topology of the space of quantum field theories · quantum field theories topology of the space of...

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