The Coulomb and Higgs branches of certain

Article funded by SCOAP3

1$ can be thought as deformations of closures of nilpotent orbits. In this paper we study one aspect of the nilpotent orbits of classical algebras that has been known for some decades among mathematicians~\cite{Kraft1982} and is starting to appear in physics~\cite{Heckman}. We call it the \emph{Kraft-Procesi transition}. We produce a systematic study of the brane realization of the phenomenon and recover one of its mathematical features: the minimal singularities that characterize each transition. In doing so we hope to bring into physics a new approach on the way of understanding geometrical spaces. This is the idea of Brieskorn~\cite{Brieskorn1970} that the structure of a variety can be understood by \emph{slicing it transversally} to a maximal subvariety. We show how this can be realized as a Higgs mechanism from the point of view of quantum field theory. From this point of view, the Hasse diagram gives an interesting view on the set of all possible mixed branches of a given quiver theory. In section~\ref{sec:S} we summarize the results of the present work. Sections~\ref{sec:M} and~\ref{sec:B} of the paper aim to serve as an introduction to the main discussion. Section~\ref{sec:M} contains an overview of the basic mathematical concepts that are needed: hyperk\"ahler singularities and nilpotent orbits. Section~\ref{sec:B} summarizes the required brane dynamics and quiver gauge theory. The reader familiarized with either of these subjects is encouraged to skip those sections and go directly to the new material in section~\ref{sec:K}. In section~\ref{sec:K} we develop the physical interpretation of the Kraft-Procesi transition. Section~\ref{sec:F} introduces a formalism which allows to perform the required computations in an efficient way. Section~\ref{sec:R} displays the results of such computations. Section~\ref{sec:C} contains some conclusions. ]]>

1/2$. \item If some of the generators of the chiral ring carry spin $s=1$, they all transform under the adjoint representation of a flavor symmetry group acting on the moduli space~\cite{Gaiotto2008}. \item Generators of spin higher than $s=1$ may be called \emph{baryons}, and their intuitive role is to increase the order of the singularity of the moduli space. It is interesting to study their role and this is left for future study. \end{itemize} ]]>

0$ and $\rho:\mathfrak{g}\mapsto \text{End}(V)$ is the adjoint representation\footnote{It can be shown that a definition with a different finite representation of the algebra is equivalent to this definition~\cite{Collingwood1993}.} of the algebra acting on a complex vector space $V$~\cite{Collingwood1993}. \boldmath ]]>

0\ \ \text{and}\\ &\sum_{i=1}^k\lambda_i=n \end{aligned} \end{align} \emph{Exponential notation} can be introduced. For example $(3^2,2,1^5)=(3,3,2,1,1,1,1,1)$ is a partition of $n=13$. We denote $\mathcal{P}(n)$ the set of all partitions of $n$, for example $\mathcal{P}(3)=\{(3),(2,1),(1^3)\}$. We remember that an \emph{elementary Jordan block} of order $i\in\mathbb{Z}^+$ is defined as the $i\times i$~matrix: \begin{align} J_i:=\left(\begin{array}{cccccc} 0&1&0&\dots&0&0\\ 0&0&1&\dots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\dots&0&1\\ 0&0&0&\dots&0&0\\ \end{array}\right) \end{align} Given a partition $\lambda=(\lambda_1,\dots,\lambda_k)$ of $n$ we can form a nilpotent endomorphism of $\mathbb{C}^n$~as: \begin{align} X_\lambda=\left(\begin{array}{cccc} J_{\lambda_1}&0&\dots&0\\ 0&J_{\lambda_2}&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\dots&J_{\lambda_k}\\ \end{array}\right) \end{align} Hence, $X_\lambda$ is a nilpotent element of the algebra $\mathfrak{sl}_n$. The $n\times n$ matrix $X_\lambda$ is in the adjoint representation of $\PSL(n)$ group.\footnote{In~\cite{Collingwood1993} the adjoint group that defines the action of elements of $\mathfrak{sl}_n$ on the algebra itself is $\PSL(n)=\SL(n)/Z$ where $Z$ is the center of $\SL(n)$, this is the group that generates the nilpotent orbits acting on the right and on the left on the matrix $X_\lambda$.} It generates an orbit under the group called the nilpotent orbit: \begin{align} \mathcal{O}_\lambda:=\PSL(n)\cdot X_\lambda \end{align} Note also that two different partitions give rise to two disjoint nilpotent orbits by the uniqueness of the Jordan normal form. Therefore, for every different partition of $n$ there is a different nilpotent orbit of $\mathfrak{sl}_n$. Furthermore, a generic nilpotent element $X\in\mathfrak{sl}_n$ has a Jordan normal form $X_\lambda$ for some $\lambda\in\mathcal{P}(n)$, i.e.\ it is $\PSL(n)$-conjugate to $X_\lambda$. Therefore it belongs to the nilpotent orbit $\mathcal{O}_\lambda$. \boldmath ]]>

,>=stealth',shorten >=0pt,shorten <=0pt,bend angle=10,auto] \tikzstyle{gauge} = [circle, draw]; \tikzstyle{flavour} = [regular polygon,regular polygon sides=4,draw]; \node (g1) [gauge,label=right:{$1$}] {}; \node (f1) [flavour,above of=g1,label=above:{$N$}] {}; \path[every node/.style={font=\sffamily\small}] (g1) edge [bend left]node {$Q_{i}$} (f1) edge [loop below] node{$\phi$} (g1) (f1) edge [bend left] node {$\tilde{Q}^j$} (g1); \end{tikzpicture} \end{subfigure} ]]>

1$ such that $j$ is neither $1$ or $N+1$ gives rise to an $A_{n}$ transition where $n=M_{1j}-1$. For example in the maximal orbit of $\mathfrak{sl}_3$, with partition $\lambda=(3)$ and matrix: \begin{align} M(\lambda)=\left(\begin{array}{cccc} 0&3&0&0\\ 0&2&1&0\\ \end{array}\right) \end{align} we find that there is an $A_2$ singularity of the form: \begin{align} M(A_2)=\left(\begin{array}{ccc} 0&3&0\\ 0&1&0\\ \end{array}\right) \end{align} embedded on the second column of $M(\lambda)$. If we remove it, by performing an $A_2$ KP transition, we find: \begin{align} M(\lambda')=\left(\begin{array}{cccc} 1&1&1&0\\ 0&1&1&0\\ \end{array}\right) \end{align} The linking numbers of the NS fivebranes have changed to $\vec{l}_s=(0,1,2)$, corresponding to $\lambda'^t=(2,1)$. Therefore the new matrix corresponds to partition $\lambda'=(2,1)$. We can easily check that this corresponds with the brane configuration in figure~\ref{fig:SU3KPa3}(a). To summarize, this matrix manipulation corresponds to the $A_2$ KP transition depicted in figure~\ref{fig:SU3KPA3}. \boldmath ]]>

1$. The minimal singularity has a matrix of the form: \begin{align} M(a_n)=\left(\begin{array}{ccccccc} 0&1&0&\dots&0&1&0\\ 0&1&1&\dots &1&1&0\\ \end{array}\right) \end{align} where there are $n-2$ columns with 0s in the first row and 1s in the second, between the two columns with $1$s in the first row. The transition is to: \begin{align} M=\left(\begin{array}{ccccccc} 1&0&0&\dots&0&0&1\\ 0&0&0&\dots &0&0&0\\ \end{array}\right) \end{align} where the $n$ D3-branes have been removed. For $a_n$ with $n>1$ to be a minimal KP singularity the NS5-branes have to be alone in their intervals, i.e.\ we cannot use a NS5-brane from an interval with $M_{1j}>1$ to perform an $a_n$ different from $a_1$ if we want to restrict the transitions to minimal KP singularities. For example, there is an $a_2$ singularity in the brane system for $\lambda=(2,1)$: \begin{align} M(\lambda)=\left(\begin{array}{cccc} 1&1&1&0\\ 0&1&1&0\\ \end{array}\right) \end{align} The result after performing an $a_2$ KP transition is: \begin{align} M(\lambda')=\left(\begin{array}{cccc} 2&0&0&1\\ 0&0&0&0\\ \end{array}\right) \end{align} The model represented by the matrix $M(\lambda')$ has linking numbers $\vec{l}_s=(0,0,3)$, corresponding to $\lambda'^t=(3)$. Therefore the new matrix corresponds to partition $\lambda'=(1^3)$. This is the trivial orbit. This corresponds to the KP transition described in figure~\ref{fig:SU3KPa3}. \boldmath ]]>

1$. In this case we can pair it up with the NS5-brane in interval 4, i.e.\ $M_{14}=1$. Hence, both NS5-branes, from intervals 2 and 4, and the three D3-branes, from intervals 2, 3 and 4, make up an $a_3$ singularity. Removing the singularity via an $a_3$ KP transition we obtain \begin{align} M=\left(\begin{array}{ccccc} 3&0&0&0&1\\ 0&0&0&0&0\\ \end{array}\right) \end{align} with linking numbers $l_s=(0,0,0,4)$, giving $\lambda^t=(4)$, $\lambda=(1^4)$. This is the minimal orbit and it marks the end of the iteration. If we make a diagram where the nodes are the orbits, and there are edges connecting them where we found a KP transition we recover the KP Hasse diagram from figure~\ref{fig:HasseSU4}(b). Note that in this formalism the quaternionic dimension is just: \begin{align} dim:=\sum_j M_{2j} \end{align} ]]>