Equivariant Cohomotopy implies orientifold tadpole cancellation

Hisham Sati; Urs Schreiber

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Equivariant Cohomotopy implies orientifold tadpole cancellation

Hisham Sati, Urs Schreiber

High Energy Physics - Theory · Submitted 26 Sep 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Here we further expand this list by proving that brane charge quantization in unstable equivariant Cohomotopy implies the anomaly cancellation conditions for M-branes and D-branes on flat orbi-orientifolds.

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There are fundamental open problems in the precise global nature of RR-field tadpole cancellation conditions in string theory. Moreover, the non-perturbative lift as M5/MO5-anomaly cancellation in M-theory had been based on indirect plausibility arguments,lacking a microscopic underpinning in M-brane charge quantization. We provide a framework for answering these questions, crucial not only for mathematical consistency but also for phenomenological accuracy of string theory, by formulating the M-theory C-field on flat M-orientifolds in the generalized cohomology theory called Equivariant Cohomotopy. This builds on our previous results for smooth but curved spacetimes, showing in that setting that charge quantization in twisted Cohomotopy rigorously implies a list of expected anomaly cancellation conditions. Here we further expand this list by proving that brane charge quantization in unstable equivariant Cohomotopy implies the anomaly cancellation conditions for M-branes and D-branes on flat orbi-orientifolds. For this we (a) use an unstable refinement of the equivariant Hopf-tom Dieck theorem to derive local/twisted tadpole cancellation, and (b) the lift to super-differential cohomology to establish global/untwisted tadpole cancellation. Throughout, we use (c) the unstable Pontrjagin-Thom theorem to identify the brane/O-plane configurations encoded in equivariant Cohomotopy and (d) the Boardman homomorphism to equivariant K-theory to identify Chan-Paton representations of D-brane charge. We find that unstable equivariant Cohomotopy, but not its image in K-theory, distinguishes D-brane charge from the finite set of types of O-plane charges.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	There are fundamental open problems in the precise global nature of RR-field tadpole cancellation conditions in string theory.	gap 0.53	0.04	0.02	asserted 1.00
s1	Moreover, the non-perturbative lift as M5/MO5-anomaly cancellation in M-theory had been based on indirect plausibility arguments,lacking a microscopic underpinning in M-brane charge quantization.	gap 0.93	0.04	0.02	asserted 0.99
s2	We provide a framework for answering these questions, crucial not only for mathematical consistency but also for phenomenological accuracy of string theory, by formulating the M-theory C-field on flat M-orientifolds in the generalized cohomology theory called Equivariant Cohomotopy.	contribution 0.98	0.93	0.14	asserted 0.99
s3	This builds on our previous results for smooth but curved spacetimes, showing in that setting that charge quantization in twisted Cohomotopy rigorously implies a list of expected anomaly cancellation conditions.	background 0.89	0.85	0.06	asserted 1.00
s4	Here we further expand this list by proving that brane charge quantization in unstable equivariant Cohomotopy implies the anomaly cancellation conditions for M-branes and D-branes on flat orbi-orientifolds.	result 0.72	0.95	0.84	asserted 1.00
s5	For this we (a) use an unstable refinement of the equivariant Hopf-tom Dieck theorem to derive local/twisted tadpole cancellation, and (b) the lift to super-differential cohomology to establish global/untwisted tadpole cancellation.	method 0.99	0.94	0.15	asserted 0.99
s6	Throughout, we use (c) the unstable Pontrjagin-Thom theorem to identify the brane/O-plane configurations encoded in equivariant Cohomotopy and (d) the Boardman homomorphism to equivariant K-theory to identify Chan-Paton representations of D-brane charge.	method 0.99	0.88	0.05	asserted 0.99
s7	We find that unstable equivariant Cohomotopy, but not its image in K-theory, distinguishes D-brane charge from the finite set of types of O-plane charges.	result 0.98	0.96	0.26	asserted 1.00

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Original abstract

There are fundamental open problems in the precise global nature of RR-field tadpole cancellation conditions in string theory. Moreover, the non-perturbative lift as M5/MO5-anomaly cancellation in M-theory had been based on indirect plausibility arguments,lacking a microscopic underpinning in M-brane charge quantization. We provide a framework for answering these questions, crucial not only for mathematical consistency but also for phenomenological accuracy of string theory, by formulating the M-theory C-field on flat M-orientifolds in the generalized cohomology theory called Equivariant Cohomotopy. This builds on our previous results for smooth but curved spacetimes, showing in that setting that charge quantization in twisted Cohomotopy rigorously implies a list of expected anomaly cancellation conditions. Here we further expand this list by proving that brane charge quantization in unstable equivariant Cohomotopy implies the anomaly cancellation conditions for M-branes and D-branes on flat orbi-orientifolds. For this we (a) use an unstable refinement of the equivariant Hopf-tom Dieck theorem to derive local/twisted tadpole cancellation, and (b) the lift to super-differential cohomology to establish global/untwisted tadpole cancellation. Throughout, we use (c) the unstable Pontrjagin-Thom theorem to identify the brane/O-plane configurations encoded in equivariant Cohomotopy and (d) the Boardman homomorphism to equivariant K-theory to identify Chan-Paton representations of D-brane charge. We find that unstable equivariant Cohomotopy, but not its image in K-theory, distinguishes D-brane charge from the finite set of types of O-plane charges.

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30 arXiv papers cite this in 53 sentences

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Papers citing it · 41 sentences without a confident label

Cites it as support 6 papers

This motivates, following [ref] , Hypothesis H: Charge quantization in M-theory happens in J-twisted Cohomotopy theory [ref] [ref] [ref] [ref] [ref] [ref] .

Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation · Introduction and overview · 0.97
The hypothesis has since been checked to rigorously imply a fair number of phenomena expected to be characteristic of M-theory [ref] [ref] [ref] [ref] [ref] [ref] [ref] [ref] (exposition in [ref] ):

Twisted Cohomotopy implies twisted String structure on M5-branes · Introduction · 0.96
Remarkably, there is an equivalence between Cohomotopy of spacetime/worldvolumes and Cobordism classes of submanifolds behaving like solitonic branes carrying the corresponding Cohomotopy charge [ref] [ref] :

Engineering of Anyons on M5-Probes via Flux Quantization · Cohomotopy Charge of Solitons · 0.97

Cites it as background 6 papers

As explained in [ref] , we have the following picture:

Lift of fractional D-brane charge to equivariant Cohomotopy theory · Fractional brane charge quantization in M-theory · 0.81
Some implications of such C-field flux quantization are surveyed in [ref] ; for more see [ref] [ref] [ref] [ref] .

Flux Quantization on 11-dimensional Superspace · Duality-Symmetry on Super-Spacetime · 0.87
Towards an understanding of the elusive mathematical formulation of M-theory [ref] [ref] [ref] the following assumption has been proposed [ref] [ref] [ref] [ref] [ref] [ref] :

Twisted Cohomotopy implies twisted String structure on M5-branes · Introduction · 0.99

What it cites · 131 sentences without a confident label

Builds on 1 paper

The argument in [ref] , together with our discussion above, suggests that the analogous statement for $O4^0$-planes is this:

On the M-theory Interpretation of Orientifold Planes · Heterotic M-theory on ADE-orbifolds · 0.96

Cites as background 47 papers

The key class of examples of interest here are finite subgroups (see, e.g., [ref] ) [equation] of the multiplicative group of unit norm elements $q ∈S(ℍ)$ in the vector space $ℍ ≃__ℝ ℝ^4$ of quaternions, and their defining 4-dimensional linear representation on this space (by left quaternion multiplication), which we denote by [equation] All of these, except the cyclic groups of odd order, contain the subgroup [equation] generated by the quaternion $-1 ∈ℍ$.

Lift of fractional D-brane charge to equivariant Cohomotopy theory · Equivariant Cohomotopy and tadpole cancellation · 0.84
The relevant morphisms between $G$-spaces are continuous functions between the underlying spaces that are $G$-equivariant: [equation] This $G$-equivariance implies that $H$-fixed points are sent to $H$-fixed points, for every subgroup $H ⊂G$, hence that every $G$-equivariant continuous function [ref] induces a system of plain continous functions $f^H := f_|X^H$ between $H$-fixed point spaces [ref], which are each equivariant with respect to the residual $W_G(H)$-action [ref] and compatible with each other with respect to inclusions $H_i ⊂H_j$ of subgroups: [equation] We will refer to the component $f^H$ here as the Elmendorf stage labeled by $H$ [ref] [ref] .

Real ADE-equivariant (co)homotopy and Super M-branes · Equivariant Cohomotopy and tadpole cancellation · 0.89
As a consequence of the above, one may consider the union of all unstable equivariant Cohomotopy sets of representation spheres in all compatible degrees, with respect to the identifications along the equivariant suspension maps [ref] (the colimit of this system): [equation] Since the resulting union/colimit is, by construction, stable under taking further such suspensions, this is called the stable equivariant Cohomotopy in degree 0 ( [ref] , see [ref] ) also called the 0th stable $G$-equivariant homotopy group of spheres or the $G$-equivariant stable 0-stem or similar (see [ref] [ref] ).

The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups · Stable equivariant Hopf degree · 0.95

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