From rarefied elliptic beta integral to parafermionic star-triangle relation
aa r X i v : . [ h e p - t h ] O c t From rarefied elliptic beta integral toparafermionic star-triangle relation
Gor Sarkissian , ∗ and Vyacheslav P. Spiridonov , † Bogoliubov Laboratory of Theoretical Physics, JINR,Dubna, Moscow region, 141980 Russia St. Petersburg Department of the Steklov Mathematical Institute of RussianAcademy of Sciences, St. Petersburg, 191023 Russia Department of Physics, Yerevan State University,Alex Manoogian 1, 0025 Yerevan, Armenia
Abstract
We consider the rarefied elliptic beta integral in various limiting forms.In particular, we obtain an integral identity for parafermionic hyperbolicgamma functions which describes the star-triangle relation for parafermionicLiouville theory. ∗ [email protected], [email protected] † [email protected] ontents Introduction
Full understanding of the properties of non-rational 2 d conformal field theoriesis one of the most important questions in string theory, quantum field theoryand mathematical physics. The most important well known examples of the non-rational conformal field theory are Liouville field theory (LFT) and its variousgeneralizations, like supersymmetric extensions, parafermionic extensions, Todafield theory, etc. Already in the seminal works [9, 36], where the three-pointfunction in LFT was constructed, an important role of some special functionΥ b emerged. Studies of the fusion matrix [24] and boundary correlation func-tions [11,25] required the use of another related function – the noncompact quan-tum dilogarithm S b [10], which is called also the hyperbolic gamma function [29](we follow the latter terminology). Both of these functions are constructed outof the Barnes double gamma function Γ b . Study of N = 1 supersymmetric LFTshowed that description of the three-point functions [22, 27], boundary correla-tion functions [12] and fusion matrix [7, 14] requires the use of supersymmetricgeneralizations of these functions: Υ i , Γ i and S i , where i = 0 ,
1. In [4], three-point functions were studied in parafermionic LFT, which is LFT coupled with Z N parafermions. It was shown there that three-point functions can be writtenusing parafermionic generalizations of Υ b : Υ k , where k = 0 , . . . , N − N = 1 supersymmetric LFT[23] showed that the corresponding relations are implied by the generalizationof considerations of [17] to supersymmetric hyperbolic gamma functions foundin [15].This state of affairs inspires us to think that attempts to find expressions forfusion matrix and boundary correlation functions in the parafermionic LFT in-evitably will require to write parafermionic generalizations of Γ b and S b functionsas well. In fact a parafermionic generalization of Γ b was introduced in [23], where3lso some properties of this function were derived. It is also natural to assumethat parafermionic generalization of S b function should possess the star-trianglerelation as well.In this paper we would like to connect mentioned topics with the subjectwhich developed over the last decade in higher dimensional superconformail fieldtheories – the theory of superconformal indices [28] described in terms of theelliptic hypergeometric integrals [31]. So, the standard elliptic gamma functioncoincides with the superconformal index of chiral superfield of theories on S × S space-time background. Consideration of superconformal indices of gauge theoryon lens space [3] leads to a particular combination of elliptic gamma functions withdifferent bases. It was called in [33] the rarefied elliptic gamma function due toits special product type representation, using which we introduce parafermionichyperbolic gamma function as a particular limit. It is built from S b functionsalong the same rules by which Υ i function in [4] is built from Υ b functions.We show that such parafermionic hyperbolic gamma functions are related totwo computable rarefied hyperbolic beta integrals, corresponding to two valuesof a parameter ǫ = 0 ,
1. The one corresponding to ǫ = 0 was found earlier in [13],and the second one ǫ = 1 is new. Degenerating these hyperbolic beta integrals weobtain the star-triangle relation for the parafermionic LFT. For the supersym-metric case we compared obtained results with those derived earlier in [15] andfound that the star-triangle relation in [15] in some cases is missing an overall sign.Thus, it appears that 4 d superconformal indices contain a lot of important infor-mation about 2 d systems – the 2 d conformal field theories discussed above andintegrable 2 d lattice spin systems, for which they describe partition functions [32].Moreover, it is known that the same hyperbolic limit of these 4 d indices describespartition functions of 3 d supersymmetric models on the squashed sphere S b [28].The present work can be considered as a complement to [8], where the transitionfrom 4 d theories to 3 d ones was reached by degenerating elliptic hypergeometricintegrals to hyperbolic integrals, – we add to such a connection a relation to theparafermionic LFT.We would like to add that, in view of the AGT relation between para-Liouvilletheory and superconformal gauge theories on C / Z r , see e.g. [1, 5, 21] and refer-ences therein, it could be expected that parafermionic hyperbolic gamma func-tions should arise from the rarefied (or lens) elliptic gamma function.The paper is organized in the following way. In section 2 we review thenecessary formulas on elliptic gamma functions and the rarefied elliptic beta4ntegral. In section 3 we consider parafermionic hyperbolic gamma functions.In section 4 we derived a hyperbolic beta integral and star-triangle relation forparafermionic hyperbolic gamma functions. In section 5 we consider in detail thestar-triangle relation for supersymmetric case, compare it with a version of thisformula obtained earlier in [15] and indicate a sign difference in them. The standard elliptic gamma function Γ( z ; p, q ) can be defined as an infiniteproduct: Γ( z ; p, q ) = ∞ Y j,k =0 − z − p j +1 q k +1 − zp j q k , | p | , | q | < , z ∈ C ∗ . (1)The lens space elliptic gamma function is defined as a product of two standardelliptic gamma functions with different bases [3]. γ e ( z, m ; p, q ) = Γ( zp m ; p r , pq )Γ( zq r − m ; q r , pq ) (2)= ∞ Y j,k =0 − z − p − m ( pq ) j +1 p r ( k +1) − zp m ( pq ) j p rk − z − q m ( pq ) j +1 q rk − zq r − m ( pq ) j q rk , m ∈ Z . As shown in [33], the function (2) can be written as a special product of thestandard elliptic gamma functions with bases p r and q r . For 0 ≤ m ≤ r it hasthe form: γ e ( z, m ; p, q ) = m − Y k =0 Γ( q r − m z ( pq ) k ; p r , q r ) r − m − Y k =0 Γ( p m z ( pq ) k ; p r , q r ) , (3)for m < γ e ( z, m ; p, q ) = Q r − m − k =0 Γ( p m z ( pq ) k ; p r , q r ) Q − mk =1 Γ( q r − m z ( pq ) − k ; p r , q r ) , (4)and for m > r γ e ( z, m ; p, q ) = Q m − k =0 Γ( q r − m z ( pq ) k ; p r , q r ) Q m − rk =1 Γ( p m z ( pq ) − k ; p r , q r ) . (5)A convenient normalization of this function was introduced in [33]Γ ( r ) ( z, m ; p, q ) = ( − z ) m ( m − p m ( m − m − q − m ( m − m +1)6 γ e ( z, m ; p, q ) , (6)5hich yields Γ (1) ( z, m ; p, q ) = Γ( z ; p, q ). It is this object that was called therarefied elliptic gamma function.Let us define a particular combination of such functions∆ ( r ) e ( z, m ; t a , n a | p, q ) = Q a =1 Γ ( r ) ( t a z, n a + m + ǫ ; p, q )Γ ( r ) ( t a z − , n a − m ; p, q )Γ ( r ) ( z , m + ǫ ; p, q )Γ ( r ) ( z − , − (2 m + ǫ ); p, q ) , (7)Ψ ( r ) e ( z, m ; t a , n a | p, q ) = Q a =1 γ e ( t a z, n a + m + ǫ ; p, q ) γ e ( t a z − , n a − m ; p, q ) γ e ( z , m + ǫ ; p, q ) γ e ( z − , − (2 m + ǫ ); p, q ) . (8)It is shown in [33] that if parameters t a , n a satisfy the constraints | t a | < Y a =1 t a = pq , X a =1 n a = − ǫ , ǫ = 0 , , (9)then one has the following integral identity κ ( r ) r − X m =0 Z T ∆ ( r ) e ( z, m ; t a , n a | p, q ) dzz = Y ≤ a ω ) > γ (2) ( y ; ω , ω ) has the integral representation γ (2) ( y ; ω , ω ) = exp (cid:18) − Z ∞ (cid:18) sinh(2 y − ω − ω ) x ω x ) sinh( ω x ) − y − ω − ω ω ω x (cid:19)(cid:19) dxx , (14)and obeys the equations: γ (2) ( y + ω ; ω , ω ) γ (2) ( y ; ω , ω ) = 2 sin πyω , γ (2) ( y + ω ; ω , ω ) γ (2) ( y ; ω , ω ) = 2 sin πyω . (15)Setting z = e − πvy/r , p = e − πvω /r , q = e − πvω /r , (16)one can write q r − m z ( pq ) k = e − πv [ yr + ω ( − mr ) +( ω + ω ) kr ] ,p m z ( pq ) k = e − πv [ yr + mr ω +( ω + ω ) kr ] . Now one can show that: γ e (cid:16) e − πvyr , m ; e − πvω r , e − πvω r (cid:17) = v → e − π (2 y − ω − ω ) / vω ω Λ( y, m ; ω , ω ) , (17)where the function Λ( y, m ; ω , ω ) is defined as follows. For 0 ≤ m ≤ r one hasΛ( y, m ; ω , ω ) = m − Y k =0 γ (2) (cid:18) yr + ω (cid:16) − mr (cid:17) + ( ω + ω ) kr ; ω , ω (cid:19) (18) × r − m − Y k =0 γ (2) (cid:18) yr + mr ω + ( ω + ω ) kr ; ω , ω (cid:19) , for m < y, m ; ω , ω ) = Q r − m − k =0 γ (2) (cid:0) yr + mr ω + ( ω + ω ) kr ; ω , ω (cid:1)Q − mk =1 γ (2) (cid:0) yr + ω (cid:0) − mr (cid:1) − ( ω + ω ) kr ; ω , ω (cid:1) , (19)and for m > r Λ( y, m ; ω , ω ) = Q m − k =0 γ (2) (cid:0) yr + ω (cid:0) − mr (cid:1) + ( ω + ω ) kr ; ω , ω (cid:1)Q m − rk =1 γ (2) (cid:0) yr + mr ω − ( ω + ω ) kr ; ω , ω (cid:1) . (20)7n fact it is enough to consider functions Λ( y, m ; ω , ω ) only for 0 ≤ m ≤ r .Recall the quasiperiodicity property [33]: γ e ( z, m + kr ; p, q ) γ e ( z, m ; p, q ) = (cid:18) − √ pqz (cid:19) mk + r k ( k − (cid:18) qp (cid:19) k (cid:16) m + mr k − + r k − k − (cid:17) , k ∈ Z . (21)In the limit (17) it impliesΛ( y, m + kr ; ω , ω ) = ( − mk + r k ( k − Λ( y, m ; ω , ω ) (22)Let us study the function Λ( y, m ; ω , ω ) for the particular choice r = 2.Eq. (22) implies that in this case we have only two functions corresponding to m = 0 ,
1. For m = 0 we have:Λ( y, ω , ω ) = γ (2) (cid:16) y ω , ω (cid:17) γ (2) (cid:18) y ω + ω ω , ω (cid:19) , (23)and for m = 1Λ( y, ω , ω ) = γ (2) (cid:16) y ω ω , ω (cid:17) γ (2) (cid:16) y ω ω , ω (cid:17) . (24)Setting ω = b and ω = b and Q = b + b and using the notation accepted inconformal field theory literature γ (2) ( z ; b, /b ) = S b ( z ) , (25)we obtain thatΛ( y, b − , b ) = S b (cid:16) y (cid:17) S b (cid:18) y Q (cid:19) ≡ S NS ( y ) ≡ S ( y ) , (26)Λ( y, b − , b ) = S b (cid:18) y b (cid:19) S b (cid:18) y b − (cid:19) ≡ S R ( y ) ≡ S ( y ) . (27)The functions S NS ( y ) and S R ( y ) appear in numerous aspects of N = 1 super-symmetric Liouville conformal field theory. Subscripts NS and R refer to theNeveu-Schwarz and Ramond sectors respectively. First defined in [12] for calcu-lation of the boundary two-point functions, they played important role in writingdown fusion and braiding matrices of conformal blocks [14]. It was suggestedin [15] to denote them as S ( y ) and S ( y ), respectively, to write in compact waythe corresponding star-triangle relation.8onsider now the functions Λ( y ; m ; ω , ω ) for arbitrary r :Λ( y, m ; b − , b ) = m − Y k =0 S b (cid:18) yr + b (cid:16) − mr (cid:17) + Q kr (cid:19) × r − m − Y k =0 S b (cid:18) yr + mr b − + Q kr (cid:19) . (28)Compare them with the Υ ( r ) m ( y ) functions defined in [4] for the purpose ofcalculation of three-point functions in the parafermionic Liouville field theory:Υ ( r ) m ( y ) = r − m Y j =1 Υ b (cid:18) y + mb − + ( j − Qr (cid:19) r Y j = r − m +1 Υ b (cid:18) y + ( m − r ) b − + ( j − Qr (cid:19) . (29)Let us replace Υ b by S b in expression (29). Then the substitution j = k + 1in its first product yields precisely the second product in (28). Similarly, thesubstitution j = k + r − m + 1 converts its second product to the first one in (28)because b − − Q = − b . So, we have intriguing exact structural correspondencebetween the functions (29) and (28).For this reason we call Λ( y ; m ; ω , ω ) the parafermionic hyperbolic gammafunction. It should play the same role in the construction of parafermionic fusionmatrices as Υ ( r ) m ( y ) serves the correlation functions. Applying the limit (17) toexpression (2) one can derive another expression for itΛ( y, m ; ω , ω ) = γ (2) (cid:18) y + mω r ; ω , ω + ω r (cid:19) γ (2) (cid:18) y + ( r − m ) ω r ; ω , ω + ω r (cid:19) , (30)which was obtained in [13, 16, 20]. Using equations (15) one can easily show that(30) satisfies (22). Now we apply the limit (17) to the rarefied elliptic beta integral evaluation (12).For that we set additionally to (16) the parameterization: t a = e − πvsar , X a =1 s a = ω + ω (31)9nd take the limit v → + . As a result, we obtain the following identity repre-senting a rarefied hyperbolic beta integral evaluation Z i ∞− i ∞ r − X m =0 Q a =1 Λ( y + s a , n a + m + ǫ ; ω , ω )Λ( − y + s a , n a − m ; ω , ω )Λ(2 y, m + ǫ ; ω , ω )Λ( − y, − (2 m + ǫ ); ω , ω ) dyi √ ω ω = 2 r ( − ǫ Y ≤ a
3, i.e. we have only one independent relation.Comparing (50) with the star-triangle relation found in [15], we see thatthey coincide in all aspects besides of the overall sign in the right-hand side( − ( P µ a )(1+ P a ( ν a + µ a )) / present in our formula. We suggest the following inde-pendent check of the presence of this multiplier in a particular case, when it isequal to −
1. Such a situation takes place only when both P µ a and (1+ P a ( ν a + µ a ))2 are odd. The following choice of the parameters obviously satisfies both condi-tions: ν = 0 , ν = 0 , ν = 0 (59)and µ = 1 , µ = 0 , µ = 0 . (60)13ubstituting these values in (50) we obtain: Z dxi [ S ( x + f ) S ( x + f ) S ( x + f ) S ( − x + g ) S ( − x + g ) S ( − x + g ) − S ( x + f ) S ( x + f ) S ( x + f ) S ( − x + g ) S ( − x + g ) S ( − x + g )]= − S ( f + g ) S ( f + g ) S ( f + g ) × S ( f + g ) S ( f + g ) S ( f + g ) S ( f + g ) S ( f + g ) S ( f + g ) . (61)Let us study this integral directly in the limit f + g → Z dxi Y j =1 S b ( x + f j ) S b ( − x + g j ) = Y j,k =1 S b ( f j + g k ) . (62)Recall that the function S b ( x ) is meromorphic with poles at x = − nb − mb − , andzeros at x = Q + nb + mb − , where n and m are non-negative integers. Aroundzero x = 0 the S b ( x ) function has the behavior:lim x → xS b ( x ) = 12 π . (63)Take the limit f + g → − f and g approach to a point A of imaginary axis ( A ∈ i R ) from different sides. Without loss of generalitywe can assume that − f moves to this point from the left side and g comesfrom the right side. This results in the pinching of the integration contour (theimaginary axis) by two poles. Consider the left-hand side integral as a functionof parameters f i and g i . Let us show that pinching of the contour results in thepole singularity of this function 1 / ( f + g ) and compute its leading asymptotics.For that we deform the integration contour and force it to cross over the point x = − f and pick up the corresponding pole residue determined by the integralover small circle around − f . The integral over deformed contour is finite and thesingularity can arise only from the taken residue. According to (63) the integrandaround the point x = − f ≈ g takes the asymptotic form:14 iπ ( x + f )( − x + g ) S b ( x + f ) S b ( x + f ) S b ( − x + g ) S b ( − x + g ) . (64)Then, by the Cauchy theorem the integral over small circle around this point isequal to 12 π ( f + g ) S b ( − f + f ) S b ( − f + f ) S b ( f + g ) S b ( f + g ) . (65)14n the other hand, we see that the right-hand side expression in (62) indeedhas the pole singularity at f + g → j = k = 1 multiplier. Therest can be seen to yield the same result due to the balancing condition, which inthis limit takes the form f + f + g + g = Q , and relation S b ( x ) S b ( Q − x ) = 1. Thesame situation will take place if we take the limit f + g → f . For instance, we may deformthe integration contour close to a fixed point − f and in the limit g → − f we come inevitably to pinching of the contour which leads to the same singularasymptotics for the integral.Now let us get back to the integral (61). First let us indicate necessaryproperties of the functions S ( x ) and S ( x ). The function S ( x ) has zeros at x = Q + nb + mb − and poles at x = − mb − nb − , where m and n are both non-negativeintegers and m + n is odd. The function S ( x ) has zeros at x = Q + nb + mb − and poles at x = − mb − nb − , where m and n are both non-negative integers and m + n is even. The function S ( x ) near zero has the behavior:lim x → xS ( x ) = 1 π . (66)Also we have S ( x ) S ( Q − x ) = 1 , S ( x ) S ( Q − x ) = 1 . (67)In the same limit f + g → S ( x ) functions in (61) do not pinchthe contour ( S (0) is regular) and the contribution from the first term in theintegrand remains finite. The pole singularity is produced only by the secondterm in the integrand. Using (66) one can see that around the point x = − f theintegrand asymptotically takes the form − iπ ( x + f )( − x + g ) S ( x + f ) S ( x + f ) S ( − x + g ) S ( − x + g ) . (68)Again, by the Cauchy theorem the integral over the small circle around x = − f is equal to − π ( f + g ) S ( − f + f ) S ( − f + f ) S ( f + g ) S ( f + g ) . (69)It is easy to see that, due to the balancing condition and properties (66), (67),the asymptotics of the right-hand side expression in (61) indeed coincides with(69) with the correct sign. 15 Conclusion
To conclude, in this work we established a link between the superconformal in-dices of 4 d SCFTs on the lens space, the corresponding rarefied elliptic hyper-geometric functions and the parafermionic Liouville model. The parafermionicstar-triangle relation (45) should play a proper role in the consideration of corre-sponding LFT fusion matrices. Following the logic of the present work it wouldbe also interesting to investigate the hyperbolic degeneration of the rarefied el-liptic hypergeometric function V ( r ) constructed in [33] and search for its properparafermionic, or supersymmetric for r = 2 interpretation.One of the relevant topics which we skipped in the present note concernspartition functions of supersymmetric 3 d field theories described by hyperbolicintegrals. Our relations (32) and (45) should describe dualities of certain modelson the manifold S / Z r similar to the r = 1 cases [28]. Indeed, in [16] a number ofsuch dualitites has been investigated, but coincidence of dual partition functionswas established only numerically. It would be interesting to analyze whether thecorresponding conjectural identities are consequences of (32) or hyperbolic limitsof other identities from [33], or they describe somewhat different systems. Acknowledgements.
This work is partially supported by the Russian Sci-ence Foundation (project no. 14-11-00598).
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