Eta-Products, BPS States and K3 Surfaces YANG-HUI HE Dept of Mathematics, City University, London; School of Physics, NanKai University; Merton College, University of Oxford University of Bath, Nov. 2014 YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 1 / 26
Acknowledgements 1308.5233 YHH, John McKay 1211.1931 YHH, John McKay, James Read; 1309.2326 YHH, James Read 1402.3846 YHH, Mark van Loon 1410.2227 Sownak Bose, James Gundry, YHH YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 2 / 26
A Pair of Classical Functions ∞ ∞ (1 − q n ) − 1 = π k q k ϕ ( q ) = � � n =1 k =0 Euler ϕ and Dedekind η : ∞ 1 1 (1 − q n ) = q 24 ϕ ( q ) − 1 � η ( q ) = q 24 n =1 Notation: upper-half plane H := { z : Im( z ) > 0 } ; nome q = exp(2 πiz ) Remarks, 24 is special mathematically and physically q-expansion π k = # integer partitions of k 1 η is modular form of weight 1 24 is crucial ( 24 comes from ζ ( − 1) through 2 : q Bernoulli B 2 and Eisenstein E 2 ( q ) ) Familiar to string theorists, bosonic oscillator partition function ∞ α − n · α n = � ∞ d n q n = ϕ ( q ) 24 = qη ( q ) − 24 ; Hardy-Ramanujan G ( q ) := Tr q � n =1 n =0 gives asymptotics ❀ Hagedorn (24 comes from conformal anomaly ζ ( − 1) ) Rmk: 24 iff modularity of 1-loop diagram YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 3 / 26
Elliptic Curves For elliptic curve y 2 = 4 x 3 − g 2 x − g 3 , two (related) functions “discriminate” – test isomorphism/inequivalent modular forms ∆ = g 3 2 − 27 g 2 Modular Discriminant: 3 j = 1728 g 3 θ E 8 Klein j-Invariant: ∆ = 2 ∆ In terms of modular parameter z , ( x, y ) = ( ℘ ( z ) , ℘ ′ ( z )) ∞ ∞ ∆( z ) = η ( z ) 24 := q (1 − q n ) 24 = � � τ ( n ) q n n =1 n =1 Similarly (only 1980’s! by Borcherds in his proof of Moonshine) � 1 p − 1 � � (1 − p n q m ) c n ∗ m j ( p ) − j ( q ) = q m,n =1 c n q n = 1 for j ( q ) = � q + 744 + 196884 q + . . . n YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 4 / 26
Multiplicativity Multiplicative Function/Sequence { a n } (with a 1 = 1 ) (Completely) Multiplicative : a m ∗ n = a m a n , m, n ∈ Z > 0 ; (Weakly) Multiplicative : a m ∗ n = a m a n , gcd( m, n ) = 1 ; a n Rmk: Dirichlet transform ❀ interesting: L ( s ) = � n s , e.g. n =1 a n = 1 ❀ L ( s ) = ζ ( s ) ∞ Ramanujan: ∆( q ) = qG ( q ) − 1 = τ ( n ) q n ❀ Ramanujan tau-function τ ( n ) � n =1 is (weakly) multiplicative: n 1 2 3 4 5 6 7 8 9 10 τ ( n ) 1 − 24 252 − 1472 4830 − 6048 − 16744 84480 − 113643 − 115920 24 is crucial in ∆( z ) = η ( z ) 24 24 24 τ ( n ) 2 ≡ c n ( j ) 2 ≡ 42(mod70) Fun fact [YHH-McKay, 2014] � � n =1 n =1 YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 5 / 26
Multiplicative Eta-Products η ( q ) 24 is multiplicative, are there others made from η ? Define Frame Shape [J. S. Frame, or cycle shape] ( t : cycle length) t t � � η ( q n i ) F ( z ) = [ n 1 , n 2 , . . . , n t ] := η ( n i z ) = i =1 i =1 Dummit-Kisilevsky-McKay (1982): a 1 = 1 and [ n 1 , n 2 , . . . , n t ] is partition of 24 Balanced: n 1 > . . . > n t , n 1 n t = n 2 n t − 1 = . . . there are precisely 30 which are multiplicative out of π (24) = 1575 each is a modular form of weight k = t/ 2 , level N = n 1 n t , Jacobi character χ d − 1 � N az + b � 2 a b ) = ( cz + d ) kχkF ( z ) , ( − 1) , d odd d ∈ Γ0( N ) F ( χ = , � d � cz + d c d , d even N in summary: YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 6 / 26
The 30 k N eta-product χ k N eta-product χ 2 15 [15 , 5 , 3 , 1] 1 [124] = ∆( q ) 12 1 1 14 [14 , 7 , 2 , 1] 1 [28 , 18] 8 2 1 24 [12 , 6 , 4 , 2] 1 [36 , 16] [112 , 12] 6 3 1 11 1 [212] [102 , 22] 4 1 20 1 � − 1 [92 , 32] [44 , 22 , 14] � 27 1 5 4 d k eta-product [82 , 42] [62 , 32 , 22 , 12] 32 1 4 6 1 “ 3 [83] 2 ” [64] [54 , 14] 36 1 5 1 � − 23 “ 1 2 ” [24] � [44 , 24] 1 23 [23 , 1] 8 1 d � − 11 � [38] 9 1 44 [22 , 2] d � − 7 � − 2 � [82 , 4 , 2 , 12] � 63 [21 , 3] 3 8 d d � − 20 � − 7 � [73 , 13] � 7 80 [20 , 4] d d � − 3 � − 3 � [63 , 23] � 108 [18 , 6] 12 d d � − 2 � − 1 [46] � � 16 128 [16 , 8] d d � − 1 [122] � 144 d Q: Do these show up as partition functions in physics? YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 7 / 26
Type II on K 3 × T 2 4-D, N = 4 theory ( ≃ heterotic on T 6 ) Any other preserving N = 4 ? cf. Aspinwall-Morrison: freely-acting quotients of K3, a total of 14, Nikulin Classification (preserves the (2 , 0) -form, 1979): Z 2 Z 3 Z 4 Z n =2 ,..., 8 , m =2 , 3 , 4 , Z 2 × Z 4 , Z 2 × Z 6 , 2 , 2 CHL orbifold [Chaudhuri-Hockney-Lykken, 1995]: S 1 × S 1 with Nikulin involution and simultaneously Type IIB on K 3 × ˜ Z t � S 1 by exp(2 πi/t ) Dual to het on T 4 × ˜ S 1 × S 1 with Z t � Γ 20 , 4 Narain lattice YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 8 / 26
Dyonic Spectrum and 1/2-BPS States Dyons ( q e , q m ) [cf. precision BH micro-state counting, Sen-David, 2006]: D5-branes wrapping K 3 × S 1 and Q 1 D1-branes wrapping S 1 S 1 with (2 − k, J ) units of ( S 1 , ˜ KK monopole for ˜ S 1 ) momentum q 2 q 2 e = 2( k − 2) /t , m = 2( Q 1 − 1) , q e · q m = J in the unorbifolded case (Het on T 6 ): 1/2-BPS states with charge n = 1 2 q 2 e has degeneracy [Sen, Dabholkar-Denef-Moore-Pioline, 2007] η ( q ) − 24 = ∆( q ) − 1 = [1 24 ] − 1 = 1 � d n q n 16 n = − 1 In general [Govindarajan-Krishna, 2009]: t η ( n i z )) − 1 = 1 � � d n q n/t ( 16 n = − 1 i =1 in summary: YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 9 / 26
The 14 k N eta-product χ Nikulin K3 [1 24 ] = ∆( q ) 12 1 1 - [2 8 , 1 8 ] 8 2 1 Z 2 [3 6 , 1 6 ] 6 3 1 Z 3 [2 12 ] 4 1 Z 2 × Z 2 � � [4 4 , 2 2 , 1 4 ] − 1 5 4 Z 4 d [6 2 , 3 2 , 2 2 , 1 2 ] 4 6 1 Z 5 [5 4 , 1 4 ] 5 1 Z 6 [4 4 , 2 4 ] 8 1 Z 2 × Z 4 [3 8 ] 9 1 Z 3 × Z 3 � � [8 2 , 4 , 2 , 1 2 ] − 2 3 8 Z 7 d � � [7 3 , 1 3 ] − 7 7 Z 8 d � � [6 3 , 2 3 ] − 3 12 Z 2 × Z 6 d � � [4 6 ] − 1 16 Z 4 × Z 4 d [11 2 , 1 2 ] 2 11 1 Z 11 Rmk: No k ≤ 1 and only one k = 2 YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 10 / 26
Special K3 Surfaces X � � � = H 2 ( X ; Z ) ∩ H 1 , 1 ( X ) = N´ eron-Severi Lattice NS ( X ) := ker γ → γ Ω Divisors/(Alg. equiv.); Picard Number ρ ( X ) = rk( NS ( X )) Mordell-Weil Lattice MW ( X ) = rk( X Q ) (cf. Birch-Swinerton-Dyer for E ) K3 Alg. K3 Elliptic K3 . . . Exceptional Generically Γ 20 ⊂ E 8 ( − 1) 2 ⊕ ( U 2 ) 3 NS ( X ) { 0 } H Z B Z ⊕ F Z . . . ρ ( X ) 0 1 2 . . . 20 Classification results (each a subset) Exceptional (“singular”) [Shioda-Inose, 1977]: top ρ ( X ) = 20 ; 1:1 with �� � a b integral binary quadratic forms SL (2; Z ) similarity; b c Extremal non-Elliptic: ??? Extremal Elliptic [Shimada-Zhang]: + finite MW ( X ) , a total of 325; Extremal Semi-Stable Elliptic [Miranda-Persson, 1988]: + Type I n fibres only, a total of 112 YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 11 / 26
Elliptic Semi-Stable Extremal K3 K 3 → P 1 with elliptic fibration: { y 2 = 4 x 3 − g 2 ( s ) x − g 3 ( s ) } ⊂ C [ x, y, s ] Only I n sing. fibres, s.t. I n partition of 24 , i.e., EssE ∼ Frame Shape Shioda-Tate: ρ = � ( n i − 1) + rk( MW ) + 2 = 26 + rk( MW ) − t ❀ t ≥ 6 i Extremal: t = 2 k = 6 Klein j-invariant is a rational function 1728 j ( s ) = g 3 g 3 2 ( s ) 2 ( s ) 1 3 ( s ) : P 1 → P 1 , J ( s ) = ∆( s ) = s − s.t. g 3 2 ( s ) − 27 g 2 8 preimages of J ( s ) = 0 all multiplicity (ramification index) 3; 12 preimages of J ( s ) = 1 with ramification index 2; t preimages of J ( s ) = ∞ , ramification indices [ n 1 , . . . , n t ] ; ? ∃ ramification points x 1 , . . . , x m � = (0 , 1 , ∞ ) but for t = 6 , no such points. YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 12 / 26
Grothendieck’s Dessin d’Enfant → P 1 ramified only at (0 , 1 , ∞ ) ı Map: rational map β : Σ − Belyˇ Theorem [Belyˇ ı]: (1980) β exists ⇔ Σ can be defined over Q ( β, Σ) Belyˇ ı Pair Dessin d’Enfants = β − 1 ([0 , 1] ∈ P 1 ) ⊂ Σ bi-partite graph on Σ : label all the preimages β − 1 (0) black and β − 1 (1) white, then β − 1 ( ∞ ) lives one per face and β − 1 ([0 , 1]) gives connectivity B blacks and W whites, with valency of each = ramification index r 0 (1) , r 0 (2) , . . . , r 0 ( B ) Ramification data / Passport: r 1 (1) , r 1 (2) , . . . , r 1 ( W ) r ∞ (1) , r ∞ (2) , . . . , r ∞ ( I ) Rmk: Dimer Models on T 2 = Quivers on Toric CY3 = Dessins [Hanany-YHH-Jejjala-Pasuconis-Ramgoolam-Rodriguez-Gomez] YANG-HUI HE (London/Tianjin/Oxford) Multiplicative Bath, Nov, 2014 13 / 26
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