Higher structures in M-theory

Due to the existence of gauge fields and of fermion fields, the geometry of physics is higher supergeometry, i.e. super-geometric homotopy theory. This is made precise via Grothendieck's functorial geometry implemented in higher topos theory. We give an introduction to the higher topos of higher superspaces and how it accommodates super-L1-algebras and higher gauge fields in the form of twisted differential cohomology. We indicate how geometric homotopy theory reveals that the superpoint emerges `from nothing', and that core structure of M-theory emerges out of the superpoint, as will be discussed in more detail in talks by Hisham Sati, Vincent Braunack-Mayer, Domenico Fiorenza, and John Huerta. Lecture notes at https://ncatlab.org/schreiber/show/Introduction+to+Higher+Supergeometry.

Recent developments in formulating higher gauge theory with Lie quasi-groupoids as gauge structure will be reviewed. The approach develops higher gauge theory from first principles, and, as such, captures a wide class of theories including ordinary gauge theories and gauged sigma models as well as their categorifications: it will be explained how these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of quasi-isomorphisms. This is based on joint work with Branislav Jurco and Christian Samann.

will review the problem of integration and differentiation in higher Lie theory, i.e. he relation between higher Lie algebroids (dg manifolds) and higher Lie groupoids (simplicial manifolds), and explain, in particular, why integration gives smooth (i.e. Lie) higher groupoids, and why differentiation is inverse to integration (results obtained with Michal Siran). I will then turn to a related, but more speculative topic|why seeing non-topological field theories as living on the boundary of topological ones (of the AKSZ type) is useful for understanding of Poisson Lie T-duality and (hopefully) of more general dualities.

We will review the concept of anomaly field theory, which encodes elegantly all the anomalies associated to a given quantum field theory. We will show how the familiar properties of anomalies can be naturally derived in this context and will illustrate the concept with concrete examples. In this framework, Hamiltonian anomalies originate from higher structures, as the shadow of a 2-functor. We will also explain how this framework provides a conceptual picture of the Green-Schwarz mechanism, and its use to study field theories with no single partition function, such as 2d chiral conformal field theories or the 6d (2,0) superconformal field theories. We will close the talk with a review of various string theory setups where the checks of anomaly cancellation are still open problems.

As it is known, L1-morphisms of L1-algebras are precisely maps of the corresponding supermanifolds which are in general non-linear. The non-linearity is what is responsible for (higher) homotopies. Therefore, in the situation of homotopy brackets on functions (such as S1 or P1) one should look for a construction giving non-linear transformations between the spaces of smooth functions. This cannot be pullbacks by smooth maps because those are algebra homomorphisms, so in particular, linear. We shall show that there are `non-linear pullbacks' of functions on supermanifolds induced by `thick morphisms' (or `microformal morphisms'), which are not ordinary maps, but rather certain relations. For S1- or P1-manifolds, pullbacks by `Poisson thick morphisms' give L1-morphisms on functions. There are two parallel constructions of thick morphisms (`bosonic' and `fermionic'). Also, there is a quantum version of thick morphisms (for the bosonic case) given, roughly, by certain type Fourier integral operators. They in particular provide L1-morphisms for `homotopy quantum brackets' generated by higher order Batalin{Vilkovisky type operators.

Recent investigations of M-branes have revealed how rich in structure they are from both a physical and mathematical points of view. It is hoped that such structures will help us shed light onto the physical and mathematical nature of M-theory itself. I will survey this area, starting with connections to generalized cohomology and homotopy theory (twisted K-theory, Morava K-theory, elliptic cohomology and the sphere spectrum) and then including higher geometry and higher structures, leading to description of multiple M-branes via String bundles with connections. I will also describe recent work on derivation of twisted K-theory from M-theory via parametrized homotopy theory and the proper topological description of M-branes at ADE singularities. Various aspects of this are part of joint work with other speakers at this meeting (with related talks): Urs Schreiber, Vincent Braunack-Mayer, Domenico Fiorenza, and John Huerta.

In this talk we will review the construction of M2-brane SCFT’s highlighting some novelties and the role of 3-algebras. Next we will discuss M5-branes: the basics, the obstacles as well as various attempts to construct the associated SCFT and potential relations between M2-branes and M5-branes.

We review old and new developments in 6d SCFTs. We first briefly discuss the constructions of (2, 0) and (1, 0) SCFTs from string theory, M-theory, F-theory. We then review microscopic methods to describe these QFTs in various set-ups, such as DLCQ, deconstruction, effective field theory methods in various limits. We shall conclude by overviewing interesting open directions.

In this talk, I will discuss a newly discovered effect in BCFT where a background gauge field strength induces a current in the vicinity of the boundary. In 6d, this give rises to a string current. Its implication will be discussed.

In this talk, we will unravel the ground states and left-excited states of the A_{k-1} N=(2,0) little string theory. Via a theorem by Atiyah, these sectors can be captured by a supersymmetric quasi-topological sigma model on CP^1 with target space the based loop group of SU(k). The ground states, described by L^2-cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. We also apply our results to unravel the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory, which spectrum we find to be captured by cousins of modular and automorphic forms.

We will review physical effects that three-form gauge fields may produce in four-dimensional field theories and gravity, and discuss possible origin of these fields from Type-II string compactifications on Calabi-Yau manifolds with Ramond-Ramond fluxes. We will also consider couplings of the three-form fields to membranes within N=1, D=4 supergravity and give examples of BPS domain wall solutions that separate supersymmetric vacua with different values of the cosmological constant.

In this talk we will discuss three (intimately related) examples of 4-manifold invariants based on higher structures: - VOA[M4] from transgression of EFTs; - SW and Donaldson invariants as chiral algebra correlators; - Massey triple products from trisections. These topics are based, respectively, on recent work with B.Feigin; paper with M.Dedushenko and P.Putrov; and a solo paper of the speaker.

In algebraic quantum field theory we assign to each spacetime an algebra. These algebras are required to behave properly under embeddings into larger spacetimes; specifically, the algebras associated to casually disjoint spacetimes must commute. We use coloured operads to formalise this property.

I will give a biased definition of a properad and recall the construction of the cobar complex and algebra over it. Equivalent description in terms of solutions of generalised master equations, which can be interpreted as homological differential operators, will be explained. This is parallel to the Barannikov's theory for modular operads.

We describe how the super Minkowski spacetimes relevant to string theory and M-theory, complete with their Lorentz metrics and spin structures, emerge from a much more elementary object: the superpoint. In the sense of higher structures, this comes from treating the superpoint as an object in a flavor of rational homotopy theory, and repeatedly constructing "maximal invariant extensions". We will fit this story into the larger picture of the brane bouquet of Fiorenza-Sati-Schreiber: string theories and membrane theories emerge from super Minkowski spacetimes in precisely the same way as the super Minkowski spacetimes themselves emerge from the superpoint.

According to the Dirac charge quantisation argument, flux forms in a quantum theory lift to define classes in generalised cohomology. The classical example of this is the refinement of the electromagnetic field strength to a class in integral cohomology. Another example is the expected refinement of Ramond-Ramond flux forms to classes in K-theory or, when the background B-field is non-trivial, to classes in twisted K-theory. In this talk, I will provide a broad overview of the true mathematical home of twisted cohomology-parametrised homotopy theory. Following this general overview, I will present recent results, https://user.math.uzh.ch/cattaneoBraunack-Mayer.pdf, in the simplified torsion-free approximation which provide a framework for explicit algebraic computations. Finally, I will explain recent work with Hisham Sati and Urs Schreiber, arXiv:1806.01115, which applies this general theory to obtain a partial solution to the problem of gauge enhancement in M-theory, which makes M-branes exhibit the twisted K-theory degrees of freedom of D-branes in type IIA string theory.

A pair of 2-cocycles on an $L_\infty$-algebra together with a trivialization of their product induce a Fourier-type transform between the twisted cohomologies of the central extensions classified by the 2-cocycles themselves. On the one hand, this can be read as the image in rational homotopy theory of topological T-duality for principal U(1)-bundles. On the other hand, the result is completely algebraic and can be easily derived without assuming the existence topological T-duality. In this sense, the algebraic result can be seen to suggest the topological one as the nontrivial globalization of a simple infinitesimal construction. Even more interestingly, the algebraic construction admits an immediate generalization to super-$L_\infty$-algebras and to higher cocycles. This allows for a derivation of Hori's formula and of spherical T-duality for M5-branes directly from the structure of super-Minkowski cocycles occurring in superstring and M-theory. Joint work with Hisham Sati and Urs Schreiber (arXiv:1611.06536; arXiv:1803.05634).

Self-dual strings are categorified analogues of monopoles. They are expected to arise as BPS states in the long-sought six-dimensional superconformal field theory known as the (2,0)-theory, which describes the effective dynamics of parallel stacks of M5-branes. While the description of abelian self-dual strings is well-known, the non-Abelian generalisation remains somewhat controversial. In this talk, I give a mathematically consistent description of non-Abelian self-dual strings. I also present a six-dimensional superconformal field theory with many of the properties of the (2,0)-theory which has these self-dual strings as a BPS solution. I conclude by discussing the open problems that need to be solved in order to arrive at a satisfying M5-brane model.

Homotopy algebra and its involutive generalisation plays an important role in the consontruction of string field theory. It ensures consistemcy and also enters crucially in deformation theory and background independence of string theory. Conversely, world sheet string theory naturally realizes a minimal model map. I will review recent progress in these applications of homotopy algebra, their operadic description and its relation to moduli spaces.

The talk will give an overview of DFT and EFT with an emphasis on recent progress and open problems.

I review L1-algebras and their role in field theories, specifically for duality-covariant formulations of string- and M-theory (double field theory and exceptional field theory).

For any Kac-Moody algebra g and any integrable highest weight representation of it, generalised diffeomorphisms can be defined, including those in ordinary, double, and exceptional geometry when g belongs to the A-, D-, or E-series (and the highest weight is the fundamental weight associated to the `outermost' node in the Dynkin diagram). In the cases where they close into themselves, the gauge structure of the transformations can be derived by extending g to a Borcherds-Kac-Moody superalgebra. Otherwise, when so called `ancillary' g transformations are present (in particular when g is infinite-dimensional), the correct structure seems rather to be a tensor hierarchy algebra. Tensor hierarchy algebras include finite-dimensional Lie superalgebras of Cartan type in the A and D cases, and have proven useful in applications to gauged maximal supergravity in the E cases. However, their general structure is still largely unexplored and many questions remain to be answered.

Courant algebroids generalize quadratic Lie algebras in a natural way and they find their application throughout the mathematical physics. In particular, an analogue of Levi-Civita connections can be used to find a geometrical description for equations of motion of string low-energy effective actions. This observation allows one to employ the tools of geometry to derive some intriguing relations of the effective theories. As an example, the supergravity analogue of Kaluza-Klein reduction and (quasi)-Poisson-Lie T-duality are shown.

Exceptional field theories are manifestly duality covariant formulations of higher-dimensional supergravity. I will review the recent construction of such theories based on infinite-dimensional affine symmetry algebras, notably the case of the affine algebra E9, which is the symmetry of maximal two-dimensional supergravity. I describe the construction of its generalised diffeomorphisms and of its dynamics.

NQ-manifolds and the derived bracket construction give a unified view on (infinitesimal) symmetries in field theory: In degree one these are Lie algebroids and in degree two Courant algebroids. They describe infinitesimal symmetries in Riemannian geometry and generalized geometry (gravity and type II/heterotic supergravity). A modified version of the degree two case gives insight into double field theory. In degree three and higher, we get degree-k Vinogradov algebroids governing certain types of exceptional field theories. We will give an overview of these cases emphasizing their relation to finite term strongly homotopy Lie algebras. As a first step towards global questions, we present an NQ-manifold description of three-dimensional nilmanifolds carrying a U(1) gerbe structure and its implications for the well-studied T-duality in this case.

I will briefly discuss the generalised geometry formulation of supergravity and some of its unresolved issues and major goals, with an emphasis towards its applications to the study of supersymmetric flux backgrounds and dimensional reductions. I will also discuss some recent work on the moduli of N = 1 heterotic geometries, featuring holomorphic generalised geometry structures and associated L1-algebras, and highlight some of the questions that emerge.

In higher gauge theory one often encounters the fake curvature condition, i.e. the requirement that the fake curvature vanishes. In this talk I will briefly discuss twisted versions of the string algebra and, using the example of the non-Abelian self-dual string, demonstrate how these allow the fake curvature condition to be lifted: under categorical equivalences gauge orbits of solutions are mapped to gauge orbits of solutions even for non-vanishing fake curvature.

The vacua obtained by compactifying the heterotic string at large radius|while also preserving a certain amount of supersymmetries|can be labelled by parameters that are coordinates for a moduli space of solutions. It is our interest to study this moduli space, both from physics and mathematics perspective. We do this by embedding the heterotic vacua and their moduli space inside a single geometry, that we name `universal geometry', and by appropriately extending the heterotic fields to this space. The deformation theory is then described via deferential geometry on a `universal bundle' and we can study natural extensions of some equations. Remarkably, these can incorporate results obtained with more conventional methods in single tensor equations

Higher spin gauge theory is a theory exhibiting higher symmetries that can become manifest at ultra high (trans-Planckian) energies. Hence, it is anticipated to be related to quantum gravity. Holographic interpretation of nonlinear higher spin gauge theory will be discussed and its main properties and structures such as unfolded formulation of dynamical equations and non-commutative higher spin algebra in the twistor-like spinor space will be reviewed. The emphasise will be on the peculiarities of the spacetime interpretation of higher spin gauge theory and its potential relation to string and M-theory via a new class of models associated with Coxeter groups and Cherednik algebras.

In this talk, we will give a short overview of results revolving around sigma models on gerbes and their appearance in string theory. We will briefly review aspects of the corresponding quantum field theories, including 2-group symmetries as well as decomposition. We will also briefly review the appearance of gerbes in moduli spaces of superconformal field theories and supergravity theories, focusing on moduli spaces of elliptic curves. In string theory, the ordinary moduli space of elliptic curves is replaced by a gerbe over that moduli space in order to make Ramond vacua well-defined in families, and we will discuss the implications for string dualities.

Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying L1-algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory. In this talk I will discuss the L1-bootstrap program: the basic ideas, construction, including the recurrence relations for Lgauge 1 -algebra, and uniqueness. As a particular examples we construct the explicit expressions for the non-commutative su(2)-like and non-associative octonionic-like deformations of the abelian gauge transformation in slowly varying field approximation. The latter is related to non-geometric backgrounds in string and M-theory

AKSZ sigma models were originally proposed to describe topological systems. In fact, an AKSZ model with finite number of fields and space-time dimension higher than one is necessarily topological. It turns out that AKSZ formalism extends to general gauge theories if one allows for infinite- dimensional target space manifolds. More specifically, a general gauge system can be cast into an AKSZ sigma model whose target space is the BRST-extended equation manifold equipped with the BRST differential and the horizontal differential. The resulting AKSZ formulation can be also considered as a far going generalization of the so-called unfolded formulation known in the context of higher-spin theories. We employ the ambient space version of the approach to (conformal) higher spin theories and their holographic relations. In particular, we show how interacting higher spin theories in anti-de Sitter space can be holographically reconstructed starting from a free CFT on the boundary. We also comment on a possible interpretation of the construction as a higher spin extension of the Fefferman-Graham approach.

Logarithmic conformal field theories are based on vertex algebras with non-semisimple representation categories. While examples are known for more than 25 years, some aspects of local logarithmic CFTs have been understood only recently, in a description of conformal blocks by non-semisimple modular functors. We present some results on bulk fields and boundary states in logarithmic theories. We then describe some recent results towards a derived modular functor. This is a summary of work with Jürgen Fuchs, Terry Gannon, Simon Lentner, Svea Mierach, Gregor Schaumann and Yorck Sommerhäuser.

AQFT is a well-established rigorous framework to describe quantum field theories on Lorentzian manifolds. Unfortunately, its traditional formulation seems to be too rigid to capture some finer (higher) aspects of quantum gauge theories. In this talk I will provide an overview of our recently initiated homotopical AQFT program whose aim is to address and solve these issues by extending AQFT to a higher categorical/homotopy theoretical framework.

We will overview some approaches to the quantisation of systems whose dynamics are governed by twisted Poisson structures, using the simple model of an electric charge in the background of a smooth monopole distribution (or dually of closed strings in locally non-geometric backgrounds) as an illustrative example. We will compare and contrast approaches to this problem based on deformation quantisation and symplectic realisation, to more geometric constructions based on weak projective 2-representations on the 2-Hilbert space of sections of a suitable gerbe and on the universal enveloping A_infinity-algebra of a twisted Lie algebroid.

We discuss the quantisation of higher gauge theories in the context of Batalin{Vilkovisky formalism which allows for treating the case of gauge theories with an open gauge algebra. In particular, we present the homotopy Maurer{Cartan theory and we discuss how the Batalin{Vilkovisky formalism applies to it. We argue that any Batalin{Vilkovisky quantisable field theory can be discussed in this setup. This is a joint work with Branislav Jurco, Christian Samann, and Martin Wolf.

Gauge theories describing the same physics are supposed to be related by a Seiberg-Witten map. It is shown that this is equivalent to having a specifc quasi-isomorphism between the underlying L1-algebras of the gauge theories. In addition, the proof suggests an extension of the definition of a Seiberg-Witten map.