The Anomaly Line Bundle of the Self-Dual Field Theory

In this work, we determine explicitly the anomaly line bundle of the abelian self-dual field theory over the space of metrics modulo diffeomorphisms, including its torsion part. Inspired by the work of Belov and Moore, we propose a non-covariant action principle for a pair of Euclidean self-dual fields on a generic oriented Riemannian manifold. The corresponding path integral allows one to study the global properties of the partition function over the space of metrics modulo diffeomorphisms. We show that the anomaly bundle for a pair of self-dual fields differs from the determinant bundle of the Dirac operator coupled to chiral spinors by a flat bundle that is not trivial if the underlying manifold has middle-degree cohomology, and whose holonomies are determined explicitly. We briefly sketch the relevance of this result for the computation of the global gravitational anomaly of the self-dual field theory, that will appear in another paper.     

The aetiology of sigma model anomalies

Certain nonlinear sigma models with fermions are ill-defined due to an anomaly which exhibits characteristics of both the nonabelian gauge theory anomaly and the SU(2) anomaly. The simplest way to diagnose the anomaly involves consideration of the global topology of the theory. We review the mathematical methods needed for this analysis and apply them to several supersymmetric sigma models. Some of these are found to be anomalous.     

Three-Generat ion Models from Global Gauge Anomaly-Free Theories

In this work, we show how a global gauge anomaly of a gauge group H can be computed from a local one of a larger gauge group G ⊃ H. We also show that the number f of generations is tied to the consistency of a gauge theory H with initially a Z_f gIobal gauge anomaly. We give some examples of SU(N) models, in different dimensions of spacetime, with three families.     

Determinants,torsion, and strings

We apply the results of [BF1, BF2] on determinants of Dirac operators to String Theory. For the bosonic string we recover the “holomorphic factorization” of Belavin and Knizhik. Witten's global anomaly formula is used to give sufficient conditions for anomaly cancellation in the heterotic string (for arbitrary background spacetimes). To prove the latter result we develop certain torsion invariants related to characteristic classes of vector bundles and to index theory.     

Hamiltonian Anomalies from Extended Field Theories

We develop a proposal by Freed to see anomalous field theories as relative field theories, namely field theories taking value in a field theory in one dimension higher, the anomaly field theory. We show that when the anomaly field theory is extended down to codimension 2, familiar facts about Hamiltonian anomalies can be naturally recovered, such as the fact that the anomalous symmetry group admits only a projective representation on the Hilbert space, or that the latter is really an abelian bundle gerbe over the moduli space. We include in the discussion the case of non-invertible anomaly field theories, which is relevant to six-dimensional (2, 0) superconformal theories. In this case, we show that the Hamiltonian anomaly is characterized by a degree 2 non-abelian group cohomology class, associated to the non-abelian gerbe playing the role of the state space of the anomalous theory. We construct Dai-Freed theories, governing the anomalies of chiral fermionic theories, and Wess-Zumino theories, governing the anomalies of Wess-Zumino terms and self-dual field theories, as extended field theories down to codimension 2.     

Generalized Chern–Simons modified gravity in first-order formalism

We propose a generalization of Chern–Simons (CS) modified gravity in first-order formalism. CS modified gravity action has a term that comes from the chiral anomaly which is Pontryagin invariant. First-order CS modified gravity is a torsional theory and in a space-time with torsion the chiral anomaly includes a torsional topological term called Nieh-Yan invariant. We generalize the CS modified gravity by adding the Nieh-Yan term to the action and find the effective theory. We compare the generalized theory with the first-order CS modified gravity and comment on the similarities and differences.     

Global gravitational anomalies

A general formula for global gauge and gravitational anomalies is derived. It is used to show that the anomaly free supergravity and superstring theories in ten dimensions are all free of global anomalies that might have ruined their consistency. However, it is shown that global anomalies lead to some restrictions on allowed compactifications of these theories. For example, in the case of O(32) superstring theory, it is shown that a global anomaly related to ₇(O(32)) leads to a Dirac-like quantization condition for the field strength of the antisymmetric tensor field.Related to global anomalies is the question of the number of fermion zero modes in an instanton field. It is argued that the relevant gravitational instantons are exotic spheres. It is shown that the number of fermion zero modes in an instanton field is always even in ten dimensional supergravity.Supported in part by NSF Grant No. PHY80-19754     

Conformally invariant Yang-Mills theory realizing the Virasoro-Kac-Moody algebra

We will discuss a conformally invariant Yang-Mills theory in two dimensions, which realizes the Virasoro-Kac-Moody algebra as the BRST symmetry. The theory will have the Virasoro anomaly with the central charge −26, but no anomaly for the Kac-Moody algebra.     

Chiral anomalies in higher-derivative supersymmetric 6D gauge theories

We show that the recently constructed higher-derivative 6D SYM theory involves internal chiral anomaly breaking gauge invariance. The anomaly is cancelled when adding to the theory an adjoint matter hypermultiplet.     

Gauge theory ofSU(2) Weyl fermion: Is it consistent?

Gauge theory ofSU(2) Weyl fermions was alleged by Witten to be inconsistent due to global anomaly. Evidences of inconsistency were also reported from contradictions between the anomalousU(1) symmetry and the fact that theSU(2) group is free of local anomaly. Here we show how the zero modes of Dirac operator, ignored by the authors of these arguments, play a decisive role and saveSU(2) Weyl fermions from inconsistency in each case. The symmetric chiral current, obtained by adding the Chern-Simons current to the fermionic chiral current, fails to be conserved precisely due to the contributions of zero modes to the ABJ anomaly equation. The Jacobian of the fermion measure under rigid chiralU(1) transformation is, however, guaranteed to be trivial by the Atiyah-Singer index theorem. Finally, a zero mode is the point of bifurcation of eigenvalue trajectory in the homotopy space. In its neighbourhood the hypothesis of adiabaticity made by Witten breaks down due to violent oscillations between levels, which makes his allegation of global anomaly untenable.     

A note on anomaly matching for finite density QCD

We note that the QCD phases at large finite density respect 't Hooft anomaly matching conditions. Specifically the spectrum of the light excitations possesses the correct quantum numbers required to obey global anomaly constraints. We argue that 't Hooft constraints can be used at finite density along with non perturbative methods to help selecting the correct phase.     

Finite-mode regularization of the fermion functional integral (II)

Using the finite-mode regularization introduced in a previous paper, we define the functional integral for a theory of Weyl fermions. We check this definition by making sure the resulting triangle anomaly satisfies the Wess-Zumino consistency conditions. We compare our result with others found in the literature. We apply the finite-mode regularization to compute the axial anomaly in any space-time dimension and to find the explicit expression of anomalous currents in terms of the gauge fields. We illustrate the phenomenon of the infrared compensation of the chiral anomaly.     

Induced quantum curvature and three-dimensional gauge theories

The effects of quantum holonomy in three-dimensional gauge theories with massless fermions is examined and different definitions of the fermion determinant are discussed. The source of a global gauge and parity anomaly is identified in Schrödinger picture quantization as an induced holonomy that arises from the fermionic sector of the theory. In certain fermion representations this holonomy leads to a global obstruction to imposing either gauge or parity invariance through the implementation of Gauss' law constraint. However, such obstructions can be removed by exploiting renormalization ambiguities inherent in the definition of composite operators.     

Lattice dynamics of molybdenum at high pressure

We have determined the lattice dynamics of molybdenum at high pressure to 37 GPa using high-resolution inelastic x-ray scattering. Over the investigated pressure range, we find a significant decrease in the H-point phonon anomaly. We also present calculations based on density functional theory that accurately predict this pressure dependence. Based on these results, we infer that the likely explanation for the H-point anomaly in molybdenum is strong electron-phonon coupling, which decreases upon compression due to the shift of the Fermi level with respect to the relevant electronic bands.     

The anomaly in the Slavnov identity for N = 1 supersymmetric Yang-Mills theories

We solve the consistency conditions of the BRS symmetry in a general N = 1 sypersymmetric Yang-Mills theory with semi-simple gauge group. As a result we find uniquely the supersymmetric extension of the chiral anomaly. Its coefficient is calculated in one loop and does not, in general, vanish. This corrects our earlier statement on the absence of this anomaly.     

The vanishing viscosity limit for a dyadic model

A dyadic shell model for the Navier-Stokes equations is studied in the context of turbulence. The model is an infinite nonlinearly coupled system of ODEs. It is proved that the unique fixed point is a global attractor, which converges to the global attractor of the inviscid system as viscosity goes to zero. This implies that the average dissipation rate for the viscous system converges to the anomalous dissipation rate for the inviscid system (which is positive) as viscosity goes to zero. This phenomenon is called the dissipation anomaly predicted by Kolmogorov’s theory for the actual Navier-Stokes equations.     

Gravitational anomaly cancellation type I superstring theory

By explicit calculations we show that the gravitational anomaly of type I superstring theory vanishes at the string level. There are contributions from four topologically different diagrams to the anomaly: annulus, Möbius strip, torus, and Klein bottle. We explicitly show how the non-trivial cancellation occurs between the open (annulus and Möbius strip) and closed (Klein bottle) sectors. The anomaly of the torus diagram has the same form of type II superstring theory and vanishes because of the modular invariance.     

Peierls-Fröhlich instability and Kohn anomaly

A mathematical basis is given to the Peierls-Fröhlich instability and the Kohn anomaly. The techniques and ideas are based on the recently developed mathematical theory of quantum fluctuations and response theory. We prove that there exists a unique resonant one-mode interaction between electrons and phonons which is responsible for the Peierls-Fröhlich instability and the phase transition in the Mattis-Langer model. We prove also that the softening of this phonon mode at the critical temperature (Kohn anomaly) is a consequence of the critical slowing down of the dynamics of the lattice distortion fluctuations. It is the result of the linear dependence of two fluctuation operators corresponding to the frozen charge density wave and the distortion order parameter.     

Trace Extensions,Determinant Bundles,and Gauge Group Cocycles

We study the geometry of determinant line bundles associated with Dirac operators on compact odd-dimensional manifolds. Physically, these arise as (local) vacuum line bundles in quantum gauge theory. We give a simplified derivation of the commutator anomaly formula using a construction based on noncyclic trace extensions and associated nonmultiplicative renormalized determinants.     

Chiral U(1) anomaly in D=4 super Yang-Mills theory coupled to supergravity

We evaluate the chiral U(1) anomaly in D=4 supersymmetric Yang-Mills theory coupled to supergravity. We consider not only the minimal coupling between the gauge fields and fermions but also the interaction term which mixes the gravitino and the gaugino. We show that the mixing interaction gives new contributions to the anomaly.