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arXiv2011

Thermodynamic limit of random partitions and dispersionless Toda hierarchy

We study the thermodynamic limit of random partition models for the instanton sum of 4D and 5D supersymmetric U(1) gauge theories deformed by some physical observables. The physical observables correspond to external potentials in the statistical model. The partition function is reformulated in terms of the density function of Maya diagrams. The thermodynamic limit is governed by a limit shape of Young diagrams associated with dominant terms in the partition function. The limit shape is characterized by a variational problem, which is further converted to a scalar-valued Riemann-Hilbert problem. This Riemann-Hilbert problem is solved with the aid of a complex curve, which may be thought of as the Seiberg-Witten curve of the deformed U(1) gauge theory. This solution of the Riemann-Hilbert problem is identified with a special solution of the dispersionless Toda hierarchy that satisfies a pair of generalized string equations. The generalized string equations for the 5D gauge theory are shown to be related to hidden symmetries of the statistical model. The prepotential and the Seiberg-Witten differential are also considered.

math-ph↗
arXiv2012

Heat Kernels and Zeta Functions on Fractals

On fractals, spectral functions such as heat kernels and zeta functions exhibit novel features, very different from their behaviour on regular smooth manifolds, and these can have important physical consequences for both classical and quantum physics in systems having fractal properties.

math-ph↗
arXiv2013

Resonances in sinh- and sine-Gordon models and higher equations of motion in Liouville theory

The notion of operator resonances was introduced earlier by Al. Zamolodchikov within the framework of the conformal perturbation theory. The resonances are related to logarithmic divergences of integrals in the perturbation expansion, and manifest themselves in poles of the correlation functions and form factors of local operators considered as functions of conformal dimensions. The residues of the poles can be computed by means of some operator identities. Here we study the resonances in the Liouville, sinh- and sine-Gordon models, considered as perturbations of a massless free boson. We show that the well-known higher equations of motion discovered by Al. Zamolodchikov in the Liouville field theory are nothing but resonance identities for some descendant operators. The resonance expansion in the vicinity of a resonance point provides a regularized version of the corresponding operators. We try to construct the corresponding resonance identities and resonance expansions in the sinh- and sine-Gordon theories. In some cases it can be done explicitly, but in most cases we are only able to obtain a general form so far. We show nevertheless that the resonances are perturbatively exact, which means that each of them only appears in a single term of the perturbation theory.

hep-th↗
arXiv2013

WKB Approximation to the Power Wall

We present a semiclassical analysis of the quantum propagator of a particle confined on one side by a steeply, monotonically rising potential. The models studied in detail have potentials proportional to $x^α$ for $x>0$; the limit $α\to\infty$ would reproduce a perfectly reflecting boundary, but at present we concentrate on the cases $α=1$ and 2, for which exact solutions in terms of well known functions are available for comparison. We classify the classical paths in this system by their qualitative nature and calculate the contributions of the various classes to the leading-order semiclassical approximation: For each classical path we find the action $S$, the amplitude function $A$ and the Laplacian of $A$. (The Laplacian is of interest because it gives an estimate of the error in the approximation and is needed for computing higher-order approximations.) The resulting semiclassical propagator can be used to rewrite the exact problem as a Volterra integral equation, whose formal solution by iteration (Neumann series) is a semiclassical, not perturbative, expansion. We thereby test, in the context of a concrete problem, the validity of the two technical hypotheses in a previous proof of the convergence of such a Neumann series in the more abstract setting of an arbitrary smooth potential. Not surprisingly, we find that the hypotheses are violated when caustics develop in the classical dynamics; this opens up the interesting future project of extending the methods to momentum space.

math-ph↗
arXiv2012

Canonical bifurcation in higher derivative, higher spin, theories

We present a non-perturbative canonical analysis of the D=3 quadratic-curvature, yet ghost-free, model to exemplify a novel, "constraint bifurcation", effect. Consequences include a jump in excitation count: a linearized level gauge variable is promoted to a dynamical one in the full theory. We illustrate these results with their concrete perturbative counterparts. They are of course mutually consistent, as are perturbative findings in related models. A geometrical interpretation in terms of propagating torsion reveals the model's relation to an (improved) version of Einstein-Weyl gravity at the linearized level. Finally, we list some necessary conditions for triggering the bifurcation phenomenon in general interacting gauge systems.

hep-th↗
arXiv2013

Modified melting crystal model and Ablowitz-Ladik hierarchy

This paper addresses the issue of integrable structure in a modified melting crystal model of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum expectation value of an operator on the Fock space of 2D complex free fermion fields. The quantum torus algebra of fermion bilinears behind this expression is shown to have an extended set of "shift symmetries". They are used to prove that the partition function (deformed by external potentials) is essentially a tau function of the 2D Toda hierarchy. This special solution of the 2D Toda hierarchy can be characterized by a factorization problem of $\ZZ\times\ZZ$ matrices as well. The associated Lax operators turn out to be quotients of first order difference operators. This implies that the solution of the 2D Toda hierarchy in question is actually a solution of the Ablowitz-Ladik (equivalently, relativistic Toda) hierarchy. As a byproduct, the shift symmetries are shown to be related to matrix-valued quantum dilogarithmic functions.

math-ph↗
arXiv2013

Hydrogen atom on curved noncommutative space

We have calculated the hydrogen atom spectrum on curved noncommutative space defined by the commutation relations $\left[ \hat {x}^{i},\hat{x}^{j}\right] =iθ\hatω^{ij}\left( \hat {x}\right) $, where $θ$ is the parameter of noncommutativity. The external antisymmetric field which determines the noncommutativity is chosen as $ω^{ij}(x) =\varepsilon^{ijk}{x}_{k}f\left( {x_i}x^{i}\right) $. In this case the rotational symmetry of the system is conserved, preserving the degeneracy of the energy spectrum. The contribution of the noncommutativity appears as a correction to the fine structure. The corresponding nonlocality is calculated: $ΔxΔy \geq \frac{θ^2}{4} |m\langle f^2\rangle| $, where $m$ is a magnetic quantum number.

math-ph↗
arXiv2020

$W$-algebras associated with centralizers in type $A$

We introduce a new family of affine $W$-algebras associated with the centralizers of arbitrary nilpotent elements in $\mathfrak{gl}_N$. We define them by using a version of the BRST complex of the quantum Drinfeld--Sokolov reduction. A family of free generators of the new algebras is produced in an explicit form. We also give an analogue of the Fateev--Lukyanov realization for these algebras by applying a Miura-type map.

math.RT↗
arXiv2010

Moduli spaces of noncommutative instantons: gauging away noncommutative parameters

Using the theory of noncommutative geometry in a braided monoidal category, we improve upon a previous construction of noncommutative families of instantons of arbitrary charge on the deformed sphere S^4_θ. We formulate a notion of noncommutative parameter spaces for families of instantons and we explore what it means for such families to be gauge equivalent, as well as showing how to remove gauge parameters using a noncommutative quotient construction. Although the parameter spaces are a priori noncommutative, we show that one may always recover a classical parameter space by making an appropriate choice of gauge transformation.

math.QA↗
arXiv2012

Numerical Regularization of Electromagnetic Quantum Fluctuations in Inhomogeneous Dielectric Media

Electromagnetic Casimir stresses are of relevance to many technologies based on mesoscopic devices such as MEMS embedded in dielectric media, Casimir induced friction in nano-machinery, micro-fluidics and molecular electronics. Computation of such stresses based on cavity QED generally require numerical analysis based on a regularization process. A new scheme is described that has the potential for wide applicability to systems involving realistic inhomogeneous media. From a knowledge of the spectrum of the stationary modes of the electromagnetic field the scheme is illustrated by estimating numerically the Casimir stress on opposite faces of a pair of perfectly conducting planes separated by a vacuum and the change in this result when the region between the plates is filled with an incompressible inhomogeneous non-dispersive dielectric.

math-ph↗
arXiv2012

WKB Analysis of PT-Symmetric Sturm-Liouville problems. II

In a previous paper it was shown that a one-turning-point WKB approximation gives an accurate picture of the spectrum of certain non-Hermitian PT-symmetric Hamiltonians on a finite interval with Dirichlet boundary conditions. Potentials to which this analysis applies include the linear potential $V=igx$ and the sinusoidal potential $V=ig\sin(αx)$. However, the one-turning-point analysis fails to give the full structure of the spectrum for the cubic potential $V=igx^3$, and in particular it fails to reproduce the critical points at which two real eigenvalues merge and become a complex-conjugate pair. The present paper extends the method to cases where the WKB path goes through a {\it pair} of turning points. The extended method gives an extremely accurate approximation to the spectrum of $V=igx^3$, and more generally it works for potentials of the form $V=igx^{2N+1}$. When applied to potentials with half-integral powers of $x$, the method again works well for one sign of the coupling, namely that for which the turning points lie on the first sheet in the lower-half plane.

hep-th↗
arXiv2012

Quantum Bose and Fermi gases with large negative scattering length in the 2-body S-matrix approximation

We study both Bose and Fermi gases at finite temperature and density in an approximation that sums an infinite number of many body processes that are reducible to 2-body scatterings. This is done for arbitrary negative scattering length, which interpolates between the ideal and unitary gas limits. In the unitary limit, we compute the first four virial coefficients within our approximation. The second virial coefficient is exact, and we extend the previously known result for fermions to bosons, and also for both bosons and fermions for the upper branch on the other side of unitarity (infinitely large positive scattering length). Assuming bosons can exist in a meta-stable state before undergoing mechanical collapse, we map out the critical temperatures for strongly coupled Bose-Einstein condensation as a function of scattering length.

cond-mat.quant-gas↗
arXiv2013

Diffraction in Time: An Exactly Solvable Model

In the recent years, mater-wave interferometry has attracted growing attention due to its unique suitability for high-precision measurements and study of fundamental aspects of quantum theory. Diffraction and interference of matter waves can be observed not only at a spatial aperture (such as a screen edge, slit, or grating), but also at a time-domain aperture (such as an absorbing barrier, or "shutter", that is being periodically switched on and off). The wave phenomenon of the latter type is commonly referred to as "diffraction in time". Here, we introduce a versatile, exactly solvable model of diffraction in time. It describes time-evolution of an arbitrary initial quantum state in the presence of a time-dependent absorbing barrier, governed by an arbitrary aperture function. Our results enable a quantitative description of diffraction and interference patterns in a large variety of setups, and may be used to devise new diffraction and interference experiments with atoms and molecules.

quant-ph↗
arXiv2011

An integrable modification of the critical Chalker-Coddington network model

We consider the Chalker-Coddington network model for the Integer Quantum Hall Effect, and examine the possibility of solving it exactly. In the supersymmetric path integral framework, we introduce a truncation procedure, leading to a series of well-defined two-dimensional loop models, with two loop flavours. In the phase diagram of the first-order truncated model, we identify four integrable branches related to the dilute Birman-Wenzl-Murakami braid-monoid algebra, and parameterised by the loop fugacity $n$. In the continuum limit, two of these branches (1,2) are described by a pair of decoupled copies of a Coulomb-Gas theory, whereas the other two branches (3,4) couple the two loop flavours, and relate to an $SU(2)_r \times SU(2)_r / SU(2)_{2r}$ Wess-Zumino-Witten (WZW) coset model for the particular values $n= -2\cos[π/(r+2)]$ where $r$ is a positive integer. The truncated Chalker-Coddington model is the $n=0$ point of branch 4. By numerical diagonalisation, we find that its universality class is neither an analytic continuation of the WZW coset, nor the universality class of the original Chalker-Coddington model. It constitutes rather an integrable, critical approximation to the latter.

cond-mat.dis-nn↗
arXiv2011

Supersymmetric K field theories and defect structures

We construct supersymmetric K field theories (i.e., theories with a non-standard kinetic term) in 1+1 and 2+1 dimensions such that the bosonic sector just consists of a nonstandard kinetic term plus a potential. Further, we study the possibility of topological defect formation in these supersymmetric models. Finally, we consider more general supersymmetric K field theories where, again, topological defects exist in some cases.

hep-th↗
arXiv2011

Properties of some (3+1) dimensional vortex solutions of the CP^N model

We construct new classes of vortex-like solutions of the CP^N model in (3+1) dimensions and discuss some of their properties. These solutions are obtained by generalizing to (3+1) dimensions the techniques well established for the two dimensional CP^N models. We show that as the total energy of these solutions is infinite, they describe evolving vortices and anti-vortices with the energy density of some configurations varying in time. We also make some further observations about the dynamics of these vortices.

hep-th↗
arXiv2012

Spin operator and spin states in Galilean covariant Fermi field theories

Spin degrees of freedom of the Galilean covariant Dirac field in (4+1) dimensions and its nonrelativistic counterpart in (3+1) dimensions are examined. Two standard choices of spin operator, the Galilean covariant and Dirac spin operators, are considered. It is shown that the Dirac spin of the Galilean covariant Dirac field in (4+1) dimensions is not conserved, and the role of non-Galilean boosts in its nonconservation is stressed out. After reduction to (3+1) dimensions the Dirac field turns into a nonrelativistic Fermi field with a conserved Dirac spin. A generalized form of the Levy-Leblond equations for the Fermi field is given. One-particle spin states are constructed. A particle-antiparticle system is discussed.

math-ph↗