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 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(2⌧+2z) E: '(⌧,z+ ⌧ + µ)=e '(⌧,z) ,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,)= 10(⌧,z,) Has zeros (divisors) 1 2⇡im = 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.