Pure Spinors, Superalgebras, and Holomorphic Twists

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.

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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

• Drazen Adamovic (University of Zagreb)
• Tomoyuki Arakawa (RIMS - Kyoto)
• Laurent Baulieu (Sorbonne University and CNRS - Paris)
• Nathan Berkovits (UNESP - São Paulo)
• Martin Cederwall (Chalmers University - Göteborg)
• Cyril Closset (University of Birmingham)
• Thomas Creutzig (University of Alberta)
• Thomas Dumitrescu (UCLA)
• Richard Eager (Yokohama)
• Konstantin Eder (Erlangen - Nürnberg)
• Matthias Gaberdiel (ETH - Zürich)
• Naoki Genra (Kavli IMPU)
• Antonio Grassi (Università del Piemonte Orientale)
• Owen Gwilliam (UMASS, Amherst)
• Thorsten Heidersdorf (University of Bonn)
• John Huerta (University of Lisbon)
• Branislav Jurco (Charles University - Prague)
• Axel Kleinschmidt (MPI - Potsdam)
• Thomas Krämer (Humboldt Universität - Berlin)
• Pavel Mnëv (University of Notre Dame)
• Jakob Palmkvist (Örebro University)
• Pavel Safronov (University of Edinburgh)
• Volker Schomerus (DESY - Hamburg)
• Vera Serganova (UC Berkeley)
• Brian Williams (University of Edinburgh)

The notion of a superalgebra has become a central part of both mathematics and physics. It provides a helpful unifying perspective on many important mathematical constructions, such as polynomial and exterior or Weyl and Clifford algebras; modern perspectives on Koszul duality and derived geometry are grounded in the philosophy of replacing naive linear or algebraic objects by differential graded analogues. A similar perspective took root independently among physicists; the BRST construction implements gauge symmetries cohomologically using a variant of the Chevalley-Eilenberg complex, whereas the BV formalism goes further and rephrases the sheaf of classical solutions to equations of motion as a derived critical locus equipped with shifted symplectic structure. In addition, since a Z/2Z grading by fermion parity is present in any physical system, the study of symmetry algebras in physics admits natural extensions to Lie superalgebras.

Supersymmetry algebras are a particularly interesting class of such Lie superalgebras, which extend the spacetime symmetries of a quantum field theory (either the affine or conformal group) by odd elements transforming in spin representations of the Lorentz group. Supersymmetric field theories admit natural deformations of the differential, known as twists, by any square-zero odd element of the supersymmetry algebra. Such twists have been the subject of intense study for many years, for a variety of reasons: They produce myriads of interesting examples of holomorphic and topological field theories, which stand in a precise relationship to protected or BPS quantities in the original field theory. They are naturally derived objects, since their sheaves of classical solutions (for example, holomorphic sections of bundles or locally constant sheaves) are resolved in smooth functions. Their algebras of local operators admit interesting higher operations, controlled by the geometry and topology of analogues of the little disks operad. In addition, it has been pointed out in recent work that holomorphic field theories naturally admit actions of centrally extended, infinite-dimensional dg Lie algebras, which enhance flavor and conformal symmetry in the twisted theory and generalize Kac-Moody and Virasoro symmetry to higher dimensions.

A central object connecting representation theory of infinite dimensional superalgebras with quantum field theory and exciting topics of pure mathematics are Vertex operator algebras (VOAs). VOAs are a mathematical notion of the symmetry algebra of a two dimensional conformal quantum field theory and as such they always have an action of the Virasoro algebra. VOAs and their representation categories appear not only as meaningful invariants of higher dimensional supersymmetric quantum field theories but they are also central in the quantum geometric Langlands program and as new invariants of three- and four-manifolds. The relevant VOAs are so-called W-superalgebras, which are obtained as certain semi-infinite cohomologies from the VOAs of affine Lie superalgebras. Their representation theory is far more complicated than the VOAs usually studied by researchers in the area and in the recent years new effective methods for their study have been introduced. This is due to a very fruitful interaction between physicists and mathematicians in many different directions of pure mathematics.

The last couple of years have seen a convergence of interest from various lines of research related to superalgebras and supersymmetry. For one thing, it was realized that “twists” are not only powerful tools for understanding sectors of specific supersymmetric theories, but can also be used to “reconstruct” such theories when considered globally in families over the space of square-zero elements, or “nilpotence variety.” These spaces are sometimes, but not always, related to the space of Cartan pure spinors, which lends its name to the pure-spinor formalism in physics. It turns out that these techniques can be applied in any dimension and with any amount of supersymmetry, and Koszul duality predicts close relations between the derived category of equivariant sheaves on the nilpotence variety and a category of supermultiplet representations of the supersymmetry algebra, which has yet to be properly understood. One expects that this geometric perspective can develop further unifying power, and that well-known folk theorems in physics (such as the absence of auxiliary-field formalisms for maximal supersymmetry) can perhaps be understood or proved in terms of the algebraic geometry of the relevant nilpotent variety. This should also offer a new perspective on the representation theory of superconformal algebras. Unitary representations of the higher Virasoro algebra and other new infinite-dimensional dg Lie algebras remain completely unexplored territory.

Our goal is to bring together experts in quantum field theory, pure spinor techniques, Lie superalgebras and representations, and derived geometry, and to foster exchange, collaboration, and the development of shared language between these diverse groups.

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.

Venue, Zoom and YouTube Links

This is a hybrid-format event: participation is available both on-line and in person in Heidelberg. Talks that will be given in person will be be held in the 'Hörsaal', on the groundfloor of the Mathematikon and livestreamed.

Zoom: https://us02web.zoom.us/j/6567207054?pwd=TFphTnpBRVJodHlRV3MvSDl4V1ZUZz09
Meeting ID: 656 720 7054
Passcode: 4AS82e

YouTube Channel: https://www.youtube.com/channel/UCsi29QiswqpMuCf1CVN4spw

Accommodation and Provisioning

The Vier Jahreszeiten hotel is located just next to the Old Bridge, one of the landmark of the city of Heidelberg, and few steps away from the main attractions of the Old Town.

For a fast connection to the university campus, board route 31 at Universität Platz, headead to Kopfklinik and get off at Bunsengymnasium. The main entrance of the Mathematikon is accross the street.

Alternatively, cross the Old Bridge and board route 37 at Alte Brücke Nord, headed to Sportzentrum Nord and get off at Technologiepark. Cross the street and head south, toward the southern wing of the Mathematikon: the main entrance is right at the next crossroad.

The pleasant walk, crossing the Old Bridge and heading East down the Neckar river, and up Berliner Strasse takes about 40/45 minutes at a chill pace.

Simone Noja, Ingmar Saberi, Johannes Walcher

           

Mathematical Institute >> Pure Spinors, Superalgebras, and Holomorphic Twists