Asymptotic M5-brane entropy from S-duality Nahmgoong June Seoul National University KIAS, December 16, 2016 ◮ Tallk based on: S.Kim and J.Nahmgoong [work in progress] 1/14
6d (0,2) SCFT ◮ Worldvolume theory of coincident N M5-branes → 6d (0,2) A N − 1 SCFT – Entropy ∝ N 3 – No known Lagrangian description ◮ Reducing 6d over S 1 : 5d N = 2 SYM + instanton (KK mode) – M-theory circle X 1 ∼ X 1 + 2 πR 1 – Hypermultiplet mass m : N = 2 → N = 1 ∗ ◮ Separating M5 branes: Coulomb phase U ( N ) → U (1) N – Giving VEV to 5d vector multiplet ( A µ , φ 6 , 7 , 8 , 9 , 10 ) – � φ 6 � = diag ( a 1 , ..., a N ) , � φ 7 ,..., 10 � = 0 2/14
5d N = 1 ∗ SYM ◮ BPS objects of 5d SYM in Coulomb phase – W-boson: F1 connected between two D4s – Instanton: D0 bounded on a D4 ◮ They form 1/4 BPS states counted by partition function ◮ Reducing 5d over temporal S 1 : Ω -background – Temporal circle X 0 ∼ X 0 + 2 πR 0 w 1 , 2 ∼ e − ǫ 1+ ǫ 2 – Twisting parameter ǫ 1 , 2 : z 1 , 2 ∼ e ǫ 1 , 2 z 1 , 2 , w 1 , 2 2 z 1 , 2 ∈ C 2 � = R 4 � ( X 2 , 3 , 4 , 5 ) , w 1 , 2 ∈ C 2 ⊥ = R 4 ⊥ ( X 7 , 8 , 9 , 10 ) 1 – Effective volume of the system V ∼ ǫ 1 ǫ 2 3/14
5d N = 1 ∗ partition function: I ◮ Partition function Z = Z pert · Z inst ◮ Instanton partition function ∞ � Z k ( a, m, ǫ 1 , 2 ) q k , q = e 2 πiτ Z inst ( τ, a, m, ǫ 1 , 2 ) = 1 + k =1 N � � � ( − 1) F q k e − β { Q,Q † } e − 2 ǫ + ( J 1 R + J 2 R ) e − 2 ǫ − J 1 L e − 2 mJ 2 L e − 2 a i Π i Z k = Tr i =1 – m, ǫ 1 , 2 : Chemical potentials on T 2 ( R 5 , 1 → R 4 × T 2 ) – a i : Coulomb VEV – τ : Complex structure of T 2 , complex gauge coupling in 5d = i 4 πR 0 τ = i R 0 g 2 R 1 Y M – q : instanton number fugacity – Z k : Witten index of 5d SYM on R 4 , 1 4/14
5d N = 1 ∗ partition function: II ◮ Instaton partition function – Z k : computed from D0-D4 ADHM QM k � Z k = 1 � dφ I � � Z vec ( φ, a, ǫ 1 , 2 ) · Z adj ( φ, a, m, ǫ 1 , 2 ) k ! 2 πi JK I =1 – φ I = − 2 πiτ ( A 1 I + iA t I ) : Complexified gauge holonomy – Contour: Jefferey-Kirwan residue – Z vec/adj : Gaussian path integral over massive modes · 2 sinh φ IJ +2 ǫ + 2 sinh φ IJ 1 � � 2 2 Z vec = · 2 sinh φ IJ + ǫ 1 · 2 sinh φ IJ + ǫ 2 2 sinh φ I − a i ± ǫ + I,J I,i 2 2 2 2 sinh φ IJ ± m − ǫ − 2 sinh φ I − a i ± m � � 2 Z adj = 2 sinh φ IJ ± m − ǫ + 2 I,J I,i 2 ◮ Perturbative partition function: Only W-boson contribution sinh m + ǫ + sinh m − ǫ + � 1 e α · a � � 2 2 Z pert ( a, m, ǫ 1 , 2 ) = PE sinh ǫ 1 2 sinh ǫ 2 2 2 α ∈ root 5/14
Prepotential ◮ We consider the limit ǫ 1 , 2 → 0 � − F ( τ, a, m ) � + O ( ǫ 0 Z ( τ, a, m, ǫ 1 , 2 ) = exp 1 , 2 ) ǫ 1 ǫ 2 ◮ Prepotential F – Effective action of Coulomb branch – Classical part + quantum corrections (pertuabative / instanton) F tot = πiτa 2 i + F, F = F pert + F inst ◮ Ω -background parameter ǫ 1 , 2 dependence is removed in prepotential. – Easier then partition function to treat 6/14
S-duality ◮ We compactified 6d (0,2) theory on R 4 × T 2 ◮ S-duality: Interchanging radii of M-theory circle ( R 1 ) ↔ temporal circle ( R 0 ) τ D = − 1 τ = i R 0 τ D = i R 1 , → R 1 R 0 τ ◮ S-dual transform of chemical potentials m D = m 1 , 2 = ǫ 1 , 2 ǫ D τ , τ ◮ S-duality of 4d N = 2 ∗ prepotential: Legendre transform 1 1 � � 1 F 4d tot ( τ D , a D , m D ) = F 4d tot ( τ, a, m ) − a i ∂ i F 4d a D 2 πiτ ∂ i F 4d , i = tot tot ǫ D 1 ǫ D ǫ 1 ǫ 2 2 � 4 πiτ ( ∂ i F 4d ) 2 � F 4d ( τ D , a D , m D ) = 1 1 1 F 4d ( τ, a, m ) + 2 πiτ ∂ i F 4d a D , i = a i + τ 2 ◮ S-duality of 5d prepotential? – Suspected to be Legendre transform. – Perturbative check is impossible: F D cannot be expanded in q – Test using Modular anomaly equation 7/14
Modular anomaly equation ◮ τ dependence of F : q -series – Eisenstein series E 2 n can be used as a basis for q -series ∞ q k 2 � k 2 n − 1 E 2 n ( τ ) = 1 + 1 − q k ζ (1 − 2 n ) k =1 4 n + 2 2 ζ (1 − 2 n ) q 2 + O ( q 3 ) = 1 + ζ (1 − 2 n ) q + – E 2 n> 2 ’s are exact-modular. Only E 2 is quasi-modular E 2 ( τ D ) = τ 2 � 6 � E 2 n> 2 ( τ D ) = τ 2 n E 2 n ( τ ) , E 2 ( τ ) + πiτ ◮ S-duality is determined by E 2 dependence: Modular anomaly equation → ∂F 4d � 4 πiτ ( ∂ i F 4d ) 2 � ( F 4d ) D = 1 1 = − 1 F 4d + 24 ( ∂ i F 4d ) 2 ← τ 2 ∂E 2 ◮ By checking E 2 dependence of 5d prepotential, we can reconstruct its S-duality transform 8/14
5d N = 1 ∗ prepotential: I ◮ Partition function � � 1 + Z 1 q + Z 2 q 2 + O ( q 3 ) Z = Z pert ◮ 5d N = 1 ∗ prepotential is given in series of q � Li 3 ( e a ij ) − 1 � � 2 Li 3 ( e a ij ± m ) F = 1 · i,j 1 − sinh 2 m � � + q · 4 sinh 2 m � � 2 T i where T i = sinh 2 a ij 2 i j � = i 2 � 8 sinh 4 a ij + 12 sinh 2 a ij − 4 sinh 2 m sinh 6 ( m � + q 2 · 2 2 2 2 ) T i T j sinh 2 a ij 2 sinh 2 a ij ± m i,j 2 � + sinh 2 ( m � 6 + 2 sinh 2 m � � i + 4 sinh 4 ( m � T i T (2) T 2 + O ( q 3 ) 2 ) 2 ) i 2 i i ◮ E 2 dependence is not obviously seen. ◮ We expand F in series of m , and resum over q 9/14
5d N = 1 ∗ prepotential: II ◮ We expand F in series of m , F = − m 2 � 1 Li 1 ( e a ij ) + N ( q + 3 � � 2 q 2 + ... ) 2 i,j − m 4 1 − 24 q − 72 q 2 + ... � � Li − 1 ( e a ij ) − N � 24 12 i,j � 1 − 48 q + 432 q 2 + ... − m 6 2304 1 + sinh 2 a ij � 1 2 + cosh a ij �� cosh a ik a kj � � + cosh �� � 2 2 2 2 × + sinh 4 a ij 2 sinh a ij sinh 3 a ik sinh 3 a kj 2 2 2 2 i,j i,j,k + 1 + 240 q + 2160 q 2 + ... 11520 ��� 1 − sinh 2 a ij � 1 2 + cosh a ij �� cosh a ik a kj + cosh � � � 2 2 2 2 − 5 × sinh 4 a ij 2 sinh a ij sinh 3 a ik sinh 3 a kj 2 2 2 i,j i,j,k 2 + O ( m 8 ) 10/14
5d N = 1 ∗ prepotential: II ◮ We expand F in series of m , and resum over q F = − m 2 � 1 � − m 4 E 2 � � Li − 1 ( e a ij ) − N � � Li 1 ( e a ij ) − N ln φ ( q ) 2 24 12 i,j i,j � 1 + sinh 2 a ij � 1 2 + cosh a ij �� cosh a ik a kj E 2 � � + cosh �� � − m 6 2 2 2 2 2 + sinh 4 a ij 2 sinh a ij sinh 3 a ik sinh 3 a kj 2304 2 2 2 i,j i,j,k 2 ��� 1 − sinh 2 a ij � 1 2 + cosh a ij �� cosh a ik a kj � � + cosh E 4 � 2 2 2 2 + − 5 sinh 4 a ij 2 sinh a ij sinh 3 a ik sinh 3 a kj 11520 2 2 2 2 i,j i,j,k checked up to q 3 for generic N , q 4 for N = 2 + O ( m 8 ) ◮ m 2 n +2 order: quasi-modular form with weight 2 n m 4 : E 2 , m 6 : E 2 m 8 : E 3 m 10 : E 4 2 , E 2 2 E 4 , E 2 E 6 , E 2 2 , E 4 , 2 , E 2 E 4 , E 6 , 4 – Only E 2 , E 4 , E 6 are independent – The number of combinations are finite → Resumming q -series into Eisenstein series can be done uniquely 11/14
S-duality of 5d N = 1 ∗ prepotential ◮ m 2 order, F = − m 2 � 1 � − m 4 E 2 � � Li − 1 ( e α · a ) − N � � Li 1 ( e α · a ) − N ln φ ( q ) + ... 2 24 12 α α ◮ Euler totient function φ ( q ) ∞ − iτ − πi ( τ D − τ ) √ � (1 − q n ) = q − 1 ln φ D = ln φ + ln 24 η ( q ) φ ( q ) = → 12 n =1 ◮ E 2 dependence of F 24 ( ∂ i F ) 2 + m 4 N 3 ∂F = − 1 checked up to m 6 for generic N , m 10 for N = 2 ∂E 2 288 ◮ S-duality of the 5d N = 1 ∗ prepotential F � � πi ( τ − τ D ) N 3 � � F D = 1 1 i 4 πiτ ( ∂ i F ) 2 + m 4 48 πiτ + m 2 N F + + ln τ 2 12 τ 12/14
Asymptotic entropy: I ◮ S-duality of the 5d prepotential F � � πi ( τ − τ D ) N 3 F D = 1 � 1 i � 4 πiτ ( ∂ i F ) 2 + m 4 48 πiτ + m 2 N F + + ln τ 2 12 τ ◮ Strong coupling for τ D ← → Weak coupling for τ q D = e 2 πiτ D = 1 − τ D = i · 0 + , q = 0 + τ = i · ∞ , ← → ◮ Weak coupling prepotential ≃ Perturbative prepotential F ≃ F pert ◮ Leading terms are Coulomb VEV independent � � N 3 m 4 + N 2 m 3 F D ≃ − 1 πi D D + Nm 2 D τ D 48 πi 12 12 � 1 1 2 Li 4 ( e Nm D ) + 1 � 2 Li 4 ( e − Nm D ) − Li 4 (1) = Nπiτ D ���� 0 < Im [ m D ] < 2 π N ◮ Polynomial expression is valid for small m D . For a generic value of m D , the expression is continuated to tetra-logarithm. 13/14
Asymptotic entropy: II ◮ Asymptotic entropy of N M5-branes � 1 1 2 Li 4 ( e Nm ) + 1 � F 2 Li 4 ( e − Nm ) − Li 4 (1) τ = i · 0 + ≃ , ǫ 1 ǫ 2 Nπiǫ 1 ǫ 2 τ ◮ Imaginary m : Periodicity m ∼ m + i 2 π N N , m D = 2 πiτ D m = i 2 π e Nm D = q D : Phase transition point → N ◮ Real m : N 3 scaling N 3 � m � m iπ 3 � � 4 � 2 � F ≃ − N ǫ 1 ǫ 2 3 ǫ 1 ǫ 2 τ 2 π 2 π 14/14
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