JHEP07(2019)026 , 1 2 47 Co · = 2 c ' Springer 0 July 4, 2019 May 3, 2019 : June 26, 2019 : : January 18, 2019 : Revised , the triple covering Published B Accepted · Received [email protected] Published for SISSA by , https://doi.org/10.1007/JHEP07(2019)026 = 12 has the Conway group Co c and multiple-covering of the second largest Conway 0 24 [email protected] Fi , · . 3 . Various twined characters are shown to satisfy generalized bilinear 2 1811.12263 Co · The Authors. = 23 ‘dual’ to the critical Ising model, the three state Potts model and c c Conformal and W Symmetry, Field Theories in Lower Dimensions

We investigate the two-dimensional conformal field theories (CFTs) of , 1+22 [email protected] 2 and · 5 116 = Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-Gu, Seoul 02455, Korea E-mail: Open Access Article funded by SCOAP ArXiv ePrint: relations involving Mckay-Thompson series.dimensional bosonic We conformal also field rediscover theoryas that of an the automorphism group. ‘self-dual’ two- Keywords: c the tensor product ofmoonshines two for Ising the models, doubleof respectively. covering the of We largest the argue Fischer baby thatgroup, group, Monster these 2 3 group, CFTs 2 exhibit Abstract: Jin-Beom Bae, Kimyeong Lee and Sungjay Lee Monster anatomy JHEP07(2019)026 ]. 6 9 ), 18 18 11 16 11 πiτ – (1.2) (1.1) 2 ) of a 3 e τ ( and the i = f q M 744 ( − ) τ ( j , ··· ] to further explore a new + 12 2 q , 3 0 24 15 744 Fi 2 · − ) Co τ ( · ] provided a (heuristic) derviation of the + 21493760 j 2 q = 24 chiral CFT, obey an intriguing bilinear relation giving a c ) = 1+22 c 0 24 τ – 1 – 2 ( i · Fi ˜ f · ) τ 18 + 196884 ( i q 1 f 1 and 2 1 -invariant, motivated the study of the so-called ‘Monstrous − =0 and 3 2 i j X n Co B · 744 = · − orbifold. Many examples of the generalizations of moonshine ) 2 τ ( Z ), j τ ( j 1 ]. Each Fourier coefficient of the modular invariant 13 1 0 24 B Fi · · . Frenkel, Lepowsky and Meurman [ M In this article, we utilize a holomorphic bilinear relation [ C.1 2 C.2 3 certain rational CFT withmodular central invariant charge group Monstrous moonshine from an explicitlattice construction followed of by the chiral a phenomena CFT with based different on sporadic the groups Leech have been uncovered inclass the of last decades moonshine [ phenomena. It has been observed recently that characters which can describe the partition function of can be decomposed into the dimension of the irreducible representation of the monster 1 Introduction Mckay and Thompson’s remarkablemodular observation objects, especially, between the theMoonshine’ monster in group [ C Generalized bilinear relations 6 Discussion A Dimension of the irreducible representations B Character tables of 2 3 Dual of the three-state4 Potts model and Dual 3 of the critical5 Ising Self-dual RCFT and 2 Contents 1 Introduction 2 Dual of the Ising model and the Baby Monster JHEP07(2019)026 2 47 1) ) of and − l ˜ = (1.5) (1.6) (1.7) (1.3) (1.4) 0 24 c ˜ c, Fi = 8 and · , n, c } , . . . , n i of primaries 2 ˜ h , i , { 1 r h , ). Namely, one 1.2 = 0 , respectively. Note ]. Further examples 2 i 13 Co )( · τ ( , i , f , 1+22 4) 6= 0 2 , · i − l 6 and a rational CFT with of primaries, the central charge n ) = 0 1 2 − , n τ ) ( -expansion of each character in two 3)( = τ q 1) f ( ) implies that a rational CFT can be c − # 2 − k τ = 2 for n 12 1.5 n D i ( ) ˜ h = ( τ n iπrE ( l ˜ 1 6 + ) ) that the conformal weights k = i + − − – 2 –  1.3 n . The MDE can be used to explore the space of τ k c 2( and a number ∂ 24 ) of a rational CFT are related to ( ] φ i ≡ 1 dual to the critical Ising model exhibits Moonshine − , h h 15 , l − =0 i τ k X n 2 h 47 D = 0  , n, c, l + 1 0 } = 24 = i n τ − ˜ =0 h c i h X n c D { = " + ˜ 0 c h of a candidate RCFT have to satisfy the relation below c ), can play a role as candidate characters Z ]. , . When ( l 14 ) can be interpreted as characters of a ‘dual’ rational CFT with central charge denotes the Serre derivative acting on a modular form of weight τ ) are modular forms of weight ( τ is a non-negative integer other than 1. ( τ i ˜ ( ). For instance, the critical Ising model with l f D k c φ We analyze rational CFTs dual to the three-state Potts model and to the product The rational CFT with − and an integer = 16 having no Kac-Moody symmetry but finite group symmetry [ rational CFT is dual to the other. of two Ising models.CFTs Interestingly show novel enough, connections our to the resultsthe triple multiple-covering can covering of of propose the the second that largest largest Fischer the Conway group, group, dual 3 2 rational and the characters of two rational CFTs can obey the bilinear relation ( where characterized by conformal weights c another rational CFT as follows and the central charge and rational CFTs. This is because solutionsresentation to of an SL(2 MDE, which furnish ain finite-dimensional a rep- rational CFT. One can show from ( where for the baby Monsterfor a group, dual second rational largestMonster CFT group. sporadic showing Moonshine group. The for searchrational the first It CFTs sporadic requires is of an groups challenging explicit other dualequation to than (MDE) pair. search the of the baby To form do below [ so, we make use of a modular-invariant differential (24 satisfies the bilinear relation.c Another example iscan be a found pair in of [ rational CFTs of where JHEP07(2019)026 , 1 . can 3 (2.1) (2.2) 1 respectively. 1 8 , in baby Monster 1 and section , B 2 · ]. Each arrow from A ]. Our observations 0 24 16 [ 23 17 Fi , · Fi 2 M 3 . We also observe that the as an automorphism group. , , 1 0 ! ! = 12 [ ) ) ) ) c τ τ τ τ ( ( ( ( 4 4 η η ϑ ϑ s s + − . 1 1 2 , , , 24 ) ) ) 1 2 2 ) ) ) τ τ τ φ φ φ τ τ τ M B ( ( ( ( · ( ( 3) describes the critical point of the second- 3) as follows 2 3 3 M · 2 , , η η η ϑ ↔ ↔ ↔ ϑ ϑ – 3 – described in figure 12 (4 (4  2 1 σ s s s M 2 M M

1 √ 1 2 1 2 whose scaling dimensions are ∆ = 0 in σ 1 16 ) = ) = ) = , τ τ τ 1 2 ( ( (  0 CFT are further supported by generalized bilinear relations , σ f f f 2 1 5 = 0 116 Co Co h · · = and the spin = 12 has the largest Conway group Co c 1+22  c = 1 superconformal extremal CFT of 2 in 1+24 · 2 N 2 0 24 of weight Fi is a maximal subgroup of 2 · , Virasoro characters for three primary fields are 2 24 0 1 2 . Partial flows of maximal subgroups from the monster group , φ Fi 1 , · = 2 c , φ 1 , 1 The Assuming three operators haveφ no spin, one can relate them with allowed primary fields 2 Dual of theThe Ising simplest model unitary and minimalorder the model phase Baby transition Monster ofthe energy the density Ising model. The critical Ising model has the identity the well-known suggest that the subgroup decompositionbe along realized the by red the and holomorphicCFT the bilinear blue and relation. arrows 3 The in moonshines figure involving for Mckay-Thompson 2 series. The details will be shown in section to B implies that B is a maximal subgroupthat of A. 3 self-dual theory of In fact, we can show that the self-dual theory can be identified as the GSO projection of Figure 1 JHEP07(2019)026 and (2.7) (2.8) (2.9) (2.3) (2.4) (2.5) (2.6) in the \ (1) 3 2 B , 31 16 ,V . , . \ (0)
B , = 0 as a finite group ··· ··· V
, h B ) + + , τ ··· 2 3 , ) ( q q . . σ τ
˜ + ˜ f, (¯ f 3        σ · ¯ ˜ ) ) ) q ) ) ) ) ··· f ) ) τ τ τ τ τ τ τ τ ( ( ( ( ( ( ( τ + ( 6   0 0 ( σ σ ˜ ˜ ˜ σ 4 f f f f f f σ E ). In particular, the modular ˜ f q f 3 Z       , π + 21493760 ) +       ) + q 2 2 τ + 9685952512 τ 2 2 ( 6912 (¯  2 √ √  ˜ √ √ ¯ ˜ f q 27025 f − − + 1829005611 · i ) 2 ) τ 2 0 2 0 q ( − τ  √ √ ( + 366845011 ˜ τ f  has baby Monster group ), the modular S-matrix of the dual RCFT 3 − − f D q Z ) 2 2 2 , 47 τ ) + 11 1 1 11 1 1 – 4 – ( ]. ) + √ √ τ 4 (¯ = τ + 744 + 196884 13 0       ( E + 420831232 ¯ ˜ c f 0 2 q 1 ˜ 1 2 1 2 q ) ] that the characters of the critical Ising model obey f + 64680601 π as follow. τ · ( q = = = 12 q ) 0 + 9646891 576 ˜ ) τ f 2 (       τ 2315 0 ) ) q ) ) ) ) ( 0 ) f 2 6 ), the modular invariant torus partition function of the baby τ + ) = /τ /τ /τ /τ /τ /τ ( E ¯ ) are identical to the characters of modules τ 1 1 3 τ 1 1 1 1 3 4 2.8 − − − D − − − − τ, E ( ( ( ( 2.7 ( ( (  3 )   0 0 σ σ 744 = ˜ . ˜ τ 2 ˜ f f 47 f f f f ( 12 2 − 3 47 4 = 1 + 96256       c 0 = ) 4371 + 1143745 96256 + 10602496 are three independent solutions to a third order modular differential E Z τ = (  47 48 ) is invariant under SL(2 ˜ c j f 25 48 23 24 τ − ) = ( q q q τ j ], because ( ( 744 is the Monster module, j and ) can be identified as three characters of conformal weights , often referred to as the baby Monster CFT, obtained from that of the critical 12 ) = ) = ) = − σ τ τ τ 2 ) ˜ 47 f ( ( ( 2.7 τ  0 σ ˜ ( , ˜ ˜ f f f = 0 j ˜ f c It is known that dual RCFT with Since the It has been shown recently in [ which agrees with numerical results in [ symmetry [ In addition, thefusion Verlinde rule formulae algebra. implies From ( Monster that CFT these has to two be RCFTs also share the same with Ising model, namely In fact, ( dual RCFT with and can be expanded in powers of Here equation an intriguing bilinear relation, where S-matrix is These characters transform into one another under SL(2 JHEP07(2019)026 . ], in B · 19 B . As 2 · (2.14) (2.11) (2.12) (2.13) (2.10) B · ∈ g . , , . Explicitly, the ··· , , C )
+ τ ( 4 = 2 , ··· C q 2 g σ ˜ 1139374 + f · 4 . 347643114 ⊕ q )
τ , ⊕ 96256 (
σ ··· ) can be expressed as a sum of f . 4371 ··· + . + 10698752 2.7 2 i + ) + 3 q , + 470099 q c τ 2 24 ( 96256 = 3 q 9550635 − q C h 2  q ⊕ ˜ , f · , e.g. · g B 1143745 = and its descendants. As illustrated in [ ) . In this way, one can check that the twined 410132480 ] to provide further supporting evidence that h · + 565248 τ i i ( ⊕ q h 63532485 + 24  + 1240002 , + 37675 + 144025 H 20 96255 f , – 5 – 2 2 q ⊕ 2048 24 q q , ⊕ 19 ) 9458750 ) can be identified as the Virasoro descendent of . To this end, we introduce the twined character, 2 24 ) + 1 ) q B ⊕ τ ( q ) = Tr · ( = ( η 2.10 τ C 0 η ( 9550635 2 0 12 and the trace is taken over all states in the Hilbert space g i ˜ 10506240 + 96256 f 2 ˜ f 1 + 2048 ⊕ B · 1139374 q 275 + 9153 2048 + 47104 · ) satisfy a bilinear relation of the form ⊕ 96255 ) + 10506240 96256 ⊕ · τ 47 48 ( ⊕ 2 24 25 48 23 24 − 0 24 ) 2.12 ) q q q f , 2 , is replaced by ⊕ q 96255 q + 4372 ( ( 1 4371 96256 η ⊕ q η 1 · · · ) = ) = ) = ) = q q q τ 4371 1 96256 ( ( ( ( = 96256 C C C A ) = 2 2 2 2 0  σ τ ˜ ˜ ˜ ), j f f f ( τ A ( , we present list of the generalized bilinear relations for various 2 σ 4371 = in the second line of ( ˜ j f C ) is Mckay-Thompson series of class 2A, ], respectively. In this paper, we conjecture that the finite group symmetry of . For instance, let us consider the twined characters for τ 1 ( 9646891 = 18 B 10602496 = 64680601 = 2 A is a group element of 2 2 RCFT can be promoted to the double covering of baby Monster group 2 366845011 = 2 420831232 = 2 CFT exhibit moonshine for 2 j in [ g 2 2 47 47 We apply a refined test proposed in [ \ (2) , that consist of primary state of weight i B = = In appendix Combined with the charactersgated constitutes of Mckay-Thompson the series of Isingtwined certain model, characters class. for all Sometimes, different the combination group of twined the elements characters and we characters investi- of the critical Ising model where Intriguingly, we find that ( appendix first term in characters are given by where H it is straightforward to obtain the twined characters using the character table of 2 Note that the vacuum. c c a demonstration, one can showdimensions that of each irreducible coefficient representations in of 2 ( V JHEP07(2019)026 r of r,s ) φ (3.2) (3.3) (3.4) (3.1) r,s ( φ for each 3 r, φ ) are consistent to Virasoro characters ), the left-hand side 4 5 . 2.10 i 2 = ) 2.13 s c +5 r . 0 24 120 2 +6

n Fi 3 , , (5) · (60 r, ) ) , . f ). The q , τ τ

) ) s ( ( 1 τ τ containing ten primary fields − 5 5 ( ( , , − (5) (5) − + 2 3 3 1 2 2 4 5 , , ) 6 f f (5) (5) 2 s 1 2 2 5 )

f f = r, s − 5 r c 5 (5) ) + ) + r, 120 5 are present in the theory. Notice also that 120 − f +6 τ τ , ) = ) = − ( ( n τ τ (5 1 1 + r ( ( , , – 6 – (60 0 0 (5) (5) 2 3 1 2 = 1 1 (6 q f f f f ' (5) s r, h 5) with f ) = Z

, ∈ symmetry under which two copies of 2 ) = ) = ) = ) = , X n (6 r, s r,s τ τ τ τ 3 ]. ( ( ( ( ) h =1 X 2 and 0 1 2 3 q Z M r 1 , 21 ( f f f f ). Note that the decompositions in ( η symmetry plays a key role to have well-defined fusion rules = τ ( = 1 3 ) are given by Z A 6 and ( Z r 2 ) = ) to define the characters of the critical three-state Potts model j q r, s ( 3.3 s < (5) r,s with f ≤ 3 r, 1 φ , 5 r < as an automorphism group. ≤ 0 24 ) still remained as It is natural from ( It is known that the three-state Potts model at the critical point can be described as ) is the partition function of the three-state Potts model, which implies that only 2.13 3.3 transform differently. This of the three-states Potts model [ as follows, ( and two copies of the three-state Potts model has Some but not all ofOne these can primary fields indeed are showrules, present that in which the a leads critical to subset three-state a of Potts non-diagonal model. the modular ten invariant partition primary function fields are closes under the fusion where 1 labelled by two integers ( a “subset” of the minimalconformal model weights We now make use ofnew the RCFTs bilinear related relations to but the withthat other different a pairs sporadic dual CFT of groups. of characters In thegroup to three-state particular, Fi look Potts we for model propose has in the triple this covering section of the largest Fischer characters to 2A(orof 2B) ( characters inbilinear the relations. right-hand side of ( 3 Dual of the three-state Potts model and 3 yields Mckay-Thompson series of identical class. More precisely, even if we replace 2C JHEP07(2019)026 , 29 15 (3.8) (3.9) (3.5) (3.6) (3.7) (3.10) (3.11) and 4 3 . , ) . 8 5 τ (          ) ˜ . , f ) ) ) ) ) ) τ   τ τ τ τ τ τ (          ) ( ( ( ( ( ( 0 = 0, 2 0 0 ) ) ) ) ) ) 0 1 2 2 3 3 ˜ τ f . τ τ τ τ τ τ ( ··· f f f f f f h ) ( ( ( ( ( ( 2 4 , . τ 0 0 2 3 3 0 1 2          + ( ˜ ˜ ˜ ˜ ˜ ˜ ,  E  f f f f f f 0 2 3
4          f q ) = 0          π 2 1 ··· ··· 2 1 , τ s s 2 1 ··· (           2 2 s s i + + ) + 2 1 ˜ ωs ωs 1 2 f ω ω + s s 4 τ 2 1 2 i 2 2 ··· ( 50625 q s s q 3 2 1 ) 2 1 2 ωs ωs -expansion q s s ω ω ˜ 2 1 τ 175769 f q + 3 2 2 ( s s ) 1 2 ωs ωs 2 a 4 2 2 1 ω ω τ s s − 2 1 q ( E 2 2 s s 1 + τ 2 1 3 ωs ωs 2 s ω ω 1 2 f , 2 µ D s 2 2 s ) + 2175548448 ) q ) 1 2 ωs s 1 ω τ 2 2 2 ωs τ s − τ 1 + 2 ω ( − ( s a 2 ( q 2 ) + s 0 − 3 i τ 2 6 ωs ˜ s . ˜ τ ω f f + 201424111 ωs − + 261202536 1 + ) have the ( ) are the solutions to a fourth order differ- ω ) characters of weights ) D 1 E − πi 3 2 s − 1 3 1 ) 2 τ τ q τ 2 q ˜ 3 s 2 2 f ( q s ( ( τ 1 e 1 2 i 3.9 ωs ) π s 0 + 19648602 4 1 ( 3 ω ˜ 2 a ωs s ˜ − f 1 2 τ 6 ω f f ) satisfy a bilinear relation − – 7 – s q 2 2 s ( = − 2 675 ωs s E + 1 ω ωs − 2 2 1 2 2 1 1 ω f ω 0 3.4 4289 − s s s s s s − ) + µ i a + 83293626 τ 1 1 1 2 1 1 2 1 2 ( ) and + q 2 2 2 − ) + s s s s s s s s s ) 3 and τ s s s ˜ ), one can read that the modular S-matrix of the dual τ 2 τ f 2 + 5477520 τ c ( − − − 1 1 1 24 ( ) -expansion of
3 2 2 2 s s s 2 D 0 D ˜ ). The fourth order differential equation of our interest q τ − ˜ f          s s s ) π q 3.6 ) f i ( 5 − − − , ) τ h τ 3 ) 12 ( ( ( 3.7 τ f 15          783 + 306936 64584 + 6789393 τ 4 2 ( q 4 ( 0   √ E 2 + E f ˜ 15 1 f 2 in ( 11 30 29 30 2 , µ ) = = q q √ π i ) τ = sin µ τ ( + 1 + 57478 225          ( i = 2 ˜ ) ) ) ) ) ) 4 1 τ 907  8671 + 1675504 f ) = ) = ˜ 744 = f , s  τ τ D          29 30 /τ /τ /τ /τ /τ /τ + ( ( ) ) ) ) ) )
h − 0 0 1 1 1 1 1 1 19 30 2 3 − π 4 τ ˜ ˜ 5 ) q q f f 6 /τ /τ /τ /τ /τ /τ − − − − − − τ D ( ( ( ( ( ( 1 1 1 1 1 1 ( 0 0 2 2 3 3 0 1  j − − − − − − f f f f f f ) = ) = ) = ) = ( ( ( ( ( ( 0 0 τ τ τ τ 0 1 2 2 3 3          ˜ ˜ ˜ ˜ ˜ ˜ ( ( ( ( f f f f f f 0 = 1 2 3 0 = sin ˜ ˜ ˜ ˜ f f f f          1 s From the bilinear relation ( One can show that the characters ( theory is to fix free parameters then becomes Now, It is easy to see that the solutions of ( Here, we denote by respectively. We further use the where the four characters ofential equation, a dual theory where The modular S-matrix of the model then becomes JHEP07(2019)026 29. are · A (3.16) (3.15) (3.14) (3.12) (3.13) 23 · is one of = 2 17 · 0 24 g ) ) 13 τ Fi τ · · ( ( A , E 11 2 2 3 2 · 0 0 ˜ 3 ˜ f f that agree with the 7 · · · ) , 2 ) τ 5 . τ ( · ( ) , , 2 0 2 0 6724809 17

f , f 3 64584 ··· ) · ⊕ , , ··· ··· + ) ) 21 ) + ) + ··· 1666833 τ q + + , , and τ ( ( , 3 ) ) 3 ··· ··· + ⊕ E q ) q A CFT and examine if they form a , 3 q 2 2 2 64584 + + ˜ ) ˜ f ··· ··· f · ··· 5 · 8671 symmetry has positive integer fusion 116 + + αq αq ) ··· ) , 8671 + read τ 4864431 τ 0 24 ( + 7833 q = ( + αq αq 2 + 20112 2 E ⊕ 2 Fi q f c 783 4158 4158 f · ) 2 q ) q τ τ − − (79 + 2808 (1352 + 27729 ( = 3 1925 ( 6789393 = ) + ) + A α α g E 11 30 29 30 τ 2 3 − τ – 8 – + 1485 + 1485 3 3 0 ( 0 q q ) can be expressed as a sum of dimensions of ( ˜ , ˜ f E α α 1675504 = f A 77 297 297 ) can be understood as the Virasoro descendant of 555611 1 + 616 · 3 1 2 ). One can also show that · 1 ˜ ˜ f ) − − − f τ ) ) = ) = ⊕ 3.10 1 + 1158 ( (54 (54 ( ( · , ( τ · q q τ 29 30 ( 3 3.12 ) ( ( ( ). For instance, the twined characters for ˜ ) 3 (351 + 11504 30 29 30 19 30 11 30 11 30 29 f 0 − 3 0 τ A A 29 30 τ q q q q q q ( f 2 2 2 3 ( f 306153 0 0 19 30 1 − 1 ˜ ˜ f q q f f 2.13 f ⊕ 57477 57477 ) + ) = ) = ) = ) = ) = ) = ) + ) and q q q q q q τ τ ⊕ ⊕ ) + ( ( ( ( ( ( ( τ ) = ) = ) = ) = ) + ( is its complex conjugate. Then, we get a new type of bilinear ( τ q q q q 1 783 1 E E E E E E τ E A 1 ( ( ( ( ( 3 3 3 3 3 3 2 3 ( 3 3 0 1 2 ˜ 3 2 3 0 0 f ˜ ˜ ˜ ˜ ˜ , the triple covering of the largest Fischer group. 3 α E ˜ A A A A f f f f f A ˜ ˜ f f f 3 2 2 2 2 0 2 0 1 2 3 · 0 0 24 · ˜ ˜ ˜ ˜ ˜ ), ˜ f f f f f f ) ) τ · · Fi τ ( τ and · ) ( 2 ) ( ˜ 3 τ f 3 τ 3 57478 = ( f ( f 306936 = √ 0 0 ) merged into the Mckay-Thompson series of class 2A. i 5477520 = f 2 + f + 1+ − 3.13 ) = ) = τ τ = ( ( A A α 3 2 j j Now we will find the twined characters of Notice that each coefficient of ( where relation of the form, As another example, twined characters for Combined with the characterscharacters ( of three-state Potts model, it turns out that the twined where the first term ofthe each corresponding line primary in field. ( bilinear relation analogous togiven ( by first coefficient of coefficients, obtained from the above S-matrix via the Verlinderepresentations formula. of 3 the maximal subgroup of the MonsterIt group, has and 256 is irreducible of representations order 2 including We can also verify that the dual theory with 3 JHEP07(2019)026 (4.5) (4.2) (4.3) (4.4) (4.1) (3.17) ). 2.2 ) τ ( 0 2 g , )˜ 1 8 , τ 2 . | ( ) dual to the critical 0 2                 = g τ , , 2 2 ) ··· ( ) ) 5 5 ) τ 0 2 − − τ τ 116 ( τ g + ( ( | ) + ( ]. This theory has nine σ   2 2 2 , h . 4 5 τ f = f 2 2 2 f 2 0 2 q ( g + | 24 · √ √ · · 9 c )˜ 2 ) 16 − √ √ – 2 ) ) ) 42 g − − τ | τ τ )˜ ( ) τ τ ( 22 = ( ( ( 5 2 2 τ 5 − τ  2 2 σ 0 2 0 0 ( g ( g 0 4 | f √ √ 2 f f 2 12 h − √ √ 2 g g − − | +  ) + + 371520 = ) 2 2 2 0 2 22 0 0 2 0 0 ) = ) ) = τ ) = + | 3 3 Co 4 ) + ( q τ ) τ τ q q √ √ √ √ 2 0 4 ( given by, ( · τ ( ( | ( τ ( 3 g 0 η 5 1 ) ( − − − − η , h 0 )˜ 1 0 g 4 A τ g τ ( 2 1 2 1 g  )˜ ( | 2 1 2 1 0 2 1 , g , g 0 4 τ √ √ = 1 1+22 g ) ) . ( | − g √ √ ) = + τ 2 2 τ 3 0 − − 1 τ + 65467 ( ( 2 g · ( + |   σ 2 2 2 1 ) f ) + – 9 – 2 2 2 f 2 0 , h q | τ · √ √ τ · ) ( − √ √ ) + ( 1 ) ) 4 16 τ − − 4 , g + 27 τ reads, ( τ g τ ) ( g = 1 CFTs studied in [ | ( 1 ( 2 2 0 2 )˜ 1 τ = 12 S 2 0 2 2 0 ) satisfy a bilinear relation g 0 σ 2 0 0 2 0 0 0 and 2 c ( g τ | + √ √ f f 0 2 )˜ + 8672 0 (  as an automorphism group. − √ √ 2 2 4 f h τ ) − − 4.2 + q | ) ( g 3 · ) q 0 24 2 1 q 2 2 = ) | ) = ) = ( τ g ( 2 2 2 ) ( τ τ τ η Fi 2 √ √ η 3 · ( τ ( ( ) + − √ √ ( 0 0 g 0 2 4 − − τ + 783 | 0 ) + f g g ( , h g 3 2 2 2 2 τ q 1 |   + 1 2 11 11 1 1 1 1 1 1 1 1 2 ( g √ √ √ √ )˜ 0 ) = ) = ) = = g τ =                 )˜ ( τ τ τ ) = ) = ( ( ( 0 1 3 τ ¯ τ τ 1 4 0 2 4 ( g h ( g g g 0 , we listed generalized bilinear relations that various twined characters τ, A g = + ( = 2 j C S 1 Z h ) is Mckay-Thompson series of class 3 744 = τ ( − 9 extended modular matrix A ) 3 τ j × ( j In appendix We assumed that the characters ( thus the product of two critical Ising model admit diagonal modular invariants of the form The 9 and their characters can be expressed by characters of the critical Ising model ( We now in turnmodel, consider the a simplest rational example CFTprimaries of dual of the conformal to weights the tensor product of the critical Ising satisfy. These suggest that thethree-state six-character Potts rational model CFT has with 3 4 Dual of the critical Ising where JHEP07(2019)026 , ), , 23 4 4.7 (4.9) (4.6) (4.7) π (4.10) ). To in ( 4.7 i . 82944 ) µ τ 598979 ( is a maximal ˜ − g
.  2 τ = 8 15 ··· 3 D τ Co · ) ) D + = τ τ ) ( ( 3 5 ) with six constraints τ 4 6 4
q (
˜ 1+22 h . 23. This multi-covering 6
) )) (4.8) E E 2 . · 4.8 , , µ , τ τ τ 9 · E 5 ··· ··· ( ( ( 3

) 6 µ ··· π 11 2 4 5 23 16 π τ π + · + g g g ( + )˜ )˜ )˜ + ··· ··· read, 4 4 7 . 2 2 = ), which eventually provide us τ τ τ 3 · q τ 2 3 q i E ( ( ( + + 0 4 q 3  4 µ D σ σ ˜ 5 h 2.4 2 2 f Co ) f f ) µ · · ) does not determine all q q τ τ ) can be expressed as a sum of the 6 = ( ( + + 4630417408 3 3 6 4 ) + 4 4.7 ) + ) + · 2 τ 2 τ 1+22 ˜ h E E τ τ q ( 4.10 D 2 ( ( 15 8 0 ) and ( 4 , ) 1 3 · µ g τ g g )˜ ( )˜ )˜ 4.5 + 155027130 τ 2 + + 178382848 4 τ τ = 1 140037228850176 ( + 155017746 ( ( 3 5 τ q E 6721786984808448 3   σ + 28051444 + 31700992 q 3495263687743883 2 3 f f f ˜ h 13443573969616896 D i q q q µ 284686915225948007 ) ) – 10 – 634818358457751325 , − iπ, i has irreducible representations that includes τ τ + ( ( ) + ) + ) + 2 31 16 2 6 = = = = 4 3 τ τ τ τ + 204855296 ( ( ( 5 7 9 2 E E combining three equations ( D q 0 0 0 1 2 = ) 7 g g g i 0 2 )˜ )˜ )˜ τ µ µ + 4310154 ( τ τ τ ˜ h , µ 6 , µ + 4311948 ( ( ( 2 6 0 0 0 5 q E q = ) + π f f f π 2 τ 2 µ ( 2048 + 565248 2300 + 529828 47104 + 4757504 ˜ h 2 , µ 6 ) = ) = ) = + 3 , E τ τ τ 23 48 13 24 47 48 π 4 6 τ 3 2 ( ( ( q q q  µ 0 σ D ˜ ˜ ˜ f f ) = f 1 + 46851 τ 0 1 23 + 46598 47104 + 5230592 ( ) = ) = ) = ) + -expansion of the characters into ( ˜ h 4 q τ τ τ τ 23 24 ( ( ( ( E 1 24 11 12 = 0 0 0 − 4 1 2 3 4 1 q g q q g g 1 µ E , µ ˜ h 5 2 + µ ) = ) = ˜ ) = ) = ) = ˜ ) = ˜ π 20165360954425344 6 τ 120992165726552064 τ τ τ τ τ τ + 779163580240684865 46679076283392 ( ( ( ( ( ( D ). In this way, we find that the nine parameters 5641332583789180993 576 i 5 1 2 3 4 0  810490346954549 ˜ ˜ ˜ ˜ ˜ ˜ g g g g g g 2647 i − − 4.7 = = = = One can easily show that every coefficient in ( 0 = 8 1 4 6 µ µ µ µ dimension of the irreduciblesubgroup of representation the of baby Monster 2 of group. the Its second order largest is 2 Conway group Co As a result, we finally get below six different characters of dual CFT. Now one can fixfrom nine ( parameters remedy it, we compared twothree bilinear additional relations constraints ( However, inserting because it give us six constraints while there are nine unfixed parameters in ( are the solutions of a differential equation, where the characters of a dual CFT with weights JHEP07(2019)026 2 ), Co 2.7 ) + 96 · τ (4.13) (4.11) (4.12) ( j 1+22 = 23 CFT 2 = 12 because · c c , 2 ), because of the | ) τ 4.3 ( 0 2 , 476928 ˜ g , , . | ) ) ) . ) obey intriguing bilinear ⊕ τ τ τ 2 , + | ( ( ( ) 2 4 5 2 , the other numbers in the | τ g g g ) (5.1) ) ( 4.10 )˜ )˜ )˜ τ 1 5 τ τ τ τ ( ˜ ( g ( ( ( 2 2 | 3238400 2 46575 50600 σ σ σ f ˜ g f f f | + 2 1 ⊕ ⊕ ⊕ 1 2 + | = 12 whose torus partition function ) ) + ) + ) + 23 2 c ) + | τ τ τ τ ) ( τ ( ( ( 0 4 ( 2300 τ 1 3 4 ) to give the baby Monster modules ( ˜ ( 2 g 1 g g g | 0 1 )˜ )˜ )˜ f 4663296 ˜ g τ τ τ 2.2 | 1024650 + ( ( ( ⊕    2 + 46598 = ⊕ ) + | f f f ) τ 2 – 11 – | ( τ ) ( 2 , 0 ) + ) + ) + 529828 = 4 τ f ˜ ( τ τ τ g | ). Referring the table ( ( ( 1 = 23 CFT is identical to the ( ) cannot be an example of Monster anatomy, because 1 0 1 2 46575 ˜ τ g , 518144 | ( + g g g c , 5.1 )˜ )˜ )˜ 5 ⊕ ) guarantees the positive integer fusion rule algebra ⊕ 2 g + | τ τ τ Co ) ( ( ( 2 · 0 0 0 | 4.3 τ 4663296 ) ( f f f ) + 96 = 46575 275 3 τ ⊕ τ ˜ ( g ( | ⊕ ⊕ 0 563200 47104 ) and ˜ j ) = ) = ) = ˜ g τ | + τ τ τ ( ⊕ ⊕ . These numbers are in perfect agreement with the first terms ( ( ( 2  0 275 253 σ ˜ g ˜ ˜ f f f 47104 ) = ⊕ ⊕ · ), satisfy a bilinear relation giving, ¯ τ ), ˜ ). Therefore, the modular invariant partition function of τ τ 1 2048 2048 1 ( τ, ( = 23, dual to the product of two critical Ising model, has 2 47104 2 ( 4.5 1 c f g Z ), ˜ and τ . ( 1 4 46851 = ) and g 565248 = τ ( Co 5230592 = 4310154 = 4757504 = 2 · 1 ), ˜ 2300 τ , , f ( We also propose that the characters of the above RCFT ( The modular S-matrix of the The RCFT of our interest is self-dual in a sense that its three characters, denoted by ) 3 Strictly speaking, a bilinear relation ( τ g 1 ( 0 is not the Monsterpartition module. function of Nonetheless, this in theory also this exhibit section, moonshine we for discuss Conway a group. self-dual RCFT of We discuss in this sectionadmits a a “self-dual” natural RCFT decomposition with group in 2 terms of dimensions of representation off the Conway 5 Self-dual RCFT and 2 coefficients. relations with those of the critical Ising model ( Also, the modular S-matrix ( scendent of the correspondingrational primary CFT state. of Thus weas conjecture an that automorphism the group. nine-character bilinear relation ( is given by where the first term in each decomposition can be again understood as the Virasoro de- of ˜ characters can be decomposed as follow, 2048 JHEP07(2019)026 = 1 (5.4) (5.5) (5.2) (5.3) N , ,   . ,  ··· ···
··· ) . + + τ + 2 2 ( / / 4 λ 5 5 q q q , ) = 0 −  16 τ 1 . ( + const. (5.6) f . ··· ) ) as follows    ,  49152 τ τ  ) ) ) (  + ( τ ]. In fact, one can understand 2 λ λ λ ···

∂ λ + 49152 − ( ( ( 3 − ) 17 q  2 2 + + − ··· τ + ) q q + 10747904 ( f ) are characters for primary states  −− − + 3 τ + f f f τ 3 − ( q ( q 4 3 + 2    q f E f

) + 2    ) and τ 13 + ( τ + 184024 + 11202 + 11202 ( 2 λ − 2

2 2 + 3 2 ) / / ) q ) and + 2624 1 0 0 0 0 1 0 1 0 − 3 3 τ τ τ + 1228800 f 2 q q ( ( − ( + 1228800    q 2 1 2 2 , + ) – 12 – 2 ), f q τ − E = q ) is nothing but the Neveu-Schwarz (NS) partition τ ( f  ( τ 2048 2

(    + 24 λ + 11202 1 , ) ) ) + 300 + 2048 − 24 . These characters are three independent solutions to + −− 3 2 q

λ λ λ 1 2 −− f q q q 2 ) )

, , f + τ τ = 12 of our interest is the GSO projection of the − − − ) ) ) + 98304 ( ( = 2 τ ) + 98304 ) + τ τ τ c ) q λ λ ∂ τ ( ( ( τ h (1 (1 (1 q τ ) ( 1 1 ( ) invariant partition function would be ( 3 + − − − τ f f 3 1 + 24 3 1 + 276 Z −− ( λ −− − + 1 + 276 1 + 276 λ  1 1 λ , 2 f  f f f − q and

  1 2 8 2 16 ) ) E ) ) ) + 2 2 /     3 2 / / 1 τ τ 1 2 τ τ τ − 1 2 1 1 ( ( ( ( ( − 0 2 0 − − λ λ = q  − f q f f q = 3 τ h = 1 extremal superconformal theory [ = = 2048 = 24 + 4096 ∂ = = 24 + 4096 = Z ) = ) = ) = and  ) = ) = ) = N τ τ τ ) ) τ τ τ ( ( ( τ τ ( ( ( 1 2 0 ( ( f f f + − 4 4 2 3 ) of θ θ − + −− τ f f f ( ) is the vacuum character while ) that the RCFT with K ) = τ ( τ ( 0 5.6 f λ Notice here that the character From the modular S-matrix of the self-dual theory, function from ( one can show that the SL(2 For later convenience, let us define Here of conformal weight a modular differential equation below, with JHEP07(2019)026 ) =  5.6 (6.2) (5.8) (5.9) (6.1) (5.7) 24 as an c/ 0 − 0
¯ L ) ¯ q , a maximal =Co . τ  1 ( 24 0 1 1 ) c/ g ··· , Co q − · (¯ = 1 CFT studied Co 0 · 1 , c L 2 ) + ˜ χ ) + ) + c.c. | q )¯ τ q ) q ( q 1+24 F (¯ (¯ τ (¯ 1 10626 1 2 2 ( 1 ˜ g 1) , 37674 5 χ χ χ ⊕ )¯ g )¯ −
| ⊕ q ( q ) (¯  (¯ τ 1 1 2 R ( + 2 χ 0 1 χ , 37674 8855 2 g  | i ) allows a non-diagonal partition ⊕ + 76176 ⊕ 27300 24 τ ( ) + 4  c/ 0 1 ⊕ τ − g , ( 0 , i 1 ) can obey a new bilinear relation + 49152 L 8855 i 1771 g 24 q  ) + 24 6.1 ⊕ ⊕ F c/ ]. + 3015300 8855 τ ) + c.c. + c/ ( − q 1)  8 1 0 ⊕ −
, (¯ g 0 ) L 1 − 3 | 299 L q ( τ χ 1771 ( q h h ) + c.c. )¯ + 3 h q q ) is related to the irreducible representations of – 13 – = 12 is well-known to have 2 g 2 ⊕ (¯ (¯ NS NS R 1771 | 0 ) + c.c. 5 2 ) c q 5.8 χ τ ⊕ χ (¯ ) + ˜ ( )¯  2 299 3 τ q 10925 = χ g ( ( ) = tr ) = tr ) = tr )¯ ⊕ 0 2 1 299 τ τ τ q ˜ g ( ( ( (¯ χ , ) + ⊕ 0 + − ,
 = 1 discussed in section τ ) ) χ ) + 276 − + −− 276 ( c τ τ q f f f  · 0 253 ( ( ( g 5 2 276 3 0 | = 12 may be enhanced to a larger group 2 ⊕ g g χ )˜ ⊕ ⊕ c )¯ τ q ) is the partition function of another example of = 1 SCFT. This SCFT was first made by Frenkel, Lepowsky and ( 1 23 1 ) = ( ) + 5 ¯ 0 τ τ g + 10925 + 1081344 6.1 N χ ( τ, 0 ( g + 2 ). ( Z = 12 Virasoro characters, 276 = ) = ¯ τ c 4.4 49152 = 76176 = τ, ( = 1 extremal SCFT with ], and revisited later by Duncan [ 720 = 2 Z . Some details of the number decomposition are presented below. 1 ]. One can show that the characters in ( N − ) Co τ 24 , ( · j The 23 . Here the each trace can be performed in either the NS Hilbert space or the Ramond 2 1+24 different from ( in [ It is known thatfunction the CFT of 6 Discussion It turns out that2 every coefficients in ( in terms of the automorphism group. After the GSOgroup projection, continues the to double covering serve of ascharacter the decomposition an largest of Conway automorphism the group partition further ofgroup gives the of new the self-dual evidence RCFT that theory. with thesubgroup The automorphism of Virasoro the Monster group. To see this, let us decompose the partition function ( and the constant term24 corresponding to the WittenHilbert index, space tr of the Meurman [ extremal SCFT where JHEP07(2019)026 . ) ) τ N ( i (6.3) f 4.10 ) can p . This T 24 3.10 M · . It would be 12 1 10 (a certain CFT)’ ). More precisely, = . and correspond to × 2 ) is also realized as | h ]. This new bilinear 2 ) CFT using the Hecke 4.10 τ 6.3 17 ( Co 5 · 10 233 ˜ g | . This is partially because = = 23. However, it turns out ]. The Hecke image )) in ( 10 1+22 + 2 233 c = 47 are exactly agree to the that is relatively prime to p τ 2 25 ( 2 p p | · 5 = ) ˜ g τ c , )-function [ ( ) τ 0 1 τ ( g ( )) with 0 1 K τ g ( ). In similar way, it is easy to see that = 29. In similar way, we found that the ) + ˜ 5 τ p ( , g ) + ˜ 2.7 ) 1 τ ˜ g τ ( | ( 1 0 1 ˜ g + g , – 14 – ) with ) 2 | τ ) ( ) + τ 3.4 3 ) are not able to realized as the Hecke images of the . It is obvious that the dual CFT also exhibit the τ ( g ( 3 and a natural number 4 1 g 4.10 , g N ) + ˜ ) τ ) + ˜ τ ( ( τ 0 3 ( g ) in ( ) cannot generate the six characters in ( 0 g = 24, respectively. We checked that the characters ( ˜ τ g = 233, however it turns out that this Hecke operator does not | ( . . This is somewhat trivial, because it shares the characters ( 4.2 4 p N 720 is not the Monster module. ) + 2 g 1 τ ) = ( − Co ¯ 0 τ · 4), can exhibit the moonshine phenomena. However, it is nontrivial ) g , ]. We tried to find the dual characters of τ τ, ( ( ) and ˜ (5 j 12 ˜ τ 1+22 ). Z ( 2 ) can be understood as a non-diagonal partition function of the rational 2 M = 30 and · g = 240 and 4.2 6.3 N N = 23 discussed in section c = 48. Then, one can show that the Hecke images for The bigger challenge is to find all the two-dimensional dual CFT pairs for the rest of the On the other hand, it is known that the tricritical Ising model is endowed with the One can also ask if the dual CFT of the tricritical Ising model, the next simplest unitary It has been proposed that the Hecke images of the vector-valued modular form can N = 1 supersymmetry. The NS partition function of the model can be contributed by the sporadic groups and understandextremely the interesting origin to of see bilinear if relations.can a show In CFT moonshine particular, dual for it the tois multiple-covering would ‘(three-state of analogous be Potts the to model) largest the Mathieuthe idea group, green 2 used arrow to in find figure the dual CFT with 2 NS superconformal characters of theinteresting vacuum to and search the for primary a statewith rational of SCFT those whose of NS the charactersrelation tricritical obey would Ising a lead to new model a bilinear to picture relation group of give decomposition. the the Conway group We leave decomposition, them instead of as the a Monster future project. known CFT data [ operator of produce admissible characters. N characters in ( minimal model to obtain the characters ofone the cannot candidate determine dual CFT all with free parameters in the corresponding MDE completely from the be considered as the Heckevector-valued images modular of form (˜ ( the Hecke images of ( that the Hecke images ofthe ( characters ˜ As an example, itby has been known thatcharacters the of conductor the of baby thethe Monster critical conductor CFT, of Ising namely three-state model Potts ( are is model given given and by the tensor product of the critical Ising model as building blocks. However, aboveanatomy, example because cannot be considered as a class of the Monster construct a set ofis admissible characterized characters by for the specific conductor RCFT [ Notice that ( CFT of moonshine for 2 from which one can read the partition function of the dual theory JHEP07(2019)026 , , , , , , · · ·} , , , , , , , , 315744 , 94875 , 37674 , , , 789360 368874 531300 } , , , · · ·} , 9625 47104 , , 673750 , , 1841840 2055625 212520 ··· , , , · · ·} , 1603525 80730 27300 313950 , 10506240 , , , , 7084 , 32715683 , 673750 312984 518144 46575 , , , , , 2877875 644644 1407126890 , , 184437 , 44275 1821600 1821600 , 17250 4025 555611 , , , , , 299000 , 74837400 , 44275 9550635 , 644644 , 284625 476928 , , , , 4622913750 25356672 8855 2300 483000 , , , 37674 , , 2816856 177100 , , , 306153 1771000 1771000 170016 37422 , 410132480 , , , , , 4576 2277 483000 253000 467775 9458750 , , 48893768 , , , , , 27300 376740 , , 129536 19034730 2464749 , 95680 31878 249458 , , 2576 2048 , , , , , 4275362520 1450449 1450449 , , , 388080 245916 462000 · · ·} – 15 – 17250 , , , , 1139374 , 356054375 345345 , ]. , 2024 2024 43779879 , 94875 31625 113850 64584 , , , , , , , 26 6724809 2464749 8855 , , , 376740 1434510 1434510 239085 442750 579600 96256 1771 1771 , , , , , , , , , 80730 313950 23000 91125 57477 Dimension of the irreducible representations 4221380670 , , , , , 1771 , , 299 275 347643114 40536925 , , , , 4864431 2417415 96255 , , 351624 871884 871884 226688 430353 563200 , 8671 299 , , , , , , 44275 98304 12650 63250 , , 276 253 , , , , , , 4371 783 24 276 23 , , , , , 1 1 1 1 1 { 63532485 3214743741 { 1666833 35873145 { 40480 345345 822250 { 98280 822250 2260440 { 10395 50600 221375 398475 558900 1 2 1 Co Co 0 24 B · · · Co Fi · · 2 . Dimension of the irreducible representation in various sporadic groups. The list of the 2 3 1+24 1+22 2 2 Here we give asporadic partial groups. list of the dimension of the irreducible representations for various Foundation of Korea (NRF) Grant NRF-2017R1C1B1011440.pen Center K.L. and for S.L. Physics thank (supported the by As- National Science Foundation grantA PHY-1607611). Dimension of the irreducible representations We would like to thank Hyunmous Kyu Kim referee for useful of discussions. JHEPKL We are for grateful is to giving the supported us anony- 2017R1D1A1B06034369. in numerous, KL invaluable part would comments like by tothe on part thank the the of the work National manuscript. colleagues is done. at Research IIP The Foundation research Natal of of Brazil S.L. where is Korea supported in Grant part by NRF- the National Research Acknowledgments Table 1 representation read from GAP package [

JHEP07(2019)026

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

0 53936390144 − 53936390144 1280 − 1280 11648 − 11648 0 1835008 1835008 − 191 56 56 − 0 0 0 0

11648 0 1835008 1835008 − 0 53936390144 − 53936390144 190 56 56 − 0 0 0 0 1280 − 1280 11648 −

36657653760 189 146016 0 8198144 − 8198144 0 36657653760 − 1960 − 1960 0 0 0 0 432 432 − 146016 −

.

8844386304 − 8844386304 188 0 840 − 840 96096 − 96096 0 3629056 − 3629056 0 1904 − 1904 0 0 0 B

·

410132480 − 410132480 187 0 0 4 4 − 23648 − 23648 0 516096 − 516096 0 880 − 880 0 0

10506240 186 108 − 3456 − 3456 0 45056 − 45056 0 10506240 − 240 − 240 0 0 0 0 108

96256 − 96256 185 0 28 − 28 352 − 352 0 2048 − 2048 0 56 − 56 0 0 0

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

90807234375 22 1815 − 10725 − 10725 − 118625 − 2498375 2498375 55087175 90807234375 0 0 325 − 3001 − 42215 9625 − 1815 −

118625 − 2498375 2498375 55087175 90807234375 90807234375 21 0 0 325 − 3001 − 42215 9625 − 1815 − 1815 − 10725 − 10725 −

169170 104960 10451200 1750 1750 0 1792 − 77056 172800 − 120 − 120 − 169170 10451200 87975680 − 80426400000 80426400000 20

3402091 56581525 − 75844139371 75844139371 19 154 − 273 − 875 49931 4235 2887 2887 1001 − 1001 − 162435 3402091 154 −

6369275 6369275 37543429 − 27416186875 27416186875 18 1925 1925 1 − 2299 32827 125125 − 1385 − 1385 − 129349 129349 44149 −

17 3512928 3512928 18677152 − 13508418144 13508418144 18272 60576 − 1488 1488 57135 57135 134368 − 694 694 0 3680

0 24 52404 65807 2075359 2075359 15693601 − 12501781215 12501781215 16 715 53 − 1057 − 11935 47201 − 1293 1293 52404 715

Fi 1896994 1896994 16720418 9287037474 9287037474 15 31522 24354 1023 1023 19734 19734 87074 99 99 26 2850

·

2011350 2011350 11656918 4622913750 4622913750 14 − 22678 57750 345 345 58422 58422 42042 − 1275 1275 78 − 1066 – 16 –

1710808 1710808 10237656 4275362520 4275362520 13 770 770 0 1240 20056 45144 594 594 42471 42471 48568

4221380670 4221380670 12 44642 − 1615934 1615934 999362 − 595 595 78 − 706 − 8130 − 8130 − 465 465 37401 37401

and 3

3214743741 3214743741 11 22933 1214653 1214653 3059133 17501 270 270 32670 32670 616 616 27 − 1469 3875 −

B

1407126890 1407126890 10 25194 789866 789866 3199638 − 18326 − 155 155 25103 25103 615 615 26 406 − 3178

·

356054375 356054375 9 1377 − 419175 419175 1901977 − 18425 − 160 − 160 − 18227 18227 650 650 27 153 − 8327

347643114 347643114 8 1274 392426 392426 1511146 14378 219 − 219 − 16224 16224 539 539 78 5341 − 4394

63532485 63532485 7 4811 − 134597 134597 468027 − 7547 − 135 135 8073 8073 385 385 1 197 2437

9550635 9550635 6 781 − 35627 35627 119339 2827 6 6 3003 3003 210 210 27 43 1163 . Partial character table of the baby Monster group 2

9458750 9458750 5 3486 37950 37950 118846 2750 25 − 25 − 2729 2729 175 175 26 − 318 1214

1139374 1139374 4 782 8878 8878 26962 − 978 − 53 − 53 − 1000 1000 99 99 26 − 82 − 558

96255 96255 3 103 2047 2047 4863 351 27 27 351 351 55 55 1 − 1 −

223 Table 2

4371 4371 2 53 − 275 275 493 − 77 − 3 − 3 − 78 78 21 21 1 − 19 51

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

C 2 B 2 A 2 A 1 n A 8 D 4 C 4 B 4 B 6 B 3 A 6 A 3 A 4 D 2 A 10 A 5 B Character tables of 2 JHEP07(2019)026 . 3 √ i 2 − 1 − = α and

3

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · √

i

2

567 − α 26377 α 26377 26377 α 6724809 α 6724809 6724809 α 393 α 393 393 α 2079 α 2079 2079 α 3861 − α 3861 − 3861 − 0 α 567 − α 567 − 118 1+

−

117 26377 26377 α 6724809 α 6724809 6724809 α 2079 2079 α 3861 − α 3861 − 3861 − 0 α 567 − α 567 − 567 − α 26377 α α 393 α 393 393 α 2079

=

2729 α 2729 2729 α 306513 α 306513 306513 116 351 α 351 351 α 1431 α 1431 1431 0 α 279 − α 279 − 279 − α α 105 α 105 105 α

α

2729 2729 α 306513 α 306513 306513 115 2729 α 351 α 351 351 α 1431 α 1431 1431 0 α 279 − α 279 − 279 − α α 105 α 105 105 α

114 306513 1431 α 1431 1431 0 α 489 α 489 489 α 3433 α 3433 3433 α 306513 α 306513 α 105 α 105 105 α 351 α 351 351 α

306513 α 306513 306513 113 α 3433 3433 α α 1431 α 1431 1431 0 α 489 α 489 489 α 3433 α 105 α 105 105 α 351 α 351 351 . Here

0 24 α 1352 1352 α 64584 α 64584 64584 112 α 72 α 72 72 α 216 α 216 216 α 297 − α 297 − 297 − 0 α 72 α 72 72 α 1352

Fi

64584 111 α 72 72 α 1352 α 1352 1352 α 64584 α 64584 α 72 α 72 72 α 216 α 216 216 α 297 − α 297 − 297 − 0 α 72 ·

783 α 783 783 110 54 0 α 15 α 15 15 α 79 α 79 79 α α 15 α 15 15 α 27 α 27 27 α 54 α 54

α 79 79 α α 54 α 54 54 0 α 15 α 15 15 α 79 783 α 783 783 109 α 15 α 15 15 α 27 α 27 27

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

13871 4864431 4864431 4864431 9 1105 − 1105 − 1105 − 13871 13871 175 175 175 867 867 867 4917 4917 4917 4836 – 17 –

11153 1666833 1666833 1666833 8 273 273 273 11153 11153 987 987 1848 − 1848 − 1848 − 3093 209 209 209 987

1925 1603525 1603525 1603525 7 315 − 315 − 315 − 1925 1925 5 5 5 140 − 140 − 140 − 275 − 275 − 275 − 6475

1925 1603525 1603525 1603525 6 315 − 315 − 315 − 1925 1925 140 − 140 − 275 − 275 − 275 − 6475 5 5 5 140 −

5083 555611 555611 555611 5 91 91 91 5083 5083 518 518 2300 2300 2300 535 − 155 155 155 518

2354 249458 249458 249458 4 370 370 370 2354 2354 167 167 869 869 869 2705 50 50 50 167

1157 57477 57477 57477 3 133 133 133 1157 1157 210 210 615 615 615 534 69 69 69 210

351 8671 8671 8671 2 33 − 33 − 33 − 351 351 85 85 77 − 77 − 77 − 247 31 31 31 85

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2A 3B 3A 1A n 6D 6C 2B 6B 6A 3H 3G 3F 3E 3D 3C 12B 12A 4A 3I . Partial character table of the Fischer group 3 Table 3 JHEP07(2019)026 0 24 Fi (C.1) (C.2) · 3 ∈ which are g ) B τ · ( g 2 2 0 ˜ f · ∈ , ) ) g τ τ with the characters of ( ( . 0 2 g σ f . 0 24 ˜ M f B · Fi · ∈ · ) ) + τ τ M ( ( M g σ g g 3B 3A 4A 2 12A f ˜ f · ) denotes Mckay-Thompson series ) denotes Mckay-Thompson series M τ ) g 8C 5A 6A 6D τ ) + ( 10A τ ( τ ( M g M ( 4A 3G putative CFT for various g 2 g 3H,3I g  j f , j ˜ g 12A,12B f 6B 5A 8A 6A ) 10A 5 · τ 116 ( ) ) + M and g 3 g 3B τ M 6C 3A 3A 0 and τ = g 4B 3B 4C 4A 4A ( ˜ ( f  24 0 g 1 B c · f ˜ f · ) – 18 – Fi g · g 4B 3B 4C 4A 4D · τ 3C 2 3D ) ( ) + 3E,3F 3 6C,6D 0 3 τ ∈ τ ( M f A A A A B ( ∈ g 3 2 2 2 2 1 g g 0 M A A A B f ˜ g g 3 2 6 2 f exhibit which twined character yields Mckay-Thompson ) + · describes. g τ 2B 2C 3A 2A 2D 5 ) ) + ( 4 τ g τ 3 g . Generalized bilinear relation for 2 ( ˜ ( 2B 2A . Generalized bilinear relation for 3 f 0 present which twined character forming Mckay-Thompson g 0 3A,3B 6A,6B · f ˜ f ) 4 · τ ) ( ) = 3 τ τ ( f Table 4 ( Table 5 0 f M + g j . Table as before. Table ) = . For instance, once we have twined characters of baby Monster CFT for M . More preciesly, general expression of generalized bilinear relation have a τ M ( M g ∈ M M g 0 24 ∈ ∈ j ) is twined character for ) is twined character for . M τ Fi B τ M g A ( M ( g · · g g g ˜ , they merge with the characters of Ising model to produce Mckay-Thompson series ˜ f f C = 2 where for class series of certain class, like table It turns out thatalso the constitute twined the characters Mckay-Thompson series of three-states of Potts certain model. class We findis that given the by explicit form of the generalized bilinear relation g of class 2 C.2 3 where for class series of type C.1 2 We find thecombined twined with characters the of charactersvarious baby of Monster Ising CFT modelsform for and of various form the Mckay-Thompson series for C Generalized bilinear relations JHEP07(2019)026 , , 3 , , (1979) . SIGMA , 11 , ]. ∼ ]. Commun. Num. Theor. SPIRE math/0502267 , IN , [ arXiv:1406.5502 . , ]. surface and the Mathieu group Exceptional algebra and sporadic 3 SPIRE K Bull. London Math. Soc. IN , arXiv:1504.0817 ][ ]. 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