We show that the partition functions which enumerate Donaldson-Thomas invariants of local toric Calabi-Yau threefolds without compact divisors can be expressed in terms of specializations of the Schur measure. We also discuss the relevance of the Hall-Littlewood and Jack measures in the context of BPS state counting and study the partition functions at arbitrary points of the Kaehler moduli space. This rewriting in terms of symmetric functions leads to a unitary one-matrix model representation for Donaldson-Thomas theory. We describe explicitly how this result is related to the unitary matrix model description of Chern-Simons gauge theory. This representation is used to show that the generating functions for Donaldson-Thomas invariants are related to tau-functions of the integrable Toda and Toeplitz lattice hierarchies. The matrix model also leads to an interpretation of Donaldson-Thomas theory in terms of non-intersecting paths in the lock-step model of vicious walkers. We further show that these generating functions can be interpreted as normalization constants of a corner growth/last-passage stochastic model.

We derive some new relationships between matrix models of Chern-Simons gauge theory and of two-dimensional Yang-Mills theory. We show that q-integration of the Stieltjes-Wigert matrix model is the discrete matrix model that describes q-deformed Yang-Mills theory on the 2-sphere. We demonstrate that the semiclassical limit of the Chern-Simons matrix model is equivalent to the Gross-Witten model in the weak coupling phase. We study the strong coupling limit of the unitary Chern-Simons matrix model and show that it too induces the Gross-Witten model, but as a first order deformation of Dyson's circular ensemble. We show that the Sutherland model is intimately related to Chern-Simons gauge theory on the 3-sphere, and hence to q-deformed Yang-Mills theory on the 2-sphere. In particular, the ground state wavefunction of the Sutherland model in its classical equilibrium configuration describes the Chern-Simons free energy. The correspondence is extended to Wilson line observables and to arbitrary simply-laced gauge groups.

The study of the average of Schur polynomials over a Stieltjes-Wigert ensemble has been carried out in [J. Math. Phys. 48, 023507 (2007)] (arXiv:hep-th/0609167), where it was shown that it is equal to quantum dimensions. Using the same approach, we extend the result to the biorthogonal case. We also study, using the Littlewood--Richardson rule, some particular cases of the quantum dimensions result. Finally, we show that the notion of Giambelli compatibility of Schur averages, introduced in [Adv. Appl. Math. 37, 209 (2006)] (arXiv:math-ph/0501123), also holds in the biorthogonal setting.

We show that matrix models in Chern-Simons theory admit an interpretation as 1D exactly solvable models, paralleling the relationship between the Gaussian model and the Calogero model. We compute the corresponding Hamiltonians, ground-state wavefunctions and ground-state energies and point out that the models can be interpreted as quasi-1D Coulomb plasmas. We also study the relationship between Chern-Simons theory on $S^3$ and a system of N one-dimensional fermions at finite temperature with harmonic confinement. In particular we show that the Chern-Simons partition function can be described by the density matrix of the free fermions in a very particular, crystalline, configuration. For this, we both use the Brownian motion and the matrix model description of Chern-Simons theory and find several common features with c=1 theory at finite temperature.

The density of states of Yang-Mills integrals in the supersymmetric case is characterized by power-law tails whose decay is independent of N, the rank of the gauge group. It is believed that this has no counterpart in matrix models, but we construct a matrix model that exactly exhibits this property. In addition, we show that the eigenfunctions employed to construct the matrix model are invariant under the collinear subgroup of conformal transformations, SL(2,R). We also show that the matrix model itself is invariant under a fractional linear transformation. The wave functions of the model appear in the trigonometric Rosen-Morse potential and in free relativistic motion on AdS space.

Employing the random matrix formulation of Chern-Simons theory on Seifert manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful in exact computations in Chern-Simons matrix models. We construct a biorthogonal extension of the Stieltjes-Wigert polynomials, not available in the literature, necessary to study Chern-Simons matrix models when the geometry is a lens space. We also study the relationship between Stieltjes-Wigert and Rogers-Szegö polynomials and the corresponding equivalence with an unitary matrix model. Finally, we give a detailed proof of a result that relates quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert ensemble.

We derive discrete and oscillatory Chern-Simons matrix models. The method is based on fundamental properties of the associated orthogonal polynomials. As an application, we show that the discrete model allows to prove and extend the recently found equivalence between Chern-Simons theory and q-deformed 2dYM. In addition, the equivalence of the Chern-Simons matrix models gives a complementary view on the equivalence of effective superpotentials in N=1 gauge theories.

We point out a precise connection between Brownian motion, Chern-Simons theory on S^3, and 2d Yang-Mills theory on the cylinder. The probability of reunion for N vicious walkers on a line gives the partition function of Chern-Simons theory on S^3 with gauge group U(N). We establish a correspondence between quantities in Brownian motion and the modular S- and T-matrices of the WZW model at finite k and N. Brownian motion probabilities in the affine chamber of a Lie group are shown to be related to the partition function of 2d Yang-Mills on the cylinder.

We study matrix models with soft confining potentials. Their mathematical characterizationc is that their weight function is not determined by its moments. Relying on the moment problem and on orthogonal polynomials, we show general features of their density of states,correlation functions and loop averages. In addition, some of these models are equivalent, by a simple mapping, to matrix models that have appeared recently in connection with Chern-Simons theory. The models can be solved with q deformed orthogonal polynomials (Stieltjes-Wigert polynomials),and the deformation parameter turns out to be the usual $q$ parameter in Chern-Simons theory. In this way, we give a matrix model computation of the Chern-Simons partition function on $S^{3}$ and show that there are an infinite number of matrix models with this partition function.

Some aspects of the multiplicative anomaly of zeta determinants are investigated. A rather simple approach is adopted and, in particular, the question of zeta function factorization, together with its possible relation with the multiplicative            anomaly issue is discussed. We look primordially into the zeta functions instead of the determinants themselves, as was done in previous work. That provides a supplementary view, regarding the appearance of the multiplicative anomaly.

 

Finally, we briefly discuss determinants of zeta functions that are not in the pseudodifferential operator framework.
                             

We study analytical aspects of a generic q-deformation with q real, by relating it with discrete scale invariance. We show how models of conformal quantum mechanics, in the strong coupling regime and after regularization, are also discrete scale invariant. We discuss the consequences of their distinctive spectra, characterized by functional behavior. The role of log-periodic behavior and q-periodic functions is examined, and we show how q-deformed zeta functions, characterized by complex poles, appear. As an application,we discuss one-loop effects in discretely self-similar space-times.