The Geometry of Quantum Field Theory
The Geometry of Quantum Field Theory
Robbert Dijkgraaf Ins$tute for Advanced Study 2014 MPS Annual Mee:ng Simons Founda:on
Some Ideology
TRUTH AND BEAUTY Seal of the Ins-tute for Advanced Study
“My work always tried to unite the true with the beau-ful, but when I had to choose one or the other, I usually chose the beau-ful.” Hermann Weyl
“It is more important for our equa-ons to be beau-ful than to have them fit experiment.” Paul Dirac
“Every law of physics, pushed to the extreme, will be found to be sta-s-cal and approximate, not mathema-cal perfect and precise.” John Wheeler
“Any theory that can account for all of the facts is wrong, because some of the facts are always wrong.” Francis Crick
Esthe:cs in Physics
Garbage
Beauty
Reduc*on
Garbage
Beauty
Standard Model
macro
micro
Emergence
Beauty
Garbage
hydrodynamics thermodynamics
Standard Model sta-s-cal mechanics
macro
micro
Renormaliza*on Group
Infrared fixed points
Ultraviolet fixed points
Λ=0
Λ=∞
Large distances Low energy
Standard Model Short distances High energy
macro
micro
Quan*za*on
Geometry
Algebra
K
Z(K) ∈ C quantum invariant
geometric object
€
€
Emergent Geometry
Geometry
effec-ve geometry
Algebra
quantum system
Synthesis
Quantum Geometry
String Theory
Classical Mechanics
B A
calculus, geometry, dynamical systems, chaos,…
Quantum Mechanics
Sum over histories
A
− Action / h e ∑
functional analysis, operator algebra, differential topology,…
B
Quantum Field Theory
A
crea-on/annihila-on of par-cles
C quantum topology: knots, 3- & 4-manifolds twistors, Grassmannians & amplitudology
B
String Theory
conformal field theory, algebraic curves, moduli spaces, mirror symmetry, quantum cohomology
Quantum Gravity
A?
non-commutative geometry, emergent geometry, automorphic forms,…
What Is A Quantum Field Theory?
Classical Field Theory Space-‐*me manifold M, field space F
φ :M →F
Ac*on, cri*cal points
S[φ ], dS = 0
Classical field equa*ons
∇ 2φ + ... = 0
Moduli space of classical solu*ons
{dS = 0} / Sym = ⨿ Modn n
M
Quantum Field Theory Euclidean path integral
Z(M ) =
−S(φ )/! D φ . e ∫
Semi-‐classical localisa*on to cri*cal points Typical form Classical limit
Z(M ) ~ ∑ e−Sn /! . ∫ n
Modn
voln
Modn
Z(M ) ~ ∑ dn q n , q ~ e−1/! n
! → ∞, q → 0
φ :M →F
Duali:es Quantum equivalences
φ : M → Rn / Λ
*
Z(M, F) = Z(M, F )
e.g. D=2 T-‐duali*es, M = Riemann surface, F = n-‐torus where
F = T n = R n / Λ, F * = T * = R n / Λ *
dφ * = *dφ
Par**on func*on = theta func*on
Z = θΛ =
Extended symmetry
∑
eπ i( p+w)τ ( p+w)−π i( p−w)τ ( p−w)
p∈Λ,w∈Λ*
SL(n, Z) ⊂ SO(n, n, Z ) = Aut(Λ ⊕ Λ * )
Space Space oof f QCuantum lassical FField ield TTheories heories classical limit !↓0
classical field theory F
Q F T
special automorphisms
Duali:es dual classical field theory F*
dual classical limit !* ↓0
self-‐dual QFTs
Duali:es of Effec:ve QFT Renormaliza*on group
Z(M, Fε )
Depends on cut-‐off distance ε or energy scale Λ=1/ε RG flow on space of regularized quantum field theories
F = F∞ ε =∞
−ε∂ ε
F * = F0 ε=0
Some*mes Z independent of ε
Z(M, F0 ) = Z(M, F∞ )
Defining Quantum Field Theory: Algebra
x1 O1 (x1 )....On (xn )
xn
Operator products of local operators ScaPering amplitudes
D=2 Conformal Field Theory Oi (x)
Operator product expansion
lim
Oi (x) O j (y)
∑ k
x→y
O j (y)
Ok (y)
Vertex algebra Associa*ve algebra
∞-‐dim vertex algebras classifica-on reps Virasoro
=
Defining Quantum Field Theory: Geometry cut & paste topological indices defect operators
Category Theory M codim 0: number, par**on func*on
Φ:M →C
M
S codim 1: vector space, Hilbert space of states
Φ:S → H
M
Φ(M ) ∈ Φ(∂M )
Gluing laws
M1
M2
Φ(M1 )! Φ(M 2 ) = Φ(M )
M
Category Theory codim 2: category of boundary condi*ons α H :α → β
β
Composi*on law of morphisms α
H1 : α → β
β
H2 : β → γ
γ
H 2 ! H1 : α → γ
D=1 Supersymmetric QM Differen*al geometry & topology *me
→X point
Hilbert space = Ω* ( X ) H = −Δ = −(dd * + d *d ) Ground states = harmonic forms
Harm* ( X ) ≅ H * ( X ) Par**on func*on = WiPen index
Tr ((−1)deg e−tH ) = Euler ( X )
D=2 Conformal Field Theory
X →X *me
S1
LX Loop Space
Quan*ze loop space Fock space created by string oscillators Infinite-‐dimensional analysis Loop space thickening X ⊂ LX
String Product *
Physical fields = cohomology classes a,b ∈ H ( X )
a a ∧b
b
H* ⊗ H* → H*
Topological String Theory
F0 (t) = ∑GWd ,0e −dt
d≥0 (pseudo) holomorphic d = deg, tcurves = Area / ℓ 2string
string
Calabi-‐Yau X
Classical Intersec:on Product ℓs = 0 dual cycles
a
c b
= X =
∫
X
a∧b∧c
Quantum Cohomology ℓs > 0
=
∑
e
− dt/ℓ2s
rat curves degree d
X for P n
x n+1 = 0 ⇒ x n+1 = e −t
smooth under flops
t ↔ −t
local P1
ℓs = 0 t 0 moduli space
ℓs > 0
t =0 singularity at zero size
Open Strings & Branes fixed time picture
brane
X
Y
open strings (brane)
closed strings (bulk)
D-‐branes: Rela:ve CFT Space-time picture
string worldsheet
brane
Y
Σ X
4D Space-time
D-‐Branes
multiplicity N
Internal space
U(N) Yang-‐Mills Theory
i
j
N×N matrix of strings Aij
U(N) Yang-‐Mills Theory
j k k i matrix multiplication Σk Aik Akj
Closed/Open Duali:es open strings
closed strings
⇔ strings inf-dim Lie algebras loop spaces g ≈ 0 s Virasoro algebra genus expansion
€
D-branes vector bundles K-theory s spaces moduli non-commutative
g ≈∞
A:yah-‐Singer Index Theorem analysis ⇔ geometry
ˆ X) Index DE ⊗E = ∫ ch(E1 ⊗ E2* ) A( 1
2
zero-‐modes Dirac operator time
Tr( −1) F = Index DE1 ⊗ E2
topology of vector bundle time
〈 E1 , E2 〉 = ∫ ch( E1 ⊗ E2* ) Aˆ ( X )
String Theory Two fundamental parameters String length (Planck’s constant on world sheet)
ℓs
String coupling (Planck’s constant in space -me)
gs
ℓ Planck = g s ℓ s
String Par::on Func:on
Z string = exp ∑ g s2 g−2 Fg t;ℓ s g≥0
gs ≈ 0
( )
genus g
Phase Diagram strings
branes
CFT strings
ℓs
par-cles
gs
fields
Quantum Geometry (emergent)
Stringy Geometry (deformed)
Classical Geometry
ℓ Planck = g s ℓ s
ℓs
smooth
Simplest Calabi-‐Yau 3-‐fold
!3
(
)
z1 , z2 , z3 ∈ ! 3
Constant Maps, d = 0
g N g ,0 =
∫λ
Mg
3 g−1
=
B2 g B2 g−2 2g(2g − 2)(2g − 2)!
3d Par::ons Z = exp ∑ N g g s2 g−g g≥0
(
= ∏ 1− q n>0
=
n
)
−n
∑
3d partitions of N
q = e− gs
qN
(
Z top = ∏ 1− q n>0
n
)
−n
=1+ q + 3q 2 + 6q3 + ...
Mel:ng Crystals
Z top = ∑ q
# atoms
(
=∏ 1− q n>0
n
)
−n
Limit Shape = Mirror Manifold
ℓ Planck
ℓ string
smooth
D=4 Gauge Theories Instantons: self-‐dual connec*on F = ∗F 4-‐manifold M
moduli space ModN ,n (M)
4
ch2 = n
€
€
€
€
Ac*on S=
4π g2
θ
∫ TrF ∧∗F + 8π ∫ TrF ∧ F = −n ⋅ 2πiτ 2
Gauge coupling τ=
4π θ +i 2 ∈ H 2π g
Duali:es of D=4 Gauge Theories Electric-‐magne*c duali*es
F(A* ) = *F(A)
Langlands dual gauge group
G ⇔ LG
Dual coupling
τ ⇔ −1 / τ
Extended symmetry SL(2, Z)
τ→
aτ + b cτ + d
N=4 SUSY Gauge Theory Vafa-‐WiPen: par**on func*on is a modular form
Z(M;q) = ∑ d(n)q n ,
d(n) = Euler ( Mod N,n )
n≥0
• SL(2,Z) S-‐duality in N=4 gauge theories ↔ modular invariance of a quantum CFT on 2-‐torus aτ + b τ→ cτ + d T2 • Related to 2d CFT characters €
Z(q) = TrV q L0 = χ (q) €
N = 4 Gauge Theories on ALE spaces • Resolved singularity MΓ → C 2 /Γ finite subgroup Γ ⊂ SU(2) € • Boundary condi*on: flat connec*on € ρ ∈ Hom(Γ,U(N)), ρ = ⊕N i ρ i • MacKay correspondence: finite groups Γ ↔ gˆ affine KM €algebras
N = 4 Gauge Theories on ALE spaces
€
• ρ rep of dim N of Γ ↔ V ρ integrable rep at level N of gˆ N0 N1 € € € € €
!
2 € of SU(k)N WZW model • A k−1 : C /Z k characters
€
Z gauge (q) ρ = TrVρ q L0 = χ ρ (q) €
• Level-‐rank duality SU(N) k ↔ SU(k) N €
N=2 SUSY Gauge Theory R4
ε1
Equivariant ac*on of SO(4) x U(N) on ModN ,n (R 4 )
(ε1,ε2 ;a) ∈ T 2 × T N
ε2
€
U(1) × U(1) ⊂ SO(4)
€ Nekrasov par**on func*on: equivariant fundamental class €
€
€ Z gauge (q;ε1,ε2 ,a) = ∑ q n n≥ 0
∫ 1(ε ,ε ,a) 1
2
M N ,n
Tradi*onal (supersymmetric) case ε1 = −ε2 = ε = gs €
Seiberg-‐Wiben solu:on Z gauge = e F ,
F = ∑ε 2g−2 Fg
ε
g
Quantum (symplec:c) invariants of the spectral curve € one-‐form (C,ω ), ω = meromorphic € ∂F0 F0 : µi = ω (moduli), = ∫ω ∂µi B i Ai € 1 F1 = logdet Δ 2 Fg : genus g graphs € € €
∫
R4
Emergent Geometry
D=6 Tensor (2-‐form) Gauge Theory D=2 CFT
×
C2
€
€
M4
D=4 Gauge Theory
Wigner’s Random Matrix Model lim Z N ,
N →∞
ZN =
∫ dΦ ⋅ e−TrΦ
2
/ gs
N ×N
Eigenvalue distribu*on in ’t Hooi limit N → ∞, gs → 0, Ngs = µ = fixed €
€
Eigenvalue Dynamics Z matrix =
∫d
N
2
λ ⋅ ∏ ( λI − λJ ) ⋅ e
−∑ λ2I / g s I
Effec*ve ac*on (repulsive Coulomb force) €
Seff = ∑ λ2I − 2gs ∑ log( λI − λJ ) I
€
I <J
W (x) = x 2
General Matrix Model Z matrix =
TrW (Φ) / g s dΦ ⋅ e ∫
’t Hooi limit N I → ∞, gs → 0, N I gs = µI = fixed € Filling frac*ons
€
N3
N1 N2
W (Φ )
Spectral Curve Hyperellip*c curve
C : y 2 = W '(x) 2 + f (x)
€
Quantum invariants of C, ω = ydx
µi =
∫
Ai
ydx
∂F0 = ∫ ydx ∂µi B i
Wigner’s Semi-‐Circle Ra*onal spectral curve
C : y2 = x 2 + µ
A 1 2 ∂ F0 = µ log µ, F0 = µ€log µ 2 ∂µ
Quantum Curves
N →∞
O(1 N )
N finite
F0
∑N g≥0
2−2 g
Fg
Categorifica:on “Miraculous” integrality
Z(M ) = ∑ dn q n n
dn ∈ Z
M
Suggests higher dimensional QFT
dn = dimVn (M )
The vector spaces Vn can carry representa*ons of algebraic structures
M ×R
Knot Polynomials Knot K in S3 in Chern-‐Simons theory
Z(K, G) =
−kCS( A) DA e . TrR Hol(K ) ∫
Knot polynomials (Jones,…)
Z(K, G) = ∑ dn q n , dn ∈ Z n
q = e 2 π i/(k+h) = e !
Khovanov cohomology
Local Defini:ons CFT D-‐branes TFT
Riemann surfaces
Gauge Theories Matrix Models
General Rela*vity
String Theory
Non-‐comm geometry
Global Defini:on?
Mathema:cs of QFT • Construc*ve and algebraic QFT, asympto*cally free theories (Clay prize). • Surprisingly rich structure in Feynman diagrams (Hopf algebra of Connes-‐Kreimer, mul*ple zeta-‐func*ons, number theory). • Twistor reformula*ons, MHV calculus, amplituhedron,... • Duali*es: rela*ng different gauge groups & dynamical variables, not always a semi-‐classical expansion. • Is the path-‐integral fundamental?
What Kind of Mathema:cal Beauty?
universal calculus, Hilbert spaces
excep:onal E8, Monster Group, Calabi-‐Yau manifolds
Plato’s Cave Mathema-cal Dream
Physical Reality
Quantum Cave Physical Dream
Mathema-cal Reality
Robbert Dijkgraaf Ins$tute for Advanced Study 2014 MPS Annual Mee:ng Simons Founda:on
Some Ideology
TRUTH AND BEAUTY Seal of the Ins-tute for Advanced Study
“My work always tried to unite the true with the beau-ful, but when I had to choose one or the other, I usually chose the beau-ful.” Hermann Weyl
“It is more important for our equa-ons to be beau-ful than to have them fit experiment.” Paul Dirac
“Every law of physics, pushed to the extreme, will be found to be sta-s-cal and approximate, not mathema-cal perfect and precise.” John Wheeler
“Any theory that can account for all of the facts is wrong, because some of the facts are always wrong.” Francis Crick
Esthe:cs in Physics
Garbage
Beauty
Reduc*on
Garbage
Beauty
Standard Model
macro
micro
Emergence
Beauty
Garbage
hydrodynamics thermodynamics
Standard Model sta-s-cal mechanics
macro
micro
Renormaliza*on Group
Infrared fixed points
Ultraviolet fixed points
Λ=0
Λ=∞
Large distances Low energy
Standard Model Short distances High energy
macro
micro
Quan*za*on
Geometry
Algebra
K
Z(K) ∈ C quantum invariant
geometric object
€
€
Emergent Geometry
Geometry
effec-ve geometry
Algebra
quantum system
Synthesis
Quantum Geometry
String Theory
Classical Mechanics
B A
calculus, geometry, dynamical systems, chaos,…
Quantum Mechanics
Sum over histories
A
− Action / h e ∑
functional analysis, operator algebra, differential topology,…
B
Quantum Field Theory
A
crea-on/annihila-on of par-cles
C quantum topology: knots, 3- & 4-manifolds twistors, Grassmannians & amplitudology
B
String Theory
conformal field theory, algebraic curves, moduli spaces, mirror symmetry, quantum cohomology
Quantum Gravity
A?
non-commutative geometry, emergent geometry, automorphic forms,…
What Is A Quantum Field Theory?
Classical Field Theory Space-‐*me manifold M, field space F
φ :M →F
Ac*on, cri*cal points
S[φ ], dS = 0
Classical field equa*ons
∇ 2φ + ... = 0
Moduli space of classical solu*ons
{dS = 0} / Sym = ⨿ Modn n
M
Quantum Field Theory Euclidean path integral
Z(M ) =
−S(φ )/! D φ . e ∫
Semi-‐classical localisa*on to cri*cal points Typical form Classical limit
Z(M ) ~ ∑ e−Sn /! . ∫ n
Modn
voln
Modn
Z(M ) ~ ∑ dn q n , q ~ e−1/! n
! → ∞, q → 0
φ :M →F
Duali:es Quantum equivalences
φ : M → Rn / Λ
*
Z(M, F) = Z(M, F )
e.g. D=2 T-‐duali*es, M = Riemann surface, F = n-‐torus where
F = T n = R n / Λ, F * = T * = R n / Λ *
dφ * = *dφ
Par**on func*on = theta func*on
Z = θΛ =
Extended symmetry
∑
eπ i( p+w)τ ( p+w)−π i( p−w)τ ( p−w)
p∈Λ,w∈Λ*
SL(n, Z) ⊂ SO(n, n, Z ) = Aut(Λ ⊕ Λ * )
Space Space oof f QCuantum lassical FField ield TTheories heories classical limit !↓0
classical field theory F
Q F T
special automorphisms
Duali:es dual classical field theory F*
dual classical limit !* ↓0
self-‐dual QFTs
Duali:es of Effec:ve QFT Renormaliza*on group
Z(M, Fε )
Depends on cut-‐off distance ε or energy scale Λ=1/ε RG flow on space of regularized quantum field theories
F = F∞ ε =∞
−ε∂ ε
F * = F0 ε=0
Some*mes Z independent of ε
Z(M, F0 ) = Z(M, F∞ )
Defining Quantum Field Theory: Algebra
x1 O1 (x1 )....On (xn )
xn
Operator products of local operators ScaPering amplitudes
D=2 Conformal Field Theory Oi (x)
Operator product expansion
lim
Oi (x) O j (y)
∑ k
x→y
O j (y)
Ok (y)
Vertex algebra Associa*ve algebra
∞-‐dim vertex algebras classifica-on reps Virasoro
=
Defining Quantum Field Theory: Geometry cut & paste topological indices defect operators
Category Theory M codim 0: number, par**on func*on
Φ:M →C
M
S codim 1: vector space, Hilbert space of states
Φ:S → H
M
Φ(M ) ∈ Φ(∂M )
Gluing laws
M1
M2
Φ(M1 )! Φ(M 2 ) = Φ(M )
M
Category Theory codim 2: category of boundary condi*ons α H :α → β
β
Composi*on law of morphisms α
H1 : α → β
β
H2 : β → γ
γ
H 2 ! H1 : α → γ
D=1 Supersymmetric QM Differen*al geometry & topology *me
→X point
Hilbert space = Ω* ( X ) H = −Δ = −(dd * + d *d ) Ground states = harmonic forms
Harm* ( X ) ≅ H * ( X ) Par**on func*on = WiPen index
Tr ((−1)deg e−tH ) = Euler ( X )
D=2 Conformal Field Theory
X →X *me
S1
LX Loop Space
Quan*ze loop space Fock space created by string oscillators Infinite-‐dimensional analysis Loop space thickening X ⊂ LX
String Product *
Physical fields = cohomology classes a,b ∈ H ( X )
a a ∧b
b
H* ⊗ H* → H*
Topological String Theory
F0 (t) = ∑GWd ,0e −dt
d≥0 (pseudo) holomorphic d = deg, tcurves = Area / ℓ 2string
string
Calabi-‐Yau X
Classical Intersec:on Product ℓs = 0 dual cycles
a
c b
= X =
∫
X
a∧b∧c
Quantum Cohomology ℓs > 0
=
∑
e
− dt/ℓ2s
rat curves degree d
X for P n
x n+1 = 0 ⇒ x n+1 = e −t
smooth under flops
t ↔ −t
local P1
ℓs = 0 t 0 moduli space
ℓs > 0
t =0 singularity at zero size
Open Strings & Branes fixed time picture
brane
X
Y
open strings (brane)
closed strings (bulk)
D-‐branes: Rela:ve CFT Space-time picture
string worldsheet
brane
Y
Σ X
4D Space-time
D-‐Branes
multiplicity N
Internal space
U(N) Yang-‐Mills Theory
i
j
N×N matrix of strings Aij
U(N) Yang-‐Mills Theory
j k k i matrix multiplication Σk Aik Akj
Closed/Open Duali:es open strings
closed strings
⇔ strings inf-dim Lie algebras loop spaces g ≈ 0 s Virasoro algebra genus expansion
€
D-branes vector bundles K-theory s spaces moduli non-commutative
g ≈∞
A:yah-‐Singer Index Theorem analysis ⇔ geometry
ˆ X) Index DE ⊗E = ∫ ch(E1 ⊗ E2* ) A( 1
2
zero-‐modes Dirac operator time
Tr( −1) F = Index DE1 ⊗ E2
topology of vector bundle time
〈 E1 , E2 〉 = ∫ ch( E1 ⊗ E2* ) Aˆ ( X )
String Theory Two fundamental parameters String length (Planck’s constant on world sheet)
ℓs
String coupling (Planck’s constant in space -me)
gs
ℓ Planck = g s ℓ s
String Par::on Func:on
Z string = exp ∑ g s2 g−2 Fg t;ℓ s g≥0
gs ≈ 0
( )
genus g
Phase Diagram strings
branes
CFT strings
ℓs
par-cles
gs
fields
Quantum Geometry (emergent)
Stringy Geometry (deformed)
Classical Geometry
ℓ Planck = g s ℓ s
ℓs
smooth
Simplest Calabi-‐Yau 3-‐fold
!3
(
)
z1 , z2 , z3 ∈ ! 3
Constant Maps, d = 0
g N g ,0 =
∫λ
Mg
3 g−1
=
B2 g B2 g−2 2g(2g − 2)(2g − 2)!
3d Par::ons Z = exp ∑ N g g s2 g−g g≥0
(
= ∏ 1− q n>0
=
n
)
−n
∑
3d partitions of N
q = e− gs
qN
(
Z top = ∏ 1− q n>0
n
)
−n
=1+ q + 3q 2 + 6q3 + ...
Mel:ng Crystals
Z top = ∑ q
# atoms
(
=∏ 1− q n>0
n
)
−n
Limit Shape = Mirror Manifold
ℓ Planck
ℓ string
smooth
D=4 Gauge Theories Instantons: self-‐dual connec*on F = ∗F 4-‐manifold M
moduli space ModN ,n (M)
4
ch2 = n
€
€
€
€
Ac*on S=
4π g2
θ
∫ TrF ∧∗F + 8π ∫ TrF ∧ F = −n ⋅ 2πiτ 2
Gauge coupling τ=
4π θ +i 2 ∈ H 2π g
Duali:es of D=4 Gauge Theories Electric-‐magne*c duali*es
F(A* ) = *F(A)
Langlands dual gauge group
G ⇔ LG
Dual coupling
τ ⇔ −1 / τ
Extended symmetry SL(2, Z)
τ→
aτ + b cτ + d
N=4 SUSY Gauge Theory Vafa-‐WiPen: par**on func*on is a modular form
Z(M;q) = ∑ d(n)q n ,
d(n) = Euler ( Mod N,n )
n≥0
• SL(2,Z) S-‐duality in N=4 gauge theories ↔ modular invariance of a quantum CFT on 2-‐torus aτ + b τ→ cτ + d T2 • Related to 2d CFT characters €
Z(q) = TrV q L0 = χ (q) €
N = 4 Gauge Theories on ALE spaces • Resolved singularity MΓ → C 2 /Γ finite subgroup Γ ⊂ SU(2) € • Boundary condi*on: flat connec*on € ρ ∈ Hom(Γ,U(N)), ρ = ⊕N i ρ i • MacKay correspondence: finite groups Γ ↔ gˆ affine KM €algebras
N = 4 Gauge Theories on ALE spaces
€
• ρ rep of dim N of Γ ↔ V ρ integrable rep at level N of gˆ N0 N1 € € € € €
!
2 € of SU(k)N WZW model • A k−1 : C /Z k characters
€
Z gauge (q) ρ = TrVρ q L0 = χ ρ (q) €
• Level-‐rank duality SU(N) k ↔ SU(k) N €
N=2 SUSY Gauge Theory R4
ε1
Equivariant ac*on of SO(4) x U(N) on ModN ,n (R 4 )
(ε1,ε2 ;a) ∈ T 2 × T N
ε2
€
U(1) × U(1) ⊂ SO(4)
€ Nekrasov par**on func*on: equivariant fundamental class €
€
€ Z gauge (q;ε1,ε2 ,a) = ∑ q n n≥ 0
∫ 1(ε ,ε ,a) 1
2
M N ,n
Tradi*onal (supersymmetric) case ε1 = −ε2 = ε = gs €
Seiberg-‐Wiben solu:on Z gauge = e F ,
F = ∑ε 2g−2 Fg
ε
g
Quantum (symplec:c) invariants of the spectral curve € one-‐form (C,ω ), ω = meromorphic € ∂F0 F0 : µi = ω (moduli), = ∫ω ∂µi B i Ai € 1 F1 = logdet Δ 2 Fg : genus g graphs € € €
∫
R4
Emergent Geometry
D=6 Tensor (2-‐form) Gauge Theory D=2 CFT
×
C2
€
€
M4
D=4 Gauge Theory
Wigner’s Random Matrix Model lim Z N ,
N →∞
ZN =
∫ dΦ ⋅ e−TrΦ
2
/ gs
N ×N
Eigenvalue distribu*on in ’t Hooi limit N → ∞, gs → 0, Ngs = µ = fixed €
€
Eigenvalue Dynamics Z matrix =
∫d
N
2
λ ⋅ ∏ ( λI − λJ ) ⋅ e
−∑ λ2I / g s I
Effec*ve ac*on (repulsive Coulomb force) €
Seff = ∑ λ2I − 2gs ∑ log( λI − λJ ) I
€
I <J
W (x) = x 2
General Matrix Model Z matrix =
TrW (Φ) / g s dΦ ⋅ e ∫
’t Hooi limit N I → ∞, gs → 0, N I gs = µI = fixed € Filling frac*ons
€
N3
N1 N2
W (Φ )
Spectral Curve Hyperellip*c curve
C : y 2 = W '(x) 2 + f (x)
€
Quantum invariants of C, ω = ydx
µi =
∫
Ai
ydx
∂F0 = ∫ ydx ∂µi B i
Wigner’s Semi-‐Circle Ra*onal spectral curve
C : y2 = x 2 + µ
A 1 2 ∂ F0 = µ log µ, F0 = µ€log µ 2 ∂µ
Quantum Curves
N →∞
O(1 N )
N finite
F0
∑N g≥0
2−2 g
Fg
Categorifica:on “Miraculous” integrality
Z(M ) = ∑ dn q n n
dn ∈ Z
M
Suggests higher dimensional QFT
dn = dimVn (M )
The vector spaces Vn can carry representa*ons of algebraic structures
M ×R
Knot Polynomials Knot K in S3 in Chern-‐Simons theory
Z(K, G) =
−kCS( A) DA e . TrR Hol(K ) ∫
Knot polynomials (Jones,…)
Z(K, G) = ∑ dn q n , dn ∈ Z n
q = e 2 π i/(k+h) = e !
Khovanov cohomology
Local Defini:ons CFT D-‐branes TFT
Riemann surfaces
Gauge Theories Matrix Models
General Rela*vity
String Theory
Non-‐comm geometry
Global Defini:on?
Mathema:cs of QFT • Construc*ve and algebraic QFT, asympto*cally free theories (Clay prize). • Surprisingly rich structure in Feynman diagrams (Hopf algebra of Connes-‐Kreimer, mul*ple zeta-‐func*ons, number theory). • Twistor reformula*ons, MHV calculus, amplituhedron,... • Duali*es: rela*ng different gauge groups & dynamical variables, not always a semi-‐classical expansion. • Is the path-‐integral fundamental?
What Kind of Mathema:cal Beauty?
universal calculus, Hilbert spaces
excep:onal E8, Monster Group, Calabi-‐Yau manifolds
Plato’s Cave Mathema-cal Dream
Physical Reality
Quantum Cave Physical Dream
Mathema-cal Reality
