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APS-JPS Joint meeting Hawaii, Oct. 30th , 2006 Symmetry and Integrability of Non-singlet Sectors in MQM Y. Matsuo (Univ. Tokyo) Based on hep-th/0607052 With Y. Hatsuda I. Introduction c=1 Quantum Gravity Liouville Theory Matrix Quantum Mechanics μ: cosmological const. short string singlet sector = free fermion Representation of wave function Action is invariant under Conserved charge : Wave function transforms as Usually we consider only singlet sector: the dynamics reduce to free fermions which correspond to short strings of Liouville theory Possible representations Since we have to construct states from X (adjoint), the representation that wave function can take is limited. Constraint: #Box=#Anti-box Boulatov-Kazakov Examples of wave function : Singlet wave function : Adjoint wave function : higher representation With appropriate (anti-)symmetrization among upper and lower indices Difficulty of non-singlet sector = “angular part” does not decouple from eigenvalue dynamics Role of non-singlet sector Vortex configuration (KT phase transition) insertion of 2D Black hole (cf. Kazakov, Kostov, Kutasov) Long string with tips (Maldacena) Physical meaning of adjoint sector has never been well-understood: difficulty in technical aspect MQM non-singlet sector as integrable system Infinite dim symmetry, infinite set of commuting charges Reduction to eigenvalue dynamics (or CalogeroSutherland system) Bosonization : interaction between tachyon and tips Exact eigenstates of the commuting charges: analogy with the matrix string theory II. Infinite dim symmetry in MQM We consider “upside-up” potential instead of “upside-down” Spectrum: continuous Parabolic cylinder function Upside-down Spectrum: discrete Hermite polynomial Upside-up Wick rotation? “discrete states” Matrix version of Harmonic Oscillator W∞ algebra Operator algebra which does not change the representation = spectrum generating algebra for specific representation U(N) invariant operator These operators define an infinite dimensional algebra which is an extension of W∞ algebra generated by Algebra for lower generators U(1) current Virasoro Higher generators Target space reprametrization symmetry Algebra becomes nonlinear: Commuting charges For singlet sector, the eigenstates of these operators are given by free fermion basis, namely Schur polynomials. We want to find generalization to nonsinglet sectors. §3 Reduction to eigenvalue dynamics In order to keep We need to put, (by expansion w.r.t. K) Transformation of adjoint wave function Action of conserved quantities The action of H1 is trivial. After rewriting , H2 becomes Hamiltonian of Calogero-Sutherland equation, How to solve it ? Numerical estimate of eigenvalues By applying bosonization, numerical analysis becomes possible. Singlet sector Adjoint sector They are identical up to the multiplicity! IV. Construction of eigenstates It turns out that to work with the original matrix variable is easier to construct explicit form of eigenstates. A generic state in the Hilbert space We rewrite it as, Action of H2 to “long strings” Splitting and joining of long strings similar to SFT Use of permutation group We write the states by using permutation group, After decomposition of σ into cycles, above wave function reduces to the product of traces. In terms of More generally, , action of H2 is much simpler Young symmetrizer Eigenstate is written as, These are eigenstates since One may take various choices for C → universal expression for eigenstates Degeneracy in adjoint sector The eigenfunction depends on the location of the box 1 in the Young frame. In fact, there are some equivalences and the number of independent states is the number of rectangles of the Young frame. It reproduces correctly the known partition function: Boulatov and Kazakov V. Conclusion Our first goal of studying non-singlet sectors from the viewpoint of integrable system is mostly completed Symmetry algebra Infinite set of conserved charges Relation with Calogero(-Sutherland) system Eigenstates of conserved charges which generalizes the free fermion system in the singlet sector Future issues Large N limit Treatment of UD potential (cf. Kostov) Interaction among vortices (or “tips”) Application to black hole physics (cf. Jevicki-Yoneya) Other applications: AdS/CFT/spin chain (non-singlet = system with impurities)