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Introduction to Stringy Moonshine ” June, 2018 Tohru Eguchi, University of Tokyo

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Introduction to Stringy Moonshine ” June, 2018 Tohru Eguchi, University of Tokyo ”Introduction to Stringy Moonshine ” June, 2018 Tohru Eguchi, University of Tokyo Monstrous Moonshine Several years ago we found some curious phenomenon in string theory [1], i.e. appearance of exotic discrete symmetries in the theory. This is now called as moon- shine phenomenon and is now under intensive study. Today I would like to give you a brief introduction to moonshine phenomena which may possibly play an in- teresting role in string theory in the future. Before going to the moonshine phenomenon in string theory let me briefly recall the story of monstrous moon- shine which is well-known. Modular J function has a q-series expansion 1 J(q) = + 744 + 196884q + 21493760q2 + 864299970q3 q +20245856256q4 + 333202640600q5 + ··· : ( ) aτ + b a b q = e2πiτ ; Im(τ ) > 0;J(τ ) = J( ); 2 SL(2;Z) cτ + d c d It turns out q-expansion coefficients of J-function are decomposed into a sum of dimensions of irreducible representations of the monster group M as 196884 = 1 + 196883; 21493760 = 1 + 196883 + 21296876; 864299970 = 2 × 1 + 2 × 196883 + 21296876 + 842609326; 20245856256 = 1 × 1 + 3 × 196883 + 2 × 21296876 +842609326 + 19360062527; ··· : 1 Dimensions of irreducible representations of monster are in fact given by f1; 196883; 21296876; 842609326; 18538750076; 19360062527 · · · g Monster group is the largest sporadic discrete group, of order ≈ 1053 and the strange relationship between modular form and the largest discrete group was first noted by McKay. To be precise we may write as X n J1(τ ) = J(q) − 744 = c(n)q ; c(0) = 0 X n=−1 n = T rV (n) 1 × q ; dimV (n) = c(n) n=−1 McKay-Thompson series is given by X n Jg(τ ) = T rV (n) g × q ; g 2 M n=−1 where T rV (n) g denotes the character of a group ele- ment g in the representation V (n). This depends on the conjugacy class g of M. If McKay-Thompson se- ries is known for all conjugacy classes, decomposition of V (n) into irreducible representations become uniquely determined. Series Jg are modular forms with respect to subgroups of SL(2;Z) and possess similar proper- ties like the modular J-function such as the genus=0 (Hauptmodul) property. Phenomenon of monstrous moonshine has been un- derstood mathematically in early 1990’s using the tech- nology of vertex operator algebra. However, we still do 2 not have a ’simple’ physical explanation of this phe- nomenon. Elliptic genus We now consider string theory compactified on K3 surface. K3 surface is a complex 2-dimensional hy- perKahler¨ manifold and is ubiquitous in string theory. It possesses SU(2) holonomy and a holomorphic 2-form. Thus the string theory on K3 has an N=4 superconfor- mal symmetry and contains the level k = 1 affine SU(2) symmetry and the central charge c = 6. Now instead of modular J-function we consider the elliptic genus of K3 surface. Elliptic genus describes the topological invariants of the target manifold and counts the number of BPS states in the theory. Using world- sheet variables it is written as F +F 4πizJ 3 L − c L¯ − c − L R L;0 0 24 0 24 Zelliptic(z; τ ) = T rHL×HR ( 1) e q q¯ Here L0 denotes the zero mode of the Visasoro operators and FL and FR are left and right moving fermion num- bers. In ellitpic genus the right moving sector is frozen to the supersymmetric ground states (BPS states) while in the left moving sector all the states in the Hilbert space HL contribute. N=4, level k theory contains a SUSY algebra i ∗ j ij k ij k fG¯ ; G¯ g = 2δ L¯0 − δ ; (i; j = 1; 2) =) L¯0 ≥ 0 0 2 4 k BPS states possess L0 = 4 3 In general it is difficult to compute elliptic genus, however, we were able to evaluate it by making use of Gepner models. Elliptic genus of K3 surface is given by [2]: " # ( )2 ( )2 ( )2 θ2(z; τ ) θ3(z; τ ) θ4(z; τ ) ZK3(z; τ ) = 8 + + θ2(0; τ ) θ3(0; τ ) θ4(0; τ ) Here θi(τ; z) are Jacobi theta functions. 1 ZK3(z = 0) = 24;ZK3(z = ) = 16 + O(q); 2 1 + τ − 1 1 ZK3(z = ) = 2q 2 + O(q 2 ) 2 It is known that the elliptic genus of a complex D-dimensional manifold is a Jacobi form of weight=0 and index=D/2. When D=2, space of Jacobi form is one-dimensional and given by the above formula. Jacobi form (weight k and index m) is defined by 2 '(τ; z + aτ + b) = e−2πim(a τ +2az)'(τ; z); 2 aτ + b z k 2πimcz '( ; ) = (cτ + d) e cτ +d '(τ; z) cτ + d cτ + d We would like to study the decomposition of the el- liptic genus in terms of irreducible representations of N=4 SCA. In N=4 SCA, hightest-weight states jh; `i are parametrized by j i j i 3j i j i L0 h; ` = h h; ` ; J0 h; ` = ` h; ` and the theory possesses two different type of represen- tations, i.e. BPS and non-BPS representations. In the 4 case of k = 1 there are representations (in Ramond sec- tor) 1 1 BPS rep. h = ; ` = 0; 4 2 1 1 non-BPS rep. h > ; ` = 4 2 Character of a representation is given by F L 4πizJ 3 T rR(−1) q 0 e 0 F L Its index is given by the value at z = 0, T rR(−1) q 0 . BPS representations have a non-vanishing index index (BPS, ` = 0) = 1 1 index (BPS, ` = ) = −2 2 Character function of ` = 0 BPS representation has the form [3] 2 R~ θ1(z; τ ) chh= 1 ;`=0(z; τ ) = µ(z; τ ) 4 η(τ )3 where πiz X 1 n(n+1) 2πinz −ie q 2 e µ(z; τ ) = (−1)n − n 2πiz θ1(z; τ ) n 1 q e On the other hand the character of non-BPS represen- tations are given by 2 ~ − 3 θ1(z; τ ) R h 8 chh> 1 ;`= 1 = q 4 2 η(τ )3 These have vanishing indices index (non-BPS rep) = 0 5 At the unitarity bound non-BPS representation splits into a sum of two BPS representations 2 − 3 θ1 ~ ~ h 8 R R lim q = chh= 1 ;`= 1 + 2chh= 1 ;`=0 ! 1 3 4 2 4 h 4 η Function µ(z; τ ) is a typical example of the so-called Mock theta functions (Lerch sum or Appell function). Mock theta functions look like theta functions but they have anomalous modular transformation laws and are difficult to handle. Recently there were developments in understanding the nature of Mock theta functions due to Zwegers [4]. He has introduced a method of regu- larization which is similar to those used in physics and improved the modular property of mock theta functions so that they transform as analytic Jacobi forms. It is possible to derive the following idenities ( ) 2 2 R~ θ2(z; τ ) θ1(z; τ ) chh= 1 ;`=0(z; τ ) = + µ2(τ ) 4 θ (0; τ ) η(τ )3 ( 2 ) 2 2 θ3(z; τ ) θ1(z; τ ) = + µ3(τ ) θ (0; τ ) η(τ )3 ( 3 ) 2 2 θ4(z; τ ) θ1(z; τ ) = + µ (τ ) 4 3 θ4(0; τ ) η(τ ) where 1 1 +τ τ µ2(τ )=µ(z= ; τ ); µ3(τ )=µ(z= ; τ ); µ4(τ )=µ(z= ; τ ) 2 2 2 πiz X 1 n(n+1) 2πinz −ie q 2 e µ(z; τ ) = (−1)n − n 2πiz θ1(z; τ ) n 1 q e 6 Then we can rewrite the ellitpic genus as X4 θ (z; τ )2 R~ − 1 ZK3 = 24chh= 1 ;`=0(z; τ ) 8 µi(τ ) 4 η(τ )3 i=2 Using q-expansion of functions µi we find X n− 1 8 (µ2(τ ) + µ3(τ ) + µ4(τ )) = −2 A(n)q 8 n=0 X R~ R~ ZK3 = 24ch 1 (z; τ ) + 2 A(n)ch 1 1 (z; τ ) h= 4 ;`=0 h= 4 +n;`= 2 n≥0 At smaller values of n, Fourier coefficients A(n) may be obtained by direct inspection. We find, A(0) = −1 n 1 2 3 4 5 6 7 8 ··· A(n) 45 231 770 2277 5796 13915 30843 65550 ::: Surprize: the dimensions of irreducible reps. of Math- ieu group M24 appear dimensions : f 45 231 770 990 1771 2024 2277 3312 3520 5313 5544 5796 10395 · · · g A(6) = 13915 = 3520 + 10395; A(7) = 30843 = 10395 + 5796 + 5544 + 5313 + 2024 + 1771 Mathieu moonshine [1] M24 is a subgroup of S24 (permutation group of 24 ob- 9 jects) and its order is given by ≈ 10 . M24 is known for 7 its many interesting arithmetic properties and in partic- ular intimately tied to Golay code of error corrections. Mathieu group appeared before in the work of Mukai on K3 surface. Mukai[5] considered K3 surfaces with finite automor- phism group. Then these groups are sugbgroups of M23. | Twisted Elliptic Genus Dimension of a representation equals the trace of the identity representation: we may identify as A(n) = T rVn 1 ∗ ∗ ∗ V1 = 45 + 45 ;V2 = 231 + 231 ;V3 = 770 + 770 ; ··· We may consider the trace of other group elements in M24 2 Ag(n) = T rVn g; g M24 T r g depends only on the conjugacy class of g. There exists 26 conjugacy classes fgg in M24 and also 26 ir- reducible representations fRg. We have the character table given by g χR = T rR g 8 1A 2A 3A 5A 4B 7A 7B 8A 6A 11A 15A 15B 14A 14B 23A 23B 12B 6B 4C 3B 2B 10A 21A 21B 4A 12A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 23 7 5 3 3 2 2 1 1 1 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 252 28 9 2 4 0 0 0 1 -1 -1 -1 0 0 -1 -1 0 0 0 0 12 2 0 0 4 1 253 13 10 3 1 1 1 -1 -2 0 0 0 -1 -1 0 0 1 1 1 1 -11 -1 1 1 -3 0 1771 -21 16 1 -5 0 0 -1 0 0 1 1 0 0 0 0 -1 -1 -1 7 11 1 0 0 3 0 3520 64 10 0 0 -1 -1 0 -2 0 0 0 1 1 1 1 0 0 0 -8 0 0 -1 -1 0 0 + − + − − + 45 -3 0 0 1 e7 e7 -1 0 1 0 0 -e7 -e7 -1 -1 1 -1 1 3 5 0 e7 e7 -3 0 − + − + + − 45 -3 0 0 1 e7 e7 -1 0 1 0 0 -e7 -e7 -1 -1 1 -1 1 3 5 0 e7 e7 -3 0 + − + − − + 990 -18 0 0 2 e7 e7 0 0 0 0 0 e7 e7 1 1 1 -1 -2 3 -10 0 e7 e7 6 0 − + − + + − 990 -18 0 0 2 e7 e7 0 0 0 0 0 e7 e7 1 1 1 -1 -2 3 -10 0 e7 e7 6 0 + − − + 1035 -21 0 0 3 2 e7 2 e7 -1 0 1 0 0 0 0 0 0 -1 1 -1 -3 -5 0 -e7 -e7 3 0 − + + − 1035 -21 0 0 3 2 e 2 e -1 0 1 0 0 0 0 0 0 -1 1 -1 -3 -5 0 -e -e 3 0 0 7 7 7 7 1035 27 0 0 -1 -1 -1 1 0 1 0 0 -1 -1 0 0 0 2 3 6 35 0 -1 -1 3 0 + − 231 7 -3 1 -1 0 0 -1 1 0 e15 e15 0 0 1 1 0 0 3 0 -9 1 0 0 -1 -1 − + 231 7 -3 1 -1 0 0 -1 1 0 e15 e15 0 0 1 1 0 0 3 0 -9 1 0 0 -1 -1 + − 770 -14 5 0 -2 0 0 0 1 0 0 0 0 0 e23 e23 1 1 -2 -7 10 0 0 0 2 -1 − + 770 -14 5 0 -2 0 0 0 1 0 0 0 0 0 e23 e23 1 1 -2 -7 10 0 0 0 2 -1 483 35 6 -2 3 0 0 -1 2 -1 1 1 0 0 0 0 0 0 3 0 3 -2 0 0 3 0 1265 49 5 0 1 -2 -2 1 1 0 0 0 0 0 0 0 0 0 -3 8 -15 0 1 1 -7 -1 2024 8 -1 -1 0 1 1 0 -1 0 -1 -1 1 1 0 0 0 0 0 8 24 -1 1 1 8 -1 2277 21 0 -3 1 2 2 -1 0 0 0 0 0 0 0 0 0 2 -3 6 -19 1 -1 -1 -3 0 3312 48 0 -3 0 1 1 0 0 1 0 0 -1 -1 0 0 0 -2 0 -6 16 1 1 1 0 0 5313 49 -15 3 -3 0 0 -1 1 0 0 0 0 0 0 0 0 0 -3 0 9 -1 0 0 1 1 5796 -28 -9 1 4 0 0 0 -1 -1 1 1 0 0 0 0 0 0 0 0 36 1 0 0 -4 -1 5544 -56 9 -1 0 0 0 0 1 0 -1 -1 0 0 1 1 0 0 0 0 24 -1 0 0 -8 1 10395 -21 0 0 -1 0 0 1 0 0 0 0 0 0 -1 -1 0 0 3 0 -45 0 0 0 3 0 ( ) 1 p Character table of the Mathieu group M24 Here we have used e = −p − 1 .
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