NUMBER THEORY IN STRING THEORY:
SKEW, MOCK, FALSE, AND QUANTUM
Sarah M. Harrison
based on work with
Cheng, Duncan, Harvey, Kachru, Rayhaun (’17)
and Cheng, Chun, Ferrari, Gukov (to appear)INTRODUCTION➤ String Theory and its friends accommodate many interesting

physical phenomena discussed at this conference, e.g.

Black Holes

Gauge theories

Holography

Cosmology

Entanglement

INTRODUCTION
Not only are there many “solutions” of string theory, but there aremany ways to describe it, e.g…

SPACETIME ACTION/SUPERGRAVITY
S=

d x

p

ge 2 (R + . . .)CONFORMAL SYMMETRY&ADS/CFT÷⇐GEOMETRY/COMPACTIFICATIONd11<latexit sha1_base64="UK404O5C1hE2QYYyTHimplJ+XHA=">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</latexit>

Z

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⇐

STRING PERTURBATION THEORYINTRODUCTION
Combinations of different viewpoints elucidate different aspects of the theory,
e.g…

PEE
t.ir#ty
Field theory dualities

Non-perturbative objects

Black hole entropy

String Dualities

INTRODUCTION
I propose the following tool has been underexploited:

NUMBER THEORY
I will discuss 2 physical examples where I think number theory has something toteach us, but I expect there are many more:

I: MSW STRINGS
II: 3D N=2 QFT

INTRODUCTION
These examples relate to the larger questions:
What are the symmetries of string theory?
Are there new ways to think about special solutions?
Can we understand large classes of them?
How does modularity, in its many forms, relate to the many
physical phenomena in QFT and string theory?

PART I:MSW STRINGS

Skew and Mock

PART I: SKEW AND MOCK
5- branes

Mf
P“MSW string”

CY 3

N=(0,4) worldsheet CFT

Index counting BPS states governs black hole entropy in M-theory

Maldacena, Strominger, Witten ‘97

PART I: SKEW AND MOCK
Let’s consider a supersymmetric index which counts worldsheet BPS states, called amodified elliptic genus:
2

F L0 c/24 L̄0 c̄/24 2⇡i⇣·Q

Z(⌧, ⇣) := Tr F ( 1) q
R0<latexit sha1_base64="+APRmMntzw1ory0+AsppyJmJuV8=">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</latexit>q̄e

In general it is not holomorphic, but has modular properties:
Z<latexit sha1_base64="6a5NiSEiXoVgjHjHfo4tfVL6Bpc=">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</latexit>✓a⌧ + b⇣,c⌧ + d c⌧ + d

◆

2

✓

2

(c⌧ + d)cz
e m|c⌧ + d|
c¯
⌧ +d

◆

= Z(⌧, ⇣)weight (-3/2,1/2)
modular form

N=4 spectral flow implies the index has a theta decomposition, which
depends on the geometry of P:

Z(⌧, ⇣) =<latexit sha1_base64="xXBSE7LjxspZPwM6wg+d4wGLVpI=">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</latexit>X⇥µ (⌧, ⇣)hµ (⌧ )µ2⇤⇤ /⇤
coefficients are
modular and mock
modular forms

Gaiotto, Strominger, Yin ’06
de Boer, Cheng, Dijkgraaf, Manschot, Verlinde ’06
Denef, Moore ‘07

PART I: SKEW AND MOCKExample: single 5-brane on 1/2-K3
1(1)
Z 1 K3 (R; ⌧, z) = 12 E4 (⌧ )⇥odd
1,1 (R; ⌧, z)2⌘ (⌧ )1
H ( K3, Z) = Z
22Minahan, Nemeschansky, Vafa, Warner ‘98

where R=size of elliptic fiber2( E8 )<latexit sha1_base64="3NUmRN/w0QG+9+zKi43bZB6QJJo=">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</latexit><latexit sha1_base64="KwUa7pRFHa4oN0deMFLHi4LDO9c=">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</latexit>

Special moduli: rational radiip(1)
Z 1 K3 ( 2m; ⌧, z) =( Z)

1
' 12 K3,m (⌧, z)6⌘ (⌧ )

weight 2 skew-hol.
Jacobi form

<latexit sha1_base64="VWAeaTWgTDyec/dyuzYq1ZLyEQE=">AAACSHicbVBPaxNBHJ1NbRujrbEevQwGYQMl7KZShVIIehF6iWDSYDZZfjuZNENndteZ3wbSZT9eLx69+Rm8eLCINyfJCjbxwcDj/WFmXpRKYdDzvjmVnQe7e/vVh7VHjw8On9SfHvVNkmnGeyyRiR5EYLgUMe+hQMkHqeagIskvo+t3S/9yzrURSfwRFykfKbiKxVQwQCuF9fDTOHf9ZhHmuR8kNknbxcVJ4Qbms8a8rYqzACE7vmme//UDjjA+dZdyswjmoNOZuN8+pqpwy1pYb3gtbwW6TfySNEiJblj/GkwSlikeI5NgzND3UhzloFEwyYtakBmeAruGKz60NAbFzShfDVHQl1aZ0Gmi7YmRrtR/GzkoYxYqskkFODOb3lL8nzfMcPpmlIs4zZDHbH3RNJMUE7pclU6E5gzlwhJgWti3UjYDDQzt9jU7gr/55W3Sb7d8r+V/eNXovC3nqJLn5AVxiU9ekw55T7qkRxi5Jd/JT3LnfHF+OL+c3+toxSk7z8g9VCp/AHgEsPQ=</latexit>

R2=3:
E4 (⌧ ) ⇣' 12 K3, 32 (⌧, z) = 5✓ 32 , 12 ( ⌧¯, z)⌘ (⌧ )✓ 32 ,12( ⌧¯, z)<latexit sha1_base64="z2iW1G8oSlQ1HKuRf2kU8evhte0=">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</latexit>

M12 moonshine function⌘

Cheng, Duncan, SH, Harvey, Kachru, Rayhaun ’17

PART I: SKEW AND MOCK
Comments:

Compare with Mathieu moonshine and K3:➤✓12 (⌧, z)
EG(K3) = 3(24µ(⌧, z) + H(⌧ ))⌘ (⌧ )H(⌧ ) = 2q 1/8 ( 1 + 45q + 231q 2 + 770q 3 + . . .)<latexit sha1_base64="3uGRLvMYRjo7QsKo2dnMAnCEJKg=">AAACHXicbVDLSgMxFM3Ud31VXboJFqFSrJOxUjeC6KbLCrYV+iKTpm1o5tHkjlCG/ogbf8WNC0VcuBH/xvSx0OqBCyfn3EvuPW4ohQbb/rISC4tLyyura8n1jc2t7dTObkUHkWK8zAIZqDuXai6Fz8sgQPK7UHHquZJX3f712K/ec6VF4N/CMOQNj3Z90RGMgpFaqXwxUwcaHV04g2Z8TE7OR5ljgrM4f4YHWeyckkHTMc9CwR40T7N12Q5AH7VSaTtnT4D/EjIjaTRDqZX6qLcDFnncByap1jVih9CIqQLBJB8l65HmIWV92uU1Q33qcd2IJ9eN8KFR2rgTKFM+4In6cyKmntZDzzWdHoWenvfG4n9eLYLOeSMWfhgB99n0o04kMQR4HBVuC8UZyKEhlClhdsWsRxVlYAJNmhDI/Ml/ScXJETtHbvLpy6tZHKtoHx2gDCKogC5REZVQGTH0gJ7QC3q1Hq1n6816n7YmrNnMHvoF6/Mbg/ycmw==</latexit><latexit sha1_base64="tD4gVLLocC1VNfEXMWJyz/GQ9Xo=">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</latexit>➤

Connection between finite groups, modular forms, and complex surfaces isseemingly pervasive once we relax rules to allow signs, e.g.
X
X
M
n
i
f
(⌧
)
:=
q
(
1)
(TrVi,n g)
V =(V0,n V1,n )g<latexit sha1_base64="fJD/KQDKqkFpEPNycXqg1Z/q3Ss=">AAACHXicbVDLSsNAFJ3UV62vqks3g0WoICWRgm6EohuXFWxaaEKYTCft0MkkzNwIJfRH3Pgrblwo4sKN+DdOHwttPTBw7jn3cueeMBVcg21/W4WV1bX1jeJmaWt7Z3evvH/g6iRTlLVoIhLVCYlmgkvWAg6CdVLFSBwK1g6HNxO//cCU5om8h1HK/Jj0JY84JWCkoFx3r7AX8n6SikwHEnuCRYCrbpDbZ3KMvamOTemY0lO8PwB8GpQrds2eAi8TZ04qaI5mUP70egnNYiaBCqJ117FT8HOigFPBxiUv0ywldEj6rGuoJDHTfj69boxPjNLDUaLMk4Cn6u+JnMRaj+LQdMYEBnrRm4j/ed0Moks/5zLNgEk6WxRlAkOCJ1HhHleMghgZQqji5q+YDogiFEygJROCs3jyMnHPa45dc+7qlcb1PI4iOkLHqIocdIEa6BY1UQtR9Iie0St6s56sF+vd+pi1Fqz5zCH6A+vrB3ZToOQ=</latexit>

➤

➤

n

n

i2{0,1}<latexit sha1_base64="BGiF6aa/XgVjSOu5dio/K+OWAEs=">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</latexit>

Special skew-holomorphic property of BPS index appears at rational and attractor
points in moduli space
Can moonshine give us a new way to organize these special solutions of string
theory?

PART II:
3D N=2 QFT

Mock, False, and Quantum

PART II: MOCK, FALSE, AND QUANTUM➤ 3d-3d Correspondence:M5-branes on 3-manifold M3<latexit sha1_base64="IBfybQLy525ZgfNpOc78lBqb4zM=">AAACBnicbVDLSgMxFM34rPU16lKEYCu4aZlpEV0W3bgpVLAPaIchk8m0oZlkSDJCKV258VfcuFDErd/gzr8xbWehrQcuHM65l3vvCRJGlXacb2tldW19YzO3ld/e2d3btw8OW0qkEpMmFkzIToAUYZSTpqaakU4iCYoDRtrB8Gbqtx+IVFTwez1KiBejPqcRxUgbybdP6helQCJOFBQcVksxMq5gISzW/WrRtwtO2ZkBLhM3IwWQoeHbX71Q4DQmXGOGlOq6TqK9MZKaYkYm+V6qSILwEPVJ11COYqK88eyNCTwzSggjIU1xDWfq74kxipUaxYHpjJEeqEVvKv7ndVMdXXljypNUE47ni6KUQS3gNBMYUkmwZiNDEJbU3ArxAEmEtUkub0JwF19eJq1K2XXK7l2lULvO4siBY3AKzoELLkEN3IIGaAIMHsEzeAVv1pP1Yr1bH/PWFSubOQJ/YH3+ADSClwM=</latexit>

M3
brakes#n

3- manifold.

G-type Chern-Simons on M3<latexit sha1_base64="xNa5Hz+5XBVa68v9Wu0O+R7CURY=">AAACB3icbVDLSgMxFM34rPVVdSlIsCO4aZmpC10Wu9CNUNE+oC0lk2ba0DyGJCOUoTs3/oobF4q49Rfc+Tem0y609UDgcM693JwTRIxq43nfztLyyuraemYju7m1vbOb29uvaxkrTGpYMqmaAdKEUUFqhhpGmpEiiAeMNIJhZeI3HojSVIp7M4pIh6O+oCHFyFipmztyr9zCxICVAVGicEe5FBpKAd2b7pnbzeW9opcCLhJ/RvJghmo399XuSRxzIgxmSOuW70WmkyBlKGZknG3HmkQID1GftCwViBPdSdIcY3hilR4MpbJPGJiqvzcSxLUe8cBOcmQGet6biP95rdiEF52Eiig2RODpoTBm0EiYZu9RRbBhI0sQVtT+FeIBUggbW13WluDPR14k9VLR94r+bSlfvpzVkQGH4BicAh+cgzK4BlVQAxg8gmfwCt6cJ+fFeXc+pqNLzmznAPyB8/kDwFuXTQ==</latexit>

3d N = 2 theory T [M3 ]<latexit sha1_base64="glkGw58iuhpS581RJFT4XLp6U6U=">AAACCnicbVDLSsNAFJ3UV62vqEs3o43gqiTtQjdC0Y0bpUJfkIYwmUzboZMHMxMhhK7d+CtuXCji1i9w5984abPQ1gMXDufcy733eDGjQprmt1ZaWV1b3yhvVra2d3b39P2DrogSjkkHRyzifQ8JwmhIOpJKRvoxJyjwGOl5k+vc7z0QLmgUtmUaEydAo5AOKUZSSa5+3PChMQiQHGPE4N1l3YByTCKeQqNt37oNx3D1qlkzZ4DLxCpIFRRoufrXwI9wEpBQYoaEsC0zlk6GuKSYkWllkAgSIzxBI2IrGqKACCebvTKFp0rx4TDiqkIJZ+rviQwFQqSBpzrzo8Wil4v/eXYihxdORsM4kSTE80XDhEEZwTwX6FNOsGSpIghzqm6FeIw4wlKlV1EhWIsvL5NuvWaZNeu+Xm1eFXGUwRE4AWfAAuegCW5AC3QABo/gGbyCN+1Je9HetY95a0krZg7BH2ifP0dGmBM=</latexit>
sha1_base64="hP+6LrUf2d3tZaldqaQQvEKMXyw=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odBu3wMYA6nMMFXEEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4</latexit>
sha1_base64="9XzpDn8JygvYMQfg5Fdo6WJPpFg=">AAAB/3icbZBLSwMxFIXv+Ky16ujWTbQVXJWZdqEbQXDjRqnQF0yHIZNm2tDMgyQjDEPXbvwrblwo4r9w578x03ahrQcCH+ck5N7jJ5xJZVnfxtr6xubWdmmnvFvZ2z8wDytdGaeC0A6JeSz6PpaUs4h2FFOc9hNBcehz2vMnN0Xee6RCsjhqqyyhbohHEQsYwUpbnnnSHKLaIMRqTDBH91eNGlJjGosM1drOndd0a55ZterWTGgV7AVUYaGWZ34NhjFJQxopwrGUjm0lys2xUIxwOi0PUkkTTCZ4RB2NEQ6pdPPZKlN0pp0hCmKhT6TQzP39IsehlFno65vF0HI5K8z/MidVwaWbsyhJFY3I/KMg5UjFqOgFDZmgRPFMAyaC6VkRGWOBidLtlXUJ9vLKq9Bt1G2rbj9YUIJjOIVzsOECruEWWtABAk/wAm/wbjwbr8bHvK41Y9HbEfyR8fkDvzqWoA==</latexit>
sha1_base64="IYwyP3e5XRwhfsGp+1GU3AhfHFo=">AAACCnicbVDLSsNAFJ3UV62vqEs3o43gqiTtQjdC0Y0bpUJfkIYwmUzboZMHMxMhhK7d+CtuXCji1i9w5984abPQ1gMXDufcy733eDGjQprmt1ZaWV1b3yhvVra2d3b39P2DrogSjkkHRyzifQ8JwmhIOpJKRvoxJyjwGOl5k+vc7z0QLmgUtmUaEydAo5AOKUZSSa5+3PChMQiQHGPE4N1l3YByTCKeQqNt37oNx3D1qlkzZ4DLxCpIFRRoufrXwI9wEpBQYoaEsC0zlk6GuKSYkWllkAgSIzxBI2IrGqKACCebvTKFp0rx4TDiqkIJZ+rviQwFQqSBpzrzo8Wil4v/eXYihxdORsM4kSTE80XDhEEZwTwX6FNOsGSpIghzqm6FeIw4wlKlV1EhWIsvL5NuvWaZNeverDavijjK4AicgDNggXPQBDegBToAg0fwDF7Bm/akvWjv2se8taQVM4fgD7TPH0ammBE=</latexit>

Topological invariants

BPS quantities

We will take G = SU (2)<latexit sha1_base64="q29DiYRKCEOv+Z6Fux99HAQrmzQ=">AAACEHicbVC7TgJBFJ31ifhCLW0mghEbskujjQnRQkuMLpAAIbPDBSbMPjJzVyUbPsHGX7Gx0BhbSzv/xuFRKHiSSU7OuffOvceLpNBo29/WwuLS8spqai29vrG5tZ3Z2a3oMFYcXB7KUNU8pkGKAFwUKKEWKWC+J6Hq9S9GfvUOlBZhcIuDCJo+6waiIzhDI7UyRw2EBxzPSRS0h0kV6L2QkiLrA81dnt24+eJxbtjKZO2CPQadJ86UZMkU5Vbmq9EOeexDgFwyreuOHWEzYQoFlzBMN2INEeN91oW6oQHzQTeT8SJDemiUNu2EyrwA6Vj93ZEwX+uB75lKn2FPz3oj8T+vHmPntJmIIIoRAj75qBOba0M6Soe2hQKOcmAI40qYXSnvMcU4mgzTJgRn9uR5UikWHLvgXBezpfNpHCmyTw5InjjkhJTIFSkTl3DySJ7JK3mznqwX6936mJQuWNOePfIH1ucPUCicCg==</latexit>

Dimofte,Gaiotto,Gukov ‘11

PART II: MOCK, FALSE, AND QUANTUM
We will consider the so-called half-index with boundary conditions labeled by abelian flat
connections on M3:Ẑa (q) = Z(D2 ⇥ S 1 ; Ba )<latexit sha1_base64="Ixq8CDfE65jGqFHrD1YguFVfm04=">AAACFnicbVDLSgNBEJyN7/iKevQyGITkYNgVQUEEiR48KpoHySZL72SSDM4+nOkVQshXePFXvHhQxKt482+cxBw0saChqOqmu8uPpdBo219WamZ2bn5hcSm9vLK6tp7Z2CzrKFGMl1gkI1X1QXMpQl5CgZJXY8Uh8CWv+LdnQ79yz5UWUXiDvZg3AuiEoi0YoJG8zJ7bBaQ1D3J3eXpCa7nz5j51UQRc0+umc0zdALDLQNKiB3kvk7UL9gh0mjhjkiVjXHqZT7cVsSTgITIJWtcdO8ZGHxQKJvkg7Saax8BuocPrhoZg9jb6o7cGdNcoLdqOlKkQ6Uj9PdGHQOte4JvO4ZF60huK/3n1BNtHjb4I4wR5yH4WtRNJMaLDjGhLKM5Q9gwBpoS5lbIuKGBokkybEJzJl6dJeb/g2AXn6iB7WhzHsUi2yQ7JEYccklNyQS5JiTDyQJ7IC3m1Hq1n6816/2lNWeOZLfIH1sc3R06cUQ==</latexit>

q=e

2⇡i⌧

<latexit sha1_base64="Tk/eOj1cFuIJ7YDtIaAXk9PtRzM=">AAAB+nicbVBNS8NAEN3Ur1q/Uj16WSyCp5IUQS9C0YvHCvYDmlg220m7dLOJuxulxP4ULx4U8eov8ea/cdvmoK0PBh7vzTAzL0g4U9pxvq3Cyura+kZxs7S1vbO7Z5f3WypOJYUmjXksOwFRwJmApmaaQyeRQKKAQzsYXU399gNIxWJxq8cJ+BEZCBYySrSRenb5/gLuspqXMMywp0k66dkVp+rMgJeJm5MKytHo2V9eP6ZpBEJTTpTquk6i/YxIzSiHSclLFSSEjsgAuoYKEoHys9npE3xslD4OY2lKaDxTf09kJFJqHAWmMyJ6qBa9qfif1011eO5nTCSpBkHni8KUYx3jaQ64zyRQzceGECqZuRXTIZGEapNWyYTgLr68TFq1qutU3ZvTSv0yj6OIDtEROkEuOkN1dI0aqIkoekTP6BW9WU/Wi/VufcxbC1Y+c4D+wPr8AW7Ok3Q=</latexit>

3d N =2
theory2d N =(0,2) boundary
condition Ba<latexit sha1_base64="L4tszY4zA2dpq0p9jfpO83j6IBU=">AAAB+HicbVC7TsMwFHXKq5RHA4wsFi0SU5V0gbEqC2OR6ENqo+jGdVqrjhPZDlKJ+iUsDCDEyqew8Tc4bQZoOZKlo3Pu1T0+QcKZ0o7zbZW2tnd298r7lYPDo+OqfXLaU3EqCe2SmMdyEICinAna1UxzOkgkhSjgtB/MbnO//0ilYrF40POEehFMBAsZAW0k367WRxHoKQGetRc+1H275jScJfAmcQtSQwU6vv01GsckjajQhINSQ9dJtJeB1IxwuqiMUkUTIDOY0KGhAiKqvGwZfIEvjTLGYSzNExov1d8bGURKzaPATOYp1bqXi/95w1SHN17GRJJqKsjqUJhyrGOct4DHTFKi+dwQIJKZrJhMQQLRpquKKcFd//Im6TUbrtNw75u1Vruoo4zO0QW6Qi66Ri10hzqoiwhK0TN6RW/Wk/VivVsfq9GSVeycoT+wPn8AHYiStw==</latexit>

where a labels solutions:

ZCS (M3 ) =<latexit sha1_base64="8RpohHn9rJwCT92snDYGInXpIIg=">AAACDnicbVDLSsNAFJ34rPUVdelmsBTaTUmqoBuhtS7cCJXaBzYxTKbTdujkwcxEKCFf4MZfceNCEbeu3fk3TtsstPXAhcM593LvPW7IqJCG8a0tLa+srq1nNrKbW9s7u/refksEEcekiQMW8I6LBGHUJ01JJSOdkBPkuYy03VFt4rcfCBc08G/lOCS2hwY+7VOMpJIcPX/nxLVGUrh2jovn0KK+hJdVSO7jshVSSOGoUagWE0fPGSVjCrhIzJTkQIq6o39ZvQBHHvElZkiIrmmE0o4RlxQzkmStSJAQ4REakK6iPvKIsOPpOwnMK6UH+wFXpe6Zqr8nYuQJMfZc1ekhORTz3kT8z+tGsn9mx9QPI0l8PFvUjxiUAZxkA3uUEyzZWBGEOVW3QjxEHGGpEsyqEMz5lxdJq1wyjZJ5c5KrXKRxZMAhOAIFYIJTUAFXoA6aAINH8AxewZv2pL1o79rHrHVJS2cOwB9onz/nEJl4</latexit>Z

DAe2⇡ikS(A)dA + A ^ A = 0<latexit sha1_base64="gaYRwjERiTl4S22EF65sAX4y/4w=">AAAB+XicbVDLSgNBEOz1GeNr1aOXwSAIQtgVQS9CohePEcwDkhBmZ3uTIbMPZmYjYcmfePGgiFf/xJt/4yTZgyYWNBRV3XR3eYngSjvOt7Wyura+sVnYKm7v7O7t2weHDRWnkmGdxSKWLY8qFDzCuuZaYCuRSENPYNMb3k395gil4nH0qMcJdkPaj3jAGdVG6tm2XyXnpNp5Qr+PpHrj9OySU3ZmIMvEzUkJctR69lfHj1kaYqSZoEq1XSfR3YxKzZnASbGTKkwoG9I+tg2NaIiqm80un5BTo/gkiKWpSJOZ+nsio6FS49AznSHVA7XoTcX/vHaqg+tuxqMk1Rix+aIgFUTHZBoD8blEpsXYEMokN7cSNqCSMm3CKpoQ3MWXl0njouw6ZffhslS5zeMowDGcwBm4cAUVuIca1IHBCJ7hFd6szHqx3q2PeeuKlc8cwR9Ynz/RO5HS</latexit>

Such functions are referred to as “homological blocks” because, e.g., of the integrality of
For certain 2d (0, 2) b.c. Ba labeled by a 2 TorH1 (M3 , Z)/Z2 , we havecoefficients in this index:Ẑa (q) = q a<latexit sha1_base64="WoEUELjJD4SMFLr3t6l4R/8WVCs=">AAACwnicbVFNb9NAELXNVzEfDXDkMiJGSqQo2OEAF0RVKtQLUpGatiJOrdnNJFm6Xju760Jkwo/k1n/Dxg1fLU9a6e3MvJ23M6yUwtg4vvCDGzdv3b6zdTe8d//Bw+3Wo8dHpqg0pyEvZKFPGBqSQtHQCivppNSEOZN0zM7erfPH56SNKNShXZY0znGmxFRwtC6UtS7eFxo4aYtCwWACUSfuDboRsD7vQ5TmaOccJexmGIFERpImwJYQIaROkFr6auvDQq/2s6TzIXvZaxSMwafui980G0Q9+EIwx3MKU0YzoWpaVI2DVaoKVeWMdJjO0bpi7Cy68AYWp3W6R9JihqvUVHmmAN1ZnKoefHdobo2HX23ClNTkz8NZqx334wZwnSQb0vY2OMhaP9JJwauclOUSjRklcWnHNWoruKRVmFaGSuRnOKORowpzMuO6WcEKnrvIBKZumNNCWWiifytqzI1Z5sxVrv2aq7l18H+5UWWnr8e1UGVlSfHLRtNKgi1gvU+YCE3cyqUjyLVwXoHPUSO3buuhG0Jy9cvXydGgn8T95OOgvbO7GceW99R75nW8xHvl7Xj73oE39Lj/1idf+UWwF3wOFoG5LA38jeaJ9w+Cbz8BKbbWDg==</latexit>Xnan q n ,an 2 Z

Gukov,Putrov,Vafa ’16
Gukov,Pei,Putrov,Vafa ‘17

PART II: MOCK, FALSE, AND QUANTUM
The Ẑa are conjectured to play a role as building blocks
in the following quantities:<latexit sha1_base64="K/elLJ3qkOY+fgiFq+GMZZFibbQ=">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</latexit>

The Chern Simons partition function (“WRT invariants”):

ZSU (2)k (M3 ) /e2⇡ikCS(a) Sab Ẑb (q)|X

Ẑa (q)Ẑa (1/q)

2⇡i
q!e k

a,b2{abelian}<latexit sha1_base64="7h7KdjVxVjmo9PcaRoWoo8LB6+w=">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</latexit>
sha1_base64="hP+6LrUf2d3tZaldqaQQvEKMXyw=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odBu3wMYA6nMMFXEEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4</latexit>
sha1_base64="dKn9tfU9poFz3bFVyz+MH7Glgsk=">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</latexit>
sha1_base64="HZ42P1dGZy5AJSF6A780qCNEwmM=">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</latexit>

for q approaching roots of unity

And the superconformal Index:
ZT [M3 ] (S 2 ⇥ S 1 ) ⇠<latexit sha1_base64="NG/rRYARerHnshzS1OdwmCiIBIA=">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</latexit>Xa2{abelian}

after defining a continuation to outside the unit circle

Gukov,Putrov,Vafa ’16
Gukov,Pei,Putrov,Vafa ‘17

PART II: MOCK, FALSE, AND QUANTUM
There exists a localization-type contour integral formula combining 2d and 3d
degrees of freedom:Ẑa ⇠<latexit sha1_base64="tiRCo4Ndh6SNruBRBYWMQO9pAbA=">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</latexit>

Z

dx
(a)
F3d (x)⇥2d (x)2⇡ix

Questions: 1. What are the modularity properties of Ẑa (q)?
<latexit sha1_base64="R0px17h7xrgxGHM5/tbEsmPe8kQ=">AAACIHicbVC7TgMxEPTxDOEVoKSxCEjQRHdpoANBQwkSIRFJFO05e8TCdz7sPaQoyqfQ8Cs0FCAEHXwNzqOAhJEsjWZm19oJUyUt+f6XNzM7N7+wmFvKL6+srq0XNjavrc6MwIrQSptaCBaVTLBCkhTWUoMQhwqr4d3ZwK8+oLFSJ1fUTbEZw20iIymAnNQqHAYlXu0AcTDIqYM81u1MgZHU5anRKRqSaLmO+G5jELtpwf79we5xq1D0S/4QfJoEY1JkY1y0Cp+NthZZjAkJBdbWAz+lZg/cfqGwn29kFlMQd3CLdUcTiNE2e8MD+3zPKW0eaeNeQnyo/p7oQWxtNw5dMgbq2ElvIP7n1TOKjpo9maQZYSJGH0WZ4qT5oC3elgYFqa4jIFwpUnDRAQOCXKd5V0IwefI0uS6XAr8UXJaLJ6fjOnJsm+2wfRawQ3bCztkFqzDBHtkze2Vv3pP34r17H6PojDee2WJ/4H3/AA7xoaM=</latexit>
sha1_base64="hP+6LrUf2d3tZaldqaQQvEKMXyw=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odBu3wMYA6nMMFXEEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4</latexit>
sha1_base64="UDGroH8HnbcqHupYSiQUD85CrS0=">AAACFXicbZA7SwNBFIXv+ozxFW1tBh+gTdi1iZ2CjaWCScQkhLuTu2ZwdmeduSuE4E+x8a/YWCiinf/GSUzh68DA4Zw7M9wvzrVyHIYfwdT0zOzcfGmhvLi0vLJaWVtqOFNYSXVptLEXMTrSKqM6K9Z0kVvCNNbUjK+PR33zlqxTJjvnQU6dFK8ylSiJ7KNupRZVRbOPLNCS4D6J1PQKjVbxQOTW5GRZkRMmEdvt0dhlF3dv9rYPu5WtsBqOJf6aaGK2YKLTbuW93TOySCljqdG5VhTm3Bmif19quiu3C0c5ymu8opa3GabkOsPxgndixyc9kRjrT8ZinH6/McTUuUEa+8kUue9+d6Pwv65VcHLQGaosL5gy+fVRUmjBRoxoiZ6yJFkPvEHpoSgpZB8tSvZMyx5C9Hvlv6axX43CanQWQgk2YBN2IYIaHMEJnEIdJNzDIzzDS/AQPAWvX7imggm3dfih4O0TIamgGg==</latexit>
sha1_base64="3v6R2qj00L6l3Y5Qt36LhMwYJUQ=">AAACIHicbVA9TwJBFNzzE/ELtbTZCCbakDsb7CTaWGIiYgRC3i3vZOPe7bn7zoQQfoqNf8XGQmO001/jghQqTrLJZGbe27wJUyUt+f6HNzM7N7+wmFvKL6+srq0XNjYvrM6MwLrQSpvLECwqmWCdJCm8TA1CHCpshDcnI79xh8ZKnZxTP8V2DNeJjKQAclKnUAnKvNED4mCQUw95rLuZAiOpz1OjUzQk0XId8VJrFLvqwN7tfumoUyj6ZX8MPk2CCSmyCWqdwnurq0UWY0JCgbXNwE+pPQC3Xygc5luZxRTEDVxj09EEYrTtwfjAId91SpdH2riXEB+rPycGEFvbj0OXjIF69q83Ev/zmhlFh+2BTNKMMBHfH0WZ4qT5qC3elQYFqb4jIFwpUnDRAwOCXKd5V0Lw9+RpcnFQDvxycOYXq8eTOnJsm+2wPRawCquyU1ZjdSbYPXtkz+zFe/CevFfv7Ts6401mttgveJ9fDlGhoQ==</latexit>➤ 3d theory trivial=>2d elliptic genus (modular)➤ 3d theory non-trivial =>modular properties spoiled➤

In large class of examples (certain Seifert manifolds), we
find “false theta functions” and mock modular forms

2. How does it encode the physics?
<latexit sha1_base64="Ixuy1mfM11jChTnOjGKbZzym6qA=">AAACHHicbVC7SgNBFJ2NrxhfUUubwSBYhd1YaGfQJmUE84AkhNnZm2TI7M4yc1cNSz7Exl+xsVDExkLwb5w8Ck08MHA4517O3OPHUhh03W8ns7K6tr6R3cxtbe/s7uX3D+pGJZpDjSupdNNnBqSIoIYCJTRjDSz0JTT84fXEb9yBNkJFtziKoROyfiR6gjO0Ujd/VirSirqngQJDBVKIuAqA4gBoG+EBpwmphmCcxoOREdyML7v5glt0p6DLxJuTApmj2s1/tgPFkxAi5JIZ0/LcGDsp0yi4hHGunRiIGR+yPrQsjVgIppNOo8f0xCoB7SltX4R0qv7eSFlozCj07WTIcGAWvYn4n9dKsHfRSUUUJ2ivngX1EklR0UlTNBAaOMqRJYxrYf9K+YBpxtH2mbMleIsnL5N6qei5Re+mVChfzevIkiNyTE6JR85JmVRIldQIJ4/kmbySN+fJeXHenY/ZaMaZ7xySP3C+fgC/iaHZ</latexit>

3. How can we define Ẑa (q 1 )?
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±2
1 of them isplumbing graph . There are various ways to |z|=1
define2⇡iz
M3 for a(zgivenqt;. q)One
n2pZ+at=q
to say that M3 is obtained by a Dehn surgery on the corresponding link L( ) of unknots(seeFigure “v.p.”5). For simplicity,
also
assume thatis connected.
Such classof 3-manifolds
wherestand weforprincipal
value
integration.
This
indeed agrees with the partition
contains Seifert fibrations over S 2 , 2which correspondtostar-shaped
plumbing
graphs.
Denote1 , cf. (3.122),
function
of
T
[M
]
on
D
⇥
S
3
by L the number of verticesof . It is equal to the number of components of the link L( ).
We will also need L ⇥ L linking matrix of L( ) which will be denoted by Mg:Y8(ui ; q)1
(ref.CS; )
b
b
>
Z
(q)
=
Z
(q,
t,
u)
·
(qt;
q)
.
(3.141)1,v,vconnected,
1 a2
a
1
<
(tu
q;
q)
⇠
i (3.142)1
Mv1 ,v2 = av , v1 = v2 = v,
vi 2 Vertices of, L}.
= {1, . . . i=1
t=q ,ui =1>: 0, otherwise.

PART II: MOCK, FALSE, AND QUANTUM

How to describe a 3-manifold?

3.4 M3 = plumbed

Now we move to consider anothera4very large class of 3-manifolds M3 ( ) associated to aa4plumbing graph . There are various ways to define aM
for a given . One of them is
9 3
a9aasurgery3
to say that 3Ma
isobtainedbyaDehn
the corresponding link L( ) of unknots
a5 aon36a56Plumbing diagram:(see Figure a5). For simplicity,
we also assume that is connected. Such class of 3-manifolds
a7
2
a7
2
2 , which aacontainsaSeifert
fibrations
overS
correspondto star-shaped plumbing graphs. Denote8
1aa8
by L the number of vertices of . It is 1equal to the numberof components of the link L( ).
We will also need L ⇥ L linking matrix of L( ) which will be denoted by M :Figure 5: An example of a plumbing graph (left) and the corresponding link L( ) of framed unknots
8in S 3 (right). The associated 3-manifold M3 ( ) can be constructed by performing a Dehn surgery onencodes topology of>< 1, v1 , v2 connected,L( ).
M3 and parameters⇠
Mv1 ,v2 = av , v1 = v2 = v,
vi 2 Vertices of(3.142)= {1, . . . , L}.
Linking matrix:
inUV
Lagrangianof>The first homology group
3 ( ) is given by the cokernel (over Z) of the linking matrix:: 0,of Motherwise.
T[M3]
H1 (M3 , Z) ⇠(3.143)= Coker M = ZL /M ZL .

3d action:L<latexit sha1_base64="r/pF5N7AW5eFfWPgaoOjHbiT/Nc=">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</latexit>Ẑa (q) ⇠ p.v.
<latexit sha1_base64="U2xXkrFYnC51VjTUSlgnjtNX858=">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</latexit>

ZY|zv |=1 v2Vertices

n Z
X

i,j=1

dzv
(zv
2⇡izv

ij
4 –M
50 –

d ✓

4⇡

Ai dAj1/zv )2 ⇥Y(v1 ,v2 )2Edges

Gadde,Gukov,Putrov ‘13

(zv1

⇥M
a (z)
1/zv1 )(zv2

1/zv2 )

Gukov,Pei,Putrov,Vafa ‘17

PART II: MOCK, FALSE, AND QUANTUM
Example:M3 = ⌃(2, 3, 7)<latexit sha1_base64="Vk9/O9/B8fExut8euPENVm1aZMs=">AAAB+nicbVDLSgMxFM34rPU11aWbYBEqlDLTCnUjFN24ESraB7TDkEnTNjTJDElGKWM/xY0LRdz6Je78G9N2Ftp64MLhnHu5954gYlRpx/m2VlbX1jc2M1vZ7Z3dvX07d9BUYSwxaeCQhbIdIEUYFaShqWakHUmCeMBIKxhdTf3WA5GKhuJejyPicTQQtE8x0kby7dyNX7no3tEBR4VysVKsnvp23ik5M8Bl4qYkD1LUffur2wtxzInQmCGlOq4TaS9BUlPMyCTbjRWJEB6hAekYKhAnyktmp0/giVF6sB9KU0LDmfp7IkFcqTEPTCdHeqgWvan4n9eJdf/cS6iIYk0Eni/qxwzqEE5zgD0qCdZsbAjCkppbIR4iibA2aWVNCO7iy8ukWS65Tsm9PcvXLtM4MuAIHIMCcEEV1MA1qIMGwOARPINX8GY9WS/Wu/Uxb12x0plD8AfW5w85eZID</latexit>

3•<latexit sha1_base64="siJ6gNbhU0JvLruze6V372nhzvU=">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</latexit>

2
•

7
•

•
1

Ẑ0 (q) = q

1/2⌃(a, b, c) := S 5 \ {(x, y, z) 2 C3 |xa + y b + z c = 0}(1q<latexit sha1_base64="XxqLLZGYc3VgJ7jHrT8wC5E9B3I=">AAACKXicbVDLSgNBEJz1bXxFPXoZDELEEHZ9oAhC0ItHRaNCNht6JpNkcHZ2mZkV1zW/48Vf8aKgqFd/xEnMwVdBQ1HVTXcXiQXXxnXfnKHhkdGx8YnJ3NT0zOxcfn7hTEeJoqxKIxGpCwKaCS5Z1XAj2EWsGIREsHNyedDzz6+Y0jySpyaNWT2EtuQtTsFYqZGv+Ce8HUIRSqREV3f38EmwhX0KMfaz4nUpLd2s+lz6IZgOIfgg2Li9DmAtDcjaTUD3XL/byBfcstsH/ku8ASmgAY4a+Se/GdEkZNJQAVrXPDc29QyU4VSwbs5PNIuBXkKb1SyVEDJdz/qfdvGKVZq4FSlb0uC++n0ig1DrNCS2s3ey/u31xP+8WmJaO/WMyzgxTNKvRa1EYBPhXmy4yRWjRqSWAFXc3oppBxRQY8PN2RC83y//JWfrZc8te8ebhcr+II4JtISWURF5aBtV0CE6QlVE0R16QM/oxbl3Hp1X5/2rdcgZzCyiH3A+PgE1WqQz</latexit>

5

q +q

10

q

11

+q

18

+q

30

+ . . .)

<latexit sha1_base64="wifXwkCnmlsn7UywvOjdmfQsB5M=">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</latexit>=q

3 dim’l SL(2,Z) irrep

42+6,14,21(⌧ )1

83/168<latexit sha1_base64="BPx5qtZX4LffwbTBhu3DR593/XQ=">AAACDXicbVDLSsNAFJ3UV62vqEs3wSpULDVTS+1GKLpxWcE+oI8wmU7aoZOHMxOhhPyAG3/FjQtF3Lp35984bbPQ1gMXDufcy7332AGjQprmt5ZaWl5ZXUuvZzY2t7Z39N29hvBDjkkd+8znLRsJwqhH6pJKRloBJ8i1GWnao+uJ33wgXFDfu5PjgHRdNPCoQzGSSrL0o8v7XlQ5P4PlStypCdqLSsXTch6W8kUYWzDXkSg8sfSsWTCnMBYJTEgWJKhZ+len7+PQJZ7EDAnRhmYguxHikmJG4kwnFCRAeIQGpK2oh1wiutH0m9g4VkrfcHyuypPGVP09ESFXiLFrq04XyaGY9ybif147lE6lG1EvCCXx8GyREzJD+sYkGqNPOcGSjRVBmFN1q4GHiCMsVYAZFQKcf3mRNIoFaBbgbSlbvUriSIMDcAhyAIILUAU3oAbqAINH8AxewZv2pL1o79rHrDWlJTP74A+0zx9uLJic</latexit>

only 1 abelian flat
connection“false theta function”
where

42+6,14,21(⌧ ) =
10<latexit sha1_base64="+FHn3yW8IB6wqXDbiO8UiX27otY=">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</latexit>

and<latexit sha1_base64="GHqCGMmit+nI5bIrLUUV4cQ5Gak=">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</latexit>Bm,r (⌧ ) := @X

42,r (⌧ )r2{1, 13, 29,41}

Xn>0
n=r mod 2m

Xn>0n= r mod 2m1

C n2 /4m
Aq

Holomorphic Eichler
integral of wt-3/2
theta series

PART II: MOCK, FALSE, AND QUANTUM“Modular properties” of false theta functions governs physics of nonabelian
flat connections

1
p
k

m+K
(1/k) ⇠
a<latexit sha1_base64="TtIrMA+1ceyzDU3mcAI/7u1SiSQ=">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</latexit>X

non-perturbative
m+K
ab

m+K
(
b

b2

S-matrix assoc. to SL2 (Z) irrep

k) +

X

perturbative
✓
◆

cn

n+1/2
k
n 0

⇡i
2m

n

m+K

<latexit sha1_base64="5yYzbJ6uJWcyjCht2TDnFdnKWWI=">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</latexit>

where forX

n 0<latexit sha1_base64="AoJibU9bq1Uot0JVTq9f6OrL2Ro=">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</latexit><latexit sha1_base64="FJx1R5k8njCo+rDHavqdQmzMtbI=">AAACAXicbVDLSgMxFM3UV62vUTeCm2ARKpY6KUVdFt24rGAf0I5DJk3b0ExmSDJCGerGX3HjQhG3/oU7/8a0nYW2HrhwOOde7r3HjzhT2nG+rczS8srqWnY9t7G5tb1j7+41VBhLQusk5KFs+VhRzgSta6Y5bUWS4sDntOkPryd+84FKxUJxp0cRdQPcF6zHCNZG8uyDTk2x+6RSPj0vokqxjMYeKqCz4Yln552SMwVcJCgleZCi5tlfnW5I4oAKTThWqo2cSLsJlpoRTse5TqxohMkQ92nbUIEDqtxk+sEYHhulC3uhNCU0nKq/JxIcKDUKfNMZYD1Q895E/M9rx7p36SZMRLGmgswW9WIOdQgnccAuk5RoPjIEE8nMrZAMsMREm9ByJgQ0//IiaZRLyCmh20q+epXGkQWH4AgUAAIXoApuQA3UAQGP4Bm8gjfryXqx3q2PWWvGSmf2wR9Ynz8QgZQM</latexit>

42+6,14,21(1/k)
1

we have

M3 = ⌃(2, 3, 7)<latexit sha1_base64="Vk9/O9/B8fExut8euPENVm1aZMs=">AAAB+nicbVDLSgMxFM34rPU11aWbYBEqlDLTCnUjFN24ESraB7TDkEnTNjTJDElGKWM/xY0LRdz6Je78G9N2Ftp64MLhnHu5954gYlRpx/m2VlbX1jc2M1vZ7Z3dvX07d9BUYSwxaeCQhbIdIEUYFaShqWakHUmCeMBIKxhdTf3WA5GKhuJejyPicTQQtE8x0kby7dyNX7no3tEBR4VysVKsnvp23ik5M8Bl4qYkD1LUffur2wtxzInQmCGlOq4TaS9BUlPMyCTbjRWJEB6hAekYKhAnyktmp0/giVF6sB9KU0LDmfp7IkFcqTEPTCdHeqgWvan4n9eJdf/cS6iIYk0Eni/qxwzqEE5zgD0qCdZsbAjCkppbIR4iibA2aWVNCO7iy8ukWS65Tsm9PcvXLtM4MuAIHIMCcEEV1MA1qIMGwOARPINX8GY9WS/Wu/Uxb12x0plD8AfW5w85eZID</latexit>

3•n! 2n 2 sinh(6z) sinh(14z)
cn
z =
2n!
cosh(21z)<latexit sha1_base64="siJ6gNbhU0JvLruze6V372nhzvU=">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</latexit>

via Borel resummation

2
•

•
1

7
•

Gukov,Mariño,Putrov ‘16

PART II: MOCK, FALSE, AND QUANTUM
What about Ẑ0 (q 1 ) ?
<latexit sha1_base64="m0qGx2WfFQnfe9Tw6CUWmqutMVo=">AAAB+XicbVBNS8NAEN3Ur1q/oh69LBahHiyJCHosevFYwX5gG8Nmu2mXbjZxd1Ioof/EiwdFvPpPvPlv3LY5aOuDgcd7M8zMCxLBNTjOt1VYWV1b3yhulra2d3b37P2Dpo5TRVmDxiJW7YBoJrhkDeAgWDtRjESBYK1geDP1WyOmNI/lPYwT5kWkL3nIKQEj+bbdHRDAD75TeXrMztzJqW+XnaozA14mbk7KKEfdt7+6vZimEZNABdG64zoJeBlRwKlgk1I31SwhdEj6rGOoJBHTXja7fIJPjNLDYaxMScAz9fdERiKtx1FgOiMCA73oTcX/vE4K4ZWXcZmkwCSdLwpTgSHG0xhwjytGQYwNIVRxcyumA6IIBRNWyYTgLr68TJrnVdepuncX5dp1HkcRHaFjVEEuukQ1dIvqqIEoGqFn9IrerMx6sd6tj3lrwcpnDtEfWJ8/s2OSYw==</latexit>

F0 (q) :=<latexit sha1_base64="kmIYZ8JPvOnNqui9UM3nmZE0xzE=">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</latexit>

1
X

q

n2

n+1 ; q)(qnn=0

Ramanujan’s order 7 mock theta function= 1 + q + q 3 + q 4 + q 5 + 2q 7 + . . .

MOCK

q$q 1
<latexit sha1_base64="nlUiTCR/uoG5/nueejYqKdlAAQI=">AAACAXicbVDLSgNBEJz1GeNr1YvgZTAIXgy7Iugx6MVjBPOAZA2zk9lkyOzMZqZXCUu8+CtePCji1b/w5t84eRw0saChqOqmuytMBDfged/OwuLS8spqbi2/vrG5te3u7FaNSjVlFaqE0vWQGCa4ZBXgIFg90YzEoWC1sHc18mv3TBuu5C0MEhbEpCN5xCkBK7Xc/T5uChaB5p0uEK3VA+7fZSf+sOUWvKI3Bp4n/pQU0BTllvvVbCuaxkwCFcSYhu8lEGREA6eCDfPN1LCE0B7psIalksTMBNn4gyE+skobR0rbkoDH6u+JjMTGDOLQdsYEumbWG4n/eY0Uoosg4zJJgUk6WRSlAoPCozhwm2tGQQwsIVRzeyumXaIJBRta3obgz748T6qnRd8r+jdnhdLlNI4cOkCH6Bj56ByV0DUqowqi6BE9o1f05jw5L8678zFpXXCmM3voD5zPH0TnlsE=</latexit>

F0⇤ (q) = F0 (1/q) =

1
X

n=0

<latexit sha1_base64="XQyPfNNrsa9ZyHHGQq/E89lSqVE=">AAACbHicbZFdaxQxFIYzUz/q+tFt7YVShOCizLrsNilKC7JQFMTLCm5b2NkZMtlMG5pJZpOMsIRc+Q+98yd4428ws90LbT1wOA/vycnHm6IW3FiEfkbxxp279+5vPug8fPT4yVZ3e+fUqEZTNqFKKH1eEMMEl2xiuRXsvNaMVIVgZ8XVx7Z/9o1pw5X8apc1m1XkQvKSU2KDlHe/f8pR9iZZ9McBErwfAKamqXInx8hnKZelXcJkiPuZTEtNqFtkzuFUhU3hgZeJHOC+9y4JckD/ftHPpYdjiOEQLtrM3sEBDMVh5IdtwX7QliMPB6mYK2vybg+N0CrgbcBr6IF1nOTdH+lc0aZi0lJBjJliVNuZI9pyKpjvpI1hNaFX5IJNA0pSMTNzK7M8fBWUOSyVDiktXKl/TzhSGbOsirCyIvbS3Oy14v9608aWRzPHZd1YJun1QWUjoFWwdR7OuWbUimUAQjUPd4X0kgRLbfifTjAB33zybTg9GGE0wl/e9o4/rO3YBHvgJUgABofgGHwGJ2ACKPgVbUXPoufR73g33otfXC+No/XMU/BPxK//AAp4tDY=</latexit>

FALSE
( 1)

n q

1
2 n(n+1)(q n+1 ; q)n=1

q 5 + q 10

q

q 11 + q 18 + . . .

= q 1/2 Ẑ0 (q)<latexit sha1_base64="ajq1hlfPD45Ncz3RoU9kpiUkUQE=">AAAB/XicbVDJSgNBEO2JW4zbuNy8NAYhHowzQdCLEPTiMYJZMBlDT6cnadKzpLtGiEPwV7x4UMSr/+HNv7GTzEETHxQ83quiqp4bCa7Asr6NzMLi0vJKdjW3tr6xuWVu79RUGEvKqjQUoWy4RDHBA1YFDoI1IsmI7wpWd/tXY7/+wKTiYXALw4g5PukG3OOUgJba5t7F4D45tk9Ko1aPAL5rW4XBUdvMW0VrAjxP7JTkUYpK2/xqdUIa+ywAKohSTduKwEmIBE4FG+VasWIRoX3SZU1NA+Iz5SST60f4UCsd7IVSVwB4ov6eSIiv1NB3dadPoKdmvbH4n9eMwTt3Eh5EMbCAThd5scAQ4nEUuMMloyCGmhAqub4V0x6RhIIOLKdDsGdfnie1UtG2ivbNab58mcaRRfvoABWQjc5QGV2jCqoiih7RM3pFb8aT8WK8Gx/T1oyRzuyiPzA+fwD3UJOa</latexit>Ẑ(q 1 ) = q 83/168 H142+6,14,21 (⌧ )<latexit sha1_base64="PGZnlFwzaqOk/wX51dtC/X/r6Co=">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</latexit>

M3 = ⌃(2, 3, 7)<latexit sha1_base64="Vk9/O9/B8fExut8euPENVm1aZMs=">AAAB+nicbVDLSgMxFM34rPU11aWbYBEqlDLTCnUjFN24ESraB7TDkEnTNjTJDElGKWM/xY0LRdz6Je78G9N2Ftp64MLhnHu5954gYlRpx/m2VlbX1jc2M1vZ7Z3dvX07d9BUYSwxaeCQhbIdIEUYFaShqWakHUmCeMBIKxhdTf3WA5GKhuJejyPicTQQtE8x0kby7dyNX7no3tEBR4VysVKsnvp23ik5M8Bl4qYkD1LUffur2wtxzInQmCGlOq4TaS9BUlPMyCTbjRWJEB6hAekYKhAnyktmp0/giVF6sB9KU0LDmfp7IkFcqTEPTCdHeqgWvan4n9eJdf/cS6iIYk0Eni/qxwzqEE5zgD0qCdZsbAjCkppbIR4iibA2aWVNCO7iy8ukWS65Tsm9PcvXLtM4MuAIHIMCcEEV1MA1qIMGwOARPINX8GY9WS/Wu/Uxb12x0plD8AfW5w85eZID</latexit>

3•<latexit sha1_base64="siJ6gNbhU0JvLruze6V372nhzvU=">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</latexit>

2
•

•
1

7
•

“mock theta
function”

PART II: MOCK, FALSE, AND QUANTUM
Quantum Modular Forms
Definition. A quantum modular form is a function Q(x) on Q such that✓◆ax + bp(x) := Q(x) Q(cx + d)kcx + d<latexit sha1_base64="6hX7mzbhwo1kBQG1Ltg2rMfM+to=">AAAB8XicbVBNSwMxEJ31s9avqkcvwSJUhLIrgh6LXjxWsB/YriWbzbah2WRJsmJZ+i+8eFDEq//Gm//GtN2Dtj4YeLw3w8y8IOFMG9f9dpaWV1bX1gsbxc2t7Z3d0t5+U8tUEdogkkvVDrCmnAnaMMxw2k4UxXHAaSsYXk/81iNVmklxZ0YJ9WPcFyxiBBsr3VfIEzpF4cnDsFcqu1V3CrRIvJyUIUe9V/rqhpKkMRWGcKx1x3MT42dYGUY4HRe7qaYJJkPcpx1LBY6p9rPpxWN0bJUQRVLZEgZN1d8TGY61HsWB7YyxGeh5byL+53VSE136GRNJaqggs0VRypGRaPI+CpmixPCRJZgoZm9FZIAVJsaGVLQhePMvL5LmWdVzq97tebl2lcdRgEM4ggp4cAE1uIE6NICAgGd4hTdHOy/Ou/Mxa11y8pkD+APn8we5u4+i</latexit>

extends to a function of R minus finitely many points and has some analyticity
or continuity property.
captures values of
CS partition
function (WRT
invariants)

x

×

×

~

×

+

x

<latexit sha1_base64="gBke5oZatKBHvQVglwHy4Jj2UiU=">AAAB/HicbVBNS8NAEN34WetXtEcvi0XwVJMi6LHoxWMF+wFNLJvtpF262cTdjRBC/StePCji1R/izX/jts1BWx8MPN6bYWZekHCmtON8Wyura+sbm6Wt8vbO7t6+fXDYVnEqKbRozGPZDYgCzgS0NNMcuokEEgUcOsH4eup3HkEqFos7nSXgR2QoWMgo0Ubq25UHT8cY7vO6lzDMsDwTk75ddWrODHiZuAWpogLNvv3lDWKaRiA05USpnusk2s+J1IxymJS9VEFC6JgMoWeoIBEoP58dP8EnRhngMJamhMYz9fdETiKlsigwnRHRI7XoTcX/vF6qw0s/ZyJJNQg6XxSmHJt/p0ngAZNANc8MIVQycyumIyIJ1SavsgnBXXx5mbTrNdepubfn1cZVEUcJHaFjdIpcdIEa6AY1UQtRlKFn9IrerCfrxXq3PuatK1YxU0F/YH3+AJmOlBM=</latexit>Ẑ0 (q)×' '@q ! e2⇡ir/n<latexit sha1_base64="fBQAZyiTooBh2aEdcOzn648KyLg=">AAAB8nicbVBNS8NAEJ34WetX1aOXxSLUS0lE0GPRi8cK9gPTUDbbTbt0s4m7E6GU/gwvHhTx6q/x5r9x2+agrQ8GHu/NMDMvTKUw6Lrfzsrq2vrGZmGruL2zu7dfOjhsmiTTjDdYIhPdDqnhUijeQIGSt1PNaRxK3gqHN1O/9cS1EYm6x1HKg5j2lYgEo2glvzOgSB66buXxrFsqu1V3BrJMvJyUIUe9W/rq9BKWxVwhk9QY33NTDMZUo2CST4qdzPCUsiHtc99SRWNugvHs5Ak5tUqPRIm2pZDM1N8TYxobM4pD2xlTHJhFbyr+5/kZRlfBWKg0Q67YfFGUSYIJmf5PekJzhnJkCWVa2FsJG1BNGdqUijYEb/HlZdI8r3pu1bu7KNeu8zgKcAwnUAEPLqEGt1CHBjBI4Ble4c1B58V5dz7mrStOPnMEf+B8/gDYSJBM</latexit>

×

×

x

X

x

X

x

x

X

x

x

⇒

xto

HE
6
xxx

a

<latexit sha1_base64="9qeh7O3NHiHek0b5aStngSmYamI=">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</latexit>

PART II: MOCK, FALSE, AND QUANTUM
Both FALSE and MOCK give rise to quantum modular forms

1. Q(x) := lim

t!0+<latexit sha1_base64="/0enf9IlqTlVphyKolAWm7NVG1o=">AAACGHicbVDLSgMxFM34rPU16tJNsAgtlTopRUUQim5cVrAP6LRDJk3bYOZBckcsQz/Djb/ixoUibt35N6a1C18HAodz7uXmHD+WQoPjfFhz8wuLS8uZlezq2vrGpr213dBRohivs0hGquVTzaUIeR0ESN6KFaeBL3nTv7mY+M1brrSIwmsYxbwT0EEo+oJRMJJnH17l7wqnZ9iVIvBScCHCTrc4dmtadNNKuXh0QCoHZTL2SP6uKKDg2Tmn5EyB/xIyIzk0Q82z391exJKAh8Ak1bpNnBg6KVUgmOTjrJtoHlN2Qwe8bWhIA6476TTYGO8bpYf7kTIvBDxVv2+kNNB6FPhmMqAw1L+9ifif106gf9JJRRgnwEP2daifSGziT1rCPaE4AzkyhDIlzF8xG1JFGZgus6YE8jvyX9Iol4hTIleVXPV8VkcG7aI9lEcEHaMqukQ1VEcM3aNH9IxerAfryXq13r5G56zZzg76Aev9E2jXnNk=</latexit>

42+6,14,21(x + it)11091 2 3902137 3 29292761483 4⇠ 2t +t +t +t + ...
x=0
84
28224
14224896

M3 = ⌃(2, 3, 7)<latexit sha1_base64="Xe6uodGf4WsAGo85VehmVcMD+z4=">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</latexit>

2. Q(x) := lim g ⇤ (xt!0+<latexit sha1_base64="QL4yg3yHcllXcP7finYug5qxATM=">AAACCHicbVDLSgMxFM34rPU16tKFwSK0imVGBEUQim5ctmAf0BeZNNOGZh4kd6Rl6NKNv+LGhSJu/QR3/o1pOwttPRA4nHMvN+c4oeAKLOvbWFhcWl5ZTa2l1zc2t7bNnd2KCiJJWZkGIpA1hygmuM/KwEGwWigZ8RzBqk7/duxXH5hUPPDvYRiypke6Pnc5JaCltnlQyg5yV9e4IbjXjqEBAbZaJ6Nu6zg7OOWQa5sZK29NgOeJnZAMSlBsm1+NTkAjj/lABVGqblshNGMigVPBRulGpFhIaJ90WV1Tn3hMNeNJkBE+0koHu4HUzwc8UX9vxMRTaug5etIj0FOz3lj8z6tH4F42Y+6HETCfTg+5kcA67rgV3OGSURBDTQiVXP8V0x6RhILuLq1LsGcjz5PKWd628nbpPFO4SepIoX10iLLIRheogO5QEZURRY/oGb2iN+PJeDHejY/p6IKR7OyhPzA+fwCrVJfV</latexit><latexit sha1_base64="Vk9/O9/B8fExut8euPENVm1aZMs=">AAAB+nicbVDLSgMxFM34rPU11aWbYBEqlDLTCnUjFN24ESraB7TDkEnTNjTJDElGKWM/xY0LRdz6Je78G9N2Ftp64MLhnHu5954gYlRpx/m2VlbX1jc2M1vZ7Z3dvX07d9BUYSwxaeCQhbIdIEUYFaShqWakHUmCeMBIKxhdTf3WA5GKhuJejyPicTQQtE8x0kby7dyNX7no3tEBR4VysVKsnvp23ik5M8Bl4qYkD1LUffur2wtxzInQmCGlOq4TaS9BUlPMyCTbjRWJEB6hAekYKhAnyktmp0/giVF6sB9KU0LDmfp7IkFcqTEPTCdHeqgWvan4n9eJdf/cS6iIYk0Eni/qxwzqEE5zgD0qCdZsbAjCkppbIR4iibA2aWVNCO7iy8ukWS65Tsm9PcvXLtM4MuAIHIMCcEEV1MA1qIMGwOARPINX8GY9WS/Wu/Uxb12x0plD8AfW5w85eZID</latexit>

3
•

it)
<latexit sha1_base64="siJ6gNbhU0JvLruze6V372nhzvU=">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</latexit>⇠

x=0<latexit sha1_base64="V+fNDBgW2IUweZMzW+4KAm2farM=">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</latexit>

1091 2
2t +
t
84

2
•

•
1

7
•

3902137 3 29292761483 4
t +t + ...
28224
14224896

where Ĥ := H142+6,14,21 + g ⇤<latexit sha1_base64="zGZ0CJVovicgLCiadf/XNE1qiWA=">AAACB3icbVDLSgMxFM34rPU16lKQYBHEljIpRUUQim66rGAf0E6HTJppQzMPkoxQhu7c+CtuXCji1l9w59+YtrPQ1gMXDufcy733uBFnUlnWt7G0vLK6tp7ZyG5ube/smnv7DRnGgtA6CXkoWi6WlLOA1hVTnLYiQbHvctp0h7cTv/lAhWRhcK9GEbV93A+YxwhWWnLMo84AK1i9uq52k3Ipf15A5UIJjR0E87DfPXPMnFW0poCLBKUkB1LUHPOr0wtJ7NNAEY6lbCMrUnaChWKE03G2E0saYTLEfdrWNMA+lXYy/WMMT7TSg14odAUKTtXfEwn2pRz5ru70sRrIeW8i/ue1Y+Vd2gkLoljRgMwWeTGHKoSTUGCPCUoUH2mCiWD6VkgGWGCidHRZHQKaf3mRNEpFZBXRXTlXuUnjyIBDcAxOAQIXoAKqoAbqgIBH8AxewZvxZLwY78bHrHXJSGcOwB8Ynz9Qt5W+</latexit>

is modular

For more complicated 3-manifolds, one may not see false/mock structure, but
we expect the structure of the quantum modular form to survive
Cheng, Chun, Ferrari, Gukov, SH

PART II: MOCK, FALSE, AND QUANTUM
Summary:➤

For many theories, “false theta functions” are the natural number theoretic objects
appearing in the half-index➤

In many cases they can be continued to the lower half plane to obtain mock modular
forms

➤

Quantum modular forms capture the value of the Chern Simons path integral at rational
points, and we expect these objects to play a role more generally

Many open questions, including:➤ Relation between modularity and resurgence➤

Appearance of many mock modular forms from umbral moonshine➤ Interpretation of the Rademacher expansion for false and mock theta functions

ありがとうございました

THANK YOU!