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 five-day workshop on recent topics at the intersection of Physics and Geometry. The workshop is intended to serve as an unofficial satellite workshop of String Math 2017, held in Hamburg July 24–29.

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.

Thursday, July 20

Time	Speaker	Title, Abstract
9:30am	Ganor	Quadratic Reciprocity and Janus Configurations
	Compactification of N=4 Super-Yang-Mills theory on a circle with various SL(2,Z) duality twists will be discussed. A technique for counting ground states on a torus will be presented, and simple Wilson loops will be analyzed as well. The partition function on a mapping torus, calculated in two different ways, leads to the Landsberg-Schaar identity for quadratic Gauss sums, and generalizations.
Coffee break
11:00am	Sułkowski	Knots are quivers?
	I will present a surprising relation between knot invariants and quiver representation theory, motivated by various string theory constructions involving BPS states. Consequences of this relation include the proof of the famous Labastida-Marino-Ooguri-Vafa conjecture, explicit (and unknown before) formulas for colored HOMFLY polynomials for various knots, new viewpoint on knot homologies, new dualities between quivers, and many others. The crucial role in this relation is play by A-polynomials -- which represent certain flat connections in Chern-Simons theory -- and their quantization.
Lunch break
1:30pm	Bullimore	The Twisted Hilbert Space of 3d N = 2 Theories
	I will discuss a description of 3d N = 2 gauge theories on R x C with topological twist on a Riemann surface C as a supersymmetric quantum mechanics on R. I will focus on a mathematical description of the Hilbert space of supersymmetric ground states, which in general contains more information than the twisted partition function on S^1 x C. In particular, I will consider some simple abelian examples and demonstrate invariance of the Hilbert space under three-dimensional mirror symmetry.
3:00pm	Yi	Index, Partition Functions, and Rational Invariants
	In recent years, varieties of index-like quantitites have been computed by exact path integral, a.k.a. the localization. For gauge theories, the path integral reduces to contour integrals, albeit with many subtleties. However, it turns out that interpretation of the result, which should be really called the twisted partition function rather than the refined index, requires even more care. After a cursory review of the recent derivations and accompanying subtleties, we consider theories with noncompact Coulomb phases. The rational nature of the twisted partition function is observed and physically explained, which also connects to the rational invariant of the Kontsevich-Soibelman wall-crossing algebra. This gives us a nontrivial tool for extracting the refined and integral index out of such rational twisted partition functions. An application to pure Yang-Mills quantum mechanics solves a decades-old problem of counting D0 bound states with an Orientifold point. Along the way, we also resolve a critical conflict between Kac/Smilga and Staudacher/Pestun, circa 1999~2002, by isolating the notion of H-saddles. The latter also proves to be a universal feature of partition functions in the high "temperature" limit.
4:15pm	del Zotto	Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States
	It is possible to understand whether a given BPS spectrum is generated by a relevant deformation of a 4D N=2 SCFT or of an asymptotically free theory from the periodicity properties of the corresponding quantum monodromy. With the aim of giving a better understanding of the above conjecture, we revisit the description of framed BPS states of four-dimensional relativistic quantum field theories with eight conserved supercharges in terms of supersymmetric quantum mechanics. We unveil aspects of the deep interrelationship in between the Seiberg-dualities of the latter, the discrete symmetries of the theory in the bulk, and quantum discrete integrable systems.

Friday, July 21

Time	Speaker	Title, Abstract
9:30am	Williams	Symmetries of higher dimensional holomorphic field theories
	The most well known examples of holomorphic field theories come from chiral CFTs in complex dimension one. Of the symmetries of these theories, a special role is played by both the Kac-Moody and Virasoro vertex algebras. In this talk I will discuss generalizations of these symmetries to higher dimensions in the language of factorization algebras. For example, holomorphic gauge theories in complex dimension d (theories depending on the data of a holomorphic connection) generically have the symmetry of a certain dg Lie algebra studied recently by Faonte, Hennion, and Kapranov. My talk will focus on this higher Kac-Moody algebra and I will relate it to a shadow of a richer OPE algebra. Time permitting, I will discuss a similar picture for the higher Virasoro algebras.
Coffee break
11:00am	Gwilliam	Chern-Simons theory and Koszul duality
	The observables of perturbative Chern-Simons theory naturally form an algebra over the little 3-disks operad, as I will explain. In consequence, line operators form a braided monoidal category. We then use higher abstract nonsense, notably Koszul duality, to recover a quantum group with formal parameter. This work in progress is joint with K. Costello and J. Francis.
Lunch break
1:30pm	Lho	Stable quotient and holomorphic anomaly equations
	I will prove holomorphic anomaly equations for the stable quotient invariant of Local P^2 in precise form predicted by B-model Physics. After that I will also discuss about the holomorphic anomaly equation for [C3/Z3] and formal quintic invariants. This talk is based on joint work with Rahul Pandharipande.
3:00pm	Li	Quantum master equation and integrable hierarchy
	We establish the Batalin-Vilkovisky quantization theory for chiral deformation of free conformal field theories. As an application, we describe a universal approach to intebrable hierarchies associated to topological B-model on Calabi-Yau geometry. The talk is based on arXiv: 1612.01292[math.QA] and a joint work in progress with Weiqiang He and Philsang Yoo.
4:15pm	Hilburn	S-duality of boundary conditions for 4d N=4 and ring objects in the geometric Satake category
	In a recent paper Braverman, Finkelberg, and Nakajima showed how one can associate a ring object in the category of equivariant D-modules on the affine Grassmannian Gr_G to a 3d N=4 gauge theory whose cohomology gives the ring of functions on the Coulomb branch. I will explain a physical context for this result in terms of boundary conditions to 4d N=4 super Yang-Mills and use this to give a mathematical formulation of some ideas of Gaiotto.

Saturday, July 22

Time	Speaker	Title, Abstract
9:30am	Hori	2d Seiberg duality, with boundary
	I will talk about 2d Seiberg duality and discuss how it may act on the B-type boundary conditions. The grade restriction rule, obtained from the hemisphere partition function, plays an important role. Based on a joint work in progress with Richard Eager.
Coffee break
11:00am	Dedushenko	Four-manifold invariants and 2d CFT
	Recently, it has been conjectured that there exists a class of 2d N=(0,2) SCFTs T[M_4, G], labeled by a 4-manifold M_4 and a Lie group G, which could provide a new type of smooth structure invariants for M_4. Such theories are given by twisted compactifications of the 6d (2,0) theory on M_4. We study one of the simplest cases when G=U(1). Even though such T[M_4, U(1)] looks trivial (in particular, it is free), from the string theory point of view, it is natural to include certain local defect operators in this theory. They are naturally defined in the UV, while in the IR description they explicitly depend on the Seiberg-Witten (SW) invariants of M_4. T[M_4, U(1)] perspective leads to the equivariant multi-monopole generalization of SW invariants. We study them from several points of view, including the 4d gauge theoretic approach, and identify the structure of the VOA T[M_4, U(1)]. We also propose some further directions for generalizations, including the case of non-abelian G.
Lunch break
1:30pm	Romo	All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern-Simons Theory
	I will review some general aspects of complex Chern-Simons theory on hyperbolic 3-manifolds, focusing on the case of gauge group G=SL(2,C). After a brief introduction to the Volume Conjecture (VC), for knot complements and, a very recent mathematical proposal, for closed hyperbolic 3-manifolds, I'll show how complex Chern-Simons theory is related with them and how this connection leads to a novel generalization of the most recently proposed VC for closed 3-manifolds.

Venue

All talks will be held in the 5th floor Conference Room (5/104) in the Southern wing of the MATHEMATIKON.

Morning and afternoon coffee/tea and lunches will be served in the 5th floor common room.

Accommodation and Provisioning

The Heidelberger Hof hotel is conveniently located midway between the Neuenheimer Feld and the main attractions of old town Heidelberg. For a fast bus connection to Campus, board route 31 at Bismarckplatz, headed to Kopfklinik (not Universitätsplatz!) and get off 5 stops later at Bunsengymnasium. The main entrance of the MATHEMATIKON is across the street. Shops and restaurants are in the middle and Northern wing.

The pleasant walk from Bismarckplatz across the bridge, down the Neckar river, and up Berliner Strasse takes about 25 minutes.

Registration and Workshop Dinner

There is no registration fee, however we ask that all those interested in attending the talks send us an Email ahead of time, so we may have an idea of the number of prospective participants. (We reserve the right to raise a guard in the unlikely event of a general stampede.) In your message, please indicate whether or not you are interested in joining us for the workshop dinner on Friday night. You will be asked for a € 20 contribution for the dinner; further information to follow.

Richard Eager, Ingmar Saberi, Johannes Walcher