Reply to the comment by D. Kreimer and E. Mielke
We respond to the comment by Kreimer et. al. about the torsional contribution to the chiral anomaly in curved spacetimes. We discuss their claims and refute its main conclusion.
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arXiv subjects
Osvaldo Chandia
Publications and source records attributed to Osvaldo Chandia.
On the N=2 Superstring BRST Operator
We write the BRST operator of the N=2 superstring as $Q= e^{-R} (\oint \frac{dz}{2πi} ~ b γ_+ γ_-)e^R$ where $b$ and $gamma_\pm$ are super-reparameterization ghosts. This provides a trivial proof of the nilpotence of this operator.
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Dynamical content of Chern-Simons Supergravity
The dynamical content of local AdS supergravity in five dimensions is discussed. The bosonic sector of the theory contains the vielbein ($e^{a}$), the spin connection ($ω^{ab}$) and internal SU(N) and U(1) gauge fields. The fermionic fields are complex Dirac spinors ($ψ^{i}$) in a vector representation of SU(N). All fields together form a connection 1-form in the superalgebra SU(2,2|N). For N=4, the symplectic matrix has maximal rank in a locally AdS background in which the dynamical degrees of freedom can be identified. The resulting efective theory have different numbers of bosonic and fermionic degrees of freedom.
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Supersymmetric Particle in a Spacetime with Torsion and the Index Theorem
The supersymmetric Lagrangian compatible with the presence of torsion in the background spacetime requires, in addition to the minimal coupling, an interaction between the spin and the torsion of the form ${1/2} N_{μνλρ}ψ^μψ^νψ^λψ^ρ$, where $N$ is the Nieh-Yan 4-form. This gives rise to a coupling between helicity ($h$) and the Nieh-Yan density of the form $hN$. The classical Lagrangian allows computing the index for the Dirac operator on a four-dimensional compact manifold with curvature and torsion using the path integral representation for the index. This calculation provides an independent check for the recent result of the chiral anomaly in spaces with torsion.
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Torsional Topological Invariants (and their relevance for real life)
The existence of topological invariants analogous to Chern/Pontryagin classes for a standard $SO(D)$ or $SU(N)$ connection, but constructed out of the torsion tensor, is discussed. These invariants exhibit many of the features of the Chern/Pontryagin invariants: they can be expressed as integrals over the manifold of local densities and take integer values on compact spaces without boundary; their spectrum is determined by the homotopy groups $π_{D-1}(SO(D))$ and $π_{D-1}(SO(D+1))$. These invariants are not solely determined by the connection bundle but depend also on the bundle of local orthonormal frames on the tangent space of the manifold. It is shown that in spacetimes with nonvanishing torsion there can occur topologically stable configurations associated with the frame bundle which are independent of the curvature. Explicit examples of topologically stable configurations carrying nonvanishing instanton number in four and eight dimensions are given, and they can be conjectured to exist in dimension $4k$. It is also shown that the chiral anomaly in a spacetime with torsion receives a contribution proportional to this instanton number and hence, chiral theories in $4k$-dimensional spacetimes with torsion are potentially anomalous.
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Topological Invariants, Instantons and Chiral Anomaly on Spaces with Torsion
In a spacetime with nonvanishing torsion there can occur topologically stable configurations associated with the frame bundle which are independent of the curvature. The relevant topological invariants are integrals of local scalar densities first discussed by Nieh and Yan (N-Y). In four dimensions, the N-Y form $N= (T^a \wedge T_a - R_{ab} \wedge e^a \wedge e^b)$ is the only closed 4-form invariant under local Lorentz rotations associated with the torsion of the manifold. The integral of $N$ over a compact D-dimensional (Euclidean) manifold is shown to be a topological invariant related to the Pontryagin classes of SO(D+1) and SO(D). An explicit example of a topologically nontrivial configuration carrying nonvanishing instanton number proportional to $\int N$ is costructed. The chiral anomaly in a four-dimensional spacetime with torsion is also shown to contain a contribution proportional to $N$, besides the usual Pontryagin density related to the spacetime curvature. The violation of chiral symmetry can thus depend on the instanton number of the tangent frame bundle of the manifold. Similar invariants can be constructed in D>4 dimensions and the existence of the corresponding nontrivial excitations is also discussed.
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