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Curvature function and coarse graining
High Energy Physics - Theory · Submitted 20 Jan 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We exhibit a discrete set of functions of the connection and prove that in the abelian case their evaluation characterizes the bundle structure (up to equivalence), and constrains the connection modulo gauge up to "local details" ignored when working at a given scale.
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A classic theorem in the theory of connections on principal fiber bundles states that the evaluation of all holonomy functions gives enough information to characterize the bundle structure (among those sharing the same structure group and base manifold) and the connection up to a bundle equivalence map. This result and other important properties of holonomy functions has encouraged their use as the primary ingredient for the construction of families of quantum gauge theories. However, in these applications often the set of holonomy functions used is a discrete proper subset of the set of holonomy functions needed for the characterization theorem to hold. We show that the evaluation of a discrete set of holonomy functions does not characterize the bundle and does not constrain the connection modulo gauge appropriately. We exhibit a discrete set of functions of the connection and prove that in the abelian case their evaluation characterizes the bundle structure (up to equivalence), and constrains the connection modulo gauge up to "local details" ignored when working at a given scale. The main ingredient is the Lie algebra valued curvature function $F_S (A)$ defined below. It covers the holonomy function in the sense that $\exp{F_S (A)} = {\rm Hol}(l= \partial S, A)$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A classic theorem in the theory of connections on principal fiber bundles states that the evaluation of all holonomy functions gives enough information to characterize the bundle structure (among those sharing the same structure group and base manifold) and the connection up to a bundle equivalence map. | background 1.00 | 0.03 | 0.04 | asserted 1.00 |
| s1 | This result and other important properties of holonomy functions has encouraged their use as the primary ingredient for the construction of families of quantum gauge theories. | background 1.00 | 0.04 | 0.02 | asserted 0.89 |
| s2 | However, in these applications often the set of holonomy functions used is a discrete proper subset of the set of holonomy functions needed for the characterization theorem to hold. | background 0.58 | 0.08 | 0.06 | asserted 0.89 |
| s3 | We show that the evaluation of a discrete set of holonomy functions does not characterize the bundle and does not constrain the connection modulo gauge appropriately. | result 0.93 | 0.91 | 0.46 | asserted 1.00 |
| s4 | We exhibit a discrete set of functions of the connection and prove that in the abelian case their evaluation characterizes the bundle structure (up to equivalence), and constrains the connection modulo gauge up to "local details" ignored when working at a given scale. | result 0.75 | 0.95 | 0.57 | asserted 1.00 |
| s5 | The main ingredient is the Lie algebra valued curvature function $F_S (A)$ defined below. | method 0.56 | 0.54 | 0.04 | asserted 0.95 |
| s6 | It covers the holonomy function in the sense that $\exp{F_S (A)} = {\rm Hol}(l= \partial S, A)$. | method 0.50 | 0.70 | 0.06 | asserted 1.00 |
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Original abstract
A classic theorem in the theory of connections on principal fiber bundles states that the evaluation of all holonomy functions gives enough information to characterize the bundle structure (among those sharing the same structure group and base manifold) and the connection up to a bundle equivalence map. This result and other important properties of holonomy functions has encouraged their use as the primary ingredient for the construction of families of quantum gauge theories. However, in these applications often the set of holonomy functions used is a discrete proper subset of the set of holonomy functions needed for the characterization theorem to hold. We show that the evaluation of a discrete set of holonomy functions does not characterize the bundle and does not constrain the connection modulo gauge appropriately. We exhibit a discrete set of functions of the connection and prove that in the abelian case their evaluation characterizes the bundle structure (up to equivalence), and constrains the connection modulo gauge up to "local details" ignored when working at a given scale. The main ingredient is the Lie algebra valued curvature function $F_S (A)$ defined below. It covers the holonomy function in the sense that $\exp{F_S (A)} = {\rm Hol}(l= \partial S, A)$.
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Keeping the Ashtekar–Lewandowski (AL) representation the construction of coherent states has been explored [ref] , also coarse graining in the kinematical Hilbert space have been considered [ref] .
The continuum limit of loop quantum gravity - a framework for solving the theory · Kinematical understanding of the continuum limit · 0.91
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We would need to compare with the work presented in [ref] .
Spinfoam 2d quantum gravity and discrete bundles · Regularization · 0.96
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