Topology of the space of Quantum Field TheoriesTopology of the space of Topology of the space of Quantum Field TheoriesQuantum Field Theories

arXiv:1811.07884

Du Pei Pavel Putrov Cumrun Vafa

Chapter One

Homology

= space of Quantum Field Theories

in D dimensions, with a given symmetry,supersymmetry, …

Charles C. Conley 1933-1984

RG Flow = Dynamical System

m=2 m=0

m=1

m=0

Chapter Two

Homotopy

families of 2d (0,1) theories parametrized by X

deformations

= space of all 2d (0,1) theories

In particular,

graded by

• Physics of 2d (0,1) theories

• Generalizations and applications

• scalar multiplet:

• Fermi multiplet:

• The (0,1) version of J-interaction:

[C.Hull, E.Witten]:

• scalar multiplet:

• Fermi multiplet:

• vector multiplet:

[C.Hull, E.Witten]:

Anomalies

*

Chapter Three

The Ising model of 2d (0,1) theories

2d N = (0,1) SQCD

SU(2) vector

– – Ncccc

gauge anomaly: 1

2

2d N = (0,1) SQCD

SU(2) vector,

2 complex fundamental chirals

gauge anomaly:

2d N = (0,2) appetizer

SQCD:

SU(2) vector,

4 fundamentals

LG model:

6 chirals1 Fermi

[S.G., M.Dedushenko]

2d N = (0,2) SQCD N = (0,2) LG model

SU(2) with N = 2ffff

2d N = (0,1)5 free scalars

2d N = (0,1) SQCD

Classical space of vacua = cone on

cf. three homomorphisms

i) (2,2)

described by how 4 of SU(4) transforms under SU(2) x SU(2)

ffffcccc

ii) (2,1) + (2,1)

iii) (2,1) + (1,1) + (1,1)

[C.Vafa, E.Witten]

Chapter Four

Modularity of the 21st century

integral weakly holomorphicmodular forms

but

~~~~

~~~~

Hurewicz homomorphism:

gen. by

“Hopf invariant”(Witten anomaly)

6d (0,1) theory

on � x M6-4

42d N N N N = (0,1) theory

T[M ]4

topological

invariant of M4

2d N = (0,1) theories from higher dimensions

“effective”

Example: 6d (0,1) free tensor

Enriques surface

M4 T[M ]4

1

3

-29

-2

-15

h

n

0

0

h . E4

D

= { 2d N = (0,1) theories w/ symmetry G }

[L.Fidkowski, A.Kitaev][A.Kapustin, R.Thorngren, A.Turzillo, Z.Wang]

[E.Witten][D.Freed, M.Hopkins]

:

cf. ( ) = ( )SPT phases

in D+1 dim

Anomalies

in D dim

graded by

[L.Fidkowski, A.Kitaev][A.Kapustin, R.Thorngren, A.Turzillo, Z.Wang]

[E.Witten][D.Freed, M.Hopkins]

:

cf. ( ) = ( )SPT phases

in D+1 dim

Anomalies

in D dim

Example (D = 1): reduction to N =1 quantum mechanics in 0+1 dimensions

Fermionic SPT and Spin(7) holonomy

Cayley 4-form

Chapter Five

Hidden Algebraic Structures in Topology

3d theory 2d theory

6d theory 6d theory

4-manifold3-manifold

T[M ]3 T[M ]4

4-manifold

3-manifold

VOA[M ]4

MTC[M ]3

Log-VOA[M ]3

TMF class [M ]4

6d N = (0,2)

4d N = 2

5d N = 1

3d N = 2on 2-manifold

VOA

MTC

TMF

MTC