequently, if setting F ( τ, z ) := Φ ( N ) ′ ( τ, N z )Φ ( N ) ( τ, N z ) , (2.40)then F ( τ, z ) is an elliptic, modular invariant function with the IR-part [ F ( τ, z )] = 1. Thus, ifwe would succeed in proving the holomorphicity of F ( τ, z ) for ∀ τ ∈ H ∪ { i ∞} , ∀ z ∈ C , we canconclude F ( τ, z ) ≡ ( N ) ′ ( τ, N z ) is obviously holomorphic for ∀ τ ∈ H ∪ { i ∞} , ∀ z ∈ C , and Φ ( N ) ( τ, N z ) hassimple zeros only at N ( N + 2) points z a,b := aτ + bN + 1 , a, b = 0 , , . . . , N, ( a, b ) = (0 , , (2.41)in the fundamental region of double periodicity of F . Hence the function F ( τ, z ) at mostpossesses simple poles at z = z a,b in the fundamental region, and the following lemma is enoughfor completing the proof; [Lemma] All the residues of F ( τ, z ) at z = z a,b vanish. ∵ ) Let us denote R a,b ( τ ) := Res z = z a,b [ F ( τ, z )] ≡ πi I C a,b ( τ ) dz F ( τ, z ) , (2.42)where C a,b ( τ ) is a small contour encircling z a,b . Because of the modular invariance of F ( τ, z ),we find the modular properties of R a,b as R a,b ( τ + 1) = R a,a + b ( τ ) , R a,b (cid:18) − τ (cid:19) = 1 τ R b, − a ( τ ) . (2.43)In fact, the first formula is trivial, and the second formula is proven as follows; R a,b (cid:18) − τ (cid:19) = 12 πi I C a,b ( − τ ) dz F (cid:18) − τ , z (cid:19) = 12 πi I C b, − a ( τ ) dz ′ τ F (cid:18) − τ , z ′ τ (cid:19) ( z ′ := τ z )= 1 τ πi I C b, − a ( τ ) dz ′ F ( τ, z ′ ) ( ∵ modular invariance of F )= 1 τ R b, − a ( τ ) . It is obvious that R a,b ( τ ) is holomorphic over H , since z = z a,b is at most a simple pole andΦ ( N ) ′ ( τ, N z ) is holomorphic for ∀ τ ∈ H , ∀ z ∈ C . Moreover, one can showlim τ → i ∞ | R a,b ( τ ) | < ∞ , ( ∀ a, b ) , (2.44)with the helps of (2.38) and (2.39). The proof of (2.44) is straightforward, and we shall presentit in Appendix C.Now, we define the ‘modular orbit’ O r ⊂ Z N +1 × Z N +1 for ∀ r ∈ D ( N + 1) ≡ { r =1 , , . . . , N ; r | ( N + 1) } as O r := (cid:8) ( a, b ) ∈ Z N +1 × Z N +1 : ( a, b ) ≡ ( r, A (mod ( N + 1) × ( N + 1)) , ∃ A ∈ SL (2 , Z ) (cid:9) . Z N +1 × Z N +1 − { (0 , } = a r ∈ D ( N +1) O r , holds, and R ( r,k ) ( τ ) := X ( a,b ) ∈O r ( R a,b ( τ )) k , (2.45)should be a modular form of weight − k . Therefore, R ( r,k ) ( τ ) has to vanish everywhere over H ∪{ i ∞} for arbitrary r ∈ D ( N +1) and k ∈ Z > . This is sufficient to conclude that R a,b ( τ ) ≡ z a,b .In this way, the identity (2.31) has been proven, leading to the main formula (2.24). (Q.E.D) In this section we try to make a physical interpretation of the main result given above, that is,the identities (2.24) or (2.31). In other words, we discuss what superconformal system leads toΦ ( N ) ( τ, z ) (2.2) as its elliptic genus. X ( N ) ( v, a ) We first note several helpful facts about the expected ‘building block’ X ( N ) ( v, a ) (2.22):
1. modular and spectral flow properties :
Once we achieve the identity (2.24), we can readily derive the formulas of modular andspectral flow transformations of X ( N ) ( v, a ) by using (2.16), (2.17) and (2.20), which arewritten as X ( N ) (cid:18) v, a ; − τ , zτ (cid:19) = e iπ ˆ cτ z N − X v ′ =0 X a ′ ∈ Z N N e πi vv ′− ( v +2 a )( v ′ +2 a ′ )2 N X ( N ) ( v ′ , a ′ ; τ, z ) ≡ e iπ ˆ cτ z N − X v ′ =0 X a ′ ∈ Z N N e − πi ( v +2 a )( v ′ +2 a ′ )2 N cos (cid:18) πvv ′ N (cid:19) X ( N ) ( v ′ , a ′ ; τ, z )(3.1) X ( N ) ( v, a ; τ + 1 , z ) = e πi aN ( v + a ) X ( N ) ( v, a ; τ, z ) . (3.2) X ( N ) ( v, a ; τ, z + rτ + s ) = ( − r + s e πi v +2 aN s q − ˆ c r y − ˆ cr X ( N ) ( v, a + r ; τ, z ) , ( ∀ r, s ∈ Z ) , (3.3)10n other words, s ( r,s ) · X ( N ) ( v, a ) = e πi v +2 a + rN s X ( N ) ( v, a + r ) , ( ∀ r, s ∈ Z ) . (3.4)As a consistency check, one may also derive these formulas directly from the ‘Fourierexpansion relation’ (2.23).
2. manifestly holomorphic expression :
In place of (2.24), X ( N ) ( v, a ) can be rewritten in a manifestly holomorphic expression; X ( N ) ( v, a ; τ, z ) = χ ( N, dis ( v, a ; τ, z ) + χ ( N, dis ( N − v, a + v ; τ, z ) . (3.5)In fact, the non-holomorphic correction terms in the R.H.S of (2.24) are found to becanceled out precisely, as in (2.36). One can achieve this identity by making the Fourierexpansion of (2.36) and recalling the identity Φ ( N ) ( τ, z ) = Φ ( N ) ′ ( τ, z ) proven in the pre-vious section.
3. interpretation as ‘analytic continuation’ of the character of N = 2 minimal model It would be worthwhile to recall basic facts of the N = 2 minimal model, namely, the SU (2) N − /U (1)-supercoset which has ˆ c = 1 − N . As we mentioned at the beginning ofthis paper, the elliptic genus of the minimal model is given as [5] Z ( N − ( τ, z ) = θ (cid:0) τ, N − N z (cid:1) θ (cid:0) τ, zN (cid:1) = N − X ℓ =0 ch ( e R) ℓ,ℓ +1 ( τ, z ) , (3.6)where ch ( e R) ℓ,m ( τ, z ) denotes the e R-character of the N = 2 minimal model which has theWitten index ch ( e R) ℓ,m ( τ,
0) = δ (2 N ) m,ℓ +1 − δ (2 N ) m, − ( ℓ +1) . (3.7)Moreover, the spectrally flowed elliptic genus is Fourier expanded in terms of the e R-characters as follows ( a, b ∈ Z N ); Z ( N − a,b ) ( τ, z ) ≡ ( − a + b q ˆ c a y ˆ ca e − πi abN Z ( N − ( τ, z + aτ + b )= N − X ℓ =0 e πi bN ( ℓ +1 − a ) ch ( e R) ℓ,ℓ +1 − a ( τ, z ) . (3.8)11n this way, by comparing (2.23) with (3.8) one would find a similarity between thefunction X ( N ) ( v, a ; τ, z ) and the minimal character ch ( e R) ℓ,m ( τ, z ) with the correspondence N −→ − N, ℓ −→ − v − , m −→ − v − a. (3.9)Furthermore, let us recall the modular transformation formula of ch ( e R) ℓ,m ( τ, z );ch ( e R) ℓ,m ( τ + 1 , z ) = e πi (cid:26) ℓ ( ℓ +2) − m N + − N − N (cid:27) ch ( e R) ℓ,m ( τ, z )= e πi ( ℓ +1)2 − m N ch ( e R) ℓ,m ( τ, z ) , (with m ≡ ℓ + 1 + 2 a ) , (3.10)ch ( e R) ℓ,m (cid:18) − τ , zτ (cid:19) = ( − i ) e iπ ˆ cτ z N − X ℓ ′ =0 X m ′∈ Z N ℓ ′ + m ′ ∈ Z +1 × r N sin (cid:18) π ( ℓ + 1)( ℓ ′ + 1) N (cid:19) √ N e πi mm ′ N ch ( e R) ℓ ′ ,m ′ ( τ, z )= e iπ ˆ cτ z N − X ℓ ′ =0 X m ′∈ Z N ℓ ′ + m ′ ∈ Z +1 N e − πi ( ℓ +1)( ℓ ′ +1) − mm ′ N ch ( e R) ℓ ′ ,m ′ ( τ, z ) . (3.11)The first line of (3.11) is familiar formula and we have made use of the property of minimalcharacter; ch ( e R) N − − ℓ,m + N ( τ, z ) = − ch ( e R) ℓ,m ( τ, z ) , (3.12)to derive the second line . The modular transformation formulas (3.2), (3.1) nicely corre-spond to (3.10), (3.11) under the formal replacement (3.9). In this way, one would regard X ( N ) ( v, a ; τ, z ) as a formal ‘analytic continuation’ of the minimal character ch ( e R) ℓ,m ( τ, z ). SL (2) /U (1) -Supercoset’ in the ( e R , e R ) -sector Now, let us discuss what is the superconformal model whose modular invariant is build up fromthe functions X ( N ) ( v, a ) (2.24).Let H (R) A be the Hilbert space of the axial supercoset of SL (2) N +2 /U (1) in the (R , R)-sector,while H (R) V be that of the vector supercoset. In other words, H (R) A corresponds to the cigar-type We have an obvious identity for X ( N ) ( v, a ; τ, z ) X ( N ) ( N − v, v + a ; τ, z ) = X ( N ) ( v, a ; τ, z ) , which again corresponds to (3.12) under (3.9) up to the overall sign. √ N α ′ , whereas H (R) V should be associatedwith [cigar] / Z N ∼ = [ N -fold cover of trumpet] , as is already mentioned in section 2. This means that the torus partition functions in the ( e R , e R)-sector of these theories are schematically written as (ˆ c ≡ N , τ ≡ τ + iτ , z ≡ z + iz , and F L , F R denotes the fermion number operators mod 2) Z ( e R) V ( τ, ¯ τ ; z, ¯ z ) ≡ e − π ˆ c z τ Tr H ( R ) V h ( − F L + F R q L − ˆ c q ˜ L − ˆ c e πizJ e πiz ˜ J i = e − π ˆ c z τ N − X v =0 X a ∈ Z N b χ ( N, dis ( v, a ; τ, z ) hb χ ( N, dis ( v, a ; τ, z ) i ∗ + [cont. terms] , (3.13) Z ( e R) A ( τ, ¯ τ ; z, ¯ z ) ≡ e π ˆ c z τ Tr H ( R ) A h ( − F L + F R q L − ˆ c q ˜ L − ˆ c e πizJ e πiz ˜ J i = e π ˆ c z τ N − X v =0 X aL,aR ∈ Z N v + a L + a R ∈ N Z b χ ( N, dis ( v, a L ; τ, z ) hb χ ( N, dis ( v, a R ; τ, z ) i ∗ + [cont. terms]= e π ˆ c z τ N − X v =0 X a ∈ Z N b χ ( N, dis ( v, − v − a ; τ, z ) hb χ ( N, dis ( v, a ; τ, z ) i ∗ + [cont. terms] . (3.14)The discrete part of (3.13) obviously looks the diagonal modular invariant, whereas that of(3.14) can be regarded as the anti-diagonal one with respect to the ‘minimal model like’ quantumnumber m ≡ v + 2 a adopted in [15] .The ‘anomaly factors’ e − π ˆ c z τ , e π ˆ c z τ , which ensures the modular invariance, originate pre-cisely from the path-integrations, and they differ due to the gauged WZW actions of vectorand axial types [1, 15]. Note that these factors just get the common form e π ˆ c z τ , if we replace z with − z in the axial model. Thus it would be useful to rewrite (3.14) as Z ( e R) A ( τ, ¯ τ ; − z, ¯ z ) = e − π ˆ c z τ N − X v =0 X a ∈ Z N b χ ( N, dis ( v, − v − a ; τ, − z ) hb χ ( N, dis ( v, a ; τ, z ) i ∗ + [cont. terms]= e − π ˆ c z τ N − X v =0 X a ∈ Z N b χ ( N, dis ( N − v, v + a ; τ, z ) hb χ ( N, dis ( v, a ; τ, z ) i ∗ + [cont. terms] , (3.15) The quantum number m labels the U (1) R -charges appearing in the spectral flow orbit defining χ ( N, dis ( v, a ).As mentioned in [15], the axial type (3.14) is anti-diagonal only in the case of integer levels ( i.e. K = 1), whilethe vector type (3.13) is always diagonal. b χ ( N, dis ( v, a ; τ, − z ) = b χ ( N, dis ( N − v, − a ; τ, z ) , (3.16)in the second line.Moreover, it is found that the continuous terms appearing in (3.13) and (3.15) are writtenin the precisely same functional form with inverse sign . We will prove this fact in the nextsubsection, and here address our main result in this section; Z ( e R) cSL (2) /U (1) ( τ, ¯ τ ; z, ¯ z ) := Z ( e R) V ( τ, ¯ τ ; z, ¯ z ) + Z ( e R) A ( τ, ¯ τ ; − z, ¯ z )= 12 e − π ˆ c z τ N − X v =0 X a ∈ Z N (cid:12)(cid:12) X ( N ) ( v, a ; τ, z ) (cid:12)(cid:12) . (3.17)Note that, even though both Z ( e R) V and Z ( e R) A include contributions of continuous characterswith non-trivial coefficients showing IR-divergence, the combined partition function (3.17) iswritten in terms only of a finite number of holomorphic building blocks X ( N ) ( v, a ; τ, z ) givenin (2.24) or (3.5). This fact strongly suggests that the modular invariant partition function(3.17) would define an N = 2 superconformal model with ˆ c = 1 + N associated with some compact background. Therefore, we tentatively call this model as the ‘compactified SL (2) /U (1)-supercoset model’ here. It is obvious that the elliptic genus of this model is given by theholomorphic function Φ ( N ) ( τ, z ) (2.2); Z cSL (2) /U (1) ( τ, z ) = Φ ( N ) ( τ, z ) ≡ θ (cid:0) τ, N +1 N z (cid:1) θ (cid:0) τ, N z (cid:1) . (3.18) In this subsection, we prove the precise cancellation of continuous terms potentially appearingin the first line of (3.17)We start with the explicit form of the partition function Z ( e R) V , which is evaluated in [15] bymeans of the path-integration ( u ≡ s τ + s ); Z ( e R) V ( τ, ¯ τ ; z, ¯ z ; ǫ ) = N e − πτ N +4 N z X n ,n ∈ Z Z Σ( z,ǫ ) d uτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ (cid:0) τ, u + N +2 N z (cid:1) θ (cid:0) τ, u + N z (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × e − π u z τ e − πNτ | n τ + n | e πiN ( n s − n s ) . (3.19)14ere we introduced the IR-regularization adopted in [1, 15], which removes the singularity ofintegrand originating from the non-compactness of target space. Namely we setΣ( z, ǫ ) ≡ Σ \ (cid:26) u = s τ + s ; − ǫ − k ζ < s < ǫ − k ζ , < s < (cid:27) , (3.20)where we set z ≡ ζ τ + ζ , ζ , ζ ∈ R , and ǫ ( >
0) denotes the regularization parameter.In the same way, the axial partition function Z ( e R) A , which describes the cigar background, iswritten as [1] Z ( e R) A ( τ, ¯ τ ; z, ¯ z ; ǫ ) = N e πτ ( ˆ c | z | − N +4 N z ) X m ,m ∈ Z Z Σ( z,ǫ ) d uτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ (cid:0) τ, u + N +2 N z (cid:1) θ (cid:0) τ, u + N z (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × e − π u z τ e − πNτ | m τ + m + u | . (3.21)As demonstrated in [1], one can make the ‘character decompositions’ of partition functions(3.19), (3.21). They are schematically expressed as Z ( e R) V ( τ, ¯ τ ; z, ¯ z ; ǫ ) = e − π ˆ c z τ " N − X v =0 X a ∈ Z N b χ ( N, dis ( v, a ; τ, z ) hb χ ( N, dis ( v, a ; τ, z ) i ∗ + X m ∈ Z N Z dp L Z dp R n ρ ( p L , p R , m ; ǫ ) χ ( N, con ( p L , m ; τ, z ) (cid:2) χ ( N, con ( p R , m ; τ, z ) (cid:3) ∗ + ρ ( p L , p R , m ; ǫ ) χ ( N, con ( p L , m ; τ, z ) (cid:2) χ ( N, con ( p R , m + N ; τ, z ) (cid:3) ∗ oi , (3.22) Z ( e R) A ( τ, ¯ τ ; z, ¯ z ; ǫ ) = e π ˆ c z τ " N − X v =0 X a ∈ Z N b χ ( N, dis ( v, − v − a ; τ, z ) h b χ ( N, dis ( v, a ; τ, z ) i ∗ + X m ∈ Z N Z dp L Z dp R n ρ ′ ( p L , p R , m ; ǫ ) χ ( N, con ( p L , − m ; τ, z ) (cid:2) χ ( N, con ( p R , m ; τ, z ) (cid:3) ∗ + ρ ′ ( p L , p R , m ; ǫ ) χ ( N, con ( p L , − m ; τ, z ) (cid:2) χ ( N, con ( p R , m + N ; τ, z ) (cid:3) ∗ oi , (3.23)where χ ( N, con ( p, m ) denotes the extended continuous character (B.1) explicitly written as χ ( N, con ( p, m ; τ, z ) = q p N Θ m,N (cid:18) τ, zN (cid:19) θ ( τ, z ) iη ( τ ) . In the continuous part, the ‘density functions’ ρ ( ρ ′ ) and ρ ( ρ ′ ) have rather complicated forms.As expected, ρ ( ρ ′ ) includes the logarithmically divergent term as the leading contribution; ρ ( p L , p R , m ; ǫ ) = C | ln ǫ | δ ( p L − p R ) + · · · , C , which corresponds to the strings freely propagating in the asymptoticregion, and C | ln ǫ | is roughly identified as an infinite volume factor. However, both ρ ( ρ ′ )and ρ ( ρ ′ ) include subleading, non-diagonal terms with p L = p R and considerably non-trivialdependence on m , as is mentioned in [1].Now, we would like to prove the equalities; ρ ( p L , p R , m ; ǫ ) = ρ ′ ( p L , p R , m ; ǫ ) , ρ ( p L , p R , m ; ǫ ) = ρ ′ ( p L , p R , m ; ǫ ) . (3.24)If this is the case, by using the ‘charge conjugation relation’ χ ( N, con ( p, m ; τ, − z ) = − χ ( N, con ( p, − m ; τ, z ) , (3.25)as well as (3.16), we obtain Z ( e R) A ( τ, ¯ τ ; − z, ¯ z ; ǫ ) = e − π ˆ c z τ " N − X v =0 X a ∈ Z N b χ ( N, dis ( N − v, v + a ; τ, z ) hb χ ( N, dis ( v, a ; τ, z ) i ∗ − X m ∈ Z N Z dp L Z dp R n ρ ( p L , p R , m ; ǫ ) χ ( N, con ( p L , m ; τ, z ) (cid:2) χ ( N, con ( p R , m ; τ, z ) (cid:3) ∗ + ρ ( p L , p R , m ; ǫ ) χ ( N, con ( p L , m ; τ, z ) (cid:2) χ ( N, con ( p R , m + N ; τ, z ) (cid:3) ∗ oi , (3.26)which leads to the desired formula (3.17). [proof of (3.24)] We start with recalling the ‘orbifold relation’ between Z ( e R) A and Z ( e R) V , which is shown in[1, 15]; Z ( e R) A ( τ, ¯ τ , z, ¯ z ; ǫ ) = e πτ ˆ c | z | N X α ,α ∈ Z N Z V, ( α ,α ) ( τ, ¯ τ ; z, ¯ z ; ǫ ) , (3.27)where we introduced the ‘twisted partition function’ in the R.H.S of (3.27); Z ( e R) V, ( α ,α ) ( τ, ¯ τ ; z, ¯ z ; ǫ ) = N e − πτ N +4 N z X n ,n ∈ Z Z Σ( z,ǫ ) d uτ e − π u z τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ (cid:0) τ, u + N +2 N z (cid:1) θ (cid:0) τ, u + N z (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × e − πkτ | ( Nn + α ) τ +( Nn + α ) | e πi { ( Nn + α ) s − ( Nn + α ) s } . (3.28) The minus sign just originates from the θ -factor. The relation of the notations here and given in [15] is as follows; Z ( e R ) A ( τ, ¯ τ ; z, ¯ z ; ǫ ) = Z reg ( τ, z ; ǫ ) , Z ( e R ) V ( τ, ¯ τ ; z, ¯ z ; ǫ ) = e Z reg ( τ, z ; ǫ ) , ,Z ( e R ) V, ( β ,β ) ( τ, ¯ τ ; z, ¯ z ; ǫ ) = e Z ( N ) reg ( τ, z | β , β ; ǫ ) .
16e can rewrite (3.28) by means of the Poisson resummation; Z ( e R) V, ( α ,α ) ( τ, ¯ τ ; z, ¯ z ; ǫ ) = e − π ˆ cτ z (cid:12)(cid:12)(cid:12)(cid:12) θ ( τ, z ) η ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) X v ∈ Z X ℓ, ˜ ℓ ∈ Z ℓ − ˜ ℓ ∈ α + N Z e πi α N ( v + ℓ +˜ ℓ ) × πi (cid:20)Z R − i dp e − ε ( v + ip ) yq ℓ h yq ˜ ℓ i ∗ − Z R + i ( N − dp e ε ( v + ip ) (cid:21) × e − πτ p v N p − iv ( yq ℓ ) vN − yq ℓ " ( yq ˜ ℓ ) vN − yq ˜ ℓ ∗ y ℓN q ℓ N h y ℓN q ˜ ℓ N i ∗ , (3.29)where we set ε := 2 π τ N ǫ . A closely related analysis is given in [1]. We obviously find Z ( e R) V, (0 , ( τ, ¯ τ , z, ¯ z ; ǫ ) = Z ( e R) V ( τ, ¯ τ , z, ¯ z ; ǫ ) , and (3.29) immediately implies the relation Z ( e R) V, ( α ,α ) ( τ, ¯ τ ; z, ¯ z ; ǫ ) = s L ( α ,α ) · Z ( e R) V ( τ, ¯ τ ; z, ¯ z ; ǫ ) , (3.30)where s L ( α ,α ) ( α i ∈ Z ) denotes the spectral flow operator (2.7) acting only on the left-mover; s L ( α ,α ) · f ( τ, ¯ τ , z, ¯ z ) := ( − α + α q ˆ c α y ˆ cα e πi α α N f ( τ, ¯ τ , z + α τ + α , ¯ z ) (3.31)Finally, by using the identity1 N X α ,α ∈ Z N s ( α ,α ) · b χ ( N, dis ( v, a ; τ, z ) = b χ ( N, dis ( v, − v − a ; τ, z ) , (3.32)1 N X α ,α ∈ Z N s ( α ,α ) · χ ( N, con ( p, m ; τ, z ) = χ ( N, con ( p, − m ; τ, z ) , (3.33)which are easily proven by direct calculations , we can achieve the wanted identities (3.24). (Q.E.D) We finally briefly discuss about other spin structures. We shall assume the diagonal spinstructures and the non-chiral GSO projection.We start with the ansatz of the total partition function; Z ( τ, ¯ τ ; z, ¯ z ) = 12 X σ =NS , f NS , R , e R h Z ( σ ) V ( τ, ¯ τ ; z, ¯ z ) + ε ( σ ) Z ( σ ) A ( τ, ¯ τ ; − z, ¯ z ) i , (3.34) (3.33) is essentially the same identity as (2.35). σ = NS , f NS , R , e R areevaluated by means of path-integration as in (3.19) and (3.21). Relevant calculations for generalspin structures as well as helpful formulas are presented in [19] .We set the sign factor ε ( e R) = 1 to reproduce (3.17) when setting σ = e R, and we have thefollowing two possibilities of ε ( σ ) for other spin structures that are compatible with the modularinvariance; (i) ε ( σ ) = 1 , ∀ σ : ‘non-geometric deformation of SL (2) /U (1) -supercoset’ With this naive choice, the continuous parts of partition functions of the vector and axialtypes are common and appear with the same sign contrary to the e R-case. This feature justoriginates from the simple fact; θ ( z ), θ ( z ), θ ( z ) are even functions of z although θ ( z )is an odd function. Thus, there are continuous excitations in the sectors σ = NS , f NS , e Ras in the standard SL (2) /U (1)-supercoset.On the other hand, the discrete part of each spin structure is described by the followingbuilding blocks ( v ∈ Z N , a ∈ + Z N for σ = NS , f NS, a ∈ Z N for σ = R , e R); X ( N ) [NS]+ ( v, a ; τ, z ) := b χ ( N,
1) [NS] dis ( v, a ; τ, z ) + b χ ( N,
1) [NS] dis ( N − v, a + v ; τ, z ) ,X ( N ) [ f NS] − ( v, a ; τ, z ) := b χ ( N,
1) [ f NS] dis ( v, a ; τ, z ) − b χ ( N,
1) [ f NS] dis ( N − v, a + v ; τ, z ) ,X ( N ) [R]+ ( v, a ; τ, z ) := b χ ( N,
1) [R] dis ( v, a ; τ, z ) + b χ ( N,
1) [R] dis ( N − v, a + v ; τ, z ) ,X ( N ) [ e R]+ ( v, a ; τ, z ) := b χ ( N,
1) [ e R] dis ( v, a ; τ, z ) + b χ ( N,
1) [ e R] dis ( N − v, a + v ; τ, z ) ( ≡ (2 . , (3.35)where we have explicitly indicated the spin structure. The explicit definitions of modu-lar completions with general spin structures are presented in [19]. We remark that thefunctions X ( N ) [ σ ] ∗ ( v, a ) appearing in (3.35) are generically non-holomorphic except for the e R-sector.This model shares the asymptotic cylindrical region with the radius N √ N α ′ ≡ q N α ′ with the standard SL (2) /U (1) (of the vector type) and aspect of propagating strings isalmost the same. However, we have non-trivial deformations in the discrete spectrum,which leads us to the holomorphic elliptic genus (3.18) and would be non-geometric sincethey are never realized only within the cigar theory (or the trumpet theory). Since we are now assuming the non-chiral GSO projection, we do not need choose the parameter as ˆ c =1 + KN , K ∈ Z > ( N and K are not necessarily co-prime) as in [19]. This assumption is necessary whenconsidering the chiral GSO projection. ii) ε ( σ ) = − for σ = NS , g NS , R, and ε ( e R ) = 1 : ‘Compactified SL (2) /U (1) -Supercoset’ This second possibility is more curious. In this case, the continuous sectors are canceledout for all the spin structures, and the discrete parts are described respectively by X ( N ) [NS] − ( v, a ; τ, z ) := b χ ( N,
1) [NS] dis ( v, a ; τ, z ) − b χ ( N,
1) [NS] dis ( N − v, a + v ; τ, z ) ,X ( N ) [ f NS]+ ( v, a ; τ, z ) := b χ ( N,
1) [ f NS] dis ( v, a ; τ, z ) + b χ ( N,
1) [ f NS] dis ( N − v, a + v ; τ, z ) ,X ( N ) [R] − ( v, a ; τ, z ) := b χ ( N,
1) [R] dis ( v, a ; τ, z ) − b χ ( N,
1) [R] dis ( N − v, a + v ; τ, z ) ,X ( N ) [ e R]+ ( v, a ; τ, z ) := b χ ( N,
1) [ e R] dis ( v, a ; τ, z ) + b χ ( N,
1) [ e R] dis ( N − v, a + v ; τ, z ) ( ≡ (2 . . (3.36)Namely, we achieve the total partition function in a very simple form ; Z cSL (2) /U (1) ( τ, ¯ τ ; z, ¯ z ) = 14 e − πτ ˆ cz X v ∈ Z N X a ∈ Z N "(cid:12)(cid:12)(cid:12)(cid:12) X ( N ) [NS] − ( v, a + 12 ; τ, z ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) X ( N ) [NS]+ ( v, a + 12 ; τ, z ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) X ( N ) [R] − ( v, a ; τ, z ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) X ( N ) [ e R]+ ( v, a ; τ, z ) (cid:12)(cid:12)(cid:12) . (3.37)This is a natural extension of (3.17) including all the spin structures, and should bedirectly compared with the N = 2 minimal model (with level N − Z min ( τ, ¯ τ ; z, ¯ z ) = 14 e − πτ ( − N ) z X σ =NS , f NS , R , e R N − X ℓ =0 X m ∈ Z N (cid:12)(cid:12)(cid:12) ch ( σ ) ℓ,m ( τ, z ) (cid:12)(cid:12)(cid:12) . (3.38)All of the building blocks X ( N ) [ σ ] ∗ ( v, a ) in (3.37) are holomorphic and can be rewritten interms only of the extended characters (that are not modular completed) as in (3.5). Infact, the functions (3.36) for σ = NS , f NS , R are reproduced by the ‘half spectral flows’ ; z z + τ +12 , z z + τ , z z + . The absence of non-holomorphic corrections meansthat they are directly associated to some infinitely reducible representations of N = 2SCA with ˆ c = 1 + N . These representations are, however, non-unitary due to the relativeminus sign appearing in the NS and R-sectors. This aspect is in a sharp contrast to the N = 2 minimal model, in which the partition function (3.38) only includes the charactersof unitary irreducible representations. The overall factor 1 / / / Z -symmetry X ( N ) [ σ ] ± ( v, a ) = ± X ( N ) [ σ ] ± ( N − v, v + a ) . Summary and Comments
In this paper, we have studied a possible ‘analytic continuation’ with N → − N of the N = 2minimal model with the central charge ˆ c = 1 − N . Namely, we have examined the problem ofwhat is the superconformal system with ˆ c = 1 + N that has (1.2) as its elliptic genus.Our main results are summarized as follows; (i) The ‘Fourier expansion’ of the function (1.2) is rewritten by a holomorphic linear combina-tion of the modular completions of the extended discrete characters of SL (2) /U (1)-model[1]. This result is exhibited in terms of the formula (2.24) or (2.31), equivalently. This issimilar to the fact that the elliptic genus of N = 2 minimal model Z ( N − ( τ, z ) (1.1) isFourier expanded by the characters associated to the Ramond ground states [5]. (ii) The superconformal system corresponding to (1.2) is identified with a ‘compactified’ modelof SL (2) /U (1)-supercoset, as is given by (3.17). (iii) Two possibilities of extending to general spin structures have been presented; One is anon-compact model regarded as a ‘non-geometric deformation’ of SL (2) /U (1)-supercoset,and the other is the natural extension of the compactified model (3.17). The latter isquite similar to the N = 2 minimal model, although it is not a unitary theory.We would like to add a few comments;The partition function (3.17) (and (3.37)) looks very like those of RCFTs. We only possessfinite conformal blocks that are holomorphically factorized in the usual sense. However, there isa crucial difference from generic RCFTs defined axiomatically. The partition function (3.17) orthe elliptic genus (3.18) does not include the contributions from the Ramond vacua saturatingthe unitarity bound Q = ± ˆ c . This implies that the Hilbert space of normalizable states doesnot contain the NS-vacuum ( h = Q = 0) which should correspond to the identity operator. Ofcourse, this feature is common with the spectrum of original SL (2) /U (1)-supercoset read offfrom the torus partition function evaluated in [9] (see also [16, 17, 18]). It may be an interestingquestion whether or not the finiteness of conformal blocks without the identity representation ,which is observed in our ‘compactified SL (2) /U (1)-model’, unavoidably leads to a non-unitarityof the spectrum in general conformal field theories.A natural extension of this work would be the study of the cases of ‘fractional levels’ ˆ c =1 + KN , ( K ≥ , GCD { N, K } = 1). In other words, one may search a theory of which elliptic20enus would be Z ( τ, z ) = K Φ ( N/K ) ( τ, z ) ≡ K θ (cid:0) τ, N + KN z (cid:1) θ (cid:0) τ, KN z (cid:1) , which has the Witten index Z ( τ, z = 0) = N + K . However, the function Φ ( N/K ) ( τ, z ) is only meromorphic with respect to the angle variable z , and such a function is not likely to be realizedas the elliptic genus of any superconformal field theory.We also point out that the cancellation of continuous parts such as (3.17) does not seem tohappen in that case. This fact suggests that the ‘compactification’ of SL (2) /U (1)-supercosetworks only for integer levels, that is, ˆ c = 1 + N . Acknowledgments
The author should thank T. Eguchi for useful discussions at the early stage of this work.This research was supported by JSPS KAKENHI Grant Number 23540322 from JapanSociety for the Promotion of Science (JSPS). 21 ppendix A: Conventions for Theta Functions
We assume τ ≡ τ + iτ , τ > q := e πiτ , y := e πiz ; θ ( τ, z ) = i ∞ X n = −∞ ( − n q ( n − / / y n − / ≡ πz ) q / ∞ Y m =1 (1 − q m )(1 − yq m )(1 − y − q m ) ,θ ( τ, z ) = ∞ X n = −∞ q ( n − / / y n − / ≡ πz ) q / ∞ Y m =1 (1 − q m )(1 + yq m )(1 + y − q m ) ,θ ( τ, z ) = ∞ X n = −∞ q n / y n ≡ ∞ Y m =1 (1 − q m )(1 + yq m − / )(1 + y − q m − / ) ,θ ( τ, z ) = ∞ X n = −∞ ( − n q n / y n ≡ ∞ Y m =1 (1 − q m )(1 − yq m − / )(1 − y − q m − / ) . (A.1)Θ m,k ( τ, z ) = ∞ X n = −∞ q k ( n + m k ) y k ( n + m k ) . (A.2)We also set η ( τ ) = q / ∞ Y n =1 (1 − q n ) . (A.3)The spectral flow properties of theta functions are summarized as follows ( m, n, a ∈ Z , k ∈ Z > ); θ ( τ, z + mτ + n ) = ( − m + n q − m y − m θ ( τ, z ) ,θ ( τ, z + mτ + n ) = ( − n q − m y − m θ ( τ, z ) ,θ ( τ, z + mτ + n ) = q − m y − m θ ( τ, z ) ,θ ( τ, z + mτ + n ) = ( − m q − m y − m θ ( τ, z ) , Θ a,k ( τ, z + mτ + n )) = q − km y − km Θ a +2 km,k ( τ, z ) . (A.4) Appendix B: Summary of Modular Completions
In this appendix we summarize the definitions as well as useful formulas for the ‘extendeddiscrete characters’ and their modular completions of the N = 2 superconformal algebra withˆ c (cid:0) ≡ c (cid:1) = 1 + k . We focus only on the e R-sector , and when treating the extended characters,we assume k = N/K , (
N, K ∈ Z > ) (but, not assume N and K are co-prime). In this paper we shall use the convention of e R-characters with the same sign as [15], and the inverse signcompared to those of [1, 9, 10], so that the Witten indices appear with the positive sign. (See (B.30) below.) xtended Continuous (non-BPS) Characters [8, 9]: χ ( N,K ) con ( p, m ; τ, z ) := q p NK Θ m,NK (cid:18) τ, zN (cid:19) θ ( τ, z ) iη ( τ ) . (B.1)This corresponds to the spectral flow sum of the non-degenerate representation with h = p + m NK + ˆ c , Q = mN ± ( p ≥ m ∈ Z NK ), whose flow momenta are taken to be n ∈ N Z . The modularand spectral flow properties are simply written as χ ( N,K ) con (cid:18) p, m ; − τ , zτ (cid:19) = ( − i ) e iπ ˆ cτ z N K Z ∞−∞ dp ′ X m ′ ∈ Z NK e πi pp ′− mm ′ NK χ ( N,K ) con ( p ′ , m ′ ; τ, z ) . (B.2) χ ( N,K ) con ( p, m ; τ + 1 , z ) = e πi p m NK χ ( N,K ) con ( p, m ; τ, z ) , (B.3) χ ( N,K ) con ( p, m ; τ, z + rτ + s ) = ( − r + s e πi mN s q − ˆ c r y − ˆ cr χ ( N,K ) con ( p, m + 2 Kr ; τ, z ) , ( ∀ r, s ∈ Z ) . (B.4) Extended Discrete Characters [8, 9, 10]: χ ( N,K ) dis ( v, a ; τ, z ) := X n ∈ a + N Z ( yq n ) vN − yq n y KN n q KN n θ ( τ, z ) iη ( τ ) , (B.5)This again corresponds to the sum of the Ramond vacuum representation with h = ˆ c , Q = vN − ( v = 0 , , . . . , N ) over spectral flow with flow momentum m taken to be mod. N , as m = a + N Z ( a ∈ Z N ). If one introduces the notation of Appell function or Lerch sum withlevel 2 k [11, 12, 13], K (2 k ) ( τ, z ) := X n ∈ Z y kn q kn − yq n , (B.6) χ dis ( v, a ) is identified as its Fourier expansion; y KN a q KN a K (2 NK ) (cid:18) τ, z + aτ + bN (cid:19) θ ( τ, z ) iη ( τ ) = N − X v =0 e πi vbN χ ( N,K ) dis ( v, a ; τ, z ) . (B.7)We also note χ ( N,K ) dis ( N, a ; τ, z ) = χ ( N,K ) dis (0 , a ; τ, z ) − Θ Ka,NK (cid:18) τ, zN (cid:19) θ ( τ, z ) iη ( τ ) . (B.8)23he modular transformation formulas of χ ( N,K ) dis ( v, a ) and K (2 k ) can be expressed as [8, 9, 10,12, 13]; χ ( N,K ) dis (cid:18) v, a ; − τ , zτ (cid:19) = e iπ ˆ cτ z " N − X v =0 X a ∈ Z N N e πi vv ′− ( v +2 Ka )( v ′ +2 Ka ′ )2 NK χ ( N,K ) dis ( v ′ , a ′ ; τ, z ) − i N K X m ′ ∈ Z NK e − πi ( v +2 Ka ) m ′ NK Z R + i dp ′ e − π vp ′ NK − e − π p ′ + im ′ K χ ( N,K ) con ( p ′ , m ′ ; τ, z ) , (B.9) χ ( N,K ) dis ( v, a ; τ + 1 , z ) = e πi aN ( v + Ka ) χ ( N,K ) dis ( v, a ; τ, z ) , (B.10) K (2 k ) (cid:18) − τ , zτ (cid:19) = τ e iπ kz τ " K (2 k ) ( τ, z ) − i √ k X m ∈ Z k Z R + i dp ′ q p ′ − e − π (cid:16) p ′√ k + i m k (cid:17) Θ m,k ( τ, z ) , (B.11) K (2 k ) ( τ + 1 , z ) = K (2 k ) ( τ, z ) . (B.12)The spectral flow property is also expressed as χ ( N,K ) dis ( v, a ; τ, z + rτ + s ) = ( − r + s e πi v +2 KaN s q − ˆ c r y − ˆ cr χ ( N,K ) dis ( v, a + r ; τ, z ) , ( ∀ r, s ∈ Z ) , (B.13) K (2 k ) ( τ, z + rτ + s ) = q − kr y − kr K (2 k ) ( τ, z ) , ( ∀ r, s ∈ Z ) , (B.14) Modular Completion of the Extended Discrete Characters:
The modular completion of the discrete character χ dis is defined as follows; b χ ( N,K ) dis ( v, a ; τ, z ) := θ ( τ, z )2 πη ( τ ) X n ∈ a + N Z r ∈ v + N Z (cid:26)Z R + i ( N − dp − Z R − i dp ( yq n ) (cid:27) e − πτ p r NK ( yq n ) rN p − ir y KN n q KN n − yq n = χ ( N,K ) dis ( v, a ; τ, z ) + θ ( τ, z )2 πη ( τ ) X n ∈ a + N Z r ∈ v + N Z Z R − i dp e − πτ p r NK p − ir ( yq n ) rN y KN n q KN n . (B.15)This expression (B.15) is obviously periodic with respect to both of v and a ; b χ ( N,K ) dis ( v + N m, a + N n ; τ, z ) = b χ ( N,K ) dis ( v, a ; τ, z ) , ( ∀ m, n ∈ Z ) . (B.16)Especially, we have b χ ( N,K ) dis ( N, a ; τ, z ) = b χ ( N,K ) dis (0 , a ; τ, z ) , (B.17)in spite of (B.8). 24he modular completion of Appell function K (2 k ) ( τ, z ) is given as [13]; b K (2 k ) ( τ, z ) := K (2 k ) ( τ, z ) − X m ∈ Z k R m,k ( τ ) Θ m,k ( τ, z ) , (B.18)where we set R m,k ( τ ) := 1 iπ X r ∈ m +2 k Z Z R − i dp e − πτ p r k p − ir q − r k , (B.19)which is generically non-holomorphic due to the τ -dependence.One can easily show R m +2 ks,k ( τ ) = R m,k ( τ ) , ( ∀ s ∈ Z ) , R m,k ( τ ) = 2 δ (2 k ) m, − R − m,k ( τ ) , (B.20)and thus R ,k ( τ ) ≡ , R k,k ( τ ) ≡ , (B.21)holds, especially.The ‘Fourier expansion relation’ (B.7) is inherited to the modular completions; y KN a q KN a b K (2 NK ) (cid:18) τ, z + aτ + bN (cid:19) θ ( τ, z ) iη ( τ ) = X v ∈ Z N e πi vbN b χ ( N,K ) dis ( v, a ; τ, z ) . (B.22)The modular transformation formulas for the modular completions (B.15), (B.18) are writtenas b χ ( N,K ) dis (cid:18) v, a ; − τ , zτ (cid:19) = e iπ ˆ cτ z N − X v ′ =0 X a ′ ∈ Z N N e πi vv ′− ( v +2 Ka )( v ′ +2 Ka ′ )2 NK b χ ( N,K ) dis ( v ′ , a ′ ; τ, z ) , (B.23) b χ ( N,K ) dis ( v, a ; τ + 1 , z ) = e πi aN ( v + Ka ) b χ ( N,K ) dis ( v, a ; τ, z ) . (B.24) b K (2 k ) (cid:18) − τ , zτ (cid:19) = τ e iπ kτ z b K (2 k ) ( τ, z ) , (B.25) b K (2 k ) ( τ + 1 , z ) = b K (2 k ) ( τ, z ) . (B.26)When compared with (B.9) and (B.11), the S-transformation formulas have been simplifiedbecause of the absence of continuous terms.Also the spectral flow property is preserved by taking the completion; b χ ( N,K ) dis ( v, a ; τ, z + rτ + s ) = ( − r + s e πi v +2 KaN s q − ˆ c r y − ˆ cr b χ ( N,K ) dis ( v, a + r ; τ, z ) , ( ∀ r, s ∈ Z ) . (B.27) b K (2 k ) ( τ, z + rτ + s ) = q − kr y − kr b K (2 k ) ( τ, z ) , ( ∀ r, s ∈ Z ) . (B.28)25ote that the IR-part of modular completions are evaluated as hb χ ( N,K ) dis ( v, a ; τ, z ) i (cid:18) ≡ lim τ →∞ b χ ( N,K ) dis ( v, a ; τ, z ) (cid:19) = δ ( N ) a, (cid:16) y − + y (cid:17) ( v = 0 , N ) δ ( N ) a, y vN − ( v = 1 , . . . , N − , (B.29)while we have h χ ( N,K ) dis ( v, a ; τ, z ) i = δ ( N ) a, y vN − , ( v = 0 , . . . , N ) . (B.30) Appendix C: Finiteness of R a,b ( τ ) at the cusp τ = i ∞ In Appendix C, we confirm the finiteness of the residue function R a,b ( τ ) (2.42) at the cusp τ = i ∞ . Namely we will prove lim τ → i ∞ | R a,b ( τ ) | < ∞ , (C.1)for ∀ ( a, b ) ∈ Z N +1 × Z N +1 − { (0 , } .We classify z a,b (2.41) into three groups and examine the behavior of F ( τ, z ) (2.40) aroundthem separately; (i) a = 0 , b = 1 , . . . , N : In this case, we have z ,b ≡ bN + 1 , ( b = 1 , . . . , N ). All of them are simple zeros of thefunction; (cid:2) Φ ( N ) ( τ, N z ) (cid:3) = h Φ ( N ) ′ ( τ, N z ) i = y − N N X j =0 y j ≡ y − N − y N +1 − y . This means lim τ → i ∞ F ( z ,b ) = 1 , ( ∀ b = 1 , . . . , N ) , and thus we obtain lim τ → i ∞ | R ,b ( τ ) | = 0 , ( ∀ b = 1 , . . . , N ) . (ii) a = 1 , . . . , N − , b = 0 , . . . , N : In this case, we first note z a,b = ξ a,b + e z a,b , ξ a,b := aτ + bN , e z a,b := − aτ + bN ( N + 1) , (C.2)26nd the term ξ a,b can be interpreted as the spectral flow caused by s ( a,b ) . Therefore, itis enough to compare the behaviors of Φ ( N )( a,b ) ( τ, N z ) and Φ ( N ) ′ ( a,b ) ( τ, N z ) around z = e z a,b inplace of examining F ( τ, z ) around z = z a,b .Recalling Φ ( N )( a,b ) ( τ, N z ) = y a θ ( τ, ( N + 1) z + ξ a,b ) θ ( τ, z + ξ a,b ) , ( a = 0)we obtain Φ ( N )( a,b ) ( τ, N z ) ∼ ( z − e z a,b ) q − a N ( N +1) + a N +1) , ( z ∼ e z a,b , τ ∼ i ∞ ) . (C.3)On the other hand, since h Φ ( N ) ′ ( a,b ) ( τ, N z ) i = y − N + a , ( a = 0) holds, we also obtainΦ ( N ) ′ ( a,b ) ( τ, N z ) ∼ (cid:0) e πi e z a,b (cid:1) − N + a ∼ q − a N ( N +1) + a N +1) , ( z ∼ e z a,b , τ ∼ i ∞ ) . (C.4)We thus concludelim τ → i ∞ | R a,b ( τ ) | < ∞ , ( a = 1 , . . . , N − , b = 0 , . . . , N ) . (iii) a = N, b = 0 , . . . , N : Around z = z N,b , we find θ ( τ, ( N + 1) z ) ∼ ( z − z N,b ) q − N + and θ ( τ, z ) ∼ q − N N +1) + ,and hence Φ ( N ) ( τ, N z ) ∼ ( z − z N,b ) q − N + N N +1) , ( z ∼ z N,b , τ ∼ i ∞ ) . (C.5)On the other hand, by using the decomposition (C.2), we can evaluate Φ ( N ) ′ ( τ, N z ) around z = z N,b as follows;Φ ( N ) ′ ( τ, N z ) = Φ ( N ) ′ ( τ, N ( ξ N,b + e z N,b )) ∼ q − N ( N +2) e − πiN ( N +2) e z N,b × h Φ ( N ) ′ ( τ, N e z N,b ) i ∼ q − N ( N +2) (cid:0) e πi e z N,b (cid:1) − N ( N +2)+ N ∼ q − N + N N +1) , ( z ∼ z N,b , τ ∼ i ∞ ) . 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