Four-point correlation numbers in super Minimal Liouville Gravity in the Ramond sector

F our-p oint co rrelation numb ers in sup er Minimal Liouville Gravit y in the Ramond secto r V. Belavin, J. Ramos Cab ezas, B. Runov Physics Dep artment, Ariel University, A riel 40700, Isr ael. E-mail: vladimirbe@ariel.ac.il, juanjose.ramoscab@msmail.ariel.ac.il, borisru@ariel.ac.il Abstra ct: In this work, we contin ue the in vestigation of correlation n umbers in N = 1 sup e r Minimal Liouville Gra vity (SMLG), with ph ysical fields in the Ramond sector. Build- ing up on our previous construction of physical op erators and the ev aluation of three-p oin t correlation functions in volving Ramond and Neveu-Sc hw arz (NS) insertions, we now turn to the analytic computation of four-point correlation n umbers. This developmen t is motiv ated b y the framework established for the b osonic Minimal Liouville Gravit y and its sup ersym- metric NS analog, where the in tegration ov er moduli space in correlation functions can be p erf ormed explicitly using the higher equations of motion (HEM) in Liouville theory . In particular, if one of the insertions corresp onds to a degenerate field, the four-p oin t ampli- tude can b e expressed in terms of b oundary con tributions obtained from the OPE structure of logarithmic counterparts of ground ring elements. W e aim to adapt and generalize this approac h to the Ramond sector.Our result is a closed-form analytic expression for four-p oin t correlation n um b ers in v olving Ramond fields. Con tents 1 In tro duction 1 2 Sup er MLG 2 2.1 Liouville sector 2 2.2 Matter sector 3 2.3 Ghost sector 4 3 Ph ysical fields 4 3.1 Ph ysical fields in the NS sector 5 3.2 Ph ysical fields in the R sector 5 4 Analytical four-p oin t function 6 4.1 Strategy and required OPE data 6 4.2 Sp ecial structure constants and auxiliary OPEs 7 4.3 O 1 , 3 R a 7 4.4 Con tour-integral reduction and assembled four-p oint formula 9 5 Discussion of other type of correlation n umber 11 6 Conclusion 12 A Conformal data 13 A.1 Sup erconformal algebra 13 A.2 OPE and structure constants 13 1 In tro duction Minimal Liouville Gravit y (MLG) and its N = 1 sup ersymmetric extension, Sup er Minimal Liouville Gra vity (SMLG), are exactly solv able laboratories for tw o-dimensional quan tum gra vity coupled to conformal matter. In b osonic MLG and in the NS sector of SMLG, analytic four-p oin t correlation n umbers were obtained using the Higher Equations of Motion (HEM) of Liouville theory [ 1 – 3 ]. F or degenerate insertions, these relations reduce the mo duli in tegral to b oundary terms fixed by OPEs of logarithmic ground-ring op erators with the remaining ph ysical fields. The analogous Ramond-sector analysis is still incomplete. Although it has long b een conjectured that the Ramond sector is captured b y the same discrete matrix-mo del frame- w ork [ 4 ], explicit mo duli-in tegrated correlators with Ramond insertions ha ve not been de- riv ed in the con tin uous approach. In our previous w ork, we solv ed the three-p oint problem – 1 – b y constructing Ramond ph ysical op erators R a and ev aluating ⟨ R a 1 R a 2 W a 3 ⟩ , where W a is an NS physical field. Here w e pro ceed to the next step and compute four-p oint correlation n um b ers with Ramond insertions, extending the metho d of [ 2 , 3 ]. Our primary setup is the case where the in tegrated insertion b elongs to the NS sector: Z d 2 z ⟨ G − 1 / 2 G − 1 / 2 U a 1 ( z , z ) R a 2 ( z 2 , z 2 ) R a 3 ( z 3 , z 3 ) W a 4 ( z 4 , z 4 ) ⟩ , (1.1) where R a = U R a ccσ σ is the BRST-inv ariant Ramond ph ysical field, while W a and f W a denote the tw o NS physical op erators. W e deriv e an explicit expression for ( 1.1 ) b elo w. W e also comment on the complementary configuration in which the in tegrated insertion is Ramond; that case con tains an imp ortant subtlet y that is not y et fully resolv ed. In both configurations, once one insertion is degenerate, the in tegrand can b e written as a total deriv ativ e mo dulo BRST-exact terms, so the mo duli integral is reduced to b oundary data. The pap er is organized as follo ws. In section 2 w e summarize the SMLG setup and notation. Section 3 presen ts the NS and Ramond physical op erators, together with ghost- n umber constraints and ground-ring ingredien ts. W e then in section 4 deriv e the required OPE data and ev aluate the corresp onding four-p oint correlation n umbers. In the final sections 5 and 6 we discuss implications, op en questions and directions. App endix A is added as a complement to section 2 . 2 Sup er MLG In this section we fix notation and summarize standard ingredients of SMLG; additional kno wn formulas are collected in App endix A . The SMLG model [ 5 – 7 ] is a tensor product of sup erconformal matter (SM), sup er Liouville [ 8 – 12 ], and sup er ghost systems [ 13 – 16 ], with the action A SLG = A SM + A SL + A SG , (2.1) eac h of whic h ob eys the symmetry ( A.1 ) with the cen tral charge parameters constrained by b c SM + b c SL + b c SG = 0 . (2.2) Eac h component theory realizes the same N = 1 sup erconformal symmetry , with b oth NS and Ramond sectors. In the contin uous form ulation we use an extension of minimal Liouville gravit y in which the matter part is a generalized minimal mo del with non-rational b c S M and a sp ectrum con- taining non-degenerate primaries. T o matc h with matrix-model observ ables, one ev entually imp oses the usual constraints on cen tral charges and conformal dimensions. 2.1 Liouville sector The Liouville central charge is parameterized by a coupling b : b c L = 1 + 2 Q 2 , Q = b − 1 + b. (2.3) – 2 – Primary fields of sup er Liouville theory are exp onential op erators. F or NS b osonic fields V a and Ramond fields R ϵ a : V a ( z ) = e aφ ( z ) , R ϵ a ( z ) = Σ ϵ e aφ ( z ) , (2.4) where Σ ϵ denotes the twist field. The conformal weigh ts of V a and R ± a are ∆ L,N S ( a ) = a ( Q − a ) 2 , ∆ L,R ( a ) = 1 16 + a ( Q − a ) 2 . (2.5) Except at ∆ = b c/ 16 , the Ramond highest-weigh t mo dules are doubly degenerate. The tw o lev el-zero states are exchanged by the fermionic zero mo de G 0 : G L 0 | R σ a ⟩ = iβ L a e − iπσ 4 | R − σ a ⟩ , β L a = 1 √ 2  Q 2 − a  . (2.6) Liouville structure constants used later are summarized in App endix A . 2.2 Matter sector In SMLG, conformal matter is described b y a sup ersymmetric generalized minimal model (GMM). Unlike rational minimal mo dels, its cen tral charge is allow ed to b e non-rational. NS primaries are denoted by Φ α (with con tinuous parameter α ), and Ramond primaries b y Θ ± α . The GMM can b e related to sup er Liouville theory through the analytic cont in uation b → ib , a → − ia , (2.7) with a different normalization chosen so that the tw o-p oint functions are unity: ⟨ Ψ a Ψ a ⟩ = G M N S ( α ) = G M R ( α ) = 1 . (2.8) Here Ψ a stands for a matter primary . Degenerate primaries are indexed by integer pairs ( m, n ) : Φ n,m = Φ α n,m , Θ ± n,m = Θ ± α n,m , α n,m = q 2 − λ − m,n , (2.9) where m − n is even for NS fields and o dd for Ramond fields. Condition ( 2.2 ) fixes the matter cen tral c harge in terms of b : b c M = 1 − 2 q 2 , q = b − 1 − b. (2.10) The corresp onding NS and Ramond conformal dimensions are ∆ M ,N S ( α ) = α ( α − q ) 2 , ∆ M ,R ( α ) = 1 16 + α ( α − q ) 2 . (2.11) The normalization of Θ − α is chosen so that the matter co efficient β M α en tering the action of G M 0 is tied to its Liouville analog β L : G M 0 | Θ σ α ⟩ = iβ M α e − iπσ 4 | Θ − σ α ⟩ , β M a − b = − iβ L a . (2.12) The sp ecial matter structure constan ts follow from the sp ecial Liouville ones b y applying ( 2.7 ) together with the normalization ( 2.8 ); explicit expressions are giv en b elow. – 3 – 2.3 Ghost sector The sup er-ghost theory A SG is a free sup erconformal system with cen tral charge b c g h = − 10 . It con tains anticomm uting fermionic ghosts ( b, c ) with spins (2 , − 1) and b osonic ghosts ( β , γ ) with spins (3 / 2 , − 1 / 2) . The field δ ( γ (0)) (of dimension 1 / 2 ) en ters in NS physical fields and amplitudes [ 2 ] discussed b elo w. The fundamental OPEs of basic fields are b ( z ) c (0) = 1 z , γ ( z ) β (0) = 1 z . (2.13) The mo des of the fermionic ghosts b n , c n are lab eled b y in tegers. The b osonic mo des β k , γ k are half-in teger mo ded in the NS sector and in teger moded in the Ramond sector. The corresp onding sup er-Virasoro generators are L g m = X n ( m + n ) : b m − n c n : + X k ( m 2 + k ) : β m − k γ k : + a g h δ m, 0 , (2.14) G g k = − X n h ( k + n 2 ) : β k − n c n : +2 b n γ k − n i , (2.15) where a g h = − 1 / 2 in the NS sector and a g h = − 5 / 8 in the R sector. In the Ramond sector w e also use the v acuum state | v , − 1 / 2 ⟩ in picture q = − 1 / 2 , with corresp onding field σ ( z ) of conformal dimension 3 / 8 . In the b osonized representation σ ∼ e − ϕ/ 2 . W e will also use the companion field σ 2 ∼ e ϕ/ 2 , with conformal dimension − 5 / 8 . Ghost counting rules References [ 2 , 17 ] show that nontrivial correlators m ust satisfy the follo wing num b er of fields rules N c − N b = 3 , (2.16) N δ ( γ ) − N δ ( β ) + N β − N γ = 2 . (2.17) Including σ and σ 2 insertions in the Ramond sector mo difies the second condition to N δ ( γ ) − N δ ( β ) + N β − N γ + N σ / 2 − N σ 2 / 2 = 2 . (2.18) If only δ ( γ ) , σ , and σ 2 app ear in the β γ sector, this reduces to N δ ( γ ) + N σ / 2 − N σ 2 / 2 = 2 . (2.19) 3 Ph ysical fields Ph ysical fields in SMLG are selected by BRST closere with resp ct to the charge Q 1 together with v anishing total conformal dimension. These requiremen ts can b e summarized as Q| Ψ ⟩ = 0 , | Ψ ⟩  = Q [ ... ] , (3.1) L 0 | Ψ ⟩ = 0 . (3.2) 1 The BRST c harge Q in ( 3.1 ) can be written as Q = X m :  L M+L m + 1 2 L g m  c − m : + X r :  G M+L r + 1 2 G g r  γ − r : + a gh 2 c 0 . – 4 – 3.1 Ph ysical fields in the NS sector In the NS sector there are t wo basic classes of ph ysical fields [ 2 ]: W a ( z , z ) = U a ( z , z ) · c ( z ) c ( z ) · δ ( γ ( z )) δ ( γ ( z )) , (3.3) and f W a ( z , z ) =  G M+L − 1 / 2 + 1 2 G g − 1 / 2  G M+L − 1 / 2 + 1 2 G g − 1 / 2  U a ( z , z ) · c ( z ) c ( z ) , (3.4) where U a ( z , z ) = Φ a − b ( z , z ) V a ( z , z ) . (3.5) The momen tum parameter a is generic. Besides these con tinuous op erators, the NS sp ectrum con tains “discrete states” built from ground-ring fields: O m,n ( z , z ) = H m,n H m,n Φ m,n ( z , z ) V m,n ( z , z ) . (3.6) The op erators H m,n are p olynomials in super-Virasoro generators, defined uniquely up to Q -exact terms. F or analytic computations, tw o key identities are QQ O ′ m,n = B m,n f W m, − n , (3.7) and G − 1 / 2 G − 1 / 2 U m, − n = B − 1 m,n ∂ ∂ O ′ m,n mo d Q , (3.8) where the logarithmic counterparts of the discrete states O m,n , O ′ m,n = H m,n H m,n Φ m,n V ′ m,n , (3.9) and B m,n are the co efficien ts arising from the higher equations of motion ( A.14 ) of SLFT [ 18 ]. In what follows we fo cus on the illustrative case O 13 : O 13 ( z ) = Φ ′ 13 ( z ) V 13 ( z ) − Φ 13 ( z ) V ′ 13 ( z ) − Ψ 13 ( z ) Λ 13 ( z )+ +  b 2 : β ( z ) γ ( z ) : +2 b 2 : b ( z ) c ( z ) :  Φ 13 ( z ) V 13 ( z ) − b 2 β ( z ) c ( z ) Ψ 13 ( z ) V 13 ( z ) − b 2 β ( z ) c ( z ) Φ 13 ( z ) Λ 13 ( z ) , (3.10) where Λ 13 = G L − 1 / 2 V 13 and Ψ 13 = G M − 1 / 2 Φ 13 . 3.2 Ph ysical fields in the R sector Bac kground on the Ramond sector of SMLG can b e found in [ 7 , 17 ]. Ph ysical states | Ψ ⟩ satisfy the cohomological constraints ab ov e, and in the Ramond sector one also imp oses β 0 | Ψ ⟩ = G 0 | Ψ ⟩ = 0 . The ph ysical field in the Ramond sector is therefore giv en by R a = U R a ccσ σ, (3.11) – 5 – where U R a = Θ − a − b R + a + i Θ + a − b R − a . (3.12) Ramond “discrete states” are constructed as O m,n = H m,n H m,n Y R m,n σ σ , Y R m,n =  Θ − m,n R + m,n + i Θ + m,n R − m,n  , (3.13) where H mn has dimension mn 2 − 1 and ghost num b ers N c = 0 , N σ = 0 , and is built from matter/Liouville sup er-Virasoro generators and ghosts. F or the simplest Ramond ground- ring elemen t one finds [ 17 ] H 1 , 2 = 1 2 + b 2 1 − 2 b 2 G M 0 β − 1 c 1 − b 2 1 + 2 b 2 G L 0 β − 1 c 1 + 4 b 2 (1 − 2 b 2 )(1 + 2 b 2 ) G L 0 G M 0 . (3.14) One can further show that physical fields satisfy QQ O ′ m,n = B m,n R m, − n . (3.15) 4 Analytical four-p oin t function The goal of this section is to derive analytically the four-p oint correlation n umber introduced in ( 1.1 ). F ollowing the same logic as in the NS case, the mo duli integral can b e reduced to b oundary terms, and these b oundary terms are fixed b y OPEs of the logarithmic ground- ring op erator with the physical fields insertions. 4.1 Strategy and required OPE data Expression ( 1.1 ) requires the OPEs of O 1 , 3 with all physical fields en tering the correlator. W e organize the required OPE data as follows: O 1 , 3 W a = X η = − 1 , 0 , 1 A N S a + η b W a + η b , O 1 , 3 f W a = X η = − 1 , 0 , 1 e A N S a + η b f W a + η b , O 1 , 3 R a = X η = − 1 , 0 , 1 A R a + η b ( a ) R a + η b . (4.1) The NS OPE data A N S a + η b , e A N S a + η b ha ve b een computed b efore [ 2 ]. Here w e only need their coun terparts for the logarithmic op erator O ′ 1 , 3 . The genuinely new ingredient is the OPE O 1 , 3 R a , whic h con trols the Ramond contribution to the boundary terms. The exp ected ph ysical structure is diagonal in the BRST cohomology: O 1 , 3 R a = X η =1 , − 1 , 0 A R a + η b R a + η b . (4.2) W e now present the explicit computation. – 6 – 4.2 Sp ecial structure constan ts and auxiliary OPEs Here w e compute the sp ecial structure constants with Ramond fields in the intermediate c hannel. W e extract them from the general expression ( A.5 ) by taking the appropriate degenerate limits, and we use the resulting constan ts in the next subsection to determine the OPE co efficients in ( 4.2 ). 4.3 O 1 , 3 R a Here w e compute the OPE O 1 , 3 R a . W e tak e O 1 , 3 from ( 3.10 ) and aim to reproduce ( 4.2 ). T o compute the OPE co efficien ts A a + η b , we need the sp ecial structure constants ( A.5 ) at the degenerate v alue a 3 = a 1 , 3 = − b . This can b e done by substituting in ( A.5 ) a 1 = a, a 2 = Q − ( a + η b ) , a 3 = − b + x, (4.3) and computing the limiting expression x → 0 (see app endix B of [ 19 ] for a more detailed explanation). The result we obtain is as follows: C [ a + b ] ,L,ϵ ( − b )[ a ] = ϵ  2 π 2 µ 2 γ  1 2  b 2 + 1  2 γ  b  a − b 2  γ  2 ab − 1 2 b 2   γ  b  a + b 2  γ  a + b − 1 2 b b  , (4.4) C [ a − b ] ,L,ϵ ( − b )[ a ] 2 = ϵ, (4.5) C [ a ] ,L,ϵ ( − b )[ a ] = 2 π b 2 µγ  b ( a − b ) − 1 2  γ ( − b 2 ) γ  ab + 1 2  . (4.6) W e will also need the corresp onding matter sp ecial structure constants. F or this w e p erform the transformations a → − ia, b → ib , divide the Liouville structure constants b y √ G R and √ G N S , and introduce the prop er normalization. As an illustration, from ( 4.4 )–( 4.6 ) w e obtain the relation C [ a + η b ] ,M ,ϵ ( b )[ a ] = C [ a − η b ] ,L,ϵ ( − b )[ a ] f 0 ( b )( G N S ( − b ) G R ( a ) G R ( Q − ( a − η b ))) 1 / 2       a →− ia b → ib , (4.7) where f 0 ( b ) is chosen according to the normalization ( 2.8 ), that is C (0) ,M ,ϵ [ b/ 2][ b/ 2] = 1 . Notice also that in the denominator the argument of G R is shifted by Q − ( a − η b ) , since it corresp onds to the upp er index. W e obtain the results: C [ a + b ] ,M ,ϵ ( b )[ a ] = − ϵ  γ  − 3 2  b 2 − 1  γ  1 2  b 2 + 1  γ  a b − 1 2 b 2 + 1  γ  1 − 2 b ( a + b ) 2 b 2  1 / 2 b 2  γ  1 2 b (2 a + b )  γ  − ab − 3 b 2 2 + 1  1 / 2 , (4.8) C [ a − b ] ,M ,ϵ ( b )[ a ] = − ϵ  γ  − 3 2  b 2 − 1  γ  1 2  b 2 + 1  γ  a b − 1 2 b 2 + 1  1 / 2 b 2 γ  ab − b 2 2   γ  − ab + b 2 2 + 1  γ  1 − 2 ab 2 b 2 + 1   1 / 2 × × s γ  1 2 b (2 a + b )  γ  1 − 2 ab 2 b 2  , (4.9) – 7 – C [ a ] ,M ,ϵ ( b )[ a ] = γ  1 2  b 2 + 1  γ  1 2 − ab  γ  b ( a + b ) − 1 2  γ ( b 2 )  γ  3 2 − 1 2 b 2  γ  1 2  1 b 2 − 1  γ  1 2 − b 2 2  γ  3 b 2 2 − 1 2  1 / 2 . (4.10) W e will also require the following sp ecial structure constan ts: e C [ a + η b ] ,L,ϵ ( b )[ a ] , e C [ a + η b ] ,M ,ϵ ( b )[ a ] , d [ a + η b ] ,L,ϵ ( − b )[ a ] , d [ a + η b ] ,M ,ϵ ( b )[ a ] . (4.11) These are computed by substituting the sp ecial v alues ( 4.3 ) in the definitions ( A.6 ) and using the expressions ( 4.4 - 4.6 ) and ( 4.8 - 4.10 ). With these data at hand, one can compute the OPE co efficients A R a + η b . The terms that pro vide nontrivial contributions are: O 13 ( z ) = ( b O 1 + b O 2 )( b O 1 + b O 2 )Φ 13 ( z ) V 1 , 3 ( z ) + ... , (4.12) b O 1 = L M − 1 − L L − 1 + 2 b 2 b − 2 c 1 + b 2 β − 1 γ 1 , b O 2 = − G M − 1 / 2 G L − 1 / 2 . (4.13) Computing the OPE O 1 , 3 R a , w e obtain the co efficients A R a + η b in the form: A R a + η b = ( c 1 − c 2 + 3 2 b 2 ) 2 C [ a − b + η b ] ,M , − ( b )[ a − b ] C [ a + η b ] ,L, + ( b )[ a ] − e C [ a − b + η b ] ,M , − ( b )[ a − b ] e C [ a + η b ] ,L, + ( b )[ a ] − i ( c 1 − c 2 + 3 2 b 2 )( d [ a − b + η b ] ,M , + ( b )[ a − b ] d [ a + η b ] ,L, − ( − b )[ a ] + d [ a − b + η b ] ,M , + ( b )[ a − b ] d [ a + η b ] ,L, − ( − b )[ a ] ) , (4.14) where the co efficients c 1 , c 2 come from deriv ativ es with resp ect to the coordinates: c 1 = ∆ M ,R ( a − b + η b ) − ∆ M ,R ( a − b ) − ∆ M ,N S ( b ) , c 2 = ∆ L,R ( a + η b ) − ∆ L,R ( a ) − ∆ L,N S ( − b ) . (4.15) Computing explicitly ( 4.14 ), we arrive at the result: A R a + η b = K ( b ) B 1 , 3 N N S (2 b ) N R ( a ) N R ( a + η b ) , (4.16) where the function K ( b ) 2 and the leg factors N N S and N R are determined as follows: K ( b ) = 1 b  γ  1 2  1 b 2 − 1  γ  1 2  b 2 + 1   1 / 2 , (4.17) N R ( a ) = s γ  a b − 1 2 b 2  γ  ab − b 2 2   π µγ  1 2  b 2 + 1   − a b , (4.18) N N S ( a ) =  π µγ  1 2 + b 2 2  − a/b  γ  ab − b 2 2 + 1 2  γ  a b − b − 2 2 + 1 2  1 / 2 . (4.19) It is w orth noting that the form of the result ( 4.16 ) is particularly conv enien t in the compu- tation of the four-p oint correlation num b ers. One can also verify that the remaining terms in the OPE ( 4.2 ) which do not contribute to the Ramond ph ysical fields v anish. W e now assemble these data to compute the four-p oin t correlation n umber. 2 Our K ( b ) differs from that in [ 2 ] by a factor of 1 / 2 , due to a different normalization of the structure constan ts. – 8 – 4.4 Con tour-integral reduction and assem bled four-p oin t formula Belo w we pro ceed with the computation of ( 1.1 ). W e follow the procedure explained in [ 1 , 2 ]. F or a 1 = a 1 , − 3 , b y using relation ( 3.8 ) and Stokes’ theorem, the moduli integral of ( 1.1 ) con v erts in to a contour integral: ⟨ ⟨ a 1 , − 3 a 2 a 3 a 4 ⟩ ⟩ SLG = 1 B 1 , 3 Z ∂ Γ ∂ z  O ′ 1 , 3 ( z ) R a 2 ( z 2 ) R a 3 ( z 3 ) W a 4 ( z 4 )  dz 2 i . (4.20) Here ∂ Γ = ∂ Γ 2 + ∂ Γ 3 + ∂ Γ 4 + ∂ Γ ∞ , with ∂ Γ i around z i ( i = 2 , 3 , 4 ) clo ckwise and ∂ Γ ∞ coun terclo c kwise. As usual, BRST-exact terms do not contribute. The finite-circle terms are controlled by the logarithmic OPE contributions: O ′ 1 , 3 ( z ) W a ( z 4 ) N N S ( a ) = log | z − z 4 | 2 K ( b ) B 1 , 3 N N S ( a 1 , − 3 ) X r,s ∈ (1 , 3) q (1 , 3) r,s ( a ) W a + λ r,s ( z 4 ) N N S ( a + λ r,s ) . (4.21) Analogously , by using the result ( 4.2 , 4.16 ), one can deriv e a similar expression for the Ramond sector, namely: O ′ 1 , 3 ( z ) R a ( z i ) N R ( a ) = log | z − z i | 2 K ( b ) B 1 , 3 N N S ( a 1 , − 3 ) X r,s ∈ (1 , 3) q (1 , 3) r,s ( a ) R a + λ r,s ( z i ) N R ( a + λ r,s ) , i = 2 , 3 , (4.22) where the co efficients q (1 , 3) r,s ( a ) are given by q (1 , 3) r,s ( a ) =     a − λ r,s − Q 2     Re − λ 1 , 3 , (1 , 3) = { (0 , − 2) , (0 , 0) , (0 , 2) } , (4.23) with | x | Re = ( x, Re x > 0 , − x, Re x < 0 . (4.24) Applying the residue identit y ( 4.25 ), one computes the finite contour integrals in ( 4.20 ). 1 2 i I ∂ Γ i ∂ z log | z − z i | 2 dz = − π , i = 2 , 3 , 4 . (4.25) The contribution from the infinit y con tour (curv ature con tribution) app ears b ecause O ′ 1 , 3 is not a scalar under transformation z → y : O ′ 1 , 3 ( y ) = O ′ 1 , 3 ( z ) − 2∆ ′ 1 , 3 O 1 , 3 ( z ) log     dy dz     , (4.26) ∆ ′ 1 , 3 = d da ∆ ( L ) a     a = a 1 , 3 = λ 1 , 3 . (4.27) Therefore, as z → ∞ ,  O ′ 1 , 3 ( z ) R a 2 ( z 2 ) R a 3 ( z 3 ) W a 4 ( z 4 )  ∼ − 2∆ ′ 1 , 3 log( z z ) ⟨ O 1 , 3 R a 2 R a 3 W a 4 ⟩ , (4.28) and 1 2 i I ∂ Γ ∞ ∂ z  O ′ 1 , 3 ( z ) · · ·  dz = − 2 π λ 1 , 3 ⟨ O 1 , 3 · · · ⟩ . (4.29) – 9 – Summing the finite b oundary terms and the curv ature contribution, we obtain ⟨ ⟨ a 1 , − 3 a 2 a 3 a 4 ⟩ ⟩ = π K ( b ) N N S ( a 1 , − 3 )    4 X i =2 X r,s ∈ (1 , 3) q (1 , 3) r,s ( a i ) + 6 λ 1 , 3    × ⟨ ⟨ a 2 a 3 a 4 ⟩ ⟩ ( R ) SLG , (4.30) where ⟨ ⟨ a 2 a 3 a 4 ⟩ ⟩ ( R ) SLG = ⟨ R a 2 R a 3 W a 4 ⟩ is the three-p oint correlation n umber given by [ 17 ] ⟨ ⟨ a 2 a 3 a 4 ⟩ ⟩ ( R ) SLG = Ω R ( b ) N R ( a 2 ) N R ( a 3 ) N N S ( a 4 ) , (4.31) where Ω R ( b ) = b − 3 ( π µ ) 1 b 2 +1 q γ  1 2 − 1 2 b 2  γ  1 2  b 2 + 1  1 b 2 +2 γ  3 2 − 1 2 b 2  q γ  1 2 ( b 2 − 1)  . (4.32) Using the result ( 4.32 ), one can rewrite ( 4.30 ) equiv alently as ⟨ ⟨ a 1 , − 3 a 2 a 3 a 4 ⟩ ⟩ = π K ( b )Ω R ( b ) N N S ( a 1 , − 3 ) N R ( a 2 ) N R ( a 3 ) N N S ( a 4 ) ×    4 X i =2 X r,s ∈ (1 , 3) q (1 , 3) r,s ( a i ) + 6 λ 1 , 3    . (4.33) F ormula ( 4.33 ) represen ts our main result and corresp onds to the direct analogue of the pure NS four-point correlation n umber found in [ 2 ]. T w o commen ts are in order. First, one exp ects that for general ( m, n ) and the parameter a m,n , the formula ( 4.33 ) will hold, so that w e hav e ⟨ ⟨ a m, − n a 2 a 3 a 4 ⟩ ⟩ = π K ( b )Ω R ( b ) N N S ( a m, − n ) N R ( a 2 ) N R ( a 3 ) N N S ( a 4 ) ×    4 X i =2 X r,s ∈ ( m,n ) q ( m,n ) r,s ( a i ) + 2 mnλ 1 , 3    . (4.34) where in the sum the fusion set is ( m, n ) = { 1 − m : 2 : m − 1 , 1 − n : 2 : n − 1 } . Second, for the purpose of comparing ( 4.33 , 4.34 ) with results from other approaches (e.g. direct computations and matrix mo del approaches), it is con v enient to rewrite the ab ov e expressions in a standard normalized form, in which one introduces renormalized fields: b R a = R a N R ( a ) , c W a = W a N N S ( a ) , b U a = U a N N S ( a ) . (4.35) Then the normalized four-p oint correlation num b er ( 4.33 ) takes the compact form: 1 Ω R ( b ) Z d 2 z D G − 1 / 2 G − 1 / 2 b U 1 , − 3 ( z ) b R a 2 ( z 2 ) b R a 3 ( z 3 ) c W a 4 ( z 4 ) E = π K ( b )    4 X i =2 X r,s ∈ (1 , 3) q (1 , 3) r,s ( a i ) + 6 λ 1 , 3    . (4.36) – 10 – 5 Discussion of other t yp e of correlation num b er In this section we discuss the complemen tary four-p oint correlator of the form Z d 2 x D U R 1 , 2 ( z , z ) σ ( z ) σ ( z ) R a 2 ( z 2 , z 2 ) W a 3 ( z 3 , z 3 ) f W a 4 ( z 4 , z 4 ) E . (5.1) T o apply the contour-reduction logic used in previous section, all the required OPEs in the c hannels of ( 5.1 ) must preserve the ph ysical short-distance pattern of Eq. ( 4.1 ). Channel O 1 , 2 R a . This OPE w as discussed previously in [ 17 ] and was shown to satisfy the required b ehavior ( 4.1 ). Sp ecial structure constan ts. In addition, the ev aluation of ( 5.1 ) requires the sp ecial structure constan ts en tering the O 1 , 2 analysis. W e collect here the relev ant formulas : C [ a + b/ 2] ,L,ϵ [ − b/ 2]( a ) = 2 π b 2 µγ  1 2  b 2 + 1  γ  ab − b 2 2 − 1 2  γ ( ab ) , (5.2) C [ a − b/ 2] ,L,ϵ [ − b/ 2]( a ) 2 = ϵ, (5.3) C [ a + b/ 2] ,M ,ϵ [ b/ 2]( a ) = ϵ  γ  1 − b 2  γ  1 2  b 2 − 1  γ  − 2 ab + b 2 − 1 2 b 2  γ  2 ab + b 2 − 1 2 b 2  1 / 2  γ  1 2  1 − 1 b 2  γ  1 2  1 b 2 − 1  γ  1 2 (2 ab + b 2 − 1)  γ (1 − b ( a + b ))  1 / 2 , (5.4) C [ a − b/ 2] ,M ,ϵ [ b/ 2]( a ) = − b 2  γ  1 − b 2  γ  1 2  b 2 − 1  γ  − 2 ab + b 2 +1 2 b 2  γ  1 2  2 ab + b 2 − 1   1 / 2 γ ( ab )  γ  1 2  1 − 1 b 2  γ  1 2  1 b 2 − 1  γ (1 − ab )  1 / 2 × ×  γ  2 ab + b 2 − 1 2 b 2  1 / 2 . (5.5) Channel O 1 , 2 f W a . The corresp onding OPE is more cumbersome and depends not only on the structure constan ts ab o v e, but also on the asso ciated e C and d, d structure con- stan ts. Nev ertheless, the general exp ectation for this OPE is according to ( 4.1 ) with the corresp onding co efficients is e A R a ± b/ 2 ha ving the form e A R a ± b/ 2 = e f ± K ( b ) B 1 , 2 N R ( a 1 , − 2 ) N N S ( a ) N R ( a ± b/ 2) . (5.6) – 11 – Organizing the computation term by term, the con tributions naturally split in to (I) = Y ( a 1 , 2 ) σ σ f W a , (I I) =  b 2 1 − 2 b 2 G M 0 − b 2 1 + 2 b 2 G L 0  β 1 c − 1 Y ( a 1 , 2 ) σ σ f W a , (I I I) =  b 2 1 − 2 b 2 G M 0 − b 2 1 + 2 b 2 G L 0  β 1 c − 1 Y ( a 1 , 2 ) σ σ f W a , (IV) =  b 2 1 − 2 b 2 G M 0 − b 2 1 + 2 b 2 G L 0  β 1 c − 1  b 2 1 − 2 b 2 G M 0 − b 2 1 + 2 b 2 G L 0  β 1 c − 1 Y ( a 1 , 2 ) σ σ f W a . (5.7) The resulting intermediate expressions are length y . As a representativ e example, w e only pro vide here the explicit form of the contribution (I): (I) =   e C [ a − b ± b/ 2] ,M , − [ b/ 2]( a − b ) C [ a ± b/ 2] ,L, + [ − b/ 2]( a ) + C [ a − b ± b/ 2] ,M , − [ b/ 2]( a − b ) e C [ a ± b/ 2] ,L, + [ − b/ 2]( a ) + i  d a − b ± b/ 2 ,M , − [ b/ 2]( a − b ) d [ a ± b/ 2] ,L, + [ − b/ 2]( a ) − d [ a − b ± b/ 2] ,M , − [ b/ 2]( a − b ) d [ a ± b/ 2] ,L, + [ − b/ 2]( a )    R a ± b/ 2 . (5.8) A direct computation of ( 5.8 ) already yields the structure ( 5.6 ), but with a co efficien t e f ( I ) ± ( a, b ) rational in a and b . The final co efficient e f ± arises only after summing (I)–(IV); although we do not determine it explicitly here, w e exp ect, b y analogy with the previous case, that the a -dependence cancels in the sum. This, together with the cancellation of unph ysical terms in this c hannel, requires further inv estigation. Channel O 1 , 2 W a . Finally , this channel contains a subtlety in the β γ ghost sector. In the b osonized language σ ∼ e − ϕ/ 2 and δ ( γ ) ∼ e − ϕ , hence σ ( z ) δ ( γ )(0) ∼ e − 3 ϕ/ 2 (0) and is not prop ortional to σ (it shifts the picture from q = − 1 / 2 to q = − 3 / 2 ). On the other hand, the correlator ⟨ σ δ ( γ ) σ ⟩ is non-v anishing. This problem requires a carefully analysis of the picture c hanging in pro duct of physical fields. W e leav e this question for future work. 6 Conclusion In this work w e computed analytically the four-p oint correlation n um b er ( 1.1 ) in sup er minimal Liouville gravit y with Ramond insertions, in the case when the nonlo cal physical field corresp onds to a degenerate insertion with parameter a 1 = a 1 , − 3 . The key new ingre- dien t is the explicit OPE data for the logarithmic ground-ring op erator O ′ 1 , 3 with Ramond ph ysical states. Combining these OPEs with the higher equations of motion reduction and the logarithmic transformation prop ert y ( 3.8 ), w e obtained a closed-form expression for the four-p oin t correlation num b er, summarized in ( 4.34 ). An important c heck of this result w ould b e a direct numerical ev aluation of the mo duli in tegral in ( 1.1 ) and a comparison with the normalized closed-form expression (see ( 4.36 )). W e leav e such a numerical test for future work. It w ould b e very in teresting to repro duce the same four-point num b er from the dual matrix-mo del description. In particular, after passing to the renormalized fields ( 4.35 ), the normalized correlator ( 4.36 ) b ecomes the natural ob ject to compare with matrix-mo del – 12 – predictions in the Ramond sector, extending the pure bosonic analysis [ 1 ] and recen t analysis in the sup ersymmetric case [ 20 ]. Finally , w e also discussed the complementary four-p oint correlator with an in tegrated Ramond insertion, ( 5.1 ). While the contour reduction again requires OPE data compat- ible with the ph ysical pattern ( 4.1 ), the necessary OPEs app ear to b e more subtle. W e presen ted the relev ant structure constants and the exp ected b ehavior of the corresp onding OPE co efficients, and we leav e a complete analytic ev aluation of ( 5.1 ) for future work. A Conformal data A.1 Sup erconformal algebra The symmetry algebra of SMLG is N = 1 sup erconformal algebra, [ L n , L m ] = ( n − m ) L n + m + b c 8 ( n 3 − n ) δ n, − m , { G r , G s } = 2 L r + s + b c 2  r 2 − 1 4  δ r, − s , [ L n , G r ] =  1 2 n − r  G n + r , (A.1) where r , s ∈ Z + 1 2 for the NS sector , r , s ∈ Z for the R sector . A.2 OPE and structure constants The basic structure constan ts in the pure NS sector C L ( a 1 )( a 2 )( a 3 ) and e C L ( a 1 )( a 2 )( a 3 ) w ere computed in [ 21 , 22 ]. The 3-point functions in volving tw o fields from the Ramond sector and the sup erfields from the NS sector can be written as follo ws [ 21 ] D R ϵ 1 α 1 ( z 1 ) R ϵ 2 α 2 ( z 2 )  V α 3 ( z 3 )+ θ G − 1 2 V α 3 ( z 3 ) + θ G − 1 2 V α 3 ( z 3 ) + θ θ G − 1 2 G − 1 2 V α 3 ( z 3 ) E = v 3 pt ( z 1 , z 2 , z 3 )  δ ϵ 1 ,ϵ 2  C L,ϵ 1 [ α 1 ][ α 2 ]( α 3 ) + | z 12 | θ θ | z 13 z 23 | e C L,ϵ 1 [ α 1 ][ α 2 ]( α 3 )  + δ ϵ 1 , − ϵ 2   z 1 2 12 θ ( z 13 z 23 ) 1 2 d L,ϵ 1 [ α 1 ][ α 2 ]( α 3 ) + z 1 2 12 θ ( z 13 z 23 ) 1 2 d L,ϵ 1 [ α 1 ][ α 2 ]( α 3 )     (A.2) with v 3 pt ( z 1 , z 2 , z 3 ) = | z 12 | − 2 γ 123 | z 13 | − 2 γ 132 | z 23 | − 2 γ 231 , (A.3) γ ij k = ∆( α i ) + ∆( α j ) − ∆( α k ) , z ij = z i − z j . (A.4) – 13 – With all the structure constants ( C , e C , d , d ) are found to b e giv en by [ 21 ] C L,ϵ [ a 1 ][ a 2 ]( a 3 ) = 1 2  π µγ  Qb 2  b 1 − b 2  Q − a b ×  Υ R ( b )Υ R (2 a 1 ) Υ R (2 a 2 ) Υ NS (2 a 3 ) Υ R ( a − Q )Υ R ( a 1 + a 2 − a 3 ) Υ NS ( a 2 + a 3 − a 1 ) Υ NS ( a 3 + a 1 − a 2 ) + ϵ Υ R ( b )Υ R (2 a 1 ) Υ R (2 a 2 ) Υ NS (2 a 3 ) Υ NS ( a − Q )Υ NS ( a 1 + a 2 − a 3 ) Υ R ( a 2 + a 3 − a 1 ) Υ R ( a 3 + a 1 − a 2 )  . (A.5) e C L,ϵ [ α 1 ] , [ α 2 ] , ( α 3 ) = i ϵ h  β 2 1 + β 2 2  C L,ϵ [ α 1 ] , [ α 2 ] , ( α 3 ) − 2 β 1 β 2 C L, − ϵ [ α 1 ] , [ α 2 ] , ( α 3 ) i , (A.6) d L,ϵ [ α 1 ] , [ α 2 ] , ( α 3 ) = i e − iπ ϵ/ 4 h β 2 C L,ϵ [ α 1 ] , [ α 2 ] , ( α 3 ) − β 1 C L, − ϵ [ α 1 ] , [ α 2 ] , ( α 3 ) i , (A.7) d L,ϵ [ α 1 ] , [ α 2 ] , ( α 3 ) = − i e iπ ϵ/ 4 h β 2 C L,ϵ [ α 1 ] , [ α 2 ] , ( α 3 ) − β 1 C L, − ϵ [ α 1 ] , [ α 2 ] , ( α 3 ) i . (A.8) The expression for Υ NS , Υ R can b e read from [ 17 , 21 ]. The normalization used in ( A.5 ) is suc h that the reflection coefficients (the Liouville tw o-p oint functions) are given by C L,ϵ [ a ][ a 2 ]( a 3 ) C L,ϵ [ Q − a ][ a 2 ]( a 3 ) = G R ( a ) = γ  ab − bQ 2 + 1 2   π µγ  1 2  b 2 + 1  Q − 2 a b γ  − a b + Q 2 b + 1 2  , (A.9) C ϵ L, [ a 1 ][ a 2 ]( a ) C ϵ L, [ a 1 ][ a 2 ]( Q − a ) = G N S ( a ) = b 2 γ  ab − bQ 2   π µγ  1 2  b 2 + 1  Q − 2 a b γ  Q 2 b − a b  , (A.10) here w e use the notation γ ( x ) = Γ( x ) Γ(1 − x ) , (A.11) where Γ( x ) is the gamma function. Degenerate fields In our discussion an imp ortan t role is pla yed b y the the degenerate fields V m,n and R ± m,n whic h one can define as follows V m,n = V a    a = a m,n , R ± m,n = R ± a    a = a m,n , a m,n = Q 2 − b − 1 m + bn 2 , (A.12) where m − n is is ev en or o dd dep ending on the sector of the fields, as discussed ab ov e. 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