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Topics in supersymmetry: from pure spinors to BPS states

Mathematisches Institut der Universität Heidelberg
Wintersemester 2018/2019

Actual Syllabus:

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.
   References:
   • C. Chevalley and S. Eilenberg, “Cohomology theory of Lie groups and Lie algebras.” Trans. Amer. Math Soc. 63, no. 1, pp. 85–124 (1948).
   • H. Georgi, Lie algebras in particle physics: from isospin to unified theories. Westview Press (1999).
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.
   References:
   • V. Bargmann, “Note on Wigner's theorem on symmetry operations.” Journal of Mathematical Physics 5, no. 7, pp. 862–868 (1964).
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.
   References:
   • M. Atiyah, R. Bott, and A. Shapiro, “Clifford modules.” Topology 3, pp. 3–38 (1964).
   • P. Deligne, “Notes on spinors.” In Quantum fields and strings: a course for mathematicians, vol. 1, pp. 2ff. American Mathematical Society (1999).
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.
   References:
   • V. Kac, “Lie superalgebras.” Advances in Mathematics 26, pp. 8–96 (1977).
   • E. Wigner, “On unitary representations of the inhomogeneous Lorentz group.” Annals of Mathematics 40, no. 1, pp. 149–204 (1939).
   • V. Bargmann and E. Wigner, “Group theoretical discussion of relativistic wave equations.” Proceedings of the National Academy of Sciences 34, no. 5, pp. 211–223 (1948).
   • S. Coleman and J. Mandula, “All possible symmetries of the S-matrix.” Physical Review 159, no. 5, pp. 1251–6 (1967).
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.
   References:
   • R. Haag, J. T. Łopuszański, and M. Sohnius, “All possible generators of supersymmetries of the S-matrix.” Nuclear Physics B 88, pp. 257–274 (1975).
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.
   References:
   • R. Eager, I. Saberi, and J. Walcher, “Nilpotence varieties.” arXiv:1807.03766.
7. (10. Januar)
   Some review on general aspects of nilpotence varieties. A general fact: for minimal supersymmetry algebras that are not dimensional reductions, the nilpotence variety is (closely related to) a standard symmetric space, the space of complex structures on V. The nilpotence variety for six-dimensional minimal supersymmetry, and its dimensional reductions to five, four, and three dimensions.
   References:
   • R. Eager et al., op. cit.
   • C. Elliott and P. Safronov, “Topological twists of supersymmetric observables.” arXiv:1805.10806.
8. (22. Januar)
   General remarks on stratified spaces; a stratification as a continuous map from a space to a poset. Dimension, and various properties of twists, as maps of (or functors from) posets. The nilpotence variety for ten-dimensional minimal supersymmetry (the original space of pure spinors), and its dimensional reductions to nine, eight, and seven dimensions. Universality of the first three dimensional reductions. Weak and strong notions of topological twist. Exceptional holonomy. Some remarks on finding coordinate charts on the nilpotence variety from the supersymmetry algebra.
   References:
   • R. Bryant, “Metrics with exceptional holonomy.” Annals of Mathematics 126 pp. 525–576 (1987).
   • D. Ayala, J. Francis, and H. Tanaka, “Local structures on stratified spaces.” Advances in Mathematics 307, pp. 903–1028 (2017).
9. (24. Januar)
   Some elementary notions of algebraic geometry. Graded rings and graded modules; graded dimension (also known as Hilbert series). The Hilbert series of a complete intersection. Varieties that are not complete intersections; the twisted cubic. Free resolutions. The Hilbert numerator as graded Euler characteristic of a free resolution. The Koszul complex.
   References:
   • J. Tate, “Homology of Noetherian rings and local rings.” Illinois Journal of Mathematics 1, no. 1, pp. 14–27 (1957).
   • H. Cartan and S. Eilenberg, Homological algebra. Reprint, Princeton Landmarks in Mathematics (1999).
10. (29. Januar)
    Tate's extension of the Koszul complex; free resolutions by R-algebras. The pure spinor superfield formalism: a canonical differential on the tensor product of an A-module and an O-module. Here A is a supersymmetry algebra, and O is the homogeneneous coordinate ring of its nilpotence variety. This differential is a scalar (as long as the O-module is equivariant). Applications: producing supermultiplets for A from equivariant O-modules, using the free superfield.
    References:
    • J. Tate, op. cit.
    • R. Eager et al., op. cit.
    • M. Cederwall, “Pure spinor superfields: an overview.” In Breaking of supersymmetry and ultraviolet divergences in extended supergravity, pp. 61–93. Springer (2014).
11. (31. Januar)
    The relation of the Berkovits complex (the output of last week's construction; a supermultiplet associated to the datum of an equivariant O-module) in terms of a minimal free resolution of O in R-modules. The E₁ page of a natural spectral sequence is the Koszul complex, in O-modules, of the maximal ideal, tensored with functions on spacetime; this is related to the free resolution of O in R-modules! Further interpretation of this formalism: what is the meaning of the differentials on the E₁ page? The return of the BRST procedure; the arrival of Batalin and Vilkovisky. A main example: ten-dimensional minimal supersymmetry and the corresponding pure Yang–Mills theory.
    References:
    • V. Alexandrov, D. Krotov, A. Losev, and V. Lysov, “On the pure spinor superfield formalism.” Journal of High Energy Physics 2007.10, p. 74 (2007).
    • I. A. Batalin and G. A. Vilkovisky, “Gauge algebra and quantization.” Physics Letters 102B, no. 1, pp. 27–31 (1981).
12. (5. Februar)
    More on the BV formalism. The classical BV complex as a model for the derived critical locus of the action functional. Integration defines a pairing between homology classes of Lagrangian submanifolds and cohomology classes of the BV Laplacian, in any shifted cotangent bundle. All “QP-manifolds” are shifted cotangent bundles.
    References:
    • A. Schwarz, “Geometry of Batalin–Vilkovisky quantization.” Communications in Mathematical Physics 155, pp. 249–260 (1993).
    • D. Fiorenza, “An introduction to the Batalin–Vilkovisky formalism.” arXiv:math/0402057 (2004).
    • O. Gwilliam and T. Johnson-Freyd, “How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism.” In Topology and quantum theory in interaction (D. Ayala, D. S. Freed, and R. E. Grady, eds.). AMS Contemporary Mathematics Series, no. 718 (2018).
    • J. Qiu and M. Zabzine, “Introduction to graded geometry, Batalin–Vilkovisky formalism, and their applications.” Archivum Mathematicum 47, no. 5, pp. 415–471 (2011).
    • P. Mnëv, “Lectures on Batalin–Vilkovisky formalism and its applications in topological quantum field theory.” arXiv:math-ph/1707.08096 (2017).
13. (7. Februar)
    Loose ends, unanswered questions, open problems, »natürliche Schwankungen«!
    A few scattered remarks in lieu of closure: The Chevalley–Eilenberg complex of the supertranslation algebra is in fact an R-algebra; it is the Koszul complex of the nilpotence variety over R (the coordinate ring of A¹). This implies that its zeroth homology is O, and the higher-degree homologies are equivariant O-modules—and therefore correspond to supermultiplets. Further, the total cohomology is isomorphic to O precisely when Y is a complete intersection. Also, an exhortation: It's profitable to think of a twisted theory as valued in chain complexes, rather than naively passing to homology. Example: there are higher operations on the local operators of a (topologically) twisted theory, coming from the homology of the little disks operad via topological descent. The homology of the little n-disks operad is the degree-(1 - d) Poisson operad. A twist is a functor from A-modules to chain complexes (which are also equipped with an action of Z(Q)); the nilpotence variety is the space of such functors.
    References:
    • C. Beem, D. Ben-Zvi, M. Bullimore, T. Dimofte, and A. Neitzke, “Secondary products in supersymmetric field theory.” arXiv:hep-th/1809.00009 (2018).

Aspirational Syllabus:

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 February 8, 2019. —