Modular Matrix Models & Monstrous Moonshine: Yang-Hui He Dept. - PowerPoint PPT Presentation

Modular Matrix Models & Monstrous Moonshine: Yang-Hui He Dept. of Physics and Math/Physics RG, Univ. of Pennsylvannia hep-th/0306092, In Collaboration with: Vishnu Jejjala Feb, 2004, Madison, Wisconsin

Modular Matrix Models and Monstrous Moonshine Motivations MATRIX MODELS • Resurrection of old matrix models; • Dijkgraaf-Vafa Correspondence; • Powerful unified view of SUSY gauge theory/2D qauntum gravity/geometry; • Geometrisation and discretisation of string theory;

MOONSHINE • One of the most amazing “coincidences” in mathematics; • McKay-Thompson: Relation of elliptic j -function and the Monster Group; • Conway-Norton: (crazy) Moonshine conjecture; • Frenkel-Lepowski-Meurman: Vertex Algebras; • Borcherds: Proof (Fields Medal 98); QUANTUM/STRINGY MOONSHINE??? • Does moonshine mean anything to String Theory? • Dixon-Ginsparg-Harvey; Craps-Gaberdiel-Harvey • Is there a quantum generalisation of moonshine?

Outline Four Short Pieces 1. The Klein Invariant j -function 2. The One-Matrix Model 3. The Master Field Formalism 4. Dijkgraaf-Vafa Modular Matrix Models • Constructing a matrix model given a modular form • The j -MMM Discussions and Prospectus • A precise program for finding quantum corrections • geometric meaning

Four Short Pieces I. The Klein Invariant • Modular Invariant: The most important (only) meromorphic function invariant under SL (2; Z ) ( z → az + b cz + d , ad − bc = 1) (profound arithmetic properties); � 4 � ϑ 2 ( q ) q := e 2 πiz , j ( e 2 πiz ) : H /SL (2; Z ) → C λ ( q ) := , ϑ 3 ( q ) (1 − λ ( q ) + λ ( q ) 2 ) 3 J ( q ) := 4 j ( q ) := 1728 J ( q ) , λ ( q ) 2 (1 − λ ( q )) 2 27 • The q -expansion q − 1 + 744 + 196884 q + 21493760 q 2 + 864299970 q 3 + j ( q ) = 20245856256 q 4 + 333202640600 q 5 + 4252023300096 q 6 . . .

• j -function and modularity known to Klein, Dedekind, Kronecker, and as far back as Hermite (1859) • Classification of Simple Groups (1970’s) Monster = Largest Sporadic Simple Group M , | M | ∼ 10 53 = 2 46 · 3 20 · 5 9 · 7 6 · 11 2 · 13 3 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808017424794512875886459904961710757005754368000000000 • Andrew Ogg (1975): H / (Γ( p ) ⊂ Γ), �� a � 1 b 0 � � � � has Γ( p ) := � ∈ SL (2; Z ) , c ≡ 0(mod p ) , c d − p 1 genus = 0 if p = 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 41 , 47 , 59 , 71 • Jacque Tits (1975): (described the order of the then-conjectural M in a lecture attended by Ogg);

• Ogg offers a bottle of Jack Daniel’s • Until JOHN McKAY (1978) Letter to Thompson: • MOONSHINE (J. Conway and S. Norton) j -function Monster 196884 = 1 + 196883 , 21493760 = 1 + 196883 + 21296876 , 864299970 = 2 · 1 + 2 · 196883 + 21296876 + 842609326 , . . . • Frenkel-Lepowski-Meurman: Vertex Algebras (1980’s); • RICHARD BORCHERDS, Proof (1986) Fields (1998)

II. The One-Matrix Model • The Hermitian one-matrix model (easily generalised to complex) � � 1 − 1 � [ D Φ] exp Z = g Tr V (Φ) , Vol( U ( N )) N � � dλ i ∆( λ ) 2 exp 1 − N � � � = V ( λ i ) Vol( U ( N )) g i =1 i Vandermonde: ∆( λ ) = � ( λ j − λ i ) i<j • At PLANAR LIMIT (higher genus ∼ 1 /N -expansions): 1 � – Eigenvalue Density: ρ ( λ ) := δ ( λ − λ i ); N i dτ ρ ( τ ) λ − τ = 1 � g V ′ ( λ ); – Saddle Point: 2 − • CUTS Branch-cuts ❀ solution of integral eq.

dλ ρ ( λ ) � • Resolvent: R ( z ) = z − λ satisies loop equation R ( z ) 2 − 1 1 g V ′ ( z ) R ( z ) − 4 g 2 f ( z ) = 0 • Purely algebraic equation. • Reverse Engineering: 1 ρ ( z ) = 2 πi lim ǫ → 0 ( R ( z + iǫ ) − R ( z − iǫ )) , − 1 g V ′ ( z ) ǫ → 0 ( R ( z + iǫ ) + R ( z − iǫ )) = lim • KEY: knowing R ( z ) ❀ knowing everything about the MM.

III. The Master-Field Formalism • A convenient (algebraic) formulation of matrix models. • In the 1MM, observables are O k := TrΦ k 1 � − N � � �O k � = Z − 1 lim [ D Φ] Tr O k exp g Tr V (Φ) , N N →∞ • Using free probability theory , Voiculescu shewed that correlators of MM are encoded in the CUNTZ algebra aa † = I , a † a = | 0 �� 0 | , with a | 0 � = 0 and there exists a Master Field ˆ M ( a, a † ) s.t. �O k � = � 0 | ˆ M ( a, a † ) k | 0 �

THM [Voiculescu]: In particular, for the 1MM ∞ ˆ � m n ( a † ) n M ( a, a † ) = a + n =0 ( m n are coefficients) • VEV’s are in Voiculsecu polynomials of m n : tr[ M ] := � ˆ M ( a, a † ) � = m 0 , �O 1 � = tr[ M 2 ] := � ˆ M ( a, a † ) 2 � = m 2 �O 2 � = 0 + m 1 , tr[ M 3 ] := � ˆ M ( a, a † ) 3 � = m 3 �O 3 � = 0 + 3 m 0 m 1 + m 2 • Write generating function ∞ K ( z ) = 1 � m n z n z + n =0 then, the resolvent is simply the inverse: R ( z ) = K − 1 ( z ) .

• To determine the Voiculescu polynomials, simply series-invert f ( z ) = 1 z + b 0 + b 1 z + b 2 z 2 + b 3 z 3 + b 4 z 4 + . . . to give + b 04 +6 b 02 b 1 +2 b 12 +4 b 0 b 2 + b 3 z 2 + b 02 + b 1 + b 03 +3 b 0 b 1 + b 2 f − 1 ( z ) 1 z + b 0 = z 3 z 4 z 5 + b 05 +10 b 03 b 1 +10 b 0 b 12 +10 b 02 b 2 +5 b 1 b 2 +5 b 0 b 3 + b 4 + . . . . z 6 Rmk: (McKay) The Voiculescu polynomials ∼ generating function for the number of Dyke paths in a 2-D grid (Catalan Numbers). • KEY POINT: Master Field ❀ Resolvent ❀ Everything about the MM Rmk: The formalism become very convenient for multi-matrix models, e.g., QCD

IV. Dijkgraaf-Vafa • Generalisation and new perspective on the Gopakumar-Vafa large N duality for the conifold. • An intricate web (from Aganagic-Klemm-Mari˜ no-Vafa 0211098) B−Brane on Chern−Simons Mirror Symmetry theory blownup CY Y Canonical quantization Large N duality Large N duality Matrix integral Planar limit Blown up CY Deformed CY ^ X ^ Y Mirror Symmetry

• an U ( n ) gauge theory, adjoint Φ and tree-level superpotential p +1 1 � k g k Tr Φ k W tree (Φ) = k =1 – Full non-pert. effective (Cachazo-Intriligator-Vafa) in 1 32 π 2 Tr W α W α glueball S = W eff ( S ) = n ∂ ∂ S F 0 ( S ) + S ( n log( S / Λ 3 ) − 2 πiτ ) – F 0 ( S ) is the planar free energy of a large N (bosonic) MM with potential W tree (Φ); identify: S ≡ gN (’t Hooft) – N = 1 thy is geometrically engineered on (local) CY3 { u 2 + v 2 + y 2 + W ′ tree ( x ) 2 = f p − 1 ( x ) } ⊂ C 4 ,

– Special Geometry: cpt A -cycles and non-cpt B -cycles, A i Ω , Π i := ∂F 0 � � identify S i = ∂ S i = B i Ω, ( Ni := Bi G 3 ) � � Ai G 3 , α := p p � � � ⇒ W eff ( S ) = G 3 ∧ Ω = S i N i Π i + α CY 3 i =1 i =1 – non-trivial geometry is the hyper-elliptic curve: y 2 = W ′ tree ( x ) 2 + f p − 1 ( x ) – 1. The Seiberg-Witten curve of the N = 1 theory (deformation of N = 2 by W tree ; 2. The spectral curve (loop eq) of MM • KEY POINT: Each (bosonic) MM actually computes non-perturbative information for an N = 1 gauge theory geometrically engineered on a CY3.

Modular Matrix Models Observatio Curiosa: ∞ f ( q ) = q − 1 + a n q n � • q -expansion: n =0 ∞ ˆ M ( a, a † ) = a + m n ( a † ) n � • Master Field: n =0 • Question: Can we consistently construct a MM whose master field is a given modular form? • Take the favourite and most important example: ∞ j ( q ) = 1 m n q n = 1 � q + q + m 0 + m 1 q + . . . n =0 { m 0 , m 1 , . . . , m 5 , . . . } = { 744 , 196884 , 21493760 , 864299970 , 20245856256 , 333202640600 , . . . }

• Procedure: 1. Identify j ( q ) ∼ K ( q ), the generating function for the Master; 2. Resolvent R ( z ) = j − 1 ( e 2 πiz ). • KEY: Find the inverse of j as a function of z . • The Inverse j -function (well-known) � � r ( z ) − s ( z ) j − 1 ( z ) z z � � � � = i , r ( z ) := ˜ r , s ( z ) := ˜ s r ( z )+ s ( z ) 1728 1728 � 5 � 1 � 2 12 , 1 12 ; 1 � r ( z ) ˜ := Γ 2 F 1 2 ; 1 − z , 12 √ � 2 √ z − 1 2 F 1 � 11 � 7 12 , 7 12 ; 3 � s ( z ) ˜ := 2( 3 − 2) Γ 2 ; 1 − z . 12 • The Branch-cuts – Two-cut: ( −∞ , 0] ∪ [1 , ∞ )

– For Hypergeometrics: ǫ → 0 2 F 1 ( a, b ; c ; z − iǫ ) = 2 F 1 ( a, b ; c ; z ) , lim ǫ → 0 2 F 1 ( a, b ; c ; z + iǫ ) = lim 2 πie πi ( a + b − c ) Γ( c ) Γ( c − a )Γ( c − b )Γ( a + b − c +1) 2 F 1 ( a, b ; a + b − c + 1; 1 − z ) + e 2 πi ( a + b − c )2 F 1 ( a, b ; c ; z ) – Discontinuity of the resolvent: πi  3 ( s − r )+( t − u ) e 3 ( s + r )+( t + u ) ± i r − s i r + s , z ∈ ( −∞ , 0);  πi  − e   R ( z + iǫ ) ± R ( z − iǫ ) = (1 ± 1) i r − s r + s , z ∈ (0 , 1);   r + s ± i r + s i r − s  z ∈ (1 , ∞ ) . r − s ,  • KEY: have analytic form for the resolvent • Recall: 1 ρ ( z ) = 2 πi lim ǫ → 0 ( R ( z + iǫ ) − R ( z − iǫ )) − 1 g V ′ ( z ) = ǫ → 0 ( R ( z + iǫ ) + R ( z − iǫ )) lim

Constructing the MMM • The eigenvalue distribution:  � � 1 st − ru z ∈ ( −∞ , 0); ,   πi π  3 ( r + s )) ( r + s )( t + u − e   ρ ( z ) = 0 , z ∈ (0 , 1);   � � 1 2 rs  , z ∈ (1 , ∞ ) .   s 2 − r 2 π • real for z ∈ [1 , ∞ ), so for convenience restrict to this range (similar restriction done in Gross-Witten model) where MM is Hermitian (Rmk: [Lazaroiu] need C -MM for DV) � a • Normalisation and regularisation: a →∞ A ( a ) lim 1 dz ρ ( z ) = 1.