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?

✓a bc d

◆2 SL(2, Z)

⌧ 2 H

f(a⌧ + b

c⌧ + d) = (c⌧ + d)kf(⌧)

Holomorphic function ,f(⌧)

Periodicity ⌧ ! ⌧ + 1 Fourier series

q = e2⇡i⌧f(⌧) =X

n

a(n) qn ,

)

Interesting numbers

Weight Ring structure

Basic examples: Eisenstein series

E6(⌧) = 1� 5041X

n=1

n5qn

1� qn= 1� 504 q � 16632 q2 � · · · ,

E4(⌧) = 1 + 2401X

n=1

n3qn

1� qn= 1 + 240 q + 2160 q2 + · · · ,

E2k(⌧) = · · ·

�(⌧) = q1Y

n=1

(1� qn)24 = q � 24 q2 + 252 q3 + ...

=�E4(⌧)3 � E6(⌧)2

�/1728 Ramanujan

tau function

j(⌧) = (7E4(⌧)3 + 5E6(⌧)2)/�(⌧)

= q�1 + 24 + 196884 q + · · ·Partition function of Leech lattice

1 + 196883 (c.f. Talks of Harvey, Hikami, Taormina, Wendland)

The space of modular forms of a given weight is finite-dimensional

(see e.g. Zagier, 1-2-3 of modular forms)

Relations to physics

1. Modular forms are generating functions of solutions to interesting counting problems

e.g.: Heterotic string theory has 16 supersymmetries

Fundamental string states with right-movers in ground state

Left-moving energy N distributed in 24 oscillators

Number of 1/2 BPS states at d(N)

(Dabholkar, Harvey ‘89)

m2 = Q2 = N � 1

1X

N=0

dmicro

(N) qN�1 =q�1

Q1n=1

(1� qn)=

1�(⌧)

= q�1 + 24 + 324 q + · · ·

=1

⌘(⌧)24

In string theory, ensembles of these microscopic excitations form black holes

Microscopic

N

gs

gsN� 1gsN� 1

Macroscopic

N

Sen ’94, Strominger-Vafa ’96 Bekenstein-Hawking ’74

(N�⇥)dmicro(N) = e��

N + · · · SclassBH =

AH

4`2Pl

= ⇡p

N

Asymptotic estimates a very useful guide for Quantum gravity: Hardy-Ramanujan-Rademacher expansion

2. CFT on a torus naturally produces modular forms

Large coordinate transformations

⌧ ! a⌧ + b

c⌧ + d

2

Vibration of a string governed by a two-dimensional CFT.

⌧

Symmetry should be reflected in the physics.

Superconformal theories produce holomorphic partition functions

N=(2,2) SCFT (L0, Q±0 , J0), (�1)F

q := e2⇡i⌧ , ⇣ := e2⇡iz .

Zell(M ; ⌧, z) = TrH(M) (�1)F+ eF qL0 qfL0 ⇣J0Elliptic

genus

(Subtlety! Troost,+Ashok, Eguchi-Sugawara, Talk of Troost)

{Q+0 , Q�0 } = L0

holomorphicZell

⌧ (and z).in (Witten)

+-

fL0

(�1) eF

Jacobi forms

Jacobi forms: basic definitions'(⌧, z) holomorphic in

Growth condition (weak Jacobi form): c(n, r) = 0 unless n � 0.

'(⌧, z) =X

n,r

c(n, r) qn ⇣r

Interesting numbersFourier expansion:

⌧ 2 H and z 2 C(Eichler-Zagier)

M:

E:

'⇣a⌧ + b

c⌧ + d,

z

c⌧ + d

⌘= (c⌧ + d)k e

2⇡imcz2c⌧+d '(⌧, z) 8

⇣ a bc d

⌘2 SL(2; Z)

Index

Weight

'(⌧, z + �⌧ + µ) = e�2⇡im(�2⌧+2�z)'(⌧, z) 8 �, µ 2 Z

Relation between Jacobi forms and modular forms

Elliptic property )

'(⌧, z) =X

`2Z/2mZh`(⌧) #m,`(⌧, z) ,

#m,`(⌧, z) :=X

r2Zr⌘ ` (mod 2m)

qr2/4m ⇣r .where

Theta expansion:

vector valued modular form

) h`(⌧) =Z

'(⌧, z) e�2⇡i`z dz

Examples of Jacobi forms

Ring of weak Jacobi forms generated by A, B, C.

C = '�1,2(⌧, z) =#1(⌧, 2z)

⌘(⌧)3=

⇣2 � 1⇣

� (⇣2 � 1)3

⇣3q + · · ·

A = '�2,1(⌧, z) =#1(⌧, z)2

⌘(⌧)6=

(⇣ � 1)2

⇣� 2

(⇣ � 1)4

⇣2q + · · ·

B = '0,1(⌧, z) =4X

i=2

#i(⌧, z)2

#i(⌧, 0)2

=⇣2 + 10⇣ + 1

⇣+ 2

(⇣ � 1)2 (5⇣2 � 22⇣ + 5)⇣2

q + · · · .

What is new?

Wall-crossing and BH phase transitions

ΔS

N

Serious problem: throwing out multi-centered BHs (Denef-Moore 2007) destroys the modular symmetry.

Phase I

(Q,P)(Q,P)+

Phase II

(Q,P)(Q,P)

Q

P

A concrete realization: N=4 string theoryPartition function of 1/4 BPS dyons

Igusa cusp form

Jacobi forms of weight -10, index m.

Z(N=4)

(dyon)

(⌧, z, �) =1

�10

(⌧, z, �)

=1X

m=�1

m(⌧, z) e2⇡im� .Has zeros (divisors) in the Siegel upper half plane.

(Dijkgraaf, Verlinde, Verlinde; Gaiotto, Strominger, Yin; David, Sen)

m(⌧, z) =X

n,r

dmicro

(n, r) qn ⇣r .?

Meromorphic(poles in z)

(c.f. talks of Hohenneger, Govindarajan, Persson, Volpato)

Questions

• What is the correct expansion of the meromorphic Jacobi forms?

• What are the modular properties of the corresponding Fourier coefficients?

• Can we extract the degeneracies of the single-centered black hole?

Mock modular forms.

Solution of BH wall-crossing problem

Multi-centers and wall-crossing info in Appell-Lerch sum.

Partition function of the isolated BH is a mock modular form.

Canonical decomposition of the partition function:

m = BHm + multi

m