The Relativistic Quantum Motions

Using the relativistic quantum stationary Hamilton-Jacobi equation within the framework of the equivalence postulate, and grounding oneself on both relativistic and quantum Lagrangians, we construct a Lagrangian of a relativistic quantum system in one dimension and derive a third order equation of motion representing a first integral of the relativistic quantum Newton's law. Then, we plot the relativistic quantum trajectories of a particle moving under the constant and the linear potentials. We establish the existence of nodes and link them to the de Broglie's wavelength.Comment: Latex, 18 pages, 3 eps figure

Nonperturbative Relations in N=2 SUSY Yang-Mills and WDVV equation

We find the nonperturbative relation between $\langle {\rm tr} \phi^2 \rangle$, $\langle {\rm tr} \phi^3\rangle$ the prepotential ${\cal F}$ and the vevs $\langle \phi_i\rangle$ in $N=2$ supersymmetric Yang-Mills theories with gauge group $SU(3)$. Nonlinear differential equations for ${\cal F}$ including the Witten -- Dijkgraaf -- Verlinde -- Verlinde equation are obtained. This indicates that $N=2$ SYM theories are essentially topological field theories and that should be seen as low-energy limit of some topological string theory. Furthermore, we construct relevant modular invariant quantities, derive canonical relations between the periods and investigate the structure of the beta function by giving its explicit form in the moduli coordinates. In doing this we discuss the uniformization problem for the quantum moduli space. The method we propose can be generalized to $N=2$ supersymmetric Yang-Mills theories with higher rank gauge groups.Comment: 12 pages, LaTex. Expanded version. New results, corrections, references and acknowledgements adde

Nonperturbative Renormalization Group Equation and Beta Function in N=2 SUSY Yang-Mills

We obtain the exact beta function for $N=2$ SUSY $SU(2)$ Yang-Mills theory and prove the nonperturbative Renormalization Group Equation $$\partial_\Lambda{\cal F}(a,\Lambda)= {\Lambda\over \Lambda_0}\partial_{\Lambda_0}{\cal F}(a_0,\Lambda_0) e^{-2\int_{\tau_0}^\tau {dx \beta^{-1}(x)}}.$$Comment: LaTex, 10 pg. Expanded introduction, references added, to appear in Phys. Rev. Let

Non-holomorphic terms in N=2 SUSY Wilsonian actions and RG equation

In this paper we first investigate the Ansatz of one of the present authors for K(\Psi,\bar\Psi), the adimensional modular invariant non-holomorphic correction to the Wilsonian effective Lagrangian of an N=2 globally supersymmetric gauge theory. The renormalisation group beta-function of the theory crucially allows us to express K(\Psi,\bar\Psi) in a form that easily generalises to the case in which the theory is coupled to N_F hypermultiplets in the fundamental representation of the gauge group. This function satisfies an equation which should be viewed as a fully non-perturbative ``non-chiral superconformal Ward identity". We also determine its renormalisation group equation. Furthermore, as a first step towards checking the validity of this Ansatz, we compute the contribution to K(\Psi,\bar\Psi) from instantons of winding number k=1 and k=2. As a by-product of our analysis we check a non-renormalisation theorem for N_F=4.Comment: 39 pages, LaTex file, no figure

Matrix Models, Argyres-Douglas singularities and double scaling limits

We construct an N=1 theory with gauge group U(nN) and degree n+1 tree level superpotential whose matrix model spectral curve develops an A_{n+1} Argyres-Douglas singularity. We evaluate the coupling constants of the low-energy U(1)^n theory and show that the large N expansion is singular at the Argyres-Douglas points. Nevertheless, it is possible to define appropriate double scaling limits which are conjectured to yield four dimensional non-critical string theories as proposed by Ferrari. In the Argyres-Douglas limit the n-cut spectral curve degenerates into a solution with n/2 cuts for even n and (n+1)/2 cuts for odd n.Comment: 31 pages, 1 figure; the expression of the superpotential has been corrected and the calculation of the coupling constants of the low-energy theory has been adde

RG Flow Irreversibility, C-Theorem and Topological Nature of 4D N=2 SYM

We determine the exact beta function and a RG flow Lyapunov function for N=2 SYM with gauge group SU(n). It turns out that the classical discriminants of the Seiberg-Witten curves determine the RG potential. The radial irreversibility of the RG flow in the SU(2) case and the non-perturbative identity relating the $u$-modulus and the superconformal anomaly, indicate the existence of a four dimensional analogue of the c-theorem for N=2 SYM which we formulate for the full SU(n) theory. Our investigation provides further evidence of the essentially topological nature of the theory.Comment: 9 pages, LaTeX file. Discussion on WDVV and integrability. References added. Version published in PR

Matrix-model description of N=2 gauge theories with non-hyperelliptic Seiberg-Witten curves

Using matrix-model methods we study three different N=2 models: U(N) x U(N) with matter in the bifundamental representation, U(N) with matter in the symmetric representation, and U(N) with matter in the antisymmetric representation. We find that the (singular) cubic Seiberg-Witten curves (and associated Seiberg-Witten differentials) implied by the matrix models, although of a different form from the ones previously proposed using M-theory, can be transformed into the latter and are thus physically equivalent. We also calculate the one-instanton corrections to the gauge-coupling matrix using the perturbative expansion of the matrix model. For the U(N) theories with symmetric or antisymmetric matter we use the modified matrix-model prescription for the gauge-coupling matrix discussed in ref. [hep-th/0303268]. Moreover, in the matrix model for the U(N) theory with antisymmetric matter, one is required to expand around a different vacuum than one would naively have anticipated. With these modifications of the matrix-model prescription, the results of this paper are in complete agreement with those of Seiberg-Witten theory obtained using M-theory methods.Comment: 40 pages; v2: reference added; v3: minor typo fixed; v4: clarifying footnote added; published versio

Abelian gerbes as a gauge theory of quantum mechanics on phase space

We construct a U(1) gerbe with a connection over a finite-dimensional, classical phase space P. The connection is given by a triple of forms A,B,H: a potential 1-form A, a Neveu-Schwarz potential 2-form B, and a field-strength 3-form H=dB. All three of them are defined exclusively in terms of elements already present in P, the only external input being Planck's constant h. U(1) gauge transformations acting on the triple A,B,H are also defined, parametrised either by a 0-form or by a 1-form. While H remains gauge invariant in all cases, quantumness vs. classicality appears as a choice of 0-form gauge for the 1-form A. The fact that [H]/2i\pi is an integral class in de Rham cohomology is related with the discretisation of symplectic area on P. This is an equivalent, coordinate-free reexpression of Heisenberg's uncertainty principle. A choice of 1-form gauge for the 2-form B relates our construction with generalised complex structures on classical phase space. Altogether this allows one to interpret the quantum mechanics corresponding to P as an Abelian gauge theory.Comment: 18 pages, 1 figure available from the authors upon reques

Instanton Calculus and SUSY Gauge Theories on ALE Manifolds

We study instanton effects along the Coulomb branch of an N=2 supersymmetric Yang-Mills theory with gauge group SU(2) on Asymptotically Locally Euclidean (ALE) spaces. We focus our attention on an Eguchi-Hanson gravitational background and on gauge field configurations of lowest Chern class.Comment: 15 pages, LaTeX file. Extended version to be published in Physical Review