There are five fundamental operations in mathematics: addition, subtraction, multiplication, division, and modular forms. — Apocryphal quote ascribed to Martin Eichler Mock modular forms and physics: an invitation Sameer Murthy King’s College London IMSc Chennai April 14, 2014 Modular forms What are modular forms? Holomorphic function f ( ⌧ ) , ⌧ 2 H a⌧ + b k ab f( )=(c⌧ + d) f(⌧) SL(2, Z) cd2 c⌧ + d ✓ ◆ Weight Ring structure Periodicity ⌧ ⌧ +1 Fourier series ! ) f(⌧)= a(n) qn , q = e2⇡i⌧ n X Interesting numbers Basic examples: Eisenstein series 1 n3qn E (⌧) = 1 + 240 = 1 + 240 q + 2160 q2 + , 4 1 qn ··· n=1 X − 1 n5qn E (⌧)=1 504 =1 504 q 16632 q2 , 6 − 1 qn − − − ··· n=1 X − E (⌧)= 2k ··· The space of modular forms of a given weight is finite-dimensional (see e.g. Zagier, 1-2-3 of modular forms) 1 ∆(⌧)=q (1 qn)24 = q 24 q2 + 252 q3 + ... − − n=1 Y = E (⌧)3 E (⌧)2 /1728 4 − 6 Ramanujan tau function 3 2 j(⌧)=(7E4(⌧) +5E6(⌧) )/∆(⌧) 1 = q− + 24 + 196884 q + Partition function ··· of Leech lattice (c.f. Talks of Harvey, Hikami, 1 + 196883 Taormina, Wendland) Relations to physics 1. Modular forms are generating functions of solutions to interesting counting problems e.g.: Heterotic string theory has 16 supersymmetries Number of 1/2 BPS states d ( N ) at m2 = Q2 = N 1 − Fundamental string states with right-movers in ground state Left-moving energy N distributed in 24 oscillators (Dabholkar, Harvey ‘89) 1 1 N 1 q− 1 1 dmicro(N) q − = = = 1 (1 qn) ∆(⌧) ⌘(⌧)24 N=0 n=1 − X 1 =Qq− + 24 + 324 q + ··· In string theory, ensembles of these microscopic excitations form black holes Microscopic Macroscopic g N 1 gsN 1 s gs N N Sen ’94, Strominger-Vafa ’96 Bekenstein-Hawking ’74 class AH π√N S = = ⇡pN d (N) = e + (N ) BH 4`2 micro ··· ⇥ Pl Asymptotic estimates a very useful guide for Quantum gravity: Hardy-Ramanujan-Rademacher expansion 2. CFT2 on a torus naturally produces modular forms Vibration of a string governed by a two-dimensional CFT. a⌧ + b ⌧ ⌧ ! c⌧ + d Large coordinate transformations Symmetry should be reflected in the physics. Superconformal theories produce holomorphic partition functions F N=(2,2) SCFT (L ,Q±,J), ( 1) 0 0 0 − F +F L0 L0 J0 Elliptic Zell(M; ⌧,z)=Tr (M) ( 1) q q ⇣ H − genus qe:= e2⇡i⌧ f, ⇣ := e2⇡iz . + Q ,Q− = L { 0 0 } 0 Zell holomorphic L0 in ⌧ (and z). (Witten) f (Subtlety! Troost,+Ashok, Eguchi-Sugawara, Talk of Troost) ( 1)F - + − e Jacobi forms Jacobi forms: basic definitions (Eichler-Zagier) '(⌧,z) holomorphic in ⌧ and z C 2 H 2 Weight 2 a⌧ + b z k 2⇡imcz ab M: ' , =(c⌧ + d) e c⌧+d '(⌧,z) SL(2; Z) c⌧ + d c⌧ + d 8 cd 2 ⇣ ⌘ ⇣ ⌘ 2⇡im(λ2⌧+2λz) E: '(⌧,z+ ⌧ + µ)=e− '(⌧,z) λ,µ 8 2 Z Index Interesting numbers Fourier '(⌧,z)= c(n, r) qn ⇣r expansion: n,r X Growth condition (weak Jacobi form): c(n, r)=0unlessn 0. ≥ Relation between Jacobi forms and modular forms Elliptic property Theta expansion: ) '(⌧,z)= h`(⌧) #m,`(⌧,z) , ` /2m 2ZX Z r2/4m r where #m,`(⌧,z):= q ⇣ . r Z r ` (mod2 2m) ⌘ X 2⇡i`z vector valued h (⌧)= '(⌧,z) e− dz ) ` modular form Z Examples of Jacobi forms 2 2 4 #1(⌧,z) (⇣ 1) (⇣ 1) A = ' 2,1(⌧,z)= = − 2 − q + − ⌘(⌧)6 ⇣ − ⇣2 ··· 4 # (⌧,z)2 B = ' (⌧,z)= i 0,1 # (⌧, 0)2 i=2 i X ⇣2 + 10⇣ +1 (⇣ 1)2 (5⇣2 22⇣ + 5) = +2 − − q + . ⇣ ⇣2 ··· 2 2 3 #1(⌧, 2z) ⇣ 1 (⇣ 1) C = ' 1,2(⌧,z)= = − − q + − ⌘(⌧)3 ⇣ − ⇣3 ··· Ring of weak Jacobi forms generated by A, B, C. What is new? Wall-crossing and BH phase transitions Phase I Phase II N (Q,P) ΔS + (Q,P) Q Serious problem: throwing out multi- centered BHs (Denef-Moore 2007) P destroys the modular symmetry. A concrete realization: N=4 string theory (Dijkgraaf, Verlinde, Verlinde; Partition function of 1/4 BPS dyons Gaiotto, Strominger, Yin; David, Sen) (N=4) 1 Igusa cusp form Z(dyon) (⌧,z,σ)= Φ10(⌧,z,σ) Has zeros (divisors) 1 2⇡imσ = m(⌧,z) e . in the Siegel upper half plane. m= 1 X− Meromorphic Jacobi forms of weight -10, index m. (poles in z) ? n r m(⌧,z)= dmicro(n, r) q ⇣ . n,r X (c.f. talks of Hohenneger, Govindarajan, Persson, Volpato) Questions • What is the correct expansion of the meromorphic Jacobi forms? • Can we extract the degeneracies of the single-centered black hole? • What are the modular properties of the corresponding Fourier coefficients? Mock modular forms. Solution of BH wall-crossing problem Canonical decomposition of the partition function: BH multi m = m + m Partition function of the isolated BH Multi-centers and is a mock modular form. wall-crossing info in Appell-Lerch sum..