We are an informal mathematics and theoretical physics discussion group at Duke University. If you want to be added to the mailing list, please send an email to dirac-request@duke.edu.

We welcome anyone who is willing to give a talk or two. The talks need not be tied to any of the topics below; we are always open to learning new things, but they must be accessible to a graduate student audience composed of both mathematicians and physicists.

This semester we are meeting on Wednesdays at 1:30 PM in Physics 298.

Talks

Fall 2019


  • Singularities in General Relativity | James Wheeler
    • October 8, 2019

      I’ll introduce the question of what a singularity in General Relativity should be. I’ll discuss the prototypical GR singularity, the Schwarzschild solution, as well as another example or two demonstrating the challenge of analyzing whether a spacetime is singular, especially via coordinate charts. To take a more general approach, I’ll discuss various properties required of a spacetime to make it a physically reasonable model, in particular time orientation, the hierarchy of causality conditions one might require, and timelike geodesic- or bundle-completeness. I’ll again look to examples to demonstrate these properties. Ultimately, I’ll arrive at a reasonable definition of a singular spacetime, and I’ll point to the questions left open by this definition.

    • October 22, 2019

      This week, I’ll begin by recalling our definition of a singular spacetime and the primary questions it leaves open that we’d like to address. In particular, I’ll discuss the question of “where” singularities are by giving a brief overview of the alphabet of boundary constructions (any one of which would take a full lecture to describe in detail) that have been attempted as well as their merits and drawbacks. Having addressed this question as best as we can, I’ll then begin the discussion of how serious we need to take singular spacetimes as general relativists, leading to the singularity theorems of Hawking and Penrose.

    • October 30, 2019

      Tomorrow, I’ll finally move past the question of what singularities are, turning instead to the question of why we indeed need to worry ourselves with them when doing General Relativity. The main feature of the discussion will be (finally) Hawking and Penrose’s singularity theorems. Depending on how the timing works out, I may go on to introduce the topic of what we mean by a black hole (more generally than our Schwarzschild prototype) and how the problem of Cosmic Censorship arises out of all that we’ve discussed so far– the bulk of that topic, however, will probably be treated in my final talk next week.


  • Gravitation | Benjamin Hamm and James Wheeler
    • August 28, 2019: Mathematical structure of general relativity I | James Wheeler

      Tomorrow, I’ll give a brief review of the mathematical structures underlying General Relativity, at each step giving a physical motivation for invoking the structure. My perspective will be from the standpoint of asking what kind of global structure we can reasonably infer from the local observations we’re limited to making. In particular, I’ll begin with the definition of a smooth manifold and discuss the construction of its tangent space and the notion of a connection. I’ll introduce tensors and why special relativity leads us to expect that a manifold model for the universe should come equipped with a metric tensor. Finally, I’ll introduce the Levi-Civita connection and use it to talk about the decomposition of a general connection into its metric compatibility and torsion tensors, demonstrating how torsion can change geodesics when metric compatibility is fixed. This will set the stage for Ben’s discussion, which will culminate in a computation of how light redshift would be different if one takes the geometric motivation of dark matter through the addition of torsion seriously.

    • September 4, 2019: Mathematical structure of general relativity II | James Wheeler

      I’ll discuss some examples together with some simple explicit computations to demonstrate concepts introduced last week. Having (hopefully) provided a bit of clarity, I’ll complete last week’s discussion by introducing the Levi-Civita connection and the decomposition of a general connection into metric compatibility and torsion tensors.

    • September 10, 2019: Einstein's Equations, Curvature, and the Hilbert Action - I | Benjamin Hamm

      This week I will start with a brief introduction of the Einstein Equations. From here, I will introduce the concepts of parallel transport and geodesic deviation, and ultimately use them to motivate a discussion of the Riemann curvature tensor. I will also provide a brief discussion and description of the stress-energy tensor. Finally, time provided, this will allow us to return to the discussion of the Einstein-Equations and motivate the form for the Einstein-Hilbert Action. Looking forward, this will motivate a discussion of how the action may be modified to include torsion, and the results of doing so.

    • September 18, 2019: Einstein's Equations, Curvature, and the Hilbert Action - II | Benjamin Hamm

      I will continue with a short recap of parallel transport and Riemann curvature. From here I will finally introduce the stress-energy tensor and the notion of energy-momentum conservation. We will then derive the Einstein equation from the Hilbert Action, and discuss the fact that the Hilbert Action is the most general action of its type. We will then be able to discuss the various ways that the action may be modified in order to attain alternative theories of Gravity. Then, given time, we will discuss a particular case of such modification, often referred to as the “Wave Dark Matter” or “Fuzzy Dark Matter” theory.

    • September 24, 2019: Einstein's Equations, Curvature, and the Hilbert Action - III | Benjamin Hamm

      This week we will finally discuss the Einstein-Hilbert Action in more detail. Starting with a brief recap of where we arrived last week, I will restate the axioms which produce the usual form of the action. From here I will point out a few ways that people have considered modifying the action, and the theories which result from doing so. I will then end with the example of Wave Dark Matter, a theory identical to General Relativity, but with a non-trivial torsion tensor.


Spring 2019


  • Projective Representations in Quantum Mechanics | Paul Aspinwall
    • March 20, 2019: Projective Representations in QM

      I will describe the issue of projective representations in QM. For finite dimensional symmetry groups this idea leads to the familiar notion of spin, but for infinite dimensional groups the issue is more subtle. This leads to a nice way of understanding why “central charges” appear in symmetry algebras and I’ll go through the example of the Virasoro algebra.

    • April 3, 2019: Projective Representations in QM

      In the last lecture I introduced the abstract idea of projective representations of a Lie algebra. This is trivial for the familiar examples of finite-dimensional simple groups (like, SU(2), SU(3), etc.). This talk I’ll go through the Virasoro algebra (key in string theory and two-dimensional conformal field theories) and show it’s not trivial, and very important!

    • April 10, 2019: Bosonization

      I’ll give a “physics proof” of the Jacobi Triple Product formula and then apply it to bosons and fermions in 2 dimensional conformal field theory. Lots of fun number theory!


  • Quantum Symmetries and Twisted Equivariant Representations | Akos Nagy

    Akos Nagy will resume his series of talks titled “Quantum Symmetries and twisted equivariant representations.”

    Talks

    • February 6, 2019

      In the first talk Akos will review what he talked about last year, namely how symmetries can be represented in QM, and how these representations canonically fall into 10=8+2 classes, labelled by Clifford algebras.

    • February 13, 2019
    • February 27, 2019


  • Intro to Topological Insulators | Alexander Watson

    Alexander Watson will finish up the series of talks on topological insulators that he started last semester.

    Notes

    Talks

    • January 16, 2019

      I will recap: (1) the strangeness of the quantum Hall and related effects (transverse conductivity proportional to an integer) (2) Bloch (band) theory (3) how the quantum anomalous Hall effect is related with integrating a curvature over a closed surface (yielding an integer: the Chern number). I will then introduce the Haldane model, a tight-binding (discrete) model of a hypothetical 2d material which exhibits the quantum anomalous Hall effect where the Chern number can be computed relatively easily and verified non-zero. If there is time I will discuss the edges of such materials where very robust currents propagate. The robustness of these currents has stimulated recent interest for applications.

    • January 23, 2019
    • January 30, 2019

      I will describe the Haldane model which describes a hypothetical material which would exhibit the quantum anomalous Hall effect, a quantum Hall effect which does not require a magnetic field through the material. Such materials are now known as topological insulators.


Fall 2018


  • Quantum Symmetries Revisited, Part I | Akos Nagy

    In the next few talks, I will be talk about symmetries of quantum (mechanical) systems and how they relate to the topological phases of matter, following the highly influential paper of Freed and Moore [1]. In the first talk, I will have a more nuanced look at what constitutes as a “quantum symmetry”, and extend the usual notion of only unitary symmetries that commute with the Hamiltonian. This is not entirely new to physicists, part of it was know since Dyson’s famous 3-fold way. Mathematically, Dyson’s 3-fold way is the trichotomy of real, complex, and quaternionic representations. The detailed analysis of Freed and Moore gives a nice and complete understanding of quantum symmetries, and generalizes Dyson’s result to a 10-fold way. The 10 = 8+2 ways correspond to the 8+2 (Morita-)equivalence classes of Clifford algebras, and gives a mathematically clear understanding of Kitaev’s periodic table [2].

    [1] D. S. Freed and G. W. Moore: Twisted equivariant matter, Annales Henri Poincare, Volume 14, pages 1927-2023, (2013)
    [2] A. Y. Kitaev: Periodic table for topological insulators and superconductors, AIP Conf. Proc., Volume 1134, pages 22-30, (2009)

    Talks

    • November 28, 2018
    • December 5, 2018


  • Intro to Topological Insulators | Alexander Watson

    The experimental observation and subsequent theoretical investigation of the quantum Hall effect showed that a surprising connection exists between condensed matter physics (roughly, the study of why matter behaves as it does) and topology. Specifically, it was found that under appropriate conditions part of the resistivity (roughly, the response of electrons in the material to an applied electric field) of certain two-dimensional materials must be exactly proportional to an integer. This integer was later understood as the first Chern number of a certain vector bundle associated to the atomic structure of the material. In these talks I will explain, starting from basic electromagnetism and quantum mechanics, how this correspondence arises. I will then discuss some of the interest in such materials (known as “topological insulators”) for applications.

    Notes

    Talks

    • November 12, 2018
    • November 19, 2018
    • November 26, 2018
    • December 3, 2018


  • String Theory | Ronen Plesser
    • October 31, 2018

      Ronen Plesser will discuss the basics of string theory. It is going to be an introduction equally accessible to both mathematicians and physicists and no background in either string theory or advanced geometry will be necessary.

    • November 7, 2018

      Ronen Plesser will continue his introduction to string theory. For those who missed last week’s talk, Ronen is basing his discussion on Edward Witten’s “What every physicist should know about string theory”

    • November 14, 2018


  • Braids and the Yang Baxter equation | Orsola Capovilla-Searle

    Orsola Capovilla-Searle will give a series of talks on “Braids and the Yang Baxter equation.” She will give an outline of how braids are related to the solution of the Yang Baxter equation. Orsola has promised that the talks would be accessible to everyone.

    Talks


  • Foundations of Quantum Mechanics | Hersh Singh

    References

    Talks

    • September 19, 2018

      I’ll start with the very basics and discuss some simple thought experiments that illustrate the weirdness of quantum mechanics and the need for it. I will then introduce the postulates of QM and dwell on them for a bit. This should set the stage for discussing the “EPR paradox” and it’s conclusion with the Bell inequality, which we will probably get to next week.

      Tomorrow’s talk will require absolutely no prior knowledge of quantum mechanics.

    • September 26, 2018

      I’ll discuss the EPR paradox and describe a simpler version of Bell’s argument (sometimes called the GHZM experiment) that shows, yet again, how quantum mechanics is weird.

    • October 9, 2018

      Tomorrow I’ll conclude my talks, on the foundations of quantum mechanics, with the GHZM experiment.


Spring 2018


  • BRST | Ronen Plesser
    • April 27, 2018

      Tomorrow I will continue the discussion of BRST quantization. I will show how BRS and T found the fermionic symmetry in the path integral approach to quantizing gauge theory, and try to compare to the symplectic approach we learned in Paul’s lectures.


  • Anomalies in Physics | Ronen Plesser
    • April 9, 2018

      I want to try to explain what anomalies tell us about physics. Some of this is a bit off the mathematical physics path we have been following, but it seems to me that it belongs in the series. In particular, I want to say something about: (a) How the chiral anomaly is related to pion decay (b) What anomalies in global symmetries tell us - ‘tHooft matching, anomaly cancellations. (c) How SM anomalies break Lepton number (d) What the anomaly has to do with the theta angle and instantons.

    • April 23, 2018


  • Feynman Diagrams from the BV Formalism | Eugene Rabinovich

    References

    Talks

    • April 6, 2018

      Eugene Rabinovich will tell us more about the Batalin-Vilkovisky formalism in QFT and show, with a simple example, how it encodes information about the path integral.


  • BRST | Paul Aspinwall

    References

    Talks

    • March 23, 2018

      I’ll do the purely classical version of BRST. Ronen will hopefully do the path integral version later. For those who’ve never heard of BRST, it’s kind of central to string theory and topological field theory, not to mention gauge theory.

    • March 29, 2018

      I’ll continue my BRST talks. This will include a description of Lie algebra cohomology and building a double complex to define BRST cohomology.

    • April 20, 2018

      Today I’ll finish my BRST talks. If you remember I’d written down two complexes. I’ll remind of these, write them as a double complex and get the BRST operator that QFT people will recognize. I’ll give some examples to show it’s not quite what the QFT people ordered.


  • Conformal Perturbation Theory | Ilarion Melnikov
    • March 2, 2018

      I will sketch out how conformal perturbation theory (CPT) should fit into the framework of renormalizable quantum field theory. CPT is expected to have some remarkable properties, especially when compared to perturbative quantum field theory computations. I will review some of these conjectured properties and then discuss steps to prove them in the context of two-dimensional theories.


  • Characteristic Classes and Index Theory | Matthew Beckett

    Thanks to Orsola Capovilla-Searle for the beautiful notes!

    Talks

    • February 16, 2018: Characteristic classes 1

      I will be giving the first few talks, with the eventual goal of talking about index theorems. This Friday I will take the first step in that direction by talking about characteristic classes.

    • February 23, 2018: Characteristic classes 2

      My plan is to wade a little deeper into the subject of characteristic classes, building on Chern classes to define Pontrjagin classes, and then talk about Chern and Pontrjagin genera.

    • March 8, 2018: Clifford Actions

      I will be discussing Clifford actions on vector bundles, Dirac operators, and, time permitting, spinor bundles.

    • March 26, 2018: Index Theorem

      I will give a bit more detail about spinor bundles and talk a little of their topology. I should then have everything I need in order to state the index theorem.


Fall 2017


  • Path Integrals in Quantum Mechanics | Paul Aspinwall

    Thanks to Mendel Nguyen for the notes!

    Talks

    • October 23, 2017: Path Integrals in Quantum Mechanics

      Tomorrow Paul Aspinwall will start his talks on quantum mechanics.

    • November 6, 2017: Path Integrals in Quantum Mechanics

      Paul Aspinwall will continue with his series on quantum mechanics.

    • November 13, 2017: Path Integrals in Quantum Mechanics

      I’ll continue with the Feynman path integral and then start talking about Norbert Wiener’s version.

    • November 27, 2017: Path Integrals in Quantum Mechanics

      Paul Aspinwall will continue with his series on quantum mechanics.


  • Anomalies | Hersh Singh

    References

    Talks

    • October 5, 2017: Path Integral Formulation of QFT

      With the goal of eventually understanding anomalies in quantum field theory, I will start talking about quantum field theory. I am still trying to figure out the minimal subset of QFT that we will need. But at the bare minimum, we will need to talk about the path integral formulation, so that’s where we will start. We will also discuss the symmetries of a Lagrangian, and how they give us conservation laws by Noether’s theorem. An important question to ask: Is a classical symmetry of the Lagrangian also a symmetry of the quantum theory? (Hint: This is how anomalies get their name.)

    • October 12, 2017: Path Integral Formulation of QFT

      Hersh will continue with his talks and will introduce the path integral formalism. Let’s meet in the faculty lounge at 5 pm as usual.

    • October 19, 2017: Path Integral Formulation of QFT

      Hersh will set up the path integral formalism in QFT and introduce the Dirac equation if he can get to it.

    • October 26, 2017: The Dirac Equation

      Hersh will discuss the Dirac equation.

    • November 9, 2017: The Abelian Anomaly

      Having delayed the goal of discussing anomalies in quantum field theory by several ‘background’ talks, tomorrow I’ll finally drop into the middle of the action and start talking about the abelian anomaly. Among other things, it will lead us into good math.

    • November 16, 2017: The Abelian Anomaly

      Tomorrow I’ll continue with the abelian anomaly and derive the index theorem.

    • November 30, 2017: The Index Theorem

      Hersh will probably complete his series on anomalies.


  • Constrained Dynamics | Travis Maxfield

    Notes

    Talks

    • September 18, 2017

      Travis Maxfield will talk about singular hamiltonians which should be the perfect continuation of what Orsola started last week.

    • September 21, 2017

      Travis will discuss some of the confusing issues about the toy model Lagrangian he constructed in Tuesday’s talk.

    • October 2, 2017

      Travis Maxfield of CGTP will talk about singular hamiltonians (constrained dynamics).


  • Symplectic Geometry | Orsola Capovilla-Searle
    • September 14, 2017: Symplectic Geometry

      I’m going to go over some of the basic definitions of symplectic geometry. I will also look at how it was first developed in Hamiltonian mechanics, and discuss Poisson brackets and Noethers theorem. I will also indicate some of the key differences between Riemannian geometry to which were are more accustomed to and symplectic geometry.


Spring 2017


  • General Relativity with Torsion | Benjamin Hamm
    • June 14, 2017

      I plan to cover the basic axioms/assumptions of General Relativity along with some Riemannian geometry. Then, given time, I will cover metric compatible, torsion-free gravitation and then a theory which uses neither assumption.


  • Principal Bundles | Matthew Beckett

    References

    • I originally learned from Spivak’s Comprehensive Introduction to Differential Geometry (although it was the very old version).
    • I think the material on principal bundles is in Volume II. Kobayashi-Nomizu’s Foundations of Differential Geometry Volume I is a very clean (but opaque) treatment.
    • Taubes also has a book called Differential Geometry: Bundles, Connections, Metrics and Curvature, although I am not as familiar with it.
    • Also, by popular demand I am attaching my Master’s Thesis. Sections 2.2 and 2.3 are more or less what we covered today. I will cover 3.3 on Friday. Also, section 4 covers a lot of what I went through before on Yang–Mills theory, in case you wanted a review of that.

    Talks

    • May 31, 2017

      I will be talking about Principal Bundles. I aim to get through the definitions and some basic examples, and to discuss Associated Vector Bundles.

    • June 2, 2017

      Matthew will talk about connections on principal bundles.


  • Supersymmetry | Hanqing Liu

    Notes

    Talks

    • April 24, 2017: Supersymmetry

      Today I will introduce supersymmetry in 4 dimension. I will first spend quite some time to motivate the supersymmetry, including Coleman-Mandula theorem and the result of Haag, et al. Then I will review the spin representation of Lorentz group (more precisely, SL(2,C)) with the Van der Waerden notation (dotted and undotted), and derive the super algebra.

    • April 27, 2017: Supersymmetry

      Today I will first make something clear which was unclear last time, then finish the representation theory of Lorentz group. The next thing is defining the supersymmetry algebra and the Witten index.

    • May 1, 2017: Supersymmetry

      Things always turn out to be slower than I expect, so today I will finish the remaining stuff: defining the supersymmetry algebra and the Witten index.


  • Yang-Mills | Matthew Beckett
    • April 13, 2017: Yang-Mills

      I will be talking this evening. I aim to give a brief introduction to Yang-Mills theory. I’m not sure how far I will get, but I aim to derive the Yang-Mills equation and define instantons. Time permitting, I will look a little at the specific case of an SU(2) bundle over S^4.

    • April 20, 2017: Yang-Mills

      I will start where I left off last week, beginning by deriving the Euler-Lagrange equation for the Yang-Mills functional. I will then start looking at some properties of instantons, including some topological considerations, and then look at some examples on S^4.


  • Instantons | Hersh Singh
    • April 6, 2017: Instantons

      Today I’ll start talking about instantons, from chapter 7 in Sydney Coleman’s “Aspects of Symmetry.” I’ll start with a bit of path integrals in quantum mechanics, and talk about how we can recover some well known results in QM by looking from the instanton perspective. The aim of course is to get to instantons in gauge theories, which is where topology comes in, but we’ll see how far I can go today.

    • April 10, 2017: Instantons

      Today I’ll continue talking about instantons. We’ll discuss a couple of somewhat more non-trivial examples in quantum mechanics. Looking at the double-well potential, in particular, will clarify the connection of instantons to “tunneling.” Time permitting, we’ll move on to gauge theories. Hopefully, today I’ll be able to set the stage up for that, so that we can discuss the important results next time.


  • Monopoles | Mendel Nguyen

    My talk discussed topological conservation laws in gauge theories. In particular, I introduced the ‘t Hooft-Polyakov monopole.

    References:

    • S. Coleman, Aspects of Symmetry, Chapter on ‘Lumps’; and
    • V. Rubakov, Classical Theory of Gauge Fields

    Talks

    • March 30, 2017: Monopoles
    • April 3, 2017: Monopoles


  • Cohomology | Arman Margaryan
    • March 23, 2017: Cohomology

      Since we have done quite a lot of bundle, connection stuff already, Arman will catch us up on the algebraic topology side of things by introducing DeRham cohomology (definition, long exact sequence etc.) and possibly some basics of homotopy today. It is designed to be an introduction for those who haven’t seen much algebraic topology.

    • March 27, 2017: Cohomology

      Arman is going to carry on about cohomology groups and so on today at 6.


  • Differential Geometry | Arya Roy
    • February 13, 2017: Vector Bundles

      Today I will cover the basics of vector bundles on manifolds. I will start with the definition of a differentiable manifold and lay out the basics of the construction of a vector bundle. Then I will give several examples of vector bundles that we will encounter, like the tangent and cotangent bundle, and how to construct new bundles like the direct sum, tensor, exterior power, Hom and the pullback bundle. All of these constructions will be useful later for our applications. Even though it sounds like a long list nothing more than a standard linear algebra background will be necessary.

    • February 20, 2017

      Today I will construct various vector bundles corresponding to the vector space constructions I discussed last Monday. Math grad students might again be bored out of their minds! Once we get past connections and curvature everyone will be on the same page and we can construct the Yang Mills Lagrangian.

    • March 9, 2017

      I will start with the theory of connections on vector bundles and talk about connections on all the bundles I gave constructions for last day. Connections have a couple of equivalent of definitions and I intend to talk about all of them, starting with the most obvious (covariant derivative) as a generalization of derivatives. Later I will talk about how connections can be viewed as a splitting on tangent bundles. I realized I was too quick in my discussion of exterior derivatives on bundles, so I will revisit bundle valued forms and exterior derivatives and finish of connections and curvature.


References