Supermatrix Models * Robbert Dijkgraaf Ins$tute for Advanced Study Walter Burke Ins.tute for Theore.cal Physics Inaugural Symposium Caltech, Feb 24, 1015 * Work with Cumrun Vafa and Ben Heidenreich, in progress
Holography/Emergent Geometry Quantum Theory in d dimensions Matrix Models (d=0) d+1 Space- Time Gravity + QFT N N dφ e TrW ( Φ )/g s Quantum Spectral Curves (Riemann Surface + CFT)
Quan.za.on Geometry Algebra K Z(K) C geometric object quantum invariant
Emergence Geometry Algebra effeccve geometry quantum system
Supersymmetric Gauge Theory: N=4 Path integral localizes to instantons Z = n 0 DA e S = d(n)q n τ aτ + b cτ + d, q = e 2πiτ,τ coupling ContribuCon of moduli space Mod N,n (M) M SL(2,Z) S- duality in N=4 gauge theories modular invariance of a quantum CFT on 2- torus T 2 Related to 2d CFT characters Z = Tr ql 0 = χ(q)
Supersymmetric Gauge Theory: N=2 Seiberg- WiSen curve genus g (Σ,ω ), ω = meromorphic one- form Effec.ve gauge coupling Nekrasov deforma.on BPS masses τ ij (x), C ω Z = e F, F = ε 2g 2 F g, g F 0 : µ i = ω, F 0 = ω µ A i i B i (ε, ε) 4 " 2 F 1 = 1 2 logdet Δ
D=6 (2,0) Tensor (2- form) SCFT D=2 CFT Σ 2 M 4 D=4 Gauge Theory
Geometry/Algebra Duality Gauge Theories Geometry 4- manifold M 4 Conformal Field Theory Algebra ˆ g affine symme.res Algebra gauge group (vector bundle) G Geometry algebraic curve Riemann surface Σ 2
Wigner s Random Matrix Model lim Z N N, Z N = Eigenvalue distribu.on in t HooY limit 1 VolU(N) N N dφ e TrΦ2 /g s N, g s 0, Ng s = µ = fixed
String ParSSon FuncSon Z = exp g 2g 2 F ( µ ) s g g 0 ribbon diagrams 1/N expansion genus g string surface
Eigenvalue Dynamics Z matrix = d N λ λ I λ J I ( ) 2 e λ I 2 / g s Effec.ve ac.on (repulsive Coulomb force) I I <J ( ) S eff = λ I 2 2g s log λ I λ J W (x) = x 2
Z matrix = General Matrix Model 1 VolU(N) N N dφ e TrW ( Φ )/g s t HooY limit N I, g s 0, N I g s = µ I = fixed Filling frac.ons (perturba.ve expansion) N 1 N 2 N 3 W ( Φ)
Resolvent ydx = ds eff = dw(x)+2g s Tr dx Φ x x eigenvalues probe
Resolvent ydx = ds eff = dw(x)+2g s Tr dx Φ x x probe branch cut N
EffecSve Geometry y2 = W n ' (x) 2 + f n 1 (x)= P 2n (x) x probe second sheet
Periods (A) A I ydx = g s N I = µ I x eigenvalues probe
Periods (B) B I ydx = F µ I x eigenvalues probe
Spectral Curve Hyperellip.c curve Σ : y2 = W '(x) 2 + f (x) B i A i x Quantum invariants of A i µ i = ydx Σ, ω = ydx F 0 = ydx µ i B i
Wigner s Semi- Circle Ra.onal spectral curve A B Σ : y2 = x 2 + µ A F = 1 0 2 µ2 log µ = log Vol( U(N) )
Quantum Curves N F 0 (µ) O(1 N) N 2 2g F g (µ) g 0 N finite non$perturbative
Complex Curves in Phase Space algebraic curve = level set Σ : H(x, y) = 0 y Σ H(x, y) = 0 Hamilton- Jacobi theory y = p(x) branch pts = turning pts Liouville form ac.on ω = ydx S(x) = x x * ω BS Quan.za.on dx dy " = g s x
Topological string on hypersurface CY 3 Calabi- Yau hypersurface in 4 X : uv + H(x, y) = 0 Period maps Ω = du u dx dy ω = ydx
Matrix models and chiral CFT Eigenvalue density/matrix resolvent = collec.ve boson field ϕ(x)= Tr Free scalar field 1 x Φ Z matrix = Dϕ e S[ϕ ], Fermions = D- branes S = ( ϕ)2 +O(1/ g s ) ϕ(x) ψ (x)= e iϕ(x ) = det( x Φ) ( ) 1 ψ * (x)= e iϕ(x ) = det x Φ
Loop EquaSons & Virasoro Constraints Invariance under diffeomorphism x x (x) Generated by stress- tensor Loop equa.ons T(x) = 1 2 ( ϕ ) 2 (x) T(x) = W '(x) 2 + f (x)
SU(2) affine KM algebra structure Currents J + = e 2iϕ =ψ 2 * ψ 1 J 3 = ϕ =ψ 1 * ψ 1 ψ 2 * ψ 2 J = e 2iϕ =ψ * 1 ψ 2 In the limit W =0 Z matrix = exp J + N Screening charge N
Nilpotent Geometry y n = 0 Σ x } n Nilpotent algebra y n = 0 ˆ y = g s x + 0 0 0 1 0 " # # 0 0 1 0
A n 1 Geometry of SU(n) Gauge Theory singularity uv + y n = 0 R 4 C 2 /Z n x S U ( n ) Calabi- Yau (internal space) Space- Time
SingulariSes & Matrix Models A 1 : y 2 = 0 dφ e atrφ2 lim a 0 { y 2 = a 2 (x 2 + µ) }
Matrix Models & Liouville Theory Z matrix = d N x ( x I x ) 2 J e W (x I CFT Coulomb gas correla.on func.on )/ g s Z matrix = d N x e 2iϕ(x i ) e ϕ W /g s N Screening charge Z matrix = e e2iϕ Large N : Coulomb gas picture. e ϕ W /g s N
Beta Ensemble General Liouville theory b i, ε 1 ε 2 Z = exp e bϕ N Generalized matrix model measure β W (x I ) g s I Z = d N x ( x I x ) 2β J e, β = b 2 β =1/2 : SO(N), β = 2 : Sp(N) Generalized loop equa.ons T(x) = 1 2 ( ϕ ) 2 + Q 2 ϕ
Eynard- OranSn: All- Genus SoluSon Recursion relasons for correla.on func.ons, completely geometric W(x 1,,x n )= Tr 1 x 1 Φ Tr 1 x n Φ con = ϕ(x 1 ) ϕ(x n ) con = g 2g 2 s W g g Lowers genus, adds more inser.ons. Reduces to Bergmann kernel g W 0 (w,z) = B(w,z)dwdz w z
Recursion RelaSons g = + g 1 h g h
Quantum Spectral Curves Random surfaces Geometry of M g y p = x q Toric Calabi- Yau y = sin x Hyperbolic knots e x + e y = 1 A(ex,e y )= 0
EffecSve Geometry probe x Are both sheets visible?
probe x
Topological D- Branes y Σ H(x, y) = 0 x 1 x n Open string par..on func.on Ψ(x 1,,x n )= ψ (x 1 )ψ (x n ) = det(φ x i ) i
Quantum Wave FuncSon Σ Ψ(x) x Single brane wave func.on ˆ H Ψ = 0, Ψ(x) ˆ H = ˆ H ( ˆ x, ˆ y ) Quan.za.on of phase space ŷ = g s x,[ ˆx, ŷ]= g s
Gaussian Matrix Model Ψ N (x)= det( Φ x) = H N N 1 (x) e x2 /2 Eigenfunc.ons of harmonic oscillator 2 2 g s x + x 2 g 2 s (2N 1) Ψ (x)= 0 N Σ H (x, y)= 0 N
Ψ N * (x)= AnSbranes 1 ( ) N det Φ x Non- normalizable eigenfunc.ons of harmonic oscillator H N Ψ * N (x)= 0 = Ψ ( y) N y x dy WKB behavior Probe other sheet Ψ* (x) e + x2 /2
Classical Spectral Curve y2 = W '(x) 2 y Σ x branes:y = W '(x) antibranes:y = W '(x)
Super gauge group Symmetries of Φ = Super Gauge Theories U(N + k k) A C Large N, M expansion B D N+k,A,Deven,B,Codd. k v = (z,θ) N+k k, v 2 2 = z i + θa θ a i=1 a=1 Str(1) = Perturba.vely N +k 1 1 i=1 k = N i j=1 U (N + k k) U (N )
Super Matrix Models Supergroup U(N M) Z N+k k = 1 VolU(N + k k) dφ e StrW ( Φ )/g s Volume Lie supergroup vanishes: need regulariza.on Diagonalize: eigenvalues & an.- eigenvalues Φ x i 0 0 y j dθ 1 = 0 N+k k S = W(x i ) W(x i ) i=1 j=1
Super Jacobian i< j ( x i x ) 2 j a<b ( ) 2 y a y b dφ d N+k x d k y ( x i y ) 2 b i,a Eigenvalues of charge +1 and - 1 (Coulomb gas model) Resolvent ydx Str dx Φ x = Same periods as U(N) model i dx x i x i dx y a x
Non- PerturbaSve CorrecSons [Vafa] U(n+k k) has more Casimirs than U(N), e.g. Δ = General deforma.on W = i,a n For example, for U(2 1) ( x i y ) a t n StrΦ n Δ = 1 2 StrΦ2 StrΦ ( ) 2 = 0forU(1)
Non- PerturbaSve CorrecSons So correlators are sensi.ve to k. Δ N+k k General pasern: calcula.on of par..on func.on and correla.on func.ons for U(N+k k) depend non- perturba.vely (order e - N ) on k. Working with supermatrix models seems to probe the full effec.ve geometry.
(Topological) Strings holomorphic curves, mirror symmetry