text:
Q: How to link to a glossary item (using package glossaries) In my document i'm using the package glossaries to create a glossary. Everything works fine except that the is no link between the words in the text and the corresponding entry in my glossary (so you can click the word to be explained and get to the glossary entry).
The most important parts of my document:
%----Header----
...
\usepackage[nonumberlist,acronym,toc,style=altlist]{glossaries}

\usepackage[
colorlinks=true,
pdfborder=0 0 0,
pdfpagelabels,
plainpages=false,
linktocpage=false,
pdfcreator={LaTeX}]{hyperref}
...
%Glossary entries
\newglossaryentry{glos:twitter}{name=Twitter,
description={Mikroblogging-Service.}}

%----Main document----
\begin{document}
\chapter{Introduction}
This text is a normal glossary item: \gls{glos:twitter}.
This text should also link to the glossary item: \glslink{glos:twitter}{Link to Twitter}
but there is no link
...
\printglossary
\end{document}

As you see i'm also using the package hyperref, but there seems to be no mechanism that automatically links words in the main text to the glossary.
I also tried to use \ref and \label, but this doesnt' work when the element that is referred is outside the main document (like my glossary is).
I'm using the makeglossaries-script coming from miktex (calling makeglossaries main on build), but this also doesn't give me a link.
Anyone knows a way to do that? Or maybe i should use another package than glossaries which supports a functionality like that?
I would also appreciate any working examples where this functionality works.
EDIT:
I just got a working minimal example where gls/glslink actually works. Seems like the linking of glossary items interfers with one of the packages im using in my document. Will have to try by adding my packages one by one to the example to see which package is the reason. The example:
\listfiles
\documentclass{article}

\usepackage[
colorlinks=true,
pdfborder=0 0 0,
pdfpagelabels,
plainpages=false,
linktocpage=false,
pdfcreator={LaTeX}]{hyperref}
\usepackage[nonumberlist,acronym,toc,style=altlist,]{glossaries}
\makeglossaries

%Glossary entries
\newglossaryentry{glos:twitter}{name=Twitter,
description={Mikroblogging-Service.}}

%----Main document----
\begin{document}

\chapter{Introduction}
Ein normales Wort aus dem Glossar: \gls{glos:twitter}.
Dieses Wort soll zum Glossar verlinkt werden: \glslink{glos:twitter}{Link to Twitter}
funktioniert aber nicht...
\clearpage
\printglossary
\end{document}

A: As not mentioned in the extract of my latex-code, in my documentclass, the draft property was set to true. When removing this property or setting it to false, the gls/glslink work fine.

redpajama_set_name: RedPajamaStackExchange


text:
How Systemic Government Failures Have Contributed to Violence in America
by Dr. Lynn S. Kahn in Public Safety Nov 16, 2017
To wrap our minds around violence in America, we must step back and look at the whole.
Violence has many forms: child neglect and abuse, spouse and partner rage, school bullying, sexual assaults on campus and in the military, violent incidents with law enforcement, harassments and humiliations at work.
At the same time, institutional violence impacts our lives as federal policies destroy local economies, poison our air and water, incarcerate unreasonable numbers of people of color, and damage millions of children and families every year through wrong-headed decisions about foster care and child custody.
Our government continually underserves our veterans, seniors, people with disabilities and still after hundreds of years violates trust and treaties with our Native American sisters and brothers.
Failed schools impose an authoritarian structure that generates resentment and destroys the self-confidence of our young people who graduate high schools without the skills needed to be successful in the 21st century.
Institutional violence impacts our lives as federal policies destroy local economies, poison our air and water, incarcerate unreasonable numbers of people of color, and damage millions of children and families every year...
Dr. Lynn S. Kahn
The media surrounds us with images of violence while industry rolls out military-grade weapons.
Stepping back and seeing the whole also takes in the full range of programs that work – programs that change perceptions and attitudes to turn lives around and reduce violence. The core of what works is gut-wrenching honesty and a range of services from different government agencies and non-government partners.
Problems cross boundaries and so must solutions.
Personal responsibility for choosing aggression is always a factor in violence, and our easy access to guns makes restraint more difficult for those unable or unwilling to control their rage.
This article holds that underneath the widespread anger of America is an extraordinary amount of pain and a pervasive sense of injustice that far too often flows from the trauma of broken government.
Reducing anger-driven violence means transforming and coordinating agencies to deliver better results.
The guiding principle is one of connection as I noted in Performance Networks: Transforming Governance for the 21st Century:
The basic building blocks of democracy in the 21st century are networks of teams and partnerships that leverage wisdom and resources to deliver better results for more of us.
Underneath the widespread anger of America is an extraordinary amount of pain and a pervasive sense of injustice that far too often flows from the trauma of broken government.
This daunting task is eased by lessons from the gold standard in complex, multi-partner planning and implementation: aviation's Next Generation Air Transportation System (NextGen).
The first practical lesson from NextGen is writing all plans and documents in plain language with lots of graphics so different agencies and various technical specialists have a common language to work together. A conceptual lesson from aviation is to separate out the strategic plan from the enterprise architecture.
A strategic plan includes mission, vision, values, core goals, strategic activities to achieve those goals and performance measures to track progress. Even with roadmaps and timelines, the plan is linear – it can be laid out step after step in a straight line. Reality is not linear and blueprints for complex social change must be multi-dimensional.
In aviation, the framework for the complicated effort to upgrade technology is called the National Airspace Enterprise Architecture. It covers every aspect of our airspace and contains multiple views to show different systems. The architecture is way more than a strategic plan; it is a framework that guides multiple partners from the current state to the future state.
Addressing violence in America calls for better agency strategic plans and a multi-level architecture that connects many different partners. We can improve individual schools or law enforcement agencies or any other agency with a strategic plan for change.
Working toward a vision for a vibrant, diverse, and violence-free America requires a larger framework — an enterprise architecture for social justice.
A basic format for a complex architecture includes levels (aspects of community life that touch violence) and views (ways to look at progress towards goals).
For teams delivering actual services, seeing the connections across agencies and solutions improves performance and allows everyone to see how their individual and group work fits into the overall plan.
A Social Architecture of Violence and Justice
We can start building this architecture by looking at government from the community perspective: what do communities need to be strong not just what one particular agency does. The basic levels of that architecture would include:
Neighborhood Wellbeing: How do we map and describe our neighborhoods? The picture includes crime, violence, poverty, employment, housing, hunger, schools, healthcare and "protective factors" such as strong community involvement.
Programs: What works? What specifically reduces and prevents violence and crime, lowers incarceration, or improves relations between law enforcement and communities. What programs restore trust?
Partners: Who is involved? Successful violence-prevention programs rely on the widest possible network of neighborhood-based groups.
Performance Measures: What does success look like? Is crime going down? Are abuse and assaults decreasing? Are fewer people going into the criminal justice system and far more coming out? Have we stopped the school-to-prison pipeline and the foster care to jail pipeline?
Roadmaps: Clear timelines with deliverables and interdependencies (such as training staff before starting a new program) must align all partners.
Budgets: In aviation, to get funded, new programs must fit into the NextGen architecture.
A Social Architecture of Violence and Justice: Start Here
Government is a living system where safety and security, violence and justice, jobs, health care, schools, resources to build stronger neighborhoods, and the will to repair our environment are all connected.
The basic assumption of this short essay is that the failures of government in the United States have contributed to an epidemic of social violence. Problems cross agency and neighborhood boundaries and so must solutions.
We know what works. Good research shows us what programs truly impact crime and violence. The challenge is to bring agencies and neighborhoods together in a multi-dimensional framework that will focus efforts and deliver results beyond expectations.
Read More Articles by Dr. Lynn S. Kahn
7 Reasons Why Independent Presidential Debates Will Save America in 2020
By Dr. Lynn S. Kahn
Betrayed and Outraged: Winning The Rural Vote In Today's Political Climate
A Common Sense Approach to Healing America's Political Divide
- 5 months ago
Connections: Severe Weather, Universal Health Care, and National Security
Dr. Lynn S. Kahn was a third party candidate for New York's 21st Congressional District in 2018. She ran for president as an independent in 2016. Her platform calls for Transforming America by Transforming Government. She is the author of two books on government reform and one on peacemaking. She received her Ph.D. in Clinical Psychology from The American University in 1977 and worked inside the federal government as an organizational psychologist for more than 32 years including six years on the White House Partnership for Reinventing Government.
Read more by Dr. Lynn S. Kahn

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: A combinatorial argument that $5$ divides $(n+2)(n+1)n(n-1)(n-2)$.
Give a combinatorial argument to prove that $5$ divides $(n+2)(n+1)n(n-1)(n-2)$.

I am definitly weakest in discrete math, but it jumps out to me to use the permutation definition that
$$P(n,m)=n\cdot(n-1)\cdot(n-2)\cdots (n-(m-1))=\frac{n!}{(n-m)!}$$
Pretty clearly here $$P(n+2,5)=(n+2)(n+1)n(n-1)(n-2)=\frac{(n+2)!}{(n+2-5)!}
$$ Where I seem to want to go from here is that $P(m,5)$ is always divisible by $5$. Heres the 'proof' that I feel a bit shakey on.
Let $m$ be arbitrary, then $$P(m,5)=\frac{m!}{(m-5)!}$$ We require $m-5\geq0$, so $m\geq5$. This means $m!$ must be divisible by $5$ and hence $P(m,5)$ must also be divisible by $5$.

A: Combinatorial argument:
$N=(n+2)(n+1)n(n-1)(n-2)$ represents the number of ways you can create a row of 5 people from a group of $n+2$ people (there are $n+2$ ways to select the first one, $n+1$ to select the second one, etc...)
Say that one row of people looks like $ABCDE$. How many rows do you have with the same set of people? It's 5! (one of five guys ABCDE can be selected to be first, one of the remaining four to be the second, etc...). So $N$ must be divisible by 5! which also means that $N$ must be divisble by 5.

A: $$(n+2)(n+1)n(n-1)(n-2)=5!\binom{n+2}{5}$$ and since $\binom{n+2}{5}$ is a natural number for $n\geq 3,$ we see that $(n+2)(n+1)n(n-1)(n-2)$ is divisible by $5$.


text:
Хоча традиційно художниці коміксів здавна становлять меншість в індустрії, вони здійснили помітний вплив з самого початку зародження коміксів. Наразі все більше і більше художниць отримують визнання в медіа. Мисткині працювали в усіх жанрах: від супергероїв до романтики, від вестерну до війни, від криміналу до жахів.

У деяких країнах, як-от Японія та Південна Корея, мисткині похитнули традиційний ринок і досягли широкого загального успіху.

Америки

Сполучені Штати

Газетні комікси
На початку ХХ століття, коли ринок газетних коміксів у США був у зародковому стані, Вільям Рендольф Херст привів художницю Нелл Брінклі з конкуруючого Denver Post, і хоча вона сама не робила комікси, її романтичні та гламурні образи стали джерелом натхнення для покоління художниць та художників коміксів.

Ще одним популярним стилем того часу були милі комікси з круглощокими дітьми, схожими на ляльку. У 1909 році Роуз О'Ніл створила The Kewpies, серію, яка триває десятиліттями і широко використовується в різних маркетингових цілях.

Інша мультиплікаторка, Грейс Відерзайм (також відома як Грейс Дрейтон і Грейс Геббі), працювала в подібному ракурсі та з 1910-х до 1930-х років створила безліч серіалів з дітьми-херувимами, які носять такі імена, як Тудлз, Дімплез, Доллі Дінгл і Дотті Дарлінг. Вона також була творчинею «Campbell kids», які Campbell Soup використовував у маркетингових цілях до 1930-х років. Її сестра Маргарет Гейс також часто співпрацювала з нею над кількома її роботами.

У 1910-х роках газетна карикатуристка Фей Кінг малювала ранні автобіографічні комікси в The Denver Post і Cartoons Magazine.

У 1918 році Едвіна Дамм створила тривалий серіал «Кеп Стаббс і Типпі» про хлопчика і собаку, хоча жвавий пес незабаром став найпопулярнішим персонажем. Серіал йшов до 1960-х років.

У 1920-х роках у США відбувся економічний бум і широкі соціальні зміни, що призвело до появи «флепперок» — жіночої субкультури, яка в той час приділяла велику увагу ЗМІ. Флепперки любили вечірки, джазову музику та безкоштовні побачення, а також кидали виклик багатьом соціальним нормам, які стосувалися жінок того часу. Кілька мультиплікаторок підхопили стереотип про флепперок, часто працюючи в стильному стилі арт-деко, в тому числі Етель Хейс (з її коміксом Маріанна та її відомим мультфільмом Флаппер Фанні), Вірджинія Х'югет ( Джентльмени віддають перевагу блондинкам, Жінки в суспільстві), Гледіс Паркер (Гей і її банда) і Марджорі Хендерсон Бьюелл (Лиха точка).

У 1930-х роках у США вразила велика депресія, і історії про бідні, але щасливі сім'ї та їх стоїчну боротьбу, щоб заробити на життя, стали популярними читачами. Марта Орр створила однією з найуспішніших серіалів, Apple Mary, про стару жінку, яка продає яблука по сусідству в 1932 році.

Розповіді про остаточну долю серіалу відрізняються. Більшість джерел стверджують, що в 1938 році вона залишила його своїй помічниці Дейл Коннер, яка перейменувала його в Мері Ворт, хоча у власному обліковому записі King Features Syndicate стверджується, що Apple Мері згорнула, а Мері Ворт стала його заміною. У 1940 році був залучений новий письменник Аллен Сондерс, і Коннер і Сондерс почали підписувати стрічку спільним псевдонімом «Дейл Аллен», який залишився після того, як Коннер залишив серіал. Mary Worth зарекомендувала себе успішною концепцією, і вона досі поширюється по всьому світу.

У 1935 році Марджорі Хендерсон Бьюелл (підпис «Мардж») створила коміксну панель «Маленька Лулу», згодом породивши успішну серію коміксів Джона Стенлі та Ірвінга Тріппа. Цей персонаж надихнув назву організації Friends of Lulu, організації, яка сприяє читанню та написанню коміксів для дівчат і жінок.

У 1940 році художниwz-ветеранrf Дейл Мессік створиkf комікс «Бренда Старр, репортерка» про гламурну репортерку з любовним життям, схожим на мильну оперу. Після того, як Мессік залишила серіал, його продовжили виключно інші художниці.

У 1941 році Тарпе Міллс створила стрічку про супергероїню «Міс Ф'юрі» для Sunday pages. Вразивши читачів, вона малювала стрічку до 1951 року.

Джекі Ормс була першою національно синдикованою темношкірою художницею коміксів зі своїм серіалом «Торчі Браун», створеним у 1937 році як пригодницька гумористична стрічка, яка тривала три роки, і знову охоплена в 1950 році як «Серцебиття Торчі Брауна», в основному перероблена як чорна версія Бренди Старр, Репортерка, молода темношкіра однойменна героїня, натикаючись на пригоду за пригодою, переходить від одного любовного інтересу до іншого, хоча серіал також розглядав більш серйозні теми, такі як расовий фанатизм та забруднення навколишнього середовища. Серіал так і не мав широкого успіху, оскільки його підхопили лише чорношкірі газети.

У 1940-х роках підліткові комікси стали популярним жанром. Це був досить приземлений жанр, здебільшого комедійний і рекламований для молодих підліток, де молоді, часто зухвалі, підлітки переживали різні проблеми з протилежною статтю та зустрічами. Серед відомих художниць, які слід згадати, є Хільда Террі (Тіна, 1941), Марті Лінкс (Еммі Лу, 1944) та Лінда Волтер (Сьюзі К. Сміт разом зі своїм чоловіком Джеррі Волтером над сценаріями). Усі ці три художниці раніше працювали у сфері моди. У 1951 році, після деяких внутрішніх суперечок всередині організації, Террі стала першою художницею, яка була прийнята до Національного товариства карикатуристів США.

Інші успішні стрічки включають напівавтобіографічну Кеті Кеті Гізвайт про невротичну міську жінку та її проблеми з покупками та романтикою та Лінн Джонстон «На краще чи на гірше» про родину Паттерсонів та їхні сімейні стосунки.

Цикл Ніколь Голландер «Сільвія» розповсюджується на національному рівні в Tribune Media Services є відверто феміністичною, вона містить багато гострих соціальних коментарів на додаток до гумору, заснованого на характерах. Цикл подано у 19 опублікованих книгах. Сильна особистість Сільвії та різко критичні погляди відрізняють її від менш наполегливих героїв мультфільмів жінок.

Через часто суворі вимоги синдикатів до повторюваних персонажів і небажання ризикувати образити читачів, деякі карикатуристки пішли в самосиндикацію, щоб зберегти контроль над своєю роботою. Деякі тривалі комікси, що самосиндикуються, — це феміністка Maxine or Laughing Gas від карикатуристки та авторки Маріан Хенлі (не плутати з персонажем Джона М. Вагнера), а також сюрреалістичний Way Lay або Story Minute, написаний ветеранкою андеґраундуКерол Лей.

Основні комікси
Книги коміксів також були створені рядом художниць.

Так,  видавництво, Fiction House, використовувало багато мультиплікаторок, як у штаті, так і через Eisner & Iger, одного з посередників виробництва коміксів тієї епохи, який постачав комікси на вимогу видавців. У той час були популярні жанри, орієнтовані на екшн та пригодницьку діяльність, і сильною стороною Fiction House були здібні та красиві героїні, які працювали пілотесами, детективками чи авантюристками в джунглях. Серед жінок, які працюють у видавництві, є Лілі Рене, Френ Хоппер з Lambiek Comiclopedia та майбутні художниці-романтикині Рут Аткінсон та Енн Брюстер. Ці оповідання також часто писала письменниця: Рут Рош, пізніше редакторка. Перш ніж здобути славу як кримінальна романістка, Патрісія Хайсміт писала для «Чорного терору» та інших коміксів.

У 1950-х роках Марі Северін, сестра художника Джона Северіна, часто була колористом EC та Atlas / Marvel, пізніше також малювала власні історії. Її стиль мультфільму зробив її частою допискою до сатиричної назви Marvel Not Brand Echh кінця 1960-х років. Іншою плідною художницею була Рамона Фрадон, яка намалювала Аквамена і була співавторкою Метаморфо.

До пізніших мисткинь та письменниць належать Енн Ноченті (творчиня Тифозної Мері та Лонгшота), Луїза Сімонсон (авторка Power Pack), Джун Брігман (художниця Power Pack), Гейл Сімон (West to Tranquility), Девін Грейсон (авторка Бетмена), Беккі Клунан, перша художниця Бетмена, Марджорі Лю (письменниця The Amazing X-Men), Сара Пічеллі (Ultimate Spider-Man artist), Г. Віллоу Вілсон (Ms. Marvel), Аманда Коннер (художниця Power Girl) і Келлі Сью ДеКонік (Pretty Deadly, Bitch Planet) у Image Comics.

Андеграундний, альтернативний і незалежний
Рух андеграундних коміксів приваблював художниць, оскільки дозволяв більш зрілі теми та особисту роботу, ніж комерційна газета та індустрія коміксів того часу. Піонеркою на цьому ринку була Тріна Роббінс, рушійна сила у створенні перших жіночих коміксів It Ain't Me, Babe і All Girl Thrills, а пізніше засновниця серії антології Wimmen's Comix. Роббінс написала кілька книг про художниць та їхні комікси.

Іншою серією коміксів, що складаються з жіночих коміксів, був Tits & Clits Comix, заснований Лін Чевелі та Джойс Фармер, які були натхненні чесністю андеграундних коміксів, але вражені частими сексистськими поглядами та ставленням чоловіків. З переконанням, що секс є політичним, серіал був створений з акцентом на секс і сексуальність з жіночої точки зору.

Художниці, які виросли з цього руху, включають Лі Маррс (Pudge Girl Blimp про повну одержиму хіпі дівчиною), Шарі Фленнікен (Тротс і Бонні про швидкостилу дівчинку та її собаку, яка намагається осмислити своє заміське життя), Алін Комінскі (The Bunch, автобіографічне зображення її найменш втішних сторін) і Дорі Седа (автобіографічні оповідання).

Після того, як андеграундна сцена перетворилася на альтернативну, художниці продовжували зосереджуватися на автобіографічних роботах, таких як Деббі Дрекслер (Дівчинка тата, 1996, про інцест і сексуальне насильство в дитинстві) і Фібі Глокнер (Щоденник підлітки, 2002).

Невибачливе ставлення сцени також надихнуло художниць за межами США, таких як канадійка Джулі Дусет, чий сюрреалістичний напівавтобіографічний серіал «Брудний сюжет» став всесвітнім культовим фаворитом у 1990-х роках.

Андеграундний/альтернативний ринок дозволив більш відкрите зображення сексуальності, і в 1970-х і 1980-х роках відкрито лесбійські та бісексуальні митці розповідали свої історії у формі коміксів, наприклад, Мері Вінгс (художниця першого коміксу Come Out Comix для всіх лесбійок). (1973)), Роберта Грегорі (Bitchy Bitch, і часто бере участь у Gay Comix) та Елісон Бечдел (Dykes to Watch Out For і графічний роман Fun Home, 2006).

На незалежному ринку, який почав з'являтися з 1970-х років, Венді Піні разом зі своїм чоловіком Річардом Піні розпочали натхненний мангою серіал Elfquest, який незабаром став великим хітом.

Коллін Доран створила свій культовий серіал космічної опери «Віддалений ґрунт», який був опублікований на початку 1980-х у невеликих фанзінах у пресі, а потім видав Доран на початку 1990-х, а потім перейшов до Image Comics у 1996 році.

Серед інших популярних художниць – Донна Барр ( Післівний персик, про вигаданого брата-гея Ервіна Роммеля), Джилл Томпсон (Scary Godmother, дружня відьма в оточенні Гелловіна) і Лінда Медлі (Castle Waiting, повсякденне життя казкових персонажів).

Вебкомікс
Багато художниць знайшли свою славу у веб- коміксах і пізніше опублікували друковані копії своїх робіт, наприклад, Кейт Бітон для Hark! A Vagrant!, і Hyperbole and a Half Еллі Брош. Інші, як Емілі Керролл (відома за веб-коміксом His Face All Red), продовжили роботу над іншими мультимедійними проектами, такими як Керролл Gone Home.

Серед художниць вебкоміксів є такі письменниці та ілюстраторки, як Кейт Лет (канадійка), Даніель Корсетто, Ремсі Бейєр, Люсі Ніслі, Еббі Говард, Мадлен Флорес (Час пригод), Дороті Гембрелл, Ліз Принс та Еріка Моен, яка працювала з Грейс Елліс та Н.Д. Стівенсон про популярну альтернативну серію принтів Lumberjanes.

Азія
Серед країн з великим відсотком авторок коміксів є Японія та Південна Корея.

Японія
Першою значною художницею манги була Мачіко Хасегава, творець сімейної Sazae-san, яка була запущена в 1946 році в газеті Асахі Сімбун. Вона діяла кілька десятиліть.

Комікси, призначені для дівчат (шодзё манга), мають довгу історію в Японії. Вони виросли з журналів про стиль життя, спрямованих на дівчат і підлітків на початку ХХ століття. У цих журналах були романтичні оповідання та модні ілюстрації під керівництвом чоловічої редакції.

У 1953 році «Бог манги» Осаму Тедзука опублікував свій класичнний твір «Принцеса-лицар» з довшим, складнішим сюжетом і гендерно неоднозначною головною героїнею. Ця манга справила великий вплив на багатьох японських творців та творчинь.

У 1950-х роках з'явилися щомісячні журнали «Рібон» і «Накайоші», а в 1963 році — тижневик «Сьодзьо Френд» і «Маргарет». Більшість із цих ранніх коміксів були написані такими людьми, як Тецуя Чіба, Міцутеру Йокояма та Фудзіо Акацука. Їм не вдалося залучити широку читацьку аудиторію.

У 1960-х роках Йошіко Нішітані створила роботи, в яких головні ролі виконали гламурні дівчата-підлітки, а центральна тема — колись табуйовані романи. Це допомогло підготувати шлях до великої хвилі в кінці 1960-х - початку 1970-х років, коли вільні зв'язки жінок, які пізніше назвали групою 24 року, об'єднали оповідання Тедзуки про "історійну мангу" з романтичним художнім стилем зі способу життя дівчат. Це відбилося у журналах у процесі революції в жанрі як у візуальних експериментах (включаючи макети сторінок, схожих на монтаж), так і в сюжетах.

Деякі з художниць, такі як Кейко Такемія та Мото Хагіо, написали історії про молодих коханців-геїв, залучених у трагічні стосунки. Ці історії виявилися надзвичайно популярними і породили жанр яой, який досі дуже популярний (Пізніше Кейко Такемія зняла популярний науково-фантастичний фільм «Назустріч Террі»).

Відтоді комікси для дівчат були процвітаючою галуззю, яку, загалом, створювали й читали жінки, займали помітну частину ринку, і, оскільки манга стає все більш популярною за кордоном, все більше й більше створює вплив на західні країни.

Пізніші популярні художниці включають дуже плідну й успішну Руміко Такахаші (малювала в основному історії для хлопчиків сьоненів), Хірому Аракава (Стальнометалевий алхімік), Кадзуе Като (Блакитний екзорцист), а також жіночий колектив Clamp.

Японія не тільки випускає комікси для дітей та підлітків і підліток, але також має сцени сейнен (дорослі чоловіки) і дзьосей (дорослі жінки), що дозволяє використовувати більш зрілі теми та сюжетні лінії.

Багато митців, які працюють на цьому ринку, отримали широке визнання серед альтернативних сцен коміксів у США та Європі, включаючи таких художниць, як Кіріко Нананан, Мойоко Анно, Джунко Мізуно та Кан Такахама  .

Південна Корея
Корейські комікси відомі як манхва (етимологічно схожий на японську мангу). Комікси стали популярними особливо в 1950-х і 1960-х роках. Жанр сунйонґ (sunjŏng)  став особливо популярним серед молодих дівчат і жінок. Корейська асоціація художниць коміксів (KWCA) обслуговувала жінок. Вона була заснована 2 грудня 1997 року, і веб-сайт був активний з 2001 по 2012 рік. У 2019 році ділянку використовували для Асоціації дерев'яної архітектури.

Уродженка Південної Кореї Кеум Сок Джендрі-Кім порушила конвенції, публікуючи важкі комікси з великим політичним змістом. Серед її графічних романів — «Пісня мого батька», «Дзисеуль», «Когаєї» та відзначений нагородами «Грасс» (2019), історія про корейську дівчину, примушену в сексуальне рабство для японських військових під час Другої світової війни. Грасс  була нагороджена премією Харві за найкращу міжнародну книгу  2020 року та номінована на кілька інших видатних нагород.

Європа
Хоча в середовищі творців коміксів художниці завжди складали меншість, вони працювали в медіа з перших днів його існування. Однією з перших художниць була Марі Дюваль, яка разом зі своїм чоловіком Чарльзом Генрі Россом була співавторкою і художницею одного з найперших персонажів сучасних мультфільмів і коміксів Еллі Слопер.

Туве Янссон найбільш відома як письменниця, але вона також писала та малювала комікси зі своїми героями мумі-тролями у 1950-х роках, які містили ті ж поетичні якості, що й її книги.

У Великій Британії Посі Сіммондс розпочала свою кар'єру в 1979 році з щотижневого комікса «иха трійка святого Ботольфа» для The Guardian про щоденне життя трьох колишніх шкільних друзів, яке тривало протягом десятиліття. Вона також написала дитячі книжки, часто у формі коміксів, такі як «Фред» (до якого пізніше зробили успішний анімаційний випуск) і «Лулу та літаючі немовлята». У 1990-х і 2000-х роках вона створила більш серйозні роботи, натхненні літературними класиками, такими як Джеммою Бовері та Тамарою Дрю.

Франція/Бельгія
Першим ветераном франко-бельгійського ринку була Ліліан Функен (уроджена Шорілс), яка, зустрівши свого чоловіка Фреда Функена (сам ветеран коміксів), об'єдналася з ним, щоб почати довготривалу кар'єру з журналом Tintin з 1950-х до 1980-х років, де пара співпрацювала над коміксами та ілюстраціями. Вони прийняли реалістичний стиль і в основному спеціалізуються на історичних роботах.

Однією з найперших успішних художниць була Клер Бретешер, яка розпочала свою кар'єру в 1960-х роках і відома своєю гумористичною серією Les Frustrés і співтворчістю журналу L'Écho des savanes разом з Готлібом і Мандрикою.

У 1976 році був заснований французький журнал Ah ! Нана. Він був натхненний феміністичним андеграундним коміксом із США, опублікованим Humanoïdes Associés, і був спробою відділитися від журналу Metal Hurlant тим самим редактором з більшістю художниць. У ньому намагалися дотримуватися рок-н-ролу колишнього журналу, а іноді показували чоловіків-виконавців з журналу, таких як Жак Тарді та Мебіус. Кожен випуск був побудований навколо теми, як-от нацизм або гомо- та транссексуальність. Випуск 7 за 1978 р. про садомазохізм вважався порнографією, і його заборонялося продавати неповнолітнім віком до 18 років, правило, яке в подальшому забороняло кіоскам рекламувати журнал, таким чином відрізавши багато ринкових точок журналу. Зрештою, це змусило видання журналу скасувати через погані продажі, через засоби, які авторство розглядало як цензуру феміністського голосу. Останній випуск був номер 9, присвячений інцесту. Відтоді на франко-бельгійському ринку не з'явилося жодного подібного журналу коміксів, але він допоміг розпочати або консолідувати кар'єру Шанталь Монтельє (феміністка, політична фантастика), Ніколь Клавелу (сюрреалістичне фентезі) та Флоренс Кестас (кумедні мультфільми).

Іншою письменницею, яка з'явилася в цей час, була Енні Гетцінгер, яка працювала в стилі реалістичного модерну та малювала пригоди з героїнями. Вона часто співпрацювала з П'єром Крістіном і отримала дві нагороди на фестивалі в Ангулемі.

На початку ХХІ століття Марджан Сатрапі випустила відомий критиками «Персеполіс» про своє дитинство та повноліття в політично турбулентному Ірані та Європі.

Див. також
 Друзі Лулу
 Список жінок-твориць коміксів
 Список феміністичних коміксів
 Зображення жінок у коміксах

Посилання

Подальше читання
 Horn, Maurice. Women in the Comics (Chelsea House; New York, London; 1977) ; (trade paperback) ISBN o-97754-205-8
 Robbins, Trina. From Girls to Grrrlz: A History of Women's Comics from Teens to Zines (Chronicle, 1999)
 Robbins, Trina. The Great Women Cartoonists (Watson-Guptill, 2001)
 Yronwode, Catherine and Robbins, Trina. Women and the Comics (Eclipse, 1983)

Фемінізм
Сторінки з неперевіреними перекладами

redpajama_set_name: RedPajamaWikipedia


text:
The Eagles/Jackson Browne/Linda Ronstadt: Live at Staples Center, Los Angeles
Erik Himmelsbach, Rolling Stone, 17 February 2000
THE KINGS of Country Rock have been on ice for the better part of two decades. Still, they earned most of their reported $7 million keep at this rare but less-than-capacity homecoming, skillfully puttin' on the hits for nearly three hours with the relaxed confidence of a band that could look back and laugh at its combustible legacy, and itself, all the way to the bank.
back to JACKSON BROWNE ARTICLES
back to EAGLES, THE ARTICLES
back to LINDA RONSTADT ARTICLES
back to ARTICLES BY ERIK HIMMELSBACH
back to ARTICLES FROM ROLLING STONE


Virginia is Publishing Assistant here at Modiphius and will be running a game of John Carter of Mars on Facebook live at Modiphius HQ from 3 PM BST on Wednesday the 17th of April.
If you've never explored the world of John Carter and want to know more watch this video and she'll take you through all the races of Barsoom, as well as the cultures and traditions of the planet, so that you can get some ideas about what you want to feature in your own game of John Carter of Mars: Adventures on the Dying World of Barsoom!
"My attention was quickly riveted by a large red star close to the distant horizon. As I gazed upon it I felt a spell of overpowering fascination—it was Mars, the god of war, and for me, the fighting man, it had always held the power of irresistible enchantment. As I gazed at it on that far-gone night it seemed to call across the unthinkable void, to lure me to it…"
These join the recent John Carter Collector's Slipcase set, miniature sets featuring the Heroes of Barsoom, Helium and Zodanga Fighting Crews and Thark Warband, plus Helium, Thark and Zodangan dice sets, which are all available as part of the John Carter of Mars Collection on Modiphius.net. You can also buy all the PDFs as part of the John Carter Collection on DriveThruRPG.com. and remember these gorgeous Slipcase and hardback books, Narrators Screen, Dice sets and Notebook are all now available in your friendly local gaming store.


text:
Immune evasion by hepatitis C virus NS3/4A protease-mediated cleavage of the Toll-like receptor 3 adaptor protein TRIF
Kui Li, Eileen Foy, Josephine C. Ferreon, Mitsuyasu Nakamura, Allan C.M. Ferreon, Masanori Ikeda, Stuart C. Ray, Michael Gale, Stanley M. Lemon
Graduate School of Medicine Dentistry and Pharmaceutical Sciences
Toll-like receptors (TLRs) bind pathogen-specific ligands early in infection, initiating signaling pathways that lead to expression of multiple protective cellular genes. Many viruses have evolved strategies that block the effector mechanisms induced through these signaling pathways, but viral interference with critical proximal receptor interactions has not been described. We show here that the NS3/4A serine protease of hepatitis C virus (HCV), a virus notorious for its ability to establish persistent intrahepatic infection, causes specific proteolysis of Toll-IL-1 receptor domain-containing adaptor inducing IFN-β (TRIF or TICAM-1), an adaptor protein linking TLR3 to kinases responsible for activating IFN regulatory factor 3 (IRF-3) and NF-κB, transcription factors controlling a multiplicity of antiviral defenses. NS3/4A-mediated cleavage of TRIF reduces its abundance and inhibits polyl:C-activated signaling through the TLR3 pathway before its bifurcation to IRF-3 and NF-κB. This uniquely broad mechanism of immune evasion potentially limits expression of multiple host defense genes, thereby promoting persistent infections with this medically important virus.
IFN regulatory factor 3
INF-κB
Protease inhibitor
10.1073/pnas.0408824102
Fingerprint Dive into the research topics of 'Immune evasion by hepatitis C virus NS3/4A protease-mediated cleavage of the Toll-like receptor 3 adaptor protein TRIF'. Together they form a unique fingerprint.
hepatitis C virus NS3 protein Medicine & Life Sciences
Toll-Like Receptor 3 Medicine & Life Sciences
Immune Evasion Medicine & Life Sciences
Interferon Regulatory Factor-3 Medicine & Life Sciences
Hepatitis C virus NS3-4A serine protease Medicine & Life Sciences
Viral Interference Medicine & Life Sciences
Li, K., Foy, E., Ferreon, J. C., Nakamura, M., Ferreon, A. C. M., Ikeda, M., Ray, S. C., Gale, M., & Lemon, S. M. (2005). Immune evasion by hepatitis C virus NS3/4A protease-mediated cleavage of the Toll-like receptor 3 adaptor protein TRIF. Proceedings of the National Academy of Sciences of the United States of America, 102(8), 2992-2997. https://doi.org/10.1073/pnas.0408824102
Immune evasion by hepatitis C virus NS3/4A protease-mediated cleavage of the Toll-like receptor 3 adaptor protein TRIF. / Li, Kui; Foy, Eileen; Ferreon, Josephine C.; Nakamura, Mitsuyasu; Ferreon, Allan C.M.; Ikeda, Masanori; Ray, Stuart C.; Gale, Michael; Lemon, Stanley M.
In: Proceedings of the National Academy of Sciences of the United States of America, Vol. 102, No. 8, 22.02.2005, p. 2992-2997.
Li, K, Foy, E, Ferreon, JC, Nakamura, M, Ferreon, ACM, Ikeda, M, Ray, SC, Gale, M & Lemon, SM 2005, 'Immune evasion by hepatitis C virus NS3/4A protease-mediated cleavage of the Toll-like receptor 3 adaptor protein TRIF', Proceedings of the National Academy of Sciences of the United States of America, vol. 102, no. 8, pp. 2992-2997. https://doi.org/10.1073/pnas.0408824102
Li K, Foy E, Ferreon JC, Nakamura M, Ferreon ACM, Ikeda M et al. Immune evasion by hepatitis C virus NS3/4A protease-mediated cleavage of the Toll-like receptor 3 adaptor protein TRIF. Proceedings of the National Academy of Sciences of the United States of America. 2005 Feb 22;102(8):2992-2997. https://doi.org/10.1073/pnas.0408824102
Li, Kui ; Foy, Eileen ; Ferreon, Josephine C. ; Nakamura, Mitsuyasu ; Ferreon, Allan C.M. ; Ikeda, Masanori ; Ray, Stuart C. ; Gale, Michael ; Lemon, Stanley M. / Immune evasion by hepatitis C virus NS3/4A protease-mediated cleavage of the Toll-like receptor 3 adaptor protein TRIF. In: Proceedings of the National Academy of Sciences of the United States of America. 2005 ; Vol. 102, No. 8. pp. 2992-2997.
@article{94c33be131c74bfa8016b4baf3ca17ec,
title = "Immune evasion by hepatitis C virus NS3/4A protease-mediated cleavage of the Toll-like receptor 3 adaptor protein TRIF",
abstract = "Toll-like receptors (TLRs) bind pathogen-specific ligands early in infection, initiating signaling pathways that lead to expression of multiple protective cellular genes. Many viruses have evolved strategies that block the effector mechanisms induced through these signaling pathways, but viral interference with critical proximal receptor interactions has not been described. We show here that the NS3/4A serine protease of hepatitis C virus (HCV), a virus notorious for its ability to establish persistent intrahepatic infection, causes specific proteolysis of Toll-IL-1 receptor domain-containing adaptor inducing IFN-β (TRIF or TICAM-1), an adaptor protein linking TLR3 to kinases responsible for activating IFN regulatory factor 3 (IRF-3) and NF-κB, transcription factors controlling a multiplicity of antiviral defenses. NS3/4A-mediated cleavage of TRIF reduces its abundance and inhibits polyl:C-activated signaling through the TLR3 pathway before its bifurcation to IRF-3 and NF-κB. This uniquely broad mechanism of immune evasion potentially limits expression of multiple host defense genes, thereby promoting persistent infections with this medically important virus.",
keywords = "IFN, IFN regulatory factor 3, INF-κB, Innate immunity, Protease inhibitor",
author = "Kui Li and Eileen Foy and Ferreon, {Josephine C.} and Mitsuyasu Nakamura and Ferreon, {Allan C.M.} and Masanori Ikeda and Ray, {Stuart C.} and Michael Gale and Lemon, {Stanley M.}",
doi = "10.1073/pnas.0408824102",
journal = "Proceedings of the National Academy of Sciences of the United States of America",
publisher = "National Academy of Sciences",
T1 - Immune evasion by hepatitis C virus NS3/4A protease-mediated cleavage of the Toll-like receptor 3 adaptor protein TRIF
AU - Li, Kui
AU - Foy, Eileen
AU - Ferreon, Josephine C.
AU - Nakamura, Mitsuyasu
AU - Ferreon, Allan C.M.
AU - Ikeda, Masanori
AU - Ray, Stuart C.
AU - Gale, Michael
AU - Lemon, Stanley M.
N2 - Toll-like receptors (TLRs) bind pathogen-specific ligands early in infection, initiating signaling pathways that lead to expression of multiple protective cellular genes. Many viruses have evolved strategies that block the effector mechanisms induced through these signaling pathways, but viral interference with critical proximal receptor interactions has not been described. We show here that the NS3/4A serine protease of hepatitis C virus (HCV), a virus notorious for its ability to establish persistent intrahepatic infection, causes specific proteolysis of Toll-IL-1 receptor domain-containing adaptor inducing IFN-β (TRIF or TICAM-1), an adaptor protein linking TLR3 to kinases responsible for activating IFN regulatory factor 3 (IRF-3) and NF-κB, transcription factors controlling a multiplicity of antiviral defenses. NS3/4A-mediated cleavage of TRIF reduces its abundance and inhibits polyl:C-activated signaling through the TLR3 pathway before its bifurcation to IRF-3 and NF-κB. This uniquely broad mechanism of immune evasion potentially limits expression of multiple host defense genes, thereby promoting persistent infections with this medically important virus.
AB - Toll-like receptors (TLRs) bind pathogen-specific ligands early in infection, initiating signaling pathways that lead to expression of multiple protective cellular genes. Many viruses have evolved strategies that block the effector mechanisms induced through these signaling pathways, but viral interference with critical proximal receptor interactions has not been described. We show here that the NS3/4A serine protease of hepatitis C virus (HCV), a virus notorious for its ability to establish persistent intrahepatic infection, causes specific proteolysis of Toll-IL-1 receptor domain-containing adaptor inducing IFN-β (TRIF or TICAM-1), an adaptor protein linking TLR3 to kinases responsible for activating IFN regulatory factor 3 (IRF-3) and NF-κB, transcription factors controlling a multiplicity of antiviral defenses. NS3/4A-mediated cleavage of TRIF reduces its abundance and inhibits polyl:C-activated signaling through the TLR3 pathway before its bifurcation to IRF-3 and NF-κB. This uniquely broad mechanism of immune evasion potentially limits expression of multiple host defense genes, thereby promoting persistent infections with this medically important virus.
KW - IFN
KW - IFN regulatory factor 3
KW - INF-κB
KW - Innate immunity
KW - Protease inhibitor
U2 - 10.1073/pnas.0408824102
DO - 10.1073/pnas.0408824102
JO - Proceedings of the National Academy of Sciences of the United States of America
JF - Proceedings of the National Academy of Sciences of the United States of America


text:
Business Information and 'Personal Data': Some Common Law Observations about the Draft EU Data Protection Directive
Iowa Law Review, Vol. 80, No. 3, 1995
20 Pages Posted: 18 Aug 2008 Last revised: 27 Feb 2014
See all articles by James R. Maxeiner
James R. Maxeiner
University of Baltimore - School of Law
Discusses the public interest in the free flow of information.
Keywords: Privacy, data protection, freedom of information, freedom of speech
Maxeiner, James R., Business Information and 'Personal Data': Some Common Law Observations about the Draft EU Data Protection Directive (May 1, 1995). Iowa Law Review, Vol. 80, No. 3, 1995, Available at SSRN: https://ssrn.com/abstract=1232754
James R. Maxeiner (Contact Author)
University of Baltimore - School of Law ( email )


Hygiene slings are specifically designed for toileting and hygiene functions. Access is unrestricted for most users, making hygiene slings ideal for independent transfer and toileting.
As the individual is lifted, his or her position will become more extended and having the ability to adjust the tension on the chest band is an important feature for the comfort and security of the individual.


text:
\section{\bf Introduction}

There is a fundamental topological distinction between an ordinary
Higgs boson and a composite \cite{Kaplan:1983fs} or Little Higgs boson
\cite{nima}.
The ordinary Higgs boson is
described by a scalar field taking values on a flat, unbounded
manifold.  The composite or Little Higgs boson, on the other hand, is a (pseudo)
Nambu-Goldstone boson (pNGB) of a spontaneously broken chiral
symmetry, the analog of a meson such as the kaon (we'll henceforth
refer to all such composite Higgs bosons as Little Higgs bosons).
It is described
by a field that is confined to a compact manifold, of ``radius''
$1/F$.  The Little Higgs is thus an angular variable and it can be
translated through $2\pi F$ circumnavigating the manifold.  As a
consequence, topologically stable configurations can exist that are
described by conserved topological currents.  These currents arise
from the full effective action only if we include a new topological
interaction, known as the Wess-Zumino-Witten (WZW)
term~\cite{Wess:1971yu,Witten}.  This term is the low-energy effective
description of anomaly physics in terms of pNGB's and gauge
fields~\cite{BJ,Adler,bardeen}.  It unifies the topological physics in
a chiral Lagrangian, and describes new physical processes that are not
expected for the ordinary Higgs boson.  It is tied to any particular
UV completion model through integer quantities such as the number of
``techni-colors'' of the constituent ``techni-quarks.''  Thus, the new
WZW interactions of Little Higgs bosons probe the underlying UV
completion theory, much like the $\pi^0\rightarrow \gamma\gamma$
interaction probes QCD.

To motivate our discussion, we recall the typical interactions
contained in the ungauged Wess-Zumino~\cite{Wess:1971yu} term of QCD,
\begin{eqnarray}
\label{onee}
\int \Tr( \tp d\tp d\tp d\tp d\tp )
&=&
\int\! d^4x\, \bigg[ 5 K^\dagger \partial_\mu\pi \partial_\nu\pi \partial_\rho\pi
\partial_\sigma K
-{5\sqrt{3}} \eta\; \partial_\mu K^\dagger \partial_\nu K
\partial_\rho K^\dagger \partial_\sigma K
\nonumber \\  && \qquad
- {5 \sqrt{3}}\eta\; \partial_\mu K^\dagger \partial_\nu\pi
\partial_\rho\pi \partial_\sigma K + \dots \bigg]\epsilon^{\mu\nu\rho\sigma}
\nonumber \\
&\propto& \int\! d^4 x \, \partial_\mu \pi^+ \partial_\nu \pi^-
\partial_\rho \pi^0 K^\dagger \partial_\sigma K
\epsilon^{\mu\nu\rho\sigma} + \dots \,.  \end{eqnarray}
Witten~\cite{Witten}
pointed out that this expression arises from a topological
construction in $D=5$.  It reflects the nontrivial homotopy group
$\pi_5(SU(3))=\bm{Z}$ and its coefficient is subject to quantization.
In QCD the WZ term locks the parity of the pion to that of spacetime, and it
describes allowed strong interactions such as $K \mybar{K} \rightarrow
\pi\pi\pi$, which are otherwise absent in a pure kinetic-term chiral
Lagrangian.

In Little Higgs models the Higgs
field $H$ is identified with an object like the kaon, $H\sim K$.
Upon electroweak
gauging, we thus expect to have a nontrivial WZW term, and by analogy
to the QCD chiral Lagrangian, new interactions can arise involving
gauge bosons and Little Higgs bosons.  In generic schemes we
might expect novel interactions like:
\begin{equation}\label{one}
\int d^4x\, W_\mu^+ W_\nu^- Z_\rho^0 H_i^\dagger
\partial_\sigma H_j \epsilon^{\mu\nu\rho\sigma} \, + ... ,
\end{equation}
with multiple Little Higgs bosons $H_i$.

The discussion of such topological
interactions requires the full gauging of the
Wess-Zumino-Witten term.  Witten originally initiated the program of
brute force gauging of the WZ term, which was developed by a number of
subsequent authors~\cite{Witten,kay,man,tye}.  For the cases where it
is applicable, we will employ the most transparent of these, the form
of Kaymacalan, Rajeev and Schechter (KRS)~\cite{kay}. The full WZW
term of KRS can be seen to descend from the Chern-Simons term and the
Bardeen counterterm \cite{bardeen} in a compactified pure Yang-Mills
theory of flavor in $D=5$~\cite{hillch}.

To retain intuition based on the familiar chiral Lagrangian of QCD, we
first consider a theory based on $SU(3)_L\times SU(3)_R \times
U(1)/SU(3)\times U(1)$, gauging an $SU(2)$ subgroup of $SU(3)_L$.
This is a simplified version of technicolor. The isovector techni-pions
are eaten
to become longitudinal $W$ and $Z$ bosons, and physical techni-kaons
($\sim H$) and a techni-eta remain in the spectrum.  The resulting WZW
term indeed yields techni-kaon interactions of the form
$ \epsilon^{\mu\nu\rho\sigma} W_\mu^+
W_\nu^- Z_\rho^0 H^\dagger \partial_\sigma H$.

We then turn to the simplest Little Higgs theories in which the
techni-pions of the previous example do not occur.  This ensures that
the $W$ and $Z$ bosons remain massless prior to electroweak symmetry
breaking.  These are generic Kaplan-Schmaltz (KS)
models~\cite{Kaplan:2003uc}.  They can be obtained by two physically
distinct approaches.  First, we consider reducing the $SU(3)_L\times
SU(3)_R \times U(1) / SU(3)\times U(1)$ QCD-like scheme to an
$SU(3)\times U(1)/SU(2)\times U(1)$ scheme by decoupling the
isovector techni-pions.
In particular, we can ``eat and decouple'' the pions by
introducing an $SU(2)$ $A_R$ gauge field in a strong coupling limit. This
converts the usual unitary matrix chiral field into a {\em nonlinear
realization} of $SU(3)$ containing only the isodoublet Higgs (kaon)
and a singlet (eta).  This construction enforces the correct gauge
transformation for a nonlinear realization, and dictates the proper form
of the covariant derivative.
Nonlinear realizations
afford an interesting point of departure for the construction of Little
Higgs models in general~\cite{2hill}.
With the correct gauging established, we can then
directly use the KRS form to obtain the full WZW
term for $SU(3)\times U(1)/SU(2)\times U(1)$ Little Higgs models.

Alternatively, we may contemplate directly the topological structure of the KS
models.  The pNGB's for $SU(3)\times U(1)/SU(2)\times U(1)$
are described by a complex triplet scalar field $\Phi$ with
$\Phi^\dagger\Phi = {\rm constant}$.  The fields thus live in a space
that is topologically equivalent to the five-dimensional unit sphere,
denoted $S^5$.  The topological interactions reflect the obvious but
nontrivial homotopy group $\pi_5(S^5) = \bm{Z}$, and are described by
the $SU(3)\times U(1)$-invariant form:~
\footnote{
Note that each term in eq.(\ref{onee}) involves five
``coordinates'' that locally parameterize an $S^5$.
The term $\sim \eta dK^+ dK^-dK^0d\bar{K}^0$
corresponds to the submanifold of $SU(3)$ that is
identified with $\Phi$.
}
\begin{equation}
\omega_{ABCDE} = -{i\over 8} \Phi^\dagger \partial_{[A } \Phi \partial_B\Phi^\dagger\partial_C\Phi\partial_D\Phi^\dagger \partial_{E] }\Phi  \,,
\end{equation}
which corresponds to the surface area of $S^5$ parameterized by
NGB's.  Under a local gauge transformation, $\delta \omega$ is a
total derivative in $D=5$ and can thus be gauged in $D=4$ to yield the
WZW term.  We describe the intricate procedure of gauging this
structure, and discuss the interesting question of equivalence of the
two approaches to the gauged topological action for $SU(3)\times
U(1)/SU(2)\times U(1)$.
We also touch on the related question of UV completions, noting that
the latter model can be viewed as arising from an underlying fermion
theory in which a triplet of techni-quarks $\psi_L$ condenses with a
singlet $q_R$, while the previous model arises from a condensate of two triplets
of techni-quarks $\psi_L$ and $\psi_R$ with the pions eaten and decoupled.

While we presently sketch how all of this works, the full details will
be presented elsewhere~\cite{2hill}.  The above constructions provide
a building block for many composite Higgs scenarios.  Gauging
$SU(2)\times U(1)$ for a single complex triplet KS $\Phi$ field yields
a simple and intriguing model of a composite Higgs boson.
Gauging $SU(2)\times U(1)$ for two (or more) $\Phi_i$ fields yields a
multi-Higgs doublet model, while gauging the full $SU(3)\times U(1)$
for two $\Phi_i$ fields describes the Kaplan-Schmaltz Little Higgs
model.

When the symmetry breaking pattern respects an internal parity
operation the WZW term takes a special form that is identical to that
obtained in the QCD chiral Lagrangian.  This occurs, for example, in
Little Higgs models with ``T parity''~\cite{Cheng:2003ju}.  As an
application, we describe the main results for an $SU(5)/SO(5)$ Little
Higgs model~\cite{Arkani-Hamed:2002qy}.  We point out that the WZW
term is odd under the internal parity, and describes interactions
between a single ``T odd'' particle and standard model particles.

The WZW term contains interactions that are quantized, subject to
Adler-Bardeen nonrenormalization~\cite{AdlerBardeen}.  They are
suppressed by factors of $1/F$, with $F\sim 1$ TeV for typical Little
Higgs models, and occur with loop-order coefficients commensurate with
the underlying anomalies. Thus, they may be hard to detect.
Nonetheless, these interactions can provide a powerful discriminant of
underlying short-distance physics, and it is worth understanding what
effects can occur and determining whether they are suited to discovery
at the next generation of colliders, the LHC and the ILC.  Much of
this phenomenology is beyond the scope of the present paper, but will be
developed elsewhere
\cite{2hill}.

\section{Illustration Based on Technicolor}

To illustrate the procedures used in this analysis, we first consider
a chiral Lagrangian based on a QCD-like strong gauge group $SU(N_c)$,
containing $SU(3)$ flavor triplets of techni-quarks, $(\Psi_L,\Psi_R)$,
transforming in the fundamental representation with $N_c$ colors.  The
strong interaction results in a condensate $\langle \psi_L^i
\bar{\psi}_R^j \rangle \sim F^3 \delta^{ij}$, leading to an
$SU(3)_L\times SU(3)_R\times U(1) / SU(3)\times U(1)$ chiral
Lagrangian described by the $3\times 3$ unitary matrix field $U^{ij}
\sim \psi_L^i \bar{\psi}_R^j$.  We parameterize the field as $U =
\exp(2i {\tilde{\pi}}/F)$, where $\tilde{\pi}=\sum_{a=1}^8
\pi^a\lambda^a/2$ are Nambu-Goldstone bosons, transforming bilinearly
under $SU(3)_L\times SU(3)_R\times U(1)$ as:
\begin{equation}\label{Utransform}
U \rightarrow e^{i\epsilon_L} U e^{-i\epsilon_R} \,.
\end{equation}
We presently turn off the standard model $U(1)_Y$
coupling constant, although it is straightforward to include it.  We
thus gauge an $SU(2)$ subgroup of $SU(3)_L$, with covariant
derivative:
\begin{equation}\label{Afield}
D_\mu U = \partial_\mu U - iA_\mu U \,, \qquad \qquad
A_\mu=\left( \begin{array}{cc} g W_\mu^a \frac{\sigma^a}{2} & 0 \\ 0 &
 0 \end{array}\right) \,.
\end{equation}
In general such ``left-side'' gauging is anomalous, but
pure $SU(2)$ gauging is always anomaly free (we either ignore the
discrete Witten anomaly presently, or choose $N_c$ even).

The anomaly physics is contained in the gauged WZW term:
\begin{equation}\label{WZW}
\Gamma_{WZW} = \Gamma(U,A)  \,.
\end{equation}
This can be obtained directly from the general form of $\Gamma(U,
A_L,A_R)$ in eq.(4.18) of KRS \cite{kay} (see eq.(\ref{WZWfull})
below), by setting $A_L = A$ and $A_R = 0$, where we use form
notation, \eg, $A = A_\mu dx^\mu$, $ABCD=
\epsilon^{\mu\nu\rho\sigma}A_\mu B_\nu C_\rho D_\sigma d^4x$, and
$dA=\frac12(\partial_\mu A_\nu - \partial_\nu A_\mu) dx^\mu dx^\nu$.
A chiral current is defined as
$\alpha = dU U^\dagger = (2i/F) d\tp + ...$ \,.  Explicitly, we have:
\begin{eqnarray}
\label{final1}
\Gamma(U,A) & = & \Gamma_{0}(U) +
\frac{  N_c}{48\pi^2} \int_{M^4} \Tr\left[
 {A}\alpha^3  -\frac{i}{2}({A}\alpha)^2
 +i(dA A + A dA)\alpha + {A}^3\alpha \right] \,,
\end{eqnarray}
where $M^4$ denotes four-dimensional spacetime ($M^5$ below denotes a
five-dimensional manifold with spacetime as its boundary).  We recall
that $\Gamma_0(U)$ is defined as the surface term of a globally
chirally invariant operator in $D=5$:
\begin{equation}\label{Gamma0}
{\Gamma}_{0}  = -{i N_c\over 240\pi^2} \int_{M^5} \Tr\left( \alpha^5\right)
= -\frac{i N_c}{240\pi^2}\int d^5x \,  \epsilon^{ABCDE}\;
\Tr \left( \alpha_A\alpha_B\alpha_C\alpha_D\alpha_E \right) \,.
\end{equation}
$\Gamma_0$ is not manifestly local in four dimensions, but since
$d\Tr\alpha^5= \Tr\alpha^6=0$, it can be written as an expansion in
mesons in $D=4$.  Under a left-handed chiral transformation, $\delta U
= i\epsilon U$, we have $\delta\alpha = id\epsilon +i[\epsilon,
\alpha]$.  Using $\alpha^4=d\alpha^3$, it follows that $\Gamma_0$
shifts by an exact differential, a $D=5$ surface term,
\begin{equation}\label{brute}
\delta{\Gamma}_{0}  =
\frac{ N_c}{48\pi^2}\int_{M^5}
d \Tr \left( \epsilon \alpha^4 \right)
=
 - \frac{N_c}{48\pi^2}\int_{M^4}
 \Tr \left( d \epsilon\; \alpha^3 \right) \,.
\end{equation}
This shift is compensated by the $D=4$ term in $\Gamma(U,A)$ involving
one gauge field.  The residual shift is cancelled by the term with two
gauge fields, and so on, leading
to eq.(\ref{final1}).~%
\footnote{ Alternatively, eq.(\ref{final1}) and eq.(\ref{WZWfull}) can
  be derived from the Yang-Mills Chern-Simons term and the Bardeen
  countertyerm in a $D=5$ Yang-Mills theory compactified on $S^1$
  where $A_5$ is identified with the mesons \cite{hillch}.  }

An important comment is in order concerning gauge invariance.  The
full WZW term of eq.(\ref{final1}) generates a vanishing anomaly in
the gauged $SU(2)$ subgroup, but nonvanishing anomalies in the global
currents, $a=4,...,8$.  That is, under $\delta U = i\epsilon U$ and
$\delta A = d\epsilon + i[\epsilon,A]$ we have $\delta \Gamma \propto
\Tr[\epsilon(dA\, dA - id A^3/2)]$.  This is the left-right symmetric
(or ``consistent'') form of the anomaly and it is not gauge covariant.
 It is always possible to add Bardeen's counterterm~\cite{bardeen}
to bring the anomaly into the ``covariant'' form,
$\delta\Gamma \propto 3\Tr[\epsilon (dA - iA^2)^2]$.  However, the
coupling to NGB's is unaffected by the Bardeen counterterm, which is a
function only of gauge fields.
Expanding eq.(\ref{final1}) to leading order in NGB's, we see
that the NGB's participate in an interaction $\propto\Tr[\tp ( dA\,dA - id
A^3/2)]+...$~ which takes the form of the
consistent anomaly, and is superficially non-gauge invariant.
 How, then, can we see that this gives
a bona-fide gauge invariant interaction?  The resolution is that, with
``left-side'' gauging of $U$, the gauge fields {\em always acquire a
  mass}.  Since the $SU(2)$ gauge anomalies vanish, we are free to
transform to unitary gauge to remove eaten NGB's.  The gauge fields
can then be expressed in the ``Stueckelberg'' form, $A^\prime =
V^\dagger ( A + id ) V$, where $V$ is the transformation to unitary
gauge that removes the NGB's from $U$.  The Stueckelberg fields
are gauge invariant. The residual physical
techni-mesons, expressed in the same gauge, couple to $A^\prime$, and
the interaction takes the form $\Tr[ \tp^\prime ( dA^\prime dA^\prime
- id A^{\prime 3}/2)] + ...$, which is manifestly gauge invariant. In
summary, ``left-side'' gauging produces NGB's coupled to the
consistent anomaly which is
a gauge invariant functional of covariant Stueckelberg fields.~%
\footnote{ For illustration, consider the hypothetical situation in
  which the usual Higgs mechanism does not operate, but where the
  left-handed quarks of some number of Standard Model generations, and
  hence the QCD chiral Lagrangian, is gauged with the usual $SU(2)$
  weak interactions.  As is well known, the $W$ and $Z$ bosons then
  eat a linear combination of the pions and acquire mass.  After
  transforming to unitary gauge, the remaining pseudoscalar mesons of
  QCD couple to the $W$ boson through the consistent, not the
  covariant, anomaly.  }

Let us examine some typical physical processes contained in
eq.(\ref{final1}).  A convenient choice of $SU(3)$ coordinates around
$U=1$ is:
\begin{equation}
\label{Umatrix}
U = \left(
\begin{array}{cc}
e^{i\eta/F} \sqrt{1 - H H^\dagger/F^2} \,\, \hat{U} & H/F \\
-e^{-i\eta/F} H^\dagger \hat{U}/F & e^{-2i\eta/F}\sqrt{1-H^\dagger H/F^2}
\end{array}
\right) \,,
\end{equation}
where $\hat{U} = \exp(2i\hat{\pi})$ is an $SU(2)$ matrix containing
the techni-pions, $\hat{\pi} = \sum_{a=1}^3 \hat{\pi}^a\sigma^a/2$,
$\eta$ is a real phase and $H$ is a complex iso-doublet (techni-kaon).
Here $H
H^\dagger $ denotes a dyadic product.  Under an $SU(2)_L$
transformation, $\epsilon_L = {\rm diag}(\hat{\epsilon}, 0)$ and
$\epsilon_R=0$ in eq.(\ref{Utransform}), we have $H\to
e^{i\hat{\epsilon}} H$, $\hat{U}\to e^{i\hat{\epsilon}} \hat{U}$ and
$\eta \to \eta$.

We presently focus attention on terms involving $H$ through second
order in $1/F$.  Thus the chiral current becomes:
\begin{equation}\label{alpha}
\alpha =  \left( \begin{array}{cc} \frac{-1}{2F^2} H
\stackrel{\leftrightarrow}{d} H^\dagger
& \frac{1}{F} dH
\\  \frac{-1}{F} dH^\dagger
 &
 \frac{-1}{2F^2} H^\dagger \stackrel{\leftrightarrow}{d} H \end{array} \right) \,,
\end{equation}
and we find the WZW interactions of eq.(\ref{WZW}) take the form:
\begin{eqnarray}
\label{result1}
\Gamma_{WZW} & = &  - { i g^2N_c\over 192 \pi^2 F^2 } \int d^4 x\;
\epsilon^{\mu\nu\rho\sigma}
 \left(
Z_\mu^0 \partial_\nu Z_\rho^0 + W_\mu^-\partial_\nu W_\rho^+ +
W_\mu^+\partial_\nu W_\rho^-
-  {3 i\over 2} g Z_\mu^0 W_\nu^+ W_\rho^- \right ) \nonumber \\
&& \qquad\qquad \times  \left(H^0 \stackrel{\leftrightarrow}{\partial}_\sigma H^{0*}
+ H^+ \stackrel{\leftrightarrow}{\partial}_\sigma H^-\right)  + \dots \,.
\end{eqnarray}
The kinetic term allowed by $SU(3)_L\times SU(3)_R \times U(1)$
invariance is $F^2 {\rm Tr}|D_\mu U|^2$.  In this ``technicolor''
scheme, the $W$ and $Z$ bosons eat the $\hat{\pi}$ degrees of freedom,
acquiring a common mass $gF$.  We can transform to unitary gauge,
$\hat{U}\rightarrow 1$, leaving physical fields $\eta$ and $H$, and
eq.(\ref{result1}) describes the anomalous interactions of these
physical fields with the massive Stueckelberg $W$ and $Z$ gauge
bosons.  The WZW term for this technicolor scheme contains the
interesting physical processes $e^+e^- \rightarrow Z^*\rightarrow
W^+W^- (H^0 H^{0*} + H^+H^-)$ and $e^+e^- \rightarrow Z^*\rightarrow
Z^0 (H^0 H^{0*} + H^+H^-)$, with amplitudes that count the number of
underlying techniquark colors, $N_c$.

Eq.(\ref{result1}) does not describe a Little Higgs theory,
but we can deform this technicolor theory
to imitate a Little Higgs scheme by restricting the
kinetic term to the form $F^2 \Tr|D_\mu U P' |^2$
with the projection matrix $P' = {\rm diag}(0,0,1)$.  Doing so
blocks the $W$ and $Z$ from acquiring mass by
projecting out their longitudinal
coupling with the isovector techni-pions, but unfortunately
leaves the $\hat{\pi}$ as nonpropagating auxiliary fields in the theory.
We will see subsequently that removing the unphysical techni-pions
enforces that $U$ transform as a {\em nonlinear realization} of $SU(3)$.
The problem for Little Higgs theories is
therefore to construct the WZW term either by attacking directly and
rederiving the full WZW action in terms of a restricted manifold of
NGB's, or by adapting the above familiar form of the WZW term to the
case of $U$ treated as a nonlinear realization.  We turn to this issue
in the next sections, and derive the topological physics of bona-fide
Little Higgs models.

\section{WZW term for models involving $SU(3)/SU(2)$}

A set of ``simple'' Little Higgs models are based on $SU(3)\times U(1)
/SU(2)\times U(1)$, (or more generally $SU(n)\times U(1)
/SU(n-1)\times U(1)$).  These were introduced by Kaplan and Schmaltz
(KS) \cite{Kaplan:2003uc} .  We begin by considering ``one half'' of
such an $SU(3)/SU(2)$ Little Higgs model, described by a single scalar
field $\Phi$ which transforms as a triplet under $SU(3)$.

We can view the KS model as arising from a UV completion scheme in
which $\Phi$ is a bound state $\phi^i \sim \psi^i_{L} \bar{q}_{R} $,
where $\Psi_L$ is a flavor triplet and $q_R$ a singlet.  The fermions
transform in the fundamental representation of a color group $SU(N_c)$
and we assume that a $(3,1)$ condensate forms with $ \VEV{\Psi_L \bar{q}_R}
\sim (0,0,F^3)^T$.  We have constructed such UV completions and will
discuss their full content elsewhere.  The unbroken $SU(2)\times U(1)$
subgroup is anomaly free, with a vector-like $U(1)$ current $\propto
\overline{\psi}_L^3\gamma_\mu\psi_L^3 + \bar{q}_R\gamma_\mu q_R $.  This
subgroup can thus be identified with the electroweak gauge group of
the standard model. We can obtain the topological interactions
by deriving the WZW term directly as a functional of $\Phi$, which amounts
to gauging the sphere $S^5$ defined by
$\Phi^\dagger \Phi = 1$.

Alternatively, the KS model can be viewed as arising from
a UV completion with an $SU(3)_L\times SU(3)_R$ chiral structure, a
$(3,\bar{3})$ condensate of the form $U\sim \psi_L\bar{\psi}_R$.
The iso-vector pions are removed by gauging on the right with
a strongly coupled iso-vector gauge field $A_R$. $A_R$ eats the pions,
and becomes a functional of $H$, $\eta$ and the remaining gauge fields $A_L$,
while $U$ becomes a nonlinear realization with a modified
covariant derivative. We begin with the nonlinear realization.

\subsection{ Nonlinear Realization}

For the low energy effective Kaplan-Schmalz
theory we can write $\Phi = U \times \langle \Phi \rangle$,
where $\langle \Phi \rangle$ is a constant vector, and
$U$ is a unitary matrix function of the five (or $2n+1$) NGB's.
A convenient choice of coordinates for $\langle \Phi \rangle$ and $U$ is
\begin{equation}
\label{higgs}
\langle \Phi \rangle = \left(\begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right) \,,
\qquad \qquad
U =
\exp\left[ {i\over F}\left(\begin{array}{cc} \eta I_2/\sqrt{3} & H
\\ H^\dagger & -2\eta/\sqrt{3}
\end{array}\right)\right] \,.
\end{equation}
$H$ has the usual standard model quantum numbers of the Higgs, and
$\eta$ is a standard model singlet.  For simplicity, we will ignore
the $U(1)$ factor in the remainder of this section, although it is
straightforward to include it.

The critical element of this formalism is that $U$ transforms as a
{\em nonlinear realization} of $SU(3)$~\cite{Coleman:1969sm}.  That
is,
\begin{equation}\label{nonlinear}
U \rightarrow e^{i\epsilon}U e^{-i\epsilon'} \,,
\end{equation}
where $\epsilon \in SU(3)$ and $\epsilon'$ is a matrix function of
$\epsilon$ and $U$, belonging to the unbroken $SU(2)$ subgroup, the
upper left block with our choice of coordinates.  Hence, while $U$
contains only the five degrees of freedom described by $H$ and $\eta$,
we can implement the full $SU(3)$ transformation, albeit nonlinearly.

The key challenge with nonlinear realizations is that the ordinary
derivative, $dU$, is not covariant under global transformations with
constant $\epsilon$, owing to the spacetime dependence of
$\epsilon'(\epsilon, U(x))$.  We therefore need to construct a
covariant derivative that involves the gauge field $A_L$ and acts on
$U$ so that $DU \rightarrow e^{i\epsilon}DU e^{-i\epsilon'}$.  Such
a covariant derivative can be written as follows:
\begin{equation}
\label{cov1}
DU = d U - i A_L U + i  U A_{R} \,,
\qquad A_{R} = P U^\dagger(A_L + i d )U P
- {1\over n-1}{\rm Tr}\left[ P U^\dagger(A_L + i d )U P \right] \,.
\end{equation}
$A_R$ is the projection of $U^\dagger(A_L+id)U$ onto the unbroken
subgroup.  Here we have defined the projection matrix
\begin{equation}
P = \left(\begin{array}{ccc} 1 \\ & 1 \\ && 0 \end{array} \right) \,.
\end{equation}
Under the gauge transformation eq.(\ref{nonlinear}), with
\begin{equation}\label{trans0}
\delta A_L = d\epsilon + i[\epsilon,A_L],
\end{equation}
$A_R$ transforms as:
\begin{equation}
\label{trans1}
\delta A_R = d\epsilon' +i[\epsilon', A_R] \,,
\end{equation}
ensuring that $DU$ is covariant.

The nonlinear realization eq.(\ref{nonlinear}) embeds the NGB's from
$SU(3) \to SU(2)$ inside a larger manifold corresponding to
$SU(3)_L\times SU(3)_R\to SU(3)$.  We can describe the removal of the
extra pionic degrees of freedom in physical terms as follows.  We begin by
treating $A_L$ and $A_R$ as independent gauge fields, where $A_L$ is a
general $SU(3)$ matrix field, and $A_R$ belongs to the unbroken
$SU(2)$ subgroup.  Correspondingly, the $\epsilon$ and $\epsilon'$ in
the transformation eq.(\ref{nonlinear}) are independent rotations.  Now
suppose that the $A_R$ kinetic term vanishes, corresponding to very
strong coupling, \ie, $ (-1/g_R^2)\Tr F_{R\mu\nu} F_{R}^{\mu\nu}
\rightarrow 0$.  $A_R$ then becomes an auxiliary field with equation
of motion determined by the $SU(3)_L\times SU(3)_R$ chiral-invariant
kinetic term, $F^2 \Tr|D_\mu U|^2$.  Noting that $P A_R P = A_R$ and
${\rm Tr}(A_R) =0$, we obtain precisely the locking condition
eq.(\ref{cov1}) as the solution for $A_R$ as a function of $A_L$ and
$U$.  This allows us to ``eat and decouple'' the unwanted
isovector NGB's
in $U$.  Using the gauging of
eq.(\ref{cov1}) and expanding the resulting kinetic term in powers of $1/F$ we
see that $(F^2/2) |D_\mu U|^2 \to |D_\mu H|^2 + (\partial_\mu \eta)^2
+ \dots$~, and we are therefore dealing with a Little Higgs theory. Thus,
in a sense, a Little Higgs theory is just a technicolor theory with
the usual chiral field $U$ replaced by a nonlinear realization.

For any chiral theory based on a unitary matrix $U$ transfoming as
$U\rightarrow e^{i\epsilon} U e^{-i\epsilon'}$ and gauge fields $A_L$
and $A_R$ that likewise transform under $\epsilon$ and $\epsilon'$ as
in eqs.(\ref{trans0}) and eq.(\ref{trans1}), the gauged WZW term is given
explicitly by KRS eq.(4.18):
\begin{align}
\label{WZWfull}
&{\Gamma}_{WZW}(U, A_L, A_R)  = \Gamma_0(U) \;  + \frac{N}{48\pi^2}\Tr\int_{M^4} \bigl\{
({A}_L\alpha^3 + {A}_R\beta^3 )
-\frac{i}{2}[({A}_L\alpha)^2 - ({A}_R\beta)^2 ]
\nonumber \\
&\quad
+i \big[ (dA_LA_L + A_L dA_L)\alpha+ (d{A}_R{A}_R +{A}_RdA_R)\beta \big]
+({A}^3_L \alpha+{A}^3_R \beta)
\nonumber \\
&\quad
+i(A_L U {A}_R U^\dagger \alpha^2 - {A}_R U^\dagger A_L U \beta^2)
+i ( dA_R dU^\dagger A_L U - dA_L dU A_R U^\dagger  )
\big]
\nonumber \\
&\quad
- (dA_L A_L + A_L dA_L )U A_R U^\dagger +(dA_R A_R + A_R d A_R) U^\dagger A_L U
\nonumber \\
&\quad
-i(A_L U  A_R U^\dagger A_L \alpha + A_R U^\dagger A_L U A_R \beta  )
+i\big[ A_L^3 U  A_R U^\dagger - A_R^3 U^\dagger A_LU -
\half (UA_RU^\dagger A_L)^2
\bigr\} \,,
\end{align}
where $N$ is an integer; \eg, in the QCD chiral Lagrangian, $N=N_c=3$
is the number of colors.  Here $\alpha = dU U^\dagger$ and $\beta =
U^\dagger dU$.  The function $\Gamma_0(U)$ is given by eq.(\ref{Gamma0}),
which in four dimensions reads
\begin{equation}
{\Gamma}_{0}(U) =
{2N\over 15\pi^2 F^5} \int_{M^4} {\rm Tr}\left[ \tp (d\tp)^4 \right]
+ ...  \,.
\end{equation}

In the present case we need only substitute the representation of $U$
given in eq.(\ref{higgs}) and the locking of $A_R$ to $A_L$ and $U$
given in eq.(\ref{cov1}).  We can then expand to a given order in
$1/F$ to obtain the topological interactions of the mesons and gauge
fields.  We are presently ignoring $U(1)$ factors, and identify the
unbroken $SU(2)$ subgroup with electroweak gauge interactions
(we'll also presently ignore the $U(1)_Y$ gauge subgroup):
\begin{equation}
A_L = \left(\begin{array}{cc} W & 0 \\ 0 & 0 \end{array} \right) \,,
\end{equation}
where $W = gW^a\sigma^a/2$.  Defining
\begin{equation}
A_R = A_L +   \left(\begin{array}{cc}
\hat{A}_R  &
0  \\  0 &  0
\end{array}\right) \,,
\end{equation}
we find to second order in $1/F$:
\begin{eqnarray}\label{U}
\hat{A}_R = - \frac{1}{2F^2} \left( \{ W, H H^\dagger \} - H^\dagger W H \right)
+\frac{i}{2F^2} \left( H \stackrel{\leftrightarrow}{d} H^\dagger
+ \frac12 H^\dagger \stackrel{\leftrightarrow}{d} H \right) \,,
\end{eqnarray}
and
\begin{eqnarray}
\label{alph}
\alpha &=& \left(\begin{array}{cc}
{i\over \sqrt{3}F}d\eta -\frac{1}{2F^2}
H \stackrel{\leftrightarrow}{d} H^\dagger \quad
& \quad {i\over F}\left( 1+{\sqrt{3}i\over {2F}}\eta\right) dH + {\sqrt{3}\over
{2F^2}} H d\eta
\\
{i\over F}\left( 1- {\sqrt{3}i\over {2F}}\eta\right) dH^\dagger
- {\sqrt{3}\over {2F^2}} H^\dagger d\eta
 &
-{2i\over \sqrt{3F}} d\eta
 -\frac{1}{2F^2} H^\dagger  \stackrel{\leftrightarrow}{d} H
\end{array}\right) \,.
\end{eqnarray}
Here $A\stackrel{\leftrightarrow}{d}B \equiv
A\stackrel{\rightarrow}{d}B - A \stackrel{\leftarrow}{d}B$.  The
expansion for $\beta$ is given (to all orders in $1/F$) by
substituting $H\to -H$ and $\eta\to -\eta$ into $-\alpha$.

The leading WZW interactions involving $W$ and NGB's appear at order
$1/F$:
\begin{equation}\label{nonlinearWZW}
\Gamma_{WZW} =
{-N \over 8\pi^2\sqrt{3}F }
\int_{M^4} {\eta} {\rm Tr}( F_W^2 ) + \dots   \,,
\end{equation}
where $F_W = dW-iW^2$.  As the result of $SU(2)$ matrix identities, no
additional interactions appear through order $1/F^2$ from the action in
eq.(\ref{WZWfull}) involving just $W$ and $H$. In more general gauging
with $U(1)_Y$, and generalizations to multiple chiral fields
$\Phi_i \sim U_i P^\prime$ there can occur in principle at this order
other interesting operators such as
$H^\dagger F_W H F_Y$ which lead
to processes such as $e^+e^- \rightarrow (\gamma^*,Z^*) \rightarrow (h^0, A^0)
+(Z,\gamma,W^+W^-)$ ($A^0$ is a CP-odd Higgs boson).
There also appear operators
at this order that are separately
gauge-invariant and chirally invariant:
\begin{equation}\label{GI}
\Gamma_{GI} = \int_{M^4} i r {\rm Tr}(F_L U F_R U^\dagger)
= \int_{M^4} - {2 i r\over F^2} H^\dagger F_W^2 H + \dots
\,,
\end{equation}
with a coefficient $r$ that is not quantized, and is sensitive to the
details of the underlying UV completion theory. Multi-Higgs topological
interactions of $W$ and $H$ occur at order $1/F^4$:
\begin{equation}\label{resultx}
\Gamma_{WZW} \supset
{N\over 96\pi^2F^4} \int_{M^4}
 H^\dagger (DH) \, H^\dagger F_W (DH) + (DH^\dagger)H\, (DH^\dagger) F_W H \,.
\end{equation}
This operator can in principle lead to processes such as:
 $e^+e^- \rightarrow (\gamma^*,Z^*) \rightarrow h^0 + (Z,\gamma) $
and will generalize in other schemes \cite{2hill}.

Additional quantized, topological, interactions will occur when the
mesons couple to gauge fields for broken symmetry generators.  This
happens in Little Higgs models, where anomaly cancellation occurs
between different sectors that are connected only by weak gauge
interactions.  Before discussing this phenomenon in more detail, we
describe in the next section an alternative, and more direct approach
to the WZW term for $SU(3)/SU(2)$.

Finally, let us remark that the construction here based on
$SU(3)_L\times SU(3)_R/SU(3)$ relied on being able to gauge the
$SU(2)$ subgroup of $SU(3)_R$ and remove the unwanted pion degrees of
freedom.~%
\footnote{ The global structure of the NGB manifold in this case (a
  submanifold of $SU(3)$) allows an arbitrary integer coefficient,
  $N$, for the WZW term.  However, the natural UV theory in this
  physical picture has $A_R$ coupled to a right-handed fermion
  doublet.  For this theory to be free of discrete
  anomalies~\cite{Witten:1982fp} there should be an even number of
  such doublets, and therefore an even number of colors.  This is
  related to a factor of 2 advocated by the authors of
  \cite{D'Hoker:1994ti} in cases where the unbroken subgroup $H$, \eg\,
  $H=SU(2)$, has $\pi_4(H)=\bm{Z}_2$.  With the global structure
  defined by the direct construction based on $S^5$, a factor of
  $2$ is enforced automatically.  }
In the more general case of
$SU(n)/SU(n-1)$ based on $SU(n)\times SU(n)/SU(n)$, the unbroken
subgroup is anomalous,
and so cannot be gauged without adding more structure.~%
\footnote{ More precisely, a gauged WZW term can be written down for these
  cases, but the variation of the action results in an anomaly
  expression containing mesons, so that the theory does not have a
  simple UV completion in terms of constituent fermions.  }
For example, if the anomaly of colored fermions is cancelled by massless
right-handed leptons,
\begin{equation}
\Delta {\cal L}_{\rm lepton} =  \bar{\ell }_R (i\slash{\partial}
+ \slash{A}_R ) \ell_R \,,
\end{equation}
then $A_R$ in eq.(\ref{cov1}) is modified to
\begin{equation}
A_{R\mu}^a  \to A_{R\mu}^a - {1\over F^2} \bar{\ell}_R t^a \gamma_\mu \ell_R \,.
\end{equation}
Substituting this new locking condition for $A_R$ as a function of
$A_L$ and $U$ into eq.(\ref{WZWfull}) and setting $A_L=0$ yields an
ungauged action involving the leptons and the remaining mesons.  The
corresponding gauged action can then be constructed by ``brute-force''
gauging, as described after eq.(\ref{brute}).

\subsection{Gauging the Five Sphere}

A more direct method for obtaining the WZW term for KS models is to
notice that when the meson fields are described by a vector $\Phi =(
\phi^1,\phi^2,\phi^3)^T$ with
\begin{equation}
\Phi^\dagger \Phi = (\Re\phi^1)^2 + (\Im\phi^1)^2 + (\Re\phi^2)^2 + (\Im\phi^2)^2 +
(\Re\phi^3)^2 + (\Im\phi^3)^2  = 1 \,,
\end{equation}
the field lives on a manifold with very simple global topological
structure, the five-dimensional sphere, $S^5$.  The relevant
topological facts are that $\pi_4(S^5)=0$, guaranteeing that the
construction of a WZW term is possible, and $\pi_5(S^5)=\bm{Z}$,
guaranteeing that the result is nontrivial.  There is a unique
five-form on the five-sphere that is invariant under global
$SU(3)\times U(1)$ rotations, namely the volume element of the sphere,
\begin{equation}\label{omega}
\omega = -{i\over 8}\Phi^\dagger d\Phi d\Phi^\dagger d\Phi d\Phi^\dagger d\Phi \,.
\end{equation}
In analogy with the construction of $\Gamma_0(U)$ in eq.(\ref{Gamma0}),
we consider the topological action
\begin{equation}\label{sphereWZ}
\Gamma_0 = {N\over \pi^2} \int_{M^5} \omega \,.
\end{equation}
Since any five-form on a five-dimensional manifold is closed,
$d\omega=0$, this action can be written as an expansion in Goldstone
bosons in $D=4$.  Using that the area of the five-sphere is $\pi^3$,
the coefficient satisfies the quantization condition displayed in
eq.(\ref{sphereWZ}), with $N$ an {\it even} integer.

Gauging the topological action is more tedious than in the familiar
$SU(N)\times SU(N)/SU(N)$ case.  At a practical level, the gauging
begins by noticing that the variation of the ungauged action yields
\begin{equation}\label{spherevar}
\delta\Gamma_0 = {N\over 8\pi^2}\int_{M^4}
\left( \Phi^\dagger d\epsilon d\Phi + d\Phi^\dagger d\epsilon \Phi -d\epsilon_0
\Phi^\dagger d\Phi
\right) d\Phi^\dagger
d\Phi \,,
\end{equation}
where $\epsilon$ and $\epsilon_0$ are the $SU(3)$ and $U(1)$
components in the variation of $\Phi$, respectively:
\begin{equation}
\Phi\to e^{i (\epsilon+ \epsilon_0) } \Phi \,.
\end{equation}
Eq.(\ref{spherevar}) has made use of the fact that for fields confined
to the
five-sphere,~%
\footnote{ For a geometrical description of such identities, see
  \cite{Hull:1990ms}.  }
\begin{equation}
\begin{array}{c}
d(\Phi^\dagger \lambda^a \Phi ) (d\Phi^\dagger d\Phi)^2 = 0 \,, \\
\left[ \Phi^\dagger \lambda^a \Phi d\Phi^\dagger d\Phi
-2 d(\Phi^\dagger \lambda^a \Phi )\Phi^\dagger d\Phi
+2 d\Phi^\dagger \lambda^a d\Phi \right] d\Phi^\dagger d\Phi = 0 \,.
\end{array}
\end{equation}
The variation eq.(\ref{spherevar}) is compensated by the term with one
gauge field,
\begin{equation}
\Gamma_1 = {N\over 8\pi^2} \int_{M^4} \left( A_0 \Phi^\dagger d\Phi - \Phi^\dagger A d\Phi - d\Phi^\dagger A \Phi \right)
d\Phi^\dagger d\Phi \,.
\end{equation}
The residual shift is cancelled by a term with two gauge fields, and
so on.  Explicit expressions for the full gauged action will be
presented elsewhere~\cite{2hill}.  Important aspects of the analysis
are the non-uniqueness of the gauged WZW term, and a restrictive
interpretation in terms of underlying fermions, which we turn to
presently.

The complete result for the gauged WZW term leads to an anomalous
gauge variation:
\begin{align}\label{eq:4dvar}
  \delta\Gamma_{WZW} &= -{N\over 24\pi^2} \int_{M^4} {\rm Tr}\left\{
  \left(\epsilon-\frac{\epsilon_0}{2} \right)
\left[ \left(dA - \frac12 dA_0 \right)^2 - {i\over 2}d\left(A-\frac12 A_0\right)^3 \right]
\right\}  + \frac{27}{8} \epsilon_0 (dA_0)^2 \,.
\end{align}
Although the construction has made no mention of fermions, if
interpreted in terms of an underlying fermion theory, this variation
corresponds precisely to a triplet of left-handed fermions and a
single right-handed fermion, transforming under $\epsilon \in SU(3)$
and $\epsilon_0 \in U(1)$ as
\begin{equation}\label{fermi}
\Psi_L \to e^{i\epsilon - {i\over 2}\epsilon_0 } \Psi_L \,,\quad
q_R \to e^{-{3i\over 2} \epsilon_0 } q_R  \,.
\end{equation}
Note that this is precisely the combination of $U(1)$ transformations
that is not broken by color anomalies.  The quantized coefficient
corresponds to an {\it even} number of colors $N=N_c$ in the color
group.

As mentioned above, the unbroken $SU(2)\times U(1)$ subgroup is
anomaly free, and can
be identified with the electroweak gauge group of the standard model.~%
\footnote{ Having identified an anomaly-free embedding of the unbroken
  subgroup, and the anomalous gauge variation of the full action, we
  can work in the opposite direction to reconstruct the gauged WZW
  term by integration~\cite{Chu:1996fr}.  The non-uniqueness of the
  WZW term will be reflected in a choice of boundary condition for
  this construction.  } The symmetry breaking corresponds to a nonzero
VEV $\langle \Psi_L \bar{q}_R \rangle \sim (0,0,F^3)^T$.  For
simplicity, we ignore the $U(1)$ factor in the remainder of this
section, and use the choice of coordinates of eq.(\ref{higgs}).  The
leading WZW interactions are identical to eq.(\ref{nonlinearWZW}).
There are again operators that are invariant under local $SU(3)\times
U(1)$ transformations.  Suppressing $U(1)_Y$,
\begin{equation}\label{GIsphere}
\Gamma_{GI} = \int_{M^4} c\, \Phi^\dagger (dA - iA^2)^2 \Phi
+ \dots  \,,
\end{equation}
where $c$ is a number sensitive to the UV completion theory, and
the ellipsis denotes additional terms
such as $[\Phi^\dagger (dA-iA^2) \Phi]^2$ relevant at $ {\cal O}(1/F^4)$.
When viewed as deriving from an $SU(3)_L\times SU(3)_R/SU(3)$ theory,
parity arguments can be invoked to single out a unique choice of these
gauge-invariant operators~\cite{kay}, \eg\, $r=0$ in eq.(\ref{GI}), leading
to particular values of the coefficients in eq.(\ref{GIsphere}).
Sensitivity to the global structure of the field space of NGB's,
equivalently, sensitivity to the UV completion theory, is reflected in
the undetermined coefficients in eqs.(\ref{GI}) and (\ref{GIsphere}).

It is interesting to investigate further what novel features of
$SU(3)/SU(2)$ allow an underlying fermion theory to produce a
low-energy theory containing just scalar NGB's and gauge fields, and
in particular why a similar construction is not possible for general
$SU(n)/SU(n-1)$.  At the meson level, the nonlinear realization
construction required an anomaly-free gauging of $A_R$ to ``eat and
decouple'' the extra pions; this singled out $SU(2)$ as the unbroken
symmetry group (with an even number of colors in the strong color
group).  The direct construction required the form $\omega$ in
eq.(\ref{omega}) to be closed, $d\omega=0$, which is true for $S^5$, but
not for general $S^{2n-1}$ (notice that $\pi_5(S^5)=\bm{Z}$, but
$\pi_5(S^{2n-1})=0$ for $n>3$).  At the fermion level, we expect in
general that the condensate $\langle \Psi_L \bar{q}_R \rangle \sim
(0,0,\dots,0,F^3)^T$ will leave $n-1$ massless fermions $\psi_L^{i}$,
$i=1\dots (n-1)$, in the low-energy spectrum.  For $n>3$, these
fermions enable the low-energy theory to reproduce the full nonabelian
anomaly of the underlying UV theory.  For the special case of $n=3$,
such fermions are not mandated by anomaly matching.  For example, when
$N_c=2$, we observe that it is possible to write the operator
\begin{equation}\label{UV}
\epsilon_{ijk} \epsilon^{ab}\epsilon^{\alpha\beta}
 \psi^i_{a\alpha} \psi^j_{b\beta} \, \epsilon^{cd}\epsilon^{\gamma\delta} \psi^k_{c\gamma} {q}_{d\delta}  + h.c. \,,
\end{equation}
where $i,j,\dots = 1..3$ are flavor indices, $a,b,\dots = 1..2$ are
color indices, and $\alpha,\beta,\dots=1..2$ are Lorentz indices in
the $(1/2,0)$ representation of the Lorentz group.  The operator
eq.(\ref{UV}) is invariant under Lorentz and color $SU(N_c=2)$
transformations, and also under the flavor $SU(3)\times U(1)$
transformation in eq.(\ref{fermi}).  When $\Psi_L \bar{q}_R $ develops a
VEV, the operator becomes a (Majorana) mass term for the remaining
$\psi_L^{i}$ fermions, removing these degrees of freedom from the
low-energy spectrum.

\section{Little Higgs Models}

Having constructed the full WZW term, we consider applications to
realistic Higgs models.  As mentioned in the introduction, the
$SU(3)/SU(2)$ symmetry breaking pattern can be applied to a number of
scenarios; we focus on the KS model here, concentrating attention on
predictions that are independent of the undetermined coefficients in
eqs.(\ref{GI}) and (\ref{GIsphere}).  We recall that this model in its
simplest form
consists of two $\Phi_i$ fields, with aligned vacuum expectation values.
The gauge fields, suppressing $U(1)$ factors, take the form
\begin{equation}
A = \left( \begin{array}{cc} W & C \\ C^\dagger & 0 \end{array} \right) \,.
\end{equation}
Concentrating on interactions involving $H$, (neglecting $\eta$), the
kinetic terms are
\begin{equation}
F^2 |D_\mu\Phi_i|^2
= |D_\mu H_i|^2 - F C_\mu^\dagger D_\mu H_i - F (D_\mu H_i^\dagger) C_\mu +
F^2 C_\mu^\dagger C_\mu + \dots
\dots \,,
\end{equation}
where $i=1,2$ and $D_\mu H = (\partial_\mu - i W_\mu)H$.  Besides the
usual kinetic term for $H$, this expression contains an $F$-scale mass
for $C$.  The NGB's from the symmetry breaking at scale $F$ are eaten
by the $C$ bosons, and the physical Higgs fields appear as, $H_1 = H$,
$H_2 = -H$ (similarly, $\eta_1= \eta$, $\eta_2 = -\eta$).  Terms
containing an odd number of meson fields thus cancel in the sum,
\begin{equation}
{\cal L}_K = (F^2/2) |D_\mu\Phi_1|^2  + (F^2/2) |D_\mu \Phi_2|^2
= |D_\mu H|^2 + F^2 C^\dagger_\mu C_\mu + \dots \,.
\end{equation}
The same cancellation will occur with the subleading even-parity
``Gasser-Leutwyler'' operators~\cite{Gasser:1984gg} (operators not
containing an epsilon symbol $\epsilon^{\mu\nu\rho\sigma}$) if the
strong-interaction physics is the same in both sectors.  In order that
anomalies cancel between the $\Phi_1$ and $\Phi_2$ sectors
(at the fermion level,
$\Phi_1 \sim \Psi_L \bar{q}_R$, $\Phi_2 \sim \Psi_R \bar{q}_L$),
the opposite cancellation must occur with odd parity operators such as the
WZW term (operators containing an epsilon symbol)---surviving
interactions involve an {\it odd} number of meson fields.  Thus for
example, the interaction eq.(\ref{nonlinearWZW}) involving $\eta$ will
survive (with a factor of $2$ from the sum of the two sectors).  The
leading terms involving $H$ also occur at order $1/F$, and can be
obtained either from the nonlinear realization approach, or from the
direct gauging of $S^5$:
\begin{equation}\label{result2}
\Gamma_{WZW} \supset {N \over 16\pi^2 F} \int_{M^4} \left[ (D H^\dagger) F_W C - C^\dagger F_W DH \right]\,.
\end{equation}
This expression is for one sector (a factor of two will appear in the
sum of the two sectors).  It is manifestly gauge-invariant under
electroweak $SU(2)$.  The $SU(3)$-invariant odd-parity
Gasser-Leutwyler operators in eqs.(\ref{GI}) or (\ref{GIsphere})
contribute only to the orthogonal combination, eq.(\ref{result2}) with
the relative minus sign replaced by a plus sign.  The new interaction
eq.(\ref{result2}) would contribute to the process
$e^+e^- \to Z^* \to h^0 C^0$ (note that this is the analog
in QCD of the process $e^+e^- \rightarrow \rho \rightarrow KK^*$).

\section{WZW term for models with an internal parity}

The Lagrangian eq.(\ref{WZWfull}) can be used to
describe general symmetry breaking patterns, via reduction to
nonlinearly realized symmetries acting on submanifolds of a larger
space.  For example, we obtained the WZW term for $SU(3)/SU(2)$
by embedding the NGB's inside a full $SU(3)\times SU(3)/SU(3)$ multiplet.
A further simplification occurs for general models in which the symmetry
breaking pattern respects an internal parity operation, and eq.(\ref{WZWfull})
applies also to these cases.  The $SU(N)_L\times SU(N)_R/SU(N)$
QCD chiral Lagrangian is one example.  Another example is the
class of Little Higgs models containing an internal parity.
We examine here the structure of the WZW term for this case.

We recall that by suitable choice of coordinates, the action of an
element of the full symmetry group can be defined to act, at least
locally, on the Nambu Goldstone bosons as~\cite{Coleman:1969sm}
\begin{equation}\label{eq:nonlinear}
e^{i\pi} \to e^{i\pi^\prime} = e^{i\epsilon} e^{i\pi} e^{-i\epsilon^\prime(\epsilon,\pi)} \,.
\end{equation}
Here $\pi = \sum_a \pi^a t^a_A$ parameterizes the spontaneously broken
``axial'' symmetry generators, and $\epsilon^\prime = \sum_a
\epsilon^{\prime a} t^a_V$ is the combination of unbroken ``vector''
symmetry generators which ensures that
$e^{i\epsilon}e^{i\pi}e^{-i\epsilon^\prime}$ can be expressed as
$e^{i\pi^\prime}$ for some $\pi^\prime$.  Now suppose that the
transformation:
\begin{equation}
t^a_V \to R(t^a_V) = t^a_V \,, \quad t^a_A \to R(t^a_A)= -t_A^a
\end{equation}
preserves the group structure, i.e. $[t_V, t_V] \sim t_V$, $[t_A,t_A]
\sim t_V$ and $[t_A,t_V] \sim t_A$.  Then by multiplying
eq.(\ref{eq:nonlinear}) on the right by the result obtained after acting
with $R$ and taking the inverse, the quantity $\Sigma \equiv
e^{2i\pi}$ is seen to obey the {\it linear} transformation law
\begin{equation}
\Sigma \to e^{i\epsilon} \, \Sigma \, e^{-i R( \epsilon) } \,.
\end{equation}
In a more familiar notation, we may write $\epsilon \equiv \epsilon_V
- \epsilon_A$ and $\epsilon_V \equiv \epsilon_V^a t_V^a$, $\epsilon_A
\equiv \epsilon_A^a t_A^a$.  Then
\begin{equation}\label{Sigma}
\Sigma \to e^{i\epsilon_L} \Sigma e^{-i\epsilon_R} \,,
\end{equation}
where $\epsilon_{L,R} \equiv \epsilon_V \mp \epsilon_A$.
This generalizes eq.(\ref{Utransform}) to the case
where the elements of $\Sigma$ do not span a full group manifold.

The choice of variables eq.(\ref{Sigma}) allows us to immediately write down the
topological interactions in the form of a WZW term for models with an
internal parity operation.
The chiral current is defined as $\alpha = d\Sigma\, \Sigma^\dagger$,
and obeys $d\alpha = \alpha^2$.
The result is simply eq.(\ref{final1}) with $U\to\Sigma$.
The anomalous gauge variation of the resulting WZW action is
\begin{align}\label{parityanomaly}
\delta \Gamma_{WZW}
&= -{N\over 24\pi^2} \int_{M^4} {\rm Tr} \bigg\{
(\epsilon_V - \epsilon_A)\left[ (dA_V-dA_A)^2 - {i\over 2}d(A_V-A_A)^3 \right] \nonumber \\
&\qquad \qquad \qquad \qquad   -
(\epsilon_V + \epsilon_A)\left[ (dA_V+dA_A)^2 - {i\over 2}d(A_V+A_A)^3 \right] \bigg\} \,.
\end{align}

As an example, we consider the interactions arising when the Higgs is
identified with a Nambu Goldstone boson of the symmetry breaking
$SU(5) \to SO(5)$.  This pattern of symmetry breaking has been incorporated
into a Little Higgs model by Arkani-Hamed, Cohen, Katz and Nelson~\cite{Arkani-Hamed:2002qy}.
Let $\Phi$ be a two-index symmetric tensor representation of $SU(5)$,
developing the VEV
\begin{equation}\label{omegavev}
\langle \Phi \rangle \equiv \Omega = \left(
\begin{array}{ccccc}
&&&& 1 \\
&&&1 \\
&&1 \\
&1 \\
1
\end{array}
\right) \,.
\end{equation}
The fourteen NGB's corresponding to
broken symmetry generators are
\begin{equation}\label{broken}
\pi = \pi^a t_A^a = \left(\begin{array}{ccc}
\chi^T + \frac12 \eta & H^* & \phi^\dagger \\
H^T & -2\eta & H^\dagger \\
\phi & H & \chi + \frac12 \eta \end{array}\right) \,,
\end{equation}
where $\chi$ is a Hermitian, traceless $2\times 2$ matrix, $\eta$ is a real
singlet, $H$ is a complex doublet and $\phi$ is a symmetric $2\times 2$ matrix.
The ten unbroken symmetry generators correspond to
\begin{equation}\label{unbroken}
A = A^a t^a_V = \left(\begin{array}{ccc}
-W^T-\frac{g_1}{2} B & C & D\sigma^2 \\
C^\dagger & 0 & -C^T \\
D^*\sigma^2 & -C^* & W + {g_1\over 2} B \end{array}\right) \,,
\end{equation}
where $W=\sum_{a=1}^3 g W^a\sigma^a/2$ is a Hermitian, traceless
$2\times 2$ matrix, $B$ is a real singlet, $C$ is a complex doublet,
and $D$ is a complex singlet.  The model
gauges two $SU(2)\times U(1)$ groups, one corresponding to the
standard model $W$ and $B$ in eq.(\ref{unbroken}), and another
corresponding to heavy partners $W^\prime$ and $B^\prime$ that will
eat $\chi$ and $\eta$ in eq.(\ref{broken}).  This implements the notion
of ``collective symmetry breaking'' to stabilize the Higgs mass.

The WZW term can be evaluated straightforwardly.  Its existence is
related to the nontrivial homotopy group $\pi_5(SU(5)/O(5)) = \bm{Z}$.
Since the full WZW term is odd under the internal parity, and
invariant under weak isospin, interactions involving only standard
model fields $W$, $B$, $H$ are forbidden (the parity operation takes
$W,B\to +W,B$ and $H\to-H$).  Interactions do occur involving heavy
partners $W^\prime$, $B^\prime$ of the standard model gauge bosons, or
the Goldstone boson field $\phi$.  For example, interactions involving
the heavy hypercharge field $B^\prime$ are
\begin{multline}\label{result4}
\Gamma_{WZW} \supset {N \over 4\pi^2 F^2 }
\int_{M^4} (v+h^0)^2 B^\prime\big[
g^2(W^+ dW^- + W^- dW^+ + W^3 dW^3) -gg_1( W^3 dB + B dW^3 ) \\
+ g_1^{2} B dB - i g^2 W^+ W^- ( 3g W^3  - g_1 B )  \big]\,.
\end{multline}
Here $B^\prime$ is written in unitary gauge, having eaten the $\eta$
meson.~\footnote{
Since the $B^\prime$
symmetry is anomalous, other structure, \eg\, leptons, must be present
to cancel this gauge anomaly.}
Similarly, $W$ and $B$ are written in unitary gauge,
having eaten the Goldstone bosons inside of $H$.  In this gauge
we write $H \sim (0,v+h^0)^T/\sqrt{2}$.
This WZW term describes ``T parity''-violating interactions,
\eg\, decays of the single ``T-odd'' field $B^\prime$ into standard model fields.

From eq.(\ref{parityanomaly}), it is straightforward to see that
the anomalous gauge variation of the WZW action
is precisely that of a fermion theory with $2N$ left-handed fermions transforming
in the fundamental representation of $SU(5)$:~%
\footnote{
As noted in \cite{Chu:1996fr}, eq.(\ref{parityanomaly})
is the gauge variation of a Lagrangian with
{\it two} sets of Weyl fermions, $\Psi_L$ and $\Psi_R$,
coupled respectively to the restricted set of gauge fields
$A_V-A_A$ and $A_V+A_A$.
However, eq.(\ref{parityanomaly}) would describe only a subset
of the anomalies for such a fermion theory.
Using trace identities arising from
the anomaly-free embedding of $SO(5)$ inside of $SU(5)$
(\eg, $\Tr[\epsilon_V (dA_V)^2]=0$, $\Tr[\epsilon_A (dA_AdA_V+dA_VdA_A)]=0$, \dots),
(\ref{parityanomaly}) actually gives precisely
the nonabelian anomaly for {\it one} set of Weyl fermions.
}
\begin{equation}
\Psi_L \to e^{i(\epsilon_V + \epsilon_A)} \Psi_L \,.
\end{equation}
In a composite theory of underlying fermions, the symmetry breaking corresponds
to a VEV for the operator
\begin{equation}
\epsilon^{\alpha\beta} \psi_{\alpha}^{a i} \psi_{\beta}^{a j} \sim F^3
\Omega^{ij} \,,
\end{equation}
where $i,j=1..5$ are flavor indices, $\Omega$ is a symmetric matrix as in
(\ref{omegavev}),
$a=1..2N$ is a summed color index, and $\alpha,\beta=1..2$
are Lorentz indices.   Such a scheme is possible for fermions transforming in a real representation
of the color group---\eg,
in the adjoint representation
of $SU(N_c)$ for an odd number of colors, $2N=N_c^2-1$, or in the fundamental representation
of $SO(N_c)$ for an even number of colors, $2N=N_c$.

\section{Summary}

We have summarized the first steps toward the theory
of topological interactions of Higgs bosons. Such
interactions are present when these fields
occur as composite pNGB's,  or more generally in
theories of extra dimensions.

While the analysis involves novel constructions based on
topological features of the NGB field manifold, we emphasize that
in the spirit of effective field theory, it is simply a mistake
to omit such interactions.
As in the familiar case of the QCD chiral Lagrangian, the WZW term
is a remnant of underlying UV physics, and modifies the predictions of
the low-energy theory.
For the case of technicolor,
we found the interactions eq.(\ref{result1}) amongst the pNGB's and gauge bosons.
For cases of interest to Little Higgs theories, we derived the
gauged WZW term by two methods, via nonlinear realization of the symmetry group
on a restricted submanifold, or via a direct construction beginning with
a topological action.  Application to specific models give predictions
such as eqs.(\ref{nonlinearWZW}) and (\ref{result2}),
which are analogs of $\pi^0\to\gamma\gamma$.   These
interactions directly probe the structure of the underlying fermion
UV completion theory, \eg,
allowing one to count the number of colors
in an underlying strong gauge group.

We also pointed out the simple structure of the gauged WZW term for
symmetry-breaking patterns that respect an internal parity.
This includes the case of the QCD chiral Lagrangian, and the case of
Little Higgs models with $T$ parity.  If such models arise from composite
fermions, we see that $T$ parity cannot be an exact symmetry.
As an example, eq.(\ref{result4}) is
an interaction between the single ``$T$ odd'' partner of the
hypercharge gauge boson and ``$T$ even'' standard model particles.

Much work remains in order to explore the phenomenological implications
of WZW interactions for Little Higgs bosons,
and to identify the most promising signatures
at future colliders such as the LHC, ILC, and even beyond to
CLIC or a muon collider.

\vskip .1in
\noindent
{\bf Acknowledgements}
\vskip .1in
We thank W. Bardeen for helpful discussions.
This work was hosted at Fermilab
in the Fermilab Theoretical Physics Department
operated by Fermi Research Alliance, LLC  under Contract No.
DE-AC02-07CH11359 with the United States Department of Energy.

redpajama_set_name: RedPajamaArXiv


filmmaking.net forums - How about a bunch of stickies?
This forum keeps getting inquiries of the same type, as in whether a newbie should major in film or go to film school. So I suggest we do a bunch of stickies to answer those questions, which are pretty routine.
Sounds like a good idea. Everyone should feel free to nominate threads which should be made sticky. Let one of the moderators, or myself, know and we'll make it so.


text:
Q: Count function for ordered set/map container.count(val) - gives the frequency of "val" in container. Why count need to traverse till end for each value "val" in ordered set. My point is for ordered set/map, elements will be in order so we may get the final value of count even before reaching the end. I understand, this doesn't apply for other containers (list, vector etc.).
Example :-
set<int, less<int>> s{1, 2, 3, 4, 5, 6};
int cnt = s.count(2);  // return 1

Since set/map can have count as 0 or 1, once i reach 2 it should break the loop rather than traversing till end. Also, compiler knows the comparator (less or greater). Same can be done for multiset/multimap, only difference is count can be >1.

A: From n4618, § 23.2.6, page 848
Associative container requirements

Expression: b.count(k)
  Return type: size_type
  Assertion/note/pre-/post-condition: returns the number of elements with key equivalent to k
  Complexity: log(b.size()) + b.count(k)

This means that, by definition, these functions cannot count every element of the container before deciding what the count of k is.