— This website is still under construction; all information is subject to change. —

ANNOUNCEMENT: No meeting 15.1. OR 17.1., contrary to prior announcements! We resume the normal schedule on 22.1.

Topics in supersymmetry: from pure spinors to BPS states

Mathematisches Institut der Universität Heidelberg
Wintersemester 2018/2019

Course: Modulkürzel MM34.
The format will be that of a standard lecture course (2 SWS), without Übungen.
There is no required homework, though some problems to further understanding may be mentioned.
Credit for the course is by arrangement; those desiring credit will sit an oral examination at the end of the term.
Time and place: Tuesdays, 11:00–13:00, Mathematikon SR 8.
Note that there will be some variability during the semester: due to travel, we will not meet beginning on 27. November and running through Christmas. The current plan is to replace those four meetings with additional sessions on four Thursdays in January, beginning on the 10th.
These meetings will be Thursdays from 14:00 to 16:00, also in Mathematikon SR 8.
Instructor: Ingmar Saberi
Mathematikon 05/208 (or by email)
Language: Lectures to be held in English.
Relevant dates: Vorlesungszeit — 15. Oktober 2018 bis 09. Februar 2019
Winterferien — 24. Dezember 2018 bis 06. Januar 2019
Abwesenheit der Lehrkräfte — 21. November 2018 bis 5. Januar 2019

Actual Syllabus:

(I'll try and keep this portion of the site updated to reflect the order and selection of topics that actually ends up happening. Discrepancies from the aspirational syllabus, preserved below, will be both numerous and profound. I'll also post references here, as well as—at least eventually—some of my lecture notes.)
  1. (16. Oktober)
    Some review of the theory of Lie algebras: Cartan's classification; fundamental weights; Dynkin labels. SO groups in odd and even dimension. Some remarks on gradings and filtrations. Building algebras from vector spaces: free tensor algebras; symmetric and exterior algebras; Clifford algebras. The Chevalley–Eilenberg construction, a.k.a. BRST procedure.
  2. (23. Oktober)
    The theorem of Wigner–Bargmann on lifting quantum-mechanical symmetries to the Hilbert space. Unitary and antiunitary symmetries. The role of central extensions/projective representations.
  3. (30. Oktober)
    Computations of Clifford algebras; Bott periodicity and the Clifford clock. Clifford modules; spinor representations. Construction of the Dirac spinor from a maximal isotropic subspace. Clifford multiplication.
  4. (6. November)
    Lie superalgebras. Definitions; some basic examples; gl(m|n), sl(m|n), and osp(m|n). The Poincaré algebra and its unitary representations, according to Wigner; the little group. The beginning of the proof of the Coleman–Mandula theorem.
  5. (13. November)
    The end of the proof of the Coleman–Mandula theorem (some details suppressed). Consequence: we can only hope to find interesting super Lie algebras that extend Poincaré symmetry. Spin and statistics implies that the odd part of the algebra lies in a spin representation of SO(V) (this is essentially Haag-Łopuszański-Sohnius). Construction of super-Poincaré algebras. The twisting construction for supersymmetric theories; a favorite example (the de Rham complex). Cohomology as the invariants of a nilpotent symmetry generator.
  6. (20. November)
    The variety of nilpotents in a super-Lie algebra. An obvious role that it plays: the moduli space of twists. Tautological data. Stratification by orbits. Some details on the procedure of dimensional reduction, and discussion of a simple example (minimal supersymmetry in four dimensions), together with its dimensional reduction to (2,2) supersymmetry in two dimensions, producing either the holomorphic or B-model twist.

Aspirational Syllabus:

(There will be a total of fifteen ninety- to one-hundred-minute lectures, in keeping with 2 SWS. However, note that we will not meet in December, and (to make up for this) twice per week in January; see remarks above.)
  1. (16. Oktober)
    Review: Lie algebras, Dynkin labels, SO groups in odd and even dimension. Clifford algebras and Clifford modules. Spin groups and spinor representations. Canonical pairings and intertwiners (gamma matrices). The Chevalley–Eilenberg construction, a.k.a. BRST procedure; the infamous "Koszul duality."
  2. (23. Oktober)
    Physics groundwork: Bargmann's theorem. Wigner's classification; the theory of the little group. The spin and statistics theorem. The theorems of Coleman–Mandula and (briefly) Haag–Łopuszański–Sohnius.
  3. (30. Oktober)
    Fermions in math: Lie superalgebras. Constructions: matrix Lie superalgebras. Super-Poincaré algebras in various dimensions; minimal and extended supersymmetry. Nahm's classification of superconformal algebras.
  4. (6. November)
    Homogeneous spaces: Orbit structure of spin representations; Igusa's stratification; Cartan pure spinors. A bit on more general homogeneous spaces in algebraic geometry; (orthogonal) Grassmannians; flag varieties. The theorem of Borel–Weil–Bott. Hilbert series of homogeneous spaces.
  5. (13. November)
    Applications of supersymmetry I: A zoology of "protected quantities" in supersymmetric field theories. BPS states and operators; BPS bounds, Q-cohomology. Supersymmetric indices. The elliptic genus. Römelsberger's index in four dimensions.
  6. (20. November)
    Applications of supersymmetry II: More on index-type constructions, both cohomological and not. The new supersymmetric index of Cecotti–Fendley–Intriligator–Vafa. Chiral rings in two and four dimensions. Topological twists. Chiral algebras from superconformal theories in four dimensions; other spatially inhomogeneous constructions. Preliminary words on wall-crossing phenomena.
  7. (8. Januar)
    Nilpotence varieties: Definitions: the moduli space of nilpotents. Basic properties. Stratification and orbit structure. Canonical data subordinate to the stratification. Examples in several dimensions. Relations to standard homogeneous spaces.
  8. (Donnerstag, 10, Januar)
    Applications of nilpotence varieties I: Parameter spaces for twists of theories. Weakly topological theories. Manifolds of special holonomy. Some of the usual yoga of supersymmetry breaking in curved backgrounds. Further twists of twisted theories. Spectral sequences in physics.
  9. (15. Januar)
    Applications of nilpotence varieties II: The pure spinor superfield technique: constructing supermultiplets from (stratified equivariant bundles over) nilpotence varieties. Nilpotence varieties from Chevalley–Eilenberg cohomology. The example of ten-dimensional Yang–Mills theory.
  10. (Donnerstag, 17. Januar)
    More on "pure spinors" in physics: The Brink–Schwarz superparticle and super Yang–Mills. The pure-spinor superparticle. Worldsheet interpretation of Berkovits' operator. The pure-spinor formulation of the superstring.
  11. (22. Januar)
    Lie algebra cohomology in string theory and supergravity: A reminder on Chevalley–Eilenberg cohomology of supertranslations. The Hochschild–Serre spectral sequence. Central extensions. "Brane scans." As much as I can fit in or process about applications in the work of d'Auria–Fré, Huerta–Schreiber, and friends.
  12. (Donnerstag, 24. Januar)
    The M5-brane theories and their uses: The (2,0) superconformal algebra in six dimensions. Its nilpotence variety (?). The superconformal theory (conjecturally) associated to a simply-laced (ADE) Lie algebra. A bit on compactifications and geometric correspondences: AGT and 3d–3d. Matching protected quantities.
  13. (29. Januar)
    BPS states in four dimensional N=2 theories I: Theories of class S. Moduli spaces; Higgs and Coulomb branches. The appearance of Hitchin's moduli space. Hyperkähler metrics.
  14. (Donnerstag, 31. Januar)
    BPS states in four dimensional N=2 theories II: The Kontsevich–Soibelman wall-crossing formula. Line operators. Framed BPS states.
  15. (5. Februar)
    Abschied: Unfinished business and loose ends (there will undoubtedly be many). Outlook and future directions. Unanswered questions.

— Last updated on November 8, 2018. —