Counting BPS states in E-string theory. Kazuhiro Sakai

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1 String Theory Group at Nagoya University December 4, 202 Counting BPS states in E-string theory Kazuhiro Sakai (YITP, Kyoto University) JHEP06(202)027 (arxiv: ) JHEP09(202)077 (arxiv: )

2 Plan. What is the E-string theory? 2. Compactification down to 4D: Nekrasov-type expression 3. Nekrasov-type expressions with Wilson line parameters

3 String Theory predicts the existence of nontrivial QFTs in 6D. The worldvolume theory of multiple M5 branes --- (2,0) SUSY (6 supercharges) What about theories with (,0) SUSY? Heterotic string theories on K3 (6 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)

4 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 K3 x 6 x 9

5 E-string theory (Ganor-Hanany 96) (Seiberg-Witten 96) (Klemm-Mayr-Vafa 96) (Ganor-Morrison-Seiberg 96) (Minahan-Nemeschansky-Vafa-Warner 98) 6D (,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 The simplest interacting QFT with (,0) SUSY in 6D!?

6 Toroidal compactification down to lower dimensions 6D N = (,0) tensor multiplet 5D N = vector multiplet 4D N = 2 vector multiplet One can study the low energy theory in the Coulomb branch by means of Seiberg-Witten theory.

7 Seiberg-Witten theory (Seiberg-Witten 94) Exact solution to the low energy theory of 4D N = 2 SYM Low energy effective Lagrangian L eff = 4π Im d 4 θ F 0(A) A i Ā i + d 2 θ 2 2 F 0 (A) A i A W i j α W αj prepotential F 0 (a,...,a n ) : holomorphic function Seiberg-Witten curve λ SW : Seiberg-Witten differential a i = α i λ SW, F a i = β i λ SW u,...,u n

8 Nekrasov partition function (for pure SU(N) theory; instanton part; = 2 = ) for 5D theory (in R 4 S ) (Nekrasov 02) Z = Z = R R Λ R N Λ R i,j= N sinh β a ij (a+ ij (µ + (µ i,k i,k µ j,l µ + j,l l+ l k) k)) k,l= i,j= k,l= sinh β a ij (a+ ij (l + (l k) k)) R =(R,...,R N ) Z = exp F g 2g 2 g=0 Ri : partition µ i, µ i,2 (a ij := a i a j ) F 0 : prepotential

9 SU(2) gauge theories with various flavor symmetries 6D Ê 8 5D ÊE 89 E 8 E 7 E 6 E 5 E 4 E 3 E 2 E E 0 Ẽ 4D E 8 E 7 E 6 (Minahan-Nemeschansky 96) D 4 D 3 D 2 D D 0 A 2 A A 0 (Argyres-Plesser-Seiberg-Witten 95) E 9 = Ê 8

10 SU(2) gauge theories with various flavor symmetries Seiberg-Witten curve --- known for all theories 6D The Ê 8 (Seiberg-Witten 94) (Minahan-Nemeschansky-Warner 97) BPS partition function (including higher genus parts): theory needs further investigation. (Eguchi-K.S. 02) 5D E 8 E 7 E 6 E 5 E 4 E 3 E 2 E E 0 Ẽ 4D E 8 E 7 E 6 (Minahan-Nemeschansky 96) D 4 D 3 D 2 D D 0 Partition functions are E 9 = Ê constructed by 8 the ruled vertex formalism A 2 A A 0 (Argyres-Plesser-Seiberg-Witten 95) Nekrasov partition functions are known for these theories. (Diaconescu-Florea 05) (Konishi-Minabe 06)

11 Gauge theories and the E-string theory --- similarities and differences --- 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)

12 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 = 2 local K3 surface (the total space of the canonical bundle of a K3 surface) 2 local 2K3 (non-compact CY3) 2 K3

13 BPS states counting in M-theory on CY3 (Gopakumar-Vafa 98) 5D N = theory in R 4 S BPS states = M2-branes wrapped over holomorphic 2-cycles in CY 3 Counting of BPS states = Counting of holomorphic 2-cycles Seiberg-Witten prepotential of 5D N = theory in R 4 S = BPS partition function = (genus-zero) topological string amplitude of the CY3

14 2K3 surfaces = rational elliptic surfaces = almost del Pezzo surfaces B 9 obtained by blowing up 9 base points of a pencil of cubic curves in P 2 del Pezzo surfaces B n (n 8) complex surfaces with c > 0 obtained by blowing up n points in or (n-) points in P 2 P P

15 del Pezzo surfaces = F = P 2 B 9 B 8 B 7 B 6 = 2 K3 M-theory on local B n B 5 B 4 B 3 B 2 B B B 0 (well-understood) = F 0 = P P 5D N = SU(2) gauge theory with an E n flavor symmetry E 9 E 8 E 7 E 6 E 5 E 4 E 3 E 2 E E 0 Ẽ = Ê 8 local mirror symmetry local del Pezzo surfaces Seiberg-Witten curves

16 2nd homology of K3 surfaces 2 h 0,0 K3 K3 2 h,0 h 0, h 2,0 h, h 0,2 h 2, h,2 h 2, : a line in P 2 : exceptional curves e 0 e j, j =,...,9 e i e j = diag(+,,..., ) H 2 ( K3, Z) =Γ9, 2 = Γ, Γ 8 odd, self-dual lattice

17 2nd homology of 2K3 surfaces H 2 ( K3, Z) =Γ9, 2 = Γ, Γ 8 odd, self-dual lattice root lattice of E8 E (fiber) B (base) B B = B E = E E =0 α i α j = C E 8 ij

18 Topological string amplitudes for the local 2K3 surface F (ϕ, τ, µ; x) = r=0 n= k=0 λ P+ w O λ ϕ τ µ N r n,k, λ m= m 2sin mx 2r 2 2πim(nϕ+kτ + w µ) e 2 N r n,k,λ (Gopakumar-Vafa 98) : multiplicity of BPS states (Gopakumar-Vafa inv.) n k : winding number : momentum λ : weight of E8 r : genus F = g=0 F g x 2g 2 F g : topological string amplitude at genus g

19 Prepotential = BPS partition function of wound E-strings F 0 (ϕ, τ )= n= k=0 N n,k m= m 3 e2πim(nϕ+kτ ) n : winding number k : momentum N n,k : BPS multiplicity (instanton number) (Klemm-Mayr-Vafa 96) k n

20 Winding number expansion of the prepotential F 0 (ϕ, τ )= n= Q n Z n (τ ) Q := e 2πiϕ+πiτ ϕ : scalar vev (tension) τ : complex modulus of T 2 Z = E 4 η 2, Z 2 = E 2E E 4E 6 24η 24, Z n = P 6n 2(E 2,E 4,E 6 ) η 2n E 2n (τ ) : Eisenstein series η(τ ) : Dedekind eta function Zn at low orders can be determined by using the Seiberg-Witten curve or the modular anomaly equation (Minahan-Nemeschansky-Warner 97)

21 Nekrasov-type expression for the prepotential (K.S. 2) F 0 = (2 2 ln Z) =0 Z = R Q R a,b,c,d (i,j) R ab ϑ ab 2π (j i), τ 2 ϑ a c, b d 2π h ab,cd(i, j), τ 2 R =(R,R 0,R 00,R 0 ) a, b, c, d =0, Rab : partition µ ab,2 µ ab, ϑ ab (z, τ ) : Jacobi theta functions h ab,cd (i, j) := µ ab,i + µ cd,j i j + (relative hook-length) i j Rcd Rab

22 The prepotential represents g = 0 topological string amplitude F 0 = F 2 K3 0 However, Z = Z 2 K3 := exp g=0 2g 2 F 2 K3 g Difference of modular anomalies E2 Z = φ Z E2 Z 2 K3 = 24 2 φ ( φ + )Z 2 K3 (Hosono-Saito-Takahashi 99) E2 F 0 = 24 ( φf 0 ) 2 φ = 2πi ϕ = Q Q

23 Compactification with nontrivial Wilson line parameters One can introduce eight Wilson line parameters µ =(m,...,m 8 ) C 8 and break the E8 global symmetry Today we consider the cases with µ =(m,m 2,m 3,m 4,m,m 2,m 3,m 4 ) µ = (0, 0, 0, 0,m + m 2,m m 2,m 3 + m 4,m 3 m 4 ) Notation m =(m,m 2,m 3,m 4 )

24 Seiberg-Witten curve Explicit forms with general µ =(m,...,m 8 ) (Ganor-Morrison-Seiberg 96) --- expressed in terms of E8-invariant Jacobi forms are known (Eguchi-K.S. 02) For the cases with µ =(m,m 2,m 3,m 4,m,m 2,m 3,m 4 ), we find a new expression based on the SW curve for the 4D SU(2) Nf = 4 theory! (K.S. 2)

25 Nekrasov-type expression Z = R (4) Q R (4) a,b,c,d (i,j) R ab ϑ ab 2π (j i) + m cd, τ ϑ ab 2π (j i) m cd, τ ϑ a c, b d 2π h ab,cd(i, j), τ 2 m =(m,m 2,m 3,m 4 )=(m,m 0,m 00,m 0 ) (K.S. 2) F 0 = (2 2 ln Z) =0 This is in perfect agreement with the prepotential computed from the Seiberg-Witten curve (checked up to order Q 0 ).

26 Elliptic analogue of the Nekrasov partition function for the SU(N) gauge theory with N f = 2N flavors (Nekrasov 02) (Hollowood-Iqbal-Vafa 03) Z SU(N) N f =2N (; ϕ, τ ; a,...,a N ; m,...,m 2N ) := R (N) e 2πiϕ R (N) N k= (i,j) R k 2N n= ϑ N l= ϑ ak + m n + (j i), τ 2π ak a l + 2π h kl(i, j), τ 2 Our formula can be expressed as a special case of this function Z = Z SU(4) N f =8 ; ϕ, τ ;0, +τ, 2 2, τ 2 ; m,m 2,m 3,m 4, m, m 2, m 3, m 4

27 Global symmetries and counting of holomorphic curves in CY3 Wilson line parameters unbroken global symmetry local geometry m = (0, 0, 0, 0) E 8 E 8 del Pezzo m = 0, 0, 0, E 7 A E 7 del Pezzo 2 m = 0, 3, 3, E 6 A 2 E 6 del Pezzo 3 m = 0, 4, 4, E 5 A 3 E 5 del Pezzo 2 m = 0, 0, 2, D 8 P P 2

28 Simplification m = (0,m,m 2,m 3 ) Z = Z SU(3) N f =6 ; ϕ, τ ; +τ, 2 2, τ 2 ; m,m 2,m 3, m, m 2, m 3 All the cases with are of this type and E n A 8 n (n =8, 7, 6, 5) D 8 m = 0, 2,m,m 2 Z = Z SU(2) N f =4 ; ϕ, τ ; +τ 2, τ 2 ; m,m 2, m, m 2 The cases with E 7 A, E 5 A 3 and D 8 are of this type

29 Example: the E 7 A Z = R (2) Q R (2) 2 k,l= case (i,j) R k ϑ k+2 2π (j i), τ 2 ϑ k l + 2π h kl(i, j), τ 2 Q = e 2πiϕ+πiτ = Z SU(2) N f =4 ; ϕ, τ ; +τ 2, τ 2 ;0, 0, 0, 0 F 0 (ϕ, τ ) = (2 2 ln Z) =0 = n= k=0 N n,k m= m 3 e2πim(nϕ+kτ ) 2 BPS multiplicities Instanton numbers of local E7 del Pezzo surface

30 Summary We have found a Nekrasov-type expression for the Seiberg-Witten prepotential for the E-string theory. We have generalized the expression to the cases with four Wilson line parameters. Our formulas give very simple, closed expressions for the genus-zero topological string amplitudes of some basic non-toric Calabi-Yau threefolds.

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