NYUAD | Jan 27-30, 2020

View the Project on GitHub Hisham-sati/M-theory-and-Mathematics

The goal of this workshop is to bring together experts on the mathematical aspects of M-theory, with implications to string theory and quantum field theory and interactions with geometry and topology.

**David Berman**(Queen Mary University of London, UK):

*Machine learning and M-theory?***Peter Bouwknegt**(The Australian National University, Australia):

*Higher spin algebras and representation theory***Martin Cederwall**(Chalmers University of Technology, Sweden):

*Supersymmetry, nilpotent varieties and superalgebras***Bianca Letizia Cerchiai**(Politecnico di Torino & Centro Fermi Roma, Italy):

*Supergravity in a Pencil***Michael Duff**(Imperial College, UK):

*Perspectives on M-theory***José Figueroa-O’Farrill**(University of Edinburgh, UK):

*Lie superalgebra deformations and d=11 supergravity backgrounds***Domenico Fiorenza**(Sapienza University of Rome, Italy):

*Twisted cohomotopy and the level quantization of the 6d Wess-Zumino term***Pietro Grassi**(University del Piemonte Orientale, Italy):

*The Integral Form of Supergravity***Fei Han**(National University of Singapore):

*Projective elliptic genera and elliptic pseudodifferential genera***Yang-Hui He**(City, University of London & Merton College, Oxford University, UK):

*Universes as Bigdata: Superstrings, Calabi-Yau Manifolds and Machine-Learning***Chris Hull**(Imperial College, UK):

*Dualities, K3, Exotic Branes and Orientifolds***Branislav Jurco**(Charles University, Czech Republic):

*Homotopy algebras in string field theory***Neil Lambert**(Kings College, UK):

*Lagrangians with (2,0) supersymmetry***William Linch III**(Texas A & M University, USA):

*Off-shell Supersymmetry and the M-theory Effective Action***Varghese Mathai**(University of Adelaide, Australia):

*T-duality, loop space and Witten gerbe modules***Christian Saemann**(Heriot-Watt University, UK):

*Towards an M5-Brane Model: Progress Report***Henning Samtleben**(ENS de Lyon, France):

*Exceptional field theories and AdS compactifications***Hisham Sati**(NYUAD):

*M-theory and cohomotopy***Urs Schreiber**(NYUAD and Czech Academy of Sciences):

*Microscopic brane physics from Cohomotopy theory***Ashoke Sen**(Harish-Chandra Research Institute, India):

*Gravitational waves from soft theorem***Eric Sharpe**(Virginia Tech, USA):

*Decomposition of two-dimensional theories with one-form symmetries***Dmitri Sorokin**(University of Padova, Italy):

*How SYM domain walls look like?***Meng-Chwan Tan**(National University of Singapore):

*Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes in String Theory*

Jan 27-30 | MON | TUE | WED | THU |
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9:00-10:00 | Duff | Hull | Lambert | Mathai |

10:10-11:10 | Figueroa-O’Farrill | Samtleben | Saemann | Han |

11:10-11:30 | Break 1 |
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11:30-12:30 | Sati | Cerchiai | Fiorenza | Tan |

12:30-2:00 | Lunch: Torch Club |
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2:00-3:00 | Grassi | Bouwknegt | Berman | Sorokin |

3:10-4:10 | Linch | Cederwall | He | Sharpe |

4:10-4:30 | Break 2 |
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4:30-5:30 | Schreiber | Jurco | Excursion |
Sen (Special lecture) |

**David Berman: Machine learning and M-theory?**

Abstract: TBD

**Peter Bouwknegt: Higher spin algebras and representation theory**

In this talk I will discuss higher spin algebras from the point of view of universal enveloping algebras and their ideals.

**Martin Cederwall: Supersymmetry, nilpotent varieties and superalgebras**

Supermultiplets can be constructed as the cohomology of a “pure spinor” superspace BRST operator. The pure spinor constraint is, in many cases, related to a superalgebra via Koszul duality. I will sketch the generalisation of this statement to situations when one instead obtains an L-infinity algebra. I will also describe how the partition function for the pure spinor encodes the full content of the supermultiplet. The same mathematics is relevant for extended geometry.

**Bianca Letizia Cerchiai: Supergraphity: Supergravity in a Pencil**

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.

**Michael Duff: Perspectives on M-theory**

Abstract: We provide a historical perspective on the development of M-theory as well as perspectives on the future of the theory and its possible relation to mathematics.

**José Figueroa-O’Farrill: Lie superalgebra deformations and d=11 supergravity backgrounds**

To every (supersymmetric) background of d=11 supergravity there is associated a filtered Lie superalgebra. For (>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.

**Domenico Fiorenza: Twisted cohomotopy and the level quantization of the 6d Wess-Zumino term**

The 6d Wess-Zumino term in the action functional for theM5-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 thatt he 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.

**Pietro Grassi: The Integral Form of Supergravity**

We discuss a new framework to formulate supersymmetric quantum field theories which encompasses all possible superspace descriptions, in any dimensions and for any extended version of supersymmetry. We will describe several examples from rigid supersymmetric models to supergravity models in higher dimensions. We also discuss their quantization and the BV-BRST formalism in this new framework. Finally, the case of 11 dimensional supergravity is also discussed.

**Fei Han: Projective elliptic genera and elliptic pseudodifferential genera**

Elliptic genera is a modular form valued topological invariant, which lies on the intersection of Atiyah-Singer index theory, loop space, infinite dimensional Lie algebra and quantum field theory. Given a projective vector bundle, we will show how to use it to twist the usual elliptic genera to get projective elliptic genera and give them analytic interpretation via the projective index theory of Mathai-Melrose-Singer. As an application, we will describe how to construct the elliptic pseudodifferential genera for any elliptic pseudodifferential operator. This represents our joint work with Mathai.

**Yang-Hui He: Universes as Big data: Superstrings, Calabi-Yau Manifolds and Machine-Learning**

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.

**Chris Hull: Dualities, K3, Exotic Branes and Orientifolds**

Abstract: TBD

**Branislav Jurco: Homotopy algebras in string field theory**

(Quantum) homotopy algebras (Lie, Associative, involutive bi-Algebras,…) and their relevance to string field theory will be reviewed.

**Neil Lambert: Lagrangians with (2,0) supersymmetry**

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

**William Linch III: Off-shell Supersymmetry and the M-theory Effective Action**

The leading part of the M-theory low-energy effective action has been conjectured to be the minimal supersymmetric completion of a certain curvature-to-the-fourth correction to eleven-dimensional supergravity. This idea is difficult to exploit, because a component Noether procedure would have an infinite number of steps of ever-higher order in the curvature expansion. In an off-shell superspace, the supersymmetry closes independently of the interactions, but any 11D, N=1 superspace would put the theory on-shell. In this talk, I will describe an ongoing program to construct an 11D, N = 1/8 supergeometry that is potentially “sufficiently off shell” to supersymmetrize the leading M-theory correction to the eleven-dimensional supergravity action.

**Varghese Mathai: T-duality, loop space and Witten gerbe modules**

I will talk about my recent joint work with Fei Han, [arXiv:2001.00322], where we extend what is now known as topological T-duality isomorphism on T-dual circle bundles with background flux, to the free (small) loop space of these T-dual circle bundles.

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

I present recent work exploring and defining higher structures required for a potential classical description of M5-branes. I start with a discussion of metric string structures, which provide good examples of higher gauge structures expected in M5-brane models. Next, I discuss an adjusted form of higher parallel transport, which does not require fake flatness. Finally, I’ll report on progress with the Lagrangian and how to replace the PST-part by Sen’s new formalism, yielding the same amount of self-duality.

**Henning Samtleben: Exceptional field theories and AdS compactifications**

Exceptional field theories are the duality covariant formulations of higher-dimensional supergravity theories. I review the formalism and its applications for the study of AdS compactifications.

**Hisham Sati: M-theory and cohomotopy**

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 theory**

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 suggest 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).

**Ashoke Sen: Gravitational waves from soft theorem**

Abstract: TBD

**Eric Sharpe: Decomposition of two-dimensional theories with one-form symmetries**

In this talk, we will discuss two-dimensional theories with discrete one-form symmetries, examples (which we have been studying since 2005), their properties, and gauging of the one-form symmetry. Their most important property is that such theories decompose into a disjoint union of theories, recently deemed ‘universes’ This decomposition has the effect of restricting allowed nonperturbative sectors, in a fashion one might deem a ‘multiverse interference effect’, which has had applications in topics including Gromov-Witten theory and gauged linear sigma model phases. After reviewing one-form symmetries and decomposition in general, we will discuss a particular example in detail to explicitly illustrate these properties and to demonstrate how gauging the one-form symmetry projects onto summands in the decomposition. If time permits, we will briefly review analogous phenomena in four-dimensional theories with three-form symmetries, as recently studied by Tanizaki and Unsal.

**Dmitri Sorokin: How SYM domain walls look like?**

We will review main features of the pure N=1, D=4 SYM and its effective description by the Veneziano-Yankielowicz generalized sigma-model. We will then argue that the construction of 1/2 BPS domain walls interpolating between different SYM vacua requires the presence of a dynamical membrane source. We will show how such an M-brane is coupled to the SYM and present the explicit form of the BPS domain walls which it creates in the Veneziano-Yankielowicz effective theory.

**Meng-Chwan Tan: Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes
in String Theory**

I will explain how, starting with a stack of D4-branes ending on an NS5-brane in type IIA string theory, one can, via T-duality and the topological-holomorphic nature of the relevant worldvolume theories, relate (i) the lattice models realized by Costello’s 4d Chern-Simons theory, (ii) links in 3d analytically-continued Chern-Simons theory, (iii) the quantum geometric Langlands correspondence realized by Kapustin-Witten using 4d N = 4 gauge theory and its quantum group modification, and (iv) the Gaitsgory-Lurie conjecture relating quantum groups/affine Kac-Moody algebras to Whittaker D-modules/W-algebras. This furnishes, purely physically via branes in string theory, a novel bridge between the mathematics of integrable systems, geometric topology, geometric representation theory, and quantum algebras.

- Michael Duff
- Chris Hull
- Neil Lambert
- Christian Saemann
- Hisham Sati
- Urs Schreiber
- Ashoke Sen
- Meng-Chwan Tan

Hisham Sati

By invitation only