REVIEW
Holonomy on D-Branes
High Energy Physics - Theory · Submitted 24 Apr 2002 · 25 references
Pith's verdict: A careful, honest bundle-gerbe construction whose Case 3 proposal has a well-definedness gap the paper does not address.
T0 · AI review
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Referee report
The paper
The paper claims
The paper constructs anomaly-free worldsheet actions for D-branes by using Deligne cohomology and bundle-gerbe holonomy, with a proposed extension to non-torsion B-fields.
The reading
What carries the argument
The load-bearing machinery is the identification of a Deligne two-class with a differential character: a holonomy map on closed surfaces plus a curvature three-form. Transgressing this class to loop space turns it into a line bundle $L_�� �� L(X)$ with connection, and a worldsheet action is a section of a tensor product of such line bundles that transforms as a scalar. Bundle gerbes give a geometric realisation of the Deligne class; bundle gerbe modules and their connections supply the extra sections needed when the $B$-field class does not match the Pfaffian's Stiefel-Whitney class. The trace of the holonomy of a module connection is the specific section that does the cancelling, and in the non-torsion case the trace-class difference of two such holonomies is the proposed replacement.
Extended reading notes
Core claim
The paper's central claim is that anomaly cancellation for D-branes can be organised around the holonomy of a Deligne two-class, and that this organisation extends to non-torsion $B$-fields. For a bundle gerbe with Deligne class $��$ and a bundle gerbe module with connection $A$, the trace of the holonomy of $A$ is a section $��$ of the line bundle $L_�� �� L(Q)$ over loop space. In the torsion case this section can be tensored with the Pfaffian section and a trivialising section to produce a function on $��_Q(M)$. In the non-torsion case the authors replace $��$ by $��$ for two trace-class module connections and assert that the resulting tensor product (2.8) is a well-defined function, hence an anomaly-free worldsheet action. The authors describe this last step as the paper's main result and as a proposal for a case not previously treated.
Load-bearing premise
The non-torsion construction assumes that the infinite-dimensional bundle gerbe module admits connections whose holonomy is a unitary differing from the identity by a trace-class operator; the paper fixes such connections rather than constructing them.
Editorial extensions
If this is right
- For torsion $B$-fields, worldsheet actions can be written without choosing an open cover, with the remaining dependence on choices isolated in gauge transformations and trivialisations.
- The Freed-Witten torsion case is recovered as a special case in which the Deligne class alone supplies the needed trivialising section.
- Kapustin's Azumaya-algebra construction is related to bundle gerbe modules, and the associated groupoid $C^*$-algebra has the same Dixmier-Douady class.
- For non-torsion $B$-fields, if the required trace-class module connections exist, the tensor product (2.8) is an anomaly-free worldsheet action.
- Twisted K-theory can be realised by bundle gerbe modules without introducing operator algebras.
Reading between the lines
- The proposal suggests a concrete existence problem: for a given non-torsion bundle gerbe over $Q$, decide whether it admits a module connection whose holonomy lies in the trace-class unitary group; a negative answer would leave Case 3 without an example.
- If the construction works, the same holonomy section could supply a regularisation-independent phase for worldsheet boundaries with several components, although the paper does not develop this extension.
- The $C^*$-algebra comparison points to a possible non-torsion analogue using continuous-trace algebras with infinite-dimensional fibres, but the paper explicitly does not attempt this.
Original abstract
This paper shows how to construct anomaly free world sheet actions in string theory with $D$-branes. Our method is to use Deligne cohomology and bundle gerbe theory to define geometric objects which are naturally associated to $D$-branes and connections on them. The holonomy of these connections can be used to cancel global anomalies in the world sheet action.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
This paper shows how to construct anomaly free world sheet actions in string theory with $D$-branes. Our method is to use Deligne cohomology and bundle gerbe theory to define geometric objects which are naturally associated to $D$-branes and connections on them. The holonomy of these connections can be used to cancel global anomalies in the world sheet action.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper shows how to construct anomaly free world sheet actions in string theory with $D$-branes. | contribution 0.89 | 0.93 | 0.84 | asserted 1.00 |
| s1 | Our method is to use Deligne cohomology and bundle gerbe theory to define geometric objects which are naturally associated to $D$-branes and connections on them. | method 0.99 | 0.83 | 0.06 | asserted 1.00 |
| s2 | The holonomy of these connections can be used to cancel global anomalies in the world sheet action. | result 0.61 | 0.70 | 0.35 | asserted 0.66 |
What was examined, and what was not
- arXiv listing abstract v3
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
How the literature uses it
32 arXiv papers cite this in 64 sentences
Papers citing it · 45 sentences without a confident label
Builds on it 2 papers
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This notion merges the holonomy for unoriented closed surfaces from [ref] with that of the holonomy for oriented surfaces with boundary from [ref] .
Bundle Gerbes for Orientifold Sigma Models · Holonomy for unoriented Surfaces · 0.87
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Our construction extends and conceptually simplifies well-known constructions of [ref] , while also providing a careful treatment of corners.
Smooth Functorial Field Theories from B-Fields and D-Branes · Introduction · 0.89
Uses its method or tool 1 paper
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In the final section, we present Carey, Johnson, and Murray's formalism [ref] for computing the holonomy of a 2-connection, which we will use in our quantization procedure for 2-plectic manifolds in Chapter [ref].
Higher Symplectic Geometry · The string Lie 2-algebra · 1.00
Compares with it 3 papers
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We can thus compare it with the holonomy of the abelian bundle gerbe $𝒢$, which has been defined in [ref] , see also [ref] .
Connections on non-abelian Gerbes and their Holonomy · Holonomy around Surfaces · 0.99
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In the case of only closed brane boundary it coincides with the holonomy on D-branes of [ref] .
Smooth Functorial Field Theories from B-Fields and D-Branes · Definition of the surface amplitude · 0.96
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The main differences are the introduction of an additional surjective submersion in [ref] , on which the vector bundle $E$ is defined, and the inclusion of infinite-rank Hilbert bundles in [ref] .
The 2-Hilbert Space of a Prequantum Bundle Gerbe · 1-morphisms · 0.99
Cites it as background 9 papers
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Our references for this section are [ref] , [ref] and [ref] .
Geometric and topological aspects of Type IIB D-branes · Cancelling the Freed-Witten anomaly · 0.89
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The non-torsion part – i.e. general classes in $^2(X,^×)$ – is elegantly realized by bundle gerbes [ref] , objects that naturally appear in conformal field theory and higher gauge theory [ref] .
Real Twistings are 2-Line Bundles · Introduction · 0.89
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We recall that the category $(𝒢,ℋ)$ of isomorphisms between two fixed bundle gerbes $𝒢$ and $ℋ$ over an arbitrary smooth manifold $X$ is a torsor over the monoidal category $X$ of principal $$-bundles over $X$, under an action functor [equation] see [ref] .
String geometry vs. spin geometry on loop spaces · From string structures to string classes · 0.94
Discussion
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