nLab M-Theory and Mathematics

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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.

M-Theory and Mathematics 2023

• on: January 12-15, 2023

  at: CQTS @ New York University, Abu Dhabi

  see: event webpage

$\phantom{-----}$ (logo adapted from JMP 62 (2021) 042301)

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

(from left to right: Valera, Jurčo, Berman, Alfonsi, Han, Schreiber, Mathai, Saemann, Sati, Voronov, Lambert, Bah, Tsimpis, Hull, Marrani, He, Singh, Giotopoulos, Szabo, )

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

• Ashoke Sen:

  D-instanton amplitudes in string theory

  slides: pdf

  video: YT

  cf. arXiv:2204.02981, arXiv:2207.07138

  D-instantons give non-perturbative contribution to string amplitudes. While the world-sheet theory of 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-instanton correction to string amplitudes.

• Neil Lambert:

  A non-Lorentzian View of the M5-brane

  slides: pdf

  video: YT

  cf.: arXiv:2011.06968, arXiv:2212.07717

  In this talk we will discuss a path-integral formulation of the M5-brane (2,0) theory based upon a Lagrangian without Lorentz invariance but a novel $SU(1,3)$ spacetime symmetry. This provides a UV complete five-dimensional field theoretic description that we will argue admits a six-dimensional Lorentz invariance at strong coupling.

• Dimitrios Tsimpis:

  Universal accelerating cosmologies from 10d supergravity

  slides: pdf

  video: YT

  cf.: arXiv:2210.10813

  4d Friedmann-Lemaître-Robertson-Walker cosmologies are obtained from time-dependent compactifications of 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 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 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 in hyperbolic slicing. We also obtain several cosmologies featuring an infinite number of cycles of alternating periods of accelerated and decelerated expansions.

• Varghese Mathai:

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

  slides: pdf

  video: YT

  cf.: arXiv:1405.1320

  We define exotic twisted $S^1$-equivariant cohomology for the 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) holonomy line bundle of a gerbe, with differential given by an equivariantly 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 $S^1$-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 T-duality in a background flux in type II String Theory from a loop space perspective. Finally we reinterpret these results as T-duality for the 1D sigma model.

• Fei Han:

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

  slides: pdf

  cf.: arXiv: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 established the 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 double loop space of spacetimes. This represents our joint work with Mathai.

• Christian Saemann:

  Geometric and Non-Geometric T-duality with Higher Bundles

  slides: pdf

  video: YT

  cf.: arXiv:2204.01783

  I will introduce the notion of adjusted curvatures for higher principal bundles. These will allow me to differentially refine the description of topological T-duality by Nikolaus and Waldorf 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 cases.

• David Berman:

  Exact Renormalisation Group and statistical Inference

  slides: pdf

  video: YT

  cf.: arXiv:2212.11379

  The exact renormalisation group is used to describe the process of statistical inference in the Bayesian sense.

• Yang-Hui He:

  Machine-Learning Mathematical Structures

  slides: pdf

  video: YT

  cf.: arXiv: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.

• Eric Sharpe:

  An introduction to decomposition

  slides: pdf

  video: YT

  cf.: arXiv:2204.09117

  In this talk I will review work on ‘decomposition’, a property of 2d theories with 1-form symmetries and, more generally, d-dim’l theories with $(d-1)$-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.

• Richard Szabo:

  D-branes and doubled geometry

  slides: pdf

  video: YT

  cf.: arXiv: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 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 generalised geometry.

• Chris Hull:

  Charges and Duality in Gravity

  slides: pdf

  video: YT

  cf. arXiv:2209.11716 and refs here

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

• Branislav Jurčo:

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

  video: YT

  cf.: arXiv:1912.06695, arXiv:2002.11168

  The quantum odd symplectic "category" is constructed from the odd symplectic “category” via an enhancement of Lagrangian relations (correspondences) by half-densities. We extend this approach to the (finite-dimensional) setting of the 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.

• Meer Ashwinkumar:

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

  video: YT

  cf.: arXiv: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. This action exhibits a reduced gauge symmetry that we discuss from the viewpoint of $L_\infty$-algebras in a slight generalization to the construction of Lie 2-algebras from Bagger-Lambert 3-algebras.

• Ibrahima Bah:

  Non-invertible symmetry, holography and branes

  video: YT

  cf.: arXiv: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 perspective from an effective theory in $AdS_5$, and from a top-down perspective in terms of D-branes in $AdS_5$ backgrounds.

• Hisham Sati:

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

  video: YT

  cf.: arXiv:2206.13563

  Various flavors of K-theory play an important role in geometric/topological perspectives on string theory. Ultimately, the twisted equivariant differential (TED) version of K-theory would be what is needed to describe 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 (topological phases) and highlight analogies with high energy physics. All of this embeds in M-theory via Hypothesis H, both from physical and mathematical perspectives.

• Urs Schreiber:

  Topological Quantum Gates from M-Theory

  notes: pdf

  video: YT

  cf.: arXiv:2203.11838

  The promising idea that strongly coupled quantum systems such as (confined QCD or) topologically ordered quantum materials are usefully modeled as worldvolume dynamics on intersecting branes in string theory has been suffering from the latter’s lack of non-perturbative formulation (M-theory), necessary in the realistic regime of small numbers of individual branes (i.e. beyond the usual holographic large-$N$ approximation). In this talk I will briefly review our “Hypothesis H” that brane charge in M-theory is quantized in a twisted equivariant non-abelian generalized cohomology theory called Cohomotopy, where M-brane quantum states are identified inside the twisted equivariant cohomology of the Cohomotopy moduli stack. Then I explain our recent derivation, from this assumption, of anyonic topological order in ground states of M5-brane intersections; and I close with an outlook on how this describes topological quantum logic gates via braiding of defect branes.

• Alexander Voronov:

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

  video: YT

  cf.: arXiv:2111.14810, arXiv:2212.13968

  Hisham Sati and I, on the way to revealing the mystery of 2001 Mysterious Duality of Iqbal, Neitzke, and Vafa, generalized hypothesis H and presented a universal target for each 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 action of the Lie algebra of type $E_k$, responsible for symmetries of $(11-k)$-dim supergravity along with its equations of motion in any spacetime.

• Alessio Marrani:

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

  slides: pdf

  video: YT

  cf.: arXiv:1906.10709

  Some time ago, 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 Horowitz and 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 Ramond‘s and Sati’s argument that M-theory has hidden Cayley plane fibers.

• Luigi Alfonsi:

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

  cf.: arXiv:2307.15106

  slides: pdf

  video: YT

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

  A derived $n$-plectic structure is a derived and categorified generalisation of a symplectic structure, where the usual closed 2-form is replaced by a shifted closed $(n+1)$-form on a 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 homotopy-algebraic structure of observables of a derived $n$-plectic space. Finally, I will point at promising applications of this formalism to non-perturbative quantisation of BV-BFV theories and to global aspects of M-theory.

---

M-Theory and Mathematics 2020

• on: January 27-30, 2020

• at: Research Institute of New York University, Abu Dhabi

• see: event webpage

• and: mtheorymath.org

• Program

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

Abstracts, Slides and Talk notes:

• Michael Duff:

  Perspectives on M-Theory

  video: YT

• José Figueroa-O’Farrill

  Lie superalgebra deformations and d=11 supergravity backgrounds

  slides: pdf

  video: YT

  To every (supersymmetric) background of d=11 supergravity there is associated a filtered Lie superalgebra. For ($\gt 1/2$)-BPS backgrounds, this Lie superalgebra determines the background up to local isometry. I will then report on an algebraic reformulation of this classification problem. This is joint work with Andrea Santi.

• William Linch III

  Off-shell Supersymmetry and the M-theory Effective Action

  video: YT

  cf.: arXiv:2101.11671

• Hisham Sati,

  M-theory and cohomotopy

  slides: pdf

  Cohomotopy theory has recently emerged as the proper generalized cohomology theory to describe the 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 C-field and its dual as well as of the M-branes, allows for cancellation of various anomalies, and provides a firm grounding for further study of M-theory. This is joint work Urs Schreiber and Domenico Fiorenza.

• Urs Schreiber:

  Microscopic Brane Physics from Cohomotopy

  slides: pdf

  video: YT

  As reviewed in H. Sati‘s talk, assuming that the C-field is charge-quantized in the generalized cohomology theory called J-twisted Cohomotopy (“Hypothesis H”) implies a list of M-theoretic anomaly cancellation conditions, such as shifted C-field flux quantization, DMW anomaly cancellation and C-field tadpole cancellation on 8-manifolds. In this talk I review the further geometric refinement of the cohomology theory to equivariant Cohomotoy theory and to differential Cohomotopy theory. Now we find that Hypothesis H implies also the Witten mechanism of multiple M5-branes on MO5-orientifolds in heterotic M-theory on ADE-orbifolds, hence RR-field tadpole cancellation in type I' string theory; as well as a multitude of effects associated with Dp/D(p+2)-brane intersections: Chan-Paton factors, BMN matrix model fuzzy funnel states and BLG 3-algebras, the Hanany-Witten rules, AdS3-gravity observables, supersymmetric indices of Coulomb branches as well as gauge/gravity duality between all these. This suggests that Hypothesis H is a correct assumption about the elusive mathematical fomulation of M-theory.

This is joint work with H. Sati (arxiv:1909.12277, arxiv:1912.10425).

• Chris Hull:

  Dualities, K3, Exotic Branes and Orientifolds

  video: YT

  cf.: arXiv:1907.04040

• Henning Samtleben:

  Exceptional Field Theories and AdS Compactifications

  video: YT

• Bianca Cerchiai

  Supergravity in a pencil

  slides: pdf slides

  video: YT

  In the 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 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 hep-th]. 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 OSp(p|2)×OSp(q|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 parameters.

• Neil Lambert,

  Lagrangians with $(2,0)$ Supersymmetry

  slides: pdf

  video: YT

  cf.: arXiv: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.

• Christian Saemann:

  Towards an M5-Brane Model: Progress Report

  video: YT

  cf.: arXiL1712.06623, arXiv:1908.08086, arXivL2012.09253

• Domenico Fiorenza:

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

  slides: pdf

  video: YT

  cf.: arXiv:1906.07417

  The 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 Wess-Zumino term. We prove that such an anomaly cancellation condition is implied by the hypothesis that the C-field is charge-quantized in twisted Cohomotopy theory. The proof follows by a twisted/parametrized generalization of the Hopf invariant, after identifying the 6d Wess-Zumino term with a twisted homotopy Whitehead integral formula, which we establish. Joint work with Hisham Sati and Urs Schreiber, arXiv:1906.07417.

• David Berman:

  Strings, Branes, and Boltzmann machines

  video: YT

• Yang-Hui He:

  Universes as Bigdata

  slides: pdf

  video: YT

  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 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 can help explore problems of physical and mathematical interest.

• Mathai Varghese:

  T-Duality, loop spaces and Witten gerbe modules

  video: YT

  cf.: arXiv:2001.00322

• Dmitri Sorokin:

  What do SYM domain walls look like?

  video: TY

  cf.: arXiv:2004.11232

• Ashoke Sen:

  Gravitational waves from soft theorem

  video: YT

  cf.: arXiv:1912.06413

  on the soft graviton theorem

Related events

• Higher Structures in M-Theory 2018

category: reference