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\usepackage{amsmath,amssymb, amscd,latexsym,amsthm}
\newtheorem*{define}{Definition}
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\def\Slash#1{#1\!\!\!\!/}
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\title{Quantum Field Theory and Representation Theory}
\author{Peter Woit}
\email{woit@math.columbia.edu}
\institution{Department of
Mathematics\\
Columbia University}
\date{June 3, 2004}
\begin{document}
\maketitle \overlays{5}{
\begin{slide}{Outline of the talk}

\begin{itemstep}
\item Quantum Mechanics and Representation Theory: Some History
\item Quantum Mechanics and Representation Theory: Some Examples
\item Quantization and Index Theory: The Dirac Operator
\item Quantum Field Theories in 1+1 dimensions
\item Twisted K-theory and the Freed-Hopkins-Teleman theorem
\end{itemstep}
\end{slide}
}
\overlays{2}{
\begin{slide}{Some History}
\ptsize{10}
Quantum Mechanics
\begin{itemize}
\item{Summer 1925: Observables are operators (Heisenberg)}
\item{Fall 1925: Poisson Bracket $\rightarrow$ Commutator (Dirac)}
\item{Christmas 1925: Representation of operators on wave-functions (Schr\"odinger)}
\end{itemize}
\fromSlide{2}{
Representation Theory
\begin{itemize}
\item{Winter-Spring 1925: Representation Theory of Compact Lie Groups (Weyl)}
\item{Spring 1926: Peter-Weyl Theorem (Peter, Weyl)}
\end{itemize}}
\end{slide}}
\overlays{2}{
\begin{slide}{Schr\"odinger and Weyl}
\ptsize{10}
\begin{figure}
\centering
\subfigure{\includegraphics[height=40mm]{schrodinger.eps}}
\fromSlide{2}{
\subfigure{\includegraphics[height=40mm]{annyschrodinger.eps}}}
\subfigure{\includegraphics[height=40mm]{weyl.eps}}
\end{figure}
\end{slide}}
\begin{slide}{Weyl's Book}
1928: Weyl's "Theory of Groups and Quantum Mechanics", with alternate chapters of group theory and quantum mechanics.
\end{slide}
\overlays{5}{
\begin{slide}{Interview with Dirac, Wisconsin, 1929}
\ptsize{10} \fromSlide{2}{ And now I want to ask you something
more: They tell me that you and Einstein are the only two real
sure-enough high-brows and the only ones who can really understand
each other. I won't ask you if this is straight stuff for I know
you are too modest to admit it. But I want to know this -- Do you
ever run across a fellow that even you can't understand?}
\fromSlide{3}{

%{\it {Yes}}}
\textcolor{yellow}{Yes.}}
\fromSlide{4}{

This will make a great reading for the boys down at the office. Do
you mind releasing to me who he is?} \fromSlide{5}{

%{\it{Weyl.}}}
\textcolor{yellow}{Weyl.}}
\end{slide}}
\begin{slide}{The Gruppenpest}
\ptsize{10}
Wolfgang Pauli: the "Gruppenpest", the plague of group theory.

For a long time physicists mostly only really needed representations of:
\begin{itemize}
\item
$\mathbf R^n,\ \ U(1)$: Translations, phase transformations. (Fourier analysis)
\item
$SO(3)$: Spatial rotations.
\item
$SU(2)$: Spin double cover of $SO(3)$, isospin.
\end{itemize}

Widespread skepticism about use of representation theory until Gell-Mann and Neeman use $SU(3)$ representations to classify strongly interacting particles in the early 60s.

\end{slide}
\begin{slide}{Representation Theory: Lie Groups}
\ptsize{10}
\begin{define}
A representation of a Lie group $G$ on a vector space $V$ is a homomorphism
$$ \rho: g\in G\rightarrow \rho(g)\in GL(V)$$
\end{define}
We're interested in representations on complex vector spaces,
perhaps infinite dimensional (Hilbert space). In addition we'll
specialize to unitary representations, where $\rho(g)\in U(V)$,
transformations preserving a positive definite Hermitian form on $V$.

For $V=\mathbf C^n$, $\rho(g)$ is just a unitary n by n matrix.
\end{slide}
\begin{slide}{Representation Theory: Lie Algebras}
\ptsize{10}
Taking differentials, from $\rho$ we get a representation of the Lie algebra $\mathfrak g$ of $G$:
$$\rho^\prime :\mathfrak g \rightarrow End(V)$$
For a unitary $\rho$, this will be a representation in terms of self-adjoint operators.
\end{slide}
\overlays{4}{
\begin{slide}{Quantum Mechanics}
\ptsize{10}
Basic elements of quantum mechanics:
\begin{itemstep}
\item
States:
vectors $|\Psi>$ in a Hilbert space $\mathcal H$.
\item
Observables: self-adjoint operators on $\mathcal H$.
\item
Hamiltonian: distinguished observable $H$ corresponding to energy.
\item
Schr\"odinger Equation: $H$ generates time evolution of states
$$i\frac{d}{dt}|\Psi>=H|\Psi>$$
\end{itemstep}
\end{slide}}
\begin{slide}{Symmetry in Quantum Mechanics}
\ptsize{10}
Schrodinger's equation: $H$ is the generator of a unitary representation of the group $R$ of time
translations.

Physical system has a Lie group $G$ of symmetries $\rightarrow$
the Hilbert space of states $\mathcal H$ carries a unitary representation $\rho$ of $G$.

This representation may only be projective (up to complex phase), since a transformation of
$\mathcal H$ by an overall phase is unobservable.

Elements of the Lie algebra $\mathfrak g$ give self-adjoint operators on $\mathcal H$, these are
observables in the quantum theory.

\end{slide}
\overlays{4}{
\begin{slide}{Standard Examples of Symmetries}
\ptsize{10}
\begin{itemstep}
\item
Time translations: Hamiltonian (Energy) $H$, $G=\mathbf R$.
\item
Space translations: Momentum $\vec P$, $G=\mathbf R^3$.
\item
Spatial Rotations: Angular momentum $\vec J$, $G=SO(3)$.

Projective representations of $SO(3)$ $\leftrightarrow$ representations of $SU(2)=Spin(3)$.
\item
Phase transformations: Charge $Q$, $G=U(1)$.
\end{itemstep}
\end{slide}}
\begin{slide}{Quantization}
\ptsize{12}
Expect to recover classical mechanical system from quantum mechanical one as $\hbar\rightarrow 0$

Surprisingly, can often \lq\lq quantize" a classical mechanical system in a unique way to get a
quantum one.
\end{slide}
\overlays{4}{
\begin{slide}{Classical Mechanics}
\ptsize{10}
Basic elements of (Hamiltonian) classical mechanics:
\begin{itemstep}
\item
States: points in a symplectic manifold (phase space) $M$, (e.g. $\mathbf R^{2n}$).
\item
Observables: functions on $M$
\item
Hamiltonian: distinguished observable $H$ corresponding to the energy.
\item
Hamilton's equations: time evolution is generated by a vector field $X_H$ on $M$ determined by
$$ i_{X_H}\omega=-dH$$
where $\omega$ is the symplectic form on $M$.
\end{itemstep}
\end{slide}}
\begin{slide}{Quantization + Group Representations}
\ptsize{12}
Would like quantization to be a functor

\center{(Symplectic manifolds $M$, symplectomorphisms)}

\center{$\downarrow$}

(Vector spaces, unitary transformations)

This only works for some subgroups of all symplectomorphisms.  Also, get projective unitary
transformations in general.
\end{slide}
\overlays{4}{
\begin{slide}{Mathematics $\rightarrow$ Physics}
\ptsize{10}
What can physicists learn from representation theory?
\begin{itemstep}
\item
Classification and properties of irreducibles.
\item
How irreducibles transform under subgroups.
\item
How tensor products behave.
\end{itemstep}
\fromSlide{4}{
Example: In Grand Unified Theories, particles form representations of groups like $SU(5)$,
$SO(10)$, $E_6$, $E_8$.}
\end{slide}}
\overlays{2}{
\begin{slide}{Physics $\rightarrow$ Mathematics}
\ptsize{12}
What can mathematicians learn from quantum mechanics?
\begin{itemstep}
\item
Constructions of representations starting from symplectic geometry (geometric quantization).
\item
Interesting representations of infinite dimensional groups (quantum field theory).
\end{itemstep}
\end{slide}}
\begin{slide}{Canonical Example: $\mathbf {R^{2n}}$}
\ptsize{10}
Standard flat phase space, coordinates $(p_i,q_i), \ i=1\ldots n$:
 $$M=\mathbf R^{2n},  \omega =\sum_{i=1}^{n}dp_i\wedge dq_i$$
Quantization:
$$[\hat p_i, \hat p_j]=[\hat q_i, \hat q_j]=0,\ \  [\hat q_i,\hat p_j]=i\hbar\delta_{ij}$$
(This makes $\mathbf R^{2n+1}$, a Lie algebra, the Heisenberg algebra)

Schr\"odinger representation on ${\mathcal H}=L^2(\mathbf R^n)$:
$$\hat q_i =\text{mult. by}\ q_i,\ \ \hat p_i= -i\hbar\frac{\partial}{\partial q_i}$$
\end{slide}
\begin{slide}{Metaplectic representation}
\ptsize{10}
Pick a complex structure on $\mathbf R^{2n}$, e.g. identify $\mathbf C^n=\mathbf R^{2n}$ by
$$z_j=q_j+ip_j$$
Then can choose ${\mathcal H}=\{\text{polynomials in}\ z_j\}$.

The group $Sp(2n,\mathbf R)$ acts on $\mathbf R^{2n}$ preserving $\omega$, $\mathcal H$ is a
projective representation, or a true representation of $Mp(2n,\mathbf R)$ a double cover.

Segal-Shale-Weil = Metaplectic = Fock or Oscillator Representation

Exponentiating the Heisenberg Lie algebra get $H^{2n+1}$, Heisenberg group (physicists call
this the Weyl group), $\mathcal H$ is a representation of the semi-direct product of $H^{2n+1}$ and
$Mp(2n,\mathbf R)$.

\end{slide}
\begin{slide}{Quantum Field Theory}
\ptsize{10}
A quantum field theory is a quantum mechanical system whose configuration space ($\mathbf R^n$,
space of $q_i$ in previous example) is infinite dimensional, e.g. some sort of function space
associated to the physical system at a fixed time.

\begin{itemize}
\item
Scalar fields: $Maps(\mathbf R^3 \rightarrow \mathbf R)$
\item
Charged fields: sections of some vector bundle
\item
Electromagnetic fields: connections on a $U(1)$ bundle
\end{itemize}

These are linear spaces, can try to proceed as in finite-dim case, taking $n\rightarrow \infty$.
\end{slide}

\begin{slide}{A Different Example: $S^2$}
\ptsize{10}
Want to consider a different class of example, much closer to what Weyl was studying in 1925.

Consider an infinitely massive particle. It can be a non-trivial projective representation of the
spatial rotation group $SO(3)$, equivalently a true representation of the spin double-cover
$Spin(3)=SU(2)$.

$\mathcal H =\mathbf C^{n+1}$, particle has spin $\frac{n}{2}$.

Corresponding classical mechanical system:  $$M=S^2=SU(2)/U(1),\ \  \omega= n\ \times \ \text{Area
2-form}$$

This is a symplectic manifold with $SU(2)$ action (left multiplication).
\end{slide}
\begin{slide}{Geometric Quantization of $S^2$}
\ptsize{10}
What is geometric construction of $\mathcal H$ analogous to metaplectic 
representation in linear case?

Construct a line bundle $L$ over $M=SU(2)/U(1)$ using the standard action of $U(1)$ on $\mathbf C$.
$$L=SU(2)\times_{U(1)} \mathbf C$$
$$ \downarrow$$
$$M=SU(2)/U(1)$$

$M$ is a K\"ahler manifold, $L$ is a holomorphic line bundle, and $\mathcal H = \Gamma_{hol}(L^n)$,
the holomorphic sections of the n'th power of $L$.
\end{slide}
\begin{slide}{Borel-Weil Theorem (1954) I}
\ptsize{10}
This construction generalizes to a geometric construction of all the representations studied by
Weyl in 1925.

Let $G$ be a compact, connected Lie group, $T$ a maximal torus (largest subgroup of form
$U(1)\times\cdots\times U(1)$).  Representations of $T$ are \lq\lq weights", letting $T$ act on
$\mathbf C$ with weight $\lambda$, can construct a line bundle
\ptsize{9}
$$L_\lambda=G\times_T\mathbf C $$
$$\downarrow $$
$$G/T$$
\ptsize{10}
$G/T$ is a K\"ahler manifold, $L_\lambda$ is a holomorphic line bundle, and $G$ acts on $\mathcal H
=\Gamma_{hol}(L_\lambda)$.
\end{slide}
\begin{slide}{Borel-Weil Theorem II}
\ptsize{10}
\begin{theorem}[Borel-Weil]
Taking $\lambda$ in the dominant Weyl chamber, one gets all elements of $\hat G$ (the set of
irreducible representations of G) by
this construction.
\end{theorem}

In sheaf-theory language $$\mathcal H = H^0(G/T, \mathcal O(L_\lambda))$$

Note: the Weyl group $W(G,T)$ is a finite group that permutes the choices of dominant Weyl chamber,
equivalently, permutes the choices of invariant complex structure on $G/T$.
\end{slide}
\begin{slide}{Relation to Peter-Weyl Theorem}
\ptsize{10}
The Peter-Weyl theorem says that, under the action of $G\times G$ by left and right translation,
$$L^2(G)=\sum_{i\in \hat G}End(V_i)=\sum_{i\in \hat G}V_i\times V_i^*$$
where the left $G$ action acts on the first factor, the right on the second.

To extract an irreducible representation $j$, need something that acts on all the $V_i^*$, picking
out a one-dimensional subspace exactly when $i=j$.

Borel-Weil does this by picking out the subspace that
\begin{itemize}
\item transforms with weight $\lambda$ under $T$
\item is invariant under $\mathfrak n_+$, where $\mathfrak g/\mathfrak t\otimes C=\mathfrak n_+
\oplus n_-$
\end{itemize}
\end{slide}
\begin{slide}{Borel-Weil-Bott Theorem (1957)}
\ptsize{10}
What happens for $\lambda$ a non-dominant weight?

One gets an irreducible representation not in
$H^0(G/T,\mathcal O(L_\lambda))$ but in higher cohomology 
$H^j(G/T,\mathcal O(L_\lambda))$.

Equivalently, using Lie algebra cohomology, what picks out the
representation is not $H^0(\mathfrak n_+,V_i^*)\neq 0$ (the $\mathfrak
n_+$ invariants of $V_i^*$), but $H^j(\mathfrak n_+,V_i^*)\neq 0$. This is
non-zero when $\lambda$ is related to a $\lambda^\prime$ in the dominant
Weyl chanber by action of an element of the Weyl group.

In some sense irreducible representations should be labeled not by a single weight, but by set of
weights given by acting on one by all elements of the Weyl group.
\end{slide}
\begin{slide}{The Dirac Operator}
\ptsize{10}
In dimension $n$, the spinor representation $S$ is a projective representation of $SO(n)$, a true
representation of the spin double cover $Spin(n)$. If $M$ has a "spin-structure", there is spinor
bundle $S(M)$, spinor fields are its sections $\Gamma(S(M))$.

The Dirac operator $\Dirac$ was discovered by Dirac in 1928, who was looking for a
\lq\lq square root" of the Laplacian.  In local coordinates
$$\Dirac=\sum_{i=1}^n e_i\frac{\partial}{\partial x_i}$$
$\Dirac$ acts on sections of the spinor bundle. Given an auxiliary bundle $E$ with connection, one
can form a \lq\lq twisted" Dirac operator
$$\Dirac_E:\Gamma(S(M)\times E)\rightarrow \Gamma(S(M)\times E)$$

\end{slide}
\begin{slide}{Borel-Weil-Bott vs. Dirac}
\ptsize{10}
\begin{itemize}
\item
Instead of using the Dolbeault operator $\overline\partial$ acting on complex forms
$\Omega^{0,*}(G/T)$ to compute $H^*(G/T,\mathcal O (L_\lambda))$, consider 
$\Dirac$ acting on
sections of the spinor bundle.
\item
Equivalently, instead of using $\mathfrak n_+$ cohomology, use an algebraic version of the Dirac
operator as differential (Kostant Dirac operator).
\end{itemize}
Basic relation between spinors and the complex exterior algebra:
$$ S(\mathfrak g/\mathfrak t)=\Lambda^*(\mathfrak n_+)\otimes\sqrt{\Lambda^n(\mathfrak n_-)}$$
Instead of computing cohomology, compute the index of $\Dirac$. This is 
independent of choice of
complex structure on $G/T$.
\end{slide}
\begin{slide}{Equivariant K-theory}
\ptsize{10}
K-theory is a generalized cohomology theory, and classes in $K^0$ are
represented by formal differences of vector bundles.  If $E$ is a vector
bundle over a manifold $M$,
$$[E]\in K^0(M)$$
If $G$ acts on $M$, one can define an equivariant K-theory, $K^0_G(M)$,
whose representatives are equivariant vector bundles.

Example: $[L_\lambda]\in K_G^0(G/T)$

Note: equivariant vector bundles over a point are just representations, so
$$K_G^0(pt.)=R(G)=\ \text{representation (or character) ring of}\ G$$
\end{slide}
\begin{slide}{Fundamental Class in K-theory}
\ptsize{10}
In standard cohomology, a manifold $M$ of dimension $d$ carries a \lq\lq
fundamental class" in homology in degree $d$, and there is an integration
map
$$\int_M: H^d(M,\mathbf R)\rightarrow H^0(pt., \mathbf R)=\mathbf R$$
In K-theory, the Dirac operator provides a representative of the
fundamental class in \lq\lq K-homology" and
$$\int_M: [E]\in K_G(M)\rightarrow \text{index} \Dirac_E
=\text{ker}\Dirac_E -\text{coker}\Dirac_E
\in
K_G(pt.)=R(G)$$
\end{slide}
\begin{slide}{Quantization = Integration in K-theory}
\ptsize{10}
In the case of $(G/T, \omega = curv(L_\lambda))$ this symplectic manifold
can be \lq\lq
quantized" by taking $\mathcal H$ to be the representation of $G$ given by
the index of $\Dirac _{L_\lambda}$.

This construction is independent of a
choice of complex structure on $G/T$, works for any $\lambda$, not just
dominant ones.

In some general sense, one can imagine \lq\lq quantizing" manifolds $M$
with vector bundle $E$, with $\mathcal H$ given by the index of
$\Dirac_E$.

Would like to apply this general idea to quantum field theory.
\end{slide}
\begin{slide}{History of Gauge Theory}
\ptsize{10}
\begin{itemize}
\item
1918: Weyl's unsuccessful proposal to unify gravity and
electromagnetism using symmetry of local rescaling of the metric
\item
1922: Schr\"odinger reformulates Weyl's proposal in terms of phase
transformations instead of rescalings.
\item
1927; London identifies the phase transformations as
transformations of the Schr\"odinger wave-function.
\item
1954: Yang and Mills generalize from local $U(1)$ to local $SU(2)$
transformations.
\end{itemize}
\end{slide}
\begin{slide}{Gauge Symmetry}
\ptsize{10} Given a principal $G$ bundle over $M$, there is an
infinite-dimensional group of automorphisms of the bundle that
commute with projection.  This is the gauge group $\mathcal G$.
Locally it is a group of maps from the base space $M$ to $G$.

$\mathcal G$ acts on $\mathcal A$, the space of connections on
$P$.
\bigskip

Example: $M=S^1$,\  $P=S^1\times G$,\  $\mathcal G=Maps(S^1,G)=LG$

$\mathcal A/\mathcal G = $ conjugacy classes in $G$,
identification given by taking the holonomy of the connection.
\end{slide}
\begin{slide}{Loop Group Representations}
\ptsize{10} See Pressley and Segal, Loop Groups.

$LG/G$ is an infinite-dimensional K\"ahler manifold, and there is
an analog of geometric quantization theory for it, with two
caveats:
\begin{itemize}
\item
One must consider \lq\lq positive energy" representations, ones
where rotations of the circle act with positive eigenvalues.
\item
Interesting representations are projective, equivalently
representations of an extension of $LG$ by $S^1$.  The integer
classifying the action of $S^1$ is called the \lq\lq level".
\end{itemize}

For fixed level, one gets a finite number of irreducible
representations.
\end{slide}
\begin{slide}{QFT in 1+1 dimensions}
\ptsize{10}
 Consider quantum field theory on a space-time
$S^1\times \mathbf R$.  The Hilbert space $\mathcal H$ of the
theory is associated to a fixed time $S^1$.

The level 1 representation for $LU(N)$ is $\mathcal H$ for a
theory of a chiral fermion with $N$ \lq\lq colors".

At least in 1+1 dimensions, representation theory and quantum
field theory are closely related.
\end{slide}
\begin{slide}{Freed-Hopkins-Teleman}
\ptsize{10} For loop group representations, instead of the
representation ring $R(LG)$, one can consider the Verlinde algebra
$V_k$.  As a vector space this has a basis of the level k positive
energy irreducible representations.

Freed-Hopkins-Teleman show:
$$V_k=K^{k+\tau}_{dim\ G, G}(G)$$
The right-hand side is equivariant K-homology of $G$ (under the
conjugation action), in dimension $dim\ G$, but \lq\lq twisted" by
the level $k$ (shifted by $\tau$).  This relates representation
theory of the loop group to a purely topological construction.
\end{slide}
\overlays{4}{
\begin{slide}{QFT Interpretation}
\ptsize{10}
\begin{itemstep}
\item
Consider the quantum field theory of chiral fermion
coupled to a connection (gauge field).
\item
Apply physicist's \lq\lq BRST" formalism.
\item
Get explicit representative of a K-homology class in \lq\lq$K_{\mathcal 
G}(\mathcal A)$".
\item
Idenitfy $K_{\mathcal G}(\mathcal A)$ with FHT's $K_G(G)$, since for
a free $H$ action on $M$, $K_H(M)=K(M/H)$.
\end{itemstep}
\end{slide}}
\overlays{4}{
\begin{slide}{Summary}
\ptsize{10}
\begin{itemstep}
\item
Quantum mechanics and representation theory are very closely linked 
subjects.
\item
Much is known about representation theory that still awaits exploitation 
by physicists. Much is known about QFT that awaits exploitation by 
mathematicians for insights into representations of infinite-dimensional 
groups.
\item
Work in Progress: 1+1 dim QFT and twisted K-theory. Relate path integrals 
and BRST formalism to representation theory and K-theory.
\item
QFT in higher dimensions?
\end{itemstep}
\end{slide}}
\end{document}
