The interaction between mathematics and physics has been increasingly thriving, leading to novel discoveries and perspectives. A deep goal of theoretical physics is to unify the four fundamental forces of nature, described by quantum field theory and general relativity. Einstein spent many years on this quest. While his attempts were not successful, his pursuit was certainly worthwhile and the mantle is being carried today by researchers from modern perspectives. Interestingly, mathematics has been playing an increasingly important role in this quest, allowing to probe deeper by studying emerging structures, classical and quantum.
This workshop brings together experts on the mathematical aspects of M-theory, with implications to string theory and quantum field theory and interactions with geometry and topology.
The workshop will cover presentations from leading researchers from Australia, Czech Republic, France, Germany, India, Singapore, Spain, the UAE, the UK, and the USA.
Title: Derived n-plectic geometry: towards non-perturbative BV-BFV quantisation and M-theory
Abstract: 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.
Title: Matrix Quantization of Classical Nambu Brackets and Super p-Branes
Abstract: 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 𝔰𝔩(N^⌈n/2⌉,ℂ)-matrices equipped with the finite bracket given by the completely anti-symmetrized matrix product, such that the classical brackets are retrieved in the N→∞ 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_∞-algebras in a slight generalization to the construction of Lie 2-algebras from Bagger--Lambert 3-algebras.
Title: Non-invertible symmetry, holography and branes
Abstract: 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)/Z_N N=1 SYM. This will be both from a bottom-up perspective from an effective theory in AdS5, and from a top-down perspective in terms of D-branes in AdS5 backgrounds.
Title: Graded T-duality with H-flux for 2d sigma models
Abstract: 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.
Yang-Hui He, London Institute for Mathematical Sciences & Merton College, Oxford
Title: Machine-Learning Mathematical Structures
Abstract: 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.
Title: Correspondences of Quantum L-infinity Algebras (BV-theories)
Abstract: 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 and the homological perturbation lemma from this point of view.
Abstract: 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.
Title: Exceptional super Yang-Mills in 27 + 3 and worldvolume M-theory
Abstract: 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 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.
Title: T-duality for loop spaces, or equivalently for the 1D sigma model
Abstract: We define exotic twisted S^1-equivariant cohomology for the loop space LZ of a smooth manifold Z via the invariant differential forms on LZ 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 S1-equivariant cohomology of LZ. We establish a localisation theorem for the completed periodic exotic twisted S1-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.
Title: Geometric and Non-Geometric T-duality with Higher Bundles
Abstract: 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. The talk is based on the paper arXiv:2204.01783.
Title: M-theory and matter via twisted equivariant differential (TED) K-theory
Abstract: 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.
Abstract: 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 observables are identified with 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.
Abstract: 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.
Abstract: 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.
Abstract: 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.
Title: Universal accelerating cosmologies from 10d supergravity
Abstract: 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 e-foldings. Moreover, we find several smooth accelerating cosmologies without Big Bang singularities: the universe is contracting in the cosmological past (T < 0), expanding in the future (T > 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.
Title: The E_k symmetry of dimensional reductions of M-theory
Abstract: 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.
Day One: Thursday January 12, 2023
Hisham Sati - Opening & Welcome Remarks
Ashoke Sen (Harish-Chandra Institute, India)
Neil Lambert (Kings College, UK)
Dimitrios Tsimpis (University of Lyon, France)
Day Two: Friday January 13, 2023
Varghese Mathai (University of Adelaide, Australia)
Fei Han (National University of Singapore)
Christian Saemann (Heriot-Watt University, Scotland)
Always be the first to know about what's going on in our community. Sign up for one of our newsletters and receive information on a wide variety of events such as exhibition, lectures, films, art performances, discussions and conferences.