Download Symmetry and Integrability of Nonsinglet Sectors in MQM

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)