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