# Pure Spinors, Superalgebras, and Holomorphic Twists

### Welcome!

From October 4 to 8, 2021, the Mathematical Physics group in the Mathematisches Institut at Universität Heidelberg will host a workshop on topics at the intersection of physics and pure mathematics, with a special emphasis on algebra, representation theory, and complex and algebraic geometry.

### Join

This is a hybrid event. Local participants and on-site speakers meet in the Hörsaal on the ground floor of the Mathematikon.

Remote speakers and external participants may Join the Zoom Meeting (ID: 656 720 7054, Passcode: 4AS82e), or follow the live Stream.

If all else fails, you may send us an email

### Program

All times are given in Central European Time (UTC+1). Talks are divided into morning / afternoon sessions. We have opted for a long (2.5 hours) lunch break to make it possible for participants from the Americas to plausibly attend most of the talks in the afternoon session and, at the same time, to allow enough time to discuss, share ideas but also rest to the in person participants.

#### Monday, October 4

Time Speaker Title, Abstract
9:30 - 10:20am Baulieu The power of the 2d Beltrami parametization in gravity and twisted supergravity and its generalization for dimensions larger than 2.
A suggestive sub-foliation of the Arnowitt-Deser-Misner leafs of d-dimensional Lorentzian manifolds is presented, Sigma^{ADM}_{d-1}= Sigma_{d-3} x Sigma_2. It defines an interesting covariant "d-dimensional Beltrami parametrization” for the d-bein and the d-metric. The "Beltrami d-bein" is parametrized by d(d+1)/2 independent fields belonging to different categories, each one with a specific interpretation. The Weyl invariant sector beautifully selects the d(d-3)/2 physical local degrees of freedom of d-dimensional gravity.
Coffee break
11:00 - 11:50am Krämer The Lie Algebra of Perverse Sheaves on Elliptic Curves
Perverse sheaves appear in many parts of mathematics and physics such as in geometric representation theory or in the Langlands program. In the case of abelian varieties, they form a tensor category which is abstractly equivalent to the representation category of some affine group scheme. This "Tannaka group of perverse sheaves" captures a lot of information, but it is huge and hard to approach explicitly, even for elliptic curves. I will discuss a new way to construct elements in its Lie algebra via a global enrichment of vanishing cycles (work in progress with Amelie Flatt).
12:00 - 12:50am Kleinschmidt The most complicated way of writing D=11 supergravity
Exceptional field theory based on the exceptional group E_n is a way of combining aspects of supergravity in a duality covariant way. It can make the Cremmer-Julia E_n symmetry in 11-n dimensions manifest but can also be seen as a way of rewriting the theory in eleven dimensions, breaking the symmetry in the course of doing so. I will review recent results where this idea is taken to the limiting case with Kac-Moody symmetry E11, highlighting interesting representation-theoretic aspects. Based on work with Guillaume Bossard and Ergin Sezgin.
Lunch break
3:30 - 4:30pm Palmkvist Nonlinear realisations of Lie superalgebras
The talk is based on 2012.10954. For any decomposition of a Lie superalgebra G into a direct sum G=H+E of a subalgebra H and a subspace E, without any further resctrictions on H and E, we construct a nonlinear realisation of G on E. The result generalises a theorem by Kantor from Lie algebras to Lie superalgebras. When G is a differential graded Lie algebra, we show that it gives a construction of an associated L-infinity-algebra.
Coffee break
5:00 - 5:50pm Mnëv Two-dimensional BF theory as a conformal field theory
We study topological BF theory on the complex plane in Lorenz gauge. In the abelian case, we find that the gauge-fixed theory is a B-twisted N=(2,2) superconformal theory - Witten's B-model with a parity-reversed target. The BV algebra structure on 0-observables is constructed explicitly using operator product expansions with the superpartner of the stress-energy tensor. In the non-abelian case, the theory becomes a logarithmic CFT with correlators given by convergent integrals (e.g., 4-point functions are expressed in terms of dilogarithms). We find vertex operators in the non-abelian theory, receiving a quantum correction to conformal dimension. This is a report on a joint work with Andrey Losev and Donald Youmans, arXiv:1712.01186, arXiv:1902.02738
6:00 - 6:50pm Creutzig Non-semisimple 3-dimensional topological field theories
A true highlight of modern mathematical physics is the construction of invariants of 3-manifolds and links using modular tensor categories associated to a compact Lie group G and a level k. Such a category has three realizations: the category of Wilson lines in Chern-Simons theory, the category of integrable modules of the WZW 2-dimensional conformal field theory (or better its underlying vertex algebra), a category of quantum group modules at root of unity. Interesting "Chern-Simons-like" 3-dimensional topological field theories can be constructed from corner considerations in 4-dimensional N=4 super Yang-Mills theory. These theories have two modern features: they are non semisimple and they can be deformed by flat connections. As in the semisimple case, the category of line operators of this theory can also be realized by some vertex algebra and some quantum group. Both can also be deformed by connections. I will try to explain this emerging picture. This is a report on joint work with T. Dimofte. N. Garner and N. Geer.

#### Tuesday, October 5

Time Speaker Title, Abstract
9:30 - 10:20am Genra Feigin-Semikhatov duality
Feigin and Semikhatov conjectured dualities between subregular W-algebras of sl_n and principal W-algebras of sl_{n|1}. In this talk, we prove these conjectures and study the structure of module categories by means of relative semi-infinite cohomology techniques. This is a report on a joint work with Thomas Creutzig, Shigenori Nakatsuka and Ryo Sato.
Coffee break
11:00 - 11:50am Adamovic Realizations of affine vertex algebras, logarithmic vertex algebras and beyond
We shall first discuss our realization of affine vertex algebra $L_k(sl(2))$ and present some applications to the representation theory. Next, we present some generalizations. We will study new realizations of certain logarithmic affine vertex algebras appearing in physical theories (jointly with T. Creutzig, N. Genra and J. Yang). We shall also study a duality between N=4 superconformal vertex algebra with central charge c=-9 and the affine vertex algebra $L_k(osp(1,2))$ at the critical level (jointly with Q. Wang).
12:00 - 12:50am Cederwall SL(5) supersymmetry
The talk is based on arXiv:2107.09037. We consider supersymmetry in five dimensions, where the fermionic parameters are a 2-form under SL(5). Supermultiplets are investigated using the pure spinor superfield formalism, and are found to be closely related to infinite-dimensional extensions of the supersymmetry algebra: the Borcherds superalgebra B(E_4), the tensor hierarchy algebra S(E_4) and the exceptional superalgebra E(5,10). A theorem relating B(E_4) and E(5,10) to all levels is given.
Lunch break
3:30 - 4:30pm Berkovits Manifest Spacetime Supersymmetry and the Superstring
The algebra of spacetime supersymmetry generators in the RNS formalism for the superstring closes only up to a picture-changing operation. After adding non-minimal variables and working in the "large" Hilbert space, the algebra closes without picture-changing and spacetime supersymmetry can be made manifest. The resulting non-minimal version of the RNS formalism is related by a field redefinition to the pure spinor formalism.
Coffee break
5:00 - 5:50pm Eager Maximally twisted eleven-dimensional supergravity
TBA
6:00 - 6:50pm Dumitrescu 2-Group Global Symmetries in Six Dimensions
I will review the notion of (continuous) 2-group global symmetries, which mix ordinary and higher-form global symmetries, and explain why they naturally arise in many 6d gauge theories. Most 6d SCFTs and little string theories have weakly-coupled infrared phases with gauge fields. While non-trivial 2-group symmetries are often present in little string theories, I will explain why they can never arise in unitary SCFTs. This observation can be used to establish a previously conjectured algorithm for computing ’t Hooft anomalies of such SCFTs. Finally, I will use this understanding to shed light on the a-theorem in six dimensions. In particular, I will show that all unitary 6d SCFTs have positive a-anomaly.

#### Wednesday, October 6

Time Speaker Title, Abstract
9:30 - 10:20am Arakawa R-filtration and Vogan filtration
The 4D/2D duality discovered by Beem et al. associates a VOA to any 4D N=2 SCFT, whose associated variety is the Higgs branch of the 4D theory. Beem and Rastelli further states that such a VOA should carry a filtration, called the R-filtration, induced from the R-charge of the 4D theory. In this talk we present a possible mathematical definition of the R-filtration. This is a joint work with Anne Moreau.
Coffee break
11:00 - 11:50am Jurco Homological perturbation and homotopy transfer
We will review the relation between the homologial perturbation, homotopy transfer for quantum homotopy algberas and the BV path integral. Time permitting we will touch on the notion of a category of quantum BV field theories suggested by this relation.
12:00 - 12:50am Grassi New Cohomologies for Lie Superalgebras
In the talk, we present new results in the computations of cohomologies for Lie Superalgebras. In particular, we will give the first explicit example of an invariant pseudoform representing a cohomology class for the Chevalley-Eilenberg cohomology for osp(2|2) superalgebra emerging in N=2 string theory.
Lunch break & Social Activity

#### Thursday, October 7

Time Speaker Title, Abstract
9:30 - 10:20am Eder Super Cartan geometry and applications to (quantum) supergravity
This talk is devoted to the geometric approach to supergravity. We interpret supergravity in terms of a super Cartan geometry which provides a link between supergravity and Yang-Mills gauge theory. To this end, we first review important aspects of the theory of supermanifolds and we establish a link between various different approaches. We then introduce super Cartan geometries using the concept of so-called enriched categories. Studying these categories turns out to be mandatory to model anticommuting classical fermionic fields in mathematical physics. Then applications of these methods in the context of supergravity will be discussed. For this purpose, we derive the so-called Holst-MacDowell-Masouri action of D=4 AdS supergravity for N=1,2 which is a 1-parameter family of deformed supergravity actions. We will show that these actions provide unique boundary terms that ensure local supersymmetry invariance at boundaries. For certain values of this parameter, we show that the action can be recast in the form of a constrained super BF theory and the boundary theory is a super Chern-Simons theory. Finally, we will give an outlook on possible applications of these results in the context of quantum supergravity.
Coffee break
11:00 - 11:50am Huerta Bundle gerbes on Lie supergroups
Bundle gerbes are analogues of line bundles important for conformal field theory, anomalies, and obstruction theory. Among bundle gerbes, a central role is played by the basic bundle gerbe, an essentially unique gerbe on any compact, simple and simply-connected Lie group. In this talk, we describe our work constructing the basic bundle gerbe for a large family of simple Lie supergroups, and show how the basic gerbe on a Lie supergroup decomposes into a tensor product of gerbes on the underlying Lie group and an auxiliary 2-form.
12:00 - 12:50am Closset The U-plane of rank-one 4d N=2 KK theories
I will revisit the Seiberg-Witten description of the Coulomb branch of rank-one 4d N=2 supersymmetric QFTs, including aspects of the global symmetry, from the point of view of rational elliptic surfaces. This will include, in particular, a detailed study of the 5d superconformal field theories with E_n symmetry, compactified on a circle. I will also sketch how to derive BPS quivers from that perspective. Interesting modular properties of the SW geometries will play a key role.
Lunch break
3:30 - 4:30pm Williams Exceptional Lie algebras from twisted supergravity
Non-topological twists of supersymmetric gauge theories have played an increasingly important role in math and physics in part due to relationships to vertex algebras and quantum groups. On the other hand, motivated by the higher genus B-model, twists of 10-dimensional theories of supergravity have been characterized. In this talk, we give a complete description of the maximally non-topological twist of 11-dimensional supergravity, the low energy limit of M-theory. I will explain the unexpected result that the global symmetry algebra of the model is equivalent to an infinite-dimensional exceptional super Lie algebra known as E(5,10). I will also explain the relationship between other exceptional algebras and extended objects such as M2 and M5 branes in the twisted setting.
Coffee break
5:00 - 5:50pm Gwilliam Higher deformation quantization for twists of N=4 supersymmetric Yang-Mills theory
In work with Elliott and Williams, we constructed the BV quantization of the holomorphic and topological twists of N=4 SYM in four dimensions, which include the Kapustin-Witten theories. The observables of these theories have interesting mathematical structure: for the topological twists, we showed they encode algebras over the framed little 4-disks operad and hence determine fully extended TFTs in the sense of Baez-Dolan-Lurie. We will discuss the construction and questions it raises.
6:00 - 6:50pm Serganova Volumes of supergrassmannians and splitting subgroups
The supergrassmannians are compact homogeneous supermanifolds with invariant volume forms. We determine in which cases the volumes of supergrassmannians are not zero. Our main tool is the Schwartz-Zaboronsky localization formula for Berezin integral. We also discuss applications of this calculation to representations of superalgebras. In particular, we generalize Green correspondence, well known theorem in representation theory of finite groups in positive characteristics, to general linear supergroups.

#### Friday, October 8

Time Speaker Title, Abstract
9:30 - 10:20am Heidersdorf On recent results in the representation theory of supergroups
I will give a (biased) survey talk about recent developments in the representation theory of supergroups and Lie superalgebras. The focus is here on "elementary" questions such as the computation of dimensions, superdimensions, character formulas and fusion rules of irreducible representations. It turns out that these have very non-elementary answers (if they are known at all).
Coffee break
11:00 - 11:50am Safronov Virtual fundamental classes and Batalin-Vilkovisky quantization from supersymmetric twists
Supersymmetric localization allows one to reduce the computation of the partition function of a supersymmetric theory to a finite-dimensional integral. In this talk I will explain how virtual fundamental classes of (-2)-shifted symplectic schemes recently introduced by Borisov-Joyce, Pridham and Oh-Thomas arise from such a supersymmetric localization in the presence of extended (i.e. 0d N=2) supersymmetry. For instance, this gives a field-theoretic origin of the DT invariants of CY4 manifolds. Similarly, I will explain that spaces of states in the presence of extended (i.e. 1d N=4) supersymmetry may be computed in terms of the cohomology of a certain perverse sheaf associated to (-1)-shifted symplectic schemes. This is a report on joint work with Brian Williams.
12:00 - 12:50am Gaberdiel The string dual of free N=4 SYM
A proposal for the worldsheet string theory that is dual to free N=4 SYM in 4d is made. It is described by a free field sigma model on the twistor space of AdS5 x S5, and it exhibits a psu(2,2|4)_1 affine symmetry. The theory is a natural generalisation of the corresponding model for tensionless string theory on AdS3 x S3 whose description involves a free field realisation of psu(1,1|2)_1. I will explain how our proposal fits into the general framework of AdS/CFT, and review the various checks that have been performed.
Lunch break
3:30 - 4:30pm Schomerus Supergroup Chern-Simons Theory
Chern-Simons gauge theory is being studied for a wide range of profound applications in mathematics and physics. It is of significant interest to consider extensions in which the gauge field takes values in a Lie superalgebra. Such supergroup Chern-Simons theories appear, for example, by twisting the low energy effective action for an intersection D3 and NS5 branes. In my talk I will review the framework of combinatorial quantization of Chern Simons theory and explain how this framework can be adapted for applications to super- algebras. This will give rise to interesting new observables which can be computed by exploiting the rich representation theory of Lie superalgebras.