Marco Matone

Instantons and recursion relations in N = 2 SUSY gauge theory

Abstract We find the transformation properties of the prepotential F of N = 2 SUSY gauge theory with gauge group SU(2). Next we show that g(a) = πi (F(a) − 1 2 a∂ a F(a)) is modular invariant. We also show that u = g(a), so that F(〈Φ〉) = 1 πi 〈 tr Φ 2 〉 + 1 2 〈Φ〉 〈Φ D 〉 . This implies that g(a) satisfies the non-linear differential equation (1 − g 2 ) g″ + 1 4 ag′ 3 = 0 . We use this equation to derive recursion relations for the instanton contributions. These results can be extended to more general cases.

The superembedding origin of the Berkovits pure spinor covariant quantization of superstrings

Abstract We show that the pure spinor formalism proposed by Berkovits to covariantly quantize superstrings is a gauge fixed, twisted version of the complexified n=2 superembedding formulation of the superstring. This provides the Berkovits approach with a geometrical superdiffeomorphism invariant ground. As a consequence, the absence of the worldsheet (super)diffeomorphism ghosts in the pure spinor quantization prescription and the nature of the Berkovits BRST charge and antighost are clarified. Since superembedding is classically equivalent to the Green–Schwarz formulation, we thus also relate the latter to the pure spinor construction.

The Equivalence postulate of quantum mechanics

The removal of the peculiar degeneration arising in the classical concepts of rest frame and time parametrization is at the heart of the recently formulated equivalence principle (EP). The latter, stating that all physical systems can be connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the quantum stationary HJ equation (QSHJE). This is a third order nonlinear differential equation which provides a trajectory representation of quantum mechanics (QM). The trajectories depend on the Planck length through hidden variables which arise as initial conditions. The formulation has manifest p-q duality, a consequence of the involutive nature of the Legendre transformation and of its recently observed relation with second order linear differential equations. This reflects in an intrinsic ψD-ψ duality between linearly independent solutions of the Schrodinger equation. Unlike Bohms theory, there is a nontrivial action even for bound states and no pilot waveguide is present. A basic property of the formulation is that no use of any axiomatic interpretation of the wave function is made. For example, tunneling is a direct consequence of the quantum potential which differs from the Bohmian one and plays the role of particles self-energy. Furthermore, the QSHJE is defined only if the ratio ψD/ψ is a local homeomorphism of the extended real line into itself. This is an important feature as the L2(ℝ) condition, which in the Copenhagen formulation is a consequence of the axiomatic interpretation of the wave function, directly follows as a basic theorem which only uses the geometrical gluing conditions of ψD/ψ at q=±∞ as implied by the EP. As a result, the EP itself implies a dynamical equation that does not require any further assumption and reproduces both tunneling and energy quantization. Several features of the formulation show how the Copenhagen interpretation hides the underlying nature of QM. Finally, the nonstationary higher dimensional quantum HJ equation and the relativistic extension are derived.

Equivalence principle, higher-dimensional Möbius group and the hidden antisymmetric tensor of quantum mechanics

We show that the recently formulated equivalence principle (EP) implies a basic cocycle condition both in Euclidean and Minkowski spaces, which holds in any dimension. This condition, that in one dimension is sufficient to fix the Schwarzian equation, implies a fundamental higher-dimensional Mobius invariance which, in turn, unequivocally fixes the quantum version of the Hamilton-Jacobi equation. This also holds in the relativistic case, so that we obtain both the time-dependent Schrodinger equation and the Klein-Gordon equation in any dimension. We then show that the EP implies that masses are related by maps induced by the coordinate transformations connecting different physical systems. Furthermore, we show that the minimal coupling prescription, and therefore gauge invariance, arises quite naturally in implementing the EP. Finally, we show that there is an antisymmetric 2-tensor which underlies quantum mechanics and sheds new light on the nature of the quantum Hamilton-Jacobi equation.

Nonperturbative renormalization group equation and beta function in N=2 supersymmetric Yang-Mills theory.

We obtain the exact beta function for

Higher genus superstring amplitudes from the geometry of moduli space

N=2

Nonperturbative Relations in N=2 Supersymmetric Yang-Mills Theory and the Witten-Dijkgraaf-Verlinde-Verlinde Equation.

SUSY

Quantum mechanics from an equivalence principle

SU(2)

Seiberg–Witten duality in Dijkgraaf–Vafa theory

Yang-Mills theory and prove the nonperturbative Renormalization Group Equation