- 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*W*_{α}and*W̅*_{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 δ
_{2}; 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;
*f*matrix and the mixing of different photons.^{ab}

Non-abelian Konishi anomaly: non-abelian*W*; origin of the non-abelian terms; indexology of the non-abelian anomaly; SQCD example.^{α}W_{α}

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*N*ratios: β(_{f}/N_{c}*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;*f*for massless photons; no exact formulae for the scattering amplitudes._{W}

Gaugino condensation in N=1 SYM: the chiral R-symmetry and its anomaly; discrete anomaly-free**Z**_{2N}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, Λ_{SYM}, 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 Λ_{low}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_{low}; 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*N*=_{f}*N*-1: effective superpotential for the moduli in the completely-Higgsed-down phase; source — instanton effects._{c}

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 (
*N*=2,_{c }*N*=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_{f }*N*>2 and_{c }*N*=_{f }*N*-1._{c }

SQCD with*N*=_{f }*N*: ‘mesonic’ and ‘baryonic’ moduli of the squark VEVs; instanton effects do not generate a_{c }*W*_{np}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
*N*=_{f }*N*; overview of SQCD with_{c }*N*=_{f }*N*+1 (the details are left for the homework)._{c }

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 S^{3}sphere; the conformal generators in H(S^{3}); 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(S^{3}); primary and descendent operators in SCFT; primary chiral operators have Δ=(^{3}⁄_{2})*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 CP^{N}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.

- 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*dã*;*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*=*n*_{el}×*a(u)*+*m*_{mag}×*ã(u)*+*M*_{bare}.

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**/**Z**_{2}; Higgs branches for several degenerate flavors. Maybe: SO(2N_{f}) symmetry for N_{f}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 N
_{c}≥3; SO(2N_{f}) for N_{c}=2; U(2) R–symmetry.

Magnetic monopoles: fermionic zero modes; gaugino zero modes and supermultiplet structure; quark zero modes and spinor multiplet of SO(2N_{f}); correlation of the L/R spinor type with the electric charge; monopoles and dyons of higher magnetic charges and their SO(2N_{f}) quantum numbers. - February 14 (Friday)
- Seiberg–Witten theory with N
_{f}=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 N
_{f}=1,2,3: SW curves for 2 or 3 massless flavors; SW curves for massive flavors.

Massless N_{f}=4 theory: scale invariance and superconformal theory; SW curve for a fixed τ modular forms. - February 28 (Friday)
- Massless N
_{f}=4 theory: Γ(2)⊂SL(2,**Z**) preserves SO(8) flavor; S_{3}=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 N_{c}=3 and N_{f}=6: hyperelliptic SW curves for N_{c}=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/g
^{2})→0; massless particles and tensionless strings; enhanced E_{n+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 mass
^{2}of H_{u}; the μ problem.

Gravitino issues: gravitino mass; cosmological problems with gravitino LSP; problems with the heavy gravitino.

Last Modified: May 2, 2014. Vadim Kaplunovsky

vadim@physics.utexas.edu