Generalised Moonshine in the elliptic genus of K3

Generalised Moonshine in the elliptic genus of K3 Daniel Persson Chalmers University of Technology Algebra, Geometry and the Physics of BPS-States Hausdorff Research Institute for Mathematics, Bonn, November 12, 2012 Based on work to appear with M. Gaberdiel, H. Ronellenfitsch & R. Volpato

In 2010, Eguchi, Ooguri, Tachikawa conjectured that there is Moonshine in the elliptic genus of K3 connected to the finite sporadic group M 24 S 24 EOT observation: Fourier coefficients of K3-elliptic genus are (sums of) dimensions of irreps of M 24

The word moonshine generally refers to surprising connections between a priori unrelated parts of mathematics and physics, involving: representation theory of finite groups generalised Kac-Moody algebras modular forms conformal field theory

Monstrous Moonshine pertains to the characters (McKay-Thompson series) 1X T g ( ) =Tr V \(gq L 0 1 )= Tr (g)q n V \ n 1M V \ = n= 1 V \ n n= 1 graded M -module 8g 2 M Moonshine conjecture (Conway-Norton): The McKay-Thompson series are modular-invariant under some genus zero g SL(2, R) g genus zero g\h The conjecture was proven by Borcherds in 1992.

In 1987 Norton proposed a vast generalisation of the monstrous moonshine conjecture, under the name of Generalised Moonshine. The aim of this talk is to address the analogue of generalised moonshine for the recently discovered connection between the K3 elliptic genus and M 24

Motivation? Of independent mathematical interest to understand new moonshine phenomena. Wall-crossing and (twisted) N=4 dyons in CHL-models. BPS-algebras....and many more

Outline 1. Generalised Monstrous Moonshine 2. Mathieu Moonshine 3. Generalised Mathieu Moonshine 4. Wall-crossing and Automorphic Lifts 5. Discussion and Outlook

Generalised Monstrous Moonshine

Generalised Monstrous Moonshine ( Maxi-Moonshine ) Norton proposed a generalisation of the monstrous moonshine conjecture: To each pair of commuting elements g, h 2 M Norton associates a holomorphic function Z g,h ( ) on the upper half plane, satisfying H Z g,h ( ) =Z xgx 1,xhx 1( ) Z ga h c,g b h d( ) = Z g,h a + b c + d a b c d 8 x 2 M 2 SL(2, Z) 24 =1 The coefficients in the q-expansion of projective representation of the centraliser Z g,h ( ) are characters of a graded C M (g) ={x 2 M xgx 1 = g}

Generalised Monstrous Moonshine ( Maxi-Moonshine ) Norton proposed a generalisation of the monstrous moonshine conjecture: To each pair of commuting elements g, h 2 M Norton associates a holomorphic function Z g,h ( ) on the upper half plane, satisfying H Z g,h ( ) =Z xgx 1,xhx 1( ) Z ga h c,g b h d( ) = Z g,h a + b c + d a b c d 8 x 2 M 2 SL(2, Z) 24 =1 The coefficients in the q-expansion of projective representation of the centraliser Z g,h ( ) Z g,h ( ) is either constant or is modular invariant under some genus zero congruence subgroup of SL(2, R) are characters of a graded C M (g) ={x 2 M xgx 1 = g}

Generalised Monstrous Moonshine ( Maxi-Moonshine ) Generalised Moonshine is so far unproven, except in special cases G. Höhn (Baby Monster) M. Tuite S. Carnahan is working towards a complete proof.

Generalised Moonshine and Holomorphic Orbifolds It was pointed out early on by Dixon-Ginsparg-Harvey and Tuite that Norton s generalised moonshine conjecture has a natural interpretation in terms of orbifolds of the Frenkel-Lepowsky-Meurman monster CFT V \. Orbifolding by some element introduces twisted sectors V \ g 2 M V \ and one may consider the twisted CFT characters g g 1 =Tr V \ g q L 0 1

Generalised Moonshine and Holomorphic Orbifolds Z g,h Most of the properties of conjectured by Norton now become natural consequences of known results in orbifold CFT. Except for the genus zero property which still lacks a satisfactory explanation.

Mathieu Moonshine

Mathieu Moonshine Non-linear sigma models with target space K3 N =(4, 4) superconformal algebra with c =6 Large moduli space of such theories: M = O(4, 20; Z)\O(4, 20; R)/(O(4) O(20)) The physical spectrum varies over moduli space, but there is a graded partition function, the elliptic genus, which is constant. The elliptic genus of RR-ground states. K3 only receives contributions from the right-moving

The elliptic genus is defined by [Witten][Kawai, Yamada, Yang] (q = e 2 i,y = e 2 iz ) K3 =Tr HRR ( 1) J 0+ J 0 q L 0 c/24 q L 0 c/24 y J 0

The elliptic genus is defined by [Witten][Kawai, Yamada, Yang] (q = e 2 i,y = e 2 iz ) K3 =Tr HRR ( 1) J 0+ J 0 q L 0 c/24 q L 0 c/24 y J 0 Cartan generator in the left SU(2) of N =(4, 4)

The elliptic genus is defined by [Witten][Kawai, Yamada, Yang] (q = e 2 i,y = e 2 iz ) K3 =Tr HRR ( 1) J 0+ J 0 q L 0 c/24 q L 0 c/24 y J 0 Holomorphic in both and z Modular and elliptic properties: K3 a + b c + d, z cz2 2 i = e c +d c + d K3(,z) a c b d 2 SL(2, Z) K3(,z + ` + `0) =e 2 i(`2 +2`z) K3(,z) `, `0 2 Z This identifies with a weak Jacobi form of weight 0 and index 1: K3 K3 = 0,1 =8 #2 (,z) 2 # 2 (,0) 2 + # 3(,z) 2 # 3 (,0) 2 + # 4(,z) 2 # 4 (,0) 2 [Eguchi, Ooguri, Taormina, Yang]

The elliptic genus further satisfies K3(,z = 0) = (K3) = 24 Now denote by H the subspace of right-moving ground states in H RR H We have a decomposition of into irreps of the left superconformal algebra, which induces a decomposition of the elliptic genus: K3(,z) = 20 h= 1 4,j=0 2 h= 1 4,j= 1 2 + 1X n=1 A n h= 1 4 +n,j= 1 2 where h,j(,z)=tr Hh,j ( 1) J 0 q L 0 c 24 y J 0

Eguchi, Ooguri, Tachikawa observed A 1 = 90 = 45 + 45 A 2 = 462 = 231 + 231 A 3 = 1540 = 770 + 770

Eguchi, Ooguri, Tachikawa observed A 1 = 90 = 45 + 45 A 2 = 462 = 231 + 231 A 3 = 1540 = 770 + 770 Dimensions of irreducible representations of the largest Mathieu group M 24! EOT conjecture: H M 24

Eguchi, Ooguri, Tachikawa observed A 1 = 90 = 45 + 45 A 2 = 462 = 231 + 231 A 3 = 1540 = 770 + 770 By now, the Mathieu Moonshine conjecture has been established in the sense that all the twining genera have been found. g [Cheng][Gaberdiel, Hohenegger, Volpato][Eguchi, Hikami][Gannon]

Generalised Mathieu Moonshine [Gaberdiel, D.P., Ronellenfitsch, Volpato] (to appear)

Twisted twining elliptic genera We now want to extend the story and consider the Norton s generalised monstrous moonshine conjecture. M 24 We are thus led to define twisted twining genera as follows: -analogue of g,h(,z)=tr h ( 1) J 0+ J 0 q L 0 Hg c/24 q L 0 c/24 y J 0 for commuting pairs g, h 2 M 24 The trace is taken over the twisted Hilbert space. If generalised Mathieu moonshine holds, g,h should be a weight 0, index 1 weak Jacobi form for some with generalised multiplier system g,h a + b c + d, z c + d g,h SL(2, Z) a b = g,h c d H g cz2 2 i e c +d g,h(,z) g,h : g,h! U(1)

Twisted twining elliptic genera g,h a + b c + d, z c + d a b = g,h c d cz2 2 i e c +d g,h(,z) g,h : g,h! U(1) In order to make sense of this we recall that the twisted twining genera can be interpreted as characters in an orbifold theory. To analyze the twisted twining genera and their connection with we must therefore bring in some orbifold machinery. M 24

Holomorphic Orbifolds and Group Cohomology Consider a holomorphic CFT V with G = Aut(V).

Holomorphic Orbifolds and Group Cohomology Consider a holomorphic CFT V with G = Aut(V). For each g 2 G there is a unique twisted sector V g with character Z g,h ( ) =g h =Tr Vg hq L 0 1 8h 2 C G (g) = Aut(V g ) The full partition function of the orbifold theory V/G is then Z( ) = X g 1 C G (g) X h2c G (g) Z g,h ( ) [Dijkgraaf, Vafa, Verlinde, Verlinde]

Holomorphic Orbifolds and Group Cohomology Fact: Consistent holomorphic orbifolds are classified by H 3 (G, U(1)). [Dijkgraaf, Witten][Dijkgraaf, Pasquier, Roche][Bantay][Coste, Gannon, Ruelle]

Holomorphic Orbifolds and Group Cohomology Fact: Consistent holomorphic orbifolds are classified by H 3 (G, U(1)). [Dijkgraaf, Witten][Dijkgraaf, Pasquier, Roche][Bantay][Coste, Gannon, Ruelle] Concretely, this works as follows. There exists an element 2 H 3 (G, U(1)) : G G G! U(1) (3-cocycle = 3-cochain + closedness) which determines a distinguished element c h 2 H 2 (C G (h),u(1)) via c h (g 1,g 2 )= (h, g 1,g 2 ) (g 1,g 2, (g 1 g 2 ) 1 h(g 1 g 2 )) (g 1,h,h 1 g 2 h)

Then, under modular transformations the twisted twining characters of a holomorphic orbifold obey the relations [Bantay][Coste, Gannon, Ruelle] Z g,h ( + 1) = c g (g, h)z g,gh ( ) Z g,h ( 1/ ) =c h (g, g 1 )Z h,g 1( ) Moreover, under conjugation of g, h one has the general relation Z g,h ( ) = c g(h, k) c g (k, k 1 hk) Z k 1 gk,k 1 hk( ) 8k 2 G

Cohomological Obstructions from H 3 (G) Z g,h ( ) = c g(h, k) c g (k, k 1 hk) Z k 1 gk,k 1 hk( ) Whenever commutes with both and one finds k g h Z g,h = c g(h, k) c g (k, h) Z g,h

Cohomological Obstructions from H 3 (G) Z g,h ( ) = c g(h, k) c g (k, k 1 hk) Z k 1 gk,k 1 hk( ) Whenever commutes with both and one finds k g h Z g,h = c g(h, k) c g (k, h) Z g,h So Z g,h =0 unless the 2-cocycle c g is regular: c g (h, k) =c g (k, h) When this is not satisfied we have obstructions! [Gannon]

Twisted Twining Genera Twisted twining genera g,h of M 24 behave similarly to the Z g,h g,h a + b c + d, z c + d a b = g,h c d cz2 2 i e c +d g,h(,z) a c b d 2 g,h SL(2, Z)

Twisted Twining Genera Twisted twining genera g,h of M 24 behave similarly to the Z g,h g,h a + b c + d, z c + d a b = g,h c d cz2 2 i e c +d g,h(,z) a c b d 2 g,h SL(2, Z) The multiplier phase g,h must be determined by some 2 H 3 (M 24,U(1))! The twisted twining genera also form a representation of the full SL(2, Z) g,h a + b c + d, z c + d = g,h ( ab cd cz 2 ) e2 i c +d a b h c g d,h a g b(,z) c d 2 SL(2, Z)

Twisted Twining Genera g,h a + b c + d, z c + d = g,h ( ab cd cz 2 ) e2 i c +d a b h c g d,h a g b(,z) c d 2 SL(2, Z) For the S- and T -transformations we then have: g,h( 1, z )=c h(g, g 1 )e 2 i z 2 h,g 1(,z) g,h( +1,z)=c g (g, h) g,gh (,z) with c a (b, c) is a 2-cocycle descending from H 3 (M 24,U(1)). Multiplier phases of twisted twining genera determined by the 3rd cohomology of. M 24

Luckily, this cohomology group has recently been computed: H 3 (M 24,U(1)) = Z 12 [Dutour Sikiric, Ellis]

Luckily, this cohomology group has recently been computed: H 3 (M 24,U(1)) = Z 12 [Dutour Sikiric, Ellis] Using GAP,with the HAP homological algebra package, we have been able to construct an explicit representative 2 H 3 (M 24,U(1)). The relevant cohomology class of is determined by the condition 1,h( ab cd 2 icd )=e Nk a c b d 2 0 (N) which is the multiplier system of the twining genera. h = 1,h

Main result: There exists a unique set of functions cohomology class conjectured properties hold. [ ] 2 H 3 (M 24,U(1)) g,h, and a unique such that all of Norton s

Main result: There exists a unique set of functions cohomology class conjectured properties hold. [ ] 2 H 3 (M 24,U(1)) g,h, and a unique such that all of Norton s Remarks: There are in total 34 genuinely twisted and twined genera. Their multiplier phases are indeed correctly determined by the 3-cocycle 2 H 3 (M 24,U(1)). Many of the twisted twining genera vanish due to the cohomological obstruction.

Example: 8A -twist and 2B -twine: 8A,2B(,z)= 2 # 1 (,z) 2 ( ) 6 # 4 (,0) 2 6

Example: 8A -twist and 2B -twine: 8A,2B(,z)= 2 # 1 (,z) 2 ( ) 6 # 4 (,0) 2 6 8A = M 24 -conjugacy class of order 8 elements.

Example: 8A -twist and 2B -twine: 8A,2B(,z)= 2 # 1 (,z) 2 ( ) 6 # 4 (,0) 2 6 8A = M 24 -conjugacy class of order 8 elements. 8A,2B(,z) is a Jacobi form of weight 0 index 1 for the group n a b o 2 SL c d 2 (Z) b 0 mod 4 = 0 (4) 8A,2B :=

Example: 8A -twist and 2B -twine: 8A,2B(,z)= 2 # 1 (,z) 2 ( ) 6 # 4 (,0) 2 6 8A = M 24 -conjugacy class of order 8 elements. 8A,2B(,z) Multiplier given by: is a Jacobi form of weight 0 index 1 for the group n a b o 2 SL c d 2 (Z) b 0 mod 4 = 0 (4) 8A,2B := 8A,2B( +4,z)= Q 3 i=0 c g(g, g i h) c g4 h(g, g 1 )c g 1(g 4 h, g 4 h) c g 1(g 4 h, k) c g 1(k, h) 8A,2B( ) = 8A,2B( ) using our result for c g1 (g 2,g 3 ) in terms of 2 H 3 (M 24,U(1))

Generalised Mathieu Moonshine All twisted twining genera g,h are either weak Jacobi forms of weight 0 index 1 for g,h SL(2, Z) with multiplier system determined by 2 H 3 (M 24,U(1)) or vanish by the cohomological obstruction. The twisted sector Hilbert space H g decompose according to: H g = M H M 24 g,r Hg,r N =4 r 0 We have found that H M 24 g,r further decomposes into H M 24 g,r = M i h i,r R i positive integer multiplicities projective representation of C M24 (g) determined by the 3-cocycle 2 H 3 (M 24,U(1))

We have found that H M 24 g,r further decomposes into H M 24 g,r = M i h i,r R i The twisted twining genera thus admit an expansion g,h(,z)=tr H M 24 g,0 (h) h= 1 4,`=0(,z)= 1X r=1 Tr H M 24 g,r (h) h= 1 4 +r,`= 1 2 (,z) where each coefficient splits into a positive sum of characters for irreducible projective representations of C M24 (g) Tr H M 24 g,r (h) =X i h i,r Tr Ri (h)

Wall-Crossing and Automorphic Lifts

Mason s M 24 -Moonshine for Eta-Products In 1990, Mason proposed a generalised moonshine for M 24 involving eta-products.

Mason s M 24 -Moonshine for Eta-Products In 1990, Mason proposed a generalised moonshine for M 24 involving eta-products. M 24 admits a permutation representation of degree 24 ( defining representation ). each conjugacy class [g] 2 M 24 can be characterised by its cycle shape: identity [g] =1A cycle shape 1 24 (1)(2)(3) (22)(23)(24) order 2 [g] =2A cycle shape 1 8 2 8 (1) (8)(9 10)(11 12) (23 24)

Mason s M 24 -Moonshine for Eta-Products Mason proposed eta-products g,h ( ) corresponding to Norton series Z g,h ( ) for all pairwise commuting elements g, h 2 M 24 Examples: 2A,2A ( ) = (2 ) 12 2A,2B ( ) = (2 ) 4 (4 ) 4 He showed that all these eta products g,h ( ) satisfy the requirements of Norton. In particular, they decompose into projective characters of C M24 (g) : g,h ( ) = 1X n=1 a n (g, h)q n/d

So, is Mason s generalised M 24 -moonshine also related to to K3?

So, is Mason s generalised M 24 -moonshine also related to to K3? Yes! They are linked via the process of automorphic lift and wall-crossing.

Multiplicative automorphic lift Lift : Jacobi forms Siegel modular forms [Borcherds]

Multiplicative automorphic lift Lift : Jacobi forms Siegel modular forms [Borcherds] For the elliptic genus of K3 this yields: [Gritsenko, Nikulin] K3(,z)= X n 0 `2Z c(4n `2)q n y` pqy Y n>0, m 0,`2Z 1 p n q m y` c(4mn `2) 1 10 is the partition function of 1/4 BPS dyons in N =4 string theory. [Dijkgraaf, Verlinde, Verlinde][Shih, Strominger, Yin]

Multiplicative automorphic lift Lift : Jacobi forms Siegel modular forms [Borcherds] For the elliptic genus of K3 this yields: [Gritsenko, Nikulin] K3(,z)= X n 0 `2Z c(4n `2)q n y` pqy Y n>0, m 0,`2Z 1 p n q m y` c(4mn `2) Alternative presentation: [Dijkgraaf, Moore, Verlinde, Verlinde] [Gritsenko, Nikulin] " # 1X 10(,z, )=A(,z, )exp p N T N K3 (,z) N=0 automorphic correction Hecke operator

The automorphic lift provides a link from Jacobi forms to eta-products.

The automorphic lift provides a link from Jacobi forms to eta-products. In the limit z! 0 one has: lim z!0 10(,, z) (2 iz) 2 = ( ) 24 ( ) 24

The automorphic lift provides a link from Jacobi forms to eta-products. In the limit z! 0 one has: lim z!0 10(,, z) (2 iz) 2 = ( ) 24 ( ) 24 Wall-crossing: 1/4-BPS 1/2-BPS 1/2-BPS

The automorphic lift provides a link from Jacobi forms to eta-products. In the limit z! 0 one has: lim z!0 10(,, z) (2 iz) 2 = ( ) 24 ( ) 24 Wall-crossing: 1/4-BPS 1/2-BPS 1/2-BPS Note: ( ) 24 ( ) 24 = 1A ( ) 1A ( ) This fact generalises to all twining genera: lift z! 0 g(,z) g(,z, ) g ( ) g ( ) [Cheng][Govindarajan][Eguchi, Hikami]

Multiplicative Lift for Generalised Mathieu Moonshine Following Dijkgraaf, Moore, Verlinde, Verlinde we define the second quantized twisted twining genus via the lift g,h(,z) g,h =exp 1 X p N (T N g,h)(,z) N=0 Here T N is the twisted equivariant Hecke operator: (T N g,h)(,z)= 1 N X a,d>0,ad=n Xd 1 b=0 g,h( ab 0 d ) a + b g d,g b h,az a d [Ganter][Carnahan]

Multiplicative Lift for Generalised Mathieu Moonshine Following Dijkgraaf, Moore, Verlinde, Verlinde we define the second quantized twisted twining genus via the lift g,h(,z) g,h =exp 1 X p N (T N g,h)(,z) N=0 By taking the z! 0 limit of g,h(,, z) we reproduce g,h ( ) all of Mason s generalised eta-products! lim z!0 p g,h(,,z) =exp[ 1X N=0 p N+1 T N g,h (, 0)] = g,h ( )

We have thus given strong evidence for the following triality of generalised M 24 -moonshines: Siegel modular forms g,h(,z, ) wall-crossing M 24 multiplicative lift ( second quantization ) g,h ( ) g,h(,z) Mason s generalised eta-products twisted twining genera (Jacobi forms)

Summary and Outlook

Summary We have established that generalised Mathieu moonshine holds by computing all twisted twining genera g,h. Twisted twining genera can be expanded in projective characters of C M24 (g). A key role is played by the third cohomology group H 3 (M 24,U(1)). The automorphic lift provides a link to Mason s generalised eta-products.

Outlook Can one construct a generalised Kac-Moody algebra for each conjugacy class [g] 2 M 24? (c.f. [Borcherds][Carnahan]) Relation to twisted dyons in CHL models, wall-crossing...? Relation with BPS-algebras à la Harvey Moore...? Generalised Umbral Moonshine...? [Cheng, Duncan, Harvey] It would be interesting to revisit the Generalised Monstrous Moonshine conjecture in light of our results for. M 24 Z g,h Some of the Norton series vanish; could this be explained via obstructions arising from H 3 (M,U(1))? H 3 (M,U(1)) not known but believed to contain Z 48. [Mason]

What does M 24 act on? Our results strongly suggests that there is a holomorphic vertex operator algebra underlying Mathieu Moonshine...