Closed Bosonic String Partition Function in Time Independent Exact PP-Wave Background

The modular invariance of the one-loop partition function of the closed bosonic string in four dimensions in the presence of certain homogeneous exact pp-wave backgrounds is studied. In the absence of an axion field the partition function is found to be modular invariant. In the presence of an axion field modular invariace is broken. This can be attributed to the light-cone gauge which breaks the symmetry in the $\sigma$-, $t$-directions. Recovery of this broken modular invariance suggests the introduction of twists in the world-sheet directions. However, one needs to go beyond the light-cone gauge to introduce such twists.Comment: 17 pages, added reference

Instantons in Four-Fermi Term Broken SUSY with General Potential

It is shown how to solve the Euclidean equations of motion of a point particle in a general potential and in the presence of a four-Fermi term. The classical action in this theory depends explicitly on a set of four fermionic collective coordinates. The corrections to the classical action due to the presence of fermions are of topological nature in the sense that they depend only on the values of the fields at the boundary points $\tau \to \pm \infty$. As an application, the Sine-Gordon model with a four-Fermi term is solved explicitly and the corrections to the classical action are computed.Comment: 8 page

The weakly coupled fractional one-dimensional Schr\"{o}dinger operator with index 1<α≤2\bf 1<\alpha \leq 2

We study fundamental properties of the fractional, one-dimensional Weyl operator $\hat{\mathcal{P}}^{\alpha}$ densely defined on the Hilbert space $\mathcal{H}=L^2({\mathbb R},dx)$ and determine the asymptotic behaviour of both the free Green's function and its variation with respect to energy for bound states. In the sequel we specify the Birman-Schwinger representation for the Schr\"{o}dinger operator $K_{\alpha}\hat{\mathcal{P}}^{\alpha}-g|\hat{V}|$ and extract the finite-rank portion which is essential for the asymptotic expansion of the ground state. Finally, we determine necessary and sufficient conditions for there to be a bound state for small coupling constant $g$.Comment: 16 pages, 1 figur

Locally Weyl invariant massless bosonic and fermionic spin-1/2 action in the (Wn(4),g)\bf (W_{n(4)},g) and (U4,g)\bf (U_{4},g) space-times

We search for a real bosonic and fermionic action in four dimensions which both remain invariant under local Weyl transformations in the presence of non-metricity and contortion tensor. In the presence of the non-metricity tensor the investigation is extended to Weyl $(W_n, g)$ space-time while when the torsion is encountered we are restricted to the Riemann-Cartan $(U_4, g)$ space-time. Our results hold for a subgroup of the Weyl-Cartan $(Y_4, g)$ space-time and we also calculate extra contributions to the conformal gravity.Comment: 16 page