Duality as a category-theoretic concept

Corfield, David

doi:10.1016/j.shpsb.2015.07.004

Cited by 12 publications

(11 citation statements)

References 7 publications

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“…Gauge/Gravity duality. Collecting the observables and states emerging in Figure 1, we observe that the mathematical duality (as an instance of the concept described in [PT91] [Co17]) between higher (co-)observables ( 21 (1) reflects the gauge/gravity duality (e.g. [DHMB15]) between observables/states of gauge theories and gravity theories on branes found in §4 -see Figure 2: Configuration spaces of intersecting branes seen in Cohomotopy.…”

Section: Introduction and Overviewmentioning

confidence: 79%

Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams

Sati¹,

Schreiber²

2019

Preprint

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We introduce a differential refinement of Cohomotopy cohomology theory, defined on Penrose diagram spacetimes, whose cocycle spaces are unordered configuration spaces of points. First we prove that brane charge quantization in this differential 4-Cohomotopy theory implies intersecting p ⊥ (p + 2)-brane moduli given by ordered configurations of points in the transversal 3-space. Then we show that the higher (co-)observables on these brane moduli, conceived as the (co-)homology of the Cohomotopy cocycle space, are given by weight systems on horizontal chord diagrams and reflect a multitude of effects expected in the microscopic quantum theory of Dp ⊥ D(p + 2)-brane intersections: condensation to stacks of coincident branes and their Chan-Paton factors, fuzzy funnel states and M2-brane 3-algebras, AdS 3 -gravity observables and supersymmetric indices of Coulomb branches, M2/M5-brane bound states in the BMN matrix model and the Hanany-Witten rules, as well as gauge/gravity duality between all these. We discuss this in the context of the hypothesis that the M-theory C-field is charge-quantized in Cohomotopy theory.

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Section: Introduction and Overviewmentioning

confidence: 79%

Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams

Sati¹,

Schreiber²

2019

Preprint

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“…Finally, we turn to duality, which in some ways presents an interesting problem for Thesis 1 in that it runs in some sense orthogonal to the n-type hierarchy, and so may fit less well with what we've discussed so far. That it is a pervasive thematic mathematical concept is clear, and that it is superbly well captured by category theory is readily established (Corfield 2015). Moreover, concerning the vertical unity of Thesis 2, duality is certainly rooted in simple, everyday concepts.…”

Section: Spaces and Dualitymentioning

confidence: 98%

Homotopy Type Theory and the Vertical Unity of Concepts in Mathematics

Corfield¹

2017

What Is a Mathematical Concept?

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The mathematician Alexander Borovik speaks of the importance of the 'vertical unity' of mathematics. By this he means to draw our attention to the fact that many sophisticated mathematical concepts, even those introduced at the cutting-edge of research, have their roots in our most basic conceptualisations of the world. If this is so, we might expect any truly fundamental mathematical language to detect such structural commonalities. It is reasonable to suppose then that the lack of philosophical interest in such vertical unity is related to the prominence given by philosophers to languages which do not express well such relations. In this chapter, I suggest that we look beyond set theory to the newly emerging homotopy type theory, which makes plain what there is in common between very simple aspects of logic, arithmetic and geometry and much more sophisticated concepts.

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“…Historically, the concepts of category theory were introduced by algebraic topologists in order to make sense of the natural transformations which they saw in cohomology theory. Similarly, mathematically inclined string theorists could have invented category theory as the minimum mathematical context for speaking about duality …”

Section: The Idea Of Higher Structuresmentioning

confidence: 99%

“…Similarly, mathematically inclined string theorists could have invented category theory as the minimum mathematical context for speaking about duality. [35] Spc functor that assigns algebras of functions…”

Section: Duality: a Categorical Point Of Viewmentioning

confidence: 99%

Higher Structures in M‐Theory

Jurčo

Sämann

Schreiber

et al. 2019

Fortschritte der Physik

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The key open problem of string theory remains its non‐perturbative completion to M‐theory. A decisive hint to its inner workings comes from numerous appearances of higher structures in the limits of M‐theory that are already understood, such as higher degree flux fields and their dualities, or the higher algebraic structures governing closed string field theory. These are all controlled by the higher homotopy theory of derived categories, generalised cohomology theories, and L∞‐algebras. This is the introductory chapter to the proceedings of the LMS/EPSRC Durham Symposium on ‘Higher Structures in M‐Theory’: We first review higher structures as well as their motivation in string theory and beyond. Then we list the contributions in this volume, putting them into context.

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Duality as a category-theoretic concept

Cited by 12 publications

References 7 publications

Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams

Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams

Homotopy Type Theory and the Vertical Unity of Concepts in Mathematics

Higher Structures in M‐Theory

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