INTEGRABILITY IN LOW‐DIMENSIONAL QUANTUM SYSTEMS 26 June - 21 July 2017 Random matrix with conformal symmetry Chaiho Rim Sogang University, Seoul Some part of presentation can be found in (arXiv:1612,00348) 2017-06-30 MATRIX @ Creswick 1 Usage of random matrix • Statistics – Whishart (1928) : variance of large samples – Wigner (1955): distribution of nuclear eigenvalues – combinatorics • topology, geometry and 2d gravity – t’hooft (1974): Feynmann diagram with topology of surface – Kazakov (1986): Ising model on fluctuating surface – Al. Zamolodchikov (2001): 2D minimal gravity on Fluctuating sphere – Penner (1988): punctured Riemann surface • Conformal symmetry – Dijkgraaf and Vafa (2009): N=2 super Yang-Mills and Liouville conformal block – Eguchi and Maruyosh (2010) : Penner-type matrix model – Choi and Rim(2014): Irregular matrix model 2017-06-30 2 MATRIX @ Creswick Plan of talk 1. Hermitian one-matrix model 2. Conformal symmetry and Matrix models 3. Application to irregular conformal block 4. Summary and outlook 2017-06-30 MATRIX @ Creswick 3 1. Hermitian one-matrix model Partition function can be evaluated using two approaches; orthogonal polynomials and conformal symmetry. We will concentrate on the method using conformal symmetry, emphasizing its generality to more complicated system. 2017-06-30 MATRIX @ Creswick 4 Hermitian one-matrix model M is a NxN Hermitian matrix. Partition function is given in terms of 휆퐼, diagonal components of M Vandermonde determinant has a special role. 2017-06-30 MATRIX @ Creswick 5 Conventional orthogonal polynomial approach • Introduce orthogonal polynomials (Mehta, 1967) • The partition is given in terms of orthogonal polynomial normalizations: 2017-06-30 MATRIX @ Creswick 6 Conformal symmetry approach • Conformal symmetry in matrix model is noted (Marshakov, Mironov, Morozov ,‘91). • Try conformal transformation: • 훿푍 = 0 leads to an identity (called loop equation), representing conformal symmetry 2017-06-30 MATRIX @ Creswick 7 Example: large N expansion • Put 푔 = 1/푁: • resolvents 푊 푧 and 푊(푧, 푧) are finite as N goes to infinite. • Resolvent has a branch cut as N goes to infinite: • For Gaussian potential V(z) =푧2/2 , 푓 푧 = −1. • Eigenvalues are distributed between 1 and -1: Wigner ‘s semi- circle law 2017-06-30 MATRIX @ Creswick 8 Conformal symmetry of loop equation • Conformal symmetry is hidden in 푓(푧). • Case of the polynomial potential: • 푣푎 represents (truncated) Virasoro generator (right action) 2017-06-30 MATRIX @ Creswick 9 2017-06-30 MATRIX @ Creswick 10 Way to find Partition function • Find 푑푎 from the analytic property of the loop equation • Maintaining filling fraction constraint: • Partition function Z is the solution of differential equation • Conformal symmetry is the key element to the partition function 2017-06-30 MATRIX @ Creswick 11 2. Conformal symmetry and Matrix models Conformal field theory allows to construct generalized matrix models. Partition function and its correlations are found from the conformal symmetry. 2017-06-30 MATRIX @ Creswick 12 Liouville conformal field theory Liouville primary field: Free field correlation: Back ground charge= Conformal dimension= Screening operator= • Neutrality condition with N screening operators: 2017-06-30 MATRIX @ Creswick 13 Beta-deformed Penner-type matrix model Dijkgraaf & Vafa (2009) Itoyama & Oota (2010) • Liouville conformal block is given as • Matrix model represents the screening operators effects (Vandermonde determinant) 2017-06-30 MATRIX @ Creswick 14 Relation with hermitian matrix model • Put 훽 = 1 • Special (colliding) limit ( 푧푎 ≪ 휆퐼, allow finite moments 푐푘) • Conformal transform (휆퐼 → 1/휆퐼): • Finally, put 푐∞ = 0. 2017-06-30 MATRIX @ Creswick 15 Conformal symmetry of the partition function For the potential of the type : 2017-06-30 MATRIX @ Creswick 16 Extension to multi-matrix model • Use Toda field theory with multi-boson Φ = ( 휙 1 , … , 휙 푠 ) • Screening operator using root vector 푒푘 ; (푒푖, 푒푗) = 퐾푖푗 =Cartan Matrix • Vertex operator with holomorphic dimension Δ훼; • 푄෠ = background charge = 푄휌; 휌 =Weyl vector • Neutrality condition 2017-06-30 MATRIX @ Creswick 17 Multi-matrix model • Partition function of s-matrix model • Δ푎푏 is the Vandermonde determinant • Potential has the generalized Penner-type matrix form • The loop equation has the power of s of 1-point resolvent and possesses 푊푠+1 symmetry; 푊2 is the Virasoro symmetry. 2017-06-30 MATRIX @ Creswick 18 3. Application to irregular conformal block A special case is the irregular conformal block: Number of Virasoro generators are truncated to finite Results in collaboration with T. Nishinaka, S. Choi, H. Zhang, D. Polyakov 2017-06-30 MATRIX @ Creswick 19 Definition of irregular conformal state • Irregular conformal state with rank 푚 is a simultaneous eigenstate of positive Virasoro generators (Gaiotto state or Whittaker state). • Irregular conformal state is not an eigenstate of 퐿0. • Irregular state is a special combination of primary and descendants. 2017-06-30 MATRIX @ Creswick 20 Free field representation of Irregular vertex operator • Define irregular operator 퐼 푚 푧 so that irregular operator satisfies H. Nagoya and J. Sun (1002.2598) J. Gomis, B. Le Floch (1407.1852) Choi, Rim and Zhang(1510.09060), Polyakov and Rim (1601.07756, 1604.08741 ) 2017-06-30 MATRIX @ Creswick 21 Irregular state as a Heisenberg coherent state • Irregular state is defined in terms of the irregular vertex operator. • The irregular state is the Heisenberg coherent state. 2017-06-30 MATRIX @ Creswick 22 Adjoint state • Conformal transform (푧 = 1/휍 ) and define adjoint operator (푛) • 푅0 is a normalization for the adjoint state to exist: 2017-06-30 MATRIX @ Creswick 23 Inner product • Irregular vertex operator and its adjoint defines inner product • Symmetric property holds due to the conformal symmetry • Inner product obeys neutrality condition (푐0Ƹ + 푐0 = 0) 2017-06-30 MATRIX @ Creswick 24 Virasoro constraint on the inner product 2017-06-30 MATRIX @ Creswick 25 Dressing with screening operator • Screening operator (operators with conformal dimension 0) commutes with positive Virasoro operators. • Screening operator with 휓 푧 conformal dimension 1. • Lioiuville screening operator with background charge 푄 = 푏 + 1/푏: • Primary operator: 2017-06-30 MATRIX @ Creswick 26 Insertion of screening operator modifies eigenvalues • Irregular vertex operator and its state is defined as in the free field case • Only eigenvalue of the irregular conformal state is modified: 2017-06-30 MATRIX @ Creswick 27 Inner product as random matrix model • Inner product between regular and irregular state • Screening operators induces Vandermonde determinants 2017-06-30 MATRIX @ Creswick 28 Symmetry of inner product 2017-06-30 MATRIX @ Creswick 29 Adjoint and scaling factor • Background charge modifies the scaling factor in the adjoint operator 2017-06-30 MATRIX @ Creswick 30 Random matrix model as inner product of two irregular states 2017-06-30 MATRIX @ Creswick 31 Symmetry of the inner product 2017-06-30 MATRIX @ Creswick 32 Equivalent picture: Colliding limit • Free partition function (‘U(1) factor’) appears at the colliding limit 2017-06-30 MATRIX @ Creswick 33 (푛;푚) Flow equation of 푍푁 (푐0; 풄ො, 풄) • Flow equations: • Consistent condition of 푑푎 : Virasoro commutation relation. 2017-06-30 MATRIX @ Creswick 34 (0;푚) Explicit example: 푍푁 • Moment 푑푎 is found from the analytic structure of the loop equation. • Partition function is the solution of flow equations. • Useful limit (Nekrasov-Shatashivili limit): ℏ → 0, ϵ = ℏ푄 =finite. • Loop equation reduces to Riccati equation (deformed spectral curve). 2017-06-30 MATRIX @ Creswick 35 Polynomial expression at NS limit • One point resolvent is given in a monic polynomial of degree N, • Filling fraction fixes the integration range (N can be finite) 2017-06-30 MATRIX @ Creswick 36 (0;2) Matrix potential of 푍푁 Rank 2 with 2-cut solution: N = 푁1+ 푁2 2017-06-30 MATRIX @ Creswick 37 • Smooth limit of 푐2 →0 exits when 푁2 →0 2017-06-30 MATRIX @ Creswick 38 Irregular state of rank m proposed by Kanno, Maruyoshi, Shiba, Taki (1301.0721) • Note that inner product: 2017-06-30 MATRIX @ Creswick 39 Inner product and irregular conformal block 2017-06-30 MATRIX @ Creswick 40 Comparison: Irregular conformal block and inner product 2017-06-30 MATRIX @ Creswick 41 Perturbative evaluation of • Perturbation potential at z=0 • Perturbation at z=∞ : change of variable 휆퐽 = 1/휇퐽 2017-06-30 MATRIX @ Creswick 42 Connection with gauge theory: Irregular state as Argyres-Douglas limit of N=2 super Yang-Mills gauge theory AGT 2d Liouville 4d, N=2 Argyres-Douglas Conformal block SU(2) SCFT gauge theory Matrix model Colliding limit Irregular conformal block Irregular matrix model of Irregular state 2017-06-30 MATRIX @ Creswick 43 Connection with gauge theory • Seiberg-Witten differential 휆 = 푥 푑푧 provides the scaling dimension so that [x]+ [z]=1. • Use the Seiberg-Witten curve (loop equation at classical limit) to find the scaling dimension. • Coulomb branch parameter: 푎푎 = 휖 푁푎 + 휖/2 2017-06-30 MATRIX @ Creswick 44 Summary and outlook • Matrix theory is a convenient tool to investigate conformal symmetry: flow equations. • Conformal symmetry: (imaginary) Liouville conformal field theory • W-symmetry: Today field theory. • Super-conformal symmetry: super-conformal field theory. • Analytic structure of partition function (tau function)? • Multi-correlation? • More physical examples? 2017-06-30 MATRIX @ Creswick 45 Thanks 2017-06-30 MATRIX @ Creswick 46.