Quantum Geons and Noncommutative Spacetimes

Physical considerations strongly indicate that spacetime at Planck scales is noncommutative. A popular model for such a spacetime is the Moyal plane. The Poincar\`e group algebra acts on it with a Drinfel'd-twisted coproduct. But the latter is not appropriate for more complicated spacetimes such as those containing the Friedman-Sorkin (topological) geons. They have rich diffeomorphism groups and in particular mapping class groups, so that the statistics groups for N identical geons is strikingly different from the permutation group $S_N$. We generalise the Drinfel'd twist to (essentially) generic groups including to finite and discrete ones and use it to modify the commutative spacetime algebras of geons as well to noncommutative algebras. The latter support twisted actions of diffeos of geon spacetimes and associated twisted statistics. The notion of covariant fields for geons is formulated and their twisted versions are constructed from their untwisted versions. Non-associative spacetime algebras arise naturally in our analysis. Physical consequences, such as the violation of Pauli principle, seem to be the outcomes of such nonassociativity. The richness of the statistics groups of identical geons comes from the nontrivial fundamental groups of their spatial slices. As discussed long ago, extended objects like rings and D-branes also have similar rich fundamental groups. This work is recalled and its relevance to the present quantum geon context is pointed out.Comment: 41 page

Novel Studies on the \eta' Effective Lagrangian

The effective Lagrangian for \eta' incorporating the effect of the QCD \theta-angle has been developed previously. We revisit this Lagrangian and carry out its canonical quantization with particular attention to the test function spaces of constraints and the topology of the \eta'-field. In this way, we discover a new chirally symmetric coupling of this field to chiral multiplets which involves in particular fermions. This coupling violates P and T symmetries. In a subsequent paper, we will evaluate its contribution to the electric dipole moment (EDM) of fermions. Our motivation is to test whether the use of mixed states restores P and T invariance, so that EDM vanishes. This calculation will be shown to have striking new physical consequences.Comment: 14 pages, 1 figure; V2: NEW TITLE; revised version to be published in JHEP; references adde

Conformal invariance in 2-dimensional discrete field theory

A discretized massless wave equation in two dimensions, on an appropriately chosen square lattice, exactly reproduces the solutions of the corresponding continuous equations. We show that the reason for this exact solution property is the discrete analog of conformal invariance present in the model, and find more general field theories on a two-dimensional lattice that exactly solve their continuous limit equations. These theories describe in general non-linearly coupled bosonic and fermionic fields and are similar to the Wess-Zumino-Witten model.Comment: 18 pages, RevTeX, 2 figures included; revision of title and introductio

Scalar Field Theory on Fuzzy S^4

Scalar fields are studied on fuzzy $S^4$ and a solution is found for the elimination of the unwanted degrees of freedom that occur in the model. The resulting theory can be interpreted as a Kaluza-Klein reduction of CP^3 to S^4 in the fuzzy context.Comment: 16 pages, LaTe

Generalized Bohr-Sommerfeld rules for anomalies with applications to symmetry breakdown and decoupling

In the presence of anomalies, the requirement that a classical symmetry group G has a proper action on the fermion measure or in the effective Lagrangian description imposes Bohr-Sommerfeld conditions on the anomalies, and often implies that G is broken to a subgroup H as well. We show these results in this paper and apply them to QCD and SU(5). In particular, constraints on the QCD order parameter are derived, and an argument is presented which suggests that the breakdown of the chiral flavor symmetry and the emergence of some sort of generation structure in QCD may be natural

Interactions of a String Inspired Graviton Field

We continue to explore the possibility that the graviton in two dimensions is related to a quadratic differential that appears in the anomalous contribution of the gravitational effective action for chiral fermions. A higher dimensional analogue of this field might exist as well. We improve the defining action for this diffeomorphism tensor field and establish a principle for how it interacts with other fields and with point particles in any dimension. All interactions are related to the action of the diffeomorphism group. We discuss possible interpretations of this field.Comment: 12 pages, more readable, references adde

The Fuzzy Disc

We introduce a finite dimensional matrix model approximation to the algebra of functions on a disc based on noncommutative geometry. The algebra is a subalgebra of the one characterizing the noncommutative plane with a * product and depends on two parameters N and theta. It is composed of functions which decay exponentially outside a disc. In the limit in which the size of the matrices goes to infinity and the noncommutativity parameter goes to zero the disc becomes sharper. We introduce a Laplacian defined on the whole algebra and calculate its eigenvalues. We also calculate the two--points correlation function for a free massless theory (Green's function). In both cases the agreement with the exact result on the disc is very good already for relatively small matrices. This opens up the possibility for the study of field theories on the disc with nonperturbative methods. The model contains edge states, a fact studied in a similar matrix model independently introduced by Balachandran, Gupta and Kurkcuoglu.Comment: 17 pages, 8 figures, references added and correcte

Boundary degrees of freedom in fractional quantum Hall effect: Excitations on common boundary of two samples

Using the Carlip's method we have derived the boundary action for the fermion Chern-Simons theory of quantum Hall effects on a planar region with a boundary. We have computed both the bulk and edge responses of currents to the external electric field. From this we obtain the well-known anomaly relation and the boundary Hall current without introducing any ad hoc assumptions such as the chirality condition. In addition, the edge current on the common boundary of two samples is found to be proportional to the difference between Chern-Simons coupling strengths.Comment: 20 pages, uses revte

Noncommutative Chiral Anomaly and the Dirac-Ginsparg-Wilson Operator

It is shown that the local axial anomaly in $2-$dimensions emerges naturally if one postulates an underlying noncommutative fuzzy structure of spacetime . In particular the Dirac-Ginsparg-Wilson relation on ${\bf S}^2_F$ is shown to contain an edge effect which corresponds precisely to the ``fuzzy'' $U(1)_A$ axial anomaly on the fuzzy sphere . We also derive a novel gauge-covariant expansion of the quark propagator in the form $\frac{1}{{\cal D}_{AF}}=\frac{a\hat{\Gamma}^L}{2}+\frac{1}{{\cal D}_{Aa}}$ where $a=\frac{2}{2l+1}$ is the lattice spacing on ${\bf S}^2_F$, $\hat{\Gamma}^L$ is the covariant noncommutative chirality and ${\cal D}_{Aa}$ is an effective Dirac operator which has essentially the same IR spectrum as ${\cal D}_{AF}$ but differes from it on the UV modes. Most remarkably is the fact that both operators share the same limit and thus the above covariant expansion is not available in the continuum theory . The first bit in this expansion $\frac{a\hat{\Gamma}^L}{2}$ although it vanishes as it stands in the continuum limit, its contribution to the anomaly is exactly the canonical theta term. The contribution of the propagator $\frac{1}{{\cal D}_{Aa}}$ is on the other hand equal to the toplogical Chern-Simons action which in two dimensions vanishes identically .Comment: 26 pages, latex fil

Modified Partition Functions, Consistent Anomalies and Consistent Schwinger Terms

A gauge invariant partition function is defined for gauge theories which leads to the standard quantization. It is shown that the descent equations and consequently the consistent anomalies and Schwinger terms can be extracted from this modified partition function naturally.Comment: 25 page