1,720,982 research outputs found
Free particles from Brauer algebras in complex matrix models
No full textThe gauge invariant degrees of freedom of matrix models based on an N x N complex matrix, with U(N) gauge symmetry, contain hidden free particle structures. These are exhibited using triangular matrix variables via the Schur decomposition. The Brauer algebra basis for complex matrix models developed earlier is useful in projecting to a sector which matches the state counting of N free fermions on a circle. The Brauer algebra projection is characterized by the vanishing of a scale invariant laplacian constructed from the complex matrix. The special case of N=2 is studied in detail: the ring of gauge invariant functions as well as a ring of scale and gauge invariant differential operators are characterized completely. The orthonormal basis of wavefunctions in this special case is completely characterized by a set of five commuting Hamiltonians, which display free particle structures. Applications to the reduced matrix quantum mechanics coming from radial quantization in N=4 SYM are described. We propose that the string dual of the complex matrix harmonic oscillator quantum mechanics has an interpretation in terms of strings and branes in 2+1 dimensions
Permutation invariant matrix quantum thermodynamics and negative specific heat capacities in large N systems
Full text linkWe study the thermodynamic properties of the simplest gauged permutation
invariant matrix quantum mechanical system of oscillators, for general matrix
size . In the canonical ensemble, the model has a transition at a
temperature given by , characterised by a sharp peak in the specific heat capacity (SHC),
which separates a high temperature from a low temperature region. The peak
grows and the low-temperature region shrinks to zero with increasing . In
the micro-canonical ensemble, for finite , there is a low energy phase with
negative SHC and a high energy phase with positive SHC. The low-energy phase is
dominated by a super-exponential growth of degeneracies as a function of energy
which is directly related to the rapid growth in the number of directed graphs,
with any number of vertices, as a function of the number of edges. The two
ensembles have matching behaviour above the transition temperature. We further
provide evidence that these thermodynamic properties hold in systems with
symmetry such as the zero charge sector of the 2-matrix model and in
certain tensor models. We discuss the implications of these observations for
the negative specific heat capacities in gravity using the AdS/CFT
correspondence.Comment: 62 pages + 8 pages appendices (22 figures); Version 2: Minor
clarifications and typos correcte
Scattering of zero-branes off elementary strings in Matrix theory
No full textWe consider the scattering of zero-branes off an elementary string in Matrix theory or equivalently gravitons off a longitudinally wrapped membrane. The leading supergravity result is recovered by a one-loop calculation in zero-brane quantum mechanics. Simple scaling arguments are used to show that there are no further corrections at higher loops, to the leading term in the large impact parameter, low velocity expansion. The mechanism for this agreement is identified in terms of properties of a recently discovered boundary conformal field theory
Gauged permutation invariant matrix quantum mechanics: Path Integrals
Full text linkWe give a path integral construction of the quantum mechanical partition
function for gauged finite groups. Our construction gives the quantization of a
system of , matrices invariant under the adjoint action of the
symmetric group . The approach is general to any discrete group. For a
system of harmonic oscillators, i.e. for the non-interacting case, the
partition function is given by the Molien-Weyl formula times the zero-point
energy contribution. We further generalise the result to a system of non-square
and complex matrices transforming under arbitrary representations of the gauge
group.Comment: 17 pages of late
Gauged permutation invariant matrix quantum mechanics: Partition functions
Full text linkThe Hilbert spaces of matrix quantum mechanical systems with matrix degrees of freedom have been analysed recently in terms of symmetric group elements acting as . Solvable models have been constructed uncovering partition algebras as hidden symmetries of these systems. The solvable models include an 11-dimensional space of matrix harmonic oscillators, the simplest of which is the standard matrix harmonic oscillator with symmetry. The permutation symmetry is realised as gauge symmetry in a path integral formulation in a companion paper. With the simplest matrix oscillator Hamiltonian subject to gauge permutation symmetry, we use the known result for the micro-canonical partition function to derive the canonical partition function. It is expressed as a sum over partitions of of products of factors which depend on elementary number-theoretic properties of the partitions, notably the least common multiples and greatest common divisors of pairs of parts appearing in the partition. This formula is recovered using the Molien-Weyl formula, which we review for convenience. The Molien-Weyl formula is then used to generalise the formula for the canonical partition function to the 11-parameter permutation invariant matrix harmonic oscillator.32 pages + 2 pages Appendices, 1 figure ; Revised version : minor typos corrected and brief remarks added ; Second revision: further minor typos correcte
Counting of surfaces and computational complexity in column sums of symmetric group character tables
Full text linkThe character table of the symmetric group , of permutations of objects, is of fundamental interest in theoretical physics, combinatorics as well as computational complexity theory. We investigate the implications of an identity, which has a geometrical interpretation in combinatorial topological field theories, relating the column sum of normalised central characters of to a sum of structure constants of multiplication in the centre of the group algebra of . The identity leads to the proof that a combinatorial computation of the column sum belongs to complexity class \shP. The sum of structure constants has an interpretation in terms of the counting of branched covers of the sphere. This allows the identification of a tractable subset of the structure constants related to genus zero covers. We use this subset to prove that the column sum for a conjugacy class labelled by partition is non-vanishing if and only if the permutations in the conjugacy class are even. This leads to the result that the determination of the vanishing or otherwise of the column sum is in complexity class \pP. The subset gives a positive lower bound on the column sum for any even . For any disjoint decomposition of as we obtain a lower bound for the column sum at in terms of the product of the column sums for and. This can be expressed as a super-additivity property for the logarithms of column sums of normalized characters.52 pages + Appendices, 9 Figure
Kronecker coefficients from algebras of bi-partite ribbon graphs
Full text linkBi-partite ribbon graphs arise in organising the large expansion of
correlators in random matrix models and in the enumeration of observables in
random tensor models. There is an algebra , with basis given by
bi-partite ribbon graphs with edges, which is useful in the applications to
matrix and tensor models. The algebra is closely related to
symmetric group algebras and has a matrix-block decomposition related to
Clebsch-Gordan multiplicities, also known as Kronecker coefficients, for
symmetric group representations. Quantum mechanical models which use
as Hilbert spaces can be used to give combinatorial algorithms
for computing the Kronecker coefficients.Comment: 13 pages, 1 figure. References updated, typos fixed. We thank the
Editors Konstantinos Anagnostopoulos, Peter Schupp, George Zoupanos, for the
invitation to contribute to this special volume on "Non-commutativity and
physics". arXiv admin note: text overlap with arXiv:2010.0405
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