### Advanced quantum field theory, 2013—2014

14:00 Fridays in room 208

### Format

This series of events introduced and reviewed central topics in modern particle theory, through pedagogical discussions facilitated by the grad students to the extent feasible.

In the fall we focused on supersymmetric gauge theories and Seiberg duality, primarily by working through Michael Peskin’s TASI96 lecture notes on “Duality in Supersymmetric Yang-Mills Theory“, with additional background from Terning’s *Modern Supersymmetry*. Peskin’s lectures follow the same general approach as Intriligator and Seiberg, while section 1 of Matt Strassler’s lecture notes on “The Duality Cascade” approaches duality from a renormalization group perspective. In 2014 I belatedly stumbled across Jose Figueroa-O’Farrill’s lecture notes titled *Electromagnetic Duality for Children*, which also appear to contain a great deal of useful information.

We started 2014 by learning about new developments in scattering amplitudes for planar *N*=4 SYM: the positive grassmannian and amplituhedron (arXiv:1212.5605, arXiv:1312.2007 and arXiv:1312.7878; arXiv:1308.1697 may be a useful introductory review). Later in the spring we explored conformal field theory and AdS/CFT duality, focusing on David Tong’s lecture notes (arXiv:0908.0333) and TASI 2001 lecture notes by D’Hoker and Freedman (hep-th/0201253), respectively.

### Schedule

**13 September 2013:** Supersymmetric gauge theory

**Facilitator:** David

**Summary:** After quickly writing down the component lagrangian for *N*=1 supersymmetric Yang—Mills theory (SYM), I presented vector and chiral superfields (without any detailed calculations), in terms of which I wrote down the Wess—Zumino model, SYM and SQCD. I didn’t quite get to the SQCD classical moduli space and super Higgs mechanism: lecture notes.

**References:** I mainly followed the notation in Terning’s *Modern Supersymmetry*, where the relevant sections are 2.5, 2.7, 3.4 and 9.1. I also consulted Srednicki’s *Quantum Field Theory* (chapter 95) and Argyres’s *Introduction to Supersymmetry* (sections 5, 6, 10 and 20).

**20 September 2013:** No meeting due to seminar by Matt Reece

**27 September 2013:** The classical moduli space

**Facilitator:** Richard

**Summary:** After reviewing the space of degenerate vacua in the linear sigma model, we turned to a simple supersymmetric “XYZ” model of three chiral superfields with three complex planes worth of vacua forming its moduli space. We concluded by reviewing SQCD and briefly considering its classical moduli space for F<N.

**References:** The linear sigma model is discussed in chapter 11 of Peskin & Schroeder, and many other places. Matt Strassler’s TASI 2001 lecture notes (“An Unorthodox Introduction to Supersymmetric Gauge Theory“) discuss the XYZ model, and the moduli space of SQCD is introduced in section 3.4 of Terning’s *Modern Supersymmetry*.

**11 October 2013:** Effective lagrangians for supersymmetric dualities

**Facilitator:** Aarti

**Summary:** We began by reviewing possible realizations of global symmetries and gauge invariance in vacua of Yang—Mills theories, considering the chiral effective lagrangian for massless QCD and generic “phases” of gauge theories (coulombic, confined or higgsed), including examples in which different phases are continuously connected as parameters vary. Adding supersymmetry to the mix immediately fixes the general form of the effective lagrangian, requiring that the superpotential and renormalized gauge coupling be holomorphic, with implications for gaugino condensation.

**References:** Sections 2—3 of hep-th/9702094, supplemented by sections 8.4–8.5 of Terning’s *Modern Supersymmetry*, on the holomorphic gauge coupling and gaugino condensation.

**18 October 2013:** No meeting due to seminar by Alexander Vikman

**25 October 2013:** Effective potentials for supersymmetric QCD

**Facilitator:** Bithika

**Summary:** After quickly reviewing supersymmetric effective superpotentials and the renormalization group flow of the holomorphic gauge coupling, we considered consequences of holomorphy on R-symmetry, gaugino condensation and holomorphic decoupling. Moving on to SQCD, we wrote down the effective superpotentials for SYM and the Affleck—Dine—Seiberg (ADS) case *F<N*.

**Reference:** Section 4 of hep-th/9702094

**1 November 2013:** No meeting due to seminar by JiJi at Cornell

**15 November 2013:** Phases and topological objects in the Seiberg—Witten model

**Facilitator:** David

**Summary:** I started with a brief review of how instantons generate the Affleck—Dine—Seiberg superpotential W_{ADS} for *F=N-1*, then focused on the Seiberg—Witten solution to *N*=2 SU(2) SYM higgsed by an adjoint scalar. This solution involves a string of conjectures and consistency checks, and I stuck with the simplest derivations that exploit ‘t Hooft—Polyakov monopoles in the dual magnetic theory. I ended with brief glances at two extensions of the Seiberg—Witten model: *N*=2–>*N*=1 susy breaking, and *N*=2 theories with matter hypermultiplets: lecture notes.

**References:** Sections 5—6 of hep-th/9702094 (skipping most of the geometric arguments). Many of the ingredients being used are discussed in more detail by Terning’s *Modern Supersymmetry*: monopoles in section 7.1, instantons in sections 7.5—7.6 and 8.4, W_{ADS} in chapter 9 and the Seiberg—Witten model in chapter 13. I also distributed a careful derivation of holomorphic decoupling that corrects some apparent typos in hep-th/9702094.

**22 November 2013:** No meeting due to seminar by John Terning at Cornell

**29 November 2013:** No meeting due to holiday

**6 December 2013:** No meeting due to Lattice Meets Experiment Workshop

**(Monday) 16 December 2013:** Non-abelian dualities

**Facilitator:** Aarti

**Summary:** After reviewing SQCD with *F<N*, we considered the new composite low-energy degrees of freedom for *F=N* and *F>N*. We discussed ‘t Hooft’s anomaly matching in some detail for the special cases *F=N* and *F=N+1*, checking that conjectured constraints and effective superpotentials for these systems reproduce the ADS superpotential upon holomorphic decoupling. We carried out the same tests for *F>N* more generally, using them to check Seiberg’s proposed duality. We concluded by considering the various phases of both SQCD and its dual theory, in particular the conformal window, the extent of which can be rigorously determined thanks to superconformal symmetries.

**Reference:** Sections 4.5 and 7—8 of hep-th/9702094

**17 January 2014:** On-shell diagrams and three-particle amplitudes

**Facilitator:** Jay

**Summary:** After some introductory remarks on locality and unitarity, we defined spinor helicity variables for massless gauge theories (especially planar *N*=4 SYM) and considered how they appear in scattering amplitudes. In particular, we discussed three-point amplitudes that can be combined to generate on-shell diagrams that represent scattering processes with no need for virtual internal particles.

**Reference:** Sections 1—2 of arXiv:1212.5605. Chapters 60 and 81 of Srednicki’s *Quantum Field Theory* provide some additional information about spinor helicity variables.

**24 January 2014:** Combinatorics for on-shell diagrams as permutations

**Facilitator:** Jay

**Summary:** Much of the meeting was spent considering how “BCFW-bridge” structures can be used to construct arbitrary on-shell diagrams from elementary three-particle amplitudes, which is related to the reconstruction of the full amplitude from its singularity structure. We next considered simple rules for manipulating and reducing on-shell diagrams, then saw how on-shell diagrams can be represented as permutations, which will lead to their connection with the positive grassmannian.

**Reference:** Sections 2—3 of arXiv:1212.5605

**31 January 2014:** The positive grassmannian and amplituhedron

**Facilitator:** Jay

**Summary:** We began by extending the idea of projective spaces to the more general grassmannian *G(k,n)*, the space of *k*-dimensional planes passing through the origin in an *n*-dimensional space. Considering elements of *G(k,n)* to be *k*x*n* rank-*k* matrices modulo the GL(*k*) group of row operations, we identified the minors (determinants of collections of *k* columns of such matrices) as containing the SL(*k*)-invariant information, with ratios of these minors being the corresponding GL(*k*) invariants. After showing how to describe *n* spinor-helicity variables as elements of *G(1, n)*, we moved on to consider the amplituhedron, introducing it as the higher-dimensional analogue of convex polygons, the same way the positive grassmannian is the analogue of triangles.

**Reference:** Sections 4.1—4.2 of arXiv:1212.5605 and sections 1—3 of arXiv:1312.2007

**7 February 2014:** No meeting due to seminar by Dragan Huterer

**14 February 2014:** On-shell diagrams and the positive grassmannian

**Facilitator:** Bithika

**Summary:** We focused on checking some results in sections 4.3 and 4.5 of arXiv:1212.5605 fairly carefully, also reviewing the general motivations, achievements and prospects of this direction of research.

**Reference:** Sections 4.3—4.5 of arXiv:1212.5605

**28 February 2014:** Constructing conformal field theory

**Facilitator:** David

**Summary:** We constructed the conformal group in *d* dimensions, starting from the defining requirement that (‘Weyl’) coordinate transformations leave angles unchanged. This left open the possibility that scale invariance need not imply conformal invariance, which we explored further through the connection to the trace of the stress—energy tensor: lecture notes.

**Reference:** Despite advertising section 4 of arXiv:0908.0333, I ended up relying more on chapter 4 of *Conformal Field Theory* by Di Francesco, Mathieu and Senechal, as well as section 1 of Paul Ginsparg’s Les Houches lecture notes hep-th/9108028. The connection to current research can be found in arXiv:1107.3987, arXiv:1204.5221 and arXiv:1208.3674 and (quite recently) arXiv:1402.6322.

**7 March 2014:** Radial quantization and the Virasoro algebra in two dimensions

**Facilitator:** Jayanth

**Summary:** Starting from a mapping between the cylinder and the plane, we used the Fourier decomposition of the stress—energy tensor to extract the generators of conformal transformations in two dimensions, which generate the Virasoro algebra. We identified raising and lowering operators within the Virasoro algebra, and used these to define primary (highest-weight) states that are annihilated by all lowering operators. Acting on the primary states with raising operators gives us infinite towers of descendants that form irreps of the Virasoro algebra.

**Reference:** Section 4.5 of arXiv:0908.0333

**14 March 2014:** No meeting due to spring break

**21 March 2014:** New developments in dynamical triangulation

**Facilitator:** Jack

**Summary:** After briefly reviewing euclidean dynamical triangulations, we heard about its lattice phase diagram and the search for a second-order critical point at which a continuum limit could be defined. While all phase transitions observed so far are first order, there are signs that right around them the long-distance physics could describe a de Sitter universe, with spectral dimension approaching 3/2 at short distances, consistent with the Bekenstein-Hawking entropy scaling.

**Reference:** arXiv:1401.3299

**28 March 2014:** Conformal ward identities and the state—operator map

**Facilitator:** Aarti

**Summary:** We started by deriving Ward identities for conformal transformations in two dimensions and using them to define primary operators. We then skipped ahead to the state—operator map that relates these primary operators to the primary states considered on 7 March.

**Reference:** Sections 4.2 and 4.6 of arXiv:0908.0333

**4 April 2014:** Superconformal symmetry in four dimensions

**Facilitator:** Bithika

**Summary:** After quickly reviewing the basic features of supersymmetry discussed previously this year, we focused on *N*=4 supersymmetric Yang—Mills, considering its gauge multiplet, central charges, and multiplet shortening resulting from saturated BPS bounds. We then spent some time discussing the structure of the *N*=4 SYM super-conformal algebra, reviewing all the generators and how they fit together into the complete structure.

**Reference:** Sections 1—3 of the TASI 2001 lecture notes by D’Hoker and Freedman, hep-th/0201253

**11 April 2014:** No meeting due to seminar by Anna Hasenfratz

**18 April 2014:** The AdS/CFT correspondence

**Facilitator:** Richard

**Reference:** Sections 4—5 of the TASI 2001 lecture notes by D’Hoker and Freedman, hep-th/0201253

**25 April 2014:** No meeting due to seminar by Abhay Ashtekar

**2 May 2014:** Gravity with torsion hidden in an extra dimension

**Facilitator:** Kamesh

**Summary:** The first topic considered how gravity can be modified by the presence of torsion, even when that torsion is confined to an extra dimension in order to ensure the theory depends only on the metric. Potential consequences include non-singular cosmologies, and spherically symmetric solutions with “naked” physical singularities and no event horizons. Following that discussion, we heard about potential application of non-commutative geometry to two-sheeted spacetime, which could provide a potential new source of CP violation.

Last modified 5 May 2014