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\title{
\textbf{
    Chern-Simons and Other Topological Field Theories
  }
    \\ {\normalsize MSRI, November 2021} \\
  }







\begin{document}

\date{}
\maketitle
\begin{flushleft}
\textit{D. Zack Garza} \\
\textit{University of Georgia} \\
  \textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\
{\tiny \textit{Last updated:} 2021-11-18 }
\end{flushleft}


\newpage

% Note: addsec only in KomaScript
\addsec{Table of Contents}
\tableofcontents
\newpage

\hypertarget{dan-freed-introduction-to-chern-simons}{%
\section{Dan Freed, Introduction to
Chern-Simons}\label{dan-freed-introduction-to-chern-simons}}

\begin{quote}
See their paper: Some Cohomology classes in Principal Fibers Bundle and
Their Applications to Riemannian Geometry.
\end{quote}

\begin{remark}

Setup: \(G\) a Lie group, \(\pi_0 G\) finite, a \(G{\hbox{-}}\)invariant
\(p{\hbox{-}}\)linear form on
\begin{align*}
{\mathfrak{g}}: \left\langle {-},{-},\cdots,{-}\right\rangle: {\mathfrak{g}}{ {}^{ \scriptstyle\otimes_{}^{p} }  } \to {\mathbb{R}}
.\end{align*}
These sit inside invariant polynomials on \({\mathfrak{g}}\),
i.e.~\((\operatorname{Sym}^p {\mathfrak{g}} {}^{ \vee })^G\).

\(P \xrightarrow{\pi} M\) is a principal \(G{\hbox{-}}\)bundle,
\(\Theta\) a connection in \(\Omega^1_P({\mathfrak{g}})\)

\end{remark}

\begin{remark}

The curvature:
\begin{align*}
\Omega = d \Theta + {1\over 2}[\Theta \wedge \Theta]
.\end{align*}
\({\mathcal{A}}_\pi\) the affine space of all connections, look for a
universal connection on the bundle
\({\mathcal{A}}_\pi \times P\to {\mathcal{A}}_\pi \times M\).

\(\Delta^1\to {\mathcal{A}}_\pi\) a simplex connecting
\(\Theta, \Theta_1\), define
\begin{align*}
\alpha(\Theta_0, \Theta_1) = \int_{\Delta^1} \omega(\Theta_\pi)
.\end{align*}
Check that by Stokes',
\(d\alpha = \omega( \Theta_1) - \omega(\Theta_0)\). Basepoint in this
affine space: a distinguished flat section (no holonomy)
\(P\to P{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {M} }{\times} }^{2} } }\)
See Chern-Weil form \(\alpha(\Theta)\), it satisfies a
\emph{transgression} \(d\alpha(\theta) = \pi^* d\omega\)?

\end{remark}

\begin{remark}

Descend forms on \(P\) to \(M\), need forms evaluated along fiber
direction to vanish in order to descent. How to descend the Chern-Simons
form? Original paper involves \(H^{2p}({{\mathbf{B}}G}; {\mathbb{R}})\)
and reducing to \({\mathbb{R}}/{\mathbb{Z}}\) to kill topological
obstructions to descent.

\end{remark}

\begin{remark}

Next steps: Cheeger-Simons define an abelian Lie group of
\emph{differential characters} by taking a pullback:

\begin{center}
\begin{tikzcd}
    {\mathrm{CS}^q(M)} && {\Omega_{\text{closed}}^q(M)} \\
    \\
    {H^q(M; {\mathbb{Z}})} && {H^q(M; {\mathbb{R}})}
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=3-3]
    \arrow[from=3-1, to=3-3]
    \arrow[from=1-1, to=3-1]
    \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMiwwLCJcXE9tZWdhX3tcXHRleHR7Y2xvc2VkfX1ecShNKSJdLFsyLDIsIkhecShNOyBcXFJSKSJdLFswLDIsIkhecShNOyBcXFpaKSJdLFswLDAsIlxcbWF0aHJte0NTfV5xKE0pIl0sWzMsMF0sWzAsMV0sWzIsMV0sWzMsMl0sWzMsMSwiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d}{Link
to Diagram}
\end{quote}

The Chern-Weil form of a \(G{\hbox{-}}\)connection is a canonical lift
to \(\mathrm{CS}^q(M)\). See Maurer-Cartan form.

\end{remark}

\begin{remark}

\href{https://math.mit.edu/juvitop/pastseminars/notes_2019_Fall/cheeger-simons.pdf}{Cheeger-Simons}
groups aren't local, i.e.~don't satisfy a sheaf condition. Write
\begin{align*}
S^1 = \colim( I\to I\times I\leftarrow I)
\leadsto
{\mathbb{R}}/{\mathbb{Z}}= \colim(0\to 0 \leftarrow 0)
,\end{align*}
since there is no holonomy on an interval. See
\href{https://arxiv.org/pdf/math/0211216.pdf}{Hopkins-Singer paper}:
pull back entire cochain complex, \emph{differential function spaces}.
Integrating this over manifold (after cutting into pieces) is packaged
as a partition function on an invertible field theory.

\end{remark}

\begin{remark}

Idea: use local Chern-Simons as a functional to integrate, e.g.~along
elements of \({\Omega}M\) for \(M\) a Riemannian manifold, or
e.g.~integrating scalar curvature. Can also use alternative cohomology
theories, e.g.~\({\mathsf{K}}{\hbox{-}}\)theory, do spin versions etc.
Their result: finding obstructions to conformally immersing Riemannian
manifolds into \({\mathbb{R}}^n\).

\end{remark}

\hypertarget{stephon-alexander-chern-simons-and-matter-antimatter-symmetry}{%
\section{Stephon Alexander, Chern-Simons and Matter-Antimatter
symmetry}\label{stephon-alexander-chern-simons-and-matter-antimatter-symmetry}}

\begin{remark}

Weak interaction: different to other interactions in that it violates
parity: experimentally, the mirror version has not been seen. Gravity
assumed left-right symmetric. Use resonance and power spectrum analysis
to determine that a majority of the energy expenditure of the universe
is in the form of dark energy.

\end{remark}

\begin{remark}

Pillars of cosmology:

\begin{itemize}
\tightlist
\item
  Large scale homogeneous, isotropic,
\item
  Solutions to Einstein field equations predict Hubble expansion law:
  further away implies moving away from each other faster.
\item
  General relativity
\end{itemize}

\end{remark}

\begin{remark}

Gravitational waves: tensorial fluctuations. Gravity waves: satisfies
perturbed wave equation. Turn into a quantum thing to get gravitons.
Ratio of matter to antimatter is a precise number, on the order of
\(10^{-10}\). In Physics: see Chern-Simons current. Modify Einstein's
equations with an scalar (axion?) term. This field is a candidate for
what causes inflation.

\end{remark}

\begin{remark}

Famous result:
\href{https://www.wikiwand.com/en/Chiral_anomaly}{ABJ/chiral anomaly},
related to an index theorem: a standard \({\operatorname{U}}_1\) current
is usually conserved, but can vanish? Chern-Simons form prevents
\(R\tilde R\) from vanishing, which if vanishing would cause equal left
and right handedness.

\end{remark}

\hypertarget{mina-aganagic-homological-knot-invariants-from-mirror-symmetry}{%
\section{Mina Aganagic, Homological Knot Invariants from Mirror
Symmetry}\label{mina-aganagic-homological-knot-invariants-from-mirror-symmetry}}

\begin{remark}

Motivation: knot categorification problem. Recall that the Jones
polynomial arises from a skein relation which depends on \(n\), taking
\(n=2\). Can take other values for \(n\). Witten: this polynomial comes
from Chern-Simons theory, expected value of a collection of Wilson loops
in a fundamental representation of a Lie algebra. Alexander polynomial:
take a Lie \emph{superalgebra} instead. Categorification starts with
quantum groups. Chern-Simons: assign a Hilbert space to
\(A\coloneqq\Sigma_{g, n}\). Finite dimensional, spanned by a basis of
conformal blocks (solutions to a linear PDE) Allow heavy particles to
move in surface, take a path integral in \(A\times I\) corresponding to
a braid? States in \({\mathcal{H}}\) are special solutions to the PDE.
Can be described by caps and cups? Links to conformal field theory:
braiding and fusion of particle trajectories/paths. Khovanov: assign a
cohomology theory, take graded Euler characteristic to recover Jones
polynomial. Euler characteristic: trace from supersymmetric QM:
\begin{align*}
\operatorname{Tr}(-1)^F e^{-\beta H}
.\end{align*}

\end{remark}

\begin{remark}

Action of supercharge on a chain complex is generated by instantons.
Khovanov grading: graded by fermion number and another grading. Ben
Webster: framework for quantum link invariants for Lie algebras, but
very inexplicit. Associate to conformal blocks a bigraded category,
braids go to functors, links go to vector spaces of morphisms. Hard to
prove they decategorify correctly: new framework makes this automatic
and uses mirror symmetry.

\end{remark}

\begin{remark}

Why Calabi-Yaus: theory of strings on a pair of dual tori. Complex is
B-type (algebraic geometry), symplectic is A-type. Counting holomorphic
maps from \({\mathbb{P}}^1\): hard, infinite series of enumerative
problems. Easier on mirror? Branes are central objects in mirror
symmetry, regard as objects in a category whose morphisms are open
branes. Insight: need mirrors to be fibered by dual tori.

\end{remark}

\begin{remark}

Category of branes: \({\mathbb{D}}{\mathsf{Coh}}X\), vs
\({\mathbb{D}}\mathsf{F}\) for \(\mathsf{F}\) the Fukaya category.
Quantum product: count rational curves. See \emph{quantum differential
equation}, central to mirror symmetry. Defined on a moduli space of
complex/symplectic structures. Solution space finite dimensional,
spanned by charges. Solutions are maps from infinite cigar to \(X\).

\end{remark}

\begin{remark}

New knot theory idea: take an infinite punctured cylinder. Langlands
dual group carries magnetic monopoles? Need to consider moduli space of
monopoles in \({\mathbb{R}}^3\), turns out to be Calabi-Yau and in fact
hyperKähler. Braids are paths in moduli of Kählers? Braid group is the
sigma model on the infinite punctured cigar.

\end{remark}

\begin{remark}

Fusion: singular monopoles coming together and bubbling. Fusion
diagonalizes the action of braiding in conformal field theory. Cups and
caps: branes supported on miniscule Grassmannians. Perverse filtrations
make certain problems much simpler here. See KLRW algebra, algebraic
invariants match up with geometric invariants coming from branes on
B-side. Homological mirror symmetry: equivalence of categories of branes
on different models. Smooth monopoles indexed by simple roots of a Lie
algebra. See Landau-Ginzburg model. Category of A-branes: morphisms come
from Floer theory, see More theory approach to QM? Instantons:
holomorphic maps from strips!! Can generalize Heegaard-Floer to Lie
superalgebras (need simply-laced Lie algebras?). Note that HF produces
DGAs. Categorifies the Alexander polynomial. Equivariant homological
mirror symmetry: relates moduli space \({\mathcal{X}}\) to a
half-dimensional ``core'' \(X\). Not an equivalence categories, comes
from an adjunction
\(\adjunction { h_*} {h^*}{\mathsf{C}} {\mathsf{C}'}\) instead. Can
compute
\(\mathop{\mathrm{Hom}}(A, B) = \mathop{\mathrm{Hom}}( h^* h_* A, B)\).
HF reduces curve counting to well-defined problems in 1-dimensional
complex analysis, e.g.~applications of Riemann mapping. Same story here,
yields hard but tractable problems.

\end{remark}

\begin{remark}

The theory on D-branes is related to 3-dimensional gauge theories of
quiver type. Deformation: reply affine Lie algebra with quantum affine
Lie algebras. See \emph{defects} of conformal field theories, Koszul
dual algebras. Yields some integrable lattice model, solves some analog
of Yang-Baxter equations? Braiding matrices are computed by partition
functions.

\end{remark}

\begin{remark}

How computable is this theory, compared to the diagram calculus of
Heegaard-Floer diagrams? Probably a few months away from being
algorithmic enough for a computer.

\end{remark}

\hypertarget{xie-chen-chern-simons-theory-and-fractons}{%
\section{Xie Chen, Chern-Simons Theory and
Fractons}\label{xie-chen-chern-simons-theory-and-fractons}}

\begin{remark}

How to use Chern-Simons in condensed matter physics. Toy model: take a
lattice and label edges with DOF given by Pauli matrices
\(X = { \begin{bmatrix}  {0} & {1} \\  {1} & {0} \end{bmatrix} }\) and
\(Z = { \begin{bmatrix}  {1} & {0} \\  {0} & {-1} \end{bmatrix} }\):

\includegraphics{figures/2021-11-16_11-53-32.png}

Write Hamiltonian as
\begin{align*}
H = -\sum A_i - \sum B_i
\end{align*}
where \(A_i\) are the crosses and \(B_i\) the boxes, gives an exactly
solvable model. Cross with time to get \(2+1\) where Chern-Simons terms
show up. Generalize to lattices in 3-space. See ground-state degeneracy,
minimal/stable if no perturbation can reduce it. Very large here, so
doesn't fit into TQFT framework.

\end{remark}

\begin{remark}

For anyons: quasiparticles that experience excitation, can move around.
For fracton models, points can get stuck. By Gauss' law, particles must
be created in pairs. Rank 1 tensor gauge theories: conserve charges. For
rank 2, additionally need to conserve dipoles, so particles must be
created in fours.

\end{remark}

\begin{remark}

Gauge theories: far simpler when gauge group is abelian,
e.g.~\({\operatorname{U}}_1\). An integer symmetric matrix shows up when
writing a Lagrangian here, whose properties inform integer/fractional
quantum Hall effects. Quasiparticles are restricted to move along
submanifolds, e.g.~an \(xy\)-plane when many 2d models are stacked in
the \(z\) direction. Growing cubic models: add a new decoupled layer,
then do a smooth deformation to get a new model with one more layer.
Ground state degeneracy scales like \(4^n\). Foliated systems have
statistics with finite range, vs some systems where statistics span
multiple layers with long tails.

\end{remark}

\begin{question}

Possibly easy to identify matrices that are
\({\operatorname{SL}}_n({\mathbb{Z}}){\hbox{-}}\)invariant for finite
\(n\), what about as \(n\to\infty\)? What if you allow adding on
\(k\times k\) (decoupled) blocks, and locally allow
\({\operatorname{SL}}_k({\mathbb{Z}})\) to act on blocks?

\end{question}

\begin{remark}

Quasilocal matrices: integer matrices with nonzero entries only on some
band about the diagonal. Open question: are there invariants that
determine when a system is or is not foliated? What are the equivalence
classes of foliated models?

\end{remark}

\begin{remark}

Why ``fracton''? The original models around 2010 had some fractal
structure, could also be related to the fractions showing up. Modern
interpretation: point particle that doesn't freely move through the
entire space.

\end{remark}

\hypertarget{simon-donaldson-the-chern-simons-functional-and-floer-homology}{%
\section{Simon Donaldson, The Chern-Simons Functional and Floer
Homology}\label{simon-donaldson-the-chern-simons-functional-and-floer-homology}}

\begin{remark}

Recent work: when a variety admits an Einstein metric, and \(G_2\)
manifolds. For an account of the Chern-Simons paper, see Chern's
\emph{Complex manifolds without potential theory}, appendix on
characteristic classes. 2+1: action functionals
\begin{align*}
\int e^{\mathrm{CS}} \mathrm{d} A
.\end{align*}
3+1: Floer homology \(\operatorname{HF}(M^3)\).

\end{remark}

\begin{remark}

Consider \(P\to Y^3 \in \mathop{\mathrm{Prin}}{\mathsf{Bun}_G}\), define
\({\mathcal{A}}\) the affine space of functions, then
\(A_1-A_2\in \Omega^1(\operatorname{ad}P)\) Curvature
\(F(A) \in \Omega^2(\operatorname{ad}P)\). Define a form \(\Theta\)
Chern-Simons functional:
\(\mathrm{CS} : {\mathcal{A}}\to {\mathbb{R}}\), descends to
\({\mathcal{A}}/{\mathcal{G}}\to S^1\) for \({\mathcal{G}}\) the gauge
group. Critical points are flat connections, Hessians at these points
are quadratic forms corresponding to a bilinear form from earlier.
Alternative definition: pick \(X^4\) such that \({{\partial}}X = Y\) and
set
\begin{align*}
 \mathrm{CS} (A) = \int_X \operatorname{Tr}(F(A)^2)
,\end{align*}
i.e.~integrate the Chern-Weil form.

\end{remark}

\begin{remark}

Examples of functionals: for \(\gamma\in {\Omega}M\),
\begin{align*}
E(\gamma) \coloneqq\int {\left\lvert {\gamma'} \right\rvert}^2
.\end{align*}
Sub-level sets will have compactness properties, Hessians are finite
index at critical points. The flow \(-\operatorname{grad}E\) is a
parabolic PDE, so a nonlinear heat equation. But the Chern-Simons
functional doesn't have these properties. Very different!

\end{remark}

\begin{remark}

For symplectic manifolds:
\begin{align*}
A(\gamma) = \int_{{\mathbb{D}}} \omega && \text{where } {{\partial}}{\mathbb{D}}= \gamma
.\end{align*}

For \(M={\mathbb{C}}^n\), write as a Fourier series
\(A( \gamma) = \sum c_k {\left\lvert {\gamma'} \right\rvert}^2\)? Fixed
points of exact Hamiltonian diffeomorphisms \(\phi\) correspond to
critical points of a deformed functional \(A_\phi\) on \(\Omega M\).
Arnold conjecture: \({\sharp} \mathrm{Fix} \phi \geq \sum \beta_i\) is
bounded below by Betti numbers.

\end{remark}

\begin{remark}

For a torus, \({\Omega}T^{2n} \cong T^{2n} \times H^- \times H^+\) where
the \(H^\pm\) are vector spaces. Arnold conjecture proved for torii by
Conley and Zehnder, 1983. Floer's insight: one can still ``do Morse
theory'' with \(A_\phi\) and \(\mathrm{CS}\): while gradient flow isn't
defined, flow lines between critical points do make sense.

\end{remark}

\begin{remark}

Introducing a Riemannian metric yields \(\star: \Omega^2\to \Omega^1\).
Get connections \(A_t\) over \(Y^3\) as solutions to
\begin{align*}
{\frac{\partial A_t}{\partial t}\,} = \star F(A_t)
.\end{align*}
Solutions are Yang-Mills instantons on \(Y\times {\mathbb{R}}\)
asymptotic to flat connections at \(\pm \infty\). These solve
anti-self-dual equations
\begin{align*}
F(A) = -\star F(A)
,\end{align*}
where here the star is a 4-dimensional version. Connected to
electromagnetism? In symplectic case, gradient flow lines in
\({\Omega}M\) are holomorphic curves in \(M\).

\end{remark}

\begin{remark}

Usual story: chain complex with generators for each critical point,
graded by index, \({\mathcal{M}}(p, q)\) the space of gradient flow
lines \(p\to q\), compute \(\dim {\mathcal{M}}(p, q) = i(p) - i(q) -1\),
and define a differential
\({{\partial}}p = \sum {\sharp}{\mathcal{M}}(p, q)\). Problem in
infinite dimensions: index isn't well-defined, but the difference is
e.g.~when \(G = {\operatorname{SU}}_2\). Linearize the instanton
equation. Euler characteristic with respect to \(\operatorname{HF}\) is
twice the Casson invariant.

\end{remark}

\begin{remark}

If \(Y \in \mathbb{Z}\operatorname{HS}^3\), then the trivial flat
connection is isolated. Otherwise, Floer's construction works when you
can avoid ``reducible'' flat connections, e.g.~a nontrivial
\({\operatorname{SO}}_3\) bundle over \(Y^3\). For general \(P\to Y\),
need an equivariant version of Floer theory -- at present, this doesn't
seem to exist.

\end{remark}

\begin{remark}

Toward a 3+1 TFT: solutions to Yang-Mills on a 4-manifold give
invariants by counting instantons in a 0-dimensional moduli space. For
\(X\) a 4-manifold with \({{\partial}}X = Y\), these invariants \(I(X)\)
take values in \(\operatorname{HF}(Y)\). Sum over all flat connections
\(C_i\), and count number of connections asymptotic to \(C_i\).

\end{remark}

\begin{remark}

Gluing formula: a type of surgery formula when
\(X = X_1 { \displaystyle\coprod_{Y} } X_2\) to compute
\(I(X) = I(X_1) I(X_2)\). This uses the pairing
\(\operatorname{HF}_*(Y) \otimes\operatorname{HF}^*(Y)\to {\mathbb{Z}}\).
Proof: stretching the neck. Analog in Morse theory moves intersections
closer to critical points.

\includegraphics{figures/2021-11-16_15-17-52.png}

\end{remark}

\begin{question}

Other Floer theories on \(Y^3\): solutions to SW, combinatorial Heegard
theory. An outstanding problem: how are all of these Floer theories
related?

\end{question}

\begin{remark}

Floer's deepest work: work leading to his exact surgery sequence.
\(K \hookrightarrow Y\in \mathbb{Q}\kern-0.5pt\operatorname{HS}^3\) a
knot, take a tubular neighborhood diffeomorphic to a torus with a
prefered meridian. Do \(+1\) surgery: cut out, glue meridian back along
the diagonal. Can also do \(0\) surgery. Get a LES in
\(\operatorname{HF}_*\) of the surgered pieces, using well-known
cobordisms.

\end{remark}

\begin{remark}

Representation variety: moduli of flat connections! Extending all of
this to include surfaces: see bordered Floer theory,
Lipshitz-Osvath-Thurston for Seiberg-Witten and Heegard cases. Floer
homotopy: see Manolescu, Bauer-Furuta.

\end{remark}

\begin{remark}

How to complexify this theory? E.g. for
\(G = {\operatorname{SL}}_2({\mathbb{C}})\) instead of
\({\operatorname{SU}}_2\), or replace \(Y\) with a Calabi-Yau threefold?

\end{remark}

\begin{remark}

For \(Z\) a Calabi-Yau, have a nonvanishing holomorphic 3-form
\(\omega\), so define a \({\mathbb{C}}{\hbox{-}}\)valued function on
\({\mathcal{A}}\) the space of connections:
\begin{align*}
F(A) = \int_Z  \mathrm{CS} (A) \wedge \omega
.\end{align*}
Critical points are connections with \(F^{0, 2} = 0\),
i.e.~\({ \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu}_A^2 = 0\).
See ``holomorphic Casson invariants'' by R. Thomas, counting holomorphic
line bundles over \(Z\). More generally, counting coherent sheaves on
\(Z\) to generalize DT invariants.

\end{remark}

\begin{remark}

See Atiyah-Floer conjecture.

\end{remark}

\hypertarget{sylvester-gates-susy-topology-chern-simons-theory}{%
\section{Sylvester Gates, SUSY, Topology, Chern-Simons
Theory}\label{sylvester-gates-susy-topology-chern-simons-theory}}

\begin{remark}

Topic: ectoplasm conjecture. Use pure group theory to study
supersymmetry and supergravity. M-theory and supergravity: 11
dimensions. Superspace: interior of a sphere, ordinary space is an
equatorial plane. Consider the weight lattice for \({\mathfrak{su}}_3\).
Define an action functional using \(\star\mathrm{d} \mkern-1mu \star\).
See nonlinear sigma model for pions? See super curvature. Covariant
derivative allows coupling to matter fields. See potential of a
connection? Given degrees of freedom, how are they represented? Scalars?
\(n{\hbox{-}}\)forms? What are the representations of
\({\mathfrak{so}}_4\)? Which ones are spinor representations? Traceless
symmetric tensors correspond to gravitons? Link between Young tableaux
and Dynkin labels? There are multiplication rules for both of these. Can
replace data of super fields with a poset of Young tableaux, stratified
by level, and even just track them in a formal polynomial. Group theory
for physicists: representation theory of of compact Lie groups!

\end{remark}

\hypertarget{charles-kane-quantized-nonlinear-conductance-in-ballistic-metals}{%
\section{Charles Kane, Quantized Nonlinear Conductance in Ballistic
Metals}\label{charles-kane-quantized-nonlinear-conductance-in-ballistic-metals}}

\begin{remark}

Applications of Chern-Simons to condensed matter physics. Intro:
relation to quantized Hall effect, and a related quantized response in
1D: the Landauer formula. How generalize: the Euler characteristic of
the Fermi sea.

\end{remark}

\begin{remark}

Integer quantized Hall effect: can confine electrons to a 2D plane.
Standard experiment: measure conductance as a function of the magnetic
field. Remarkably, nearly a sum of step functions! Measures fundamental
constant \(h/e^2\) extremely accurately, NIST declared it an
\emph{exact} number which in turn defines the Ohm and kilogram. Low
energy produces a Chern-Simons TFT.

\end{remark}

\begin{remark}

Topological band theory: a mean field theory that reduces QM particles
to understanding single particles. Band structure: collection of energy
eigenvalues/eigenvectors parametrized by momenta (\(S^1\) or more
generally a torus due to periodicity). If occupied/empty states are
separated by a gap, the \(N\) occupied bands forum a
\({\operatorname{U}}_n\) bundle over \(T^d\) a torus. Problem: classify
vector bundles over \(T^d\). In dimension 2, the first Chern class
\(c_1\) yields a number:
\(n = {1\over 2\pi }\int_{T^2} \operatorname{Tr}(F), F = da + A \wedge A\)
for \(A = i{\left\langle {u_i},~{du_j} \right\rangle}\) the Berry
connection.

\end{remark}

\begin{remark}

In metals, electron states are occupied for \(E<E_F\) a constant, so
define Fermi sea as the region in momentum space bounded by the
associated Fermi surface. Why topology shows up: integrate the
connection associated to the above Chern class to get something gauge
invariant. This is something that can be measured in a lab! Chern
numbers measured how twisted the vector bundle is.

\end{remark}

\begin{remark}

Seemingly tough problem (to me): put a sphere inside of a regular
polyhedron. Puncture ``symmetrically'' corresponding to opposite faces,
add a tube passing through the face, and then glue the resulting faces
together. What surface do you get? Shows that Fermi surfaces can have
nontrivial genus.

\end{remark}

\begin{remark}

For 1D: see Landauer conductance. With assumptions (no backscattering of
electrons, no reflectance), can get exact model
\begin{align*}
I = {1\over 2\pi } \int \mathrm{d}k \, ev_k = {e\over h}\int \mathrm{d}E = \qty{e^2\over h}V
.\end{align*}
What is the homeomorphism type of the 1D Fermi sea? Use quantized
Landauer conductance experiments to probe \({\sharp}\pi_0\). Idea:
\(\chi_F\) carries more information than \(\chi_{{{\partial}}F}\) for
\(F\) the Fermi sea and \({{\partial}}F\) its boundary surface. For
\(F\), there is a natural Morse functional given by measuring electron
energy. Passing through critical points is like a phase transition?

\end{remark}

\begin{remark}

How to explain the quantization in the Landauer formula: consider
dimension \(D=1\). Apply a voltage impulse to move electrons:

\includegraphics{figures/2021-11-17_11-42-54.png}

See chiral anomaly. Use Morse theory to compute \(\chi\).

\end{remark}

\begin{remark}

See conformal field theories, e.g.~\(1+1\) boson for Fermi liquids?
Where else might \(\chi_F\) show up? Generalizing chiral anomalies? A
universal way to characterize quantum entanglement? Generalizations to
higher dimensions or non-Fermi liquids?

\end{remark}

\hypertarget{john-lott-chern-simons-differential-mathsfkhbox-theory-operator-theory}{%
\section{\texorpdfstring{John Lott, Chern-Simons, Differential
\({\mathsf{K}}{\hbox{-}}\)theory, Operator
Theory}{John Lott, Chern-Simons, Differential \{\textbackslash mathsf\{K\}\}\{\textbackslash hbox\{-\}\}theory, Operator Theory}}\label{john-lott-chern-simons-differential-mathsfkhbox-theory-operator-theory}}

\begin{remark}

Chern character form: \(M\) smooth, \(E\) a Hermitian vector bundle,
\(\nabla\) a compatible connection, get a closed form
\begin{align*}
\operatorname{ch}\nabla = \operatorname{Tr}e^{-\nabla^2} \in \Omega^{\text{even}}(M)
.\end{align*}
For two connections \(\nabla_i\), take a homotopy
\(\nabla_s = s\nabla_1 + (1-s)\nabla_0\), the Chern-Simons form is
\begin{align*}
 \mathrm{CS} (\nabla_0, \nabla_1)\coloneqq\int_0^1 
\operatorname{Tr}\qty{ {\frac{\partial \nabla_s}{\partial s}\,} e^{-\nabla_s^2} } \,ds
.\end{align*}
Alternative construction of Chern character and Chern-Simons form due to
Quillen: instead of interpolating between \(\nabla_0, \nabla_1\),
interpolate between
\({ \begin{bmatrix}  {\nabla_0} & {0} \\  {0} & {\nabla_1} \end{bmatrix} }\)
and \(\infty\).

\end{remark}

\begin{remark}

Recall
\begin{align*}
{\mathsf{K}}^0 = { {\mathbb{Z}} { \left[ { \left\{{0\to A\to B \to C\to 0}\right\}} \right] }  \over  \left\langle{[B] = [A] + [C]}\right\rangle}
.\end{align*}
Differential \(\check{\mathsf{K}}{\hbox{-}}\)theory: for
\({\mathsf{K}}^0\), quadruples \((E, h^E, \nabla^E, \omega)\) with
\(E\in { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }_{/ {M}}\), \(h^E\) a
Hermitian metric, \(\nabla\) a Hermitian connection, \(\omega\) an
auxiliary form. Quotient by relations
\({\mathcal{E}}_2 = {\mathcal{E}}_1 + {\mathcal{E}}_3\) when the \(E_i\)
form a SES of vector bundles,
\(\omega_2 = \omega_1 + \omega_3 + {\varepsilon}\) where
\({\varepsilon}\) is a correction term. Forget everything but the vector
bundle to get a map \(\check {\mathsf{K}}^0 \to {\mathsf{K}}^0\) and a
Chern character
\(\check {\mathsf{K}}^0(M) \to \Omega_K^{\text{even}}(M)\). Kernel is
\({\mathsf{K}}^{-1}\) and this forms a SES, so this mixes
\({\mathsf{K}}\) and \(\Omega\).

\end{remark}

\begin{remark}

Integration over the fiber in \({\mathsf{K}}{\hbox{-}}\)theory: index
maps \({\mathsf{K}}(M) \to {\mathsf{K}}(B)\), see Atiyah-Singer index
formula. Manifests as an equality of numerical indices. Joint work with
Dan Freed establishes a similar result for \(\check K\) using local
index theory. Can compute certain \(\eta\) invariants without using
analysis.

\end{remark}

\begin{remark}

Idea: \({\mathsf{K}}{\hbox{-}}\)theory of finite dimensional vector
bundles with connections. For today, how to generalize to infinite
dimensions with super connections. Twisted
\({\mathsf{K}}{\hbox{-}}\)theory: use elements in \(H^3\). Problem:
can't define \(\operatorname{Tr}e^{-A^2}\) for \(A\) an operator,
expanding and taking the 0th term already yields \(\infty-\infty\)! Fix:
super connections, a sum \(A = \sum A_{[i]}\), and
\(\operatorname{Tr}_s\) a super trace. Idea: \(A_{[i]}\) are in
\(\Omega^i(M; ?)\) with restrictions. See \(C_2{\hbox{-}}\)graded
Hilbert bundles over a manifold \(M\).

\end{remark}

\begin{remark}

What should the structure group \(G\) be..? For fibers \(H\), should
have a subgroup \(G\leq {\operatorname{U}}(H)\) (the unitary
transformations). Needs to be general enough to include
\(\operatorname{Diff}(Z)\) for \(Z\) the fibers. Construct using a type
of Dirac operator, pseudo differential operators. Define structure group
as even unitary transformations intersected with a certain space of 0th
order pseudo differential operators \(\mathrm{op}^0\). See
Bismut-Cheeger \(\eta\) form, \(\eta(A, \infty)\) which ``interpolates
to \(\infty\)'' as before.

\end{remark}

\begin{remark}

Generators for \(\check {\mathsf{K}}^0\): triples
\(({\mathcal{H}}, A, \omega)\) with \({\mathcal{H}}\to M\) a
\(C_2{\hbox{-}}\)graded Hilbert bundle, \(A\) a super connection, plus
conditions. Turns out to be isomorphic to standard
\(\check {\mathsf{K}}\) from before. Unclear if the Hopkins-Singer model
is isomorphic. Standing assumptions: compact fibers, fiberwise tangent
bundle is \(\mathrm{Spin}^{{ \scriptscriptstyle \mathbb C} }\). Need a
horizontal distribution. Define an index as the image of a map
\(\check {\mathsf{K}}^0(M)\to \check {\mathsf{K}}^0(B)\) constructed by
pushforwards.

\end{remark}

\begin{remark}

Twisted \({\mathsf{K}}{\hbox{-}}\)theory: \(H^3(M; {\mathbb{Z}})\)
classifies \({\operatorname{U}}_1\) gerbes on \(M\). Gerbes:
\({\mathcal{U}}\rightrightarrows M\), line bundles
\({\mathcal{L}}_{ab}\) on double intersections, on triples
\({\mathcal{L}}_{ab}\otimes{\mathcal{L}}_{bc} { { \, \xrightarrow{\sim}\, }}{\mathcal{L}}_{ac}\),
and on quadruples a cocycle condition. Can define
\({\operatorname{U}}_1\) connections on a gerbe. Twisted Hilbert
bundles: maps
\(\phi_{ab}: {\mathcal{H}}_a \otimes{\mathcal{L}}_{ab}\to {\mathcal{H}}_b\).
Define super connection on the open cover, plus compatibility on double
overlaps to define globally. Closed:
\((d + H\wedge)\operatorname{ch}(A) = 0\). Take same generators,
quotient by new relations. Theorem: this new twisted
\(\check {\mathsf{K}}\) only depends on the class of the gerbe in
\(H^3\), and is independent of choices for connective structures and
curving.

\end{remark}

\begin{remark}

Open question: showing this is isomorphic to other models.

\end{remark}

\hypertarget{seiberg-lattice-vs-continuum-qft}{%
\section{Seiberg, Lattice vs Continuum
QFT}\label{seiberg-lattice-vs-continuum-qft}}

\begin{remark}

QFT: successful but still not mathematically rigorous! Regularize by
discretizing space to a lattice, then the functional integral is
well-defined. Then take a continuum limit taking spacing to zero but
fixing lengths, compute correlation functions. For condensed matter,
find low-energy/long-distance limit. Expect it to be described by an
effective continuum field theory.

\end{remark}

\begin{remark}

Challenges with just regularizing to a lattice and taking a limit:

\begin{itemize}
\tightlist
\item
  Does the limit exist?
\item
  Is the limit independent of details at finite levels?
\item
  Lattice theory doesn't necessarily capture topological features
  (\(\pi_1\), characteristic classes, Chern-Simons terms, etc). These
  are tied to global symmetries, anomalies.
\item
  Some QFTs (e.g.~with self-dual forms or fermions) don't allow putting
  DOF on the lattice, can't write an action.
\item
  Some QFTs may not have a continuum Lagrangian (e.g.~\(2,0\) theory),
  so can't do a lattice Lagrangian.
\end{itemize}

\end{remark}

\begin{remark}

Opposite problem: given an actual physical lattice, find the low energy
continuum model. Often solvable, so can couple with whatever, but
continuum limits have divergence issues. E.g. \(XY{\hbox{-}}\)plaquette,
fracton models. Some issues:

\begin{itemize}
\tightlist
\item
  Separate symmetric group elements for different subspaces
\item
  Observables vary at lattice scale \(a\), and can become discontinuous
  in the limit
\item
  Infinite ground state degeneracy in the limit
\end{itemize}

\end{remark}

\begin{remark}

Very well understood model: \(1+1\) boson. Take a 2d lattice with
periodic boundary conditions, action
\begin{align*}
S = -\beta \sum \cos(\Delta_\mu \phi)
,\end{align*}
where \(e^{i\phi}\) are phases on the lattice points. Sum over ``links''
(edge..?), \(\Delta_\mu\) is like a discrete derivative. Take limit to
get
\begin{align*}
S = {\beta\over 2}\int \qty{ {\frac{\partial \phi}{\partial \mu}\,} }^2 \,d\tau\,dx
.\end{align*}
A symmetry emerges in the continuum: the winding number of \(\phi\), and
a certain 't Hooft anomaly. How much is present in the original lattice?

\end{remark}

\begin{remark}

Try to make those symmetries appear on the lattice. Idea: replace
\(\cos\) by a linear function, add a correction term that sums over
plaquettes that is roughly curvature, scale by a constant that forces
curvature to be zero. See \emph{Villain form}. This kills the vorticity,
which helps make the winding number show up on the lattice. Use Poisson
resummation to show self-duality.

Suspected causes of anomalies: infinities, fermions, issues with
invariance of measure. But none are present here!

\end{remark}

\begin{remark}

A slightly more complex model: \(XY{\hbox{-}}\)plaquette in \(2+1\)
dimensions. Put phases \(e^{i\phi}\) on nodes with action
\begin{align*}
S = -\beta_0 \sum_{\text{edges}} \cos(\delta_{\tau} \phi) - \beta \sum_{\text{plaqs}} \cos(\Delta_x \Delta_Y \phi)
.\end{align*}
Has a global \({\operatorname{U}}_1\) symmetry, take continuum limit to
get
\begin{align*}
S = \int \,d\tau\,dx\,dy\, {\mu_0\over 2 }\qty{ {\frac{\partial \phi}{\partial \tau}\,} }^2 + {1\over 2\mu}\qty{ {\partial^2 \over \partial_x \partial_y} \phi }^2
.\end{align*}

\end{remark}

\begin{remark}

Other questions to ask to compare lattices to continuum limit:

\begin{itemize}
\tightlist
\item
  What are the (operator) spectra of the Hamiltonians?
\item
  What are the correlation functions?
\end{itemize}

Importantly: limits don't commute!

\end{remark}

\begin{remark}

Interesting generalizations:

\begin{itemize}
\tightlist
\item
  Replace \({\operatorname{U}}_1\) with \(C_n\)
\item
  Go to \(3+1\) dimensions
\item
  Look at subsystem symmetries
\end{itemize}

Modify the Villain term, compute the spectra and correlation functions
to study. QCD: infinite number of states (vs finite number of states in
these kinds of models?) There's a difference between global and gauge
symmetries, exotic models land somewhere in between. See UV/IR mixing
(short/long distances).

\end{remark}

\hypertarget{xiao-gang-wen-chern-simons-nonabelian-topological-order}{%
\section{Xiao-Gang Wen, Chern-Simons, Nonabelian Topological
Order}\label{xiao-gang-wen-chern-simons-nonabelian-topological-order}}

\begin{remark}

Goal: classify all possible phases of matter. Can model \emph{spin
liquids}, someone wrote a wave function for it. Quantum Hall states: 2d
electron gas. Fractional quantum Hall states have different phases even
when there is no symmetry (and thus no symmetry to break). See energy
gaps, literally show up as gaps in the spectrum of Hamiltonians.
Topological order: having a specific type of ground state degeneracy.
``Topological'' usually means robust against small perturbations,
particularly ones that break symmetries.

\end{remark}

\begin{remark}

Conjecture: chiral spin states and quantum Hall states are described at
low energies by a path integral over an action which includes a
Chern-Simons term corresponding to a \({\operatorname{U}}_1\) gauge.
Theme: things locally look the same, but can differ globally.

\end{remark}

\begin{remark}

Cut electron into partons, write wave function for each and take product
(gluing). Degeneracies: locally the same wave functions but globally
different wave functions. See nonabelian statistics: modular tensor
categories. Monopoles: defects or punctures in spacetime.

\end{remark}

\begin{remark}

Looking for a mathematical language to describe long-range entanglement.
Current directions: category theory, Chern-Simons theory. Framing
anomaly: path integral depends on the framing of spacetime. Believe the
path integral in Chern-Simons leads to a gravitational Chern-Simons term
\(\omega_{ \mathrm{CS} }^{\text{grav}}\).

\end{remark}

\hypertarget{simons-origins-of-chern-simons-theory}{%
\section{Simons, Origins of Chern-Simons
Theory}\label{simons-origins-of-chern-simons-theory}}

\begin{remark}

Motivations: a combinatorial formula for the signature of a 4-manifold.
Triangulate and integrate the first Pontryagin class, yields a term that
doesn't vanish coming from the 3-manifold surrounding a vertex. Setup:
\(M^3\) closed oriented Riemannian, then
\(\mathop{\mathrm{Frame}}(M)\to M \in \mathop{\mathrm{Prin}}{\mathsf{Bun}}_{{\operatorname{SO}}_3({\mathbb{R}})}\).
Write \(\theta_{ij}\) for the connection form and \(\Omega_{ij}\) its
curvature form. Get a 3-form \(Q\) which integrates to zero along fibers
(since curvature terms are horizontal). 3-manifolds are parallelizable,
yielding sections of the frame bundle. Any two sections \(\chi, \chi'\)
differ by an integer, so get a well-defined
\(\Phi(M) \coloneqq\int_\chi Q \operatorname{mod}{\mathbb{Z}}\)

\end{remark}

\begin{remark}

Theorems: \(\Phi(M)\) is a conformal invariant, and \(\Phi(M) = 0\) is
necessary for \(M\) to admit a conformal immersion into
\({\mathbb{R}}^4\). Application: \(\Phi({\mathbb{RP}}^3) = 1/2\), so
although immersible and isometrically embeddable into
\({\mathbb{R}}^4\), not conformally.

Interpret \(\Phi\) as a map from the space of conformal structures to
\({\mathbb{R}}/{\mathbb{Z}}\), whose critical points are locally
conformally flat structures. Well-known fact: a locally conformally flat
simply-connected 3-manifold is diffeomorphic to \(S^3\). Could lead to a
shorter proof of 3d smooth Poincaré. Chern got this invariant to work in
all dimensions, led to Annals paper.

\end{remark}

\hypertarget{nico-yunes-astrophysical-observational-signatures-for-dynamical-chern-simons-gravity}{%
\section{Nico Yunes, Astrophysical Observational Signatures for
Dynamical Chern-Simons
Gravity}\label{nico-yunes-astrophysical-observational-signatures-for-dynamical-chern-simons-gravity}}

\begin{remark}

Question: what observable signatures are produced by gravitational
waves, black holes, neutron stars? Why are there more baryons than
anti-baryons in the universe? General relativity predicts singularities
-- problematic due to infinities. Unclear if Chern-Simons

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  Has nothing to do with these phenomena, or
\item
  Explains them entirely.
\end{enumerate}

\end{remark}

\begin{remark}

New data: gravitational wages at LIGO in 2015, wavelike perturbations of
the metric tensor that decay like \(1/r\). After traveling \(\sim 10^3\)
megaparsecs, yields a very weak observable. Around 50 similar events in
the years after, high level of confidence in measurements
(\(>5\sigma\)). Signatures help us figure out what to sift for in
experimental data.

\end{remark}

\begin{remark}

Pontryagin invariant: \(R^*R\)? Lagrangian density: perturbed Einstein
equations for GR where tensor and scalar field are coupled? Recovers
classical GR in a limit:
\begin{align*}
L\sim R - {1\over 2} (\nabla_a \nu) (\nabla^a \nu) + \alpha_{\mathrm{dCS}} \nu R^*R
,\end{align*}
for \(R\) the Ricci tensor. Now vary with respect to degrees of freedom.
Currents: something whose divergence is the original thing? Third
derivatives: not good in physics, causes ghosts, unbounded Hamiltonians.
Metric is GR metric plus order \(\alpha^2\) terms. Need to reduce order
to get rid of third derivatives, unphysical unstable modes, due to
insistence on treating this as an exact theory.

\end{remark}

\begin{remark}

Chern-Simons form here:
\begin{align*}
\operatorname{Tr}(dA \wedge A + c A\wedge A \wedge A), && c=2/3
.\end{align*}

\end{remark}

\begin{remark}

Spherically symmetric black holes: assuming static and vacuum, the
unique solution is the Schwarzschild metric. See box operator,
d'Alembertian?
\begin{align*}
\Box  \propto {\frac{\partial ^2}{\partial t^2}\,} - \Delta
&=
\operatorname{diag}(1,-1,-1,-1)
\nabla^2(t,x,y,z) \\
&=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 
\end{bmatrix}  
\Delta(t,x,y,z) \\
&= 
{\frac{\partial ^2}{\partial t^2}\,}
-{\frac{\partial ^2}{\partial x^2}\,}
-{\frac{\partial ^2}{\partial y^2}\,}
-{\frac{\partial ^2}{\partial z^2}\,}
.\end{align*}

Effective field theory treatment for axially symmetric black holes,
yields 2nd order PDE, no exact solution. Can take first order solution,
not bad to do by hand. Second and fifth order, much more complicated but
easily handled by a computer, no clear obstructions to higher order
expansions. Yields ``hairy'' black hole solutions: a \(1/r^2\) scalar
field, a perturbed horizon and ergosphere, no naked singularities, polar
``caps'': regions where geodesics need not focus. No killing tensor, so
no 4th constant of motion. Are there observational signatures for the
caps? Could there be chaos in geodesic motion for test particles in
orbit around dynamical Chern-Simons black holes? Chaos in this dynamical
system, say around supermassive black holes, should imprint on
gravitational waves. Recent results: quasinormal modes carry such a
dynamical signature.

\end{remark}

\begin{remark}

Summary:

\begin{itemize}
\tightlist
\item
  Spinning black holes are not Kerr, bc they excite a scalar field
\item
  Binary black hole space time has two scalar fields anchored with each
  black hole.
\item
  Dynamical CS induces a \(2PN\) correction to the orbital evolution.
\end{itemize}

\(\mathrm{dCS}\) is a \(v/4\) correction to GR. Plot frequencies of
orbit, play as sound -- collision causes a chirp as amplitude and
frequency spike, correction causes a different chirp. Now add the
sinusoids to obtain beats/nodes in the waves, LIGO looks for these!

\includegraphics{figures/2021-11-18_09-56-35.png}

\end{remark}

\begin{remark}

Problem with wave detection on Earth: seismic noise, creates a frequency
wall. Future projects: labs in space in 2035-2045! Things that haven't
been worked out in dCS yet:

\begin{itemize}
\tightlist
\item
  AdS black holes?
\item
  Exact rotating solutions
\item
  Gravitational collapse
\item
  Singularity theorems
\item
  Area theorems
\end{itemize}

\end{remark}

\hypertarget{kevin-costello-chern-simons-in-dimensions-456}{%
\section{Kevin Costello, Chern-Simons in Dimensions
4,5,6}\label{kevin-costello-chern-simons-in-dimensions-456}}

\begin{remark}

Why holomorphic CS is interesting: mirrors counts of curves through
mirror symmetry, and shows up in construction of 4d integrable field
theories.z

\end{remark}

\begin{remark}

In higher dimensions: non-topological field theories. Setup: for \(X\) a
Calabi-Yau 3-fold, \(A\in \Omega^{0, 1}(X, {\mathfrak{g}})\) a
\({ \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu}\)
connection, write a Lagrangian
\begin{align*}
\mathrm{hCS}(A) = \int_X \Omega_X \wedge  \mathrm{CS} (A)
.\end{align*}
Equations of motion imply flat connection, here \(F^{0, 2}(A) = 0\), so
a holomorphic instead of flat bundle. Translation
\({\frac{\partial }{\partial \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu_i}\,}\)
involves a BRST term. For 4-manifolds, take \(\Sigma_1 \times \Sigma_2\)
a bundle of Riemann surfaces, where \(\omega \in \Omega^1(\Sigma_2)\)
has no zeros, and
\(A\in \Omega^1(\Sigma_1 \times \Sigma_2) \operatorname{mod}\Omega^{1, 0}( \Sigma_2)\).
The Lagrangian is \(\int \omega \wedge \mathrm{CS} (A)\) for \(\omega\)
a 1-form, the equations of motion become \(\omega F(A) = 0\). Locally
looks like \(\int zF\wedge F\) if one writes \(\omega = dz\).

\end{remark}

\begin{remark}

4d CS unifies integrable PDEs and field theories. Basic example of
integral PDEs: \(G\in \operatorname{Lie}{\mathsf{Grp}}\) compact,
\(\sigma: {\mathbb{R}}^2\to G\), asking \(\sigma\) to be harmonic is an
integrable PDE. Importantly \emph{lax}, so can build a connection that
is flat iff the harmonic equation holds. An important generalization:
include a WZW term. Interpret maps \({\mathbb{R}}\times S^1\to G\) as
\({\mathbb{R}}\to {\Omega}G\), where the phase space is
\({\mathbf{T}} {}^{ \vee }{\Omega}G\). Conservation:
\({\left\{ {H},~{m(z)} \right\} } = 0\).

\end{remark}

\begin{remark}

Given a Riemannian manifold with a closed 3-form, when is the harmonic
map equation on \(M\) integrable? Traditional examples: Riemannian
symmetric spaces and their deformations. One example is \(S^2\) with a
specific metric, which also satisfies a Ricci flow condition.

\end{remark}

\begin{remark}

4d Chern-Simons: equations of motion become maps
\(\sigma: {\mathbb{R}}^2\to { {\mathsf{Bun}}\qty{G} }_{/ {\Sigma}}\), a
moduli of bundles over a Riemann surface trivialized at the poles of
\(\omega\) a 1-form. This has a canonical metric and 3-form. Theorem:
the harmonic map equation for these maps is always integrable. Proved
painfully without reference to 3D CS.

\end{remark}

\begin{remark}

Ricci flow is closely tied to integrability. For Riemannian manifolds
with closed 3-form, modify the flow
\(\delta g_{uv} = \operatorname{Ric}_{uv} - ?\). The ``ancient''
solutions correspond to something unique. Consider the moduli of Ricci
flows. There is a natural flow on the moduli space
\({\mathcal{M}}(\Sigma, \omega)\) given by computing periods of the form
between its zeros? Turns out to be proportional to a Ricci flow on this
space.

\end{remark}

\begin{remark}

For 4d, mostly classical at this point. Higher dimensions: get a similar
Lagrangian
\begin{align*}
\int \,dz_1\,dz_2 \operatorname{Tr}(\operatorname{Ad} A) + c \operatorname{Tr}(A* A * A) && A*B \coloneqq A\wedge B + c {\frac{\partial A}{\partial z_1}\,} \wedge {\frac{\partial B}{\partial z_2}\,} + \cdots
.\end{align*}
5d nonabelian CS is a supersymmetric sector of \(M{\hbox{-}}\)theory.
Holography: can match supersymmetric things for \(N\, M2\) branes with
computations in this 5d CS theory. These are QM particles moving on a
moduli space of rank \(k\) instantons on \({\mathbb{R}}^4\) with charge
\(N\).

\end{remark}

\begin{remark}

Most natural variant: holomorphic CS on a Calabi-Yau with
\(A\in \Omega^{0, 1}(X)\):
\begin{align*}
\int \Omega_X \wedge  \mathrm{CS} (A)
.\end{align*}
Problem: doesn't exist as a quantum theory due to gauge anomalies. Apply
Grothendieck-Hirzebruch-Riemann-Roch to show the canonical doesn't
vanish:
\begin{align*}
c_1 { {\mathsf{Bun}}\qty{G} }(X) = \int_X \operatorname{Td} ({\mathbf{T}}X) \operatorname{ch}(\operatorname{Ad} _{{\mathfrak{g}}}) 
.\end{align*}
Do some anomaly cancellation. Famously not renormalizable. Restricts
gauge groups to \({\operatorname{SO}}_8\) or \(G_2\times G_2\)? For
\(\sigma: {\mathbb{R}}^4\to {\operatorname{SO}}_8\), Lagrangian involves
Kähler potential.

\end{remark}

\hypertarget{minhyong-kim-arithmetic-field-theories-and-invariants}{%
\section{Minhyong Kim, Arithmetic Field Theories and
Invariants}\label{minhyong-kim-arithmetic-field-theories-and-invariants}}

\begin{remark}

Problem: classify principal bundles over a point
\(\operatorname{Spec}k \in \operatorname{Spec}(\mathsf{Field})\).
Classified by
\begin{align*}
\mathop{\mathrm{Prin}}{ {\mathsf{Bun}}\qty{G} }_{/ {\operatorname{Spec}F}}  { { \, \xrightarrow{\sim}\, }}H^1(\pi_1 \operatorname{Spec}F; G)
.\end{align*}
Here \(G\) could be a \(p{\hbox{-}}\)adic Lie group, e.g.~a Tate module
\(\cocolim_{n} A[n]\) for \(A\in {\mathsf{Ab}}{\mathsf{Var}}\). If \(G\)
acts trivially, this becomes a representation variety
\(\mathop{\mathrm{Hom}}(\pi_1 \operatorname{Spec}F, G)/\mathop{\mathrm{Inn}}(G)\),
quotienting by conjugation. A complete description is essentially the
Langlands reciprocity conjecture!

\end{remark}

\begin{remark}

Idea: replace \(F\) be \({\mathcal{O}}_F\) its ring of integers, so
\(X\coloneqq\operatorname{Spec}{\mathcal{O}}_F\) with the étale
topology. Behaves like a compact closed 3-manifold. For
\(v\in\operatorname{mSpec}{\mathcal{O}}_F\),
\({ k_{\widehat{v}} }= {\mathcal{O}}_F/v\) is a finite field, so
\(\operatorname{Spec}{ k_{\widehat{v}} }\hookrightarrow X\) is like an
embedding of a knot. Completion at \(v\),
\({\mathcal{O}}_{F, v} \coloneqq({\mathcal{O}}_F){ {}^{ \widehat{v} } }\)
is like a formal tubular neighborhood. Completing the original ring at
\(v\), so \(F{ {}^{ \widehat{v} } }\) is like a tubular neighborhood
with the knot deleted \(X_B\). For \(B\) a finite set of primes, set
\({\mathcal{O}}_{F, B}\) to be the set of \(B{\hbox{-}}\)integers,
i.e.~almost algebraic but allowing denominators from \(B\). Then
\(\operatorname{Spec}{\mathcal{O}}_{F, B}\) is like a 3-manifold with
boundary, so
\begin{align*}
{{\partial}}X = \displaystyle\coprod_{v\in B} \operatorname{Spec}F_v \to X_B \hookrightarrow X
.\end{align*}

\end{remark}

\begin{remark}

What are these \(\pi_1\)? Simple structure in nice cases,
\(\pi_1 \operatorname{Spec}{ k_{\widehat{v}} }= { \widehat{ \mathbb{Z} } }\),
the profinite completion of \({\mathbb{Z}}\). A finite field extension
\(K_{/ {F}}\) is \textbf{unramified} over
\({\mathfrak{p}}\in \operatorname{Spec}{\mathcal{O}}_F\) if the prime
decomposition \({\mathfrak{p}}{\mathcal{O}}_K = \prod {\mathfrak{q}}_i\)
has no primes with multiplicity. There is a maximal unramified
extension, just take the compositum of all unramified extensions. Can
restrict this to just be unramified over primes not in \(B\) Note that
\({\mathsf{Ab}}(\pi_1 X) = {\operatorname{Pic}}{\mathcal{O}}_F = { \operatorname{cl}} (F)\)
is the ideal class group of \(F\). Old results:
\(\pi_1 \operatorname{Spec}{\mathbb{Z}}, \pi_1 \operatorname{Spec}{\mathcal{O}}_F = 0\)
for \(F\) imaginary quadratic with \({ \operatorname{cl}} (F) = 1\).
Assume GRH to get \(\pi_1 \operatorname{Spec}{\mathcal{O}}_{K} = A_5\)
for \(K = {\mathbb{Q}} { \left[ {\sqrt{653}} \right] }\), or
\({\operatorname{PSL}}_2({\mathbb{F}}_8)\times C_{15}\) for
\(K = {\mathbb{Q}} { \left[ {\sqrt{-1567}} \right] }\).

\end{remark}

\begin{question}

Central problem: classify arithmetic 3-folds, or more generally
understand \(H^1(\pi_1 X_B; G)\), isomorphism classes of principal
\(G{\hbox{-}}\)bundles over \(X_B\). Write as \({\mathcal{M}}(X_B, R)\)
for \(R\) the group, breaks into
\(\prod_{b\in B} {\mathcal{M}}(X_b, R)\).

\end{question}

\begin{remark}

Assume \(F\) is complex, so
\(F\cong {\mathbb{Q}}[x]/\left\langle{f}\right\rangle\) where \(f\) has
no real roots. For gauge theory interpretations, need to write an action
\begin{align*}
S: {\mathcal{M}}(X_B, R) = H^1(\pi_1 X_B, R)\to K
\end{align*}
and path integral
\begin{align*}
\int_{\rho\in {\mathcal{M}}(X_B, R)} \exp( \mathrm{CS} (\rho)) \,d\rho
.\end{align*}
Actions take the form of \(L{\hbox{-}}\)functions:
\begin{align*}
L: {\mathcal{M}}(X_B, \operatorname{GL}(V))\to {\mathbb{C}}\text{ or } { {\mathbb{C}}_p }
.\end{align*}
To a representation \(\pi_1(X_B) \to \operatorname{GL}(V)\) assign
\begin{align*}
L(\rho) = \prod_{v\in \operatorname{Spec}{\mathcal{O}}_F} {1\over \operatorname{det}\qty{I-\mathop{\mathrm{Frob}}_v }V^{I_v} }
.\end{align*}
Importantly, this product may not always make sense! It's a big problem
to regularize this to get convergence. Hasse-Weil conjecture says one
can renormalise to a function of \(s\) in \({\mathbb{C}}\), get
convergence for \(\Re(s) \gg 0\), and analytically continue to \(s=0\)
or \(s=1\) to recover original product. Write
\(r\coloneqq{\operatorname{Ord}}_{s=0} L(\rho(s))\) for the order of
vanishing. For the trivial representation \(\phi\),
\(r = \operatorname{rank}{\mathcal{O}}_{F}^{\times}\). For
\(\rho = T_p E\) for an elliptic curve,
\(r = \operatorname{rank}E({\mathbb{Q}})\) (the Mordell-Weil group)
assuming BSD. Néron-Tate height pairing is a metric, its determinant
appears in \(L\) function for \(E\).

\end{remark}

\begin{remark}

Need arithmetic orientations, problematic since dualizing sheaves are
often \(\mu_n\). Fact:
\begin{align*}
H^3(X; \mu_n) = H^3(\operatorname{Spec}{\mathcal{O}}_F; \mu_n) = {{1\over n}{\mathbb{Z}}\over {\mathbb{Z}}}
.\end{align*}
More well-known that
\begin{align*}
H^3(X; \mu_n) = H^3(X; {\mathbb{G}}_m)[n] \cong ({\mathbb{Q}}/{\mathbb{Z}})[n]
.\end{align*}
Local CFT yields
\(H^2(F_v; {\mathbb{G}}_m) \cong {\mathbb{Q}}/{\mathbb{Z}}\) by
classification of gerbes. Follows from SES in global CFT:

\begin{center}
\begin{tikzcd}
    0 && {H^2(F; {\mathbb{G}}_m)} && {\bigoplus_v H^2(F_v; {\mathbb{G}}_m)} && {{\mathbb{Q}}/{\mathbb{Z}}} && 0
    \arrow[from=1-1, to=1-3]
    \arrow["{\text{loc}}", from=1-3, to=1-5]
    \arrow["{\text{sum}}", from=1-5, to=1-7]
    \arrow[from=1-7, to=1-9]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwwLCIwIl0sWzIsMCwiSF4yKEY7IFxcR0dfbSkiXSxbNCwwLCJcXGJpZ29wbHVzIEheMihGX3Y7IFxcR0dfbSkiXSxbNiwwLCJcXFFRL1xcWloiXSxbOCwwLCIwIl0sWzAsMV0sWzEsMiwiXFx0ZXh0e2xvY30iXSxbMiwzLCJcXHRleHR7c3VtfSJdLFszLDRdXQ==}{Link
to Diagram}
\end{quote}

Assume \(\mu_n \subseteq F\), get a map
\begin{align*}
\mathrm{inv}: H^2(\pi_1 X; C_n) \to H^3(X;\mu_n) \cong{ {1\over n} {\mathbb{Z}}\over {\mathbb{Z}}}
.\end{align*}
Use this to build a Chern-Simons function
\(\mathrm{CS} : {\mathcal{M}}(X; R)\to { {1\over n} {\mathbb{Z}}\over {\mathbb{Z}}}\),
essentially by pulling back cocycles along the representation. Use
Bockstein to define a ``path integral'' defined as a finite sum over
representations. Closed form solution involves Legendre symbol, very
nice! Also a determinant of a quadratic form.

\end{remark}

\begin{remark}

Bockstein is common in arithmetic geometry,
\(d:H^1(X; C_n)\to H^2(X; C_n)\), used to construct de Rham-Witt complex
for crystalline cohomology. On general, having a SES like
\(0\to V\to E\to V\to 0\) allows defining such arithmetic functionals.
Can define a bilinear pairing
\begin{align*}
\mathrm{BF}: H^1(X; C_n) \times H^1(X; \mu_n) &\to 1/n {\mathbb{Z}}/{\mathbb{Z}}\\
a\times b &\mapsto \mathrm{inv}(da \smile b)
.\end{align*}

Nice closed form solutions for the ``path integral'':
\begin{align*}
\sum_{(a, b)\in H^1(X; C_n) \times H^1(X; \mu_n) } \exp(2\pi i \mathrm{BF}(a, b)) = {\sharp}{\operatorname{Pic}}(X)[n] \cdot {\sharp}( {\mathcal{O}}_X^{\times}/({\mathcal{O}}_X^{\times})^n)
.\end{align*}
Compare to orders of \(L\) functions.

\end{remark}

\begin{remark}

This setup occurs for Néron models of elliptic curves: for \(n\gg 0\),
there is a SES
\begin{align*}
0\to {\mathcal{E}}[n] \to {\mathcal{E}}[n^2] \to {\mathcal{E}}[n] \to 0
.\end{align*}
Path integral evaluates to
\({\sharp}\Sha(A)[n] \cdot ({\sharp}E(F)/n)^2\).

\end{remark}

\begin{remark}

For boundaries:
\(X_B = \operatorname{Spec}{\mathcal{O}}_F { \left[ { \scriptstyle \frac{1}{B} } \right] }\)
for \(B\) a finite set of primes. Start with \({\mathcal{M}}(X_B, R)\),
get a local version \({\mathcal{M}}({{\partial}}X_B, R)\). Here
\(H^2(\pi_v; C_n) \cong 1/n{\mathbb{Z}}/{\mathbb{Z}}\) and vanishes for
\(i\geq 3\), get global CFT SES. Pulling back cocycles lands in \(Z^3\),
but \(H^3\) is trivial, so look at space trivialization (torsors for
\(H^2\)). Get a bunch of local torsors, then sum to get a single
\(1/n{\mathbb{Z}}/{\mathbb{Z}}\) torsor. Do some local/global and
push/pull gymnastics to get a \({\operatorname{U}}_1\) bundle over
\({\mathcal{M}}({{\partial}}X_B, R)\). Space of sections breaks up as a
tensor product of local spaces of sections, can cook up a way to land in
a Hilbert space, and this is the ``state'' you assign to the 3-manifold.

\end{remark}

\begin{remark}

Recent work: entanglement of primes.

\end{remark}

\hypertarget{khovanov-categorification}{%
\section{Khovanov, Categorification}\label{khovanov-categorification}}

\begin{remark}

CS functional critical for 3 and 4 dimensional TQFTs. In 3d, path
integral in Witten's construction of WRT invariants. In 4d, in instanton
Floer homology. Generally a tensor functor from 3-manifolds with 4d
cobordisms to some algebraic category like
\({\mathsf{Ab}}{\mathsf{Grp}}\). Motivational problem: construct a 4D
TQFT that categorifies the WRT invariant for 3-manifolds. Look at tensor
functors from the categories of links in \({\mathbb{R}}^3\) and link
cobordisms in \({\mathbb{R}}^3\times I\) to some algebraic category.

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{44pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/999_Chern/sections/figures}{2021-11-18_15-43.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\end{remark}

\begin{remark}

Components in links colored by irreps of a simple Lie algebra
\({\mathfrak{g}}\). Categorifying: recover polynomials as graded Euler
characteristics. In some cases, not clear how to extend link homology to
link cobordisms. Link homology is best understood with miniscule
representations, e.g.~\({\mathfrak{g}}= {\mathfrak{sl}}_n\), labeled by
\(\bigwedge\nolimits^k V\) for \(V\cong {\mathbb{C}}^N\) the fundamental
representation. For various \(N\), recovers Jones, Alexander, HOMFLYPT
polynomials. Skein relation: assign a complex to resolutions, find an
exact triangle, set the original complex to be the cone of the map
between the complexes of the two resolutions. Why this is a good idea:
reduces 3d diagram projections to actual 2d planar graphs.

\end{remark}

\begin{remark}

WRT is an invariant of a planar graph, homology supported in a single
degree. Maps between graph homology induced by foams. Look for functors
that assign objects to vector spaces and foams to morphisms. A foam: for
\({\operatorname{SL}}_3\), a 2d combinatorial CW complex with generic
singularities embedded in \({\mathbb{R}}^3\). Examples:

\begin{itemize}
\tightlist
\item
  \(\Sigma_{g, n}\)
\item
  \(S^2\) with a marked point and an equatorial disc attached.
\item
  \(S^1\times S^1\) with an equatorial disc.
\end{itemize}

\includegraphics{figures/2021-11-18_15-42-39.png}

Introduce Tait coloring: any two facets sharing an edge must have
distinct colors. E.g. the sphere above admits 6 such colorings over
\({\mathbb{F}}_3\), since the 3 facets (north/south hemispheres and
equatorial disc) share an edge (equator). Theorem: can produce a closed
orientable surface in \({\mathbb{R}}^3\) from this. The procedure is
roughly numbering the facets, then taking \(F_{ij}\) to be the surface
obtained by leaving \(\left\{{i,c}\right\}^c\) facets out. E.g. for
index set \(\left\{{1,2,3}\right\}\), you get
\(F_{12}, F_{13}, F_{23}\). Construct an evaluation map by constructing
local evaluations with respect to an admissible coloring, then sum over
such colorings to get a symmetric function in
\(k[x_1, \cdots, x_{n}]^{S_n}\). Initially defined to be rational,
i.e.~\(\left\langle{F}\right\rangle \in k(x_1,x_2,x_3)\), but it turns
out that denominators cancel and
\(\left\langle{F}\right\rangle \in k[x_1,x_2,x_3]^{S_3}\). Can pair
foams by flipping and gluing along boundary to get a closed foam, then
apply evaluation. Yields a bilinear pairing, consider the kernel to get
interesting skein-type relations.

\end{remark}

\begin{remark}

General construction yields a lax tensor functor: just a map, not an
isomorphism.

\end{remark}

\begin{remark}

Conjecture: the state space \(\left\langle{\Gamma}\right\rangle\) for
\(\Gamma\) a graph is a free \(R{\hbox{-}}\)module of rank \(r\) the
number of Tait colorings. Known for reducible graphs, i.e.~skein
relations can eventually break them into empty graphs. Theorem is true
after a certain modification to
\(\left\langle{\Gamma}\right\rangle_\phi\)? Motivations coming from
Kronheimer and Mrowka 2019, studying \({\operatorname{SO}}_3\) instanton
Floer homology over \({\mathbb{F}}_2\) for 3-orbifolds using a CS
functional for orbifolds, primarily
\({\mathbb{R}}^3/(C_2{ {}^{ \scriptscriptstyle\times^{2} } })\) to get a
trivalent vertex in a graph. Yields a homology theory for trivalent
graphs in \({\mathbb{R}}^3\). Might give a new way to think about the
four color theorem! Reductions: reduce links in \({\mathbb{R}}^3\) to
diagrams in \({\mathbb{R}}^2\), set up evaluation, then take cones?

\end{remark}

\begin{remark}

Current work: how to categorify \(\zeta^p = 1\)? Possibly use cyclotomic
rings in characteristic \(p\).

\end{remark}

\hypertarget{freedman-universe-from-a-single-particle-metric-crystallization}{%
\section{Freedman, Universe from a Single Particle, Metric
Crystallization}\label{freedman-universe-from-a-single-particle-metric-crystallization}}

\begin{remark}

Fun fact: Freedman did work on topological quantum computing.
Interesting fun example: cone on
\(S^3\times S^3 \hookrightarrow{\mathbb{R}}^7\). Idea for today: break
metric symmetry on a Lie group.

\end{remark}

\begin{remark}

Setup toy model for the beginning of the universe: finite dimensional
Hilbert space \(X\cong {\mathbb{C}}^n\) with symmetry group
\(G\coloneqq{\mathfrak{su}}_n\), and a metric \(g_{ij}\) on
\({\mathfrak{g}}\coloneqq{\mathfrak{su}}_n\). This gives \(G\) a
left-invariant metric. Now add a probability distribution on
\({\mathfrak{g}}\), which are basically Hermitian matrices, whose draws
are random Hamiltonians.

\includegraphics{figures/2021-11-18_16-40-21.png}

Several layers of randomness: choose metric in a Boltzmann manner, then
choose a Hamiltonian using a Gaussian based on the metric.

\end{remark}

\begin{remark}

Natural metric: Killing form, \(\operatorname{ad}{\hbox{-}}\)invariant
and a local (global) extremum in the space of metrics. Other metrics
yield unit ellipsoids instead of unit spheres. Write an energy
functional on this space, essentially scalar curvature (i.e.~Ricci
scalar curvature). Can extract an explicit formula from a paper of
Milnor from the 70s on left-invariant metrics on Lie groups. Need these
invariant metrics to sort out how to actually compile a quantum circuit.
Nielsen-Brown-Susskind define a word metric
\(g_{ij} = \delta_{ij} e^{? w(i)}\) where \(w({-})\) counts the number
of letters, e.g.~
\begin{align*}
w(1\otimes x\otimes 1 \otimes 1 \otimes y \otimes z \otimes 1\otimes 1 \otimes 1) = 3
.\end{align*}
This makes travelling along long words or words with capitals
exponentially more difficult, rethinks the combinatorial problem of
building a circuit as a differential geometric problem.

\end{remark}

\begin{remark}

Qubit structure:
\(J: (C^z){ {}^{ \scriptstyle\otimes_{}^{n} } }{ { \, \xrightarrow{\sim}\, }}C^{z^N}\).
Define \(g_{ij}\) to be KAQ (``knows about qubits'') if it has a basis
of principle axes \(\left\{{H_a}\right\}\) which admit a tensor product
decomposition. Forms about a \(\sqrt{d}\) dimension subvariety in a
\(d{\hbox{-}}\)dimensional space -- codimension 1 is already very thin,
so these are exceedingly rare. However, about 30\% of critical points
for the action functional satisfy this property. Try to do perturb a
Gaussian and expand/truncate. Slight issue: get an integral like
\(\int e^{g_{ij}\cdots + c_{ij} x^i x^j}\), but the \(x^i, x^j\) are
commuting variables and the structure constants \(c_{ij}\) anticommute,
so this term vanishes after integrating. Trying to integrate
fermionically cancels the first term. See Majorana operators.

\end{remark}

\begin{remark}

How they studied various metrics: gradient flow with respect to the
action functional. I wonder how one actually does this in practice..?

\end{remark}

\begin{remark}

Conclusions: nature abhors naked Hilbert spaces and attempts to equip
them with tensor or Majorana structures. In \(D=2^n\), we see Majorana
degree groupings and Brown-Susskind exponential penalty factors. For
random initial dimensions, conjecture all but log many dimensions
develop such a structure. A few exceptions: leaky universe scenario. Not
good! Setup is well-suited to studying pairs of an initial Hamiltonian
and initial state to model near-beginning universe scenarios. Consider
the configuration space of these pairs and see how entropy evolves.
Outstanding problem: where does ``space'' come from? Finding ``space''
in a model means finding the Leech lattice, wild! But checking for this
by brute force requires \(10^{12}\) qubits.

\end{remark}

\begin{remark}

Feynman diagrams: if no thickness, kind of old school. Modern treatments
involve ribbon diagrams.

\end{remark}


\printbibliography[title=Bibliography]


\end{document}
