We study matrix integration over the classical Lie groups $U(N),Sp(2N),O(2N)$ and $O(2N+1)$, using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz$\pm$Hankel matrices. We establish a number of factorizations and expansions for such integrals, also with insertions of irreducible characters. As a specific example, we compute both at finite and large $N$ the partition functions, Wilson loops and Hopf links of Chern-Simons theory on $S^{3}$ with the aforementioned symmetry groups. The identities found for the general models translate in this context to relations between observables of the theory. Finally, we use character expansions to evaluate averages in random matrix ensembles of Chern-Simons type, describing the spectra of solvable fermionic models with matrix degrees of freedom.

We study the partition function of a TT-deformed version of Yang-Mills theory on the two-sphere. We show that the Douglas-Kazakov phase transition persists for a range of values of the deformation parameter, and that the critical area is lowered. The transition is of third order and also induced by instantons, whose contributions we characterize.

We study the phase transitions of three-dimensional N=2 U(N) Chern-Simons theory on S3 with a varied number of massive fundamental hypermultiplets and with a Fayet-Iliopoulos parameter. We characterize the various phase diagrams in the decompactification limit, according to the number of different mass scales in the theory. For this, we extend the known solution of the saddle-point equations to the setting where the one cut solution is characterized by asymmetric intervals. We then study the large N limit of Wilson loops in antisymmetric representations, with the additional scaling corresponding to the variation of the size of the representation. We give explicit expressions, both with and without FI terms, and study 1/R corrections for the different phases. These corrections break the perimeter law behavior, as they introduce a scaling with the size of the representation. We show how the phase transitions of the Wilson loops can either be of first or second order and determine the underlying mechanism in terms of the eventual asymmetry of the support of the solution of the saddle-point equation, at the critical points.

We show that a version of Abelian gauge theory on R3λ, when restricted to a single fuzzy sphere, reduces in the large N limit to the Langmann-Szabo-Zarembo (LSZ) matrix model, which originally emerges in the study of scalar field theory on the Moyal plane. We then prove that the LSZ matrix model is actually equivalent to the matrix model of U(N) Chern-Simons theory on S3. The correspondence holds in a generalized sense: depending on the spectra of the two external matrices of the LSZ model, the Chern-Simons matrix model either describes the Chern-Simons partition function, the unknot invariant, given by quantum dimensions, or the Hopf link invariant. Equivalently, the evaluation of the LSZ model can be written in terms of the S and T modular matrices of the WZW model.

We show that U(N) 3d $\mathcal{N}=4$ supersymmetric gauge theories on S^3 with N_{f} massive fundamental hypermultiplets and with a Fayet-Iliopoulos (FI) term are solvable in terms of generalized Selberg integrals. Finite $N$ expressions for the partition function and Wilson loop in arbitrary representations are given. We obtain explicit analytical expressions for Wilson loops with symmetric, antisymmetric, rectangular and hook representations, in terms of Gamma functions of complex argument. The free energy for orthogonal and symplectic gauge group is also given. The study of asymptotic expansions of the free energy then leads to the emergence of exponentially small contributions for $N_{f}, which corresponds to bad theories. Duality checks are also explicitly performed and we show how the exponential asymptotics is understood from the point of view of the duality between good and bad theories.

We express minors of Toeplitz matrices of finite and large dimension, including the case of symbols with Fisher-Hartwig singularities, in terms of specializations of symmetric functions. By comparing the resulting expressions with the inverses of some Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials.

We study a model of quantum mechanical fermions with matrix-like index structure (with indices $N$ and $L$) and quartic interactions, recently introduced by\ Anninos and Silva. We compute the partition function exactly with $q$-deformed orthogonal polynomials (Stieltjes-Wigert polynomials), for different values of $L$ and arbitrary $N$. From the explicit evaluation of the thermal partition function, the energy levels and degeneracies are determined. For a given $L$, the number of states of different energy is quadratic in $N$, which implies an exponential degeneracy of the energy levels. We also show that at high-temperature we have a Gaussian matrix model, which implies a symmetry that swaps $N$ and $L$, together with a Wick rotation of the spectral parameter. In this limit, we also write the partition function, for generic $L$ and $N,$ in terms of a single generalized Hermite polynomial.

We study N=4 supersymmetric QED in three dimensions, on a three-sphere, with 2N massive hypermultiplets and a Fayet-Iliopoulos parameter. We identify the exact partition function of the theory with a conical (Mehler) function. This implies a number of analytical formulas, including a recurrence relation and a second-order differential equation, associated with an integrable system. In the large N limit, the theory undergoes a second-order phase transition on a critical line in the parameter space. We discuss the critical behavior and compute the two-point correlation function of a gauge invariant mass operator, which is shown to diverge as one approaches criticality from the subcritical phase. Finally, we comment on the asymptotic 1/N expansion and on mirror symmetry.

We give explicit analytical expressions for the partition function of $ U(N)_{k}\times U(N+M)_{-k}$ ABJ theory at weak coupling ($k\rightarrow \infty )$ for finite and arbitrary values of $N$ and $M$ (including the ABJM case and its mass-deformed generalization). We obtain the expressions by identifying the one-matrix model formulation with a Meixner-Pollaczek ensemble and using the corresponding orthogonal polynomials, which are also eigenfunctions of a $su(1,1)$ quantum oscillator. Wilson loops in mass-deformed ABJM are also studied in the same limit and interpreted in terms of $su(1,1)$ coherent states.

We solve for finite $N$ the matrix model of supersymmetric $U(N)$ Chern-Simons theory coupled to $N_{f}$ fundamental and $N_{f}$ anti-fundamental chiral multiplets of $R$-charge $1/2$ and of mass $m$, by identifying it with an average of inverse characteristic polynomials in a Stieltjes-Wigert ensemble. This requires the computation of the Cauchy transform of the Stieltjes-Wigert polynomials, which we carry out, finding a relationship with Mordell integrals, and hence with previous analytical results on the matrix model. The semiclassical limit of the model is expressed, for arbitrary $N_{f},$ in terms of a single Hermite polynomial. This result also holds for more general matter content, involving matrix models with double-sine functions.

We solve, for finite $N$, the matrix model of supersymmetric $U(N)$ Chern-Simons theory coupled to $N_{f}$ massive hypermultiplets of $R$-charge $\frac{1}{2}$, together with a Fayet-Iliopoulos term. We compute the partition function by identifying it with a determinant of a Hankel matrix, whose entries are parametric derivatives (of order $N_{f}-1$) of Mordell integrals. We obtain finite Gauss sums expressions for the partition functions. We also apply these results to obtain an exhaustive test of Giveon-Kutasov (GK) duality in the $\mathcal{N}=3$ setting, by systematic computation of the matrix models involved. The phase factor that arises in the duality is then obtained explicitly. We give an expression characterized by modular arithmetic (mod 4) behavior that holds for all tested values of the parameters (checked up to $N_{f}=12$ flavours).

We study N=2 supersymmetric U(N) Chern-Simons with N_{f} fundamental and N_{f} antifundamental chiral multiplets of mass m in the parameter space spanned by (g,m,N,N_{f}), where g denotes the coupling constant. In particular, we analyze the matrix model description of its partition function, both at finite N using the method of orthogonal polynomials together with Mordell integrals and, at large N with fixed g, using the theory of Toeplitz determinants. We show for the massless case that there is an explicit realization of the Giveon-Kutasov duality. For finite N, with N>N_{f}, three regimes that exactly correspond to the known three large N phases of theory are identified and characterized.

We construct a 1d spin chain Hamiltonian with generic interactions and prove that the thermal correlation functions of the model admit an explicit random matrix representation. As an application of the result, we show how the observables of $U(N)$ Chern-Simons theory on $S^{3}$ can be reproduced with the thermal correlation functions of the 1d spin chain, which is of the XX type, with a suitable choice of exponentially decaying interactions between infinitely many neighbours. We show that for this model, the correlation functions of the spin chain at a finite temperature $\beta =1$ give the Chern-Simons partition function, quantum dimensions and the full topological $S$-matrix.

We give, using an explicit expression obtained in [V. Jones, Ann. of Math. 126, 335 (1987)], a basic hypergeometric representation of the HOMFLY polynomial of (n,m) torus knots, and present a number of equivalent expressions, all related by Heine's transformations. Using this result the (m,n)↔(n,m) symmetry and the leading polynomial at large N are explicit. We show the latter to be the Wilson loop of 2d Yang-Mills theory on the plane. In addition, after taking one winding to infinity, it becomes the Wilson loop in the zero instanton sector of the 2d Yang-Mills theory, which is known to give averages of Wilson loops in N=4 SYM theory. We also give, using matrix models, an interpretation of the HOMFLY polynomial and the corresponding Jones-Rosso representation in terms of q-harmonic oscillators.

We develop Coulomb gas pictures of strong and weak coupling regimes of supersymmetric Yang-Mills theory in five and four dimensions. By relating them to the matrix models that arise in Chern-Simons theory, we compute their free energies in the large N limit and establish relationships between the respective gauge theories. We use these correspondences to rederive the N^3 behaviour of the perturbative free energy of supersymmetric gauge theory on certain toric Sasaki-Einstein five-manifolds, and the one-loop thermal free energy of N=4 supersymmetric Yang-Mills theory on a spatial three-sphere.

By using random matrix models we uncover a connection between the low energy sector of four dimensional QCD at finite volume and the Heisenberg XX model in a 1d spin chain. This connection allows to relate crucial properties of QCD with physically meaningful properties of the spin chain, establishing a dictionary between both worlds. We predict for the spin chain a third-order phase transition and a Tracy-Widom law in the transition region. We finally comment on possible numerical implications of the connection as well as on possible experimental implementations.

We characterise the quantum group gauge symmetries underlying q-deformations of two-dimensional Yang-Mills theory by studying their relationships with the matrix models that appear in Chern-Simons theory and six-dimensional N=2 gauge theories, together with their refinements and supersymmetric extensions. We develop uniqueness results for quantum deformations and refinements of gauge theories in two dimensions, and describe several potential analytic and geometric realisations of them. We reconstruct standard q-deformed Yang-Mills amplitudes via gluing rules in the representation category of the quantum group associated to the gauge group, whose numerical invariants are the usual characters in the Grothendieck group of the category. We apply this formalism to compute refinements of q-deformed amplitudes in terms of generalised characters, and relate them to refined Chern-Simons matrix models (...).

We use the Villain approximation to show that the Gross-Witten model, in the weak- and strong-coupling limits, is related to the unitary matrix model that describes U(N) Chern-Simons theory on S³. The weak-coupling limit corresponds to the q→1 limit of the Chern-Simons theory while the strong-coupling regime is related to the q→0 limit. In the latter case, there is a logarithmic relationship between the respective coupling constants. We also show how the Chern-Simons matrix model arises by considering two-dimensional Yang-Mills theory with the Villain action. This leads to a U(1)^{N} theory which is the Abelianization of 2d Yang-Mills theory with the heat-kernel lattice action. In addition, we show that the character expansion of the Villain lattice action gives the q deformation of the heat-kernel as it appears in q-deformed 2d Yang-Mills theory. We also study the relationship between the unitary and Hermitian Chern-Simons matrix models and the rotation of the integration contour in the corresponding integrals.

We show that the chiral partition function of two-dimensional Yang-Mills theory on the sphere can be mapped to the partition function of the homogeneous six-vertex model with domain wall boundary conditions in the ferroelectric phase. A discrete matrix model description in both cases is given by the Meixner ensemble, leading to a representation in terms of a stochastic growth model. We show that the partition function is a particular case of the z-measure on the set of Young diagrams, yielding a unitary matrix model for chiral Yang-Mills theory on the sphere and the identification of the partition function as a tau-function of the Painleve V equation. We describe the role played by generalized non-chiral Yang-Mills theory on the sphere in relating the Meixner matrix model to the Toda chain hierarchy encompassing the integrability of the six-vertex model (...).

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 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 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.