We draw attention to a class of generalized global symmetries, which we call "Chern-Weil global symmetries," that arise ubiquitously in gauge theories. The Noether currents of these Chern-Weil global symmetries are given by wedge products of gauge field strengths, such as $F_2 \wedge H_3$ and $\text{tr}(F_2^2)$, and their conservation follows from Bianchi identities. As a result, they are not easy to break. However, it is widely believed that exact global symmetries are not allowed in a consistent theory of quantum gravity. As a result, any Chern-Weil global symmetry in a low-energy effective field theory must be either broken or gauged when the theory is coupled to gravity. In this paper, we explore the processes by which Chern-Weil symmetries may be broken or gauged in effective field theory and string theory. We will see that many familiar phenomena in string theory, such as axions, Chern-Simons terms, worldvolume degrees of freedom, and branes ending on or dissolving in other branes, can be interpreted as consequences of the absence of Chern-Weil symmetries in quantum gravity, suggesting that they might be general features of quantum gravity. We further discuss implications of breaking and gauging Chern-Weil symmetries for particle phenomenology and for boundary CFTs of AdS bulk theories. Chern-Weil global symmetries thus offer a unified framework for understanding many familiar aspects of quantum field theory and quantum gravity.
In this paper, we proved that the Weil-Petersson volume of Calabi-Yau moduli is a rational number. We also proved that the integrations of the invariants of the Ricci curvature of the Weil-Petersson metric with respect to the Weil-Petersson volume form are all rational numbers.
We prove that Chern-Weil forms are the only natural differential forms associated to a connection on a principal G-bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof. Other arguments come from classical invariant theory. We identify the Weil algebra as the de Rham complex of a specific simplicial sheaf, and similarly give a new interpretation of the Weil model in equivariant de Rham theory. There is an appendix proving a general theorem about set-theoretic transformations of polynomial functors. This paper is dedicated to …
A strong from of invariance under a group G is manifested in a family over the classifying space BG. We advocate a differential-geometric avatar of BG when G is a Lie group. Applied to G-equivariant connections on smooth principal or vector bundles, the equivariance-->families principle converts the G-equivariant extensions of curvature and Chern-Weil forms to the standard nonequivariant versions. An application of this technique yields the moment map of the determinant line of a G-equivariant Dirac operator, which in turn sheds light on some anomaly formulas in quantum field theory.
Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. Kotov and Strobl have observed that this naturally generalizes from Lie algebras to dg-manifolds and dg-bundles and that the Chern-Simons action functional associated this way to an $n$-symplectic manifold is the action functional of the AKSZ $σ$-model whose target space is the given $n$-symplectic manifold (examples of this are the Poisson sigma-model or the Courant sigma-model, including…
In this paper, we proved the Gauss-Bonnet-Chern theorem on moduli space of polarized Kahler manifolds. Using our results, we proved the rationality of the Chern-Weil forms (with respect to the Weil-Petersson metric) on CY moduli. As an application in physics, by the Ashok-Douglas theory, counting the number of flux compactifications of the type IIb string on a Calabi-Yau threefold is related to the integrations of various Chern-Weil forms. We proved that all these integrals are finite (and also rational).
DiSIT, Università del Piemonte Orientale（皮埃蒙特东方大学 DiSIT）；INFN, Sezione di Torino（都灵 INFN 分部）；Dipartimento di Fisica, Università degli Studi di Milano–Bicocca（米兰-比可卡大学物理系）；INFN, Sezione di Milano–Bicocca（米兰-比可卡 INFN 分部）
We prove that the universal Teichmuller space T(1) carries a new structure of a complex Hilbert manifold. We show that the connected component of the identity of T(1), the Hilbert submanifold T_{0}(1), is a topological group. We define a Weil-Petersson metric on T(1) by Hilbert space inner products on tangent spaces, compute its Riemann curvature tensor, and show that T(1) is a Kahler-Einstein manifold with negative Ricci and sectional curvatures. We introduce and compute Mumford-Miller-Morita characteristic forms for the vertical tangent bundle of the universal Teichmuller curve fibration ove…
In this paper we show how to translate into tensorial language the Chern-Weil theorem for the Lorentz symmetry, which equates the difference of the Euler densities of two manifolds to the exterior derivative of a transgression form. For doing so we need to introduce an auxiliary, hybrid, manifold whose geometry we construct explicitely. This allows us to find the vector density, constructed out of spacetime quantities only, whose divergence is the exterior derivative of the transgression form. As a consequence we can show how the Einstein-Hilbert, Gauss-Bonnet and, in general, the Euler scalar…
Using the well-known Chern-Weil formula and its generalization, we systematically construct the Chern-Simons forms and their generalization induced by torsion as well as the Nieh-Yan (N-Y) forms. We also give an argument on the vanishing of integration of N-Y form on any compact manifold without boundary. A systematic construction of N-Y forms in D=4n dimension is also given.
We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive structure (topological, smooth, synthetic differential, supergeometric, etc.) and equipped with connections, hence higher gauge fields. We discuss various models of the axioms and applications to fundamental notions and constructions in quantum field theory and string theory. In partic…
A direct relation between two types of topological field theories, Chern-Simons theory and BF theory, is presented by using ``Generalized Differential Calculus'', which extends an ordinary p-form to an ordered pair of p and (p+1)-form. We first establish the generalized Chern-Weil homomormism for generalized curvature invariant polynomials in general even dimensional manifolds, and then show that BF gauge theory can be obtained from the action which is the generalized second Chern class with gauge group G. Particularly when G is taken as SL(2,C) in four dimensions, general relativity with cosm…
In the context of the non-relativistic theories, a generalization of the Chern--Weil-theorem allows us to show that extended Chern--Simons actions for gravity in d=4 invariant under some specific non-relativistic groups lead to modified Poisson equations. In some particular cases, these modified equations have the form of the so-called MOND approach to gravity. The modifications could be understood as due to the effects of dark matter. This result could leads us to think that dark matter can be interpreted as a non-relativistic limit of dark energy.
A representation of general translation-invariant star products in the algebra of M(C) = lim_N\to \infty M_N (C) is introduced which results in the Moyal-Weyl-Wigner quantization. It provides a matrix model for general translation-invariant noncommutative quantum field theories in terms of the noncommutative calculus on differential graded algebras. Upon this machinery a cohomology theory, the so called star-cohomology, with groups H^k_*(C), is worked out which provides a cohomological framework to formulate general translation-invariant noncommutative quantum field theories based on the achie…
Lebedev Physical Institute, Moscow, Russia（俄罗斯莫斯科列别捷夫物理研究所）；Institute for Theoretical and Mathematical Physics, Lomonosov MSU, Moscow, Russia（俄罗斯莫斯科罗蒙诺索夫莫斯科大学理论与数学物理研究所）
A gauge invariant action principle, based on the idea of transgression forms, is proposed. The action extends the Chern-Simons form by the addition of a boundary term that makes the action gauge invariant (and not just quasi-invariant). Interpreting the spacetime manifold as cobordant to another one, the duplication of gauge fields in spacetime is avoided. The advantages of this approach are particularly noticeable for the gravitation theory described by a Chern-Simons lagrangian for the AdS group, in which case the action is regularized and finite for black hole geometries in diverse situatio…
Lorentz violation through a radiatively induced Chern-Simons-like term in a fermionic theory embedded in linearized quantum gravity with a Lorentz- and CPT-violating axial-vector term in the fermionic sector proportional to a constant field $b_μ$ has been recently studied. In a similar fashion as for the extended-QED model of Carroll-Field-Jackiw, we explicitly show that neither gauge invariance nor the more stringent momentum routing invariance condition on underlying Feynman diagrams fix the arbitrariness inherent to such induced term at one loop order. We present the calculation in a nonper…
We explore higher-form symmetries of M- and F-theory compactified on elliptic fibrations, determined by the topology of their asymptotic boundaries. The underlying geometric structures are shown to be equivalent to known characterizations of the gauge group topology in F-theory via Mordell--Weil torsion and string junctions. We further study dimensional reductions of the 11d Chern--Simons term in the presence of torsional boundary $G_4$-fluxes, which encode background gauge fields of center 1-form symmetries in the lower-dimensional effective gauge theory. We find contributions that can be int…
The Abelian Chern-Simons gauge theory is constructed on the three-dimensional spacetime lattice. This proposal introduces both lattice and dual lattice, and the gauge field on the dual lattice is expressed in terms of the gauge field on the original lattice. This treatment circumvents the issue of forward/backward difference, which is the common problem that many previous proposals have, and also avoids the duplication problem, which prevents people from introducing the dual lattice. The form of the lattice action is very simple, and is symmetric with respect to the three spacetime dimensions.…
