text:
\begin{document}
\newtheorem{lfact}{Little Fact}[section]

\newtheorem{theorem}{Theorem}[section]

\newtheorem{proposition}{Proposition}
\newtheorem{conjecture}{Conjecture}

\newcounter{def}[section]

\newcounter{defin}[section]
\newenvironment{defin}

{\vskip .3cm \noindent \stepcounter{defin}{\bf Definition}
{\bf {\arabic{section}.{\arabic{defin}} }}

\sl
}

\newenvironment{fact}

{\vskip .3cm \noindent \stepcounter{def}{\bf Fact}
{\bf {\arabic{section}.{\arabic{def}} }}

\sl
}

\newcounter{eg}[section]
\newenvironment{eg}
{\vskip .3cm \noindent \stepcounter{def}{\bf Example }
{\bf {\arabic{section}.{\arabic{def} }}}}

\newcounter{lemma}[section]

\newenvironment{lemma}

{\vskip .3cm \noindent \stepcounter{def}{\bf Lemma}
{\bf {\arabic{section}.{\arabic{def}} }}
\sl
}

\newenvironment{para}[1]
{\vskip .3cm {\bf #1.}\hskip .4cm}

\newcommand{\qed}{\nobreak \ifvmode \relax \else
      \ifdim\lastskip<1.5em \hskip-\lastskip
      \hskip1.5em plus0em minus0.5em \fi \nobreak
      \vrule height0.75em width0.5em depth0.25em\fi}

\renewcommand{\baselinestretch}{1.5}
\normalsize

\thispagestyle{empty}

\Large

\begin{center}

{\bf {Orbifold Singularities,

the LATKe,

and Pure Yang-Mills with Matter}}

\end{center}
\vskip 1cm
\large
\begin{center} {\tt Tamar Friedmann}\footnote{E-mail: tamarf@mit.edu}\end{center}

\normalsize
\centerline{\it Massachusetts Institute of Technology}
\centerline{\it Cambridge, MA 02139, USA}

\vskip 1.5cm

\abstract{
We discover the unique, simple Lie Algebra of the Third Kind, or LATKe, that stems from codimension 6 orbifold singularities and gives rise to a  kind of Yang-Mills theory which simultaneously is pure and contains matter.
The root space of the LATKe is 1-dimensional and its Dynkin diagram consists of one point.
The uniqueness of the LATKe is a vacuum selection mechanism.
}

\vskip 1cm

\thispagestyle{empty}
\newpage

\begin{center}
\Large

The World in a Point ?
\end{center}
\begin{quote}\end{quote}

\vskip 4.5cm

\renewcommand{\baselinestretch}{1.2}
\normalsize

\begin{center}
\hskip 2cm Blow-up  of $\cz$ $|$  Dynkin diagram  of the LATKe \hskip 1cm
\end{center}

\vskip -0.2cm
\begin{center}
\Huge
\[ \bullet \]
\end{center}

\normalsize

\vskip 2.1cm

\begin{center}
Pure Yang-Mills
with matter

\end{center}
\normalsize

\renewcommand{\baselinestretch}{1.4}
\normalsize

\thispagestyle{empty}

\newpage

\setcounter{page}{1}

\section{Introduction}

The interpretation of codimension 4  orbifold singularities  as ADE gauge theories, which arose within string theory in the mid 90's \cite{Hull:1995mz,Witten:1995ex}, has been extended to the case of M--theory compactifications, where codimension 4 orbifold singularities in $G_2$ spaces were also interpreted as ADE gauge theories  \cite{Acharya:1998pm,Acharya:2000gb,Atiyah:2001qf}. Further orbifolding the already-singular $G_2$ spaces led us to the first manifestation via M--theory of Georgi-Glashow grand unification:  from an $A_4$ singularity of the $G_2$ space, an $SU(5)$ gauge theory broken by Wilson lines  precisely to the gauge group of the standard model $\sm$ arose naturally, with no extraneous gauge fields \cite{Georgi:1974sy,Friedmann:2002ct,Friedmann:2003cd}. A precise relation between the energy scale of grand unification ($M_{GUT}$) and certain volumes inside the $G_2$ space was also obtained \cite{Friedmann:2003cd,Friedmann:2002ty}.

In the process of constructing the $G_2$ spaces, orbifold singularities of codimension 6 arose as well \cite{Friedmann:2002ct}. However, there was no analog of the interpretation of a codimension 4 singularity as an ADE gauge theory for the case of codimension 6. We set out to find such an analog.

To our delight and surprise, we discover far more than we  expected, both mathematically speaking and physically speaking.

On the math side, we introduce a new set of relations, which we call the Commutator-Intersection Relations, that illuminate
the connection between codimension 4 singularities and Lie algebras. These relations pave the way to construct Lie Algebras of the Third Kind, or LATKes, a
kind of algebras that arise from codimension 6 orbifold singularities. We also learn and prove the existence and uniqueness of  a simple LATKe.

On the physics side, we discover a
kind of Yang-Mills theory, called "LATKe Yang-Mills," which arises from the LATKe.  Unlike any known Yang-Mills theory, the LATKe Yang-Mills theory in its purest form automatically contains matter. We also propose that the uniqueness of the simple LATKe is a vacuum selection mechanism. The selected vacuum theory is  an $SU(2)\times SU(2)$  gauge theory with matter in the $(2,2)$ representation, and the  corresponding singularity is $\cz$. The algebra $\su \times \su$ is protected by the LATKe from being broken. The selected singularity  $\cz$ is one of those which arose in the $G_2$ spaces of \cite{Friedmann:2002ct}, and which at the time we  put on hold in anticipation of  the outcome of this investigation.

\section{The Codimension 4 Case}\label{codim4}

In this section, we review the correspondence between codimension 4 orbifold singularities and ADE Lie algebras, introduce the Commutator-Intersection Relations, and review the relation between physical interactions on the one hand and commutators and intersections on the other hand.

\subsection{DuVal-Artin correspondence}\label{DuVA}
The interpretation of codimension 4 orbifold singularities as ADE gauge theories is mathematically rooted in the work of DuVal and of Artin \cite{duval,artin,steinberg}, who  pointed out a correspondence between certain singularities and their blow-ups on the one hand and certain Lie algebras on the other hand.

Before we state the correspondence, we provide below the necessary ingredients.

The singularities in question are those that appear at the origin of $\C ^2$ under the orbifold action of finite discrete subgroups of $ SU(2)$.
These subgroups, denoted $\G$,  had been classified as early as 1884 by F. Klein \cite{klein}. They consist of the cyclic groups $\Z _n$, also denoted $A_{n-1}$; the dihedral groups ${\bf D}_n$; and three "exceptional" groups: the tetrahedral ${\bf T}$, octahedral ${\bf O}$, and icosahedral ${\bf I}$, also denoted $E_6, E_7$, and $E_8$, respectively. Such a classification is known as an ADE classification.

Each of these subgroups of $SU(2)$ has a natural action on $\C ^2$. For example, $\Z_n$ is generated by the $SU(2)$ matrix
\be \label{Zn} \left ( \begin{array} [h] {cc} e^{2\pi i/n}&0 \\ 0&e^{-2\pi i/n} \end{array}\right )
\ee
and acts on $(x, y)\in \C ^2$ via the two--dimensional representation
\be \label{2Drep} (x, y)\longmapsto (e^{2\pi i/n}x, e^{-2\pi i/n}y)~.
\ee

The singularity at the origin of $\C ^2$ is analyzed by blowing up:  the singular space is replaced by a smooth manifold that looks just like $\C^2/\G$ everywhere except at the origin, and the origin itself is replaced by a smooth space of real dimension 2. This 2-dimensional space, known as the exceptional divisor, turns out to be a union of intersecting $\P ^1$'s, or 2--spheres $\S ^2$.

In the ambient four-dimensional space, the intersection of any two $\P ^1$'s is zero dimensional, i.e. it is a set of points. Counting those points gives an intersection number. The set of all intersection numbers forms the intersection matrix of the exceptional divisor, which we denote $\{ I_{ij}\}$. The indices $ij$ run from 1 to $b_2$, where $b_2$ is the number of independent 2--cycles in the exceptional divisor.

As it happens, the intersection matrix of the exceptional divisor of $\C^2/\G$ is equal to the negative of the Cartan matrix of the A, D, or E Lie algebra corresponding to $\G$:
\be \label{cequalsi} C_{ij}=-I_{ij} ~.\ee
In addition, the exceptional divisor itself becomes the Dynkin diagram of the corresponding Lie algebra when we replace each component $\P ^1$ of the exceptional divisor by a node and connect a pair of nodes by an edge when their corresponding $\P ^1$'s intersect; the components of the exceptional divisor therefore correspond to the simple positive roots of the Lie algebra. For example, when $\G = \Z _3$, the exceptional divisor is two intersecting $\P ^1$'s and the Dynkin diagram consists of two connected nodes:
\be \bullet \frac{ \mbox{  \hskip 1cm }  }{\mbox{  \hskip 1cm            }   } \bullet ~.
\ee

Now we are ready to state the DuVal-Artin correspondence:

\bea \label{reptop}
\left \{
\begin{array}{rr} \label{divDynk}
\text {Exceptional divisor}\\
\text{of blow-up of $\C ^2/\G$}
\end{array}
\right \}
&=&
\left \{
\begin{array}{ll}
\text {Dynkin diagram of}\\
\text {ADE Lie algebra}
\end{array}
\right \}
\\
\nonumber &&\\
 \left \{
\begin{array}{rrc} \label{intCart}
\text {Intersection matrix of }\\
\text{exceptional divisor} \\
I_{ij} \hskip 1cm
\end{array}
\right \}
&=&
\left \{
\begin{array}{llc}
\text {Negative Cartan matrix}\\
\text {of ADE Lie algebra}\\
\hskip 1cm -C_{ij}
\end{array}
\right \}
\\ \nonumber &&
\eea

\subsection{Commutator-Intersection Relations}\label{CIsection}
Here we rephrase the DuVal-Artin correspondence  as a set of relations between commutators of the Lie algebra and intersection numbers of the exceptional divisor, as follows.

A complex simple Lie algebra is generated by $k$ triples $\{X_i, Y_i, H_i \}_{i=1}^k$ with their commutators determined by the following relations:

\bea \label{Serre}&&[H_i, H_j]=0\; ; \non
\\&& [X_i, Y_j] = \delta _{ij} H_j \; ;\non
\\
&&[H_i, X_j]=C_{ij}X_j \; ;
\\&&  [H_i, Y_j]=-C_{ij}Y_j \; ;\non
\\
&&\mbox{ad}(X_i)^{1-C_{ij}}(X_j)=0\; ; \non
\\
&&\mbox{ad}(Y_i)^{1-C_{ij}}(Y_j)=0\; .\non
\eea
Here, the $H_i$ form the Cartan subalgebra, the $X_i$ are simple positive roots, the $Y_i$ are simple negative roots, $k$ is the rank of the Lie algebra, $C_{ij}$ is the Cartan matrix, and  ad$(X_i)(A)=[X_i, A]$.
These equations are the familiar Chevalley-Serre relations.

Now recall from equations (\ref{cequalsi}) and (\ref{intCart}) that $C_{ij}=-I_{ij}$. If we replace  $C_{ij}$ in equations (\ref{Serre}) by $-I_{ij}$, we get a new set of relations:
\bea \label{CI}&&[H_i, H_j]=0\; ; \non
\\&& [X_i, Y_j] = \delta _{ij} H_j \; ;\non
\\
&&[H_i, X_j]=-I_{ij}X_j \; ;
\\&&  [H_i, Y_j]=I_{ij}Y_j \; ;\non
\\
&&\mbox{ad}(X_i)^{1+I_{ij}}(X_j)=0\; ; \non
\\
&&\mbox{ad}(Y_i)^{1+I_{ij}}(Y_j)=0\; .\non
\eea

These relations demonstrate that the intersection numbers of the exceptional divisor
completely determine the commutators of the corresponding Lie algebra.

This role of the intersection numbers in the structure of the Lie algebra will be central for us in what follows, and we will refer to  the relations (\ref{CI}) as the {\sl Commutator-Intersection Relations}, or the {\sl CI Relations}.

\subsection{Interactions, commutators, and intersections}\label{GE}

Physically speaking, there are relations between interactions and commutators, and interactions and intersections.

Here, we explain  roughly  how ADE gauge fields  arise from the codimension 4 singularities \cite{Mayr:1998tx,Katz:1996fh,vafa}.
First,  Kaluza Klein reduction of 3-form C--fields on 2--cycles of the exceptional divisor provides the gauge fields corresponding to the Cartan subalgebra.
Second, D2--branes wrapped on 2--cycles provide the "charged" gauge fields, forming the rest of the Lie algebra.

For example, in the $\C^2/\Z_2$ case, where the exceptional divisor is a single $\P ^1$, three fields arise: a 3--form field reduced on the $\P^1$, which gives rise to the Cartan element denoted $A_\mu$; and D2--branes wrapped on the $\P^1$, which give rise to two oppositely charged particle states denoted $W^+$ or $W^-$ depending on orientation.

\newpage
The interaction among these fields  can be pictorialized in the following way:
\be W^+ \hskip 3cm \hskip .7cm W^+ \hskip 3.8cm A_\mu \hskip 1.6cm \ee
\vskip -.7cm
\begin{figure}[h]
\centering
\includegraphics{3pt}$A_\mu$ \hskip 2cm  \includegraphics{3pt}$W^+$ \hskip 2cm  \includegraphics{3pt}$W^-$
\end{figure}
\vskip -1.7cm
\be
W^- \hskip 3cm  \hskip .7cm A_\mu \hskip 4cm W^-  \hskip 2cm \non
\ee
The three interactions depicted above, $W^+ W^-\rightarrow A_\mu$, $W^+ A_\mu \rightarrow W^+$, and $A_\mu W^-\rightarrow W^-$, all arise from the same interaction vertex and are related to each other by CPT symmetry.

These interactions can then be manifested as Lie algebra commutators:
\be [W^+, W^-]=A_\mu \; , \hskip .8cm  \hskip .5cm [W^+, A_\mu]=W^+ \, , \hskip .8cm  \hskip .5cm [A_\mu , W^-]=W^- \; .  \hskip -.5cm\ee
These commutators form precisely the $\mathfrak{su}(2)$ Lie algebra.

 In addition to the manifestation of interactions as commutators,
when the singularities are such that the exceptional divisor contains more than  a single cycle, the intersections between the cycles can be interpreted as interactions between fields wrapped or reduced on those cycles \cite{Mayr:1998tx}.

\section{Lie Algebras of the Third Kind (LATKes)} \label{L3}
In this section we define the algebraic objects that are related to codimension 6 orbifold singularities in a way analogous to the relation between Lie algebras and codimension 4 singularities.

Recall from Section \ref{CIsection} that in the correspondence between codimension 4 singularities and Lie algebras, the intersection numbers of pairs of cycles in the exceptional divisor determine the Lie algebra commutators via the CI Relations (equations \ref{CI}). In particular, the intersection numbers enter the following commutators:
\bea \label{CI2}&&[H_i, X_j]=-I_{ij}X_j \; ;
\\&&  [H_i, Y_j]=I_{ij}Y_j \; .\non
\eea
Also recall the physics interpretation of commutators and intersections as interactions between two fields (Section \ref{GE}).

Now, for codimension $2n$ singularities for any $n$, the components of the exceptional divisor are $(2n-2)$--cycles, and the intersection of a pair of those has dimension
\be \dim (C_1 \cap C_2)=\dim C_1+\dim C_2-2n=2n-4.\ee
Therefore, for codimension 6 orbifolds, the components of the exceptional divisor are 4--cycles, and the intersection of any pair $C_1, C_2$ of 4--cycles does not yield a number but a two-dimensional space:
\be \dim (C_1 \cap C_2)=4+4-6=2.\ee

However, the intersection of {\it three} 4--cycles in a six dimensional ambient space is zero--dimensional, yielding intersection numbers with three indices: $I_{ijk}$.  On the physics side, these triple intersections should lead to interactions among three fields.

Bringing together all the above leads us to a generalization of the CI Relations to the codimension 6 case. Equations (\ref{CI2}) become commutators of three objects:
\bea \label{CI3}&&[H_i, H_j, X_k]=-I_{ijk}X_k \; ;
\\&&  [H_i, H_j, Y_k]=I_{ijk}Y_k \; .\non
\eea
At this stage, we do not yet know what the $H_i$, $X_k$, or $Y_k$ are. However, equation (\ref{CI3}) provides the fundamental ingredient in the algebraic objects we have been searching for: a commutator with three entries. It is now clear that the sought-after algebraic  objects are the following natural generalizations of Lie algebras:

\begin{defin} A {\textit {\textbf {Lie algebra of the third kind (a "LATKe")}}} $\L$ is a vector space
equipped with a  commutator of the third kind, which is a trilinear anti-symmetric map
\be [\hskip .1cm\cdot \hskip .1cm , \hskip .1cm\cdot \hskip .1cm, \hskip .1cm\cdot \hskip .1cm]: \Lambda ^3 \L \rightarrow \L \ee
  that satisfies the  Jacobi identity of the third kind:
\be \label{3JI} [X,Y,[Z_1, Z_2, Z_3]] =[[X,Y,Z_1],Z_2, Z_3]+[Z_1, [X,Y,Z_2], Z_3]+[Z_1, Z_2, [X, Y, Z_3]]
\ee
for $X, Y, Z_i\in \L$.
\end{defin}

We will refer to the commutator of the third kind as a "LATKe commutator", and to the Jacobi identity of the third kind as the "LATKe Jacobi identity." The LATKe Jacobi identity is simply a generalization of the standard Jacobi identity, which says that the adjoint action is a derivation.

For codimension $2n$ singularities, it is $n$--fold intersections of $(2n-2)$--cycles that give intersection numbers. So the CI Relations for that case have commutators of $n$ objects, leading to the following generalization of a LATKe:

\begin{defin} A {\textbf {\textit  {Lie algebra $\L$ of the $n$-th kind (a "LAnKe")}}}   is a vector space equipped with a  commutator  of the $n$-th kind, which is an n-linear, totally antisymmetric  map
\be [\cdot, \cdot , \hskip .7cm , \cdot] : \wedge ^n \L \rightarrow \L ~,\ee
that satisfies the Jacobi identity of the $n$-th kind:
\be [X_1, \ldots X_{n-1},[Z_1, \ldots Z_n]]=\sum _{i=1}^n [Z_1,\ldots , [X_1, \ldots X_{n-1},Z_i], \ldots Z_n]~,\ee
for $X_i, Z_j \in \L$.
\end{defin}

Before we go any further, we should satisfy ourselves that LATKes  actually exist. Therefore, we now construct an example.
\vskip .3cm
\no \begin{underline} {\tt The algebra $\L_3$}\end{underline}
Recall that in the codimension 4 case, each component of the exceptional divisor corresponds to a node in the Dynkin diagram of the corresponding Lie algebra. Therefore, each component corresponds to a simple root of the Lie algebra.
Furthermore, the simplest singularity is $\C^2/\Z_2$, whose blow-up is a single $\P ^1$, and the simplest (non-abelian) Lie algebra is the corresponding $\mathfrak{su}(2)$.

It is reasonable to presume that similarly,  the simplest example of a LATKe  would correspond to the codimension 6 orbifold singularity with the simplest exceptional divisor. Consider the singularity $\cz$, where the action of a generator $\epsilon$ of $\Z_3$ is given by
\be \label{epsilonaction}\epsilon : (x, y, z)\longmapsto (\epsilon x , \epsilon y, \epsilon z)\; ; \mbox{ where }\;  \epsilon ^3=1 , (x,y,z)\in \C^3.\ee
The blow-up of this singularity is a single $\P^2$ (see Appendix \ref{blow}). The cycle $\P ^2$ corresponds to a node in the yet-to-be-defined   Dynkin diagram of $\L_3$. That node, in turn, should correspond to a yet-to-be-defined "root" of $\L _3$.

We must now define the notion of a root for LATKes. For standard Lie algebras, a root $\alpha$ is in the dual space of the Cartan subalgebra $\h $:
\be \alpha : \h \longrightarrow \C \; .\ee
So, to define a  root for $\L$, we must first define a  Cartan subalgebra for $\L$.

In the standard Lie algebra case, one can think of the Cartan subalgebra as a set of operators acting on the Lie algebra $\g$: for a root vector $X_\alpha \in \g$ and for $H\in \h$, we have
\bea \hskip 2cm H&:&X_\alpha \longmapsto [H, X_\alpha ]=\alpha (H)X_\alpha ~.\eea
For a LATKe $\L$, there is no natural action of a subalgebra. However, if $\h _\L \subset \L$ is a  subalgebra of $\L$ (a subspace closed under the commutator), then $\Lambda ^2 \h _\L$ does act on $\L$ naturally:  if $H_1, H_2\in \h_\L$ and $X\in \L$, then $H_1\wedge H_2\in \Lambda ^2\h_\L$ acts on $\L$ via:
\bea \label{CartanAct}
H_1\wedge H_2 &:& X \longmapsto [H_1, H_2, X] \; .
\eea
We may now define $\h_\L \subset \L$ the same way it is defined for Lie algebras:
\begin{defin} A {\textit {\textbf {Cartan subalgebra}}} $\h _\L$ of $\L$ is a maximal commuting subalgebra of $\L$ such that $\Lambda ^2\h _\L$ acts diagonally on $\L$.
\end{defin}
Finally, following the standard Lie algebra case, we have
\begin{defin} Let $\L$ be a LATKe and let $\h _\L$ be a  Cartan subalgebra of $\L$. A {\textit {\textbf {root}}} $\alpha$ of $\L$  is a map in the dual space of $\Lambda ^2 \h _\L$:
\be \alpha : \Lambda ^2 \h _\L \longrightarrow \C \; . \ee
\end{defin}
Now we may also define a generalization for Dynkin diagrams:
\begin{defin} \label{3Dynk} A {\textit {\textbf {Dynkin diagram}}} of $\L$ is a graph with a vertex for each simple positive root of $\L$.
\end{defin}
By "positive," we imply that we have chosen a direction in $(\Lambda ^2\h _\L)^*$ and ordered the roots, as is done for conventional Lie algebras. Note that while this  definition  for a Dynkin diagram may appear to be only a partial  one because it says nothing about edges, it will become clear later that this definition is
complete.

We now construct a LATKe with a single root and a single node in its  Dynkin diagram corresponding to the cycle $\P ^2$ in the blow-up of $\cz$.

For the  root  space to be one-dimensional, the  Cartan subalgebra must be two-dimensional. Let $\h _\L = \{H_1, H_2 \}$ so that $\Lambda ^2 \h _\L$ is spanned by $H_1 \wedge H_2$, and introduce candidates for one positive and one negative root vector, named $X$ and $Y$, respectively. All of the above lead to the following two equations:
\bea \;  [H_1, H_2, X]&=&\alpha (H_1\wedge H_2)X \;  ;  \\
\; [H_1, H_2, Y]&=&-\alpha (H_1\wedge H_2)Y \;  .
\eea
where $\alpha$ is the single root associated with $\Lambda ^2\L$. Normalizing $H_1$ and/or $H_2$ so that $\alpha (H_1\wedge H_2) =1$ (we should really be normalizing to  $\alpha (H_1\wedge H_2) =-I_{111}$, where $I_{111}$ is the triple intersection number of the $\P ^2$; however, the resulting algebra would be equivalent) gives
\bea \label{rootcomm}\; \hskip -2cm [H_1, H_2, X]&=& X \; ;  \\
  \; \hskip -2cm [H_1, H_2, Y]  &=&  -Y \; . \non
\eea

All that is left is determining $[H_1, X, Y]$ and $[H_2, X, Y]$ such that the LATKe Jacobi identity would be satisfied. This identity requires, among other things, that $[H_i, X, Y], i=1,2$ are zero roots:
\bea [H_1, H_2,[H_i, X, Y]]&=&[[H_1, H_2, H_i], X, Y]+[H_i, [H_1, H_2, X], Y]\non \\
&& \hskip 4cm + [H_i, X, [H_1, H_2, Y]]   \non \\
&=&0+[H_i, X, Y]+[H_i, X, -Y]=0
\eea
Therefore, $[H_i, X, Y]\in \h _\L$.

We now restrict the possibilities for $[H_i, X, Y]$ by limiting ourselves to {\sl simple} LATKes, which we now define (recall that in the codimension 4 case, all the Lie algebras corresponding to orbifolds are simple).
 First we need the notion of ideals.
\begin{defin} An {\textit {\textbf {ideal}}} of $\L$ is a subalgebra ${\cal I}$ that satisfies
\be  [\L, \L, {\cal I}]\subset {\cal I} \; .\ee
\end{defin}
\begin{defin} A  LATKe is {\bf {\textit {simple}}} if it is non-abelian and has no non-trivial ideals.
\end{defin}

Requiring that our example would be simple means that $[H_1, X, Y]$ and $[H_2,X,Y]$ must be linearly independent, otherwise there would be a non-trivial ideal. It also requires that $[H_i, X, Y]$ and $H_i$ be linearly independent for each $i$.

So $[H_i, X, Y]$ are linearly independent combinations of $H_1$ and $H_2$. Any such linear combination can be transformed into another by  linear redefinitions that do not affect the two commutators we already fixed in equation (\ref{rootcomm}). Therefore, we may make a choice and we let
\bea \label{restcomm}\; \hskip -2cm [H_1,  X, Y]&=& H_2 \; ;  \\
  \; \hskip -2cm [H_2, X, Y]  &=&  H_1 \; . \non
\eea
No more commutators are needed to be determined -- all other commutators are defined by the requirement that the commutator map is trilinear and totally antisymmetric. One can easily verify that the LATKe Jacobi identity  is satisfied and that there are no non-trivial ideals. Therefore, we have
\begin{eg} \label{L3eg} The four-dimensional algebra spanned by $\{ H_1, H_2, X, Y\}$ with the  commutators
\bea \label{L3com}\; \hskip -2cm [H_1, H_2, X]&=& X \;  ,\non \\
  \; \hskip -2cm [H_1, H_2, Y]  &=&  -Y \;  ,  \\
\; \hskip -2cm [H_1,  X, Y]&=& H_2 \;  , \non \\
  \; \hskip -2cm [H_2, X, Y]  &=&  H_1 \;  ,\non
\eea
is a simple LATKe. We name it $\L _3$. Its Cartan subalgebra is 2-dimensional,
its root space is 1-dimensional, and its Dynkin diagram consists of a single point:

\[ \bullet  \]
\vskip .5cm
\end{eg}

\section{Uniqueness of the LATKe}\label{1-proof}
We have constructed the LATKe $\L_3$ corresponding to the simplest codimension 6 orbifold singularity, $\cz$.
If the definition of LATKes has planted seeds for  a generalization of the DuVal-Artin correspondence, then there should be a simple  LATKe  for each of  the orbifolds $\C ^3/\G$, where $\G$ is any discrete, finite subgroup of $SU(3)$.

Our goal now, therefore, is to classify all simple LATKes. In doing so, we would also learn more about their structure, which should help us in eventually constructing a physics theory appropriate to these algebras.

We begin  by recalling that we can think of $\Lambda ^2 \h _\L$ as a set of operators acting on $\L$, as in equation (\ref{CartanAct}). More generally, we note that $\Lambda ^2 \L$, not just $\Lambda ^2 \h _\L$, acts on $\L$ as follows: if $X\wedge Y \in \Lambda ^2 \L$ then its action on $Z\in \L$ is given by
\bea
\label{adaction} X\wedge Y &:& Z \longmapsto [X, Y, Z] \; .
\eea
This action is called {\textit {\textbf {the adjoint action}}} on $\L$ and we denote it ad$(X\wedge Y)$.

The LATKe Jacobi identity, equation (\ref{3JI}), indicates that $\Lambda ^2 \L$ acts as an {\textit {inner derivation}} of $\L$:
\begin{defin} An operator $D$ on $\L$ is a {\textit {\textbf {derivation}}} of $\L$ if it satisfies
\be D [X,Y,Z]=[DX,Y,Z]+[X,DY,Z]+[X,Y,DZ] \; . \ee
The set of derivations of $\L$ forms a Lie algebra with bracket given by
\be [D_1, D_2]=D_1D_2-D_2D_1~.\ee
\end{defin}
\begin{defin} An {\textit {\textbf {inner derivation}}} of $\L$ is a derivation of $\L$ which is a linear combination of operators of the form given in equation (\ref{adaction}).
\end{defin}
These two definitions are analogous to the ones given for conventional Lie algebras.

We shall denote the algebra of derivations of $\L$ by Der$(\L)$ and the inner derivations by IDer$(\L)$. Note that there is an isomorphism $\Lambda ^2 \L \simeq$ IDer$(\L)$, which is given by
\be \label{adgen}\mbox {ad}: X\wedge Y \in \Lambda ^2 \L\longmapsto \mbox{ad}(X\wedge Y)\in IDer(\L) \ee
where
\be \label{adspec}\mbox{ad}(X\wedge Y)(Z)=[X,Y,Z]~.\ee

The  LATKe Jacobi identity can be rewritten to show that $\Lambda ^2 \L$  itself comes equipped with  a well-defined commutator and therefore it forms a Lie algebra:
\bea \label{JII}[X_1, X_2 , [X_3, X_4, X_5]]-[X_3, X_4,[X_1, X_2, X_5]]=\hskip 2cm \non \\ \hskip 2cm = [[X_1, X_2, X_3], X_4, X_5]+[X_3, [X_1, X_2, X_4], X_5]
\eea
so that
\be \label{gLcomm} [X_1 \wedge X_2, X_3 \wedge X_4]=[X_1, X_2, X_3]\wedge X_4 + X_3\wedge [X_1, X_2, X_4]~.
\ee
This commutator clearly satisfies the standard Jacobi identity. Equivalently, in IDer$(\L)$ this commutator takes the form
\be \label{gLcommad}[\mbox {ad}(X_1 \wedge X_2),\mbox {ad}(X_3 \wedge X_4)]=\mbox {ad}([X_1, X_2, X_3]\wedge X_4)+\mbox {ad}(X_3\wedge [X_1, X_2, X_4])~.
\ee
\begin{defin} \label{gL} For any LATKe $\L$, the {\textit {\textbf {Lie algebra of $\L$}}} is the space $\Lambda ^2\L\simeq IDer(\L)$ with the commutator given by equation (\ref{gLcomm}) or (\ref{gLcommad}); we  denote this Lie algebra $\g _\L$.
\end{defin}

\vskip .3cm
\no \begin{underline}{\tt  $\L$ simple $\Longrightarrow \ggg$ semi-simple}\end{underline} We have shown that every LATKe $\L$ has associated with it a Lie algebra $\ggg$. We now prove a series of lemmas culminating in the result that if $\L$ is a simple LATKe then its Lie algebra $\ggg$ is semi-simple. Consequently,
 we will be able to utilize the well-known classification of semi-simple Lie algebras as a tool for classifying the simple LATKes.

\begin{lemma} \label{Lirrep} If $\L$ is simple then $\L$ is irreducible as a representation of $\Lambda ^2 \L \simeq $ IDer$(\L)$.
\end{lemma}
\no {\bf Proof} Let $W\subset \L$ be an invariant subspace, i.e. IDer$(\L): W\rightarrow W$. Then $[\L, \L, W]\subset W$ (see equations (\ref{adgen}) and (\ref{adspec})). So $W$ is an ideal of $\L$. Since $\L$ is simple, $W=0$ or $\L$.
\begin{lemma} \label{idealIDer} IDer$(\L)$ is an ideal of Der$(\L)$.
\end{lemma}
\no {\bf Proof} Let  $D\in$ Der$(\L)$ and let $X\wedge Y\in  \Lambda ^2 \L$ so ad$(X\wedge Y)\in$ IDer$(\L)$. Then we have
\be  [D, \mbox{ad}(X\wedge Y)]\cdot Z = D[X,Y,Z] -[X,Y,DZ]=[DX,Y,Z]+[X,DY,Z]~, \ee
so
\be [D,\mbox{ad}(X\wedge Y)]=\mbox{ad}(DX\wedge Y)+\mbox{ad}(X\wedge DY)~, \ee
which is  an inner derivation.
\begin{fact} The space $\L$ is a representation of Der$(\L)$ and it is faithful by definition.
\end{fact}
\begin{lemma} If $\L$ is simple then $\L$ is irreducible as a representation of Der$(\L)$.
\end{lemma}
\no {\bf Proof} Any subspace $W\subset \L$ that is invariant under Der$(\L)$ is also invariant under its ideal IDer$(\L)$. By the proof of Lemma \ref{1-proof}.1, $W=0$ or $\L$.
\begin{lemma} If $\L$ is simple then Der$(\L)$ is reductive.
\end{lemma}
\no {\bf Proof} This follows from the fact that Der$(\L)$ has  a finite dimensional, faithful, irreducible representation, namely $\L$.
\begin{lemma} \label{derIder}If $\L$ is simple then any derivation of $\L$ is an inner derivation, i.e. Der$(\L)=$IDer$(\L)$.
\end{lemma}
\no {\bf Proof} Let $D$ be any derivation of $\L$. Since Der$(\L)$ is reductive, it has the form of a direct sum of commuting ideals. We already know that IDer$(\L)$ is an ideal of Der$(\L)$. Therefore, all we need to show is that if  $D$ commutes with IDer$(\L)$, then $D=0$.
Assume
\be [D, \mbox{ad}(X\wedge Y)]\cdot Z=0 \hskip 1cm \forall X,Y,Z\in \L~.\ee
Expand this equation to
\be D(\mbox{ad}(X\wedge Y)\cdot Z) - \mbox{ad}(X\wedge Y)(D\cdot Z)=D[X,Y,Z]-[X,Y,DZ]=0.\ee
Repeating this for permutations of $X,Y$, and $Z$ and using the definition of derivations, we find that
\be [DX,Y,Z]=[X,DY,Z]=[X,Y,DZ]=0 \hskip 1.5cm \forall X,Y,Z\in \L~.\ee
Therefore, $DX$ is in the center of $\L$ for any $X\in \L$. But since $\L$ is simple, it has no center. Therefore, $DX=0 ~\forall X$ so $D=0$.
\begin{lemma} If $\L$ is simple, then the center of IDer$(\L)$ is trivial.
\end{lemma}
\no {\bf Proof} The argument in the proof of Lemma \ref{1-proof}.6 shows that any derivation that commutes with IDer$(\L)$ is zero.
\begin{lemma} If $\L$ is simple then IDer$(\L)$ is semi-simple.
\end{lemma}
\no {\bf Proof}  Since IDer$(\L)$ is an ideal in a reductive Lie algebra, it is itself reductive. Any reductive Lie algebra is the direct sum of a semi-simple part and its center. Since IDer$(\L)$ has no center, it is semi-simple.

\vskip .3cm
So we have shown that IDer$(\L)\simeq \Lambda ^2 \L = \ggg$ is a semi-simple Lie algebra when $\L$ is simple. We show next that there are very strong constraints on the roots and weights of $\ggg$ that substantially limit the number of potential candidates for $\ggg$.

\vskip .3cm

\no \underline{{\tt  Constraints on roots and weights of $\ggg$}}
Let the highest weight of $\L$ as an irreducible representation of $\ggg$ be denoted $\Lambda$. Since $\L$ is a representation of $\ggg$, so is $\Lambda ^2\L$. In fact, since $\Lambda ^2\L \simeq \ggg$,  it is actually the adjoint representation of $\ggg$, whose weights are the roots of $\ggg$. Let $\h$ be the Cartan subalgebra of $\ggg$, let $H\in \h$, and let $X_{\beta  _i}\in \L$ be the weight vectors of $\L$ with $\beta _i$ denoting weights of $\L$ so that
\be H(X_{\beta  _i})= \beta  _i(H)   X_{\beta  _i}~.\ee
Then
\bea H \cdot (X_{\beta  _i}\wedge X_{\beta  _j})&=&(H \cdot X_{\beta  _i})\wedge X_{\beta  _j} +  X_{\beta  _i}\wedge (H \cdot X_{\beta  _j}) \non \\ &=&(\beta _i (H)+\beta _j (H))(X_{\beta  _i}\wedge X_{\beta  _j})\;  .
\eea
Since the weights of $\Lambda ^2\L$ are the roots of $\ggg$, it follows that any root of $\ggg$ has the form $\beta _i + \beta _j$ where $i\neq j$ (we do not allow $i=j$ because then $X_{\beta  _i}\wedge X_{\beta  _j}=X_{\beta  _i}\wedge X_{\beta  _i}=0$ so there is no root vector with weight $2\beta _i$).

More particularly, we have
\begin{lemma} The highest root $\theta$ of any simple component of $\ggg$ is the sum of $\Lambda$, the highest weight of $\L$, and a next-to-highest weight $\Lambda -\alpha$, where $\alpha$ is a simple positive root of a simple component of $\ggg$:
\be \theta = \Lambda + (\Lambda -\alpha) \ee
or
\be \label{condition}\theta + \alpha = 2\Lambda ~.\ee
\end{lemma}
Only semi-simple Lie algebras which have an irreducible representation $V$ whose highest weight $\Lambda$ satisfies
condition  (\ref{condition}) are potential candidates to be Lie algebras of some $\L$. Our next step, therefore, is to find all semi-simple $\g$ and highest weights $\Lambda$ satisfying this condition.

As it turns out, this same condition appeared in an entirely different context in Kac's work on Lie superalgebras \cite{kac}, where all pairs of semi-simple Lie algebras $\g$ and irreducible representations $V$ with highest weight $\Lambda$ satisfying this condition are classified.
The resulting list of pairs is the following.

If $\g$ is not simple then $\g=\mathfrak{so}_4 \simeq \mathfrak{sl}_2 \times \mathfrak{sl}_2   $ and $V$ is the standard four-dimensional (vector) representation.

If $\g$ is simple then the following table constitutes the complete list of all pairs of $\g$ and $V$ that satisfy the condition:
\be \label{list} \begin{array} [h] {llll} \g&\dim V& \dim \g & \dim \Lambda ^2 V \\ \hline
A_1 &3 &3 &3 \\
G_2&7&14&21\\
A_3&6&15&15\\
B_3&8&21&28\\
B_{r\geq 2}\hskip .5cm &2r+1\hskip .3cm &r(2r+1)\hskip .3cm &r(2r+1)\\
D_{r\geq 4}&2r&r(2r-1)&r(2r-1)
\end{array}
\ee
The dimensions in the table uniquely identify each representation, except that for $D_4$ there are three different representations of dimension 8, which are related to each other by the triality symmetry of $\mathfrak{so}_8$.

\vskip .3cm
\no \begin{underline}{\bf {\tt {Further conditions on $\g _\L$ and the construction of all $\L$ }}}\end{underline}
The condition $2\Lambda = \theta + \alpha$ is necessary, but not sufficient, to ensure that $\g$ is the Lie algebra of some $\L$. There are further requirements.

First, $\Lambda ^2V$ must be equal to the adjoint representation of $\g$. This condition eliminates the $G_2$ case as well as the $B_3$ case with $\dim V=8$, as is easily seen by comparing $\dim \g$ to $\dim \Lambda ^2 V$ in table (\ref{list}).

Second, the intertwining map $\omega : \Lambda ^2 V \rightarrow \g $ must yield a  LATKe commutator in the following way.
Let $v_1, v_2, v_3\in V$; then we define
\be  [v_1, v_2, v_3]=(\omega (v_1\wedge v_2))\cdot v_3~.\ee
The expression on the right hand side must be antisymmetric in all three variables for it to define a LATKe commutator. Since it is already automatically antisymmetric in the first two variables, the only remaining requirement is
\be \label{antisym123} (\omega (v_1\wedge v_2))\cdot v_3=-(\omega (v_3\wedge v_2))\cdot v_1 \hskip 1cm \forall  v_1, v_2, v_3\in V \, ,\ee
which is equivalent to requiring
 \be \label{antisym}\omega (v_1\wedge v_2))\cdot v_1=0~ \hskip .5cm \forall v_1, v_2\in V~.\ee

We first prove the following:
\begin{lemma} None of the pairs of $\g$ and $V$ in  table \ref{list} yields a LATKe  commutator.
\end{lemma}
\no {\bf Proof} There are two steps to the proof. First, we construct explicitly the intertwining map $\omega : \Lambda ^2 V \rightarrow \g $ for all these pairs (other than $G_2$ and $B_3$ with $\dim V=8$, which we have already eliminated). Since the Lie algebras are simple, their adjoint representation -- and therefore $\Lambda ^2 V$ -- is irreducible, so by Schur's lemma the map $\omega$ is unique. Second, we show that $\omega$  does not satisfy equation (\ref{antisym}), so it does not result in a  LATKe commutator.

We can construct the map $\omega$ for all the pairs simultaneously because all of them have the form $\mathfrak{so}_n$ with $V$ the standard $n$-dimensional representation. That is obvious for $B_r$ and $D_r$, which stand for $\mathfrak{so}_{2r+1}$ and $\mathfrak{so}_{2r}$, respectively. For $A_1$ and $A_3$, recall that $A_1$ stands for $\mathfrak{sl}_2 \simeq \mathfrak{so}_3$ and $A_3$ stands for
$\mathfrak{sl}_4 \simeq \mathfrak{so}_6$.

Let $\mathfrak{so}_n$ be spanned by antisymmetric matrices $e_{ab}$, $a,b=1, \ldots , n$, $a<b$, such that
\be \label{eab}\{ e_{ab} \}_{\alpha \beta}=\delta_{a\alpha}\delta _{b\beta}-\delta_{a\beta}\delta _{b\alpha}~, \ee
i.e. $e_{ab}$ has $+1$ in the $ab$-th entry and $-1$ in the $ba$-th entry, with all other entries equal zero; if $a>b$ then $e_{ab}$ is defined by $e_{ab}=e_{ba}$. Let the standard representation $V$ be spanned by the standard basis $\{ e_a \}$, $a=1, \ldots ,n$, where
\be \label{ea}( e_a )_\alpha=\delta _{a\alpha} ~,\ee
i.e. $e_a$ has $+1$ in the $a$-th entry and zero elsewhere.
It is straightforward to check that
\bea \label{socomm}[e_{ab} , e_{cd}]&=& \delta _{bc}e_{ad}+ \delta _{ad}e_{bc}-\delta _{ac}e_{bd}-\delta _{bd}e_{ac}~; \\
e_{ab}\cdot ( e_c )&=&\delta _{bc}(e_a)-\delta _{ac}(e_b)~.
\eea

We want to construct $\omega : \Lambda ^2 V \rightarrow \g $ which is intertwining, i.e.
\bea [X, \omega (v_1\wedge v_2)]&=&\omega (X(v_1\wedge v_2)) \non \\
&=&\omega ((X\cdot v_1)\wedge v_2+v_2\wedge (X\cdot v_2)) \hskip .3cm \forall X\in \g, v_i\in V  ,
\eea
so we require
\be \label{twining} [e_{ab}, \omega (e_c\wedge e_d)]=\omega ((e_{ab}\cdot e_c)\wedge e_d+e_c\wedge (e_{ab}\cdot e_d))\hskip .3cm \forall a,b,c,d ~.\ee

We now show that the map defined by
\be \label{thepsi}\omega (e_a\wedge e_b)=e_{ab} \ee
satisfies this property.

Computing the right hand side of equation (\ref{twining}) for this map gives
\bea  \omega ((e_{ab}\cdot e_c)\wedge e_d+e_c\wedge (e_{ab}\cdot e_d))= \hskip 5cm \non \\
  \hskip 2cm =\omega ((\delta _{bc} (e_a)-\delta _{ac}(e_b))\wedge e_d)+\omega (e_c \wedge (\delta _{bd}(e_a)-\delta _{ad}(e_b)))\\
 = \delta_{bc}e_{ad}-\delta_{ac}e_{bd}+\delta_{bd}e_{ca}-\delta_{ad}e_{cb}~. \hskip 3.7cm \non
\eea
Comparing this with equation (\ref{socomm}) proves that equation (\ref{thepsi}) is the desired map.

Now we check whether the condition (\ref{antisym}) is satisfied. It is not:
\be \omega (e_a\wedge e_b)\cdot e_a=e_{ab}\cdot e_a=-e_b\neq 0\; .\ee
Therefore, we do not get a LATKe from any of the simple Lie algebras on the list, and the proof of the lemma is complete. \hfill $\Box$

\vskip .3cm
We are now left with only one candidate: $\mathfrak{so}_4$ with its standard representation. Recall that $\mathfrak{so}_4$ is not a simple Lie algebra but has two simple factors, $\mathfrak{so}_4\simeq \mathfrak{sl}_2 \times \mathfrak{sl}_2$. So its adjoint representation is not irreducible, Schur's lemma does not apply, and the intertwining map $\omega$ constructed above for  $\mathfrak{so}_n$ is not unique for this case. We can construct another one.

We will show that the other map  does lead to a  LATKe commutator and in fact yields the LATKe $\L_3$ which we constructed in Section \ref{L3}.

Using the same notation as before, the basis for $\mathfrak{so}_4$ is
\be \{ e_{12}, e_{13},e_{14},e_{23},e_{24},e_{34} \} \ee
and the basis for $V$ is
\be \{ e_1, e_2, e_3, e_4 \}~.\ee
We define $\phi: \Lambda ^2 V \rightarrow \g $ explicitly by
\bea \phi (e_1\wedge e_2)&=&e_{34} \non \\
\phi (e_1\wedge e_3)&=&-e_{24} \non \\
\phi (e_1\wedge e_4)&=&e_{23} \non \\
\phi (e_2\wedge e_3)&=&e_{14} \label{phi4}\\
\phi (e_2\wedge e_4)&=&-e_{13} \non \\
\phi (e_3\wedge e_4)&=&e_{12} \non
 \eea
 It is straightforward to check that this map satisfies the intertwining condition, equation (\ref{twining}).

 We now use the map $\phi$ to construct the  LATKe commutator. The dimension of the candidate for $\L$ is $\dim V=4$, so there are only four commutators to calculate:
 \bea \; [e_1, e_2, e_3]&=&\phi(e_1\wedge e_2)\cdot e_3=e_{34}\cdot e_3=-e_4 \; ; \non \\
 \; [e_1, e_2, e_4]&=&\phi(e_1\wedge e_2)\cdot e_4=e_{34}\cdot e_4=e_3 \; ; \non \\
\;  [e_1, e_3, e_4]&=&\phi(e_1\wedge e_3)\cdot e_4=-e_{24}\cdot e_4=-e_2 \; ;  \label{L3pretty}\\
 \; [e_2, e_3, e_4]&=&\phi(e_2\wedge e_3)\cdot e_4=e_{14}\cdot e_4=e_1 \; .\non
 \eea
 With the following change of variables
 \be X=\frac{1}{\sqrt 2}(e_1+ie_4)\; ; Y=\frac{1}{\sqrt 2}(e_1-ie_4)\; ; H_1=-ie_2\; ; H_2=e_3 \; ,
 \ee
 we see that this algebra is precisely $\L_3$ as given in equations (\ref{L3com}).

 This construction of $\L_3$ generalizes to a LAnKe $\L _n$, as shown in Appendix \ref{lanke}.

 We summarize the results of this section in one theorem:
 \begin{theorem} \label{bigtheorem} There is precisely one simple LATKe, namely $\L_3$; it is four dimensional, it corresponds to the singularity $\cz$, its Lie algebra $\g_{\L_3}$ is $\mathfrak{so}_4\simeq \mathfrak{sl}_2\times \mathfrak{sl}_2$, and its Dynkin diagram consists of one node. Its commutators are given by equation (\ref{L3com}) or (\ref{L3pretty}).
 \end{theorem}

 We comment that the theorem is equivalent to the following statement. Let $V$ be a vector space with a non-degenerate symmetric  bilinear form so that $\Lambda ^2V\simeq \mathfrak{so}(V)$. There is  a natural action of $\Lambda ^2V$ on $V$ which induces a map $\omega _3: V\otimes V  \otimes V \rightarrow V$, antisymmetric in the first two factors. Let $\bar \omega _3: \Lambda  ^3V\rightarrow V$ be the fully-antisymmetric version of $\omega _3$. The theorem says that, if we assume that the action of $\mathfrak{so}(V)$ on $V$ was irreducible, then there is a unique vector space $V$ for which $\bar \omega _3$ is both non-trivial and satisfies the Jacobi identity of the third kind (equation (\ref{3JI})). That vector space is four dimensional and $\Lambda ^2V$ consists of skew-symmetric 4-matrices.

\section{The Physics of LATKes}\label{physics}
We have now reached an important juncture. Having found the unique, simple LATKe, we ask ourselves: are there any applications of the LATKe to physics?

The first type of physical theory that comes to mind in applying Lie algebras to physics is gauge theory. Can we generalize gauge theory for LATKes?

The answer is "yes," and we do so in the context of particle physics.

\subsection{LATKe representations and LATKe groups}

\vskip .3cm
\no \begin{underline}{\texttt  {Representations}}\end{underline} Whenever Lie algebras are applied to particle physics, particles are viewed as basis vectors of representations of the Lie algebra. For example,  gauge fields form the adjoint representation of the gauge group; quarks form the three-dimensional representation of color-$SU(3)$; electrons form the two-dimensional spin representation of $SU(2)$ \cite{Georgi:1982jb}. Therefore, in order to apply LATKes to particle physics,  we must define a "representation" for LATKes.

To do so, we review the standard Lie algebra case. A representation of a Lie algebra $\g$ is a map from $\g$ to operators on some vector space $V$:
\be \rho:\g \longrightarrow \mbox{End}(V) ~,\ee
 and it respects the commutator in the following way:
\be \label{repcond2}[\rho (X), \rho (Y)]=\rho ([X, Y]) ~.\ee
A particular representation that utilizes the commutator in a natural way is
the adjoint representation, given by the following map:
\bea \mbox{ad}&:&\g \longrightarrow \mbox{End}(\g) \\
\mbox{ad}(X)&:&Y\longmapsto [X,Y] ~.
\eea
This map satisfies the condition
\be \label{repsrel}[\mbox{ad}(X), \mbox{ad}(Y)] =\mbox{ad}([X,Y])\; , \ee
which is equivalent to the  standard Jacobi identity. The condition in equation (\ref{repcond2}) is a generalization of the relation given in equation (\ref{repsrel}) and it reduces to it when $\rho = $ad.

Now, we define a representation for a LATKe. We begin by defining the analog of the adjoint representation: it is also a map that utilizes the commutator in a natural way, and we have in fact seen it before (equation (\ref{adaction})):
\bea \label{adL}\mbox{ad}&:&\L\wedge \L\longrightarrow \mbox{End}(\L)\\
\mbox{ad}(X\wedge Y) &:&Z\longmapsto [X,Y,Z]~.
\eea
The map ad satisfies  the condition
\be \label{3reps}[\mbox{ad} (X_1 \wedge X_2),\mbox{ad}( X_3 \wedge X_4)]=\mbox{ad}([X_1, X_2, X_3]\wedge X_4) +\mbox{ad}( X_3\wedge [X_1, X_2, X_4])~,
\ee
which is equivalent to the  LATKe Jacobi identity.

If we generalize equations (\ref{adL})  and (\ref{3reps}), we have
\begin{defin} A {\textit {\textbf {representation}}} of a LATKe $\L$ is a map
\be \label{latkerep} \rho: \Lambda ^2 \L\longrightarrow \mbox{End}(V) \ee
for some vector space $V$ subject to the condition
\be  [\rho (X_1 \wedge X_2),\rho( X_3 \wedge X_4)]=\rho([X_1, X_2, X_3]\wedge X_4) +\rho( X_3\wedge [X_1, X_2, X_4])~.
\ee
\end{defin}
This condition is analogous to equation (\ref{repcond2}) and it generalizes the  LATKe Jacobi identity (see equations (\ref{JII}) and (\ref{gLcomm})).
\vskip .3cm
\no \begin{underline}{\tt {Groups}}\end{underline}  Another fundamental ingredient whenever Lie algebras are applied to particle physics is the Lie group, which plays the role of a symmetry of the physical system.

Therefore, in order to apply LATKes to particle physics, we also should define a "Lie group of the third kind," or a  "LATKe group," which would be related to the LATKe in a way analogous to the relation between an ordinary Lie group and its Lie algebra. The LATKe group could then play the role of some kind of generalized symmetry in the yet-to-be constructed physics of LATKes.

Here, we run into trouble: we have found it impossible to generalize the notion of a Lie group to something we might have called a  LATKe group. While there does happen to be a Lie group associated with the LATKe, namely the exponential of $\g _\L$, it is not in any way an exponential of the LATKe itself. So it is not a "LATKe group."

Since we have no LATKe analog of a Lie group, it would be impossible to generalize in a natural way any application of Lie algebras to physics in which the Lie group is an indispensable ingredient. We are limited to applications in which the only necessary mathematical ingredients are those for which we do have a LATKe analog.

Since we wish to generalize gauge theory for LATKes, we must check whether
the Lie group itself, which plays the role of the gauge group, is an indispensable ingredient in the construction of  gauge theory. If it is, we would be unable to generalize it for LATKes.
 In the next section we demonstrate that the gauge group {\sl is not} indispensable in gauge theory by rephrasing the original theory of Yang and Mills \cite{Yang:1954ek} so that all group tranformations are re-written as  Lie algebra actions. That sets the stage for a natural generalization of Yang-Mills theory to LATKes, which we construct in Section \ref{ttYM}.

\subsection{Traditional Yang-Mills theory}\label{YMsans}

Let $\psi$ be a wave function describing a field in some representation $\rho$ of a Lie algebra $\g$. Let $\g$ be spanned by basis elements $T^i$ with $i=1, \ldots , \dim \g$ (these are analogues of Pauli spin matrices in the $\mathfrak{su}(2)$ case). A gauge transformation acts via
\be \label{deltapsi}\delta \psi = -2i\Theta ^iT^i\psi \; ,\ee
where each $\Theta ^i$, $i=1, \ldots , \dim \g$ is a space-time dependent field, and each $T^i$ acts on $\psi$ via the representation $\rho$.

Invariance under such transformations is preserved only if we require derivatives of $\psi$ to appear in the combination
\be \label{covder} (\partial _\mu -ig B_\mu)\psi \; ,\ee
where $g$ is a coupling constant and
\be B_\mu = 2b_\mu^j T^j \ee
with $b_\mu ^j$  space-time dependent. The combination in equation (\ref{covder}) is the well-known covariant derivative, and $B_\mu$ is the well-known gauge field. Under the gauge transformation, $B_\mu$ transforms via
\be \delta B_\mu = 2i[B_\mu , \Theta ^i T^i]-\frac{2}{g}(\partial _\mu \Theta ^i )T^i \; .\ee
With these transformations, we have
 \be \label{inv3}\delta [(\partial _\mu -ig B_\mu)\psi] =-2i\Theta ^i T^i ((\partial _\mu -ig B_\mu)\psi) \; ,\ee
 as would be expected from equation (\ref{deltapsi}).

 We  define the field strength $F_{\mu \nu}$ by
 \bea F_{\mu \nu}&=&\partial _\nu B_\mu - \partial _\mu B_\nu + ig [B_\mu , B_\nu] \non \\
 &=&2f_{\mu \nu}^iT^i \; ,
 \eea
 where $f_{\mu \nu}^i$ are space-time dependent and  the commutator $[B_\mu , B_\nu]$ is the one defining the Lie algebra $\g$. Under the gauge transformation, $F_{\mu \nu}$ transforms by
 \be \delta F_{\mu \nu}=2i[F_{\mu \nu}, \Theta ^iT^i] \; .\ee
 Now we have all the necessary ingredients to write the Lagrangian:
 \be \label{YMlang}L=-\frac{1}{4}f_{\mu \nu}f_{\mu \nu}-\bar \psi \gamma _\mu (\partial _\mu -2ig b_\mu^iT^i)\psi -m\psi \bar \psi \; .\ee
 The Lagrangian is invariant under
 gauge transformations. From this Lagrangian, the equations of motion of the gauge theory are derived.

\newcommand{\eab}{e_a\wedge e_b}

 \subsection{LATKe Yang-Mills, or pure Yang-Mills with matter}\label{ttYM}

 We can now generalize  gauge theory by replacing Lie algebras with LATKes and replacing representations of Lie algebras with representations of LATKes.

That means  we now let $\psi$ be a field in a representation $\rho$ of a LATKe as defined in equation (\ref{latkerep}), and we let $\Theta _{ab} (\eab)\in \Lambda ^2\L$ act on $\psi$ via that representation  in lieu of the action of $\Theta ^iT^i$  of the standard case of Section \ref{YMsans}. Here, $\Theta _{ab}$, $a,b=1, \ldots , \dim \L $ are space-time dependent fields antisymmetric in $a$ and $b$.
We replace every occurrence of $\Theta ^iT^i$  in Section \ref{YMsans} by $\Theta _{ab} (\eab)$, and every index $\{ i\}$  by an antisymmetric double-index $\{ ab \}$.

If we now inspect  the resulting equations -- which are the equations of LATKe Yang-Mills theory --  we find that the LATKe $\L$ appears  only through $\Lambda ^2 \L$. This is so because we were using representations of the LATKe in the construction, and those involve  $\Lambda ^2 \L$ rather than $\L$ (see Definition 5.1).

We may now recall that $\Lambda ^2\L$ is in fact the Lie algebra $\g _\L$ (Definition \ref{gL}.3). And, we also observe that  the way in which $\Lambda ^2\L$ appears in our LATKe Yang-Mills theory is  precisely the same as the way ordinary Lie algebras appear in traditional Yang-Mills theory, i.e. exactly as in Section \ref{YMsans}.
It turns out that the definition of representations of a LATKe has conspired with the structure of the Lie algebra $\g _\L$ of the LATKe to turn LATKe Yang-Mills theory into a conventional Yang-Mills theory with Lie algebra $\g _\L$!
And, now it is inevitable that we would think of LATKe Yang-Mills theory as a conventional Yang-Mills theory with gauge group $\exp (\g _\L)$.

Yet, there is an essential and crucial difference between conventional Yang-Mills and LATKe Yang-Mills:
in conventional Yang Mills theory, we have what is known as "pure Yang-Mills theory," where the gauge fields $B_\mu$, which live in the adjoint representation of the gauge group, are the only fields. There are no matter fields -- that is, no field $\psi$ appears -- and the Lagrangian consists only of the first term of equation (\ref{YMlang}). In general, for physical theories to include matter fields they typically have to be put in by hand.

But in the LATKe Yang-Mills theory, this is not the case. Built into the theory is not just the adjoint representation $\Lambda ^2\L$ of $\g_\L$, but also the adjoint representation of the LATKe itself, i.e. $\L$. This representation is in fact a matter representation of $\g _\L$ and an inseparable part of {\sl pure} LATKe  Yang-Mills theory.

Therefore, unlike pure Yang-Mills theory, pure {\sl {LATKe}} Yang-Mills theory {\sl {automatically includes matter}}, without the need to put it in by hand.

\subsection{Gauge theory for $\L_3$}
For  $\L=\L_3$, we have $\g_\L =\mathfrak{so}_4\simeq \mathfrak{sl}_2\times \mathfrak{sl}_2$  and $\L$ forms the $(2,2)$ representation (see Theorem \ref{bigtheorem}). The unitary version of  $\exp (\g _\L)$ is   $SO(4)$ or $SU(2)\times SU(2)$. The pure LATKe Yang-Mills theory for $\L _3$ is therefore an $SO(4)$ or $SU(2)\times SU(2)$ gauge theory with matter in the $(2,2)$ representation.

\subsection{LATKe Yang-Mills theory from $G_2$ manifolds} \label{YMG2}

Here we show that $\cz$, which is the singularity corresponding to the LATKe $\L _3$, indeed arises in a $G_2$ space, as we encountered in \cite{Friedmann:2002ct}.

Let $X$ be the manifold of $G_2$ holonomy which is asymptotic at infinity to a cone over $Y=\S^3\times \S ^3=SU(2)^3/SU(2)_\Delta$, where the equivalence relation $\Delta$ is $(g_1, g_2, g_3)\sim (g_1h, g_2h, g_3h)$, $g_i, h\in SU(2)$ \cite{bryant, Gibbons:1989er,Atiyah:2001qf,Friedmann:2002ct}.

Let $\gamma \in \Z_3$ act on $Y$ as follows:
\be \gamma : (g_1, g_2, g_3) \longmapsto (\gamma g_1, \gamma g_2, g_3)\;\;  ;\; \; \;  \gamma = \left ( \begin{array} [h] {cc} e^{2\pi i/3}&0 \\ 0&e^{-2\pi i/3} \end{array}\right ) .\;
\ee
The metric of $X$ is preserved under this action \cite{Atiyah:2001qf,Friedmann:2002ct}. Using the equivalence relation $\Delta$ to set $g_1=1$, we rewrite this action as
\be (1, g_2, g_3) \longmapsto (1, \gamma g_2\gamma ^{-1}, g_3\gamma ^{-1}) \; .
\ee
We can obtain $X$ from $Y$ by filling in one of the $SU(2)$ factors to a ball that includes the origin (recall that $SU(2)\sim \S^3$). Let us fill in the third $SU(2)$ factor, and study the singularity at the origin.

We may write
 \be g_2=\left ( \begin{array} [h] {cc} z_1&z_2 \\ -\bar z _2&\bar z _1 \end{array}\right )  , \; \; |z_1^2|+|z_2^2|=1\; ; \; \; \; \; \;
         g_3=\left ( \begin{array} [h] {cc} w_1& -\bar w _2\\ w _2&\bar w _1 \end{array}\right ) \; ,
 \ee
where $z_i$ and $w_i$ are complex variables.  Then the action of $\gamma$ becomes
 \be \gamma : (z_1, z_2, w_1, w_2)\longmapsto (z_1,  e^{4\pi i/3}z_2, e^{4\pi i/3}w_1, e^{4\pi i/3}w_2)\; . \ee
 This action is singular at $z_2=w_1=w_2=0$. The locus of the singularity is the circle $z_1=e^{i\theta}$.

 The singularity itself is equivalent to the one which we used in Section \ref{L3} to construct $\L_3$, with the $\C ^3$ given by coordinates $\{ z_2, w_1, w_2\}$ and $\epsilon = e^{4\pi i/3}$ (see equation (\ref{epsilonaction})).

 Therefore, we argue that the theory obtained from M-theory compactified on a $G_2$ space with a circle of $\cz$ singularities is governed by the LATKe Yang-Mills theory we constructed in Section \ref{ttYM}, i.e. it is an $SU(2)\times SU(2)$ or $SO(4)$ gauge theory on $M^4\times \S^1$ with one matter field in the $(2,2)$ representation of $SU(2)\times SU(2)$, which is the vector representation of $SO(4)$. Since this compactification is supersymmetric (it has $G_2$ holonomy), the LATKe Yang-Mills is also supersymmetric (${\cal N} = 1$).

 Similarly, we argue that the same physics would result if a $\cz$ singularity appears in a Calabi-Yau space on which a string theory is compactified.

\subsection{Unbreakability of $\g _\L$}\label{unbroken}

We have shown that $\L_3$ leads to an $\su \times \su$ gauge theory with matter. A physicist conditioned to search for the standard model is immediately led to the following question: can we break $\su \times \su$  to $\su \times \mathfrak{u}(1)$, which is the gauge symmetry of electroweak theory and part of the standard model?

Recall that the commutators of $\L_3$ were determined from the map  $\phi : \Lambda ^2 V \rightarrow \g$, defined in equation (\ref{phi4}), which was required to satisfy the antisymmetry condition in equation (\ref{antisym123}) or (\ref{antisym}). Now we show that if $\su \times \su$ is broken to $\su \times \mathfrak{u}(1)$, {\it none} of the commutators of $\L_3$ are well-defined; in other words, they disappear.

Let $\{ u_1, \ldots , u_6 \}$ be a basis for $\mathfrak{so}(4)$ for which the $\su \times \su$ structure is explicit:
\bea u_1=e_{12}+e_{34}&&u_2=-e_{13}+e_{24} \hskip .8cm u_3=e_{14}+e_{23} \\
u_4=e_{12}-e_{34}&&u_5=e_{13}+e_{24} \hskip 1cm u_6=e_{14}-e_{23}
\eea
Here, $\{u_1, u_2, u_3\}$ span one $\su$ factor, and $\{u_4, u_5, u_6\}$ span the other $\su$ factor. In this basis, the map $\phi$ of equation (\ref{phi4}) becomes
\bea \phi (e_1\wedge e_2)&=&(u_1-u_4)/2 \non \\
\phi (e_1\wedge e_3)&=&-(u_2+u_5)/2 \non \\
\phi (e_1\wedge e_4)&=&(u_3-u_6)/2 \non \\
\phi (e_2\wedge e_3)&=&(u_3+u_6)/2 \label{uphi}\\
\phi (e_2\wedge e_4)&=&(u_2-u_5)/2 \non \\
\phi (e_3\wedge e_4)&=&(u_1+u_4)/2 \non
 \eea

Now let $u_4=u_5=0$ so that we are left with $\{u_1, u_2, u_3, u_6\}$ which forms a basis for $\su \times \mathfrak{u}(1)$. Then a fundamental requirement for the LATKe commutator is violated:
\be  (\phi (v_i\wedge v_j))\cdot v_k\neq -(\phi (v_k\wedge v_j))\cdot v_i \hskip 1cm \forall  v_i, v_j, v_k\in V \, .\ee
For example,
\be \phi (e_1\wedge e_2)\cdot e_3=\frac{u_1}{2} \cdot e_3=\frac{(e_{12}+e_{34})}{2} \cdot e_3=\frac{e_4}{2} \ee
while
\be \phi (e_3\wedge e_2)\cdot e_1=-\frac{(u_3+u_6)}{2} \cdot e_3=-e_{14} \cdot e_1=-e_4 \, ,\ee
so $\phi (e_1\wedge e_2)\cdot e_3\neq -\phi (e_3\wedge e_2)\cdot e_1$. The same can be checked for other combinations of $e_i$.

So there is no longer a well-defined LATKe commutator and not even a sub-LATKe remains.

Another way of stating this result is that $\g_\L$ is unbreakable as long as $\L$ is present; or, that $\L$ protects $\g _\L$ from being broken.
This unbreakability of $\g_{\L}$ may remind one of some global symmetries which may not be broken under certain conditions \cite{Vafa:1983tf}.
 One may be tempted to interpret the LATKe to be a manifestation of these conditions.

 \section{Vacuum Selection Mechanism}\label{sum}

 \renewcommand{\baselinestretch}{1.3}
\normalsize

There was great excitement in the physics community in the 1980's when it was discovered, through a study of anomaly cancellation, that string theory came along with gauge groups -- either $E_8\times E_8$ or $SO(32)$; this discovery allowed for the hope that string theory might have some applications to phenomenology, which is governed by gauge theories \cite{Green:1984sg,Gross:1984dd}.

In the decades that followed,
a great number of attempts at engineering a Calabi-Yau or $G_2$ space were carried out with the purpose of obtaining  a  theory in four dimensions that is as close as possible to the standard model.
As it happened, orbifolds were employed in  Calabi-Yau compactifications of heterotic string theory for this purpose, since they induced gauge symmetry breaking by Wilson lines \cite{Hosotani:1983xw,Hosotani:1983vn,Candelas:1985en}, making the gauge group  closer to the standard model group. They also reduced the number of fermion generations that arise from the compactification, bringing that number closer to the phenomenological value of three.

Since then, it has been realized \cite{Douglas:2003um} that there is a staggering number of Calabi-Yau or $G_2$ spaces, making up what is called today the  "string landscape". Therefore, the idea of  a "vacuum selection mechanism," which is some principle  that  would  single out one vacuum or at least narrow down the choices considerably, has been sought after.

The uniqueness of the LATKe is  a vacuum selection mechanism. The selected compactification space is a Calabi-Yau or $G_2$ space with a $\cz$ singularity, and the selected vacuum theory is a supersymmetric $\su \times \su $ gauge theory with matter in the $(2,2)$ representation.

While it has been accepted  that no vacuum selection mechanisms have as yet been proposed \cite{Douglas:2003um}, in retrospect we claim that before the present work, there did exist a  vacuum selection mechanism: anomaly cancellation. It selected a string theory with gauge group either $E_8\times E_8$ or $SO(32)$.

While neither the uniqueness of the LATKe nor anomaly cancellation actually selects the standard model itself, our unique, simple LATKe Yang-Mills is tantalizingly close to the standard model.

 \renewcommand{\baselinestretch}{1.4}
\normalsize

 \vskip 1cm
\no \begin{underline}{\texttt  {Acknowledgements}} \end{underline}
I am deeply indebted to I. M. Singer for inspiration and advice during the various stages of this work and for reading and commenting on an earlier draft. I am deeply indebted to F. Wilczek for discussions,  inspiration, nurturing, and mentoring.
I wish to thank R. Jackiw, P. Deligne, V. Kac, M. Goresky, D. Kazhdan, R. MacPherson, D. Freedman,  J. Harris, M. Artin, P. Kronheimer, X. de la Ossa, J. de Jong, R. L. Jaffe, B. Zwiebach, S. Steadman, E. Farhi, W. Taylor, and C. Vafa for helpful discussions. I am very grateful to Y.P. Lee for helpful discussions and for reading and commenting on a draft.
Last, but not least, I am grateful to A. Wiles for hospitality and encouragement. This work was supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement DE-FC02-94ER40818.
\vskip 1cm
\appendix

\section{Blow-up of $\CZ$}  \label{blow}

Let $\Z_n$ be the multiplicative group generated by $\epsilon _n I_n$, where $\epsilon _n=e^{2\pi i/n}$ and $I_n$ is the $n\times n$ identity matrix. Let $\epsilon _n\in \Z_n$ act on $(z_1, \ldots ,z_n)\in \C^n$  by
\be \label{Xn} (z_1, \ldots ,z_n) \to (\epsilon _n z_1, \ldots ,\epsilon _n z_n).\ee
Denote the equivalence classes in the quotient  $X_n=\C^n/\Z_n$ by $[z_1, \ldots ,z_n]$.

The origin of $\C^n$ is fixed under this action. The resolution of the singularity at the origin is given as follows.

Let $Y_n=(\C^{n+1}-\{0\})/\C^*$, where $\lambda \in \C^*$ acts via
\be \label{lambda} (w_1,\ldots , w_n,w_{n+1})\to (\lambda w_1,\ldots , \lambda w_n,\lambda ^{-n}w_{n+1}) .\ee
Denote equivalence classes in $Y_n$ by $[w_1,\ldots , w_n, w_{n+1}]$. Then $\pi:Y_n\rightarrow X_n$ is given by
\bea \label{1to1} \pi([w_1,\ldots , w_n,1])&=&[w_1, \ldots , w_n] \\
\label{exceptional}\pi([w_1,\ldots , w_n,0)&=& [0,\ldots , 0]
\eea
Equation (\ref{1to1}) is one-to-one: the equivalence class $[w_1, \ldots , w_n,1]\in Y_n$ is determined by setting $w_{n+1}=1=\lambda ^{-n}w_{n+1}$ so now the $\lambda$ appearing in equation (\ref{lambda}) is any $n^{th}$ root of unity, leading to the same quotient action as the one defining  $X_n$ in equation (\ref{Xn}).

Equation (\ref{exceptional}) provides us with the exceptional divisor: $\pi ^{-1}([0,\ldots ,0])$ is the set $[w_1, \ldots ,w_n,0]$, which is just $\P^{n-1}$ given the action in  equation (\ref{lambda}).

\section{The  LAnKe $\L _n$}\label{lanke}

Let $V_{n+1}$ be the standard $(n+1)$-dimensional vector representation of $\mathfrak{so}(n+1)$, and let $e_{ij}$ and $e_i$ be defined as in equations (\ref{eab}) and (\ref{ea}). Generalizing equation (\ref{phi4}), we define a map $\phi _n: \Lambda ^{n-1}V_{n+1}\rightarrow \mathfrak{so}(n+1)$ as follows:
\be \label{Ln} \phi _n (e_1\wedge \cdots \wedge \hat e_i \wedge \cdots \wedge \hat e_j \wedge \cdots \wedge e_{n+1})=(-1)^{i+j+1}e_{ij}\; , \ee
where a hat over $e_i$ means that it is omitted.
This map yields a commutator of the $n$-th kind:
\bea [e_1 \wedge \cdots \wedge \hat e_i \wedge \cdots \wedge e_n]&=&\phi _n (e_1\wedge \cdots \wedge \hat e_i \wedge \cdots \wedge e_{n-1})\cdot e_n\nonumber \\&=&(-1)^{i+n+1}e_{in}\cdot e_n=(-1)^{i+n+1}e_i~, \hskip .1cm i<n
\eea
\bea
[e_1 \wedge \cdots \wedge  e_{n-1}]&=&\phi _n (e_1\wedge \cdots  \wedge e_{n-2})\cdot e_{n-1} \nonumber \\ &=& (-1)^{(n-1)+n+1}e_{(n-1)n}\cdot e_n=-e_n ~.
\eea
It satisfies the requirements for a  Lie algebra of the $n$-th kind (Definition 5.2).

After a change of variables, one can show that for $\L _n$, there is a Cartan subalgebra $\mathfrak{h}_{\L_n}$ of dimension $n-1$ so $\Lambda ^{n-1}\mathfrak{h}_{\L_n}$ is one dimensional and there is a one-dimensional root space, where a root is in the dual space of $\Lambda ^{n-1}\mathfrak{h}_{\L_n}$:
\be \alpha : \Lambda ^{n-1}\mathfrak{h}_{\L_n}\longrightarrow \C \; . \ee
The  Dynkin diagram  of $\L_n$ has one node corresponding to the single cycle $\P^{n-1}$ in the exceptional divisor of the  singularity $\CZ$.

All the mathematical definitions related to LATKes  generalize quite naturally to  LAnKes. In addition, LATKe Yang-Mills easily generalizes to LAnKe Yang-Mills, and for $\L _n$, LAnKe Yang-Mills theory is an $\mathfrak{so}(n+1)$ gauge theory with matter in the $(n+1)$-dimensional vector representation. It would be interesting to see whether the uniqueness of simple LATKes generalizes to LAnKes as well.

config: arxiv


text:
\begin{document}

\flushbottom
\maketitle

\thispagestyle{empty}

\section*{Introduction}
Epidemic modeling and forecasting has gained renewed interest since late 2019 when the world was affected by the novel coronavirus pandemic (named Covid-19). Several computational studies to predict the human-to-human spread of Covid-19 have been reported~ \cite{singh2020age,harsha2020covid,pandey2020seir,ranjan2020predictions}. Most of these efforts have been based on compartmental models and stochastic models (including agent-based models)  \cite{vynnycky2010introduction}. In compartmental models (e.g., SIR, SEIR, SEIRS, DELPHI~ \cite{vynnycky2010introduction,mitdelphi}), the population is divided into different compartments and the dynamics of the different compartments are modeled by a system of coupled ordinary differential equations (ODE). Here, the interaction among compartments is usually deterministic, whereas random processes are used to model the spread of infections in stochastic models. Agent-based models are stochastic models that undertake a bottom-up approach of modeling individual members of a population and the dynamics of their interaction in terms of probabilities of movement and contact.

More than the total number of infections, it is essential to have more insightful predictions, e.g., infected population distribution across their age and level of infection severity for science-informed policy intervention and public health planning. The population distribution over the duration of infection is crucial for planing antiviral treatments, quarantine, ventilator support and contact tracing. This requirement necessitates a comprehensive and computationally efficient predictive modeling framework. Even though these features could be incorporated in ODE-based compartmental and stochastic agent-based models, it is very complex and computationally expensive. To overcome these challenges, we propose a novel partial differential equation-based spatio-temporal predictive modeling framework for forecasting the spread of infectious disease in heterogeneous populations in open geographies. The roots for our model lie in the population balance equations that are popular in chemical engineering and process studies  \cite{ramkrishna2002population}.

In the proposed model, the infected population density is defined as a scalar field on a high-dimensional space. Specifically for predicting the spread of Covid-19, a six-dimensional model is presented. The first three dimensions are the space and time, and the other three are the infection severity, duration of the infection (i.e., time since infection), and age of the population. New infections, impact of quarantine, testing, contact tracing, immunity, intervention policy impact, health infrastructure, recovery, and death are all modeled on this six-dimensional space based on data-driven functions (where available), and/or simple algebraic and integral functions.
Notably, our PDE-based model in the present paper is more compact and a versatile description of the spread of the disease compared to compartmental models, and computationally efficient compared to agent-based models. To showcase the capability of our distribution-based predictive modeling framework for infectious disease spread, we apply it to model and predict the spread of Covid-19 in India. Further, we present a scenario analysis, which could be used to draw insights for policy interventions.

\section*{Results}

\subsection*{The Population Balance Model}
Let $T_\infty$ be a given final time and $\Omega:=\Omega_x\otimes\Omega_\ell$ be the computational domain of interest. Here, $\Omega_x\subset\mathbb{R}^2$ is the spatial domain defining the geographical region of interest and $\Omega_\ell\subset\mathbb{R}^n$, where $n$ is the number of internal directions. Each of the $n$-internal directions represents the property of the population on which a distribution needs to be predicted. Suppose the properties of interest are the infection severity, duration of the infection and age of the population, then a model with three internal directions could be used as follows. Let  $\Omega_\ell:=L_v\times L_d\times L_a$ be the internal domain, where $L_v=[0, 1]$ denotes the infection severity interval, $L_d=[0,d_{\infty}]$ denotes the duration of infection, $d_{\infty}$ is the maximum duration of infection,  $L_a=[0, a_{\infty}]$ denotes the age interval and $a_{\infty}$ is the maximum age of the population. The infection index $\ell_v\in L_v$ quantifies the severity of the infection among the infected population. Specifically, the population with infection index $\ell_v=0$ is completely disease-free, with $\ell_v = 1$ has maximum severity, with $\ell_v\ge v_{\textrm {sym}}$ shows symptoms and those with $\ell_v<v_{\textrm {sym}}$ are asymptomatic. The duration of infection index $\ell_d\in L_d$ quantifies the time since a population has been exposed to and contracted the disease. Specifically, the population that just contracted the disease has $\ell_d=0$. Typically, a person is asymptomatic until they reach $\ell_v=v_{\textrm {sym}}$, and the duration elapsed $\ell_d$ is the incubation period in which the disease is sub-clinical and that population is actively spreading the disease. After recovery, a population doesn't necessarily go to $\ell_v=0$, rather they reach $\ell_v<v_{\textrm reco}$.

Let $I(t,\mbx,\mbl)$, where $t\in (0,T_\infty],~ \mbx \in \Omega_x$ and $\mbl \in \Omega_\ell$, be the infected number density function of the population.
To describe the evolution of the active infected population size distribution, we propose the population balance equation  in the time interval $(0,T_\infty]$
\begin{equation}\label{model}
 \displaystyle\frac{\partial I}{\partial t} + \nabla\cdot(\textbf{u} I) +\nabla_\ell\cdot(\textbf{G} I) + CI = F  \quad {\textrm  {in} }  \quad \Omega_x\times\Omega_\ell\,,
\end{equation}
with initial conditions
\begin{equation}\label{model2}
 \begin{array}{rcll}
 I(t,\mbx,\mbl) &=&g_n    &{\textrm {in} }  \quad  \partial\Omega^{-}_{ x}\times \Omega_\ell\,,\\
 I(t,\mbx,(\ell_v,0,\ell_a))  & =& B_{\textrm nuc} \quad  &{\textrm {in} }  \quad  \Omega_{x}\times L_v\times L_a \,,\\
 I(t,\mbx,(0,\ell_d>0,\ell_a))  & =& 0 \quad  &{\textrm {in} }  \quad  \Omega_{x}\times L_d\times L_a \,, \\
 I(0,\mbx,\mbl) & =& I_0 &{\textrm {in} }  \quad  \Omega_{x}\times\Omega_\ell \,.
\end{array}
\end{equation}
Here, $\bu$ denotes the advection vector that quantifies the multiscale spatial movement of the population in a differential neighbourhood of $\Omega_{x}$ (e.g., migrant laborers, daily commute for work, logistics-related travel, periodic gathering for religious and social events), $\bn$  is the outward unit normal vector to $\Omega_x$, $\partial{\Omega}^{-}:= \{\mbx \in \partial{\Omega_{x}} \  |~ \bu\cdot \bn <0 \  \}$,
$g_n$ is the flux that quantifies the net addition of the infected population into $\Omega_{\textrm x}$ from outside (the spatial movement of the population across the border of the domain $\partial\Omega_{\textrm x}$), and $I_0$ is the initial distribution of the infected population.
Further,
$
\bG = (G_{\ell_v}, G_{\ell_d}, G_{\ell_a})^T
$
is the internal growth vector, where
\begin{gather}
  G_{\ell_v} = \frac{d \ell_v}{d t} = G_{\ell_v}(\ell_a, \beta, \gamma(\ell_a), \alpha(\mbx) ),  \quad
  G_{\ell_d} = \frac{d \ell_d}{d t} = 1, \quad
  G_{\ell_a} = \frac{d \ell_a}{d t} = 1.
\end{gather}
Here, $\beta$ is the immunity of the infected population, $\gamma$ is the pre-medical history of the infected population and $\alpha$ is the effective treatment index.  Next, we define the rate term
$
 C =  C_R + C_{ID},
$
where $C_R(t,\mbx,\mbl)$ is a recovery rate function that quantifies the rate of recovery of the population from the infection,
and $C_{ID}(t,\mbx,\mbl)$ is the infectious death rate.
We also define a source term $F = C_T(t,\mbx,\mbl)$ that quantifies the point-to-point movement of infected population (e.g., by air, train etc) within $\Omega_x$, which are not included in $\bu$ and $g_n$.
Moreover,  $C_T$ and $\bu$ need to be defined in such a way that the net internal movement of infected population within $\Omega_{x}$ is conserved.
Moreover, $B_{\textrm nuc}$ is the nucleation function that quantifies the infection transmission from the infected to the susceptible population and it is a function of several parameters as follows

\begin{equation}
  B_{\textrm nuc} = B_{\textrm nuc}\left(t, \textbf{X}, \sigma, H, S_D, N_S, N_Q, I\right).
\end{equation}
Here, $\textbf{X}\in\Omega$, $\sigma, H$ and $S_D$ are the interactivity, hygiene and social distancing indices respectively.
Finally, the total population $N(t)$ at a given time $t\in(0,T_\infty]$ is defined by
\begin{align*}
  N(t) &= N_S(t) + N_R(t) + N_I(t)  + N_Q(t) - N_{ID}(t) + N_B(t) - N_D(t),  \quad
  N_Q(t)  = \int_{\Omega}  \gamma_Q(t,\mbx,\mbl)I(t,\mbx,\mbl) \,dX,~\\
 N_I(t)  &= \int_{\Omega} I(t,\mbx,\mbl)\,dX, \quad
 N_{R}(t)   =  \int_{\Omega} C_RI(t,\mbx,\mbl)\,dX,\quad
 N_{ID}(t)   =  \int_{\Omega} C_{ID}I(t,\mbx,\mbl)\,dX.
\end{align*}
Here, $N_S$, $N_B$, $N_R$, $N_I$, $N_Q$ $N_{ID}$ and $N_D$ are the number of susceptible, newborn, recovered, infected (symptomatic/asymptomatic), quarantined, infectious death and natural death populations, respectively. The given initial and boundary conditions and the above defined parameters close the population balance system.

\subsection*{Modeling of Parameters}
The proposed population balance model~\eqref{model} is comprehensive and built on the basis of several parameters as defined above. In this section, we describe the modeling of each parameter.

\subsubsection*{Nucleation}
The nucleation term $B_{\textrm nuc}$ quantifies the new infection number density that is added to the system at $\ell_d=0$ for all $t$, $\mbx$, $\ell_v$, and $\ell_a$.
Depending on how the susceptible population interacts with the infected population, new infections are added to the system. We call this addition as \textit{nucleation} (borrowing the terminology from process engineering), which is modelled as
\begin{align}
	B_{\textrm nuc} &=  R\int_{\Omega_\ell}[1-\gamma_Q]I(t,\mbx,\mbl)\,d\ell, \label{eq:Bnuc_eq2}\\
	R &= R_0  f_1(t,\sigma)f_2(t,H)f_3(t,S_D)f_4(t,\mbx,\ell_a)f_5(\ell_v)\,,\label{eq:Bnuc_eq3}\\
	\gamma_Q &= \frac{1}{1+\exp\left(-(\ell_v-v_{\textrm {sym}})/b_v\right)}
	  \frac{1}{1+\exp\left(-(\ell_d-d_{\textrm {sym}})/b_d\right)}
	  \frac{1}{1+\exp\left(-(\ell_a-a_{\textrm risk})/b_a\right)}. \label{eq:Bnuc_eq4}
\end{align}
Here, $\gamma_Q\in[0,1]$ in~\eqref{eq:Bnuc_eq2} determines the fraction of the infected population in quarantine and it can be modeled as in equation~\eqref{eq:Bnuc_eq4}. Further,  the factor $\gamma_Q$ is dependent on the level of screening including testing, strictness of enforcing isolation and compliance of susceptible general public. Suppose $\gamma_Q = 1$, i.e., if the entire infected population is kept under strict isolation, newly infected population will be zero and eventually there will be no spread of disease. However, due to economic, social and democratic reasons, implementing such a strategy is nearly impossible and there is bound to be spread, i.e., $\gamma_Q<1$.
Moreover, the integral on the right-hand side of equation~\eqref{eq:Bnuc_eq2} is the total non-quarantined number density of the infected population at $(t,\mbx)$, and

$R$ is the rate at which the non-quarantined population infects the susceptible population.
The factor $R$ is modelled as in equation~\eqref{eq:Bnuc_eq3}, where $R_0$ is the basic reproduction rate,
\begin{align*}
    f_1(t,\sigma) &= \left[\frac{1}{1+\exp\left(-(\sigma(t)-\sigma_c)/b_\sigma\right)}\right], \quad
    f_2(t,H)  =  \left[1-\frac{1}{1+\exp\left(-(H(t)-H_c)/b_H\right)}\right], \\
    f_3(t,S_D) &=  \left[1-\frac{1}{1+\exp\left(-(S_D(t)-S_{D_c})/b_{S_D}\right)}\right],\quad
    f_4(t,\mbx,\ell_a)  =  a_4\exp\left(-\frac{(\ell_a-b_4)^2}{c_4^2}\right), \\
    f_5(\ell_v) &=  \left\{ \begin{array}{cc}
       3\sqrt{\frac{2}{\pi}}\exp\left({\frac{-(\ell_v-v_{\textrm {sym}})^2}{2(v_{\textrm {sym}}/3)^2}}\right) & 0\leq v<v_{\textrm {sym}} \,\\
       3\sqrt{\frac{2}{\pi}}\exp\left({\frac{-(\ell_v-v_{\textrm {sym}})^2}{2((1-v_{\textrm {sym}})/3)^2}}\right)  & v_{\textrm {sym}}\leq v \leq 1.
    \end{array}   \right.
\end{align*}

\noindent Here, the interactivity index $\sigma\in[0,1]$, hygiene index $H\in[0,1]$, and social distancing index $S_D\in[0,1]$. Suppose $\sigma=0$ then everything is under perfect lockdown and $R \rightarrow 0$. In case $S_D=1$, everyone is following perfect social distancing and $R \rightarrow 0$. Moreover, the newly infected population has to be added at different age ($\ell_a$) and  infection $(\ell_v)$ levels for which the factors $f_4$ and $f_5$ are introduced.
We propose to use logistic functions fitted to data from literature for $f_1,~f_2,~f_3$,; the normalized demography at $(t,\mbx)$ for $f_4(t,\mbx,\ell_a)$, and a Gaussian mixture with two components so that maxima is at $v_{\textrm {sym}}$ and tails are proportional to the interval length over $[0,v_{\textrm {sym}}]$ and $[v_{\textrm {sym}},1]$ for $f_5(\ell_v)$.
In addition, the constant in $f_5$ is chosen such that the integral of $f_5$ over its support is one. This condition is imposed to ensure that $R_0$ can be interpreted as the basic reproduction rate used in standard epidemiological models  \cite{grassly2008mathematical}.
The parameters $v_{\textrm {sym}}$, $d_{\textrm {sym}}$, $a_{\textrm risk}$, $b_v$, $b_d$, $b_a$, $b_{\sigma}$, $b_H$, $b_{S_D}$, $\sigma_c$, $H_c$, $S_{D_c}$ can be estimated from experimental and clinical evidence. Furthermore, in light of new evidence, the functional forms of $f_1$ to $f_5$ can easily be modified. Finally, $\sigma(t)$, $S_D(t)$ and $H(t)$ change over time due to increased awareness, government measures and compliance by people.
\subsubsection*{Growth Factor}
The growth factor $G_{\ell_v}$ quantifies how the infected number density is advected along the direction of $l_v$, that is, how the infection becomes  mild to severe/critical and vice-versa in the infected population. We can model it as a function of the medical history, immunity of the population, which in turn are functions of the age $l_a$, treatment and socio-economic status. Nevertheless, as a simple first order model, we propose a nonlinear function of the age,
\begin{align}
G_{\ell_v}(\ell_a) = K_g(\ell_a-a_{\textrm risk})^p\,,
\end{align}
where $K_g$ is a non-dimensionalization factor, $p$ is a power of nonlinearity and $a_{\textrm risk}$ is the age offset.

\subsubsection*{Recovery Rate and Infectious Death Rate}
In general, the recovery rate $C_R$ and infectious death rate $C_{ID}$ depend on $\ell_v$, and in turn are functions of hospital facilities, age, and health state of the population. These rates can be modeled directly from clinical data for all ordinates $\ell_v,~\ell_d,~\ell_a$. For the exact functional forms refer to the Supplementary Information Appendix.

\subsubsection*{Initial Infection Number Density}
The initial number density $I_0(\mbx,\mbl)$ can be estimated directly from available official data at the day of starting the simulation. However, the data is available only in-terms of total number of tested and confirmed cases at a $\mbx$-location and the dependence on $\mbl$ needs to be estimated via appropriate data-driven and analytical functions. As such, first we utilize data from a period of 14 days, along with the log-normal distribution of incubation period  \cite{lauer2020incubation} to calculate the initial number density $N_D(\mathbf{x})$ at all the spatial points $\mathbf{x}$, but integrated over the three internal ordinates ($\ell_v,\ell_d,\ell_a$), i.e.,
 \begin{equation}
     N_D(\mathbf{x}) = \sum_{i=1}^{i=14} N_i \frac{1}{i a_2\sqrt{2\pi}}\exp\left(-\frac{(\log i-b_2)^2}{2a_2^2}\right)\,, \label{eq:NDxdist}
 \end{equation}
 where $N_i(\mathbf{x})$ is the data of tested and positive. For the distribution along the internal ordinates, we propose to use the following initial infection number density distribution
\begin{align}

    I_0 &=
    N_D(\mbx)[f_5(\ell_v)]\left[a_1\exp\left(-\frac{(\ell_a-b_1)^2}{c_1^2}\right)\right]
     \left[\frac{1}{\ell_d a_2\sqrt{2\pi}}\exp\left(-\frac{(\log(\ell_d)-b_2)^2}{2a_2^2}\right)\right]. \label{eq:I0dist}
\end{align}
Here, the first term in the square brackets is the normalized demography function (same as $f_4$), second term is the log-normal incubation period function with fitted \cite{lauer2020incubation} $a_2=0.42$ and $b_2=1.62$, and $f_5$ is same as before.

\subsection*{Covid-19 Epidemic Spread Predictions}
To exhibit the capabilities of the proposed model, the forecast of Covid-19 spread in India is presented here. The numerical scheme and the fitted model parameters are given in the Supplementary Information Appendix. The proposed model and numerical schemes are implemented in our in-house finite element package \cite{PAR01,PAR02} and have been verified in our earlier studies with applications to process engineering \cite{GAN13,GANW10}.

With the spread of Covid-19 in India, the federal government imposed a nation-wide lockdown from March 25, 2020. To simulate the spread of infections starting from March 23, 2020, the initial distribution of infected population is estimated using the data of active cases from March 23 to April 5 according to equations~\ref{eq:NDxdist} and \ref{eq:I0dist}. Then the infection spread forecast for one year is computed by solving the PBE system (equation~\ref{model}). Further, data until June 21, 2020 is utilized to select the parameters (e.g., $S_D$, $C_R$, $C_{ID}$, $\gamma_Q$) that best explains the actual data. Thereafter, the control parameter $S_D$ is varied to perform scenario analyses as presented next.

\subsubsection*{Scenario Analysis}
\begin{figure}[tb!]
\begin{center}
\unitlength5mm
\begin{picture}(12,13)
\put(6.5,6.75){\makebox(0,0){\includegraphics[clip, trim=0 0 0 50, width=18.5cm]{TimeSeriesIND}}}
\end{picture}
\end{center}
\caption{Time series forecast of active, total infections, recovered and deceased cases of Covid-19 in India from Mar 23, 2020 to Mar 22, 2021. The inset shows a zoom with comparison of the model forecast with the data until July 01, 2020 \label{fig:TimeSeriesIND}}
\end{figure}

Different future scenarios are predicted by varying $S_D(t)$ based on the anticipated individual behavior (social distancing, hygiene practice, compliance to government rules etc.) and government policies (quarantine rules, lockdown rules etc.). The first scenario, named \textit{Current Trend} follows business as usual assuming further relaxation to lockdown rules. A second variant named \textit{Better Scenario} assumes better compliance in the social distancing and other measures to control the spread of the disease. Sunday, and Sunday \& Wednesday lockdowns are imposed on the \textit{Current Trend} scenario to formulate the third and fourth scenarios respectively. These lockdown scenarios are introduced to measure the impact of periodic lockdowns on the effectiveness of these strategies to control the disease spread.
The active ($N_I$), recovered (cumulative $N_{R}$), deaths (cumulative $N_{ID}$) and total (sum of active, recovered and deaths) predicted by the four scenarios for the duration between March 23, 2020 and March 22, 2021 are shown as time-series plots in Fig.~\ref{fig:TimeSeriesIND}. In the \textit{Current Trend}, a peak of 0.975 million ‘Active Cases’ is predicted in the last week of October 2020, and there will be around 21 million ‘Active Cases’, 450,000 deaths and 9.1 million total cases at the end of March 2021. The peak of the \textit{Better Scenario} is predicted in the second week of September 2020 with 0.478 million 'Active Cases', which is lower than the \textit{Current Trend}. Further, there will be around 14,200 ‘Active Cases’, 0.188 million deaths and 3.74 million total cases at the end of March 2021.

The weekly lockdown scenarios assume that a complete lockdown is imposed on Sunday or Sunday and Wednesday. During this lockdown, there is a complete restriction of people's movement similar to the nationwide lockdown imposed between Mar 25 and April 14 in India. With Sunday Lockdown, a peak of 0.365 million ‘Active Cases’ is sustained for about two weeks during 5-20 September 2020, and there will be around 30,200 thousand ‘Active Cases’, 0.167 million deaths and 3.32 million total cases at the end of March 2021. With Sunday and Wednesday lockdown, a peak of 0.197 million 'Active Cases' is sustained for the period 27 June to 15 July 2020, and there will around 2,800 ‘Active Cases', 70,300  deaths and 1.39 million total cases at the end of March 2021. The insets in each panel of Fig.~\ref{fig:TimeSeriesIND} show the comparison with actual data and thereby validate the model. In addition, the time series plots for other scenarios including a worse-case scenario can be found at IISc-Model website \cite{IIScModel}.

In order to compare the performance of all states in India with the national trend, a uniform set of parameters is used for state-wide computations. In particular, the parameters are fitted by minimizing the error between the national data and the sum  of the respective state predictions.

Fig.~\ref{fig:recovisoMH} shows the actual data and the computed distribution using the above set of  parameters for the states of Karnataka and Maharashtra. We can see that Karnataka has done better, whereas Maharashtra has done worse compared to the national trend. These insights can be used by the authorities to introduce state-wise lockdown policies and to plan infrastructure for quarantine, treatments etc. The performance of  other states can be seen at IISc-Model website \cite{IIScModel}.
\begin{figure}[tb!]
\begin{center}
\unitlength5mm
\begin{picture}(12,8)
\put(6.5,4.25){\makebox(0,0){\includegraphics[clip, trim=0 0 0 50, width=18.5cm]{TimeSeriesStatesMhKa}}}
\end{picture}
\end{center}
\caption{The actual data and the predictions computed with the national trend based parameters for the states of Karnataka and Maharashtra. in India. \label{fig:recovisoMH}}
\end{figure}

\subsubsection*{Population distribution}
\begin{figure}[t!]
\begin{center}
\unitlength5mm
\begin{picture}(12,18.5)
\put(-6,3){\makebox(0,0){\includegraphics[clip, trim=0 0 0 200,width=5.5cm]{activet=240}}}
\put(6,3){\makebox(0,0){\includegraphics[clip, trim=0 0 0 200,width=5.5cm]{activet=300}}}
\put(17.5,3){\makebox(0,0){\includegraphics[clip, trim=0 0 0 200,width=5.5cm]{activet=365}}}
\put(-6,13){\makebox(0,0){\includegraphics[clip, trim=0 0 0 200, width=5.5cm]{activet=060}}}
\put(6,13){\makebox(0,0){\includegraphics[clip, trim=0 0 0 200,width=5.5cm]{activet=120}}}
\put(17.5,13){\makebox(0,0){\includegraphics[clip, trim=0 0 0 200,width=5.5cm]{activet=180}}}
\put(-8,18.25){(a)}
\put(3.5,18.25){(b)}
\put(15.5,18.25){(c)}
\put(-8, 8.){(d)}
\put(3.5, 8.){(e)}
\put(15.5, 8.){(f)}
\end{picture}
\end{center}
\caption{Distributions of Covid-19 population at different time instances, (a)~$t$=60, (b)~$t$=120, (c)~$t$=180, (d)~$t$=240, (e)~$t$=300 and (f)~$t$=365. \label{activeiso}}
\end{figure}
\begin{figure}[hb!]
\begin{center}
\unitlength5mm
\begin{picture}(12,18.5)
\put(-6,3){\makebox(0,0){\includegraphics[clip, trim=0 0 0 200,width=5.5cm]{recovered=240}}}
\put(6,3){\makebox(0,0){\includegraphics[clip, trim=0 0 0 200,width=5.5cm]{recovered=300}}}
\put(17.5,3){\makebox(0,0){\includegraphics[clip, trim=0 0 0 200,width=5.5cm]{recovered=365}}}
\put(-6,13){\makebox(0,0){\includegraphics[clip, trim=0 0 0 200, width=5.5cm]{recovered=060}}}
\put(6,13){\makebox(0,0){\includegraphics[clip, trim=0 0 0 200,width=5.5cm]{recovered=120}}}
\put(17.5,13){\makebox(0,0){\includegraphics[clip, trim=0 0 0 200,width=5.5cm]{recovered=180}}}
\put(-8,18.25){(a)}
\put(3.5,18.25){(b)}
\put(15.5,18.25){(c)}
\put(-8, 8.){(d)}
\put(3.5, 8.){(e)}
\put(15.5, 8.){(f)}
\end{picture}
\end{center}
\caption{Distributions of recovered  population from Covid-19 at different time instances, (a)~$t$=60, (b)~$t$=120, (c)~$t$=180, (d)~$t$=240, (e)~$t$=300 and (f)~$t$=365. \label{recoviso}}
\end{figure}

Our PBE model in fact predicts the distribution of the infected population over all the internal ordinates $\ell_v,~\ell_d,~\ell_a$. In the previous section, we have shown only the total number of infected, recovered and deceased populations. Now, to showcase one of the unique features of the model, we present and discuss the predicted population distribution for the Sunday lockdown scenario.
Figure \ref{activeiso} shows the predicted distribution of active Covid-19 infected population over the ordinates $\ell_v$, $\ell_d$ and $\ell_a$ at different time instances (day 60, 120, 180, 240, 300 and 365) with their corresponding dates.

Predicting the severity of the infected population is crucial to plan the hospital requirements including antiviral treatments, quarantine, hospitalisation, ventilator support, and oxygen support. In particular, the information of asymptomatic and symptomatic infected population helps the policymakers to plan quarantine rules. Moreover, the death rate is a function of the severity of infection and is crucial to predict the causalities arising from the infection spread.
The duration of infection plays a key role in epidemic modeling. Classical models usually assume a constant duration. However, the recovery and the death of the patient depend on the immunity, age and health of the patient, medical treatments etc and thus the duration of infection need not be a constant. In the proposed model, the duration of infection is considered as an independent internal ordinate. The recovered population distribution over the duration of infection and other internal ordinates for days 60, 120, 180, 240, 300 and 365 is shown in Figure \ref{recoviso} along with their corresponding dates.
Crucially, the predicted distribution with duration of infection, especially at initial stages, is key to plan for testing and to make effective decisions on quarantine, hospitalization and discharging from hospitals.
Finally, the age of the population is pivotal in epidemics like Covid-19 since it affects children and aged population severely. Therefore, it is incorporated into the proposed model as another independent ordinate. In fact, the newly infected population is added from the susceptible population across the age distribution through the nucleation term.
Moreover, the response to the antiviral treatment, death and recovery rates depend on age-specific health complications such as diabetics, cardiovascular disease, can also be incorporated in the PBE model with appropriate functions that depend on the age of the population.

\section*{Discussion}
Our spatio-temporal modeling framework is the first comprehensive partial differential equation model for predicting infectious disease spread. Computationally, our model is efficient compared to agent-based stochastic models. Mathematically, our PDE system is more compact and comprehensive compared to ODE-based compartmental models. Specifically, the PDE is a continuum description of the infected population whereas the compartmental models are a discrete representation. Crucially, in contrast to the existing models, our model provides an insight into the distribution of infected population (presented in previous sections). This information is important to plan policy interventions, especially in Covid-19 like pandemics. Not only prognostic estimates, but also diagnostic estimates for more detailed analysis using distribution can be performed with the proposed framework.

With more data and employing data-driven and machine learning approaches, we could further refine the parameters and functional forms of different model components to derive more insightful predictions. For example, to derive insights into the reopening of the workplace and educational institutions, the nucleation and advection vector could be modeled to account for interactions between different age groups and movement of people between homes and these places. The potential options for refining the model are virtually endless. In particular, there is no restriction on the choice of number of internal coordinates. For example, in addition to $\ell_v, \ell_d, \ell_a$, profession, mobility history, etc can also be added as internal coordinates.

Even though we have emphasized Covid-19 pandemic in the present paper, the proposed model can readily be used for forecasting any other infectious disease spread. In future, a  data assimilative framework for a real-time update of forecasts can also be implemented.

\bibliography{lit}

\section*{Acknowledgements}
DS acknowledges the partial support from IISc Start-up grant, DST-INSPIRE Faculty Research Grant (04/2018/003591), and Arcot Ramachandran Young Investigator Award. SG acknowledges the support from SERB and DRDO for the grants that supported development of ParMooN.

\section*{Author contributions statement}
Both authors contributed equally and reviewed the manuscript.

\section*{Appendix}
As the focus of the present paper is to introduce the spatio-temporal predictive modeling framework for infectious disease spread, we use simple estimates and algebraic relations for certain parameters. Further, a few more assumptions are made on some parameters due to lack of actual data. Nevertheless, the features of the model and the insights into the prediction of the spread can be seen even with these assumptions.

\subsection*{Numerical Scheme}
 Let $T_\infty=365$ days, $d_\infty=400$ days, $a_\infty=45,625$ days  and $\Omega_{\rm x}:=\cup\Omega_k$, $k=1,\ldots,M$, where $M$ being the   number of states and union territories in India. We assume that there will be no inter- or intra-state and international movements, that is,  $\bu = 0 $,  no growth in level of infection, that is, $G_{\ell_v}=0$, the growth age is negligible and no source. Nevertheless, population distribution in all three internal ordinates $\ell_v$, $\ell_d$ and $\ell_a$ are described and tracked. Hence, the PBE in model~\eqref{model} becomes
\begin{equation}\label{lockmodel}

 \ds\frac{\partial I}{\partial t}     +   \frac{\partial I}{\partial \ell_d}  +C I  = 0  \quad  \rm{ in }  \quad  (0,T_\infty]\times\Omega_{x}\times\Omega_\ell.

\end{equation}

\subsubsection*{Operator splitting finite element scheme}
A finite element scheme~\cite{GTBook16} based on operator splitting~\cite{GAN13,GT11a,GANW10} is used to solve the high-dimensional PBE model~\eqref{lockmodel}.  Applying, operator splitting to~\eqref{lockmodel}, we get
\\

\noindent \textbf{Step 1. ($x$-direction)}
For given $I(t^a,x,\ell)$ with $I(t^a=0,x,\ell)=I_0$, find $\tilde I(t^b,x,\ell)$ in $(t^a , t^b)$ for all $\ell\in\Omega_\ell$ such that
\begin{equation}\label{x-lockmodel}
 \begin{array}{l}
 \ds\frac{\partial \tilde I}{\partial t} + C I   = 0,\quad
  \tilde I(t^a,x,\ell) = I(t^a,x,\ell)   \quad \rm{ in }  \quad  \Omega_{x}  \,,
\end{array}
\end{equation}
\noindent \textbf{Step 2. ($\ell_d$-direction)}
For given $\tilde I(t^b,x,\ell)$, find $  I(t^b,x,\ell)$ in $(t^a , t^b)$  for all $x\in\Omega_{\rm x}$   $\ell_v \in L_v$ and  $\ell_a \in L_a$ such that
\begin{equation}\label{s2-lockmodel}
 \begin{array}{l}
 \ds\frac{\partial  I}{\partial t}   +  \frac{\partial  I}{\partial \ell_d}    = 0,\quad I(t^a,x,\ell) = \tilde I(t^b,x,\ell) \quad \rm{ in }  \quad   \Omega_\ell \,;\qquad
   I(t,\ell_v, 0, \ell_a)  =B_{nuc} \quad   \rm{ in }  \quad   L_a ,\\

\end{array}
\end{equation}
In the $x$-direction, the evaluation equation~\eqref{x-lockmodel} has to be solved for every $\ell\in\Omega_\ell$ by considering $\ell$ as a parameter. Similarly, the system \eqref{s2-lockmodel}  has to be solved in $\ell_d$-direction for every $x\in\Omega_{\rm x}$   $\ell_v \in L_v$ and  $\ell_a \in L_a$ by considering these variables as parameters. The backward Euler   and discontinues Galerkin with upwind  methods are used for the temporal and the spatial discretisation, respectively. The implementation of the splitting algorithm in the finite element context has been presented in these papers~\cite{GAN13,GT11a,GANW10}.

\subsubsection*{Parameters for  Covid-19  predictions}

The nucleation  model $B_{\textrm nuc}$  defined in \eqref{eq:Bnuc_eq2} is considered with $R_0=3.35$, $f_1=1$, $f_2=1$, $f_4=1$ and
 \[
f_3(t,S_D) = 1. -  1./(1.+ \exp(-(S_D(t) - 0.5)/0.1)), \quad
     S_D(t) =
        \left\{
            \begin{array}{lc}
               0.7 +  0.001333~t & 0\leq t<15  \,\\
               0.72 + 0.004285(t-15) & 15\leq t<36  \,\\
                0.81 - 0.004(t-36) & 36\leq t<51  \,\\
                0.75 + 0.0012(t-51.) & 51\leq t<72  \,\\
                0.3 + ds(t-72) & \textrm{else}
            \end{array}
        \right.
\]
Moreover, the values given in Table~\ref{dsvalues} are used for $ds$ and $f_1$ to perform scenario analysis.
 \begingroup
\setlength{\tabcolsep}{10pt}
\renewcommand{\arraystretch}{1.25}
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|l|}
 \hline
  Scenarios  & Current Trend & Better Trend & Periodic lockdowns\\ \hline
  $ds$  &  0.000333  & 0.0005 & 0.000333 \& \\
  & &  & $f_1 = 0.01$ on lockdown days \\ \hline
\end{tabular}
\caption{Parameter values used in scenario analysis.}
\label{dsvalues}
\end{table}
\endgroup

Furthermore, the quarantine function
\[
\gamma_Q(\ell_d) =
\left\{
    \begin{array}{lc}
       1 & 0\leq t<1  \,\\
       0.9(1./(1.+ \exp(-(l_v-0.4)/0.1)))*(1./(1.+ \exp(-(l_d-5.1)/2)))   & \textrm{else,}
    \end{array}
\right.
\]
is used in all scenarios.
Finally, the recovery and death rate functions are fitted as
\[
C_{ID}(t,\ell_v) =
\left\{
    \begin{array}{ll}
       0 & \ell_v<0.64  \,\\
        0.0475 - 0.000357~t   & \ell_v\geq 0.64 \textrm{ and }  t<21 \\
       0.0475 + 0.000208(t-21)  & \ell_v\geq 0.64 \textrm{ and }  21\leq t<36 \\
       0.0475  & \textrm{else,}
    \end{array}
\right., \quad
C_{R}(t) =
\left\{
    \begin{array}{ll}
        0.01 + t*0.00058 & t<65  \,\\
        0.0475  & \textrm{else.}
    \end{array}
\right.
\]

\noindent\textbf{Remark:} A factor $N_S(t)/N(t)$ needs to be introduced in $B_{nuc}$  when herd immunity  develops.

\subsubsection*{Initial infection number density}

 For India, the data to estimate $N_D(\mathbf{x})$ is downloaded from a publicly sourced database \cite{covid19tracker}.
 To distribute the number density among the internal ordinates, we employ distribution fits as given in equation~\eqref{eq:I0dist}. Data set of age downloaded from Statista \cite{statista} is used to fit $a_1,b_1,c_1$ of equation \eqref{eq:I0dist}.

\end{document}


TITLE: Modules over a monoid: trouble with the definition.
QUESTION [2 upvotes]: I'm having trouble with a definition.
We're working in the category of monoids. Take $A\in \mathfrak{Mon}$ and define a module over $A$ to be a set $M$ with an action:
$A\times M \rightarrow M$ such that the pair $(a,m)\mapsto am$. The question is: how do I interprit  $am$? A set doesn't usually come with an operation defined on it.
Sorry for the silly question, but I really can't make any sense of it. Thank you

REPLY [4 votes]: The specific function $f:A\times M\to M$ is the action. That is, we define the expression $am$ to mean the element $f(a,m)\in M$.
Note also that the definition should include the requirement that $f$ is compatible with the monoid operation of $A$; i.e., you need
$$f(a_1,f(a_2,m))=f(a_1a_2,m)\qquad f(1,m)=m$$
or, written with the more concise notation,
$$a_1(a_2m)=(a_1a_2)m\qquad 1m=m$$
To give another approach, you can view an action as a function $\varphi:A\to \mathrm{End}(M)$ that is a monoid homomorphism, where $\mathrm{End}(M)$ is the monoid of all functions from the set $M$ to itself. The correspondence of this notion with the other notion of action is
$$\varphi(a)\text{ is the function from $M$ to $M$ defined by }\quad \varphi(a)(m)=f(a,m)$$