Supersymmetry: Lecture Log

This is the lecture log for the Supersymmetry class PHY 396 T as taught in Fall 2025 by Professor Vadim Kaplunovsky.

Most lectures should be video recorded and the records available on Canvas. For the few lectures that did not get recorded because of technical glitches, I shall scan the notes I have used in class and links the scans to this page.

August 26 (Tuesday):

Syllabus and admin: textbooks and notes; prerequisite knowledge; class focus; lectures, homeworks, exams, and logistics.
Why supersymmetry: hierarchy and naturalness; GUTs, etc., exact non-perturbative results; use in string theory.
Conventions (quick overview).
Supersymmetric Higgs mechanism: each massive vector multiplet eats a whole scalar multiplet; SQED example in components and in superfields; moduli spaces beyond the Goldstone theorem.

August 28 (Thursday):

Intro to SQCD: the NA vector superfield; the NA gauge symmetry; the NA tension superfields W_α and W̅_α̇; complex gauge coupling and SQCD Lagrangian.
Higgs regime for the 1–flavor SQCD: VEVs, modulus, and eaten quark superfields; SUSY unitary gauge; vector masses in the superfield formulation.
SQCD with several flavors: sequential Higgs mechanism; moduli matrix M; rank(M) and the unbroken gauge group; moduli space geometry. Introduction to the Supersymmetric non-renormalization theorem: no perturbative corrections to the superpotential; consequences for the beta-functions.

September 2 (Tuesday):

Holomorphy: no counterterms except for δ^Z; consequences for beta-functions; holomorphy and chiral superfields; holomorphy and non-renormalization of the superpotential.
Effective classical action as generating functional of 1PI Feynman graphs (briefly).
Superfield Feynman rules for the Wess–Zumono model: superfield propagators and vertices; simple 1-loop example; delta-functions and their derivatives; 4-point example diagram; reading the operators graphically; higher derivatives of delta-functions.
non-renormalization theorem from superfield Feynman rules: evaluating the superspace integrals for a single loop; multi-loop diagrams; general form of superfield amplitudes; local versus nonlocal operators; no loop corrections to the superpotential. Started Infrared troubles: Infrared divergences leading to apparent δW.

September 4 (Thursday):

Infrared troubles: Infrared divergences leading to apparent δW; two-loop example; Wilsonian RG avoids the IR troubles.
Brief overview of the Wilsonian RG (cf. Peskin abd Schroeder, §12.1).
Holomorphy of the gauge couplings: Moduli dependent gauge couplings; holomorphic τ(M); harmonic 1/α and Θ; Wilsonian renormalization stops at 1 loop; conventional RG has higher-loop terms in beta-functions due to IR effects; Wilsonian coupling must be for the cutoff preserving SUSY and 4D gauge invariance; trouble with the dimensional reduction cutoff.
Begin SQED superfield Feynman rules: Massive propagators of chiral SF.

Extra lecture on September 5 (Friday):

Extended SUSY in 4D: Overview of extended supersymmetries in 4D: SUSY algebras; rigid N=4 and N=4 theories; supergravities with N=4=1,2,4,8; exotic theories with N=4=3,6.
Rigid N=4=2 multiplets: short and long multiplets; massless and short hypermultiplets; massless and short vector multiplets; long vector multiplets; Higgs mechanism and supermultiplets.

September 9 (Tuesday):

Corrected superspace Feynman rules for the WZ model; rule for chiral SF propagators: arrow heads come with D̅² operators, arrow tails come with D² operators.
SQED Feynman rules: gauge fixing for the photon propagator; superfield Landau gauge; superfield Feynman gauge; ghosts; electron propagators and vertices; simple diagram examples.
SQED one-loop beta-function: the diagrams; evaluating the superspace integrals; the net momentum integral; the δ₃ counterterm, the anomalous dimension, and the beta-function; comparing to the component-field beta; generalizing to other theories.
Introduction to Ward–Takahashi identities of SQED: quick overview of WT identities in the ordinary QED; WT identities and renormalizability of SQED.

September 11 (Thursday):

Ward–Takahashi identities for SQED: current superfield J; WT identities for purely photonic amplitudes; gauge symmetry and derivatives of external V superfields; no vector counterterms besides δ₂; WT identities for amplitudes with 2 electron lines; WT identities for the 1PI vertices and propagator dressings; the Ward identities for SQED.
Konishi anomaly for the axial current: supersymmetrizing the axial anomaly; UV divergences and regulators; the HD regulator and no anomalies beyond one loop; the PV regulator and the one-loop anomaly.
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.

September 16 (Tuesday):

Anomalies, gauge couplings, and NSVZ equations: coupling SQED to moduli; modilus-vector-vector amplitudes, in components and in superfields; chiral field redefinition, Konishi ampmaly, and adjusting the the Wilsonian gguge coupling; moduli-dependence of the physical gauge coupling; Novikov–Shifman–Vainstein–Zaharov (NSVZ) equations; beta-function to all look orders; generalizing to multiple charged fields.
Fayet–Iliopoulos term and its divergence; trace anomaly.
NSVZ equations for SQCD: Konishi anomalies for SQCD; one-loop beta function; CP-odd gluons+modulus amplitudes; CP-even amplitudes and moduli-dependence of the gauge coupling; the NSVZ beta-function for SQCD.

September 18 (Thursday):

Deep IR limits of QCD and SQCD with different N_f/N_c 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.
NSVZ beta-functions for general SUSY gauge theories.
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 regimes of N=4 SYM AdS/CFT duality (very briefly).

Extra lecture on September 19 (Friday):

N=2 extended SUSY: hyper-Kähler and special Kähler geomenties of the fiels spaces; chiral superspace for the vector supermultiplets; prepotential and gauge couplings; no quantum corrections to the hypermultiplets; renormalization of gauge couplings stops at one loop.
N=4 SYM theory: R-symmetries for N=2 and for N=4; SYM field content and interactions; conformal and superconformal symmetries; AdS/CFT duality.

September 23 (Tuesday):

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.
Possibility of spontaneous supersymmetry breaking: unbroken SUSY requires vacuum states with exactly zero energy; Witten index as a criterion; Witten's calculation of SQCD index=N_c; SQED example of index=0: SUSY may be spont broken or unbroken, depending on the FI parameter.
Allowed VEVs of local operators: only the lowest components of gauge-invariant superfields which are not derivatives of other gauge-invariant superfields.

September 25 (Thursday):

Survey of exactly-computable nonperturbative properties in SUSY vacua: holomorphic functions of parameters can be calculated exactly, the non-holomorphic cannot; VEVs of scalars in chiral SF are holomorphic and calculable; ditto lowest components of composite chiral SF, forex gaugino condensates; in low-energy EFTs, the superpotential and the Wilsonian gauge couplings are calculable but the Kähler functions and higher-derivative couplings are not; scalar potentials are not calculable but the locus of V=0 is calculable; scattering amplitudes are never exactly calculable.
Gaugino condensate in SYM: analogue to quark-antiquark condensate in QCD; chiral R-symmetry (of the SYM theory) and its anomaly; discrete anomaly-free Z_2N symmetry and its spontaneous breakdown by ⟨λλ⟩; N SUSY vacuum states and phases of gaugino condensate in these vacua; gauge coupling renormalization, dimensional transmutation, and the magnitude of the condensate; holomorphic Λ_SYM; normalization of the condensate; Veneziano–Yankielowicz superpotential for δS.

Plan for September 30 (Tuesday):

SQCD with one heavy quark flavor: Θ̅ angle in the ordinary QCD and its SQCD analogue; RG flow through a heavy quark threshold and the matching conditions for the low-energy effective theory; the effective holomorphic Λ_SYM 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 Λ_SYM; the gaugino condensate and the effective superpotential for the `meson' modulus of the Higgs VEVs; the effective scalar potential for the squarks and the runaway VEV for m=0.
Higgs-confinement complementarity in SQCD: common holomorphic formulae for ⟨S⟩ and ⟨M⟩ in both Higgs and confinement regimes; a smooth crossover between the two regimes instead of a phase transition; explanation of the smooth crossover.
SQCD with several massive flavors: confinement regime when all quarks are heavy; calculating the ⟨S⟩ and ⟨M⟩ VEVs; in the low quark mass limit, the Higgs regime obtains only for N<sub>f</sub><N<sub>c</sub>; formulae for the gaugino condensate and `meson' VEVs for N_f≤N_c−2; Veneziano–Yankielowicz effective superpotential; instanton origin of the non-perturbative W(M) for N_f=N_c−1.

Tentative plan for October 2 (Thursday):

Quick review of Yang–Mills instantons and fermionic zero modes.
Instanton effects in SQCD (N_c =2, N_f =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 N_c >2 and N_f =N_c -1.