text:
\section{Introduction}

A quantum field theory with a given spectrum and algebra of local operators may admit distinct ``global structures" encoded in a choice of extended (line, surface, etc.) ``probe" operators.  In the case of 4d Yang-Mills (YM) theories these global structures are identified with the global form of the gauge group together with additional discrete theta angles \cite{Aharony:2013hda}.  The choice of global structure is also related to the higher-form symmetries of the theory \cite{Gaiotto:2014kfa}.

In this paper we explain how this global structure is reflected in the geometry of the Coulomb branch of vacua of 4d ${\mathcal N}{=}2$ field theories.
We will focus here on only the line operator spectrum and the 1-form symmetry of these theories.
A simple example to keep in mind is 4d ${\mathcal N}{=}4$ supersymmetric YM (sYM) with gauge algebra $\mathfrak{su}(2)$, which admits three compatible assignments of line-operator charges corresponding to three different global forms ${\rm SU}(2)$, ${\rm SO}(3)_+$ and ${\rm SO}(3)_-$.
This theory has an exactly marginal coupling taking values in a ``conformal manifold".
The three global structures form a single orbit under S-duality, meaning that upon traversing non-trivial loops in the conformal manifold, they are interchanged.  Finally, this theory has a ${\mathbb Z}_2$ 1-form symmetry under which the line operators may be charged.

This global structure data is completely encoded in the  \emph{charge lattice} and its \emph{Dirac pairing} which are physical attributes of the low energy theory in a Coulomb vacuum. One aim of this paper is to point out that then the global structure is encoded in the Coulomb branch geometry, and to explain this encoding.
This point has already been made in the context of lagrangian theories of class S, see \cite [sections 6.3 and 7.3.1]{Gaiotto:2010be}, \cite[section 5.4]{Tachikawa:2013hya}, and for ${\rm SU}(2)$ sYM  \cite[section 4.5]{Tachikawa:2013kta}.
We emphasize, however, that the notion of global structure interpreted in this way applies just as well to non-lagrangian theories where there is no gauge group.
We also point out that the encoding of the global structure in the Coulomb branch geometry cannot be as straightforward in general as it is in the class S examples where to each distinct global structure (i.e., ones not related by S-duality) corresponds a distinct Coulomb branch geometry.
In particular, there are examples of theories whose different global structures are reflected in the same choice of CB geometries.

A characteristic feature of ${\mathcal N}{=}2$ QFTs is that they have a Coulomb branch, an $r$-complex-dimensional manifold of $\mathfrak{u}(1)^r$ gauge theory vacua with massive charged fields.
The finite-energy states in these vacua carry electric and magnetic charges with respect to each of these $\mathfrak{u}(1)$'s whose values form a rank-$2r$ lattice, the \emph{charge lattice} ${\Lambda}$ of the theory.
Choose some basis of this lattice with respect to which a charge $Q\in{\Lambda}$ is represented by a $2r$-tuple of integers, $Q = ( p_1, \ldots, p_r , q_1 , \ldots , q_r)$.

The Dirac quantization condition \cite{Dirac:1931kp} implies that there is a non-degenerate integral pairing on the charge lattice, $J:{\Lambda}\times{\Lambda}\to{\mathbb Z}$, which can be written in a basis as $J(Q_1,Q_2) = Q_1 J Q_2^t$ where the right side is matrix multiplication with $Q_1$ and $Q_2$ interpreted as row vectors and $J$ as a $2r\times2r$ integer antisymmetric matrix.
If there is a basis of ${\Lambda}$ in which $J$ has the $r\times r$ block form
\begin{align}\label{princ}
    J = \begin{pmatrix} & \phantom{-}1 \\ -1 & \end{pmatrix} ,
\end{align}
i.e., a basis in which the Dirac pairings take their minimal allowed values, then the Dirac pairing is \emph{principal}.
But there is no reason that such a basis needs to exist, in which case the pairing is non-principal.
For example, most ${\mathcal N}{=}4$ sYM theories have non-principal Dirac pairings.

The possible choices of global structure of the theory, and their associated 1-form symmetries, can be unambiguously read off from knowledge of the Dirac pairing on the charge lattice.
Indeed the different global structures correspond to the possible inequivalent refinements of the charge lattice on which the Dirac pairing extends to a principal pairing.
These refined lattices, ${\mathcal L}$, are possible charges of a maximal mutually local set of line operators \cite{Gaiotto:2010be}, so we refer to them as \emph{line lattices}.
This description of the global structures of an ${\mathcal N}{=}2$ field theory does not depend on its having a gauge theory description, so applies equally well, for example, to isolated strongly-coupled ${\mathcal N}{=}2$ superconformal field theories (SCFTs) for which there is no gauge group.

For ${\mathcal N}{=}2$ SCFTs with exactly marginal couplings, the S-duality group acts on the global structures, organizing them into distinct orbits.
In the $\mathfrak{su}(2)$ ${\mathcal N}{=}4$ example mentioned above, all three global structures form a single S-duality orbit, but, by contrast, the 7 global structures of the $\mathfrak{su}(4)$ ${\mathcal N}{=}4$ theory form 2 separate S-duality orbits \cite{Aharony:2013hda}.

A feature of ${\mathcal N}=2$ field theories, which was not particularly used in the original analysis of global structures \cite{Aharony:2013hda}, is that the low energy effective theory in the Coulomb vacua can be geometrically encoded in the special K\"ahler (SK) geometry of the theory's Coulomb branch \cite{Seiberg:1994aj, Seiberg:1994rs, Donagi:1995cf, Freed:1997dp}.
The Seiberg-Witten curve --- a family of Riemann surfaces with a certain 1-form varying holomorphically over the Coulomb branch --- is a way of describing the SK structure.

The main goal of this paper is to clarify the extent to which the SK structure is sensitive to the global structures of the field theory.
We will illustrate this in the relatively simple case of ${\mathcal N}=4$ sYM theories, but our analysis applies generally to all ${\mathcal N}=2$ theories.
More specifically:
\begin{itemize}
    \item We clarify the distinction between two objects which are often conflated, the charge lattice ${\Lambda}$ of the theory and the homology lattice ${\Lambda}_X$ of the Seiberg-Witten curve.  In particular, if the intersection pairing on ${\Lambda}_X$ is principal, then ${\Lambda}_X$ is naturally identified with a choice of line lattice ${\mathcal L}$ rather than with the charge lattice ${\Lambda}$.

    \item We show that a given Coulomb branch K\"ahler geometry can admit a discrete set of distinct SK ``models'' which differ by the symplectic pairing on their homology lattices.

    \item We point out that the S-duality group which is discernible from the Coulomb branch geometry depends on the SK model.  In particular, the S-duality group ``visible" in the SK model in which the homology lattice is the charge lattice is larger than the physical S-duality group as it does not distinguish between the different global structures of the theory.  Thus, for example, these ``coarse" S-duality groups are ${\rm SL}(2,{\mathbb Z})$ for all the ${\mathcal N}{=}4$ sYM theories.
\end{itemize}
The second point, in particular, has been made before in the context of class-S examples in  \cite{Gaiotto:2010be, Tachikawa:2013hya, Tachikawa:2013kta}, as mentioned above.
This point would seem to predict that (a) there are different principally-polarized ``SK models'' of a Coulomb branch corresponding to the different  global structure S-duality orbits; (b) there is a unique model with maximally non-principal pairing whose homology lattice equals the charge lattice; and (c) there will exist models corresponding to non-maximal choices of line lattices with pairings whose invariant factors divide those of the maximally non-principal pairing model.
In fact, however, it seems that prediction (a) is incorrect, as we will point out a counter-example shortly.

As part of the exposition of the above points, we re-derive some results which, while known to experts, may not be more widely appreciated.  In particular, we derive from well-known semiclassical field theory facts the (non-principal) Dirac pairing on the charge lattice of ${\mathcal N}=4$ sYMs. These results are consistent with those of \cite{DelZotto:2022ras}.
Also, we show how to re-derive the results of \cite{Aharony:2013hda} by counting maximal symplectic sublattices, and present simple algorithms for doing so.

In this work we only apply our analysis to re-derive the global structures and one-form symmetries of ${\mathcal N}=4$ sYM theories as examples, but our techniques apply more generally to ${\mathcal N}=2$ theories for which many results are by now known \cite{Closset:2020scj, DelZotto:2020esg, Closset:2020afy, Bhardwaj:2021pfz, Hosseini:2021ged, Buican:2021xhs, Closset:2021lwy, Bhardwaj:2021mzl, Carta:2022spy, DelZotto:2022ras}.
In particular, an immediate implication of the known classification of rank-1 Coulomb branch geometries \cite{Argyres:2015ffa, Argyres:2015gha, Argyres:2016xua, Argyres:2016xmc} is that since --- with one exception --- all have only principally polarized homology lattices, it follows that all rank-1 SCFTs have principal Dirac pairings, their charge and line lattices coincide, and they have no 1-form symmetries.
This agrees with the predictions from string constructions and BPS quivers \cite{Closset:2020scj, DelZotto:2020esg, Closset:2020afy, Bhardwaj:2021pfz, Hosseini:2021ged, Buican:2021xhs, Closset:2021lwy, Bhardwaj:2021mzl, Carta:2022spy, DelZotto:2022ras}.
The one exception is the Coulomb branch geometry of the ${\mathcal N}{=}2^*$ $\mathfrak{su}(2)$ sYM theory which also has a model with non-principally polarized homology lattice, which is consistent with the field theory expectation.
We describe this example in detail in section \ref{su2curves} below.

\paragraph{${\mathcal N}{=}3$ Coulomb branches which to not admit principal polarizations.}

A class of theories for which the derivations presented here could be even more directly applied are ${\mathcal N}{=}3$ theories \cite{Garcia-Etxebarria:2015wns, Aharony:2015oyb, Aharony:2016kai}.
In fact the moduli spaces of ${\mathcal N}{=}3$ theories closely resemble those of the ${\mathcal N}{=}4$ theories described here, since both, as complex geometries, are complex orbifolds by reflection groups.
For ${\mathcal N}{=}4$ theories the orbifold groups are the Weyl groups of the gauge algebras (which are crystallographic real reflection groups), which are substituted by the more general crystallographic complex reflection groups in ${\mathcal N}{=}3$ theories \cite{Caorsi:2018zsq,  Bonetti:2018fqz,  Argyres:2019ngz}.
This suggests that a perhaps straightforward generalization of the analysis in this paper will allow the computation of one-form symmetries of all ${\mathcal N}{=}3$ theories in a uniform way.

Not all crystallographic complex reflection group orbifolds admit SK models for which the homology lattice is principally polarized.
For example, at rank 2 the exceptional complex reflection group $G_8$ in the Shephard-Todd classification \cite{Shephard:1954} (with Coulomb branch scaling dimensions 8 and 12) only admits integer symplectic forms with minimum invariant factors (1,2), so are never principal; see section 3.2 of \cite{Caorsi:2018zsq}.
Furthermore, this Coulomb branch arises as the moduli space of an ${\mathcal N}{=}3$ SCFT with a known M-theory construction \cite{Kaidi:2022lyo, Garcia-Etxebarria:2016erx}.
The counting of global structures of theories with non-principal Dirac pairing, reviewed below, implies that this theory has at least 3 distinct global structures.
But the absence of an SK model with principal polarization means that no Coulomb branch geometries exist which realize a choice of maximal line lattice for this theory.
It remains to be understood what the implications of this fact are and whether the absence of a principally polarized model for the Coulomb branch of this theory has an interpretation in terms of its spectrum of extended operators or its generalized symmetry. This questions certainly deserves further study.\\

The rest of the paper is organized as follows.  Section \ref{sec2} reviews the definitions of and relations between the charge lattice, the Dirac pairing, and the possible line lattices.  We also review how these are related to the 1-form symmetry.  Section \ref{sec3} begins the discussion of ${\mathcal N}=4$ sYM theories by deriving their charge lattices and Dirac pairings. Finally, section \ref{sec4} discusses how the global structure (or, choice of line lattice) appears in the SK structure of the Coulomb branch.  We conclude the section with an explicit discussion of low-rank examples.  There are also three technical appendices summarizing some basic properties of simple Lie algebras, discussing how to bring symplectic pairings to canonical form, and showing how to count inequivalent symplectic sublattices.

\section{Charge lattices in 4d QFTs}\label{sec2}

Associated to each quantum field theory with a Coulomb phase, there is a linear space of allowed possible deconfined gauge charges of finite energy states.
In 4d these are electric and magnetic charges.
A vacuum is a Coulomb phase if there is a low energy ${\rm U}(1)^r$ gauge theory with mass gap for all charged states; $r$ is the \emph{rank} of the vacuum.
Furthermore, we assume that there are states carrying electric and magnetic charges with respect to each ${\rm U}(1)$ factor, so the linear space of possible charges has real dimension $2r$.%
\footnote{We exclude IR free non-abelian vacua or ones with non-compact ${\rm U}(1)$ gauge factors from our definition of Coulomb phase.}

Then the set of charges of finite-energy states which occur in the theory form a rank-$2r$ {\bf charge lattice}, ${\Lambda} \simeq {\mathbb Z}^{2r}$.
These include the charges of single-particle states, though there is no requirement that all lattice charges correspond to single-particle states.
The physically occurring charges form a lattice because if localized states%
\footnote{By localized in this context we mean only that the energy density falls off at least as fast as ${\rho}(t,\vec x) \sim |\vec x|^{-4}$ as $|\vec x|\to\infty$.}
with charges $p$ and $q$ occur, then there is a localized finite energy state with charge $p+q$ (e.g., an approximate 2-particle state with the particles sufficiently far apart).
Furthermore, all charges are commensurate by the Dirac quantization condition \cite{Dirac:1931kp, Schwinger:1966nj, Zwanziger:1968ams, Zwanziger:1968rs}, which implies there is a \emph{Dirac pairing}, a non-degenerate, antisymmetric, bilinear map $J:{\Lambda} \times {\Lambda} \to{\mathbb Z}$.
It supplies the charge lattice with a symplectic structure.

Note that the normalization of $J$ is not a matter of convention.
For instance, the value of the Dirac pairing between a pair of (sufficiently localized) dyonic states measures the angular momentum carried by their electromagnetic field \cite{Coleman:1982cx} in units of $\hbar/2$.
With a common definition of electric and magnetic charges --- $e = \int_{S^2} *F$ and $g = (4\pi)^{-1} \int_{S^2} F$ --- the Dirac quantization condition reads $eg=n/2$, $n\in{\mathbb Z}$.
We are implicitly using a different normalization of charges --- e.g., by defining magnetic charge to be $g = (2\pi)^{-1} \int_{S^2} F$ --- in which $J$ is integral.  This  does not change the fact that the normalization of $J$ has a definite physical meaning.

With respect to a particular basis of the lattice, $\{ e_1, \ldots, e_{2r} \}$, the Dirac pairing is represented by a non-degenerate antisymmetric $2r\times 2r$ matrix $J_{ab} = J(e_a, e_b) \in{\mathbb Z}$.
There exists a \emph{symplectic basis} in which $J$ is skew-diagonal, i.e., is of the form
\begin{align}\label{sympbas}
    J &= \begin{pmatrix} & \phantom{-}D \\ -D & \end{pmatrix} &
    &\text{with}&
    D &= {\rm diag} \{d_1, \ldots, d_n\} , &
    d_i &\in \N.
\end{align}
Furthermore, one can choose this basis so that
\begin{align}\label{ddiv}
    d_i \mid d_{i+1} .
\end{align}
In this case the $d_i$ are the unique \emph{invariant factors} of the Dirac pairing, though the symplectic basis is not unique.
If $d_i=1$ for all $i$ we say the Dirac pairing is \emph{principal}.
The existence and uniqueness of this invariant factor decomposition is ensured by the structure theorem for finitely generated modules over a principal ideal domain, and is reviewed --- together with an algorithm for computing the invariant factors --- in appendix \ref{invfacapp}.

For any electric and magnetic charges, whether they are in the charge lattice or not, a ``probe'' Wilson-'t Hooft line operator can be defined by specifying appropriate boundary conditions on the behavior of the electric and magnetic fields as one approaches the line \cite{Kapustin:2005py}.
The electric and magnetic charges of Wilson-'t Hooft line operators are $\mathfrak{u}(1)_E \oplus \mathfrak{u}(1)_M$ 1-form symmetry charges for each IR gauge $\mathfrak{u}(1)$ factor \cite{Gaiotto:2014kfa}.
The set of these line operator charges should include charges in the charge lattice, ${\Lambda}$, since in the IR limit any massive charged state can be approximated by the insertion of a line operator located at the world-line of its center of mass \cite{Wilson:1974sk}.
The charges of probe lines are restricted to lie in a lattice by Dirac quantization, which, for line operators, is the condition that they are mutually local or ``genuine" line operators \cite{Kapustin:2014gua}, i.e., that they are not the boundary of a topological surface operator that can be detected by other genuine line operators.
(For example, these topological surface operators are the world-volume of the Dirac strings emanating from the world-line of a magnetic monopole.)
Thus the charges of genuine probe lines must have integral Dirac pairing with charges in ${\Lambda}$ \cite{Gaiotto:2010be}.

This suggests defining the Dirac-dual lattice, ${\Lambda}^J$, with the interpretation as the lattice of ``possible'' probe Wilson-'t Hooft line operator 1-form charges.
It is the maximal lattice in $\R\otimes_{\mathbb Z} {\Lambda}$ such that \begin{align}\label{dualL}
    J({\Lambda}^J,{\Lambda}) \in {\mathbb Z}.
\end{align}
In other words it is integrally ``dual" to ${\Lambda}$ with respect to the Dirac pairing.
The charge lattice is a sublattice of the dual lattice of index
\begin{equation}
|{\Lambda}^J/{\Lambda}| = ({\rm Pf} J)^2\ {\rm with}\ {\rm Pf} J = \det D = \prod_i d_i,
\end{equation}
so ${\Lambda}$ is a proper sublattice of ${\Lambda}^J$ if and only if $J$ is non-principal.
In terms of the symplectic basis $\{ e_a, \ a=1, \ldots, 2r \}$ in \eqref{sympbas}, the symplectic basis basis of ${\Lambda}^J$ is $\{ d_i^{-1} e_i, d_i^{-1} e_{i{+}r}, \ i=1, \ldots, r \}$

If ${\Lambda}$ is a proper sublattice of ${\Lambda}^J$ then the extension of the Dirac pairing to ${\Lambda}^J$ will not be integral, i.e., $J({\Lambda}^J,{\Lambda}^J) \notin {\mathbb Z}$.
Indeed, in the symplectic basis given above,
\begin{align}
    J = \begin{pmatrix} &\phantom-D^{-1} \\ -D^{-1} & \end{pmatrix}
    \qquad \text{on \ \  ${\Lambda}^J$.}
\end{align}
Define a {\bf line lattice}, ${\mathcal L} \subset {\Lambda}^J$, to be a maximal sublattice of ${\Lambda}^J$ such that $J({\mathcal L},{\mathcal L}) \in {\mathbb Z}$ \cite{Gaiotto:2010be}.
A line lattice is thus interpreted as a maximal lattice of mutually local Wilson-'t Hooft probe line operators of the low energy ${\rm U}(1)^r$ gauge theory.
It follows from its definition that the extension of the Dirac pairing to ${\mathcal L}$ is principal, and that
\begin{equation}
|{\Lambda}^J  / {\mathcal L}| = |{\mathcal L} / {\Lambda}| = {\rm Pf} J.
\end{equation}
There can be finitely many inequivalent line sublattices; the number depends in a complicated way on the prime decomposition of the invariant factors, $d_i$, of the Dirac pairing; some simple examples are computed in appendix \ref{appC}.

In the case of gauge theories, the choice of line lattice is associated to the choice of global structure (including ``discrete theta angles'') of the theory \cite{Aharony:2013hda}.
The connection between the choice of line lattices and this global structure arises as follows.
In gauge theories for which there exist local operators in a Coulomb vacuum which create states carrying (gauge) charges in ${\Lambda}$, Wilson-'t Hooft line operators carrying charges in ${\Lambda}$ can be screened.
That is, there are gauge-invariant line segment operators consisting of Wilson-'t Hooft lines which end on these local operators.
In that case the $(U(1)_E \times U(1)_M)^r$ 1-form group symmetry is broken to a discrete subgroup consisting of those group elements which act trivially on all lines in ${\Lambda}$, since the $S^2$'s which carry the 1-form topological symmetry operators no longer link the line segments.
If the minimum charge in ${\Lambda}$ with respect to a given $U(1)$ 1-form symmetry factor is $d$, then that factor is broken (at least) to its ${\mathbb Z}_d$ subgroup.
Therefore, in a basis corresponding to the invariant factor decomposition \eqref{sympbas}, the 1-form symmetry group is broken (at least) to the finite subgroup, known as the 1-form defect group \cite{Tachikawa:2013hya,DelZotto:2015isa,Albertini:2020mdx}: $\DD^{(1)} = \bigoplus_{i=1}^r \left( ({\mathbb Z}_{d_i})_E \oplus ({\mathbb Z}_{d_i})_M \right)$, where, again, the $d_i$s are the invariant factors in \eqref{sympbas}. But, like the Dirac-dual lattice ${\Lambda}^J$, the charges allowed in this group generally do not have integral Dirac pairing, so the actual 1-form group is a maximal subgroup of $\DD^{(1)}$ mutually local charges (which has cardinality $\prod_i d_i$).
The choice of this subgroup is the global structure computed in \cite{Aharony:2013hda,DelZotto:2022ras}.
The above description in terms of maximal Dirac-local subgroups of the finite group $\DD^{(1)}$ can be recast in terms of ${\mathcal L} \subset {\Lambda}^J$ maximal Dirac-local sublattices.
Since ${\Lambda} \subset {\mathcal L} \subset {\Lambda}^J$, we have the associated finite group inclusions ${\mathcal L}/{\Lambda} \subset {\Lambda}^J/{\Lambda}$, and, by working in the symplectic basis, it is easy to see that  ${\Lambda}^J/{\Lambda} = \DD^{(1)}$.

An immediate consequence is that if there is a non-trivial global structure or 1-form symmetry, then the Dirac pairing on the charge lattice is not principal.  So from this perspective, and the results of \cite{Aharony:2013hda}, it is now obvious that ${\mathcal N}=4$ sYM theories will generally have non-principal Dirac pairings. In the next section we show how to compute the Dirac pairings directly in the field theory.

The results derived in the next section agree with those which appeared in \cite{DelZotto:2022ras}.
In the derivation of \cite{DelZotto:2022ras}, though, the Dirac pairing appears in a more indirect way, via the \emph{symmetry TFT} \cite{Apruzzi:2021nmk,Apruzzi:2022dlm}, i.e., the topological sector of the non-invertible field theory whose boundary theory is the four dimensional theory.

\section{Non-principal Dirac pairings for ${\mathcal N}{=}4$ super Yang-Mills} \label{sec3}

In the case of an ${\mathcal N}{=}4$ sYM theory with simple gauge algebra ${\mathfrak g}$, in a Coulomb vacuum the electric charges span the root lattice ${\Gamma}_r$ of ${\mathfrak g}$ since at weak coupling all fields are in the adjoint representation.
A semiclassical analysis shows that magnetic monopole charges span the co-root lattice ${\Gamma}_r^\vee$, which is the ``magnetic" root lattices of of ${\mathfrak g}^\vee$, the GNO or Langlands dual of ${\mathfrak g}$ \cite{Goddard:1976qe, Witten:1978mh, Osborn:1979tq, Kapustin:2005py}.
(See appendix \ref{Cartmat} for a review of simple Lie algebra definitions.)
So the charge lattice is
\begin{align}\label{chrglatt}
    {\Lambda} = {\Gamma}^\vee_r \oplus {\Gamma}_r ,
\end{align}
the span of the electric and magnetic sublattices.
These sublattices are each lagrangian with respect to $J$, so the Dirac pairing is determined by the $r\times r$ pairing $B := J({\Gamma}_r,{\Gamma}^\vee_r)$ between the two.
And any integral non-degenerate $B$ defines a potential Dirac pairing on ${\Lambda}$.

The Dirac pairing is fixed up to normalization by demanding it be Weyl invariant.
Recall that the Weyl group is a discrete subgroup of the gauge group which acts on the charge lattice, so any physical observable --- such as the value of the Dirac pairing between a pair of (sufficiently localized) charged states --- must be invariant under the Weyl group.

The Weyl group, $W$, of ${\mathfrak g}$ and ${\mathfrak g}^\vee$ are the same.
Given its linear action on ${\Gamma}_r$, the magnetic Weyl action is determined by the semiclassical description of magnetic monopoles \cite{Goddard:1976qe, Kapustin:2005py}.
Indeed, by GNO duality, the magnetic root lattice is the dual of the ``electric" weight lattice, ${\Gamma}^\vee_r \simeq ({\Gamma}_w)^*$,%
\footnote{Here the dual $({\Gamma}_w)^*$ means the space of linear maps of ${\Gamma}_w$ to ${\mathbb Z}$, and should not be confused with the notion of ``dual" with respect to the Dirac pairing used in \eqref{dualL}.}
and so inherits a dual action of the Weyl group.%
\footnote{Some Weyl groups have non-trivial outer automorphisms so one might worry that there are inequivalent choices of how the electric and magnetic Weyl actions are put into correspondence.
Weyl groups are comprised of rotations and reflections with respect to the Killing metric on the weight space, which can be characterized as the elements having positive and negative determinants, respectively, when realized as matrices in some basis.
Thus the dual of reflections are also reflections.
All automorphisms of Weyl groups which preserve the set of reflections are either inner or are Dynkin diagram automorphisms \cite{Franzsen:2001}.
In either case they are equivalent to a permutation of a basis of simple roots, and so are equivalent up to a choice of lattice basis.}
Let ${\mathfrak t}\subset{\mathfrak g}$ be a Cartan subalgebra and ${\mathfrak t}^*\subset{\mathfrak g}^\vee$ the dual Cartan subalgebra (or weight space); so ${\Gamma}_r \subset {\mathfrak t}^*$ and ${\Gamma}^\vee_r \subset {\mathfrak t}$.
Then the $W$ action on these lattices is generated by reflections ${\sigma}_k$ and ${\sigma}_k^\vee$ associated to each simple root ${\alpha}_k$, $k=1,\ldots,r$, whose actions are given in \eqref{Weylact}, so ${\sigma}_k\in W$ acts on ${\Lambda} = {\Gamma}^\vee_r \oplus {\Gamma}_r$ as ${\sigma}^\vee_k \oplus {\sigma}_k$.

Invariance of the Dirac pairing is the condition that
\begin{align}\label{Cartan1}
    B_{ij} := J({\alpha}_i,{\alpha}^\vee_j) &= J({\sigma}_k({\alpha}_i),{\sigma}^\vee_k({\alpha}^\vee_j)) \quad \text{for all} \quad k = 1, \ldots, r .
\end{align}
Using the action \eqref{Weylact} then implies that
\begin{align}\label{Cartan2}
    {\alpha}_i({\alpha}^\vee_k) \, B_{kj} + B_{ik} \, {\alpha}_k({\alpha}^\vee_j)
    = {\alpha}_i({\alpha}^\vee_k) \, B_{kk} \, {\alpha}_k({\alpha}^\vee_j)
\end{align}
for all $i$, $j$, $k$.  Choosing $i=j$ and $i=k$ then implies that $B_{ii}=2n$ for some $n$ and for all $i$, and that $B_{ij} = n \ {\alpha}_i({\alpha}^\vee_j)$.
The Cartan matrix of ${\mathfrak g}$, $A_{\mathfrak g}$, is defined to have matrix elements $(A_{\mathfrak g})_{ij} = {\alpha}_i({\alpha}^\vee_j)$.
The Cartan matrices of simple Lie algebras are recalled in appendix \ref{Cartmat}.

Thus invariance under the Weyl action determines the Dirac pairing up to an overall normalization.
And we have found that in a particular basis (corresponding to simple co-roots and roots) the Dirac pairing of the ${\mathcal N}{=}4$ sYM with (electric) gauge algebra ${\mathfrak g}$ is
\begin{align}\label{Jfg}
    J_{\mathfrak g} &=
    n\, \begin{pmatrix} 0 & (A_{\mathfrak g})^t \\ - A_{\mathfrak g} & 0 \end{pmatrix}  = n\, \begin{pmatrix} 0 & A_{{\mathfrak g}^\vee} \\ - A_{\mathfrak g} & 0 \end{pmatrix}, &
    n &\in \N ,
\end{align}
since $A_{{\mathfrak g}^\vee} = (A_{\mathfrak g})^t$, and $n$ is a non-zero integer because of the integrality of the Dirac pairing and the fact that the entries of the Cartan matrix have no common divisor.  (The sign of $J_{\mathfrak g}$ is conventional.)

The normalization can be determined by constructing electrically and magnetically charged states at weak coupling (i.e., semiclassically) with minimum non-zero Dirac pairing.
In the case of Yang-Mills theory with adjoint matter the Dirac pairing between the W-boson and 't Hooft-Polyakov monopole corresponding to an $\mathfrak{su}(2)$ subalgebra associated to a root is 2 in the units used here \cite{tHooft:1974kcl, Polyakov:1974ek, Coleman:1982cx}.
This implies that $n=1$ in \eqref{Jfg}, since diagonal elements of $A_{\mathfrak g}$ are 2.

This result also agrees with that found from the  BPS quiver description of the BPS spectrum as reported in \cite{DelZotto:2022ras}.%
\footnote{The Dirac pairing shown there has the form $J = \left(\begin{smallmatrix} A-A^t & A^t\\ -A & 0\end{smallmatrix}\right)$ which is related to \eqref{Jfg} by a change of lattice basis by the $r\times r$ block $\GL(2r,{\mathbb Z})$ matrix $\left(\begin{smallmatrix} 1&0\\1&1\end{smallmatrix}\right)$.}

With the Dirac pairing in hand, we now determine its invariant factors --- the $d_i$ in \eqref{sympbas} and \eqref{ddiv} --- that invariantly characterize it.
One can determine them by direct computation, as outlined in appendix \ref{invfacapp}.
They are simply the diagonal elements of the Smith normal form of the Cartan matrix, which are
\begin{align}\label{N4invfac}
\begin{array}{ll|cccccc}
{\mathfrak g} & & d_1 & d_2 & \cdots & d_{r-2} & d_{r-1} & d_{r} \\
\hline
\mathfrak{su}(r{+}1) & r\ge1 & 1 & 1 & \cdots & 1 & 1 & r{+}1 \\
\mathfrak{so}(2r{+}1)\ & r\ge1 & 1 & 1 & \cdots & 1 & 1 & 2 \\
\mathfrak{sp}(2r) & r\ge1 & 1 & 1 & \cdots & 1 & 1 & 2 \\
\mathfrak{so}(2r) & r\ge3 \ \text{odd} & 1 & 1 & \cdots & 1 & 1 & 4 \\
\mathfrak{so}(2r) & r\ge2 \ \text{even} & 1 & 1 & \cdots & 1 & 2 & 2 \\
E_r & r=6,7,8 & 1 & 1 & \cdots & 1 & 1 & 9{-}r \\
F_4 & & 1 & 1 & \cdots & 1 & 1 & 1 \\
G_2 & & 1 & 1 & \cdots & 1 & 1 & 1
\end{array}
\end{align}
Note that the only Dirac pairings which are principal are those of the $E_8$, $F_4$, and $G_2$ gauge theories.
By the structure theorem (see appendix \ref{invfacapp}) which ensures the existence and uniqueness of the canonical symplectic structure \eqref{sympbas} with invariant factors satisfying the divisibility conditions \eqref{ddiv}, and by the fact that from \eqref{Jfg} $\det A_{\mathfrak g} = \prod_i d_i$, the determinants of the Cartan matrices, listed in appendix \ref{Cartmat}, completely determine the invariant factors listed in \eqref{N4invfac} without further computation except in the cases of ${\mathfrak g}=\mathfrak{su}(r{+}1)$ for those $r{+}1$ not square-free and of $\mathfrak{so}(2r)$.
We determined these last two cases by direct computation for $r < 20$, but have not worked out a proof for all $r$.

As noted in the last section, the number of inequivalent line lattices following from these Dirac pairings should equal the number of different global structures for these Yang-Mills theories as computed in \cite{Aharony:2013hda}.
This equality follows from the observation that $\oplus_{i=1}^r {\mathbb Z}_{d_i}$ in \eqref{N4invfac} is the center of the simply connected Lie group of ${\mathfrak g}$, and from the connection between the counting of maximal Dirac-local symplectic sublattices of the Dirac-dual lattice and the counting of maximal Dirac-local subgroups of the defect 1-form group.
In appendix \ref{appC} we do the sublattice counting directly, verifying the equality.

\section{Comparison to CB geometry constructions}\label{sec4}

We now discuss how non-principal Dirac pairing on the charge lattice and the choice of line lattice appear in the Coulomb branch geometry of ${\mathcal N}{=}2$ field theories.

The Coulomb branch ${\mathcal C}$ --- the moduli space of Coulomb vacua --- is a special K\"ahler (SK) space \cite{Donagi:1995cf, Freed:1997dp}.
The SK structure on a rank-$r$ Coulomb branch can be expressed in terms of a family of rank-$r$ abelian varieties $X_u$, $u\in{\mathcal C}$, varying holomorphically over ${\mathcal C}$.
$X_u$ often appears as a sub-variety of the Jacobian variety of the Seiberg-Witten curve ${\Sigma}_u$ in terms of which some SK geometries are written \cite{Seiberg:1994rs, Seiberg:1994aj, Donagi:1995cf}.
The abelian varieties carry a choice of polarization, which can be thought of as a choice of integral symplectic pairing $P:{\Lambda}_X\times{\Lambda}_X\to{\mathbb Z}$ on their lattice of homology 1-cycles ${\Lambda}_X := H_1(X_u,{\mathbb Z}) \simeq {\mathbb Z}^{2r}$.
$P$ is the pairing induced by the intersection pairing on homology 1-cycles on the Seiberg-Witten curve.
We will refer to ${\Lambda}_X$ as the {\bf homology lattice} and $P$ as its {\bf polarization} which should not be confused with the charge and line lattice and their Dirac pairings. In fact our focus will be on understanding the relationship among these objects.

The main conclusion is that the homology lattice and polarization appearing in the SK structure of the Coulomb branch need not coincide with the charge lattice and Dirac pairing of the field theory.
Indeed, there is a discrete set of closely-related SK structures compatible with a given Coulomb branch K\"ahler geometry, in which (homology lattice, polarization) can take values ranging from the (charge lattice, Dirac pairing) to various (line lattices, principal pairings) as well as intermediate possibilities.
This choice of SK structure can thus be used to encode a choice of global structure of the field theory.

We illustrate this in the context of ${\mathcal N}{=4}$ sYM theories. Specifically, we examine some examples of Coulomb branch geometries of ${\mathcal N}{=}2^*$ theories appearing in the literature and show how to  determine how their homology lattices and polarizations are related to the field theory charge lattices and Dirac pairings.
These cases have an exactly marginal coupling and so S-duality groups related to the topology of their conformal manifolds.
We discuss how the S-duality group visible from the Coulomb branch geometry is related to that of the field theory.

\subsection{Review of ${\mathcal N}=4$ moduli space geometry}\label{sec41}

The moduli space of ${\mathcal N}{=}4$ sYM with gauge algebra ${\mathfrak g}$ is the flat orbifold geometry \cite{Seiberg:1997ax}:
\begin{equation}
\mathcal{M}_{\mathfrak g}\equiv\C^{3r}/W_{\mathfrak g}.
\end{equation}
Here ${\sigma}_k\in W_{\mathfrak g}$ acts as ${\sigma}_k: \vec z\otimes{\mu} \mapsto \vec z\otimes {\sigma}_k({\mu})$ where $\vec z\otimes{\mu} \in \C^3 \otimes_\R {\mathfrak t}^* \simeq \C^{3r}$ with ${\mathfrak t}\subset {\mathfrak g}$ is a Cartan subalgebra and the ${\sigma}_k$ action on ${\mathfrak t}^*$ given by \eqref{Weylact}.
Choosing an ${\mathcal N}{=}2 \subset {\mathcal N}{=}4$ subalgebra, the associated Coulomb branch ``slice", ${\mathcal C}_{\mathfrak g}$, is the orbifold ${\mathcal C}_{\mathfrak g} \doteq \C^r/W_{\mathfrak g}$ where ${\sigma}\in W_{\mathfrak g}$ acts on $\C^r \simeq \C \otimes_\R {\mathfrak t}^*$ as above.
The complex scaling action of the spontaneously broken scale plus R-symmetries is diagonalized by a (any) algebraic basis of Weyl-invariant polynomials in the complex coordinates on $\C^r$, $\{ u_1, \ldots, u_r\}$, whose degrees are the exponents plus one of the Weyl group.%
\footnote{The exponents of Weyl group $W_{\mathfrak g}$ are the $r$ integers $0<e_i<h$ with $\gcd(e_i,h)=1$ where $h$ is the Coxeter number of ${\mathfrak g}$ \cite{Humphreys:1990}.}
Since Weyl groups act as (real) complex reflection groups on $\C^r$, the Chevalley-Shepherd-Todd theorem \cite{Shephard:1954, Chevalley:1955} implies that ${\mathcal C}_{\mathfrak g} \simeq \C^r \ni (u_1,\ldots,u_r)$ as a complex space.

This metric and complex structure by themselves do not specify the SK geometry on ${\mathcal C}_{\mathfrak g}$.
An SK structure on ${\mathcal C}_{\mathfrak g}$ can be specified by choosing a holomorphic section, $s$ --- the ``special section" --- of a flat rank $2r$ complex vector bundle over ${\mathcal C}_{\mathfrak g}$ with structure group ${\rm Sp}_P(2r,{\mathbb Z})$ \cite{Freed:1997dp, Argyres:2019ngz, Argyres:2019yyb}.
Its fibers are the complexification of the (linearly) dual homology lattice, $\C \otimes_{\mathbb Z} {\Lambda}_X^* \simeq \C^{2r}$, so inherit a constant symplectic form from the polarization $P$ of ${\Lambda}_X$.
$s$ satisfies the integrability condition $J(ds \, \overset{\^}{,} \, ds)=0$ where $d$ is the exterior derivative on ${\mathcal C}_{\mathfrak g}$, and the (positive) metric on ${\mathcal C}_{\mathfrak g}$ is given by $ds^2 = i J(ds , d\overline s)$.
In the ${\mathcal N}{=}4$ case the components of $s$ are locally flat coordinates on ${\mathcal C}_{\mathfrak g}$ which vanish at the origin (the conformal vacuum) \cite{Argyres:2019ngz, Argyres:2019yyb}.
The physical significance of the special section is that the dual pairing ${\Lambda}_X^* \times {\Lambda}_X \to {\mathbb Z}$ induces the central charge map $Z: {\mathcal C}_{\mathfrak g} \times {\Lambda}_X \to \C$ where $Z(u,q) := s(u)(q)$ and whose norm is the BPS mass of a state of EM charge $q$ in the vacuum $u$.

Choose a basis $(m_i, e^i)$, $i=1,\ldots,r$, of ${\Lambda}_X$ such that the $e^i$ and $m_i$ span lagrangian sublattices with respect to $P$.  Write $P$ in this basis as the non-degenerate integral $2r\times 2r$ matrix
\begin{align}\label{JB}
    P &= \begin{pmatrix} & -B^t \\ B & \end{pmatrix} ,
\end{align}
where $B^i_{\ j} := P(e^i,m_j)$.
In the dual basis $(m^i,e_i)$ of ${\Lambda}_X^*$ (so $m^i(m_j)={\delta}^i_j$, $m^i(e_j)=0$, etc.) the special section is $s := a^D_i m^i + a^i e_i$, and $P(e_i,m^j) = -(B^{-1})^j_{\ i}$ so the induced pairing, $P^*= - P^{-1}$, is the inverse transpose of \eqref{JB} in the dual basis.
The $a_i$ are ``special coordinates" and the $a^D_i$ are ``dual special coordinates".
The metric is $ds^2 = 2 {\rm Im} ( da^D_i (B^{-1})^i_{\ j} d\overline a^j)$.
Flatness of $ds^2$ and the special coordinates, and the SK integrability of $s$ then imply that $a^D_j = {\tau}_{jk} a^k$,
where ${\tau}$ is a constant complex $r\times r$ matrix satisfying
${\tau} B = ({\tau} B)^t$.
Positivity of the metric implies ${\rm Im}({\tau} B)>0$.
These also imply that $a^D$ and $a$ are separately good holomorphic coordinates on the regular points of ${\mathcal C}_{\mathfrak g}$.

In terms of the holomorphic family $X_u$ of abelian varieties over ${\mathcal C}_{\mathfrak g}$ mentioned above, ${\Lambda}_X$ and $P$ are its homology lattice and polarization, and $B{\tau}$ is its complex modulus.

A monodromy $M \in {\rm Sp}_P(2r,{\mathbb Z})$ of the EM charge lattice satisfies $MJM^t=J$, which implies that $M^{-t} P^* M^{-1} = P^*$, so ${\rm Sp}_{P^*}(2r,{\mathbb Z}) = {\rm Sp}_P(2r,{\mathbb Z})$.
We define the dual EM monodromy by $M^* := M^{-t}$.
Upon continuing the special section along a closed path ${\gamma}$ in ${\mathcal C}_{\mathfrak g} \setminus \{\text{singular locus}\}$, it can suffer a monodromy $M^*({\gamma}) \in {\rm Sp}_{P^*}(2r,{\mathbb Z})$.
This defines a monodromy map ${\mu}^*: \pi_1({\mathcal C}_{\mathfrak g} \setminus \{\text{singular locus}\}) \to {\rm Sp}_{P^*}(2r,{\mathbb Z})$, and we call ${\rm im}{\mu}^* \subset {\rm Sp}_{P^*}(2r,{\mathbb Z})$ the \emph{EM duality group} of the theory in question.
Constancy of ${\tau}_{jk}$ then implies it is fixed by the EM duality group, so $m^*{\tau}+n^* =  {\tau}(p^*{\tau}+q^*)$ for all $\left(\begin{smallmatrix} m^* & n^* \\ p^* & q^* \end{smallmatrix}\right) \in {\rm im}{\mu}^*$.

In the ${\mathcal N}{=}4$ case where the Coulomb branch is an orbifold by the Weyl group action, the EM duality group is the image of the Weyl group in ${\rm Sp}_{P^*}(2r,{\mathbb Z})$ given by its action on the special section.
Furthermore, in this case ${\tau}$ is only by EM duality invariance up to an overall complex constant which is the exactly marginal coupling constant of the ${\mathcal N}{=}4$ sYM theory.

\subsection{Connection between special K\"ahler structures and Dirac pairing}\label{orbifoldSK}

To start with, let us choose as special coordinates $a^k = {\alpha}_k^*$ where ${\alpha}_k^*$ are is the dual basis of simple roots on $\C \otimes {\mathfrak t}^*$.
(Dual simple roots are linearly independent real linear functions on ${\mathfrak t}^*$, so extend by linearity to good complex coordinates in neighborhoods of ${\mathcal C}_{\mathfrak g}$ away from orbifold fixed points.)
This corresponds to a choice of ``electric" lagrangian sublattice basis $\{e^k\}$ of ${\Lambda}_X$ with $e^k = {\alpha}_k$, the simple roots.
Thus we are choosing the electric homology sublattice to be the root lattice.
Likewise we can choose as dual special coordinates $a^D_k = ({\alpha}^\vee_k)^*$ (the dual basis of the simple co-roots), corresponding to a the choice of ``magnetic" lagrangian sublattice basis the simple co-roots $\{m_k = {\alpha}^\vee_k\}$, so the magnetic homology sublattice is the co-root lattice.
In this case it is clear that we have chosen the SK homology lattice to be
\begin{align}\label{vr-r}
    {\Lambda}_X = {\Gamma}^\vee_r \oplus {\Gamma}_r,
\end{align}
and so it coincides with the charge lattice \eqref{chrglatt} of the field theory.

A Weyl element ${\sigma}\in W_{\mathfrak g}$ acts on the electric charge lattice in this simple root basis by multiplication by an integral $r\times r$ matrix which we denote by the same symbol, ${\sigma}$.
Denote the matrix representation of the Weyl group element ${\sigma}\in W_{\mathfrak g}$ on this basis by ${\sigma}^\vee$, as in \eqref{Weylact}.
Then $M_{\sigma} := \left(\begin{smallmatrix} {\sigma}^\vee & 0\\ 0& {\sigma} \end{smallmatrix}\right) \in \GL(2r,{\mathbb Z})$ for ${\sigma}\in W_{\mathfrak g}$ preserve the symplectic form $P$ \eqref{JB} with $B=-A_{\mathfrak g}$, the Cartan matrix of ${\mathfrak g}$.
This is just a restatement of the calculation of the last section around equations \eqref{Cartan1} and \eqref{Cartan2}.
Thus the monodromies $M_{\sigma}$ are in ${\rm Sp}_P(2r,{\mathbb Z})$, so the above choices of special and dual special coordinates give a consistent SK structure on ${\mathcal C}_{\mathfrak g}$ with polarization $P$, and this polarization coincides with Dirac pairing \eqref{Jfg} of the field theory.

But it is also clear that other, inequivalent, SK structures can be put on the ${\mathcal N}{=}4$ Coulomb branch orbifold.
The main consistency requirement is that the special section be chosen so that there is a basis in which its monodromies are given by integer matrices which preserve an integer symplectic form.
Since the monodromies are induced by the action of the orbifolding Weyl group, $W_{\mathfrak g}$, such special sections will be related to sublattices of the weight plus co-weight lattice ${\Gamma}_w^\vee\oplus {\Gamma}_w$ for ${\mathfrak g}$ which are preserved under $W_{\mathfrak g}$.
As reviewed in appendix \ref{Cartmat}, for ${\Gamma}_w$ these sublattices are the group lattices, which are in  1-to-1 correspondence with the different possible subgroups of the center of the simply connected gauge group; and similarly for ${\Gamma}^\vee_w$ as summarized in \eqref{g-lattices}.

So, for example, one can take special coordinates $a^k = {\alpha}_k^*$, the dual simple roots, as in the previous example, but choose the dual special coordinates differently to be $a^D_k = ({\omega}^\vee_k)^*$, where ${\omega}_k$ are a basis of fundamental co-weights.
This corresponds to choosing the homology lattice
\begin{align}\label{vw-r}
    {\Lambda}_X = {\Gamma}^\vee_w \oplus {\Gamma}_r .
\end{align}
From the definition \eqref{Weylact} of the Weyl group actions it follows that if ${\sigma}_i$ is the matrix representation of the ${\sigma}_i$ action in the simple root basis, then ${\sigma}^\vee_i$ acts in a fundamental co-weight basis as matrices $({\sigma}_i)^{-t}$.
Since the generating ${\sigma}_i\in W_{\mathfrak g}$ are reflections, $({\sigma}_i)^{-t} = ({\sigma}_i)^t$, making their integrality apparent.
Thus the monodromy matrices $M_{\sigma} := \left(\begin{smallmatrix} {\sigma}^{-t} & 0\\ 0& {\sigma} \end{smallmatrix}\right) \in \GL(2r,{\mathbb Z})$,and, furthermore, it follows immediately that they preserve a principal polarization
\begin{align}\label{Pvw-r}
    P = \bpm0 & 1 \\ -1  & 0 \end{pmatrix}.
\end{align}
This gives an example of an SK structure on ${\mathcal C}_{\mathfrak g}$ in which the homology lattice is not the charge lattice.
Since the homology lattice is principally polarized, and since the charge lattice \eqref{vr-r} is a sublattice of \eqref{vw-r}, it is natural to guess that this SK structure corresponds to a choice of line lattice, and so to a choice of global structure of the field theory.

Indeed, it is easy to see that \eqref{vw-r} and \eqref{Pvw-r} correspond to the global structure $({\til G}/Z({\til G}))_0$ in the notation of \cite{Aharony:2013hda}, where the electric gauge group is the ``adjoint" group with trivial center, and the magnetic gauge  group is the simply-connected group $\widetilde G^\vee$.
This follows because ${\Gamma}_r$ and ${\Gamma}^\vee_w$ are the group lattices of these two groups.
The zero subscript refers to the fact that the polarization \eqref{Pvw-r} on ${\Lambda}_X$ is in block-skew form.
It is not hard to see that there are other SK structures in which,  for instance, the dual special coordinates are the fundamental co-weights shifted by multiples of weights according to their charges under the center $Z({\til G})$ of the gauge group.
This effectively shifts the generating monodromy matrices $M_{\sigma}$ to block triangular form which preserve a principal but non-block-skew polarization and which correspond to global forms $({\til G}/Z({\til G}))_n$ with $0\neq n\in Z({\til G})$.

SK structures corresponding to the other global forms can be constructed along similar lines.
They correspond to choices of Weyl-invariant sublattices of ${\Gamma}^\vee_w\oplus {\Gamma}_w$ which preserve a principal integral symplectic pairing.
As was discussed in the last two sections and in appendix \ref{appC}, this is equivalent to the classification of global forms given in \cite{Aharony:2013hda}.

Note that it is also possible to construct SK structures ``intermediate" to the principally polarized line lattices and the ones with the physical Dirac pairing on the charge lattice.
The homology lattices which  occur in these cases could be interpreted as ``non-maximal" choices of mutually local line operators.
Their polarizations will be non-principal, and will have invariant factors which are divisors of those of the Dirac pairing.

Finally, ${\mathcal N}{=}4$ theories have a 1-dimensional conformal manifold and an associated S-duality group.
As an abstract group, the S-duality group is the fundamental group of the conformal manifold in the orbifold sense.
Orbifold fixed points on the conformal manifold can be detected as  those values of the coupling where the effective theories on the Coulomb branch have an enhanced finite global symmetry group.
If we take the coupling (a coordinate on the conformal manifold) to be ${\tau}$ in the complex upper half plane, then the conformal manifolds for ${\mathcal N}{=}4$ sYM theories are fundamental domains of the m\"obius action of a finite-index subgroup of ${\rm SL}(2,{\mathbb Z})$ (or of a closely related group in the cases of ${\mathfrak g}=G_2$ or $F_4$ \cite{Argyres:2006qr}).

In general, the S-duality group cannot be unambiguously reconstructed from the SK geometry of the Coulomb branch.
The reason is simply that the Coulomb branch geometry only captures partial information about the field theory and so might present two Coulomb branch vacua with distinct physics as having isomorphic SK geometries.
So, if we try to reconstruct the conformal manifold by identifying values of ${\tau}$ in the upper half plane with SK-isometric Coulomb branches, we will generally make mistaken identifications, leading to too small a conformal manifold.
Equivalently this will give too large an S-duality group, i.e., one which is a smaller-index subgroup of ${\rm SL}(2,{\mathbb Z})$ than the physical S-duality group.

The choice of global structure (or choice of line lattice) of a field theory is an important datum distinguishing them.
As emphasized and explored in \cite{Aharony:2013hda}, the global structures of a sYM theory with given gauge algebra ${\mathfrak g}$ form orbits under S-duality transformations.
The S-duality group and conformal manifold thus depend on the global structure orbit.
For instance, if a given global structure formed a single orbit by itself, the S-duality group would be the full ${\rm SL}(2,{\mathbb Z})$ and the conformal manifold would be a punctured sphere with a ${\mathbb Z}_2$ and a ${\mathbb Z}_3$ orbifold point.
(The puncture is the weak-coupling limit.)
An orbit involving more global structures will give a smaller S-duality group and a conformal manifold with a different set of orbifold points and punctures.

Since we have seen how to encode the choice of global structure in a choice of SK structure on the Coulomb branch, it follows that the S-duality groups and conformal manifold topologies that can be accessed from the Coulomb branch geometries with principally polarized homotopy lattices should correspond to the orbits found in \cite{Aharony:2013hda}.
The following subsections discuss from this point of view examples of Coulomb branch geometries for ${\mathcal N}{=}2^*$ theories that have appeared in the literature.

\subsection{${\mathcal N}{=}2^*$ Coulomb branch geometries}\label{uNcurves}

We now discuss ${\mathcal N}{=}2^*$ sYM Coulomb branch geometries from the perspective of understanding whether their homology lattices are charge lattices, line lattices, or something intermediate.  These geometries have been given in terms of Seiberg-Witten curves derived as spectral curves of integrable systems for ${\mathfrak g}=\mathfrak{su}(N)$ in \cite{Donagi:1995cf} and for other simple ${\mathfrak g}$ in \cite{DHoker:1997hut, DHoker:1998xad}.

Naively, the Jacobian variety of a Riemann surface is principally polarized by its intersection pairing, and so one might think that the homology lattice derived from a Seiberg-Witten curve (family of Riemann surfaces) will be principally polarized.
By the dictionary worked out in the last subsection, this would seem to imply that the homology lattice of these SK geometries correspond to line lattices.
But this is not necessarily the case.
The Seiberg-Witten curves which appear in various constructions often have genus greater than the rank of the Coulomb branch.
As a result their homology lattices have too large a rank to be interpreted directly as either charge or line lattices.
Instead, an extra condition picking out an appropriate-rank sublattice of the homology lattice must be imposed.
The principal polarization of homology lattice, restricted to this sublattice, need no longer be principal.

For example, the integrable system \cite{Donagi:1995cf}, IIA/M theory brane \cite{Witten:1997sc}, and S-class $A_{N-1}$ \cite{Gaiotto:2009we, Gaiotto:2009hg} constructions of the ${\mathcal N}{=}4$ $\mathfrak{su}(N)$ sYM theory all describe a Seiberg-Witten curve, ${\Sigma}_N$, which is a bouquet of $N$ tori all of complex modulus ${\tau}$ and all identified at a marked point.
This degenerate genus-$N$ curve is interpreted as the SW curve of the $\mathfrak{u}(N)$ theory, whose Weyl group, $S_N$, acts by permuting the tori in ${\Sigma}_N$.
The rank of the Coulomb branch is $N{-}1$, so the homology lattice of $S_N$ has rank 2 greater than what is desired.

Indeed, the lattice ${\Lambda}_X$ of homology 1-cycles on ${\Sigma}_N$ is interpreted as the lattice of (possible) charges for the $\mathfrak{u}(N)$ theory and its intersection form is the Dirac pairing.
The basis $\{m_i, e^i,i=1,\ldots,N\}$ of the homology lattice, where $\{m_i,e^i\}$ is a canonical basis of the $i$th torus, has intersection form $P(m_i, e^j) = {\delta}_i^j$.

Now restrict to the sublattice of the homology lattice which corresponds to states neutral under the central $\mathfrak{u}(1)$ factor of the $\mathfrak{u}(N)$ gauge algebra, thereby identifying the effective homology lattice of the $\mathfrak{su}(N)$ gauge theory SW curve.
We do this by identifying the rank-$2(N-1)$ symplectic sublattice of the rank-$2N$ homology lattice which is invariant under the Weyl group action.

An element $\pi\in S_N$ of the Weyl group acts as $\pi: \{m_i \, , \, e^i\} \mapsto \{m_{\pi(i)} \, , \, e^{\pi(i)}\}$.
Decompose $\R^{2N} =  \R\otimes_{\mathbb Z} {\Lambda}_X$ into invariant subspaces $\R \oplus \R \oplus \R^{N-1} \oplus \R^{N{-}1}$ under this action, where the first two factors are generated by $\sum_{i=1}^{N} m_i$ and $\sum_{i=1}^{N} e^i$, respectively, and the second two factors have bases $\{m_i {-} m_{i+1}\}$ and $\{e^i {-} e^{i+1}\}$ for $i=1, \ldots, N{-}1$, respectively.
Though this decomposition is not unique, if we require that the decomposition is into symplectic subspaces with respect to the Dirac pairing, then we do get a unique decomposition $\R^{2N} \simeq \R^2 \oplus \R^{2(N-1)}$ where the $\R^{2(N-1)}$ factor is the sum of the last two factors of the previous decomposition.
Then the rank-$2(N{-}1)$ sublattice invariant under the Weyl group action is ${\Lambda}_{\mathfrak{su}(N)} := {\Lambda}_X \cap \R^{2(N-1)}$ which has basis $\{ m_i {-} m_{i+1} \, , \, e^i {-} e^{i+1}\}$ for $i=1,2,\ldots,N{-}1$.
The pairing induced on the $\mathfrak{su}(N)$ sublattice from the intersection pairing on ${\Lambda}_X$, $P(m_i {-} m_{i+1} , e^j {-} e^{j+1} ) = 2{\delta}_{i,j} - {\delta}_{i+1,j} - {\delta}_{i,j+1}$, is the Cartan matrix for $\mathfrak{su}(N)$.

Thus the homology (sub)lattice and (induced) polarization of this SW curve are precisely those of the $\mathfrak{su}(N)$ sYM charge lattice and Dirac pairing.
As a result, these SK structures do not encode any choice of global structure of the field theory.
We therefore expect that the S-duality group that is visible from this curve should be the full ${\rm SL}(2,{\mathbb Z})$, and not one of the more refined subgroups associated to a given orbit of line lattices described in \cite{Aharony:2013hda}.
Indeed, the ${\rm SL}(2,{\mathbb Z})$ S-duality of the curve is obvious from its initial description as a bouquet of identical tori all with one marked point and the same complex modulus.

The integrable system spectral curves for other simple Lie algebras ${\mathfrak g}$ \cite{DHoker:1998xad} have a similar structure.
They give Riemann surfaces whose genus is the dimension of a non-trivial irreducible representation of the Lie algebra, so is always greater than the rank of the Lie algebra.
So it seems likely that their associated homology sublattices have non-principal induced polarizations.
It would be interesting to check whether these coincide with the charge lattices and Dirac pairings computed in the last section.

\subsection{Other curves for $\mathfrak{su}(2)$ and $\mathfrak{su}(3)$ ${\mathcal N}{=}4$ sYM} \label{su2curves}

In low ranks the possible SK geometries are better understood.
In rank 1 a full classification of these geometries is known \cite{Argyres:2015ffa, Argyres:2015gha, Argyres:2016xua}, and among them there are two corresponding to the $\mathfrak{su}(2)$ ${\mathcal N}{=}2^*$ theory.
At rank 2 much less is known \cite{Martone:2021ixp}, but a second curve --- besides the one discussed in the previous subsection --- describing the $\mathfrak{su}(3)$ ${\mathcal N}{=}2^*$ theory is known.
These additional curves are precisely the ones expected encoding the global structure of the field theory.

\paragraph{su(2).}

One curve is the $N=2$ specialization of the $\mathfrak{su}(N)$ curves described in the last subsection.
It was written directly in terms of a genus-1 Riemann surface in the original paper \cite{Seiberg:1994aj} of Seiberg and Witten.
As argued in detail in \cite{Argyres:2015ffa, Argyres:2015gha}, this curve has non-principal homology lattice polarization with invariant factor $2$.
(At rank 1 the invariant factor appears as an overall normalization of the polarization so is somewhat subtle to identify correctly.)
This coincides with the Dirac pairing on the charge lattice.
And, indeed, upon turning on the ${\mathcal N}{=}2^*$ mass deformation, the conformal singularity on the Coulomb branch splits into three singularities corresponding to IR free $\mathfrak{u}(1)$ gauge theories with massless hypermultiplets of (magnetic, electric) charges $(1,0)$, $(1,-1)$, and $(0,1)$.
These charges span the whole homology lattice of the curve, and so show that the homology lattice and charge lattice coincide.
This is also consistent with the S-duality group visible from this curve being the full ${\rm SL}(2,{\mathbb Z})$ group.

In \cite{Argyres:2015gha} a second curve with Coulomb branch consistent with the $\mathfrak{su}(2)$ ${\mathcal N}{=}2^*$ theory was constructed.
The homology lattice of this curve has principal polarization, so we expect it should be identified with the ${\rm SU}(2)$ with a choice of line lattice.
In fact, the three inequivalent choices of line lattices form a single orbit under S-duality, so there should only be a single such principally polarized SK geometry.
This interpretation is borne out by a closer comparison of the charge and homology lattices.
Upon turning on the ${\mathcal N}{=}2^*$ mass deformation, the conformal singularity splits into three IR free singularities with massless hypermultiplet charges $(1,0)$, $(1, -2)$ and $(0, 2)$ which span the charge lattice.
This is an index-2 sublattice of the homology lattice, with induced polarization with invariant factor 2.
Furthermore, the S-duality group visible in this SK structure is the index-3 subgroup ${\Gamma}^0(2) \subset {\rm SL}(2,{\mathbb Z})$ generated by $T^2$ and $TS$ \cite{Argyres:2015gha}.
This is the S-duality group predicted by the line lattice analysis \cite{Aharony:2013hda}.

\paragraph{su(3).}

Similarly, our analysis shows that in addition to the $\mathfrak{su}(3)$ curve with homology lattice equal to the charge lattice and ${\rm SL}(2,{\mathbb Z})$ S-duality group, there should be a second Seiberg-Witten curve with principally polarized homology lattice corresponding to the single S-duality orbit of line lattices of this theory.
In fact, such a curve can be constructed \cite{AM22}.
Despite the simplicity of the orbifold analysis given in section \ref{orbifoldSK}, the description in terms of a Seiberg-Witten curve is quite complicated.
The Coulomb branch has complex coordinates $(u,v) \in\C^2$ with scaling dimensions $(2,3)$, respectively.
The curve is a family of genus-2 Riemann surfaces depending holomorphically on these vevs and on an exactly marginal (dimensionless) coupling parameter, ${\tau}$, given in hyperelliptic form by
\begin{align}
  y^2&=-\frac{1}{576(u^3-v^2)}\Big(729(2+\tau)u^6+972(\tau-10)u^5 x^2+864 u\,v^2\, x(27v+2\tau x^3)\\\nonumber
  &+144 u^2\,v\,x^2(27(5+\tau) v+4(2+\tau) x^3)+108u^4(27(-6+\tau)v\, x+4(10+\tau)x^4)+\\\nonumber
  &16v^2(729v^2+108\tau v\,x^3+16 x^6)-16u^3(729 v^2-54(10+3\tau)v\, x^3-4(\tau-2)x^6)\Big).
\end{align}
The Seiberg-Witten 1-form is
\begin{align}
{\lambda} &= (u x+v) \frac{dx}y .
\end{align}
It can be checked that the periods of this 1-form define a special section satisfying the SK integrability condition, and give rise to metric non-analyticities along a single irreducible $u^3=v^2$ subvariety of the Coulomb branch, matching the geometry expected from the orbifold construction.
Since it is given as a family of genus-2 curves, its homology lattice is principally polarized.
However, the ${\mathcal N}{=}2^*$ mass-deformed version of this curve is not known, so we cannot directly verify that the charge lattice is an index-3 sublattice of the homology lattice.

\acknowledgments
We would like to thank Michele Del Zotto, I\~naki Garcia-Etxebarria, and Yuji Tachikawa for enlightening conversations and comments.
The work of PCA is partially supported by DOE grant DE-SC0011784.
The work of MR is partially supported by a U.\ Cincinnati Joiner Fellowship.
The work of MM is supported by STFC grant ST/T000759/1.

redpajama_set_name: RedPajamaArXiv


Last Friday Dries announced that Steven Peck (sepeck) had decided to step back from his leadership role on the documentation team. Steven and Dries asked me to step into those big shoes and I accepted. All I can say is that I am glad that Steven isn't going away. 🙂 I'm honored that two people I respect so much have thought me worthy of this title. I will definitely hearken back to the many lessons I have learned from them both and strive to not disappoint.
The Getting Started guide: This guide was given an amazing foundation, laid by sepeck. He spent many, many days, weeks, months forging the initial guide for Drupal 5. He saw a great need for a tightly organized document to lead people through those first, sometimes harrowing, steps of using Drupal. We need to continue the care and feeding for this most invaluable resource. We need to keep it up-to-date and I'd love to see more of this central documentation directly accessible from within Drupal itself; maybe something like a new Help system in core or even a documentation module that uses the Advanced Help module in Drupal 6.
Improving tools on Drupal.org: The redesign of Drupal.org is going to change a lot about our home for the better. It is long overdue and the excitement is palpable. As a community we need to pay keen attention to the changes we make and I am particularly attuned to a documentation wish list. How can we use this opportunity to not only make it easier to read the docs, but also make it easier for everyone to help tend our garden? The easier it is for us to maintain and care for our handbooks, the nicer it will be to stroll amongst them.
These are three salient goals right now. Underlying all of these is another, which is not so much task oriented as it is attitude. I am buoyed not only by the greater changes in the Drupal community right now, but the real energy, insight and desire of the documentation team I get to work with every day. I feel that we have great opportunities before us and I am inspired to plow ahead with gusto. My current working mantra is "Throw it on the wall and see if it sticks." By this, I mean that I want to try new ideas, try old ideas in new ways, stretch boundaries and see what works. If things don't work, look at why, but then move on. Drupal has thrived on this attitude for many, many years. It is certainly nothing new that I bring, but I want to really encourage people to play in the finger paints – get your hands dirty and see what beautiful and sometimes unexpected things you can make. We can not only make Drupal more accessible and understandable, but we have the community – the people, knowledge and desire – to make real change in what the world expects of Open Source documentation. Feel free to join in the fun.


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Roman J. Israel, Esq. – A Spiritual Journey
7 April 2018 by Daniel Dercksen·Comments Off on Roman J. Israel, Esq. – A Spiritual Journey
"Audiences are hungry for a story that resonates in real life, that's relevant."
In Roman J. Israel, Esq., writer-director Dan Gilroy teams with two-time Academy Award winner Denzel Washington to create the portrait of a layered, complex man whose life has been spent fighting for others' civil rights – and paid a price for his activism.
Gilroy wrote the film on spec specifically for Washington, feeling he was the only actor who could bring the character to life. "I wrote this movie for Denzel because of his talent and because Denzel is a man who believes in human dignity and the human spirit. Knowing who Denzel is in real life, he brings that part of himself to this character."
Set in the underbelly of the overburdened Los Angeles criminal court system. Denzel Washington stars as a driven, idealistic defense attorney whose life is upended after his mentor, a civil rights icon, dies. When Roman is recruited to join a firm led by one of the legendary man's former students – the ambitious lawyer George Pierce (Colin Farrell) – and begins a friendship with a young champion of equal rights (Carmen Ejogo), a turbulent series of events ensue that will put the activism that has defined Roman's career to the test.
"For Roman the world is a war-zone and he's never left the front lines. It's a blessing and a burden," he says. "Activism can take an enormous emotional toll, but on the other side of the balance is the knowledge that you're becoming the change you want to see in the world – that you're making the world a better place. That's one of the key elements of the film: the importance of believing in something and the burden that often comes with that belief."
For Gilroy, the dual burden and blessing of belief are exemplified by a quote from civil rights pioneer Bayard Rustin. "Roman has pictures of his heroes on his wall, and Rustin is represented along with his quote, 'Let us be enraged about injustice, but let us not be destroyed by it,'" Gilroy notes. "That's the question Roman faces in the film – to what extent has he been saved or consumed by his fight against injustice? Anyone who hears a call to do more will see themselves in Roman."
More than simply hiring an actor for a role, Washington came on as a producer of the film, along with Jennifer Fox and Todd Black. Washington collaborated with Gilroy for a year before cameras rolled. During pre-production, Gilroy and Washington would meet in a recreation of Roman's apartment set, talking about the script, music and other aspects of Roman's life that would help Washington create the character. After production, Washington spent two weeks in the edit bay with Gilroy, helping to fine-tune the film.
"Writing the script on spec for Denzel was a huge leap of faith for Dan," says Fox. "If Denzel said no, there'd have been no movie. No one else could play Roman Israel. Denzel did say yes, Dan's bet paid off, and Dan and Denzel formed an incredible collaboration."
"Dan and I talked extensively about the spiritual aspects of his storytelling. He's a very spiritual man, and I work at it every day as well," says Washington. "Roman is on a spiritual journey, but he doesn't know it. Until now, he's had only two things in his life – the law and his boss – and now his boss is gone."
With the loss of William Henry Jackson, a leading legal activist, civil rights icon, and Roman's boss, Roman begins a crisis of conscience that leads to a loss of faith in everything he's held important until now – and after a tragic mistake, he comes around to finding that faith again. "There's a reason that the first and last lines of the movie are the same," says Fox. "Roman makes a complete arc in this film – a fall from grace, and a return to faith."
Well-known as the screenwriter of Hollywood spectacles like Kong: Skull Island and The Bourne Legacy, Gilroy made his directorial debut in 2014 with a very different kind of film – Nightcrawler – which earned him an Academy AwardÒ nomination for Best Original Screenplay. "I can write spectacle-driven, entertaining films, but what I learned, coming off Nightcrawler is that audiences love to watch something character-based," says Gilroy. "Audiences are hungry for a story that resonates in real life, that's relevant."
That is what inspired Gilroy to create the character Roman J. Israel, whose life mission has been to wield the law on behalf of the disenfranchised. "The name of the character was one of the building blocks of the story – of a man in conflict with himself," says Gilroy. "The Esquire at the end of his name is of great importance to him. It's a mark of distinction which, as Roman says, is in the same realm as knight. As for the J., it's emblematic of his idiosyncratic nature."
Gilroy took his screenplay to Jennifer Fox, who also produced Nightcrawler. "Both Nightcrawler and Roman J. Israel have elements in common – they are opposite sides of the same coin, outsiders who don't conform, both incapable of navigating the world. Though the films have some obvious differences – Nightcrawler had a level of cynicism that Roman J. Israel does not – they are both criticisms of a system, led by characters who battle the system every step of the way," says Fox. "Roman pays a great price for that battle – his belief that justice and mercy cannot be divorced nearly wrecks him."
Gilroy heavily researched the legal profession, in particular civil rights and activist attorneys. He found an overburdened justice system in which the housing of inmates has been privatized and monetized, and one which disproportionately affects African-Americans. "The criminal courts and prisons are wildly out of balance and greatly in need of reform," he says.
"When you go to a courthouse or a prison, you see people struggling to prove their worth, to prove their story – not even necessarily their innocence; they just want their story to be heard," says Todd Black. He says that many lawyers in the system don't hear that, because they take a more clinical view of their cases. "They're cut and dry – does it fit the law or doesn't it fit the law? – and they put zero moral value on right or wrong."
Into this world, enter Roman J. Israel, Esq., a character whose "moral compass always points north," Black continues. "Denzel has played a lot of morally corrupt people and a lot of morally right people, but this character is like no one he's ever played before."
A Unique Character
Roman Israel is a character who continually bucks what he sees as a rigged system. "Roman's spent 40 years in a back room, fighting for the dispossessed and under-represented," Gilroy explains. "It's been his all-consuming passion."
Roman has dedicated his life to public service, but he's uncomfortable being in public – so uncomfortable, in fact, that he often finds himself unable to stop himself from going too far. His awkwardness is in his walk, the way he speaks – every ounce of his being. "He has no filter," says producer Jennifer Fox. "No ability to compromise. He's going to tell you the truth – the ugly truth – and you may not want to hear it. He's so idealistic, so pure, so morally forthright that he has no patience for people who do not see justice his way, and can't negotiate at all. He ends up hurting the very people he's fighting to protect."
"He is inexperienced in how to convey his knowledge and goes too far – he gets himself in trouble," says Washington.
That combination of fighting for the disenfranchised and never knowing when to quit – even at great personal risk – has cost Roman much. He lives a meager life in a tiny apartment on LA's Skid Row. For his entire career, he has worked in the back room of a firm headed by William Henry Jackson, a famous civil rights giant who got all the publicity arguing cases Roman meticulously prepared. But when Jackson suffers a stroke, Roman's world is upended.
Gilroy says he was inspired to create the role by seeing the spirit of activism of the 1960s and 70s in the current grassroots engagement. "We're living in a time when people on all sides of the political spectrum are seeing a need to take action and stand up for what they believe in," says Gilroy. "It reminds me of the 1960s, when there were large-scale demonstrations and protests – an incredible spirit of activism. I began to wonder: what would it be like if someone committed their entire life to those ideals and stayed with it for decades? It turns out, there are a small handful of people who never did leave the activism that drove the 1960s – and they seem to gravitate toward law."
Washington agrees that Roman Israel's fight feels very current and modern. "People like Roman – they're here now, doing good work," he says.
There are aspects to the character that Washington has never portrayed before. "His social skills are lacking; he doesn't have the ability to read the signs that the other person is giving him," says Washington. "He's a bit of a savant who's been protected by his boss while he did all of the groundwork. He's been comfortable in his small apartment, his small world, fighting the good fight, and is not really capable of going out there and being in front of people. I said that Mr. Jackson protected him from other people, but another way of looking at it is that Mr. Jackson protected other people from him."
That protection has also made Roman something of a relic. "Roman's out of step with his time," says Gilroy. "Everything about him – his clothes, his music, his apartment, his mannerisms – haven't really changed since the 70s." For example, at one point in the film, he excoriates young men for taking seats while women (played memorably by jazz musician Esperanza Spalding and Jessica Camacho) stand in the back of the room – which doesn't go well when the women don't feel a need to be 'rescued.' "It's coming from the right place but shows how the world has moved on without him."
Washington says that the film isn't preachy or showy – rather, if the film moves people or changes minds, it's because that's what the character does. "The film isn't an indictment of the system – I don't look at it in those terms," Washington says. "I look at it as an actor, through the character's eyes. He doesn't come into a courtroom to say the things he says. He's not there to change anybody. But he can't help but to speak the truth. This is who he is, and that changes people."
Black says that Roman Israel's struggle is a universal one. "As we become adults, we try to stay true to ourselves and be guided by our consciences to do the right thing, and also try to fit in by doing what one needs to do to succeed in a place of business," he says. "That's a real conflict for Roman."
With Roman's life in upheaval, he begins to question everything he thought he believed in. "Roman's self-sacrifice and idealism have taken a toll. He makes a choice with life-altering ramifications," says Gilroy. "Coming to grips with that, making peace with it, applying the same principles to himself he applies to others, that's the essence of the film."
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huf sweatshirt cheap accessories cap ,huf baseball cap beige men huf hat,shop best sellers product details cap by huf cotton twill panelled crown eyelet vents curved peak adjustable strap wipe clean 100% cotton brand huf owner keith hufnagel grew up skateboarding in the streets of new york before moving to san francisco to pursue his passion. opening his first store in 2002, huf aims to bring together streetwear, footwear and skateboarding, incorporating an ever progressive and evoing skate culture into their designs and practices. look after me wipe clean only about me body: 100% cotton.


Today, the National Center on Education Statistics (NCES) released The Nation's Report Card: 2015 Mathematics and Reading The results from the 2015 assessment are compared to those from previous years to describe change in fourth and eighth-grade students' performance in mathematics and reading over time. Performance results are presented as NAEP scale scores and as percentages of students at the Basic, Proficient, and Advanced achievement levels. The report also includes information about the performance of different student groups, as well as performance gaps by gender and race/ethnicity. NAEP results date back to the early 1990s.
Generally speaking, the trend in NAEP performance has been one of significant growth since the early 1990s. There are small variations each year, and this year's data is garnering a lot of headlines over a potential stalling in student growth. In mathematics, the 2015 average scores were 1 and 2 points lower in grades 4 and 8, respectively, than the average scores in 2013. These small declines remain well above initial scores, as scores at both grades remain higher than those from the earliest mathematics assessments in 1990 by 27 points at grade 4 and 20 points at grade 8. In reading, the 2015 average score was not significantly different at grade 4 and was 2 points lower at grade 8 compared to 2013. As was the case in mathematics, scores at both grades were higher in 2015 than those from the earliest reading assessments in 1992 by 6 points at grade 4 and 5 points at grade 8.
We must be responsible in our consumption of this data and resist the urge for drastic changes. This could be a one-year anomaly or it could be something more significant. However our students perform, we must remain focused on supporting their growth and learning, and resist the urge t point fingers and shift blame.
"The headlines today write themselves and cover all the usual angles: Our schools are failing. Our students are failing. We need more tests. We need fewer tests. We need better tests. Common Core is working. Common Core is failing. We need more school choice.
"We have had—and continue to engage in—these conversations, all of which have their time and place. But today, in this moment, when NAEP—widely regarded as the Nation's Report Card—indicates that our students aren't making the growth and achievement we would expect, perhaps the conversation isn't about what we are doing as much as what we are not doing. And in this instance, we must consider the extent to which this set of NAEP data was impacted by the significant cuts to education investment at the local, state and federal level stemming from the great recession and held in place by continued poor policy.
"When it comes to our nation's schools and the students they serve, we know that education cuts do not heal. Though we're past the end of the great recession, education investment has yet to reach pre-recession levels. That means that our nation's K-7th graders have spent the entirety of their K-12 educational experience to date under a post-recession funding climate, and that our 12th-graders have spent half of their educational experience in that underfunded environment.


text:
Modified Atmosphere Packaging
Rubbish reincarnated
According to Stats SA, 10.8 % of South African household's sort refuse for recycling and 60% is the current average level of landfill sites around S.A. Gauteng is forecast to run out of landfill space by 2022
Mandatory recycling has come to Johannesburg and households are said to be showing a "keen interest".
Albi Modise, spokesperson for the Department of Environmental Affairs, said that landfill sites are quickly filling up. Gauteng, the smallest and most populated province has only 13% of its urban households sort their waste for recycling.
Recycling rates are dismal everywhere.: 1% in Limpopo, 6% in Kwazulu Natal and 20% in the Western Cape.
Since mandatory at-source sorting of recyclable items was introduced in the greater Johannesburg area on July 1, the process has been outsourced to private companies.
Johannesburg councillor, Nico de Jager, who announced the initiative on World Environment Day, said residents would not be initially penalised for failing to sot their recyclable rubbish, but it may cost those who don't separate waste in the future.
For our children and grandchildren futures
There has been debate about the actual environmental benefit of recycling, the high cost of the new venture, and that it takes a bit of work, but according to Jaco Botha, operations manager for Phambili Waste Management Services, the costs are recoverable via sale of the material for recycling and the truth is, there aren't many options left in a growing metropolis.
In the Midrand, after a door-to-door education and awareness campaign during a testing phase last year, there are now 48 764 households in the area receiving bags and in July of 2018, 257 tonnes of sorted, recyclable waste was collected with 80% of that material successfully recycled.
There has been controversy around waste-pickers livelihoods been threatened by the recycling project. But Phambili logistics manager, Mpho Boshomane says they service areas which the project has not reached.
Boshomane works with the informal waste-pickers,"Started with 40 people who come from Alex and Joburg; now there are 150 or more and we pay them for what they bring, even if they have separated it from the rubbish bags people put out." He said
On a national level, Plastic SA's executive director, Anton Hanekom, comments: "South Africa is doing phenomenally well with recycling" More than 334 727 tonnes of plastics are recycled back into raw materials. Our country's input recycling rate is 43,7%, which is well above Europe's recycling rate of 31,1%
Cans fetch the highest price at resale – about R 11/kg after being processed into cubes. Plastic goes for about R 4/kg depending on the density.
What we recycle:
Metals (73,3%)
Paper (67,2%)
Plastic (44,6%)
Glass (42,5%)
Source DEA
Extracted from articles by:
Shanthini Naaidoo, Writer- Sunday Times, published 26th August 2018
Staff Writer, Food Review, published 20th August 2018
By Keith Coleman|2018-09-15T19:53:53+02:00September 15th, 2018|
Who is Foodtech Fresh
Foodtech Fresh is an expert consultancy in the field of Food and Food Packaging Technical Services. We have a focused spectrum of skills, background and industry experience, a network of associates and specialists, combined with the ethic of partnering with our customers.
keith@foodtechfresh.com
11 Juniper Street
Randparkridge
© Copyright 2018. Foodtechfresh


Accredited Drug Testing in Parkerton WY provides same day drug testing, alcohol testing, DOT physicals, DOT drug testing and DNA testing at locations in Parkerton Wyoming and the surrounding area. Drug and alcohol testing services are provided for employers in need of a pre-employment, random and post accident test. As well as individuals needing a test for court ordered, probation or family purposes. Same day service is available and testing centers are within minutes of your home or office.
To schedule any type of drug test at our Parkerton locations, call our scheduling department at (800) 221-4291. You can also schedule your drug test online utilizing our express scheduling registration by selecting your test and completing the Donor Information/Registration Section. You must register for the test prior to proceeding to the testing center.
Our local Parkerton Wyoming drug testing centers provide drug and alcohol tests that are utilized for a wide variety of court ordered proceedings, including probation and other types of government required tests. Observed tests are available and tests results can be submitted for a wide variety of court ordered proceedings including probation hearings, child custody cases, litigation and all manner of government required tests.
Our local Parkerton WY drug testing centers provide drug and alcohol tests that are utilized for a wide variety of court ordered proceedings, including probation and other types of government required tests. Observed tests are available and tests results can be submitted for a wide variety of court ordered proceedings including probation hearings, child custody cases, litigation and all manner of government required tests.
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Providing exceptional customer service and being available when you need us for drug testing in Parkerton WY, is our #1 Priority and we look forward to serving the needs of our clients.
A portion of Converse County territory was annexed for the formation of Niobrara County in 1911. Converse County was slightly enlarged with territory from Albany County in 1955 after a special election.
An elected commission governs each Wyoming county. Commissioner David R. Edwards resigned in January 2012 after three years of service because of health problems. He was a Republican who had served in the Wyoming House of Representatives from 2001 to 2008.
As of the 2000 United States Census, there were 12,052 people, 4,694 households, and 3,407 families in the county. The population density was 3 people per square mile (1/km²). There were 5,669 housing units at an average density of 1 per square mile (1/km²). The racial makeup of the county was 94.72% White, 0.15% Black or African American, 0.91% Native American, 0.27% Asian, 0.02% Pacific Islander, 2.46% from other races, and 1.47% from two or more races. 5.48% of the population were Hispanic or Latino of any race. 26.6% were of German, 13.5% American, 12.2% English and 11.1% Irish ancestry.
Drug Testing in Parkerton WY reviewed by Susana P. on Sep 13.
Thanks for building such a wonderful site. I was able to research the different DNA testing in my area and find one that fit my schedule. They even gave me a chance to apply for a Drug Test which will make things much easier.


I found this recipe in Dreena Burton's book Plant-Powered Families. I love that book and love this recipe. Luckily, she also posted it on her website so I can repost it here. These are super easy to make, healthy and delicious. What a great snack/dessert!
* To make homemade macadamia nut butter, place 2 cups of unroasted macadamia nuts into the food processor and let it rip until it is butter.
** Agar powder really helps to hold the squares together. I made them once without it and they kind of didn't stay together too well. They are still delish, though!
Line a 9×9 brownie pan with parchment paper. Pour the cereal into a big mixing bowl.
Place the macadamia butter, brown rice syrup, coconut sugar, vanilla, salt, and agar powder into a small saucepan. Over medium heat, use a whisk to stir the mixture until it starts pulling away from the sides of the pot.
Remove from heat and pour over the cereal. Use a wooden mixing spoon to fully incorporate with nut butter mixture.
Turn the whole deal out into the middle of parchment paper in the brownie pan. Use the wooden spoon to spread it to the edges and then the back of a regular spoon to press it down evenly.
Place in the refrigerator to cool and set. Cut into squares and enjoy!
Using agar powder is a great improvement over the usual marshmallow glue. And the other ingredients are equally good. I presume peanut butter would work too?


Massage Therapy is a manual therapy which manipulates soft tissue providing therapeutic, pain relieving benefits as well as promoting and maintaining a healthy body. Massage Therapy used to be considered a luxury, something that was reserved for pampering or an event that allowed one to splurge. Our high tech, high stress life styles have dramatically changed this trend. Our health care system has a tendency to treat the symptom with medication rather then find the cause of our problem. Since muscular pain is generally activity and stress related, medication will rarely get to the cause of the problem. The therapists at Verde Valley Myofascial Release are experts therapeutic massage techniques aimed at getting to the source of the problem. Massage Therapy has become very specialized and has a variety of different modalities including: Swedish massage, trigger point therapy, Myofascial Release, shiatsu and reflexology. Therapeutic massage can utilize any of the above modalities and is designed to treat specific conditions such as pain and dysfunction from accidents or injuries as well as chronic muscular pain and dysfunction. The therapists at Verde Valley Myofascial Release specialize in therapeutic massage with advanced level training in Myofascial Release. Myofascial Release specifically targets connective tissue, the most abundant tissue in the body. Connective tissue will shorten in response to trauma, repeated postures held over time, inflammatory conditions, just as an example. Regular massage techniques will not correct tightened connective tissue and will only provide temporary relief. As a Physical Therapist for over 28 years, I was fortunate to find Myofascial Release very early in my career. I have seen its benefits over and over for both acute and chronic conditions. People are sent to Physical Therapy by their doctors when they have muscular conditions which require skilled intervention. However, most Physical Therapy clinics treat their clients with pain reducing modalities such as electric stimulation and ultrasound and put them on exercise programs without ever addressing their muscular imbalances. If they work on their clients directly, it's generally for a very short 10-15 minute period. The problem is still present in their body and the results are either ineffective or very short lived. It's not uncommon for me to hear someone talk about going to Physical Therapy for a particular problem and not getting the relief that they were seeking. This trend is causing more people to turn to Massage Therapists for both stress reduction and pain remediation. Diagnoses such as Fibromyalgia and Chronic Myofascial Pain have become very common. I believe this is due to the problems never being appropriately addressed and masked by pain relievers and anti- inflammatory medication. Massage Therapists need to have an advanced level of training beyond their Massage Therapy license or certification in order to have the skills necessary to properly assess and treat pain and muscular imbalances effectively. The Massage therapists at Verde Valley Myofascial Release have worked closely with Physical Therapy and Occupational Therapy in a therapeutic environment. This relationship has provided our therapy staff with the skills necessary to be successful treating a vast variety of acute and chronic muscular conditions. Verde Valley Myofascial Release has two locations to serve you; Sedona and Cottonwood, AZ. We also provide mobile services. Get to the root of your problem rather then chasing it around your body or settling for temporary relief.
Check our our website for valuable articles and pain reducing techniques (www.verdevalleymfr.com).
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