G-Frobenius manifolds

A new method is developed for the treatment of molecular interactions, and is applied to a system consisting of a hydrogen atom in a 2p state and a hydrogen molecule in the ground state. The interaction of these two species is calculated using ordinary crystal field theory and also the new method. A comparison of the results shows some of the shortcomings of the conventional crystal field theory, and provides corrections to it. The new method consists of (1) expanding all electron terms of the total Hamiltonian for the system which involve interactions between the atom and the molecule, thus transforming the interaction Hamiltonian into sums of products of one-electron operators, and (2) of using properly antisymmetrized wave functions made up of products of atom and molecule eigenfunctions. The calculations show the effect of the neglect of overlap and exchange in ordinary crystal field theory.

One of the early concerns in quantum field theory was causality — can one ensure that measurements at spacelike separation do not interfere? General issues of this type were important in the days when little was understood about the underlying dynamics of particle interactions; the hope was to place constraints on the types of allowed dynamics and interactions on the basis of cherished tenets of kinematics. There were in fact two ways to approach the problem. First, one could take the point of view that only the results of scattering experiments had physical content; then causal behavior means that the scattered wave cannot reach the detector before the incident wave strikes the target. We might describe this as global causality, and it is obeyed by tachyon-free string theory S-matrices. However there is a second, local version of causality, which asks whether local measurements commute at spacelike separation. Local causality ensures global causality in ordinary field theory, and typically one can construct global violations from local ones, so it would seem that the two are equivalent. But in quantum field theory the local question is more fundamental, and can be resolved without solving any complicated global problem such as long-distance signal propagation. Local causality rests on the universal properties of field theory, e.g. that any manifold locally looks the same, rather than their implementation in any particular spacetime background. For instance we can be sure that an interacting scalar field propagating in a Schwarzschild geometry obeys the axioms of local field theory without having to solve for the full S-matrix of the problem.

In this paper we compare the construction of equilibrium states at finite temperature for self-interacting massive scalar quantum field theories on Minkowski spacetime proposed by Fredenhagen and Lindner with results obtained in ordinary thermal field theory, by means of real time and Matsubara formalisms. In the construction of this state, even if the adiabatic limit is considered, the interaction Lagrangian is multiplied by a smooth time cut-off. In this way the interaction starts adiabatically and the correlation functions are free from divergences. The corresponding interaction Hamiltonian is a local interacting field smeared over the interval of time where the chosen cut-off is not constant. In order to cope with this smearing, the Matsubara propagator needs to be modified. We obtain an expansion of the correlation functions of the interacting equilibrium state as a sum over certain type of graphs with mixed edges, some of them correspond to modified Matsubara propagators and others to propagators of the real time formalism. An integration over the adiabatic time cut-off is present in every vertex. However, at every order in perturbation theory, the final result does not depend on the particular form of the cut-off function. The obtained graphical expansion contains in it both the real time formalism and the Matsubara formalism as particular cases. Finally, we show that a particular factorisation which is used to derive the ordinary real time formalism holds only in special cases and we present a counterexample. We conclude with the analysis of certain correlation functions and we notice that corrections to the self-energy in a $\lambda\phi^4$ at finite temperature theory are expected.

A loop expansion is implemented based on the path integral quantization of the light-cone $\phi^4$ field theory in 1+1 dimensions. The effective potential as a function of the zero-mode field $\omega$ is calculated up to two loop order and its derivative with respect to $\omega$ is used to determine the vacuum expectation value of the field $\phi$. The critical coupling constant at the spontaneous symmetry breakdown is consistent with that obtained in the ordinary instant-form field theory. The critical exponents which describe the behavior of the susceptibility and the vacuum expectation value of $\phi$ near the critical point are evaluated from the effective potential. The one loop diagrams for the connected Green's function are calculated in momentum space. The relevant equal-time correlation function is shown to be closely related.

The link between Frobenius manifolds and singularity theory is well known, with the simplest examples coming from the simple hypersurface singularities. Associated with any such manifold is a function known as the $G$-function. This plays a role in the construction of higher-genus terms in various theories. For the simple singularities the G-function is known explicitly: G=0. The next class of singularities, the unimodal hypersurface or elliptic hypersurface singularities consists of three examples, \widetilde{E}_6,\widetilde{E}_7,\widetilde{E}_8 (or equivalently P_8, X_9,J_10). Using a result of Noumi and Yamada on the flat structure on the space of versal deformations of these singularities the $G$-function is explicitly constructed for these three examples. The main property is that the function depends on only one variable, the marginal (dimensionless) deformation variable. Other examples are given based on the foldings of known Frobenius manifolds. Properties of the $G$-function under the action of the modular group is studied, and applications within the theory of integrable systems are discussed.

We construct N=1 supersymmetric (SUSY) field theory in 4+2 dimensions compatible with the theoretical framework of two-time (2T) physics and its gauge symmetries. The fields are arranged into 4+2 dimensional chiral and vector supermultiplets, and their interactions are uniquely fixed by SUSY and 2T physics gauge symmetries. In a particular gauge the 4+2 theory reduces to ordinary supersymmetric field theory in 3+1 dimensions without any Kaluza-Klein remnants, but with some additional constraints in 3+1 dimensions of interesting phenomenological relevance. This construction is another significant step in the development of 2T physics as a structure that stands above 1T physics.

Global conformal invariance in D dimensions and logarithmic correlation functions

We discuss the renormalizability of finite temperature field theory. A proof is given with inductive method that for any n-loop Feynman diagram, contributions originated from finite temperatures are always finite. Therefore a finite temperature field theory is always renormalizable provided that the corresponding ordinary field theory is renormalizable.

The well-known difficulties inherent in the negative energy Bose fields limit the unambiguous application of such fields in the ,ordinary quantum field theory. A correct incorporation of the negative energy Bose fields into the ordinary theory of the positive energy Bose fields, however, seems to give a promising method in order to solve the convergence difticulties of the quantum field theory. For examples, in Bopp's field(l) or Pauli's regulator methodj(2) it seems to be an essential point of the theory that the negative energy fields or the fields having the imaginary constants of coupling with the real fields are mixed with the positive energy fields. As stated by' Pauli, the current mixed theory of ordinary fields may stand on the realistic point of view, on the other hand his regulator method may stand on the formalistic point of view. Although the regulator method could be considered to be a generalized mixed field theory as mentioned above, it will be necessary that such fictitious fields have to be eliminated by some limiting procedures in the final results which make them to be only formal auxiliary fields and to have no correspondece to any observable one. The introduction of a new hypothesis into the fictitious fields, however, will allow us to infer that it would be possible to bring about the reconciliation of the realistic point of view and the formalistic one, and the particles being accompanied by the field would appear to be as real an existence as any other particles known in physics. Recently, the p(esent author tried to intlOduce a hypothesis into the theory of negative energy Bose fields(3,. A brief consideration was presented, but it is noy yet to give a full account on it. Now, we shall give detailed accounts of the underlying idea, which is a close analogon to Dirac's positron theory.

In this paper, we review Teleman’s work on lifting Givental’s quantization of ℒ + ( 2 ) ⁢ GL ⁢ ( H ) {\mathcal{L}^{(2)}_{+}{\rm GL}(H)} action for semisimple formal Gromov–Witten potential into cohomological field theory level. We apply this to obtain a global cohomological field theory for simple elliptic singularities. The extension of those cohomological field theories over large complex structure limit are mirror to cohomological field theories from elliptic orbifold projective lines of weight ( 3 , 3 , 3 ) (3,3,3) , ( 2 , 4 , 4 ) (2,4,4) , ( 2 , 3 , 6 ) (2,3,6) . Via mirror symmetry, we prove generating functions of Gromov–Witten cycles for those orbifolds are cycle-valued (quasi)-modular forms.

We apply the spectral curve topological recursion to Dubrovin’s universal Landau–Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.

It was proved recently that the correlation functions of a semisimple cohomological field theory satisfy the so-called local Eynard–Orantin topological recursion. We prove that in the settings of singularity theory, the local Eynard–Orantin recursion is equivalent to N copies of Virasoro constraints for the total ancestor potential. The latter follow easily from some known properties of the period integrals in singularity theory. Our approach generalizes easily to an arbitrary semisimple cohomological field theory, which yields a simple proof of the local Eynard–Orantin recursion for an arbitrary semisimple cohomological field theory.

An asymptotically flat spacetime (AFST) at spatial infinity is studied. The technique of the field formulation of general relativity developed earlier is used. The properties of the latter are similar to those of an ordinary gauge field theory in a fixed background spacetime. We give a definition of an AFST. The role of a background is played by Minkowski space. Integrals of motion are defined with the use of a stress-energy tensor of the gravitational field together with its sources and Killing vectors of the background spacetime. For each of the following AFST characteristics, namely the definition and the values of the integrals of motion and the action integral, we have found the weakest asymptotics for gauge transformations such that this characteristic does not change under the gauge transformations. These three kinds of falloff conditions are compared. The suggested approach is discussed and compared with some known results and methods.

We describe an approach to logarithmic conformal field theories as limits of sequences of ordinary conformal field theories with varying central charge c. Logarithmic behaviour arises from degeneracies in the spectrum of scaling dimensions at certain values of c. The theories we consider are all invariant under some internal symmetry group, and logarithmic behaviour occurs when the decomposition of the physical observables into irreducible operators becomes singular. Examples considered are quenched random magnets using the replica formalism, self-avoiding walks as the n → 0 limit of the O(n) model, and percolation as the limit Q → 1 of the Potts model. In these cases we identify logarithmic operators and pay particular attention to how the c → 0 paradox is resolved and how the b-parameter is evaluated. We also show how this approach gives information on logarithmic behaviour in the extended Ising model, uniform spanning trees and the O( − 2) model. Most of our results apply to general dimensionality. We also consider massive logarithmic theories and, in two dimensions, derive sum rules for the effective central charge and the b-parameter.

Operator product expansion in two-dimensional quantum gravity