Progress in Quantum Field Theory and String Theory Osaka City University, 06 April 2012 Counting BPS states in E#string theory Kazuhiro Sakai!YITP, Kyoto University" arxiv:1111.3967 arxiv:1203.2921
String theory predicts the existence of nontrivial QFTs in 6D. The worldvolume theory of multiple M5 branes ### (2,0) SUSY!16 supercharges" What about theories with (1,0) SUSY? Heterotic string theories on K3!16 8 supercharges" What happens when instantons in K3 shrink to zero size? Small SO(32) instantons extra Sp(n) gauge symmetry worldvolume theory of n type I D5#branes What about small E8 instantons?!witten 95"
M#theory description of the E 8 E 8 heterotic string theory on K3 9-brane M5-brane 9-brane M2-brane x 0 x 5 E-string a zero#size E 8 #instanton x 11 K3 x 6 x 9
E#string theory!ganor#hanany 96"!Seiberg#Witten 96" 6D (1,0)#supersymmetric local QFT decoupled from gravity no vector multiplets Coulomb branch ### a tensor multiplet Higgs branch = the moduli space of an E8 instanton fundamental excitations ### strings global E8 symmetry E#string = 1 heterotic E 2 8 E 8 string
Toroidal compactification down to 4D a 4D N = 2 theory described by Seiberg#Witten theory SU(n) gauge theories E#string theory Seiberg#Witten curves are known realized by String Theory on certain CY3 Lagrangian description toric CY3 no Lagrangian description non#toric CY3 Nekrasov partition functions are known Any analogue? Yes!!at least partly"
Heterotic ## F#theory duality!morrison#vafa 96" Heterotic on K3 E#strings on R 6 F#theory on CY3 E#strings on R 6 M#theory on CY3 E#strings on R 5 S 1 = 1 2 local K3 surface!the total space of the canonical bundle of a K3 surface" 1 2 1 local 2K3!non#compact CY3" 1 2 K3
Partition function of BPS wound E#strings! = prepotential" F 0 (ϕ, τ )= n=1 k=0 N n,k m=1 p mn q mk m 3 p := e 2πiϕ, q := e 2πiτ n : winding number k : momentum N n,k : BPS multiplicity!instanton number"!klemm#mayr#vafa 96" k 0 1 2 3 4 5 n 1 1 252 5130 54760 419895 2587788 2 0 0 9252 673760 20534040 389320128 3 0 0 0 848628 115243155 6499779552 4 0 0 0 0 114265008 23064530112 5 0 0 0 0 0 18958064400....
Winding number expansion F 0 (ϕ, τ )= n=1 Q n Z n (τ ) Q := q 1/2 p, p := e 2πiϕ, q := e 2πiτ 1 Z n : partition function of N = 4 U(n) topological SYM on 2K3 Z 1 = E 4 η 12, Z 2 = E 2E 2 4 +2E 4E 6 24η 24, Z n = P 6n 2(E 2,E 4,E 6 ) η 12n!Minahan#Nemeschansky#Vafa#Warner 98" E 2n (τ ) : Eisenstein series η(τ ) : Dedekind eta function
The expansion coe$cients Zn at low orders can be computed either from the Seiberg#Witten curve or from!minahan#nemeschansky#warner 97" y 2 =4x 3 1 12 E 4(τ )u 4 x 1 216 E 6(τ )u 6 +4u 5 the modular anomaly equation E2 Z n = 1 24 and the gap condition n 1 k=1 k(n k)z k Z n k q n/2 Z n = 1 n 3 + O(qn )
Nekrasov#type expression for the prepotential!k.s. 12" F 0 = (2 2 ln Z) =0 Z = R Q R a,b,c,d (i,j) R ab ϑ ab ( 1 2π (j i), τ ) 2 ϑ 1 a c,1 b d ( 1 2π h ab,cd(i, j), τ ) 2 R =(R 11,R 10,R 00,R 01 ) a, b, c, d =0, 1 Rab : partition µ ab,2 µ ab,1 ϑ ab (z, τ ) : Jacobi theta functions h ab,cd (i, j) := µ ab,i + µ cd,j i j +1!relative hook#length" i j Rcd Rab
The expression coincides with a special case of!k.s. 12" the elliptic generalization of the Nekrasov partition function for SU(4) SYM with Nf = 8 massless fundamental hypermultiplets!nekrasov 02"!Hollowood#Iqbal#Vafa 03" Z = R ( p) R 4 k=1 (i,j) R k ( ϑ 1 ak + 1 2π (j i), τ ) 8 4 l=1 ϑ ( 1 ak a l + 1 2π h kl(i, j), τ ) 2 a 1 =0, a 2 = 1 2, a 3 = 1+τ 2, a 4 = τ 2 R =(R 1,R 2,R 3,R 4 )=(R 11,R 10,R 00,R 01 )
The prepotential represents g = 0 topological string amplitude F 0 = F 1 2 K3 0 However, Z = Z 1 2 K3 ( ) := exp g=0 2g 2 F 1 2 K3 g Di%erence of modular anomalies E2 Z = 1 12 2 2 φ Z E2 Z 1 2 K3 = 1 24 2 φ ( φ + 1)Z 1 2 K3!Hosono#Saito#Takahashi 99" E2 F 0 = 1 24 ( φf 0 ) 2 φ = 1 2πi ϕ = Q Q
Part II 1 Topological string amplitudes for the most general local 2K3
Topological string amplitudes at higher genus Holomorphic anomaly equation!bershadsky#cecotti#ooguri#vafa 93" īf g = 1 ( ) g 1 2 Cī j ke 2K G j j G k k D j D k F g 1 + D j F h D k F g h Recent developments F g are quasi#automorphic forms!aganagic#bouchard#klemm 06" For SU(2) gauge theories: F g are polynomials in h=1 F g are polynomials in a finite number of generators Ẽ 2 := E 2 ( τ ) Direct integration method!grimm#klemm#mariño#weiss 06" HAE: Ẽ2 F g =!Yamaguchi#Yau 04"!Alim#Länge 07"
1 Topological string amplitudes for the most general local K3 Z 1 2 K3 (ϕ, τ, µ; ) = exp [ g=0 2g 2 n=1 For low g and n, Zg,n can be determined by symmetry T g,n (τ, µ) : Z g,n (τ, µ) = holomorphic anomaly equation eπinτ η(τ ) 12n T g,n(τ, µ) E2 Z 1 2 K3 = 1 24 2 φ ( φ + 1)Z 1 2 K3 2 e 2πinϕ Z g,n (τ, µ) W (E 8 )#invariant quasi#jacobi form of weight 2g 2+6n and index gap condition No BPS states for k < n!except for n =1"!Minahan#Nemeschansky#Vafa#Warner 98"!Hosono#Saito#Takahashi 99" n ]
Seiberg#Witten curve for the E#string theory!eguchi#k.s. 02" y 2 =4x 3 fx g f = 4 a n u 4 n, g = 6 b n u 6 n. n=0 n=0 a n (τ, µ), b n (τ, µ) are expressed in terms of W (E 8 )#invariant Jacobi forms A n (τ, µ), B n (τ, µ) a 0 = 1 12 E 4, a 1 =0, a 2 = 6 ( ) E 4 A 2 + A 2 1, E 4 b 0 = 1 216 E 6, b 1 = 4 E 4 A 1, b 2 = 5 6E 2 4 ( E 2 4 B 2 E 6 A 2 1 ),
W (E 8 )#invariant Jacobi forms!k.s. 11" Θ(τ, µ) = 1 2 4 8 ϑ k (µ j, τ ) k=1 j=1 Theta function associated with the E 8 root lattice A 2 (τ, µ) = 8 H {Θ(2τ, 2 µ)}, 9 A 1(τ, µ) =Θ(τ, µ), A 3 (τ, µ) = 27 28 H {Θ(3τ, 3 µ)}, A 4(τ, µ) =Θ(τ, 2 µ), A 5 (τ, µ) = 125 H {Θ(5τ, 5 µ)}, 126 B 2 (τ, µ) = 32 5 H {e 1(τ )Θ(2τ, 2 µ)}, B 3 (τ, µ) = 81 80 H { h(τ ) 2 Θ(3τ, 3 µ) }, B 4 (τ, µ) = 16 15 H { ϑ 4 (2τ ) 4 Θ(4τ, 4 µ) }, B 6 (τ, µ) = 9 10 H { h(τ ) 2 Θ(6τ, 6 µ) }. H { } : sum of all possible SL(2, Z) transforms e 1 (τ )= 1 12 ( ϑ3 (τ ) 4 + ϑ 4 (τ ) 4), A n (τ, 0) = E 4, h(τ )=ϑ 3 (2τ )ϑ 3 (6τ )+ϑ 2 (2τ )ϑ 2 (6τ ). B n (τ, 0) = E 6.
F 0 = 2 φ t, F 1 = 1 2 ln ω 1 12 ln + 1 12 ln 1 2 φ, F 2 = 1 96 E 2 + 1 2 48Ẽ2 φ ln ω + 1 96Ẽ2( φ ln ω) 2 1 576 (Ẽ2 2 Ẽ 4 ) φ t φ ln ω F 3 = t := 2πi( τ τ ) Ẽ 2n := E 2n ( τ ) := η( τ ) 24!K.S. 11" 1 1920 (5Ẽ2 2 +3Ẽ 4 ) 2 φ t 1 207360 (35Ẽ3 2 + 51Ẽ 4 Ẽ 2 86Ẽ 6 )( φ t) 2, Structure of F g (g 2) 1 2 Amplitudes for the local K3 2πω τ 0 2πω polynomial in Ẽ 2k, m φ ln ω, n φ t each term contains 2g 2 φ s quasi-modular form of weight 2g 2 ([Ẽ 2k ]=2k, [ mφ ln ω] =0, [ nφ t]= 2 )
Summary We have found a Nekrasov#type expression for the Seiberg#Witten prepotential for the E#string theory. We have constructed topological string amplitudes 1 for the most general local K3!up to genus three". 2
SU(2) Seiberg#Witten theories with various flavor symmetries Seiberg#Witten curves ### constructed for all theories!seiberg#witten 94"!Minahan#Nemeschansky 96"!Ganor#Morrison#Seiberg 96"!Minahan#Nemeschansky#Warner 97"!Eguchi#K.S. 02" All#genus partition functions: 6D The Ê 8 #string theory needs further investigation! 5D E 8 E 7 E 6 E 5 E 4 E 3 E 2 E 1 E 0 Ẽ 1 4D E 8 E 7 E 6 D 4!Minahan#Nemeschansky 96" D 3 D 2 D 1 D 0 Partition functions were constructed E 9 = Êby 8 the ruled vertex formalism!diaconescu#florea 05" A 2 A 1 A 0!Argyres#Plesser#Seiberg#Witten 95" Nekrasov partition functions are available for these theories!nekrasov 02"