M-Theory and Mathematics

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### Context
#### Mathematics
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#### String theory
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[[!include string theory - contents]]
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#### Quantum systems
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[[!include quantum systems -- contents]]
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This page collects material related to:

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A **conference series on [[M-theory]] and [[mathematics]]**.

> These workshops bring together experts on the mathematical aspects of [[M-theory]], the main candidate for a [[theory of everything]], with implications to [[string theory]] and [[quantum field theory]] and interactions with [[geometry]] and [[algebraic topology]].

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\tableofcontents

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## M-Theory and Mathematics 2024
 {#2024}

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on January 15 - 17,  2024

at [[CQTS]] @ New York University, Abu Dhabi

* program: [[MTheoryAndMath2024-Program.pdf:file]]

* abstracts: [[MTheoryAndMath2024-Abstracts.pdf:file]]

* poster: [[MTheoryAndMath2024-Poster.pdf:file]]

{#Speakers2024} **Speakers:**

* [[Luigi Alfonsi]]

* [[Chris Blair]]

* [[Mario Garcia-Fernandez]]

* [[Grigorios Giotopoulos]]

* [[Fei Han]]

* [[Yang-Hui He]]

* [[Olaf Hohm]]

* [[Chris Hull]]

* [[Neil Lambert]]

* [[Emanuel Malek]]

* [[Ruben Minasian]]

* [[Christian Saemann]]

* [[Hisham Sati]]

* [[Urs Schreiber]]

* [[Carlos Shahbazi]]

* [[Eric Sharpe]]

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[[Luigi Alfonsi|Alfonsi]]$\,$[[Grigorios Giotopoulos|Giotop.]] [[Emanuel Malek|Malek]] [[Christian Saemann|Saemann]] [[Ruben Minasian|Minasian]] [[Neil Lambert|Lambert]] [[Chris Hull|Hull]] [[Urs Schreiber|Schreiber]] [[Yang-Hui He|He]] [[Hisham Sati|Sati]] X [[Fei Han|Han]] [[Meng-Chwan Tan|Tan]] [[Vivek Kumar Singh|Singh]] [[Carlos Shahbazi|Shahbazi]] [[David Jaz Myers|Myers]] [[Olaf Hohm|Hohm]]

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{#Talks2024} **Talks:**

* {#Lambert2024} 15 Jan 2024

  [[Neil Lambert]]:

  **Non-Relativistic M2-Branes and AdS/CFT**

  slides: [[Lambert-NonRelM2BranesAndADSCFT.pdf:file]]

  cf.: [arXiv:2401.14955](https://arxiv.org/abs/2401.14955)

  > We discuss a peculiar limit of [[M2-branes]] that leads to a non-relativistic [[ABJM model|Chern-Simons-matter theory]] with an infinite dimensional spacetime [[symmetry group]] and whose dynamics leads to [[quantum mechanics]] on a [[Hitchin moduli space]]. We also discuss the corresponding limit in the [[AdS-CFT correspondence|gravitational dual]] which is described by an eleven-dimensional Membrane-Newton-Cartan theory about a [[background field|background]] with an [[anti de Sitter spacetime|$AdS_2$]] factor.

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* {#Shahbazi2024} 15 Jan 2024

  [[Carlos Shahbazi]]:

  **The Differential Geometry and Topology of Four-
Dimensional Universal Supergravity**

  slides: [[Shahbazi-MTheoryAndMath2024.pdf:file]]

  cf.: [arXiv:2101.07236](pre-metric+electromagnetism#LazaroiuShahbazi23)

  > Using the [[sheaf cohomology|cohomology]] of the appropriate [[locally constant sheaf]] I will explain how to implement the [[Dirac charge quantization|Dirac-Schwinger-Zwanziger integrality condition]] on [[D=4 supergravity|four-dimensional]] [[classical field theory|classical]] ungauged [[supergravity]] and how to interpret it geometrically in order to obtain its [[U-duality|duality]]-covariant, [[gauge theory|gauge-theoretic]], [[differential geometry|differential-geometric]] global model. Using this construction, I will prove that [[D=4 supergravity|four-dimensional]] bosonic un[[gauged supergravity|gauged]] [[supergravity]] is completely determined by a choice of polarized Siegel bundle defined over the total space of a vertically [[Riemannian manifold|Riemannian]] [[submersion]] equipped with a complete [[Ehresmann connection]], showing that its gauge sector reduces to a polarized [[self-dual higher gauge theory|self-duality condition]] for connections on the underlying polarized Siegel bundle. Furthermore, I will explore the continuous and arithmetic [[U-duality]] groups of the theory, characterizing them through [[short exact sequences]] and realizing the latter through the [[automorphism group]] of the underlying Siegel bundle acting on its [[adjoint bundle]]. This elucidates the geometric origin of [[U-duality]] and justifies the miraculous existence of U-dualities by describing them as a [[gauge transformation]] of the appropriately defined [[principal bundle]].

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* 15 Jan 2024

  [[Mario Garcia-Fernandez]]:

  **Gauge Theory for String Algebroids**

  cf.: [arXiv:arXiv:2004.11399](string+algebroid#GarciaFernandezRubioTipler)

  > In this talk I will explain a [[moment map]] construction for [[string algebroids]], a special type of [[Courant algebroids]] which arise as [[Atiyah algebroids]] of [[principal 2-bundle|higher principal bundles]]. The zero locus of our moment map is given by the solutions of the Calabi system, a coupled system of equations which provides a unifying framework for the classical Calabi problem and the Hull-Strominger system in [[heterotic string theory]]. Our main results are concerned with the geometry of the [[moduli space]] of solutions, and assume a technical condition which is fulfilled in examples. We prove that the [[moduli space]] carries a pseudo-Kähler metric with Kähler potential given by the [[dilaton]] functional, a topological formula for the metric, and an infinitesimal Donaldson-Uhlenbeck-Yau type theorem. Based on joint work with Rubio and Tipler in [arXiv:arXiv:2004.11399](string+algebroid#GarciaFernandezRubioTipler) (to appear in JDG) and ongoing joint work with Álvarez-Cónsul and Tellez.

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* {#Giotopoulos2024} 15 Jan 2024

  [[Grigorios Giotopoulos]]:

  **Towards Non-Perturbative Lagrangian Field Theory via
the Topos of Smooth Sets**

  cf.: [arXiv:2312.16301](smooth+set#GiotopoulosSati23)

  > Any notion of [[non-perturbative field theory|non-pertubative]] ([[prequantum geometry|pre]])-[[quantization]] of [[classical field theories]], as in particular expected in [[M-theory|M-theoretic]] contexts, presupposes a [[convenient category of topological spaces|convenient category]] within which non-pertubative classical field theory may be rigorously formalised. In this talk, I will describe [[smooth sets]] as a [[category]] of [[generalized smooth spaces]], completely determined by "how they may be smoothly probed by finite dimensional [[smooth manifold|manifolds]]". Formally, this is the "[[topos]] [[category of sheaves|of sheaves]] over the [[site]] of [[Cartesian spaces]]". I will then explain how the [[variational calculus|variational algorithm]] of ([[bosonic field|bosonic]]) [[classical field theory]] and the space of [[on-shell]] [[field (physics)|fields]] naturally take place in [[smooth sets]], along with many more field theoretic concepts. Time permitting, I will indicate how the setting naturally generalizes to include the description of [[synthetic differential geometry|infinitesimal]] (pertubative) structure, [[fermionic fields]], and ([[gauge field|gauge]]) fields with [[internal symmetries]].

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* {#Alfonsi2024} 15 Jan 2024

  [[Luigi Alfonsi]]:

  **Towards Non-Perturbative BV-Theory via Derived
Geometry and the Puzzle of Quantisation**

  cf.: [arXiv:arXiv:2307.15106](BV-BRST+formalism#AlfonsiYoung23)

  > In this talk I will introduce and discuss a global geometric framework which allows one to encode a natural [[non-perturbative field theory|non-perturbative]] generalisation of classical [[BV-formalism|Batalin–Vilkovisky(BV-)theory]]. First, I will set the stage by briefly describing the current state of the art of [[perturbative quantum field theory|perturbative]] [[BV-theory]]. Then, I will introduce a concrete model of [derived differential geometry]], whose geometric objects are [[formal geometry|formal]] [[derived geometry|derived]] [[smooth stacks]] (i.e. [[stacks]] on formal [[derived smooth manifolds]]), and which is obtained by applying [Töen-Vezzosi's homotopical algebraic geometry](derived+algebraic+geometry#ToenVezzosi04) to the theory of [[derived manifolds]] of [Spivak](derived+smooth+manifold#Spivak08) and [Carchedi-Steffens](derived+smooth+manifold#CarchediSteffens19). I will explain how derived differential geometry is able to capture non-perturbative classical BV-theory by means of examples: [[scalar field theory]] and [[Yang-Mills theory]]. Finally, I will discuss some open questions, most importantly on [[quantisation]] and on applications to global aspects of [[M-theory]].

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* {#Sharpe2024} 15 Jan 2024

  [[Eric Sharpe]]:

  **Decomposition of 2D Pure Yang-Mills and the Gross-
Taylor String Theory**

  slides: [[Sharpe-MTheoryMath2024.pdf:file]]

  cf.: [arXiv:2307.08729](https://arxiv.org/abs/2307.08729)

  > In this talk, we will attempt to reconcile two different results on [[D=2 Yang-Mills theory|two-dimensional pure Yang-Mills theory]]. Specifically, we will discuss how the fact that [[D=2 Yang-Mills theory|2d pure Yang-Mills]] is equivalent to a disjoint union of theories, is related to the Gross-Taylor description of 2d pure Yang-Mills as the [[target space|target-space]] field theory of a [[string theory]]. The Gross-Taylor picture can be understood by first rewriting the Yang-Mills partition function (in a [[large N limit|large $N$ limit]]) as a sum of [[correlation functions]] in [[Dijkgraaf-Witten theories]] for the [[symmetric group]] $S_n$, and then interpreting those Dijkgraaf-Witten correlation functions in terms of [[branched covers]], which leads to the string theory description. We first observe that the decomposition of the pure Yang-Mills aligns perfectly with the decomposition of $S_n$ Dijkgraaf-Witten theory, and then discuss decomposition and the branched covers interpretation. We encounter two puzzles, and to solve them, propose that the Gross-Taylor string theory has a [[higher form symmetry|higher-form symmetry]].

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* {#Minasian2024} 16 Jan 2024

  [[Ruben Minasian]]:

  **Constraining and Un-constraining Supergravities**

  slides: [[Minasian-MTheoryAndMath2024.pdf:file]]

  > I will review various aspects and somewhat surprising consequences of cancellation of (different types of) [[anomalies]] in [[supergravity]] theories in [[D=8 supergravity|eight]] and [[D=6 supergravity|six]] dimensions. I will also discuss appearance and importance of exotic (singular, non-spin, non-orientable) [[background field|backgrounds]].

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* {#Malek2024} 16 Jan 2024

  [[Emanuel Malek]]:

  **Kaluza-Klein Spectrometry for String Theory
Compactifications**

  slides: [[Malek-MTheoryMath2024.pdf:file]]

  cf.: [arXiv:2212.01135](exceptional+field+theory#DuboeufMalekSamtleben23)

  > I will present a powerful new method that for the first time allows us to compute the [[Kaluza-Klein theory|Kaluza-Klein]] spectrum of a large class of [[string theory]] [[KK-compactification|compactifications]], including those arising in maximal [[gauged supergravities]] and beyond. This includes geometries with little to no remaining ([[supersymmetry|super]]-)[[symmetries]], completely inaccessible by previous methods. I will show how these insights can be used to [[holographic principle|holographically]] compute the anomalous dimensions of protected and unprotected operators in [[non-perturbative quantum field theory|strongly-coupled]] [[CFTs]], as well as to study global properties of their conformal manifolds. I will also show how the method can be used to determine the perturbative stability of non supersymmetric [[anti-de Sitter spacetime|AdS]] [[string theory vacuum|vacua]]. We will see the importance of higher Kaluza-Klein modes to the physics of string compactifications, e.g. in realising the compactness of moduli spaces, restoring [[supersymmetry]] that is lost in a consistent truncation, and in destabilising vacua that appear to stable in lower-dimensional supergravities.

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* {#Han2024} 16 Jan 2024

  [[Fei Han]]:

  **Cubic Forms, Anomaly Cancellation and Modularity**

  cf.: [arXiv:2005.02344](https://arxiv.org/abs/2005.02344)

  > [Freed and Hopkins developed](D=11+N=1+supergravity#FreedHopkins21) an algebraic theory of cubic forms, which is an analogy to the theory of [[quadratic forms]] in topology. They are motivated by the Witten-Freed-Hopkins anomaly cancellation formula in [[M-theory]], which equals a cubic form arising from an [[E8|$E_8$]] [[principal bundle|bundle]] over a 12 dimensional [[spin manifold]] to the [[index of a Dirac operator|indices]] of twisted [[Dirac operators]] on the manifold. In this talk, we will first review the Witten-Freed-Hopkins anomaly cancellation formula and the algebraic theory of cubic forms, and then show that the cubic forms as well as the anomaly cancellation formula can be naturally derived from [[modular forms]] that we construct inspired by the [[Witten genus]] and the basic representation of [[affine Lie algebra|affine]] [[E8|$\mathfrak{e}_8$]]. Following this approach, we obtain new cubic forms and anomaly cancellation formulas on non-spin manifolds and thus provide a unified way to obtain anomaly cancellation formulas of this type. This is based on our [joint work](D=D11+N=1+supergravity#HanHuangLiuZhang20) with Prof. [[Ruizhi Huang]], Prof. [[Kefeng Liu]] and Prof. [[Weiping Zhang]].

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* {#Tan2024} 16 Jan 2024

  [[Meng-Chwan Tan]]:

  **Topological-Holomorphic $\mathcal{N}=4$ Gauge Theory: From Langlands Duality of Holomorphic Invariants to Mirror Symmetry of Quasi-Topological Strings**

  slides: [[Tan-MTheoryAndMath2024.pdf:file]]

  cf.: [arXiv:2305.15965](https://arxiv.org/abs/2305.15965)

  > We perform a topological-holomorphic [[topological twist|twist]] of [[D=4 N=4 SYM|$\mathcal{N}=4$]] [[super Yang-Mills theory|supersymmetric gauge theory]] on a [[4-manifold|four-manifold]] of the form $M^4 = \Sigma_1 \times \Sigma_2$, and unravel the mathematical implications of its physics. In particular, we consider the cohomology of different linear combinations of the resulting scalar supercharges under [[S-duality]], whence we would be able to derive novel topological and holomorphic invariants of $M^4$ and their [[geometric Langlands duality|Langlands duals]]. As the twisted theory can be topological along $\Sigma_1$ such that we can [[KK-reduction|dimensionally reduce]] it to 2d, via the [[effective field theory|effective]] [[sigma-model]] on $\mathbb{C}_2$, we can also relate these 4d invariants and their Langlands duals to the [[mirror symmetry]] of [[Higgs bundles]] and that of quasi-topological strings described by the [[sheaf]] of [[chiral differential operators]].

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* {#Hohm2024} 16 Jan 2024

  [[Olaf Hohm]]:

  **Double Copy, Double Field Theory & Homotopy
Algebras**

  > The [[double copy]] denotes a technology to relate the [[scattering amplitudes]] of [[Yang-Mills theory]] to those of [[gravity]]. While enormously successful at the level of [[scattering amplitudes]], until recently there was no first-principle understanding of how to derive such relations. Such an understanding would be needed in order to describe, for instance, a double copy of classical solutions. I present an approach based on [[homotopy algebras]] such as [[L-infinity algebras|L-infinity algebras]] that allows one to provide such a first-principle derivation, at least to some finite order in [[perturbation theory]]. To this end I review how to formulate [[Yang-Mills theory]] as an [[L-infinity algebra]], how to "strip off" [[color charge|color]] to obtain a different kind of homotopy algebra and, finally, how to combine two copies of these exotic algebras to obtain the L-infinity algebra of gravity in the form of [[double field theory]].

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* {#Saemann2024} 16 Jan 2024

  [[Christian Saemann]]:

  **Atiyah Algebroids for Higher and Groupoid Gauge
Theories**

  slides: [[Saemann-AtiyahAlgebroids.pdf:file]]

  > We present an [[Atiyah Lie algebroid|Atiyah algebroid]] picture for [[higher gauge theory|higher]] and groupoid [[gauge theories]]. Common to both is the fact that straightforward definitions of [[curvatures]] are only suitable for partially [[flat bundle|flat]] cases. Instead, one has to adjust the underlying [[cocycle]] relations, leading to new curvatures and gauge transformations. The Atiyah algebroid picture I sketch provides a good idea about the origin of [[adjusted Weil algebra|adjustments]] and why they are required even in the relative conventional case of groupoid gauge theories.

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* {#Hull2024} 17 Jan 2024

  [[Chris Hull]]:

  **Self-Dual $p$-Form Gauge Fields and the Topology of the
Graviton**

  slides: [[Hull-MTheoryAndMath2024.pdf:file]]

  cf.: [arXiv:2307.04748](self-dual+higher+gauge+theory#Hull23)

  > [Sen's action](self-dual+higher+gauge+theory#Sen20) for a [[higher gauge field|$p$-form gauge field]] with [[self-dual higher gauge theory|self-dual]] [[field strength]] coupled to a [[spacetime]] [[Riemannian metric|metric]] involves an explicit [[Minkowski metric]] and the presence of this raises questions as to whether the action is [[coordinate]] independent and whether it can be used on a general spacetime [[manifold]]. A natural generalisation of [Sen's action](self-dual+higher+gauge+theory#Sen20) is presented in which the Minkowski metric is replaced by a second metric on spacetime. The theory is covariant and can be formulated on any spacetime. The theory describes a physical sector, consisting of the [[self-dual higher gauge theory|chiral $p$-form gauge field]] coupled to the dynamical metric $g$, plus a shadow sector consisting of a second chiral $p$-form and the second metric. The resulting theory is covariant and can be formulated on any spacetime. A spacetime with two metrics has some interesting geometry and some of this is explored here and used in the construction of the interactions. The action has two [[diffeomorphism]]-like symmetries, one acting only on the physical sector and one acting only on the shadow sector, with the spacetime diffeomorphism symmetry arising as the diagonal subgroup.

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* {#He2024} 17 Jan 2024

  [[Yang-Hui He]]:

  **The AI Mathematician**

  > We summarize how [[artificial intelligence|AI]] can approach mathematics in three ways: [[proof assistant|theorem-proving]], [[conjecture]] formulation, and [[natural language|language]] processing. Inspired by initial experiments in [[geometry]] and [[string theory]], we present a number of recent experiments on how various standard [[machine learning|machine-learning]] [[algorithms]] can help with pattern detection across disciplines ranging from [[algebraic geometry]] to [[representation theory]], to [[combinatorics]], and to [[number theory]].

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* 17 Jan 2024

  [[Chris Blair]]:

  **Geometry and Dualities of Decoupling Limits in String
Theory and M-Theory**

  cf.: [arXiv:2311.10564](https://arxiv.org/abs/2311.10564)

  > Our understanding of [[M-theory]] is based on a [[duality in string theory|duality web]] connecting different limits of the theory. I'll discuss the extension of this duality web to a wide variety of decoupling limits related by duality to the null reduction of M-theory (and hence to the proposal that M-theory can be described by [[BFSS matrix model|Matrix theory]]). From a modern perspective, these limits involve non-[[relativistic field theory|relativistic]] geometries, leading to new variants of [[supergravity]] in [[D=11 supergravity|11-]] and [[D=10 supergravity|10-dimensions]]. I'll discuss how to systematically explore these corners of M-theory, following the roadmap of [arxiv.org/abs/2311.10564](https://arxiv.org/abs/2311.10564).

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* {#Sati2024} 17 Jan 2024

  [[Hisham Sati]]:

  **M-Theory and Hypothesis H**

  > I will survey the ([[cohomotopy|co]])[[homotopy theory|homotopical]] perspective on the [[field (physics)|fields]] and [[branes]] in [[M-theory]], showcasing several recent developments. This talk highlights the [[classical field theory|classical]]/[[prequantum field theory|prequantum]] aspects, while the [talk by Urs Schreiber](#Schreiber2024) will highlight the [[quantum physics|quantum]] aspects.

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* {#Schreiber2024} 17 Jan 2024

  [[Urs Schreiber]]:

  **[[schreiber:Introduction to Hypothesis H|Introduction to Quantum Hypothesis H]]**

  > A famous [[hypothesis]] in [[string theory]] [[D-brane charge quantization in K-theory|says]] that the [[RR-fields]] in [[10d supergravity]] are subject to "[[flux quantization]]" in [[topological K-theory|topological K-cohomology]] theory. From a modernized point of view of [[rational homotopy theory]], analogous reasoning applies to the [[supergravity C-field|C-field]] in [[11d supergravity]] and suggests that its flux should be quantized in the "[[Cohomotopy|unstable CoHomotopy]]" [[cohomology theory]] of [Borsuk](cohomotopy#Borsuk36), [Pontrjagin](cohomotopy#Pontryagin38a) and [Spanier](#Spanier49). I'll survey this "[[schreiber:Hypothesis H]]" with focus on its implications for [[quantum observables]] on [[intersecting branes]]. This is joint work with [[Hisham Sati]]. Notes are available at: *[[schreiber:Hypothesis H|ncatlab.org/schreiber/show/Introduction+to+Hypothesis+H]]*.

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## M-Theory and Mathematics 2023
 {#2023}

* on: January 12-15, 2023

  at: [[CQTS]] @ New York University, Abu Dhabi

  see: [event webpage](https://nyuad.nyu.edu/en/events/2023/january/m-theory-and-mathematics-conference-2023.html)

<center>
<a href="https://ncatlab.org/nlab/show/M-Theory+and+Mathematics#2023">
<img src="https://ncatlab.org/nlab/files/MTheoryAndMathematics2023TitleFramed.jpg" width="700"/>
</a>
</center>

> $\phantom{-----}$ (logo adapted from [JMP **62** (2021) 042301](/schreiber/show/Twisted+Cohomotopy+implies+twisted+String+structure+on+M5-branes))

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{#Speakers2023} **Speakers:**

* [[Luigi Alfonsi]]

* [[Meer Ashwinkumar]]

* [[Ibrahima Bah]]

* [[David Berman]]

* [[Michael Duff]] (had to cancel last minute)

* [[Fei Han]]

* [[Yang-Hui He]]

* [[Chris Hull]]

* [[Branislav Jurčo]]

* [[Neil Lambert]]

* [[Alessio Marrani]]

* [[Varghese Mathai]]

* [[Christian Saemann]]

* [[Hisham Sati]]

* [[Ashoke Sen]]

* [[Urs Schreiber]]

* [[Eric Sharpe]]

* [[Richard Szabo]]

* [[Meng-Chwan Tan]] (had to cancel last minute)

* [[Dimitrios Tsimpis]]

* [[Alexander Voronov]]

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<img src="/nlab/files/MTheoryAndMathematics2023-GroupPhoto.jpg">

> {#GroupPhoto2023} (from left to right: [[Sachin Valera|Valera]], [[Branislav Jurco|Jurčo]], [[David Berman|Berman]], [[Luigi Alfonsi|Alfonsi]], [[Fei Han|Han]], [[Urs Schreiber|Schreiber]], [[Varghese Mathai|Mathai]], [[Christian Saemann|Saemann]], [[Hisham Sati|Sati]], [[Alexander Voronov|Voronov]], [[Neil Lambert|Lambert]], [[Ibrahima Bah|Bah]], [[Dimitrios Tsimpis|Tsimpis]], [[Chris Hull|Hull]], [[Alessio Marrani|Marrani]], [[Yang-Hui He|He]], [[Vivek Kumar Singh|Singh]], [[Grigorios Giotopoulos|Giotopoulos]], [[Richard Szabo|Szabo]], )

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{#Talks2023} **Titles, Abstracts, and Slides** for *M-Theory & Math 2023* (in chronological order of talks delivered):

* {#Sen2023} [[Ashoke Sen]]:

  **D-instanton amplitudes in string theory**

  slides: [[Sen-MTheoryMath2023.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=I9_OaFW9KHw)

  cf. [arXiv:2204.02981](https://arxiv.org/abs/2204.02981), [arXiv:2207.07138](https://arxiv.org/abs/2207.07138)

  >  [[D-brane instanton|D-instantons]] give [[non-perturbative contribution]] to [[string amplitudes]]. While the [[world-sheet]] theory of [[open string|open]] and [[closed strings]] gives a systematic procedure for computing these amplitudes, the analysis runs into [[infrared divergences]] that do not have any known cure within the [[world-sheet]] formalism. We describe how [[string field theory]] cures these divergences and gives an unambiguous procedure for computing [[D-brane instanton|D-instanton]] correction to [[string amplitudes]].

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* {#Lambert2023} [[Neil Lambert]]:

  **A non-Lorentzian View of the M5-brane**

  slides: [[Lambert-MTheoryMath2023.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=SlEYRaq2AO8)

  cf.: [arXiv:2011.06968](https://arxiv.org/abs/2011.06968), [arXiv:2212.07717](https://arxiv.org/abs/2212.07717)

  > In this talk we will discuss a [[path-integral]] formulation of the [[M5-brane]] [[D=6 N=(2,0) SCFT|(2,0) theory]] based upon a [[Lagrangian density|Lagrangian]] without [[Lorentz group|Lorentz]] [[gauge invariance|invariance]] but a novel $SU(1,3)$ spacetime symmetry. This provides a [[UV completion|UV complete]] five-dimensional [[quantum field theory|field theoretic]] description that we will argue admits a six-dimensional Lorentz invariance at [[non-perturbative quantum field theory|strong coupling]].

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* {#Tsimpis2023} [[Dimitrios Tsimpis]]:

  **Universal accelerating cosmologies from 10d supergravity**

  slides: [[Tsimpis-MTheoryMath2023.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=LEiqYz-G34g)

  cf.: [arXiv:2210.10813](https://arxiv.org/abs/2210.10813)

  > 4d [[FRW model|Friedmann-Lemaître-Robertson-Walker cosmologies]] are obtained from time-dependent [[KK-compactification|compactifications]] of [[type II supergravity|Type IIA 10d supergravity]] on various classes of 6d manifolds. These cosmologies are universal in that they do not depend on the detailed features of the compactification manifold, but only on the properties which are common to all the manifolds belonging to that class. Once the [[equations of motion]] are rewritten as an appropriate [[dynamical system]], the existence of solutions featuring a phase of accelerated [[cosmic inflation|expansion]] is made manifest. Some of the resulting cosmologies exhibit eternal or semi-eternal acceleration, whereas others allow for a parametric control on the number of $\mathrm{e}$-foldings. Moreover, we find several smooth accelerating cosmologies without [[Big Bang]] [[singularities]]: the [[observable universe|universe]] is contracting in the cosmological past ($T \lt 0$), expanding in the future ($T \gt 0$), while in the vicinity of $T = 0$ it becomes [[de Sitter spacetime|de Sitter]] in hyperbolic slicing. We also obtain several cosmologies featuring an infinite number of cycles of alternating periods of accelerated and decelerated expansions.

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* {#Mathai2023} [[Varghese Mathai]]:

  **T-duality for loop spaces, or equivalently for the 1D sigma model**

  slides: [[Mathai-MTheoryMath2023.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=--CFSc6VVqI)

  cf.: [arXiv:1405.1320](https://arxiv.org/abs/1405.1320)

  > We define exotic [[twisted de Rham cohomology|twisted]] [[circle group|$S^1$]]-[[equivariant de Rham cohomology|equivariant cohomology]] for the [[free loop space|loop space]] $\mathcal{L}Z$ of a [[smooth manifold]] $Z$ via the [[invariant]] [[differential forms]] on $\mathcal{L}Z$ with [[coefficients]] in the (typically non-flat) [[higher holonomy|holonomy]] [[complex line bundle|line bundle]] of a [[bundle gerbe|gerbe]], with differential given by an equivariantly [[flat connection|flat]] [[superconnection]]. We introduce the twisted [[Bismut-Chern character]] form, a loop space refinement of the [[twisted Chern character]] form, which represent classes in the completed periodic exotic twisted [[circle group|$S^1$]]-[[equivariant de Rham  cohomology|equivariant cohomology]] of $\mathcal{L}Z$. We establish a localisation theorem for the completed periodic exotic twisted $S^1$-equivariant cohomology for loop spaces and apply it to establish [[topological T-duality|T-duality]] in a [[background field|background]] [[flux]] in [[type II String Theory]] from a [[loop space]] perspective. Finally we reinterpret these results as [[topological T-duality|T-duality]] for the 1D sigma model.

\linebreak

* [[Fei Han]]:

  **Graded T-duality with H-flux for 2d sigma models**

  slides: [[Han-MTheoryMath2023.pdf:file]]

  cf.: [arXiv:2207.03134](https://arxiv.org/abs/2207.03134)

  > [[T-duality]] in [[string theory]] can be realised as a transformation acting on the [[worldsheet]] fields in the two-dimensional nonlinear [[sigma model]]. [Bouwknegt-Evslin-Mathai](topological+T-duality#BouwknegtEvslinMathai04) established the [[topological T-duality|T-duality in a background flux]] for the first time upon compactifying spacetime in one direction to a principal circle by constructing the T-dual maps transforming the twisted cohomology of the dual spacetimes. In this talk, we will describe our recent work on how to promote the T-duality maps of Bouwknegt-Evslin-Mathai in two aspects. More precisely, we will introduce (1) graded T-duality, concerning the graded T-duality maps of all levels of twistings; (2) the 2-dimensional sigma model picture, concerning the [[iterated loop space|double]] loop space of spacetimes. This represents our joint work with Mathai.

\linebreak

* {#Saemann23} [[Christian Saemann]]:

  **Geometric and Non-Geometric T-duality with Higher Bundles**

  slides: [[Saemann-MTheoryMath2023.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=i4-ohJ_Z-t8)

  cf.: [arXiv:2204.01783](https://arxiv.org/abs/2204.01783)

  > I will introduce the notion of [[adjusted Weil algebra|adjusted curvatures]] for [[principal infinity-bundle|higher principal bundles]]. These will allow me to [[differential cohomology|differentially refine]] the description of [[topological T-duality]] [by Nikolaus and Waldorf](T-fold#NikolausWaldorf18) in terms of [[principal 2-bundles]], leading to a description of full [[T-duality]]. I also discuss interesting generalizations that seem to capture the [[non-geometric string vacuum|non-geometric cases]].

\linebreak

* {#Berman2023} [[David Berman]]:

  **Exact Renormalisation Group and statistical Inference**

  slides: [[Berman-MTheoryAndMath2023.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=h5x-aNiokLg)

  cf.: [arXiv:2212.11379](https://arxiv.org/abs/2212.11379)

  > The [[exact renormalisation group]] is used to describe the process of [[statistical inference]] in the [[Bayesian reasoning|Bayesian sense]].

\linebreak

* {#He2023} [[Yang-Hui He]]:

  **Machine-Learning Mathematical Structures**

  slides: [[He-MTheoryAndMath2023.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=SlF2T4fxroQ)

  cf.: [arXiv:2101.06317](https://arxiv.org/abs/2101.06317)

  > We present a number of recent [[experiments]] on how various standard [[machine-learning]] [[algorithms]] can help with pattern detection across disciplines ranging from [[string theory]], to [[algebraic geometry]], to [[representation theory]], to [[combinatorics]], and to [[number theory]].

  > We speculate on whether there is an inherent hierarchy of "difficulty" in mathematics reflected by data. At the heart of the programme is the question how does AI help with mathematical discovery.

\linebreak

* {#Sharpe2023} [[Eric Sharpe]]:

  **An introduction to decomposition**

  slides: [[Sharpe-MTheoryAndMath2023.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=rUpEQniWho4)

  cf.: [arXiv:2204.09117](https://arxiv.org/abs/2204.09117)

  > In this talk I will review work on 'decomposition', a property of 2d [[field theory|theories]] with [[2-group|1-form symmetries]] and, more generally, d-dim'l theories with [[higher gauge theory|$(d-1)$-form symmetries]]. Decomposition is the observation that such [[quantum field theories]] are equivalent to ('decompose into’) [[disjoint unions]] of other QFTs, known in this context as "universes".

  > Examples include two-dimensional gauge theories and [[orbifolds]] with matter invariant under a [[subgroup]] of the [[gauge group]]. Decomposition explains and relates several physical properties of these theories -- for example, restrictions on allowed [[instantons]] arise as a "multiverse interference effect" between contributions from constituent universes.

  > First worked out in 2006 as part of efforts to understand [[string]] propagation on [[stacks]], decomposition has been the driver of a number of developments since. I will walk through general aspects of 2d gauge theories that exhibit decomposition, dive into specifics in orbifold examples, and as time permits, discuss recent work in progress.

\linebreak

* {#Szabo23} [[Richard Szabo]]:

  **D-branes and doubled geometry**

  slides: [[Szabo-MTheoryAndMath2023.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=D1E4Tlq7zf0)

  cf.: [arXiv:1910.09997](https://arxiv.org/abs/1910.09997)

  > We formulate the [[open string]] version of the Born [[sigma-model]] and describe its conformal [[boundary conditions]]. This leads to a general characterisation of [[D-branes]] in [[doubled geometry]] as maximally [[isotropic subspace|isotropic]] sub-bundles on a [[para-Hermitian manifold]]. The reduction of D-branes to a physical spacetime is obtained by describing them as [[Dirac structures]]. We also introduce a para-complex counterpart of the generalised complex branes of [[generalized complex geometry|generalised geometry]].

\linebreak

* {#Hull2023} [[Chris Hull]]:

  **Charges and Duality in Gravity**

  slides: [[Hull-MTheoryAndMath2023.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=7ho3n3tMTOI)

  cf. [arXiv:2209.11716](https://arxiv.org/abs/2209.11716) and refs [here](https://ncatlab.org/nlab/show/D%3D6+supergravity#ReferencesFourZero)

  > on exotic $\mathcal{N} = (4,0)$ [[D=6 supergravity]]

\linebreak

* {#Jurco2023} [[Branislav Jurčo]]:

  **Correspondences of Quantum $L_\infty$ Algebras (BV-theories)**

  video: [YT](https://www.youtube.com/watch?v=eOLyUqyKS0A)

  cf.: [arXiv:1912.06695](https://arxiv.org/abs/1912.06695), [arXiv:2002.11168](https://arxiv.org/abs/2002.11168)

  > The quantum odd [[symplectic category|symplectic "category"]] is constructed from the odd symplectic "category" via an enhancement of [[Lagrangian correspondence|Lagrangian relations]] (correspondences) by [[canonical Hilbert space of half-densities|half-densities]]. We extend this approach to the (finite-dimensional) setting of the [[BV-BRST formalism|Batalin-Vilkovisky quantum field theories]] (quantum L-infinity algebras). We discuss the effective [[observables]] of [[perturbative quantum field theory]] and the [[homological perturbation lemma]] from this point of view.

\linebreak

* {#Ashwinkumar2023} [[Meer Ashwinkumar]]:

  **Matrix Quantization of Classical Nambu Brackets and Super $p$-Branes**

  video: [YT](https://www.youtube.com/watch?v=FP_gHnSOBDk)

  cf.: [arXiv:2103.06666](https://arxiv.org/abs/2103.06666)

  > We will present an explicit matrix algebra quantization of the algebra of volume-preserving diffeomorphisms of the $n$-torus. That is, we approximate the corresponding classical [[Nambu brackets]] using $\mathfrak{sl}\big(N^{\lceil n/2\rceil}, \mathbb{C}\big)$-matrices equipped with the finite bracket given by the completely anti-symmetrized matrix product, such that the classical brackets are retrieved in the $N \to \infty$ limit. We then apply this approximation to the [[super 4-brane in 9 dimensions]] and give a regularized action in analogy with [the matrix quantization of the supermembrane](membrane#LightConeQuantizationOfM2BranesToBFSSMatrixModel). This action exhibits a reduced gauge symmetry that we discuss from the viewpoint of [[L-infinity algebras|$L_\infty$-algebras]] in a slight generalization to the construction of [[Lie 2-algebras]] from [[M-brane 3-algebra|Bagger-Lambert 3-algebras]].

\linebreak

* {#Bah2023} [[Ibrahima Bah]]:

  **Non-invertible symmetry, holography and branes**

  video: [YT](https://www.youtube.com/watch?v=yOBkH-n0JBY)

  cf.: [arXiv:2208.07373](https://arxiv.org/abs/2208.07373)

  >  In this talk I will discuss some general methods for realizing topological symmetry generators in [[holography]]. In particular I will present a specific realization of non-invertible symmetry generators and their fusion rules in holographic duals of $SU(N)/\mathbb{Z}_N$ [[SYM]]. This will be both from a [[bottom-up and top-down model building|bottom-up perspective]] from an [[effective field theory|effective theory]] in [[anti de Sitter spacetime|$AdS_5$]], and from a [[bottom-up and top-down model building|top-down perspective]] in terms of [[D-branes]] in $AdS_5$ backgrounds.

\linebreak

* {#Sati2023} [[Hisham Sati]]:

  **M-theory and matter via Twisted Equivariant Differential (TED) K-theory**

  video: [YT](https://www.youtube.com/watch?v=MQiGYl-KKfA)

  cf.: [[schreiber:Anyonic topological order in TED K-theory|arXiv:2206.13563]]

  > Various flavors of [[topological K-theory|K-theory]] [[D-brane charge quantization in K-theory|play an important role]] in geometric/topological perspectives on [[string theory]]. Ultimately, the [[twisted equivariant differential K-theory|twisted equivariant differential (TED) version of K-theory]] would be what is needed to describe [[field (physics)|fields]] and [[branes]] in the most general backgrounds. However, this theory has not been constructed yet, but major progress has been made in that direction requiring highly nontrivial machinery. I will describe the main conceptual points here. I will also explain how TED K-theory enters naturally into [[condensed matter theory|condensed matter]] ([[topological phases of matter|topological phases]]) and highlight analogies with [[high energy physics]]. All of this embeds in [[M-theory]] via [[schreiber:Hypothesis H]], both from physical and mathematical perspectives.

\linebreak

* {#Schreiber2023} [[Urs Schreiber]]:

  **Topological Quantum Gates from M-Theory**

  notes: [pdf](/schreiber/show/Topological+Quantum+Gates+from+M-Theory#Notes)

  video: [YT](https://www.youtube.com/watch?v=Ob9tOgWumPI)

  cf.: [[schreiber:Anyonic defect branes in TED-K-theory|arXiv:2203.11838]]

  >  The [[AdS-QCD correspondence|promising idea]] that [[non-perturbative field theory|strongly]] [[coupling|coupled]] [[quantum systems]] such as ([[confinement|confined]] [[QCD]] or) [[topological order|topologically ordered]] [[quantum materials]] are [[AdS-CMT duality|usefully modeled]] as [[worldvolume]] [[dynamics]] on [[intersecting branes]] in [[string theory]] has been suffering from the latter's lack of [[nLab:non-perturbative effect|non-perturbative]] formulation ([[M-theory]]), [necessary](https://ncatlab.org/nlab/show/AdS-CFT#SmallNCorrections) in the realistic regime of [[large 1/N limit|small numbers]] of individual [[branes]] (i.e. beyond the usual [[holography|holographic]] [[large N limit|large-$N$ approximation]]). In this talk I will briefly review our "[[schreiber:Hypothesis H]]" that [[brane]] [[nLab:charge]] in [[nlab:M-theory]] is [[nLab:Dirac charge quantization|quantized]] in a [[nlab:twisted cohomology|twisted]] [[nLab:equivariant cohomology|equivariant]] [[nLab:non-abelian cohomology|non-abelian]] [[nLab:generalized cohomology]] theory called *[[nLab:Cohomotopy]]*, where [[nLab:M-brane]] [[nLab:quantum states]] are identified inside the [[nLab:twisted de Rham cohomology|twisted equivariant cohomology]] of the [[nLab:Cohomotopy]] [[nLab:moduli stack]]. Then I explain our recent derivation, from this assumption, of [[nLab:anyon|anyonic]] [[nLab:topological order]] in [[nLab:ground states]] of [[nLab:M3-brane|M5-brane intersections]]; and I close with an outlook on how this describes [[nlab:topological quantum computation|topological]] [[nLab:quantum logic gates]] via [[nlab:braid group statistics|braiding]] of [[nLab:defect branes]].

\linebreak

* {#Voronov2023} [[Alexander Voronov]]:

  **The $E_k$ symmetry of dimensional reductions of M-theory**

  video: [YT](https://www.youtube.com/watch?v=otpT7bQwiqQ)

  cf.: [arXiv:2111.14810](https://arxiv.org/abs/2111.14810), [arXiv:2212.13968](https://arxiv.org/abs/2212.13968)

  > [[Hisham Sati]] and I, on the way to revealing the mystery of 2001 [[Mysterious Duality]] of [Iqbal, Neitzke, and Vafa](mysterious+duality#INV01), generalized [[schreiber:Hypothesis H|hypothesis H]] and presented a universal target for each [[double dimensional reduction|dimensional reduction]] of [[M-theory]]. Each of these targets has the [[dynamics]] of [[supergravity]] embedded into its structure. The target for $(11-k)$-dimensional [[M-theory]] also has an [[Lie algebra representation|action]] of the [[Lie algebra]] of [[exceptional Lie algebra|type $E_k$]], responsible for [[U-duality|symmetries]] of $(11-k)$-dim [[supergravity]] along with its [[equations of motion]] in any spacetime.

\linebreak

* {#Marrani2023} [[Alessio Marrani]]:

  **Exceptional super Yang-Mills in $27 + 3$ and worldvolume M-theory**

  slides: [[Marrani-MTheoryAndMath2023.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=yTpiqozLuXY)

  cf.: [arXiv:1906.10709](https://arxiv.org/abs/1906.10709)

  > Some time ago, [[D=14 supersymmetry|Bars and Sezgin proposed]] a [[super Yang-Mills theory]] in $D=s+t=11+3$ space-time dimensions, with an electric 3-brane that generalizes the 2-brane of M-theory. More recently, Rios, Chester and the speaker found an infinite family of "exceptional" super Yang-Mills theories in $D=s+t=(8n+3)+3$. A particularly interesting case occurs for $n=3$, namely in signature $D=27+3$, in which the superalgebra is centrally extended by an electric 11-brane and its 15-brane magnetic dual. The worldvolume symmetry of the 11-brane has signature $D=11+3$ and can reproduce super Yang-Mills theory in D=$11+3$. Upon reduction to $D=26+2$, the 11-brane reduces to a 10-brane with $10+2$ worldvolume signature. A single time projection gives a $10+1$ worldvolume signature, and it can serve as a model for $D=10+1$ [[M-theory]] as a reduction from the $D=26+1$ signature of the [[bosonic M-theory]] of [[Gary Horowitz|Horowitz]] and [[Leonard Susskind|Susskind]]; this is further confirmed by the reduction of chiral $(1,0)$, $D=11+3$ superalgebra to the $\mathcal{N}=1$ superalgebra in $D=10+1$, as found by Rudychev, Sezgin and Sundell some time ago. Extending previous results of Dijkgraaf, Verlinde and Verlinde, I will also put forward the realization of spinors as total cohomologies of (the largest spatially extended) branes which centrally extend the $(1,0)$ superalgebra underlying the corresponding "exceptional" super Yang-Mills theory. Moreover, by making use of an “anomalous” Dynkin embedding, I will present further evidence about [[Pierre Ramond|Ramond]]'s and [[Hisham Sati|Sati]]'s argument that M-theory has hidden Cayley plane fibers.

\linebreak

* {#Alfonsi2023} [[Luigi Alfonsi]]:

  **Derived $n$-plectic geometry: towards non-perturbative BV-BFV quantisation and M-theory**

  cf.: [arXiv:2307.15106](https://arxiv.org/abs/2307.15106)

  slides: [[Alfonsi-MTheoryAndMath2023.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=O7pQkyqCiA4)

  > In this talk I will propose the notion of [[derived geometry|derived]] [[n-plectic geometry|$n$-plectic geometry]] and show that a classical [[BV formalism|BV-BFV theory]] can be globally reformulated in this framework.

  > A derived $n$-plectic structure is a [[derived geometry|derived]] and [[categorification|categorified]] generalisation of a [[symplectic structure]], where the usual [[closed differential form|closed]] [[differential 2-form|2-form]] is replaced by a shifted closed $(n+1)$-form on a [[derived manifold|derived space]]. I will argue that derived $n$-plectic geometry is to classical BV-BFV theory as ordinary $n$-plectic geometry is to ordinary [[Lagrangian field theory]]. I will also discuss the rich [[homotopical algebra|homotopy-algebraic]] structure of [[observables]] of a derived $n$-plectic space. Finally, I will point at promising applications of this formalism to [[non-perturbative quantum field theory|non-perturbative]] [[quantisation]] of [[BV formalism|BV-BFV theories]] and to global aspects of [[M-theory]].

\linebreak

\linebreak

\linebreak

***

\linebreak

## M-Theory and Mathematics 2020
 {#2020}

<div style="float:right;margin:0 10px 10px 0;">
<a href="https://nyuad.nyu.edu/en/institute-events.html">
<img src="https://ncatlab.org/nlab/files/NYUADInstitute.jpg" width="170"/>
</a>
</div>

* on: January 27-30, 2020

* at: [Research Institute](https://nyuad.nyu.edu/en/research/research-institute-centers.html) of New York University, Abu Dhabi

* see: [event webpage](https://nyuadi.secure.force.com/Events/NYUEventRegistration?event=u2f8V99W_2FoUW_2FK0f7HydVQ_3D_3D)

* and: [mtheorymath.org](http://mtheorymath.org)

* [Program](https://hisham-sati.github.io/M-theory-and-Mathematics/)

<img src="https://ncatlab.org/nlab/files/MTheoryAndMathematics2020Program.jpg" width="450">

**Confirmed speakers:**

* [[David Berman]]

* [[Bianca Cerchiai]]

* [[Martin Cederwall]]

* [[Michael Duff]]

* [[José Figueroa-O’Farrill]]

* [[Domenico Fiorenza]]

* [[Pietro Grassi]]

* [[Fei Han]]

* [[Yang-Hui He]]

* [[Chris Hull]]

* [[Branislav Jurčo]]

* [[Neil Lambert]]

* [[William Linch III]]

* [[Varghese Mathai]]

* [[Du Pei]]

* [[Christian Saemann]]

* [[Henning Samtleben]]

* [[Hisham Sati]]

* [[Urs Schreiber]]

* [[Ashoke Sen]]

* [[Eric Sharpe]]

* [[Dmitri Sorokin]]

* [[Meng-Chwan Tan]]

\linebreak

\linebreak

**Abstracts, Slides and Talk notes:**
 {#Abstracts2020}

\linebreak

* {#Duff2020} [[Michael Duff]]:

  **[[Duff interview at M-Theory-Mathematics 2020|Perspectives on M-Theory]]**

  video: [YT](https://youtu.be/aHj_Thu8d54)

\linebreak

* {#FigueroaOFarrill2020} [[José Figueroa-O’Farrill]]

  **Lie superalgebra deformations and d=11 supergravity backgrounds**

  slides: [pdf](https://ncatlab.org/nlab/files/FigueroaOFarSlidesAtMTheoryAndMathematics2020.pdf)

  video: [YT](https://www.youtube.com/watch?v=e49vkX6HVQ0)

  > To every ([[supersymmetry|supersymmetric]]) background of [[d=11 supergravity]] there is associated a [[filtered object|filtered]] [[super Lie algebra|Lie superalgebra]]. For ($\gt 1/2$)-[[BPS state|BPS backgrounds]], this Lie superalgebra determines the background up to local [[isometry]]. I will then report on an [[Spencer cohomology|algebraic reformulation]] of this classification problem. This is [joint work with Andrea Santi](https://ncatlab.org/nlab/show/Spencer+cohomology#FFS17).

\linebreak

* {#Linch2020} [[William Linch III]]

  **Off-shell Supersymmetry and the M-theory Effective Action**

  video: [YT](https://www.youtube.com/watch?v=w50XhsA4AYo)

  cf.: [arXiv:2101.11671](https://arxiv.org/abs/2101.11671)

\linebreak

* {#Sati2020} [[Hisham Sati]],

  **M-theory and cohomotopy**

  slides: [pdf](https://ncatlab.org/nlab/files/Sati_MTheoryCohomotopy_2020.pdf)

  > [[Cohomotopy theory]] has recently emerged as the proper [[generalized cohomology|generalized cohomology theory]] to describe the [[field (physics)|fields]] in [[M-theory]]. It is hoped that viewing M-theory from a mathematical perspective will shed the light on the nature of the theory and will allow for progress. I will survey this area, illustrating how cohomotopy captures the nature of the [[nLab:supergravity C-field|C-field]] and its dual as well as of the [[M-branes]], allows for cancellation of various [[nLab:quantum anomaly|anomalies]], and provides a firm grounding for further study of M-theory. This is joint work [[Urs Schreiber]] and [[Domenico Fiorenza]].

\linebreak

* {#Schreiber2020} [[Urs Schreiber]]:

  **[[schreiber:Microscopic Brane Physics from Cohomotopy]]**

  slides: [pdf](https://ncatlab.org/schreiber/files/Schreiber-MTheoryMathematics2020-v200126.pdf)

  video: [YT](https://www.youtube.com/watch?v=PAo4cCqQhks)

  > As reviewed in [[nLab:Hisham Sati|H. Sati]]'s talk, assuming that the [[nLab:supergravity C-field|C-field]] is [[nLab:Dirac charge quantization|charge-quantized]] in the [[nLab:generalized cohomology theory]] called [[nLab:J-homomorphism|J-]][[nLab:twisted cohomotopy|twisted]] [[nLab:Cohomotopy]] ("[[Hypothesis H]]") implies a list of [[nLab:M-theory|M-theoretic]] [[nLab:anomaly cancellation]] conditions, such as [[nLab:shifted C-field flux quantization]], DMW anomaly cancellation and [[nLab:C-field tadpole cancellation]] [[nLab:M-theory on 8-manifolds|on 8-manifolds]]. In this talk I review the further [[nLab:(infinity,1)-topos|geometric]] refinement of the [[nLab:cohomology|cohomology theory]] to [[nLab:equivariant|equivariant Cohomotoy theory]] and to [[nLab:differential cohomotopy|differential Cohomotopy theory]]. Now we find that [[Hypothesis H]] implies also the Witten mechanism of multiple [[nLab:M5-branes]] on [[nLab:MO5]]-[[nLab:orientifolds]] in [[nLab:heterotic M-theory on ADE-orbifolds]], hence [[nLab:RR-field tadpole cancellation]] in [[nLab:type I' string theory]]; as well as a multitude of effects associated with [[nLab:Dp-D(p+2) brane bound state|Dp/D(p+2)-brane intersections]]: [[nLab:Chan-Paton factors]], [[nLab:BMN matrix model]] [[nLab:fuzzy funnel]] states and [[nLab:BLG 3-algebras]], the [[nLab:Hanany-Witten rules]], [[nLab:AdS3/CFT2|AdS3-gravity]] observables, [[nLab:supersymmetric indices]] of [[nLab:Coulomb branches]] as well as [[nLab:AdS/CFT|gauge/gravity duality]] between all these. This suggests that [[Hypothesis H]] is a correct assumption about the elusive mathematical fomulation of [[nLab:M-theory]].
This is joint work with [[nLab:Hisham Sati|H. Sati]] ([[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation|arxiv:1909.12277]], [[schreiber:Differential Cohomotopy implies intersecting brane observables|arxiv:1912.10425]]).

\linebreak

* {#Hull2020} [[Chris Hull]]:

  **Dualities, K3, Exotic Branes and Orientifolds**

  video: [YT](https://www.youtube.com/watch?v=9hfym2ACPo0)

  cf.: [arXiv:1907.04040](https://arxiv.org/abs/1907.04040)

\linebreak

  * {#Samtleben2020} [[Henning Samtleben]]:

    **Exceptional Field Theories and AdS Compactifications**

    video: [YT](https://www.youtube.com/watch?v=JSSmUAHYdBs)

\linebreak

* {#Cerchiai2020} [[Bianca Cerchiai]]

  **Supergravity in a pencil**

  slides: [[CerchiaiSlidesAtMTheoryAndMathematics2020.pdf:file]]

  video: [YT](https://www.youtube.com/watch?v=xE7TmwyqqaU)

  cf.: [arXiv:1910.03508](AdS-CFT+in+condensed+matter+physics#ACDGNT20)

  > In the [[AdS/CFT in condensed matter physics|spirit of the gauge-gravity correspondence]], we derive a 2+1 dimensional model with "unconventional" [[supersymmetry]] at the boundary of a 4-dimensional [[anti de Sitter spacetime|Anti de Sitter]] $N$-extended [[supergravity]], which in the case $N=2$ reproduces the AVZ model [P.D. Alvarez, M. Valenzuela, J. Zanelli, JHEP 1204 (2012) 058, [arXiv:1109.3944](https://arxiv.org/abs/1109.3944)]. The extended supersymmetry of the boundary model is instrumental to describe the electronic properties of [[graphene]], in particular at the two [[Dirac points]]. The two valleys correspond to the two independent sectors of the [[orthosymplectic super Lie group|$OSp(p\vert 2)\times OSp(q\vert 2)$]] boundary model in the $p=q$ case, which are related by a parity transformation. The Semenoff and the Haldane masses entering the corresponding [[Dirac equations]] for the graphene pseudoparticles are identified with [[supergravity torsion constraints|supergravity torsion]] parameters.

\linebreak

* {#Lambert2020} [[Neil Lambert]],

  **Lagrangians with $(2,0)$ Supersymmetry**

  slides: [pdf](https://ncatlab.org/nlab/files/LambertSlidesAtMTheoryMathematics2020.pdf)

  video: [YT](https://www.youtube.com/watch?v=36SQG-o9b9k)

  cf.: [arXiv:1908.10752](https://arxiv.org/abs/1908.10752)

  > We will discuss free and interacting six-dimensional actions which admit $(2,0)$ supersymmetry including their application to abelian and non-abelian [[M5-branes]].

\linebreak

* {#Saemann2020} [[Christian Saemann]]:

  **Towards an M5-Brane Model: Progress Report**

  video: [YT](https://www.youtube.com/watch?v=EjolwjlgTK4)

  cf.: [arXiL1712.06623](https://arxiv.org/abs/1712.06623),  [arXiv:1908.08086](https://arxiv.org/abs/1908.08086), [arXivL2012.09253](https://arxiv.org/abs/2012.09253)

\linebreak

* {#Fiorenza2020} [[Domenico Fiorenza]]:

  **Twisted Cohomotopy implies level quantization of the 6d WZ term of the M5-brane**

  slides: [pdf](https://ncatlab.org/nlab/files/FiorenzaSlidesAtMTheoryMathematics2020v2.pdf)

  video: [YT](https://www.youtube.com/watch?v=1wjD4ilp2qU)

  cf.: [arXiv:1906.07417](https://arxiv.org/abs/1906.07417)

  > The [[Hopf-Wess-Zumino term|6d Wess-Zumino term]] in the [[action functional]] for the [[M5-brane]] is anomalous as traditionally defined. What has been missing is a condition implying a higher analogue of level quantization familiar from the 2d [[WZW term|Wess-Zumino term]]. We prove that such an anomaly cancellation condition is implied by the [[Hypothesis H|hypothesis]] that the [[supergravity C-field|C-field]] is [[Dirac charge quantization|charge-quantized]] in [[twisted cohomotopy|twisted]] [[Cohomotopy theory]]. The proof follows by a twisted/parametrized generalization of the [[Hopf invariant]], after identifying the [[Hopf-Wess-Zumino term|6d Wess-Zumino term]] with a twisted [[Whitehead integral formula|homotopy Whitehead integral formula]], which we establish. Joint work with Hisham Sati and Urs Schreiber, [[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization|arXiv:1906.07417]].

\linebreak

* {#Berman2020} [[David Berman]]:

  **Strings, Branes, and Boltzmann machines**

  video: [YT](https://www.youtube.com/watch?v=s4eLW84tNQs)

\linebreak

* {#He2020} [[Yang-Hui He]]:

  **Universes as Bigdata**

  slides: [pdf](https://ncatlab.org/nlab/files/HeSlidesAtMTheoryAndMathematics2020.pdf)

  video: [YT](https://www.youtube.com/watch?v=i-kuoXjcKls)

  > We review how historically the problem of [[string phenomenology]] lead theoretical physics first to algebraic/differential geometry, and then to computational geometry, and now to data science and AI. With the concrete playground of the [[Calabi-Yau manifold|Calabi-Yau]] [[landscape of string theory vacua|landscape]], accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in [[machine learning|machine-learning]] can help explore problems of physical and mathematical interest.

\linebreak

* {#Varghese2020} [[Mathai Varghese]]:

  **T-Duality, loop spaces and Witten gerbe modules**

  video: [YT](https://www.youtube.com/watch?v=JpFjtL14exA)

  cf.: [arXiv:2001.00322](https://arxiv.org/abs/2001.00322)

\linebreak

* {#Sorokin2020} [[Dmitri Sorokin]]:

  **What do SYM domain walls look like?**

  video: [TY](https://www.youtube.com/watch?v=yHgBmTgfVoM)

  cf.: [arXiv:2004.11232](https://arxiv.org/abs/2004.11232)

\linebreak

* {#Sen2020} [[Ashoke Sen]]:

  **Gravitational waves from soft theorem**

  video: [YT](https://www.youtube.com/watch?v=FgESLruEeN0)

  cf.: [arXiv:1912.06413](https://arxiv.org/abs/1912.06413)

  > on the [[soft graviton theorem]]

\linebreak

\linebreak

## Related events

* *[[Higher Structures in M-Theory 2018]]*

\linebreak

category: reference


M-theory lift of gauge enhancement on D6-branes

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* table of contents
{:toc}

## Idea

It is well understood that in [[type IIA string theory]] there appears

* $SU(N)$ [[enhanced gauge symmetry]] on $N$-coincident [[D-branes]];

* $SO(2N)$ enhanced gauge symmetry on $N$-coincident [[D-branes]] at an [[orientifold plane]].

The physical picture of this effect is that the gauge bosons of these [[gauge fields]] are the modes of the [[open strings]] stretching between these D-branes which become massless as the branes coincide.

Now, as the situation is lifted to [[M-theory]], the [[D0-branes]], [[D2-branes]] and [[D4-branes]] lift to [[M2-branes]] and [[M5-branes]], and the gauge enhancement is thought to be similarly reflected on these [[M-branes]] (as exhibited for the [[M2-branes]] by the [[BLG-model]] and [[ABJM-model]]).

But for the [[D6-brane]] the situation is different: the [[D6-brane]] lifts not to an [[M-brane]], but to a configuration of the [[field (physics)|field]] of [[11-dimensional supergravity]]: the 11d [[Kaluza-Klein monopole]].

{#Picture} Here we discuss the picture of how [[gauge enhancement]] on [[D6-branes]] in [[type IIA string theory]] is reflected on [[Kaluza-Klein monopoles]] at [[ADE-singularities]]. The explicit realization reviewed below is due to ([Sen 97](#Sen97)).

> graphics grabbed from [HSS18](http://ncatlab.org/schreiber/show/Equivariant+homotopy+and+super+M-branes)

## Construction

### $A_{N-1}$-singularity and $SU(N)$-gauge enhancement

The 11d multi-centered [[Kaluza-Klein monopole]] [[spacetime]] has [[metric tensor]] of the form

$$
  ds^2
     \coloneqq
    - d t^2
    + \underoverset{n = 5}{10}{\sum} d y^n d y^n
    + ds^2_{TN}
  \,,
$$

where

$$
  ds^2_{TN}
    \coloneqq
  V^{-1}( d x^4 + \vec \omega \cdot d \vec r )^2 + V d \vec r^2
$$

is the metric tensor of the multi-centered [[Taub-NUT space]], with $x^4$ the canonical coordinate on a [[circle]] and with

$$
  V \coloneqq 1 + \underoverset{I = 1}{N}{\sum} \frac{4m}{{\vert \vec r - \vec r_I\vert}}
$$

and

$$
  \vec \nabla \times \vec \omega = - \vec \nabla V
$$

for $\{\vec r_I\}_{I =1}^N$ the set of positions of the KK-monopoles of mass $m$.

Notice that the radius $16 \pi m V^{-1/2}$ of the $x^4$-circle vanishes precisely at the positions $\vec r_I$. Nevertheless, as long as  all the $\vec r_I$ are distinct, then the above is a smooth spacetime also at the positions of the $\vec r_I$ -- if the periodicity of $x^4$ is taken to be $16 \pi m$.
But with this periodicity fixed, then as the $N$ monopole positions $\vec r_I$ coincide, then the resulting metric has a [[conical singularity]] at that point, of [[ADE-singularity]] type $A_{N-1}$.

To see where the gauge enhancement arises from in this singular case, observe that in the non-singular configuration there are $N-1$ linearly indepent 2-[[cycles]] $S_{i j}$ in the multi-center KK-monopole spacetime, represented by the the [[2-spheres]] that are swept out by the circle fiber as it moves from $\vec r_i$ to $\vec r_j$ (remembering that the circle fiber radius vanishes precisely at the positions $\vec r_I$).

For the canonical choice of straight path between $\vec r_i$ and $\vec r_j$ (and arbitrary fixed position in the remaining 7 dimensions) then the [[surface]] [[area]] of these [[2-spheres]] is

$$
  vol(S_{i j}) = 16 \pi m  {\vert \vec r_i - \vec r_j\vert}
  \,.
$$

For any other choice of path the surface area will be larger. Hence an [[M2-brane]] with tension $T_{M2}$ [[wrapped brane|swrapping]] the 2-[[cycle]] $S_{i j}$ has minimal tension energy when in the configuration of these spheres, namely

$$
  m_{i j} = 16 \pi m T_{M2} {\vert \vec r_i - \vec r_j\vert}
  \,.
$$

The type IIA limit is given by $m \to 0$. In this limit the [[M2-branes]] wrapping the above cycles become the type IIA [[superstring]] by [[double dimensional reduction]], the [[KK-monopoles]] become the [[D6-branes]], and it is evident from the geometry that the membrane warpping $S_{i j}$ becomes an [[open string]] strentching between the $i$th and the $j$th D6-brane.

In the limit $m \to 0$ the D6-branes coincide,the strings stretching between them become massless (in accord with the above formula for the wrapped M2-brane mass), and become the gauge bosons of an $SU(N)$ [[Chan-Paton gauge field]].

([Sen 97, section 2](Sen97))

### $D_{N}$-singularity and $SO(2N)$-gauge enhancement

Now consider the above setupmodified by replacing the [[Taub-NUT space]] with coordinates $(\vec r, x^4)$ by its $\mathbb{Z}/2$-[[orbifold]] given by the $\mathbb{Z}/2$-[[action]] with nontrivial operation given by

$$
  (\vec r, x^4) \mapsto (- \vec r, - x^4)
$$

and in the Taub-NUT metric replace $V$ by

$$
  V
    \coloneqq
  1
    -
  \frac{16m}{r}
    +
  \underoverset{i = 1}{N}{\sum}
  \left(
    \frac{4m}{\vert \vec r - \vec r_i\vert}
    +
    \frac{4m}{\vert \vec r + \vec r_i\vert}
  \right)
  \,.
$$

The type IIA image of the origin of this configuration is an [[orientifold plane]].

Now as the $\vec r_i$ all approach 0 we get an 11d spacetime with a $D_N$-type [[ADE-singularity]], whose type IIA image is $N$ D6-branes coincident on an [[orientifold plane]].

([Sen 97, section 3](Sen97))

## Related concepts

* [[heterotic M-theory on ADE-orbifolds]]

* [[membrane triple junction]]

[[!include F-branes -- table]]

## Refrences

* {#Sen97} [[Ashoke Sen]], _A Note on Enhanced Gauge Symmetries in M- and String Theory_, JHEP 9709:001,1997 ([arXiv:hep-th/9707123](http://arxiv.org/abs/hep-th/9707123))


M-theory on 8-manifolds

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[[!include string theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The [[KK-compactification]] of [[M-theory]] on [[fibers]] which are [[8-manifolds]]. In the low-energy limiting  [[11-dimensional supergravity]] this is KK-compactification to [[3d supergravity]].

Typically this is considered with a [[reduction of the structure group]] on the compactification fiber from [[Spin(8)]] to [[Spin(7)]], in which case one speaks of _M-theory on [[Spin(7)-manifolds]]_ (see the references [below](#ReferencesWithSpin7Structure)). Further reduction to [[G2-structure]] yields [[M-theory on G2-manifolds]].

## Properties

### C-field tadpole cancellation condition

In [[M-theory]] [[KK-compactification|compactified]]  [[compact topological space|compact]] [[8-manifold]] [[fibers]],  [[tadpole cancellation]] for the [[supergravity C-field]] (see also at _[[C-field tadpole cancellation]]_) is equivalently the condition

$$
  N_{M2}
  \;+\;
  \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2
  \;=\;
  \underset{
    I_8(X^8)
  }{
    \underbrace{
      \tfrac{1}{48}\big( p_2 - \tfrac{1}{2}p_1^2  \big)[X^{8}]
    }
  }
  \;\;\;\;
  \in
  \mathbb{Z}
  \,,
$$

where

1. $N_{M2}$ is the net number of [[M2-branes]] in the spacetime (whose [[worldvolume]] appears as points in $X^{(8)}$);

1. $G_4$ is the [[field strength]]/flux of the [[supergravity C-field]]

1. $p_1$ is the [[first Pontryagin class]] and $p_2$ the [[second Pontryagin class]] combining to [[I8]], all regarded here in [[rational homotopy theory]].

If $X^{8}$ has

* [[Spin(7)-structure]] (hence in particular if it is a [[Calabi-Yau manifold]], which has $SU(4) = $ [[Spin(6)]]-structure)

or

* [[Sp(2).Sp(1)-structure]]

then

$$
  \tfrac{1}{2}\big( p_2 - \tfrac{1}{4}(p_1)^2  \big)
  \;=\;
  \chi
$$

is the [[Euler class]] (see [this Prop.](Spin7-manifold#CharacteristicClassesForSpinStructure) and [this Prop.](quaternion-Kähler+manifold#CharacteristicClassesForSpin5Spin3Structure), respectively), hence in these cases the condition is equivalently

$$
  N_{M2}
  \;+\;
  \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2
  \;=\;
  \tfrac{1}{24}\chi[X^8]
  \;\;\;\;
  \in
  \mathbb{Z}
  \,,
$$

where $\chi[X]$ is the [[Euler characteristic]] of $X$.

For references see [there](C-field+tadpole+cancellation#References).

\linebreak

### Relation to F-theory
 {#RelationToFTheory}

If the 8-dimensional [[fibers]] themselves are [[elliptic fibrations]], then [[M-theory]] on these 8-manifolds is supposedly [[T-duality|T-dual]] to [[F-theory]] [[KK-compactification|KK-compactified]] to $3+1$ [[spacetime]]-[[dimensions]].

In particular, if there is an [[M2-brane]] [[wrapped brane|filling]] the base 2+1-dimensional spacetime, this is supposedly [[T-duality|T-dual]] to a 3+1-dimensional spacetime filling [[D3-brane]] in [[F-theory]] (e.g. [Condeescu-Micu-Palti 14, p. 2](#CondeescuMicuPalti14))

For more on this see at _[[F/M-theory on elliptically fibered Calabi-Yau 4-folds]]_ and at _[[F-theory on Spin(7)-manifolds]]_ and at _[[Witten's Dark Fantasy]]_.

([Bonetti-Grimm-Pugh 13a](#BonettiGrimmPugh13a), [Bonetti-Grimm-Pugh 13b](#BonettiGrimmPugh13b), reviewed in [Pugh](#Pugh))

\linebreak

### Black M2-branes and Exotic 7-spheres
 {#BlackM2BranesAndExotic7Spheres}

The discovery of [[exotic 7-spheres]] proceeded via [[8-manifolds]] $X$ [[manifold with boundary|with boundary]] [[homeomorphism|homeomorphic]] to the [[7-sphere]] $\partial X \simeq_{homeo} S^7$, but not necessarily [[diffeomorphism|diffeomorphic]] to $S^7$ with its canonical [[smooth structure]] (for more see [there](8-manifold#ExoticBoundary7Spheres)).

Hence when regarded from the point of view of [[M-theory on 8-manifolds]], exotic 7-spheres arise as [[near horizon limits]] of peculiar [[black brane|black]] [[M2-brane]] [[spacetimes]] $\mathbb{R}^{2,1} \times X$.

See also [Morrison-Plesser 99, section 3.2](#MorrisonRPlesser99).

### Relation to J-twisted Cohomotopy
 {#RelationToJTwistedCohomotopy}

On a [[spin-manifold]] [[8-manifold|of dimension 8]] a choice of topological [[Spin(7)]]-[[G-structure|structure]] is equivalently a choice of [[cocycle]] in [[J-homomorphism|J-]][[twisted Cohomotopy cohomology theory]]. This follows ([[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation|FSS 19, 3.4]]) from

1. the standard [[coset space]]-[[structures]] on the [[7-sphere]] (see [here](7-sphere#CosetSpaceRealization))

1. the fact that [[coset spaces]] $G/H$ are the [[homotopy fibers]] of the maps $B H \to B G$ of the corresponding [[classifying spaces]] (see [here](coset#ForInfinityGroups))

\begin{imagefromfile}
  "file_name": "ExceptionalSpheres.jpg",
  "width": 730
\end{imagefromfile}

## Related concepts

[[!include KK-compactifications of M-theory -- table]]

* [[M2-M5 brane bound state]]

## References

### General

* {#Witten95} [[Edward Witten]], _Strong Coupling and the Cosmological Constant_, Mod. Phys. Lett.A10:2153-2156, 1995 ([arXiv:hep-th/9506101](https://arxiv.org/abs/hep-th/9506101))

  (on possible relation to the [[cosmological constant]]: [[Witten's Dark Fantasy]])

* [[Katrin Becker]], [[Melanie Becker]], _M-Theory on Eight-Manifolds_, Nucl. Phys. B477 (1996) 155-167 ([arXiv:hep-th/9605053](https://arxiv.org/abs/hep-th/9605053))

* [[Savdeep Sethi]], [[Cumrun Vafa]], [[Edward Witten]], _Constraints on Low-Dimensional String Compactifications_, Nucl. Phys. B480: 213-224, 1996 ([arXiv:hep-th/9606122](https://arxiv.org/abs/hep-th/9606122))

### In terms of $G$-structure

Discussion in terms of [[G-structures]]:

* {#IshamPope88} [[Chris Isham]], [[Christopher Pope]], _Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the NSR and GS Superstrings_, Class. Quant. Grav. 5 (1988) 257 ([spire:251240](http://inspirehep.net/record/251240), [doi:10.1088/0264-9381/5/2/006](https://doi.org/10.1088/0264-9381/5/2/006))

  (focus on [[Spin(7)-structure]])

* {#IshamPopeWarner88} [[Chris Isham]], [[Christopher Pope]], [[Nicholas Warner]], _Nowhere-vanishing spinors and triality rotations in 8-manifolds_,  Classical and Quantum Gravity, Volume 5, Number 10, 1988 ([cds:185144](http://cds.cern.ch/record/185144), [doi:10.1088/0264-9381/5/10/009](https://iopscience.iop.org/article/10.1088/0264-9381/5/10/009))

  (focus on [[Spin(7)-structure]])

* {#CondeescuMicuPalti14} Cezar Condeescu, [[Andrei Micu]], [[Eran Palti]], _M-theory Compactifications to Three Dimensions with M2-brane Potentials_, JHEP 04 (2014) 026 ([arXiv:1311.5901](https://arxiv.org/abs/1311.5901))

* Daniël Prins, [[Dimitrios Tsimpis]], _IIA supergravity and M-theory on manifolds with $SU(4)$ structure_, Phys. Rev. D 89.064030 ([arXiv:1312.1692](https://arxiv.org/abs/1312.1692))

* [[Elena Babalic]], [[Calin Lazaroiu]], _Foliated eight-manifolds for M-theory compactification_, JHEP 01 (2015) 140 ([arXiv:1411.3148](https://arxiv.org/abs/1411.3148))

* C. S. Shahbazi, _M-theory on non-Kähler manifolds_, JHEP 09 (2015) 178 ([arXiv:1503.00733](https://arxiv.org/abs/1503.00733))

* [[Elena Babalic]], [[Calin Lazaroiu]], _Singular foliations for M-theory compactification_, JHEP 03 (2015) 116 ([arXiv:1411.3497](https://arxiv.org/abs/1411.3497))

* [[Elena Babalic]], [[Calin Lazaroiu]], _The landscape of $G$-structures in eight-manifold compactifications of M-theory_, JHEP 11 (2015) 007 ([arXiv:1505.02270](https://arxiv.org/abs/1505.02270))

* [[Elena Babalic]], [[Calin Lazaroiu]], _Internal circle uplifts, transversality and stratified $G$-structures_, JHEP 11 (2015) 174 ([arXiv:1505.05238](https://arxiv.org/abs/1505.05238))

### With $Spin(7)$-structure
  {#ReferencesWithSpin7Structure}

Discussion of [[KK-compactification]] on 8-dimensional [[Spin(7)-manifolds]] (see also at _[[M-theory on G2-manifolds]]_ and at _[[F-theory on Spin(7)-manifolds]]_):

* [[Mirjam Cvetic]], [[Gary Gibbons]], H. Lu, [[Christopher Pope]], _New Complete Non-compact Spin(7) Manifolds_, Nucl. Phys. B620: 29-54, 2002 ([arXiv:hep-th/0103155](https://arxiv.org/abs/hep-th/0103155))

* Jaydeep Majumder, _Type IIA Orientifold Limit of M-Theory on Compact Joyce 8-Manifold of Spin(7)-Holonomy_, JHEP 0201 (2002) 048 ([arXiv:hep-th/0109076](https://arxiv.org/abs/hep-th/0109076))

* [[Ralph Blumenhagen]], [[Volker Braun]], _Superconformal Field Theories for Compact Manifolds with Spin(7) Holonomy_, JHEP 0112:013, 2001 ([arXiv:hep-th/0111048](https://arxiv.org/abs/hep-th/0111048))

* {#GukovSparks02} [[Sergei Gukov]], [[James Sparks]], _M-Theory on $Spin(7)$ Manifolds_, Nucl. Phys. B625 (2002) 3-69 ([arXiv:hep-th/0109025](https://arxiv.org/abs/hep-th/0109025))

* [[Sergei Gukov]], [[James Sparks]], [[David Tong]], *Conifold Transitions and Five-Brane Condensation in M-Theory on $Spin(7)$ Manifolds*, Class. Quant. Grav. **20** (2003) 665-706 [[arXiv:hep-th/0207244](https://arxiv.org/abs/hep-th/0207244)]

* [[Melanie Becker]], Dragos Constantin, [[Sylvester James Gates Jr.]], William D. Linch III, Willie Merrell, J. Phillips, _M-theory on $Spin(7)$ Manifolds, Fluxes and 3D, $\mathcal{N}=1\,$ Supergravity_, Nucl. Phys. B683 (2004) 67-104 ([arXiv:hep-th/0312040](https://arxiv.org/abs/hep-th/0312040))

* Dragos Constantin, _M-Theory Vacua from Warped Compactifications on $Spin(7)$ Manifolds_, Nucl. Phys. B706: 221-244, 2005 ([arXiv:hep-th/0410157](https://arxiv.org/abs/hep-th/0410157))

* Dragos Constantin, _Flux Compactification of M-theory on Compact Manifolds with $Spin(7)$ Holonomy_, Fortsch. Phys. 53 (2005) 1272-1329 ([arXiv:hep-th/0507104](https://arxiv.org/abs/hep-th/0507104))

* [[Dimitrios Tsimpis]], _M-theory on eight-manifolds revisited: $N=1$ supersymmetry and generalized $Spin(7)$ structures_, JHEP 0604 (2006) 027 ([arXiv:hep-th/0511047](https://arxiv.org/abs/hep-th/0511047))

* S. Salur, O. Santillan, _New $Spin(7)$ holonomy metrics admiting $G_2$ holonomy reductions and M-theory/IIA dualities_, Phys. Rev. D79: 086009, 2009 ([arXiv:0811.4422](https://arxiv.org/abs/0811.4422))

* Adil Belhaj, Luis J. Boya, Antonio Segui, _Holonomy Groups Coming From F-Theory Compactification_, Int J Theor Phys (2010) 49: 681.  ([arXiv:0911.2125](https://arxiv.org/abs/0911.2125))

* Thomas Bruun Madsen, _$Spin(7)$-manifolds with three-torus symmetry_, J. Geom. Phys. 61: 2285-2292, 2011 ([arXiv:1104.3089](https://arxiv.org/abs/1104.3089))

* [[Federico Bonetti]], [[Thomas Grimm]], [[Tom Pugh]], _Non-Supersymmetric F-Theory Compactifications on $Spin(7)$ Manifolds_, JHEP 01 (2014) 112 ([arXiv:1307.5858](https://arxiv.org/abs/1307.5858))

* [[Federico Bonetti]], [[Thomas Grimm]], [[Eran Palti]], [[Tom Pugh]], _F-Theory on $Spin(7)$ Manifolds: Weak-Coupling Limit_, JHEP02(2014)076 ([arXiv:1309.2287](https://arxiv.org/abs/1309.2287))

* {#Pugh} [[Tom Pugh]], _M-theory on $Spin(7)$-manifolds and their F-theory duals_ ([[PughMTheoryOnSpin7.pdf:file]])

* [[Andreas Braun]], [[Sakura Schaefer-Nameki]], _$Spin(7)$-Manifolds as Generalized Connected Sums and 3d $N=1$ Theories_, JHEP 06 (2018) 103 ([arXiv:1803.10755](https://arxiv.org/abs/1803.10755))

  (generalization of [[compact twisted connected sum G2-manifolds]])

See also

* [[Mariana Graña]], C. S. Shahbazi, [[Marco Zambon]], _$Spin(7)$-manifolds in compactifications to four dimensions_, JHEP 11 (2014) 046 ([arXiv:1405.3698](http://arxiv.org/abs/1405.3698))

Relating [[M-theory on Spin(7)-manifolds]] with [[F-theory on Spin(7)-manifolds]] via [[Higgs bundles]]:

* {#CHRTZ20} [[Mirjam Cvetic]], [[Jonathan Heckman]], Thomas B. Rochais, Ethan Torres, [[Gianluca Zoccarato]], _Geometric Unification of Higgs Bundle Vacua_ ([arXiv:2003.13682](https://arxiv.org/abs/2003.13682))

### With $Sp(2)\cdot Sp(1)$-structure

M-theory on [[HP^2]], hence on a [[quaternion-Kähler manifold]] of dimension 8 with [[holonomy]] [[Sp(2).Sp(1)]], is considered in

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]], p. 75 onwards in  _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

and argued to be [[duality in string theory|dual]] to [[M-theory on G2-manifolds]] in three different ways, which in turn is argued to lead to a possible [[proof]] of [[confinement]] in the resulting 4d [[effective field theory]] (see [there](M-theory+on+G2-manifolds#Confinement) for more).

See also

* {#FSS19b} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 4 of _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]_ ([arXiv:1904.10207](https://arxiv.org/abs/1904.10207))

### Witten's Dark Fantasy

For more on the following see at _[[Witten's Dark Fantasy]]_:

An argument for [[non-perturbative effect|non-perturbative]] non-[[supersymmetry|supersymmetric]] 4d [[string phenomenology]] with fundamentally vanishing [[cosmological constant]], based on 3d [[M-theory on 8-manifolds]] decompactified at strong coupling to 4d via [[duality between M-theory and type IIA string theory]] (recall the [[super 2-brane in 4d]]):

* {#Witten00} [[Edward Witten]], p. 7 of: _The Cosmological Constant From The Viewpoint Of String Theory_, lecture at [DM2000](http://inspirehep.net/record/972507), in: David Kline (ed.) _Sources and detection of dark matter and dark energy in the universe 2000_, Springer 2001. 27-36. ([arXiv:hep-ph/0002297](https://arxiv.org/abs/hep-ph/0002297), [doi:10.1007/978-3-662-04587-9](https://link.springer.com/book/10.1007/978-3-662-04587-9))

  (see p. 7)

* [[Edward Witten]], _Strong coupling and the cosmological constant_, Mod. Phys. Lett. A 10:2153-2156, 1995 ([arXiv:hep-th/9506101](https://arxiv.org/abs/hep-th/9506101))

* [[Edward Witten]], Section 3 of _Some Comments On String Dynamics_, talk at [Strings95](https://cds.cern.ch/record/305869) ([arXiv:hep-th/9507121](http://arxiv.org/abs/hep-th/9507121))

The realization of this scenario in [[F-theory on Spin(7)-manifolds]]:

* [[Cumrun Vafa]], Section 4.3 of: _Evidence for F-Theory_, Nucl. Phys. B469:403-418, 1996 ([arxiv:hep-th/9602022](https://arxiv.org/abs/hep-th/9602022))

* {#BonettiGrimmPugh13a} [[Federico Bonetti]], [[Thomas Grimm]], [[Tom Pugh]], _Non-Supersymmetric F-Theory Compactifications on $Spin(7)$ Manifolds_, JHEP 01 (2014) 112 ([arXiv:1307.5858](https://arxiv.org/abs/1307.5858))

* {#BonettiGrimmPugh13b} [[Federico Bonetti]], [[Thomas Grimm]], [[Eran Palti]], [[Tom Pugh]], _F-Theory on $Spin(7)$ Manifolds: Weak-Coupling Limit_, JHEP 02 (2014) 076 ([arXiv:1309.2287](https://arxiv.org/abs/1309.2287))

* [[Jonathan Heckman]], Craig Lawrie, Ling Lin, [[Gianluca Zoccarato]], _F-theory and Dark Energy_, Fortschritte der Physik  ([arXiv:1811.01959](https://arxiv.org/abs/1811.01959), [doi:10.1002/prop.201900057]( https://doi.org/10.1002/prop.201900057))

* [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Jeremy Sakstein, [[Gianluca Zoccarato]], _Pixelated Dark Energy_ ([arXiv:1901.10489](https://arxiv.org/abs/1901.10489))

### M2-brane spacetimes

* {#MorrisonPlesser99} [[David Morrison]], [[M. Ronen Plesser]], section 3.2 of _Non-Spherical Horizons, I_, Adv. Theor. Math. Phys.3:1-81, 1999 ([arXiv:hep-th/9810201](https://arxiv.org/abs/hep-th/9810201))

### M-theory on $K3 \times K3$

Discussion of M-theory [[KK-compactification|KK-compactified]] on the [[product manifold]] of two [[K3]]s:

* [[Paul Aspinwall]], _An Analysis of Fluxes by Duality_,([arXiv:hep-th/0504036](https://arxiv.org/abs/hep-th/0504036), [spire:679724](https://inspirehep.net/literature/679724))

* [[Paul Aspinwall]], [[Renata Kallosh]], _Fixing All Moduli for M-Theory on $K3 \times K3$_, JHEP 0510:001, 2005 ([arXiv:hep-th/0506014](https://arxiv.org/abs/hep-th/0506014))

  > ([[moduli stabilization]])

* [[Andreas Braun]], Arthur Hebecker, Christoph Ludeling, Roberto Valandro, _Fixing D7 Brane Positions by F-Theory Fluxes_, Nucl. Phys. B815:256-287, 2009 ([arXiv:0811.2416](https://arxiv.org/abs/0811.2416))

Under [[duality between M-theory and type IIA string theory]] this translates to [[D6-branes]] [[wrapped brane|wrapped]] on [[K3]] (enhancon mechanism):

* [[Clifford Johnson]], [[Amanda Peet]], [[Joseph Polchinski]], _Gauge Theory and the Excision of Repulson Singularities_, Phys. Rev. D61:086001, 2000 ([arXiv:hep-th/9911161](https://arxiv.org/abs/hep-th/9911161))

Review:

* Satoshi Yamaguchi, _Enhancon and Resolution of Singularity_ ([arXiv:gr-qc/0108084](https://arxiv.org/abs/gr-qc/0108084))

* Laur Järv, _The enhancon mechanism in string theory_, Durham 2002 ([spire:899784](https://inspirehep.net/literature/899784), [etheses:3981](http://etheses.dur.ac.uk/3981), [[JaervEnhanconMechanism.pdf:file]])

[[!redirects M-theory on Spin(8)-manifolds]]
[[!redirects M-theory on Spin(7)-manifolds]]


M-theory on Calabi-Yau manifolds

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#Contents#
* table of contents
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## Idea

The [[KK-compactification]] of [[11-dimensional supergravity]]/[[M-theory]] on [[Calabi-Yau manifold]] fibers.

### On $CY_3$

For CY fibers of complex dimension 6, this yields the strong coupling version of CY-compactification of [[type II string theory]]/[[type II supergravity]] (see [[supersymmetry and Calabi-Yau manifolds]]).

Leads to [[5d supergravity]] [[effective field theory]] and [[5d super Yang-Mills theory]].

See at _[[5d supergravity]]_ the section _[Via KK-Compactification of 11d Supergravity](5-dimensional+supergravity#ViaCYCompactificationOf11dSupergravity)_

See also at _[[F/M-theory on elliptically fibered Calabi-Yau 3-folds]]_.

### On $CY_4$

* [[F/M-theory on elliptically fibered Calabi-Yau 4-folds]]

## Related concepts

[[!include KK-compactifications of M-theory -- table]]

## References

* [[Sergio Ferrara]], [[Ramzi Khuri]],  [[Ruben Minasian]], _M-theory on a Calabi-Yau manifold_, Phys. Lett.B 375:81-88, 1996 ([arXiv:hep-th/9602102](https://arxiv.org/abs/hep-th/9602102))

* {#CDS18} Cyril Closset, Michele Del Zotto, Vivek Saxena, _Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective_ ([arXiv:1812.10451](https://arxiv.org/abs/1812.10451))

[[!redirects M-theory on CY-manifolds]]

[[!redirects M-theory on Calabi-Yau 3-folds]]