Supersymmetry: Lecture Log


Regular class, Fall 2013

August 29 (Thursday):
Syllabus and admin. Why supersymmetry: hierarchy and naturalness; GUTs, etc., exact non-perturbative results; use in string theory.
Supersymmetric Higgs mechanism: each massive vector multiplet eats a whole scalar multiplet; SQED example in components.
September 3 (Tuesday):
Supersymmetric unitary gauge; SQED moduli space.
Intro to non-abelian SYM: the NA vector superfield, the NA gauge symmetry, and the NA Wα and α̇.
September 5 (Thursday):
SYM: the Lagrangian, the θ angle, and the holomorphic gauge coupling.
SU(2) with one chiral doublet: no Higgs mechanism. SU(2) with two doublets: the scalar potential, the Higgs mechanism, the SUSY unitary gauge, and the moduli space.
SQCD with one flavor: the scalar potential and the Higgs mechanism.
September 10 (Tuesday):
SQCD with several flavors: Higgsing in sequence, and the surviving gauge group; the classical geometry of the moduli space.
Superfield Feynman rules for the Wess–Zumino model: the (massless) chiral propagator; the Yukawa vertex; examples of tree diagrams; general rules.
September 12 (Thursday):
Loop graphs for superfields: the fermionic δ function and its derivatives; example of a one-loop graph; general rules for loop graps; momenta in the numerators; counting the derivatives. Propagators for massive chiral superfields. Non-renormalization of the superpotential. Renormalizability of the WZ model.
September 17 (Tuesday):
Holomorphy: moduli-dependent couplings; holomorphic superpotential couplings; relation to the non-renormalization theorem; integration out of heavy fields from the superpotential.
Infared problems: derivative couplings mascarading as non-derivative; D. R. T. Jones example.
Wilsonian renormalization versus conventional renormalization (overview).
September 19 (Thursday):
Wilsonian running couplings versus ‘physical’ running couplings; only the Wilsonian Yukawa couplings are holomorphic. Holomorphic gauge couplings: Wilsonian f(φ) are holomorphic; no (Wilsonian) renormalization beyond one loop; physical beta-functions have higher-loop terms due to IR effects.
SQED superfield Feynman rules: vector propagator; vertices; examples of one-loop graphs; wave-function renormalization of the charged fields.
September 24 (Tuesday):
SQED: errata for the photon propagator and for the δ2; counterterm vertices; Ward–Takahashi identities and renormalizability of SQED; Ward identities for the counterterms; conserved electric current superfield.
Calculating the SQED beta function to one-loop order.
September 26 (Thursday):
Conserved current and Ward identities; current superfields for global symmetries.
Konishi anomaly for the axial current. Adler–Bardeen theorem for QED and SQED (the anomaly comes from the one-loop diagrams only).
October 1 (Tuesday):
Adler–Bardeen theorem and the θ angle: dressing up the anomaly at higher loops; axionic couplings of moduli scalars; moduli-dependent redefinition of the fermionic fields; canceling the anomaly by changing the θ angle.
Coupling SQED to moduli; modili-vector-vector amplitudes; redefinition of the charged superfields and the Konishi anomaly; canceling the anomaly by adjusting the Wilsonian gaguge coupling.
Relation between the Wilsonian and the physical gauge couplings; NSVZ (Novikov+Shifman+Vainshtein+Zakharov) formula; NSVZ beta function to all loop orders; generalization to SQCD with multiple charged fields.
October 3 (Thursday)
Generalized Konishi anomaly: multiple chiral superfields; several vector fields; fab matrix and the mixing of different photons.
Non-abelian Konishi anomaly: non-abelian WαWα; origin of the non-abelian terms; indexology of the non-abelian anomaly; SQCD example.
Gauge anomalies: the cubic anomaly; the trace anomaly WRT gravity; the Fayet–Illiopoulos term and its UV divergence.
Cancellation of massive fields from the anomalies.
October 8 (Tuesday)
SQCD beta function: the one-loop beta-function; g(φ) and the axionic coupling; NSVZ formula for the g(ΦΦ̅); NSVZ all-loop beta function; beta-functions of general gauge theories.
Deep IR limits of QCD and SQCD with different Nf/Nc ratios: β(g) and fixed points; Banks–Zaks fixed point in QCD; the conformal window; other IR regimes of QCD; Banks–Zaks in SQCD; Seiberg limit and SQCD conformal window; other IR regimes of SQCD.
October 10 (Thursday)
N=4 SYM and its N=1 deformation; RG flow in the deformed theory; IR-attractive fixed line and a whole family of SCFT in deep IR; weak-coupling and strong-coupling limits of N=4 SYM; AdS/CFT duality.
Klebanov–Witten model: the models and its quiver diagram; renormalization of the `non-renormalizable' quartic coupling; renormalization of the gauge coupling; the IR-attractive fixed line of RG flow and the family of strongly-coupled SCFT in deep IR.
Survey of exactly-computable nonperturbative effects in SUSY theories: the low-energy EFT and its couplings; the holomorphic couplings can be computed exactly, the non-holomorphic couplings cannot; the SUSY vacua and the exact scalar VEVs in those vacua can be computed exactly; composed chiral operators and the gaugino condensate example; fW for massless photons; no exact formulae for the scattering amplitudes.
Gaugino condensation in N=1 SYM: the chiral R-symmetry and its anomaly; discrete anomaly-free Z2N symmetry and its spontaneous breakdown by ⟨λλ⟩; N SUSY vacua.
October 15 (Tuesday)
SUSY vacua: only lowest components of superfields have VEVs; SQCD examples; Witten index and the number of SUSY vacua.
Gaugino condensation in N=1 SYM: the θ angle, the anomaly, and the phase of the ⟨λλ⟩ condensate; dimensional transmutation, ΛSYM, and the magnitude of the condensate; normalization of the condensate and the holomorphic formula for the ⟨λλ⟩; Veneziano–Yankielowicz effective superpotential for the condensate.
October 17 (Thursday)
The Θ̅ angle in QCD and its invariance under chiral redefinitions of the quark fields.
Integrating out a heavy flavor from SQCD: the holomorphic invariant of anomalous field redefinitions; RG flow through a threshold and the matching conditions for the low-energy effective theory; the effective Λlow and the gaugino condensate.
Higgs regime of SQCD: integrating out the massive vector superfields; RG flow though the vector threshold; holomorphic formula for the effective Λlow; the gaugino condensate and the effective superpotential for the modulus of the Higgs VEVs; SUSY vacua of SQCD with 1 massive flavor; runaway squark VEVs for a massless flavor.
SQCD with several flavors and the Veneziano–Yankielowicz–Taylor superpotential are worked out in the homework set #8 rather than in class.
October 22 (Tuesday)
Higgs-confinement complementarity in SQCD: common holomorphic formulae for ⟨S⟩ and ⟨M⟩ in both Higgs and “confined heavy quarks” regimes; a smooth crossover between the confining and the Higgs phases of SQCD instead of a phase transition; explanation of the smooth crossover.
SQCD with Nf=Nc-1: effective superpotential for the moduli in the completely-Higgsed-down phase; source — instanton effects.
Overview of instantons in YM theories: topological sectors of the euclidean path integral; the net instanton number; the (anti) self-dual gauge fields minimize the euclidean action; single instantons and the multi-instanton solutions.
October 24 (Thursday)
Instantons and fermionic zero modes: the fermionic path integral and the zero modes of the Dirac operator; Atiyah–Singer theorem for the zero modes; zero modes and expectation values of fermionic operators; integration over instanton's collective coordinates and the effective operators; loop effects of the non-zero modes; SUSY instantons and 2 unbroken supercharges.
Instanton effects in QCD and in the Standard Model: instantons flip quarks' chiralities; one-instanton versus multi-instanton effects in QCD; breaking the axial symmetry by instantons; the electroweak instantons and the baryon number violation.
Gave out the mid-term exam.
October 29 (Tuesday)
Instanton effects in SQCD (Nc =2, Nf =1 example): zero modes of quarks ang gauginos; zero modes in the Higgs phase; instanton action in the Higgs phase; instanton-induced quark mass and the Affleck–Dine–Seiberg effective superpotential; generalization to Nc >2 and Nf =Nc -1.
SQCD with Nf =Nc : ‘mesonic’ and ‘baryonic’ moduli of the squark VEVs; instanton effects do not generate a Wnp but deform the geometry of the moduli space; special points in the moduli space where the chiral symmetry remains unbroken.
October 31 (Thursday)
't Hooft's anomaly matching conditions for unbroken chiral symmetries: conditions for a general gauge theory; the proof; a non-SUSY SU(5) example; checking the conditions for SQCD with Nf =Nc ; overview of SQCD with Nf =Nc +1 (the details are left for the homework).
Collected the mid-term exams.
November 5 (Tuesday)
Chiral rings: chiral local operators and their correlation functions; the Q̅–cohomology and the chiral ring; chiral ring and VEVs in SUSY vacua.
Chiral rings in gauge theories: restriction of Q̅–cohomology to gauge-invariant operators; examples of gaugino and gaugino-bilinear operators; chiral ring of SQED and its generators; chiral ring of SYM and its deformation by instantons; chiral ring of SQCD.
November 7 (Thursday)
On-shell chiral rings: equations of motion in the chiral ring language; chiral ring of a critical theory; chiral ring equations in gauge theories; chiral ring equations of QCD.
Conformal symmetry: scale invariance and conformal symmetry; geometric definition; conformal generators and their algebra; relation to Lorentz symmetry in d+2 dimensions.
The superconformal symmetry: the special conformal supercharges and their (anti) commutators; the R symmetry is a part of superconformal algebra; the PSU(2,2|N) algebra for extended SUSY.
November 12 (Tuesday)
Radial quantization of conformal field theories: mapping local operators at x=0 to states in the Hilbert space of the S3 sphere; the conformal generators in H(S3); multiplets of the conformal algebra; primary and descendent states and operators; unitarity bounds on operators' dimensions.
Radial quantization of superconformal theories: the supercharges in H(S3); primary and descendent operators in SCFT; primary chiral operators have Δ=(32)R; chiral rings in SCFT.
November 14 (Thursday)
SQCD conformal window and the R-charges.
Seiberg duality for the conformal SQCD: IR dualities in general; the A theory, the B theory, and their chiral rings; other checks on the duality; anomaly matching (homework); moduli spaces of the A and B theories; deformations of the A and B theories by masses and O'Raifeartaigh terms(homework).
November 19 (Tuesday)
Seiberg duality: renormalized gauge couplings of the A and B theories; duality below the conformal window — A theory in the UV, B theory in the IR; tests of Seiberg duality.
Unexpected gauge symmetry: the CPN example.
Magnetic monopoles: `hedgehog' monopole in SU(2)→U(1); magnetic charge; Dirac quantization condition; BPS lower bound on the mass.
November 21 (Thursday)
Supersymmetric monopoles and dyons: supercharges and fermionic zero modes; degenerate dyon states; supermultiplets; generalization to extended SUSY.
Electric/Magnetic duality: Maxwell eqs in presence of monopoles; duality of fields and charges; α↔1/α.
Θ angle in QED and the electric charges of the dyons; the complex charge lattice.
S–duality: E/M duality for Θ≠0.
November 26 (Tuesday)
S duality: canonical EM for Θ≠0; the charge lattice and the SL(2,Z) duality group; transformation law for the τ; T and S generators of the duality.
Intro to Seiberg–Witten theory: SU(2) with a triplet; the U modulus; the massive theory and the singulatities of the moduli space; singularities and massless particles; singularities of gauge couplings.
SW theory of τ(U): physical meaning; behavior for τ→∞; Imτ>0 and singularities at finite U; branch cuts and S dualities.
December 3 (Tuesday)
Seiberg–Witten theory: preliminaries; singularities of τ(U) due to charge particles becoming massless; un-Higgsing not allowed; massless monopoles and dyons; singularities and monodromies; group theory of mondoromies; monodromies in the Seiberg–Witten model.
Confinement: massive Φ leads to monopole condensation; dual Higgs mechanism, magnetic superconductivity, and confinement of the electric charges; relation to quark confinement in QCD; dyon condensation and oblique confinement.
December 5 (Thursday)
Elliptic curves: definition; relation to tori; period lattice and SL(2,Z) dualities; periods and contour integrals; degenerate curves and singularities of τ(U).
Seiberg–Witten curves: the SW curve of the quarkless model and its singularities; checking monodromies; adding quarks; Argyres–Douglas points.
Gave out the final exam.

Extension, spring 2014

January 17 (Friday)
N=2 SUSY: central charge; short and long multiplets; gauge couplings and their N=2 superpartnets; no renormalization beyond one loop.
N=2 SUSY non-linear sigma models: NLSM without SUSY and for N=1; Kähler geometry; separate moduli spaces for vector and hyper multiplets for N=2; hyper–Kähler geometry for hypermultiplets; chiral N=2 superfields for vector multiplets; holomorphic prepotential and its relation to the gauge couplings; special Kähler geometry for vector multiplets.
January 24 (Friday)
Canceled due to campus closure
January 31 (Friday)
Seiberg–Witten Theory: scalar superpartners a(u) and ã(u) of the vector and dual vector fields; symplectic metric in terms of a(u) and ã(u) scalars; SL(2,Z) dualities for the da and ; a(u) and ã(u) as contour integrals on the Seiberg–Witten curve; the meromorphic differential; masses of quarks, monopoles, and dyons; central charge Z=nel×a(u)+mmag×ã(u)+Mbare.
Quark hypermultiplets and Higgs branches of the moduli space: no Higgs branches for non-degenerate quark masses; Higgs branch for 2 degenerate flavors has moduli space=H/Z2; Higgs branches for several degenerate flavors. Maybe: SO(2Nf) symmetry for Nf massless quark flavors.
February 7 (Friday)
Canceled due to campus closure and rescheduled for February 10 (Monday).
February 10 (Monday)
Flavor symmetries of N=2 SQCD: no chiral symmetry for Nc≥3; SO(2Nf) for Nc=2; U(2) R–symmetry.
Magnetic monopoles: fermionic zero modes; gaugino zero modes and supermultiplet structure; quark zero modes and spinor multiplet of SO(2Nf); correlation of the L/R spinor type with the electric charge; monopoles and dyons of higher magnetic charges and their SO(2Nf) quantum numbers.
February 14 (Friday)
Seiberg–Witten theory with Nf=1,2,3: singulatities of the Coulomb branch for 3, 2, or 1 massless flavors; singularities of the SW curve; R–symmetry of the SW curve; SW curve for 1 massless flavor.
February 21 (Friday)
Seiberg–Witten curves for Nf=1,2,3: SW curves for 2 or 3 massless flavors; SW curves for massive flavors.
Massless Nf=4 theory: scale invariance and superconformal theory; SW curve for a fixed τ modular forms.
February 28 (Friday)
Massless Nf=4 theory: Γ(2)⊂SL(2,Z) preserves SO(8) flavor; S3=SL(2,Z)/Γ(2) exchanges flavor vectors and spinors; fundamental domains of the SL(2,Z) and the Γ(2); 3 distinct weak coupling limits of SU(2) with 4 massless flavors.
N=4 SYM theories: 2 weak coupling limits of the SU(2) theory; generalization to other gauge groups.
Overview of S-duality for N=2 conformal theories.
March 7 (Friday)
Kodaira ADE classification of singularities; flavor symmetries of SW curves and their deformation by masses; exotic superconformal theories with E6, E7, E8 symmetries.
N=2 SQCD with Nc=3 and Nf=6: hyperelliptic SW curves for Nc=3; curve degeneration at weak coupling; degeneration at strong coupling; E6 symmetry of the degenerate curve for u=0; dual theory of the strongly coupled SU(3).
March 21 (Friday)
Overview of SUSY in different dimensions: form fields and their dualities; maximal supergravities (32 supercharges); maximal rigid SUSY (16 supercharges); theories with 8 supercharges in d=3,4,5,6; theories with 2, 4, and 6 supercharges in d=3; dimensional reduction and quantum corrections.
SUSY in d=5: vector multiplets, pre-potential, and Chern-Simons interactions; SQED and discontinuities of CS coefficients.
March 28 (Friday)
Guest lecture by professor Jacques Distler on d=6 soperconformal theories and their compactifications.
April 4 (Friday)
SUSY in d=5: central charges; instantons particles and their flavor QN; Coulomb branch of SU(2); superconformal limit (1/g2)→0; massless particles and tensionless strings; enhanced En+1 flavor symmetry of the superconformal regime; exotic SCFTs.
April 11 (Friday)
Introduction to MSSM (minimal supersymmetric standard model): MSSM fields and particles; soft SUSY breaking; spurion origin of non-supersymmetric terms; two Higgs doublets and physical Higgs scalars; charginos and neutralinos; LSP dark matter; Yukawa couplings and fermionic masses and CKM matrix; neutrino masses.
April 18 (Friday)
MSSM: masses of squarks and sleptons; GIM; super-GIM; flavor violation and squark degeneracy; slepton masses and slepton flavor violation.
April 25 (Friday)
Guest lecture by professor Can Kilic on MSSM phenomenology.
May 2 (Friday)
Gauge mediation of SUSY breaking: hidden sector and the mediator fields; one loop gaugino masses; two loop scalar masses; RG flow of SUSY breaking; top Yukawa coupling and the negative mass2 of Hu; the μ problem.
Gravitino issues: gravitino mass; cosmological problems with gravitino LSP; problems with the heavy gravitino.

Last Modified: May 2, 2014.
Vadim Kaplunovsky