text:
User talk:Naif

First ever comment
You, sir, are a genius. —Preceding unsigned comment added by 68.255.232.59 (talk • contribs)

Comment of unknown provenance
What on Earth? Not wikilinking the word god is not a POV. -- Ec5618 23:19, 24 November 2005 (UTC)

Ambi and Naif's Great Debate
Erm, please go to school. Before making a statement about gender, please learn what the word means. You could start with the Wikipedia article or, should you not believe that, look at some journal articles, books, or something else involving actually learning something. You may, on the other hand, be referring to sex. And on that very issue, there are many institutions that would disagree with you - from professional bodies in the United States and elsewhere, to the AP Stylebook, the world's foremost guide for journalism on issues such as pronouns, to a significant number of courts worldwide. What you think of or call Brandon Teena is irrelevant. Ambi 11:55, 3 December 2005 (UTC)
 * As a matter of fact I went to a very good school, one whose name you know. But I don't boast. Professional bodies and the AP stylebook are irrelevant.  Sex and gender are biological facts.  I know this truth is painful, but part of life is accepting painful truths.  Peace to you, fellow human being. -Naif 11:57, 3 December 2005 (UTC)

 * Facts according to who? Just because you assert something does not make it so. We have many neutral articles on Wikipedia which make quite clear where science stands, where the APA stands, where religious organisations stand, where the courts stand and where governments stand. Where you personally stand, however, really isn't important to anyone. Please cease breaking Wikipedia naming conventions, and find somewhere else to push your mysognist agenda. Ambi 12:02, 3 December 2005 (UTC)

 * So I'm a misogynist now? That's news to me considering I have been nothing but a gentleman to every woman I have ever met.  APA is irrelevant.  (You will note that it even includes the name of ONE culture in its title.)  What you and your confused allies have managed to get courts and governments to declare is also irrelevant.  Simple observation of the way animals treat each other based on gender shows that there is a simple biological fact in operation that all the spinny thinking in the world will never get around. -Naif 12:21, 3 December 2005 (UTC)

 * Unfortunately, in Wikipedia, you cannot just declare something "fact"; you have to cite sources for them. We know by your userpage that you have strongly-held minority views and quite an agenda; I understand you must find transsexualism quite threatening, but this is not the place to push your views. Perhaps you might try a blog? Ambi 12:36, 3 December 2005 (UTC)

 * Who the devil is "we?" Last time I checked I was having a chat with a very intelligent if confused individual.  My views, alas, are not minority-views, but majority.  The majority of the human race does not participate on wikipedia, however, and therefore any conclusions you might have drawn as to widespread opinion may reflect systematic bias.  If we transcend specism as well, we will discover that just about every animal as well recognizes the concept of gender.  In any case, I don't find "transexualism" threatening; I deny the legitimacy of its existence.  Ambi, I don't know if we're going to resolve this issue between the two of us.  All I can ask is that you start speaking with people outside your normal intellectual orbit.  A few working-class truckers and farmers might be able to sort things out for ya.  Thank you (quite sincerely) for your very quick replies and spirited defense of your opinions.  Once again, peace, fellow Wikipedian, and good night.-Naif 12:46, 3 December 2005 (UTC)

 * Your views may be the majority in a small town in Texas, but that's really not of much concern; we're interested in being neutral here, and the general international consensus of what is appropriate in these circumstances is to refer to transgendered people by their actual pronouns - which is why you will so rarely find the opposite done in any reputable publication. We are not dictated to by the tyranny of the majority, particularly by those who have no inclination to study the evidence for either side at all.

 * I think you seem to be misunderstood about both what transsexualism and gender are. I don't for a minute dispute the existence of distinct sexes; that is as evident in the animal community as it is in humans. Neither, for that matter, do most people. That, however, has nothing to do with transsexualism; it does not dispute that sex exists, but rather that chromosomes can in rare cases "get it wrong", as such, and people apparently one sex at birth can indeed prove to be the other. There's a litany of scientific evidence to back the case, which is why it is accepted by courts, journalism manuals, and the like. I think it says something that you rely on the opinions of people who, by their very nature, are unlikely to know anything at all about this area. Ambi 13:45, 3 December 2005 (UTC)

Once again, I know you have strong opinions, but please try to keep them in check. It's obvious that there is someone from the other side pushing a quite extreme agenda in street harrassment, and I've tried to deal with this by reverting to your version from a week or two ago where you'd tried to clean most of it up, and attempting to remove the POV material from both sides. Just because one party is doing it does not give you an excuse to do it too. Ambi 11:50, 6 December 2005 (UTC)

 * Please don't. This is considered wikistalking (c.f. Harassment) and is grounds for ultimately getting banned from Wikipedia. This is particularly clear when the edits are done without looking into the situation at all, such as Articles for deletion/Nick Rose Day, a vote which appears to have been made without even reading the article simply because I voted there. Ambi 04:25, 16 December 2005 (UTC)

 * Even though I've responded on your Talk page as per wikitteque, I happen to not like unchallenged name calling on my talk page, so I'll point out that any wikistalking has started on your end, and that on three votes we're "battling" on, I've disagreed with you on two, and agreed with you on one. Hardly voting "without looking into the situation at all." - Naif 05:16, 16 December 2005 (UTC)

I'm not specifically checking your articles. I just have a lot of pages in this area on my watchlist (for the record, I'm a gender studies major). I responded here because you had stated that you would and I quote "check all my edits" because you disagreed with some of my views. This is the very definition of wikistalking. As to your comment on my talk page, for the record, there is no such thing as a "moderator", and I have no special powers over the article.

Nor am I taking sides over the article, as there are clearly two extremes, and I've spent the last couple of weeks trying to keep either of them from slanting the article their particular way. The present draft of the article stems from a revert I made to one of your (previously reverted by someone else) versions from several weeks ago, as it was the most neutral version I could find in the history. I then tried to chop the bias of both sides out of the article, and while it may not be perfect, is a damned sight better than it was when I first saw the article.

Finally, I really must urge you to take heed of Neutral point of view. I've mentioned this to you a few times, but a great many of your edits in this area continue to be quite definitely biased. I know a lot of these articles aren't brilliant to begin with, but instead of fixing them, you tend to try to fill them with your own bias instead. All of us have topics we're particularly passionate about, and sometimes it can be hard to edit neutrally on those topics. That means that we either need to be very careful that our bias isn't creeping in there, or if we can't write neutrally - even if that means "writing for the other side", as such" - then seriously considering whether we should be editing that particular topic. There are several topics which I just won't edit because they rile me too much; though it would be better if you could learn to write neutrally, if you cannot, perhaps you should think about doing the same thing here. Ambi 05:27, 16 December 2005 (UTC)

 * For Chrissakes, wikistalking is specifically defined to "not include checking up on an editor to fix errors or violations of Wikipedia policy, nor does it mean reading a user's contribution log; those logs are public for good reason." But shucks, if you do it, it's fine; if somebody checks your work, it must be the "definition of wikistalking."  But I'm willing to compromise.  Back off on stalking me, and I won't muck around with your edits, even when they include such concepts as defining "David and Goliath" as a clothing line rather than a Biblical legend. - Naif 05:31, 16 December 2005 (UTC)

 * I'm not stalking you. As I've told you before, I major in this area, so I unsurprisingly have many articles in it on my watchlist. Moreover, it is hardly surprising that I'm likely to revert biased edits to them - from either extreme. Please don't make this personal. I'm not going to let you have free reign in exchange for your stopping wikistalking me; all I'm asking is that you work on your editing and try to do so neutrally. Ambi 05:37, 16 December 2005 (UTC)

 * I love the "your stopping wikistalking me." Reminds me of the question "have you stopped beating your wife?"  Unfortunately for that sly little charge, the edit record clearly shows you following me around Wikipedia and not the reverse.  And again, 2 disagreements and 1 agreements on 3 of your recent votes does not a wikistalking make.  I will point out that "majoring" in an area is part of the activism.  There is so "masculinist studies," after all, so by definition "feminist studies" is an activist major.  I'll ask you for your own part to try to see things from all POVs and not just repeat verbatim the same cant you learned from extremist professors. - Naif 05:40, 16 December 2005 (UTC)

 * For the umpteenth time, reverting some (not all) of your edits to street harassment and Holla Back NYC does not make wikistalking. Threatening to follow me around and "check my edits" to unrelated articles does. Moreover, I take umbrage at your comments above. I've been trying to keep the extremist rubbish out of the article as well, such as getting rid of that terrorism hyperbole - yet you continue to accuse me of "repeating verbatim material from extremist professors." All I want is a genuinely neutral article on this topic, and I'm getting pretty damned tired of being attacked for trying to keep both sides from getting carried away with their biases. Ambi 05:53, 16 December 2005 (UTC)

 * And for the umpteenth time, "checking your edits" is explicitly declared in the wikipedia wikistalker page to NOT be wikistalking. Nowhere do I threaten to follow you around.  That said, sorry if my comments offend you, but in case you haven't noticed, I seem to have picked up my own stalker from a misguided "white knight" by the name of Calton... and he's been editing all MY edited pages. Who's the one getting stalked and creeped out?? -Naif 05:57, 16 December 2005 (UTC)

Re Articles for deletion/Nick Rose Day
Some Wikipedia policies you need to reacquaint yourself with:

 * Verifiability
 * Vanity
 * Vanity information is considered to be any information that was placed in any Wikipedia article that might create an apparent conflict of interest, meaning any material that presents the appearance of being intended to in any way promote the personal notoriety of the author, or one of the close family members or associates of the author.

If you want to wikistalk User:Ambi, I personally can't stop you, but don't violate WP:POINT conducting your vendetta. I mean, at least take the time to make your edits halfway plausible, so ArbCom's future case against you will be less open-and-shut. --Calton | Talk 05:34, 16 December 2005 (UTC)

 * So the big-bad name-calling "wikistalker" comes out... but instead of just allying with your political friends, perhaps check the reversion histories... you'll notice this curious pattern of Ambi checking and deleting my edits for more than two weeks now... I'll respond more to your specific comments shortly. --Naif 05:36, 16 December 2005 (UTC)

 * I don't know Ambi from Adam: what I *do* know I found out after investigating your obviously nonsensical vote on the AfD, when I checked to see whether you were a sockpuppet of one of the HLS clowns. No, but examination of your edit history -- including your antics on Sadness and its AfD -- pretty much told me what level of sincerity you bring to this project.

 * You may have noted that motto at the top of my talk page: It's clean-up duty, mopping up after the dishonest, incompetent, and fanatical. Can't imagine why you'd have a problem with that. Members of those groups, unsurprisingly enough, have a problem with that. If you're not a member of any of those groups, you shouldn't have a problem with that either, now should you? --Calton | Talk 05:55, 16 December 2005 (UTC)

 * You know, Calton, it's a little scary how you've jumped into my editing history the way you have. But fine, I'll assume good faith. That said, your Talk page speaks for itself.  Clearly I'm not the only one who's a little creeped out by your editing decisions.  Ever heard of this guy called Nietzsche?  He said something about those who hunt monsters. -Naif 06:01, 16 December 2005 (UTC)

 * ...fine, I'll assume good faith It's good to know that you've decided to start, considering that you began by sayin that I was "allying with [my] political friends"

 * it's a little scary how you've jumped into my editing history the way you have You mean doing research before acting? Yeah, I'm crazy that way.

 * Oh, and name-dropping stuff you picked up from philosophy class doesn't impress me, so don't bother. --Calton | Talk 03:30, 17 December 2005 (UTC)

Thanks
Thanks for fixing the vandalism on my userpage. #wikipedia-en-vandalism on irc.freenode.net is a good place to monitor vandals. See: IRC channels. Cheers! Jacoplane 09:35, 16 December 2005 (UTC)

Curious
Out of curiosity, why the latest change to your userpage? Ambi 09:58, 16 December 2005 (UTC)

 * It would seem that my proposals to add content are unwelcome, so I will probably focus on counter-vandalism for the time-being. Cheers, - Naif 09:59, 16 December 2005 (UTC)

 * Not unwelcome. Just like all of us, you need to watch your edits for neutrality. We have some people who hold some very extreme views, yet have few issues working on articles because they've mastered keeping their biases out of their writing. If you get the hang of that, there really isn't much of an issue. Ambi 10:06, 16 December 2005 (UTC)

Thanks
Thanks for reverting the vandalism on my talk page. Cheers. --Lee Hunter 18:12, 16 December 2005 (UTC)

yes sir
—Preceding unsigned comment added by 130.113.111.203 (talk • contribs)

Thanks for the props!
Thanks for the appreciation. It's hard to be sure what approach is best - believe me, I am no sympathizer to Aidan Work's political or religious beliefs! At the same time, I know we can't and shouldn't make this Left-Wing-Pedia. Best, FreplySpang (talk) 23:10, 16 December 2005 (UTC)

Affect
I noticed your edit to Affect and thought I'd ask. Do you think there is room/need for the article Affect (psychology)?--Commander Keane 07:06, 17 December 2005 (UTC)

thanks + question
I wanted to say thanks for removing the annoying vandalism from the vegetarianism page. But when I received that message about gratuitous profanity, I got a little worried. Surely I'm not getting blamed for them, right? —Preceding unsigned comment added by 24.131.68.15 (talk • contribs)

Homosexuality
For the umpteenth time, please familiarise yourself with the policy against original research. You hold some minority views, and while these may warrant mentioning in the article depending on their prevalance, you cannot state them as definitive fact. Moreso, you absolutely cannot do so without providing any sources.

Thirdly, if you seriously expect anyone to take you seriously, please stop making threats - especially ones that are utterly toothless. If you make biased edits, they will be reverted. If you make neutral edits, they will be happily supported. I'm not going to disappear and let you push your ideological views in articles because of some random threat or rant. Instead, please try to familiarise yourself with the way things work around here.

We've had many people with some extreme views - heck, Jimmy Wales has some - but those people have learned to write neutrally (even if that means "writing for the enemy"), rather than dedicating all their efforts towards pushing an agenda. If you do the former, you'll find yourself getting a vastly warmer reception. Ambi 10:49, 23 December 2005 (UTC)

Pedophilia
From User_talk:Haiduc: "Age disparity in sexual relationships Nice work on this article. I'd appreciate your keeping an eye on ephebophilia and pedophilia as well for attempts by pedophiles to legitimize their behavior. Cheers, -Naif 05:06, 21 December 2005 (UTC)"

 * Um, just so you know, think you probably misdirected that kind of comment. He is very much a pedophile trying to legitimize pedophilia: Look at User talk:Mistress Selina Kyle. -- Mistress Selina Kyle  (  Α⇔Ω ¦  ⇒✉  )

a goodbye of sorts
I don't know if you're going to accept this, but that wasn't about you. Still, when I saw the response, I figured out why you reacted the way you did. Thanks (sincerely). Anyway, this is still a goodbye of sorts b/c I'm busy now with some very important projects--things that are going to make an impact, on me if nobody else (and what better things can we aspire to?). Maybe we can talk later. In fact, I bet it will happen.

Look, friend (I never use that word lightly), there's something about thought itself that I've come to realize--and it didn't come about through thought, because that's an impossibility. It only came about through sheer terror. Have you ever considered not thinking a problem through to completion but wondering why you thought that original thought? You can control the direction of your thought, make it come to a logical conclusion, but can you control the initial thought to begin with? Spun out, yeah, I know. But trust me on this: there's definitely something below free will and above karma. There's just no word for that concept, that's all.

Final concept (I know I'm rambling): those people (there had to be more than one) who gave you the initial thought are clowns. I met one of them lately, and his life is just on slow, slow decline. I showed him a Japanese art house film, and said, "yo, Dick take a look at this five minute scene--it's considered groundbreaking in cinematics." He looked for like three seconds, and then turned back to the internet pornsites he wanted to look at and the J he was smoking. Mind, this is a guy I've had plenty of good times with and who trusts/likes me. It's just that he has no curiosity whatsoever. Those people in your past--I guarantee you, they're exactly the same. Complete clowns. They could never have a conversation like this.

gender =! sex
gender is in fact a social construct, it is sex that is the biological fact. and with that you must also understand there are more than xx and xy (i.e. x, xxx, xyy) but, even those "normal" people will occasionally have physical or hormonal irregularities.

AfD for Right- and Left- wing terrorism articles - have your say
Please take a look at Articles for deletion/Right-wing terrorism and also Articles for deletion/Left-wing terrorism and have your say, if possible. Thanks.Xemoi 01:19, 14 May 2006 (UTC)

Your article on Affect is under attack
The APA dictionary (2006) labels the concept you called "Affect" as "Affect display". Accordingly, I have created a new article with that name, copied your text to it, and have begun editing it. Affect will become difficult to distinguish from "feeling", IMHO, but that's the way the APA wants it and I can't find good sources for the use of the term "affect" as you had defined it. DCDuring 01:15, 14 September 2007 (UTC)

Research survey invitation
Greetings Naif-

My name is Randall Livingstone, and I am a doctoral student at the University of Oregon, studying digital media and online community. I am posting to invite you to participate in my research study exploring the work of Wikipedia editors who are members of WikiProject: Countering Systemic Bias. The online survey should take 20 to 25 minutes to complete and can be found here:

https://oregon.qualtrics.com/SE/?SID=SV_cSHzuwaQovaZ6ss

Your responses will help online communication researchers like me to better understand the collaborations, challenges, and purposeful work of Wikipedia editors like you. In addition, at the end of the survey you will have the opportunity to express your interest in a follow-up online interview with the researcher.

This research project has been reviewed and approved by the Wikimedia Research Committee as well as the Office for Protection of Human Subjects at the University of Oregon. For a detailed description of the project, please visit its Meta page. This survey is voluntary, and your confidentiality will be protected. You will have the choice of using your Wikipedia User Name during the research or creating a unique pseudonym. You may skip any question you choose, and you may withdraw at any time. By completing the survey, you are providing consent to participate in the research.

If you have any questions about the study, please contact me via my Talk Page (UOJComm) or via email. My faculty advisor is Dr. Ryan Light. If you have any questions regarding your rights as a research participant, please contact the Office for Protection of Human Subjects at the University of Oregon.

Thank you very much.

Sincerely,

Randall Livingstone School of Journalism & Communication University of Oregon UOJComm (talk) 04:17, 22 June 2011 (UTC)

ArbCom elections are now open!
MediaWiki message delivery (talk) 12:53, 23 November 2015 (UTC)

title: User talk:Naif


text:
Privacy laws of the United States

Privacy laws of the United States deal with several different legal concepts. One is the invasion of privacy, a tort based in common law allowing an aggrieved party to bring a lawsuit against an individual who unlawfully intrudes into their private affairs, discloses their private information, publicizes them in a false light, or appropriates their name for personal gain.

The essence of the law derives from a right to privacy, defined broadly as "the right to be let alone". It usually excludes personal matters or activities which may reasonably be of public interest, like those of celebrities or participants in newsworthy events. Invasion of the right to privacy can be the basis for a lawsuit for damages against the person or entity violating the right. These include the Fourth Amendment right to be free of unwarranted search or seizure, the First Amendment right to free assembly, and the Fourteenth Amendment due process right, recognized by the Supreme Court of the United States as protecting a general right to privacy within family, marriage, motherhood, procreation, and child rearing.

Attempts to improve consumer privacy protections in the U.S. in the wake of the 2017 Equifax data breach, which affected 145.5 million U.S. consumers, failed to pass in Congress.

Early years
The early years in the development of privacy rights began with English common law, protecting "only the physical interference of life and property". The Castle doctrine analogizes a person's home to their castle – a site that is private and should not be accessible without permission of the owner. The development of tort remedies by the common law is "one of the most significant chapters in the history of privacy law". Those rights expanded to include a "recognition of man's spiritual nature, of his feelings and his intellect." Eventually, the scope of those rights broadened even further to include a basic "right to be let alone," and the former definition of "property" would then comprise "every form of possession – intangible, as well as tangible." By the late 19th century, interest in privacy grew as a result of the growth of print media, especially newspapers.

Between 1850 and 1890, U.S. newspaper circulation grew by 1,000 percent – from 100 papers with 800,000 readers to 900 papers with more than 8 million readers. In addition, newspaper journalism became more sensationalized, and was termed yellow journalism. The growth of industrialism led to rapid advances in technology, including the handheld camera, as opposed to earlier studio cameras, which were much heavier and larger. In 1884, Eastman Kodak company introduced their Kodak Brownie, and it became a mass market camera by 1901, cheap enough for the general public. This allowed people and journalists to take candid snapshots in public places for the first time.

Privacy was dealt with at the state level. For example, Pavesich v. New England Life Insurance Company (in 1905) was one of the first specific endorsements of the right to privacy as derived from natural law in US law. Judith Wagner DeCew stated, "Pavesich was the first case to recognize privacy as a right in tort law by invoking natural law, common law, and constitutional values."

Samuel D. Warren and Louis D. Brandeis, partners in a new law firm, feared that this new small camera technology would be used by the "sensationalistic press." Seeing this becoming a likely challenge to individual privacy rights, they wrote the "pathbreaking" Harvard Law Review article in 1890, "The Right to Privacy". According to legal scholar Roscoe Pound, the article did "nothing less than add a chapter to our law", and in 1966 legal textbook author, Harry Kalven, hailed it as the "most influential law review article of all". In the Supreme Court case of Kyllo v. United States, 533 U.S. 27 (2001), the article was cited by a majority of justices, both those concurring and those dissenting.

Brandeis and Warren article
The development of the doctrine regarding the tort of "invasion of privacy" was largely spurred by the Warren and Brandeis article, "The Right to Privacy". In it, they explain why they wrote the article in its introduction: "Political, social, and economic changes entail the recognition of new rights, and the common law, in its eternal youth, grows to meet the demands of society". More specifically, they also shift their focus on newspapers:

"The press is overstepping in every direction the obvious bounds of propriety and of decency. Gossip is no longer the resource of the idle and of the vicious, but has become a trade, which is pursued with industry as well as effrontery. To satisfy a prurient taste the details of sexual relations are spread broadcast in the columns of the daily papers. ... The intensity and complexity of life, attendant upon advancing civilization, have rendered necessary some retreat from the world, and man, under the refining influence of culture, has become more sensitive to publicity, so that solitude and privacy have become more essential to the individual; but modern enterprise and invention have, through invasions upon his privacy, subjected him to mental pain and distress, far greater than could be inflicted by mere bodily injury."

They then clarify their goals: "It is our purpose to consider whether the existing law affords a principle which can properly be invoked to protect the privacy of the individual; and, if it does, what the nature and extent of such protection is".

Warren and Brandeis write that privacy rights should protect both businesses and private individuals. They describe rights in trade secrets and unpublished literary materials, regardless whether those rights are invaded intentionally or unintentionally, and without regard to any value they may have. For private individuals, they try to define how to protect "thoughts, sentiments, and emotions, expressed through the medium of writing or of the arts". They describe such things as personal diaries and letters needing protection, and how that should be done: "Thus, the courts, in searching for some principle upon which the publication of private letters could be enjoined, naturally came upon the ideas of a breach of confidence, and of an implied contract". They also define this as a breach of trust, where a person has trusted that another will not publish their personal writings, photographs, or artwork, without their permission, including any "facts relating to his private life, which he has seen fit to keep private". And recognizing that technological advances will become more relevant, they write: "Now that modern devices afford abundant opportunities for the perpetration of such wrongs without any participation by the injured party, the protection granted by the law must be placed upon a broader foundation".

There have been many laws related to privacy and data protection in recent years that have been enforced as a result of the rapid technological advancements. However, critics and scholars have argued that these guidelines are usually focused on legal factors, rather than technical details, which make it difficult for engineers and developers to ensure that new designs meet the guidelines stated in privacy laws.

Modern tort law
In the United States,"invasion of privacy" is a commonly used cause of action in legal pleadings. Modern tort law, as first categorized by William Prosser, includes four categories of invasion of privacy:
 * Intrusion of solitude: physical or electronic intrusion into one's private quarters
 * Public disclosure of private facts: the dissemination of truthful private information which a reasonable person would find objectionable
 * False light: the publication of facts which place a person in a false light, even though the facts themselves may not be defamatory
 * Appropriation: the unauthorized use of a person's name or likeness to obtain some benefits

Intrusion of solitude and seclusion
Intrusion of solitude occurs where one person intrudes upon the private affairs of another. In a famous case from 1944, author Marjorie Kinnan Rawlings was sued by Zelma Cason, who was portrayed as a character in Rawlings' acclaimed memoir, Cross Creek. The Florida Supreme Court held that a cause of action for invasion of privacy was supported by the facts of the case, but in a later proceeding found that there were no actual damages.

Intrusion upon seclusion occurs when a perpetrator intentionally intrudes, physically, electronically, or otherwise, upon the private space, solitude, or seclusion of a person, or the private affairs or concerns of a person, by use of the perpetrator's physical senses or by electronic device or devices to oversee or overhear the person's private affairs, or by some other form of investigation, examination, or observation intrude upon a person's private matters if the intrusion would be highly offensive to a reasonable person. Hacking into someone else's computer is a type of intrusion upon privacy, as is secretly viewing or recording private information by still or video camera. In determining whether intrusion has occurred, one of three main considerations may be involved: expectation of privacy; whether there was an intrusion, invitation, or exceedance of invitation; or deception, misrepresentation, or fraud to gain admission. Intrusion is "an information-gathering, not a publication, tort ... legal wrong occurs at the time of the intrusion. No publication is necessary".

Restrictions against the invasion of privacy encompasses journalists as well:

"The First Amendment has never been construed to accord newsmen immunity from torts or crimes committed during the course of newsgathering. The First Amendment is not a license to trespass, to steal, or to intrude by electronic means into the precincts of another's home or office."

Public disclosure of private facts
Public disclosure of private facts arises where one person reveals information which is not of public concern, and the release of which would offend a reasonable person. "Unlike libel or slander, truth is not a defense for invasion of privacy." Disclosure of private facts includes publishing or widespread dissemination of little-known, private facts that are non-newsworthy, not part of public records, public proceedings, not of public interest, and would be offensive to a reasonable person if made public.

False light
False light is a legal term that refers to a tort concerning privacy that is similar to the tort of defamation. For example, the privacy laws in the United States include a non-public person's right to privacy from publicity which creates an untrue or misleading impression about them. A non-public person's right to privacy from publicity is balanced against the First Amendment right of free speech.

False light laws are "intended primarily to protect the plaintiff's mental or emotional well-being". If a publication of information is false, then a tort of defamation might have occurred. If that communication is not technically false but is still misleading, then a tort of false light might have occurred.

The specific elements of the Tort of false light vary considerably even among those jurisdictions which do recognize this tort. Generally, these elements consist of the following:
 * A publication by the defendant about the plaintiff;
 * Made with actual malice (very similar to that type required by New York Times v. Sullivan in defamation cases);
 * Places the plaintiff in a false light; and
 * Highly offensive (i.e., embarrassing to reasonable persons)

Thus in general, the doctrine of false light holds:

"One who gives publicity to a matter concerning another before the public in a false light is subject to liability to the other for invasion of privacy, if (a) the false light in which the other was placed would be highly offensive to a reasonable person, and (b) the actor had knowledge of or acted in a reckless disregard as to the falsity of the publicized matter and the false light in which the other would be placed."

For this wrong, money damages may be recovered from the first person by the other.

At first glance, this may appear to be similar to defamation (libel and slander), but the basis for the harm is different, and the remedy is different in two respects. First, unlike libel and slander, no showing of actual harm or damage to the plaintiff is usually required in false light cases, and the court will determine the amount of damages. Second, being a violation of a Constitutional right of privacy, there may be no applicable statute of limitations in some jurisdictions specifying a time limit within which period a claim must be filed.

Consequently, although it is infrequently invoked, in some cases false light may be a more attractive cause of action for plaintiffs than libel or slander, because the burden of proof may be less onerous.

What does "publicity" mean? A newspaper of general circulation (or comparable breadth) or as few as 3–5 people who know the person harmed? Neither defamation nor false light has ever required everyone in society be informed by a harmful act, but the scope of "publicity" is variable. In some jurisdictions, publicity "means that the matter is made public, by communicating it to the public at large, or to so many persons that the matter must be regarded as substantially certain to become one of public knowledge."

Moreover, the standards of behavior governing employees of government institutions subject to a state or national Administrative Procedure Act (as in the United States) are often more demanding than those governing employees of private or business institutions like newspapers. A person acting in an official capacity for a government agency may find that their statements are not indemnified by the principle of agency, leaving them personally liable for any damages.

Example: If someone's reputation was portrayed in a false light during a personnel performance evaluation in a government agency or public university, one might be wronged if only a small number initially learned of it, or if adverse recommendations were made to only a few superiors (by a peer committee to department chair, dean, dean's advisory committee, provost, president, etc.). Settled cases suggest false light may not be effective in private school personnel cases, but they may be distinguishable from cases arising in public institutions.

Appropriation of name or likeness
Although privacy is often a common-law tort, most states have enacted statutes that prohibit the use of a person's name or image if used without consent for the commercial benefit of another person.

Appropriation of name or likeness occurs when a person uses the name or likeness of another person for personal gain or commercial advantage. Action for misappropriation of right of publicity protects a person against loss caused by appropriation of personal likeness for commercial exploitation. A person's exclusive rights to control their name and likeness to prevent others from exploiting without permission is protected in similar manner to a trademark action with the person's likeness, rather than the trademark, being the subject of the protection.

Appropriation is the oldest recognized form of invasion of privacy involving the use of an individual's name, likeness, or identity without consent for purposes such as ads, fictional works, or products.

"'The same action – appropriation – can violate either an individual's right of privacy or right of publicity. Conceptually, however, the two rights differ.'"

Fair Credit Reporting Act
The Fair Credit Reporting Act became effective on April 25, 1971 and implemented limitations on the information that could be collected, stored, and utilized by agencies such as credit bureaus, tenant screenings, and health agencies. The law also defined the rights granted to individuals in regards to their financial information including the right to obtain a credit score; the right to know what information is in your financial file; the right to know when your information is being accessed and used; and the right to dispute any inaccurate or incorrect information.

Video Privacy Protection Act
The Video Privacy Protection Act of 1988 (VPPA) was signed into law by President Ronald Reagan to preserve the privacy of people's information collected when they rented, purchased, or delivered audio visual materials, and specifically videotapes. The law arose out of the Bork tapes controversy surrounding the Washington City Paper's publication of a list of films rented by Robert Bork, a politician who was running for a spot on the United States Supreme Court at the time. The law prohibits the disclosure of personal information collected by video tape service providers unless it falls under certain exceptions. The VPPA became a focus of attention in the legal industry once again around 2022. Its revival came as part of a larger trend in consumer class actions filed based on privacy law violations, both through new laws like the California Consumer Privacy Act and older laws like the VPPA and wiretapping statutes.

Health Insurance Portability and Accountability Act
Signed in law on August 21, 1996, Health Insurance Portability and Accountability Act (HIPAA) is a piece of legislation passed in the United States that limits the amount and types of information that can be collected and stored by healthcare providers. This includes limits on how that information can be obtained, stored, and released. HIPAA also developed data confidentiality requirements that are a part of "The Privacy Rule."

Gramm-Leach-Bliley Act
The Gramm-Leach-Bliley Act (GLA) is a federal law that was signed into effect on November 12, 1999. This act placed increased limits and requirements for data collection by financial institutions, as well as limited how that information could be collected and stored. It focused on requiring financial institutions to take specific measure to increase the safety and confidentiality of the information being collected. In addition to this, the law also put limitations on what type of data could be collected by financial institutions and how they could use that information. The act strives to protect NPI, or nonpublic personal information, which is any information that is collected regarding an individual's finances that is not otherwise publicly available.

Children's Online Privacy Protection Act
The Children's Online Privacy Protection Act (COPPA), passed on April 21, 2000, is a federal law in the United States that puts severe restrictions on what data companies can collect, share, or sell about children who are under the age of 13. A core provision under COPPA is that a website operator must "obtain verifiable parental consent before any collection, use, or disclosure of personal information from children."

Federal
Although the word "privacy" is actually never used in the text of the United States Constitution, there are Constitutional limits to the government's intrusion into individuals' right to privacy. This is true even when pursuing a public purpose such as exercising police powers or passing legislation. The Constitution, however, only protects against state actors. Invasions of privacy by individuals can only be remedied under previous court decisions.

The First Amendment protects the right to free assembly, broadening privacy rights. The Fourth Amendment to the Constitution of the United States ensures that "the right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no warrants shall issue, but upon probable cause, supported by oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized." The Fourth Amendment was the Framers' attempt to protect each citizen's spiritual and intellectual integrity. A government that violates the Fourth Amendment in order to use evidence against a citizen is also violating the Fifth Amendment. The Ninth Amendment declares that "The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people."

Alaska
On August 22, 1972 the Alaska Right of Privacy Amendment, Amendment 3, was approved with 86% of the vote in support of the legislatively referred constitutional amendment. Article I, Section 22 of Alaska's constitution states, "The right of the people to privacy is recognized and shall not be infringed. The legislature shall implement this section."

California
The California Constitution articulates privacy as an inalienable right.

CA SB 1386 expands on privacy law and guarantees that if a company exposes a Californian's sensitive information this exposure must be reported to the citizen. This law has inspired many states to come up with similar measures.

California's "Shine the Light" law (SB 27, CA Civil Code § 1798.83), operative on January 1, 2005, outlines specific rules regarding how and when a business must disclose use of a customer's personal information and imposes civil damages for violation of the law.

California's Reader Privacy Act was passed into law in 2011. The law prohibits a commercial provider of a book service, as defined, from disclosing, or being compelled to disclose, any personal information relating to a user of the book service, subject to certain exceptions. The bill would require a provider to disclose personal information of a user only if a court order has been issued, as specified, and certain other conditions have been satisfied. The bill would impose civil penalties on a provider of a book service for knowingly disclosing a user's personal information to a government entity in violation of these provisions. This law is applicable to electronic books in addition to print books.

The California Privacy Rights Act created the California Privacy Protection Agency, the first data protection agency in the United States.

Florida
Article I, §23 of the Florida Constitution states that "Every natural person has the right to be let alone and free from governmental intrusion into the person's private life except as otherwise provided herein. This section shall not be construed to limit the public's right of access to public records and meetings as provided by law."

Montana
Article 2, §10 of the Montana Constitution states that "The right of individual privacy is essential to the well-being of a free society and shall not be infringed without the showing of a compelling state interest".

Washington
Article 1, §7 of the Washington Constitution states that "No person shall be disturbed in his private affairs, or his home invaded, without authority of law".
 * Law enforcement are required to obtain a warrant before using IMSI-catcher technology.
 * Private individual's text messages are protected from warrantless searches.

State privacy laws
The right to privacy is protected also by more than 600 laws in the states and by a dozen federal laws, like those protecting health and student information, also limiting electronic surveillance.

As of 2022 however, only five states had data privacy laws.

"Opt-out" requirements
Several of the US federal privacy laws have substantial "opt-out" requirements, requiring that the individual specifically opt-out of commercial dissemination of personally identifiable information (PII). In some cases, an entity wishing to "share" (disseminate) information is required to provide a notice, such as a GLBA notice or a HIPAA notice, requiring individuals to specifically opt-out. These "opt-out" requests may be executed either by use of forms provided by the entity collecting the data, with or without separate written requests.

Privacy of health-related information
The Health Information Technology for Economic and Clinical Health Act (HITECH Act) is an important piece of legislation in the United States that relates to the privacy of health-related information. Enacted as part of the American Recovery and Reinvestment Act of 2009, the HITECH Act addresses the privacy and security concerns associated with the electronic transmission of health information.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Privacy laws of the United States


text:
User talk:Will T Brown

Welcome!

Hello, and welcome to Wikipedia. Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers: I hope you enjoy editing here and being a Wikipedian! By the way, you can sign your name on Talk and vote pages using three tildes, like this: ~ ~ ~. Four tildes (~ ~ ) produces your name and the current date. If you have any questions, see the help pages, add a question to the village pump or ask me on my Talk page. Again, welcome! -- Graham ☺ | Talk 02:19, 17 Dec 2004 (UTC)
 * How to edit a page
 * Editing tutorial
 * Picture tutorial
 * How to write a great article
 * Naming conventions
 * Manual of Style

title: User talk:Will T Brown


text:
User talk:Nike Missile

Welcome!

Hello, and welcome to Wikipedia. Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers: I hope you enjoy editing here and being a Wikipedian! By the way, you can sign your name on Talk and vote pages using three tildes, like this: ~ ~ ~. Four tildes (~ ~ ) produces your name and the current date. If you have any questions, see the help pages, add a question to the village pump or ask me on my Talk page. Again, welcome! -- Graham ☺ | Talk 02:20, 17 Dec 2004 (UTC)
 * How to edit a page
 * Editing tutorial
 * Picture tutorial
 * How to write a great article
 * Naming conventions
 * Manual of Style

title: User talk:Nike Missile


text: User talk:131.123.176.238

title: User talk:131.123.176.238


text:
Karl Hudson-Phillips

Karl Terrence Hudson-Phillips, ORTT, QC (20 April 1933 – 16 January 2014) was an Attorney-General of Trinidad and Tobago and a judge of the International Criminal Court. He was also lead counsel in the murder trial of Grenadian Prime Minister Maurice Bishop.

Biography
Hudson-Phillips attended Tranquillity School and Queen's Royal College in Port of Spain, Trinidad, before going to England to read law at Selwyn College, Cambridge. In 1959, he was called to the bar at Gray's Inn, London. He returned to Trinidad and Tobago where he established a distinguished legal practice and was conferred the honour of Queen's Counsel - the mark of professional eminence in the British Commonwealth in 1971. He was the youngest Attorney General within the Commonwealth at age 36. Hudson-Phillips graduated from the University of Cambridge, England (1952-1956), M.A, LLB. He is a member of the Bar of the following countries: United Kingdom, Trinidad and Tobago, Guyana, Barbados, St Kitts, Antigua, The British Virgin Islands and Jamaica.

He entered active politics in 1966 when he was elected a Member of Parliament for the constituency of Digeo Martin East. Prior to that, he had been active in student politicsand social activities and was the President of the Cambridge University West Indian Society in 1945-1956. He was also Chairman of the first Overseas Party Group of the ruling People's National Movement (PNM) Party in 1958, soon after Dr Eric Williams came into power.

Hudson-Phillips returned to Trinidad in 1959 and commenced his law practicewith his father, who was then a prominent member of the Trinidad Bar. He became active in politics of Trinidad at a Party level and served on several party committees being the Party delegate to the conference on the Constitution for an independent Trinidad and Tobago in 1961. He was appointed a member of the Board of the Trinidad and Tobago Electricity Commission on which he served for four years from 1962-1966.

On entering Parliament in 1966, he continued his law practice while being introduced into active politics at Parliamentary level. In 1968, he made his first major speech in the House when he pioleted the first amendment to the Independent Constitution of Trinidad and Tobago to provide a widening of the provisions permitting Trinidad and Tobago citizens to have dual citizenship and to extend the date of application for citizenship by certain classes of Trinidad residents.

Between the period 1966 and 1969, he was the Government's delegate to several international conferences including, Conference of non-nuclear states -Geneva 1968 and Law of the Sea Conference 1969. Hudson-Phillips became a minister with Cabinet rank on 29 September 1969 and, on 24 December 1969, at age 36, the youngest serving Attorney General in the British Commonwealth.

Soon after his appointment to Attorney General, civil disturbances started in Trinidad and Tobago. The Black Power riots and Army Mutiny of 1970.

In 1973, he fell out of favour with then Prime Minister Eric Williams (for openly campaigning to replace Williams, who had spoken about retirement). This ended his political career in the PNM.

In 1974, Hudson-Phillips founded the National Land Tenants and Ratepayers Association of Trinidad and Tobago. In 1980, he founded the Organisation for National Reconstruction (ONR), a political party which contested the 1981 General Elections. Despite getting the second-highest vote tally in the election, the ONR failed to secure a single seat in Parliament. The ONR went on to form an accommodation with the National Alliance to contest the 1983 local government elections, and went on the merge with those parties to form the National Alliance for Reconstruction (NAR). Hudson-Phillips and Basdeo Panday, as the leaders of the two largest factions in what became the NAR, decided to step aside and allow A. N. R. Robinson, leader of one of the smaller parties, to become party leader. The NAR contested the 1986 general elections and won 33 of the 36 seats in Parliament. Hudson-Phillips did not take an active role in the party after the election.

In addition to the Grenada murder trial, Hudson-Phillips has been involved in many high-profile cases throughout the Caribbean, as both Prosecutor and Defender. In 1999, he was elected President of the Law Association of Trinidad and Tobago.

In February 2003, Hudson-Phillips was elected to the first ever bench of International Criminal Court judges. As "dean of the judges", he chaired the first meetings of judges before the election of the Presidency. He also contributed actively to the drafting of the Regulations of the Court. He resigned from the court for personal reasons on 14 March 2007.

On 23 July 2010, Sihasak Phuangketkeow, President of the UN Human Rights Council, announced that Hudson-Phillips would head a panel of experts to investigate whether Israel's Gaza flotilla raid on 31 May 2010 breached international law. Along with Hudson-Phillips, the panel includes Briton Desmond Lorenz de Silva and Malaysian Mary Shanthi Dairiam.

Hudson-Phillips died in London on 16 January 2014.

Publications

 * Hudson-Phillips, K. 1968. The Historical Development of the Settlement of Trade Disputes in Trinidad and Tobago. I.L.O. Monograph.
 * Hudson-Phillips, K. 1987. A Case for Greater Public Participation in the Legislative Process. Statute Law Review.
 * Hudson-Phillips, K. 1997. Law and Practice of Arbitration in Trinidad and Tobago. International Commercial Arbitration Bulletin.

title: Karl Hudson-Phillips


text:
Mario's Cement Factory

Mario's Cement Factory (マリオズ・セメント・ファクトリー) is a 1983 LCD game developed and published by Nintendo under their Game & Watch series. It follows earlier Mario games, like the arcade and Game & Watch versions of Donkey Kong. Players control Mario as he navigates elevators and funnels cement through a factory, while trying to prevent the cement from crushing his fellow workers. Two versions of the game were released — a Table Top unit and a handheld game akin to most other Game & Watch titles. Development was headed by Nintendo R&D1, led by engineer Gunpei Yokoi.

The game has been re-released several times; it was featured as part of Game Boy Gallery for the Game Boy, Game & Watch Gallery 4 for the Game Boy Advance, and as a digital download for the Nintendo DSi. It has been described by critics as one of the strangest games in the Mario franchise.

Gameplay
Mario's Cement Factory puts players in control of Mario, who works at a cement factory where he funnels cement into cement trucks. Mario must navigate two dangerous elevators and avoid falling or being crushed and losing a life. He must also continually empty cement into the trucks, or else the cement will overflow and crush one of the workers below, which costs players a life. There are two game modes: A and B, the latter being faster paced and more difficult.

Development
Mario's Cement Factory was developed by Nintendo R&D1, which at the time was led by Gunpei Yokoi, and published by Nintendo. Like all Game & Watch releases, each unit is a standalone portable device that doubles as a clock and can only play the one game. Hirokazu Tanaka composed the game sounds.

Two versions of the game were released. The first is part of the Game & Watch Table Top series and debuted on April 28, 1983. It has a full color illuminated screen, and approximately 250,000 models were produced. The Table Top series did not sell as well, leading to Mario's Cement Factory being one of only four Table Top units ever produced. A smaller handheld version was later released on June 8 that year as part of the New Wide Screen series. It has a monochrome screen with a color overlay, and approximately 750,000 units were produced.

The game was released the same year that Nintendo's Famicom system debuted in Japan, and two years after the first Mario title (the arcade game Donkey Kong).

Re-releases
Mario's Cement Factory has been re-released in various forms. It was included in the 1995 Game Boy Gallery for Game Boy, featuring updated graphics. It was also re-released in the Nintendo Mini Classics line, which repackaged Game & Watch games in small Game Boy-like devices. Both the New Wide Screen and an updated version were included in the 2002 Game & Watch Gallery 4 for the Game Boy Advance. In 2009, the game was re-released for the Nintendo DSi's DSiWare download service (along with other Game & Watch games). The DSi version was released in Japan on August 18. It was released in North America and Europe in March 2010. The DSi version was also given as a reward on Nintendo's now-defunct Club Nintendo service.

Reception
Mario's Cement Factory received generally positive reception, called one of the best Game & Watch games, and praised for its relative complexity and fun factor. It was identified as a favorite by Wired writer Andy Robertson, and in a preview of Game & Watch Gallery 4, NintendoWorldReport writer Ben Kosmina expressed that he loved Mario's Cement Factory. Staff for the magazine Video Games regarded it as a more difficult game than many before it. It was also named the 10th best Game & Watch game in a survey of ITMedia readers. Despite this positive reception, it received some criticism, considered primitive by the standards of today, In a review by IGN writer Lucas M. Thomas felt that the controls were too "picky and precise". NintendoLife writer Corbie Dillard noted it as being among the most popular Game & Watch titles, feeling that it's more playable and enjoyable thanks to the diversity of its gameplay. He noted that there may be nostalgia involved in the enjoyment of Game & Watch games, he legitimately had fun with the game. It has been called one of the stranger entries in the Mario series. Mario's role as a cement factory worker has been mentioned in multiple articles that cover the array of professions Mario has undertaken.

The original units have become collector's items and, like many Game & Watch titles, a complete-in-box unit can sell for over US$100. The game was featured in a Gunpei Yokoi exhibit in Harajuku in 2010.

title: Mario's Cement Factory


text:
Vertex operator algebra

In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric Langlands correspondence.

The related notion of vertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Lie algebra due to Igor Frenkel. In the course of this construction, one employs a Fock space that admits an action of vertex operators attached to elements of a lattice. Borcherds formulated the notion of vertex algebra by axiomatizing the relations between the lattice vertex operators, producing an algebraic structure that allows one to construct new Lie algebras by following Frenkel's method.

The notion of vertex operator algebra was introduced as a modification of the notion of vertex algebra, by Frenkel, James Lepowsky, and Arne Meurman in 1988, as part of their project to construct the moonshine module. They observed that many vertex algebras that appear 'in nature' carry an action of the Virasoro algebra, and satisfy a bounded-below property with respect to an energy operator. Motivated by this observation, they added the Virasoro action and bounded-below property as axioms.

We now have post-hoc motivation for these notions from physics, together with several interpretations of the axioms that were not initially known. Physically, the vertex operators arising from holomorphic field insertions at points in two-dimensional conformal field theory admit operator product expansions when insertions collide, and these satisfy precisely the relations specified in the definition of vertex operator algebra. Indeed, the axioms of a vertex operator algebra are a formal algebraic interpretation of what physicists call chiral algebras (not to be confused with the more precise notion with the same name in mathematics) or "algebras of chiral symmetries", where these symmetries describe the Ward identities satisfied by a given conformal field theory, including conformal invariance. Other formulations of the vertex algebra axioms include Borcherds's later work on singular commutative rings, algebras over certain operads on curves introduced by Huang, Kriz, and others, D-module-theoretic objects called chiral algebras introduced by Alexander Beilinson and Vladimir Drinfeld and factorization algebras, also introduced by Beilinson and Drinfeld.

Important basic examples of vertex operator algebras include the lattice VOAs (modeling lattice conformal field theories), VOAs given by representations of affine Kac–Moody algebras (from the WZW model), the Virasoro VOAs, which are VOAs corresponding to representations of the Virasoro algebra, and the moonshine module V♮, which is distinguished by its monster symmetry. More sophisticated examples such as affine W-algebras and the chiral de Rham complex on a complex manifold arise in geometric representation theory and mathematical physics.

Vertex algebra
A vertex algebra is a collection of data that satisfy certain axioms.

Data

 * a vector space $$V$$, called the space of states. The underlying field is typically taken to be the complex numbers, although Borcherds's original formulation allowed for an arbitrary commutative ring.
 * an identity element $$1\in V$$, sometimes written $$|0\rangle$$ or $$\Omega$$ to indicate a vacuum state.
 * an endomorphism $$T:V\rightarrow V$$, called "translation". (Borcherds's original formulation included a system of divided powers of $$T$$, because he did not assume the ground ring was divisible.)
 * a linear multiplication map $$Y:V\otimes V\rightarrow V((z))$$, where $$V((z))$$ is the space of all formal Laurent series with coefficients in $$V$$. This structure has some alternative presentations:
 * as an infinite collection of bilinear products $$ \cdot_n : u \otimes v \mapsto u_n v$$ where $$n \in \mathbb{Z}$$ and $$ u_n \in \mathrm{End}(V)$$, so that for each $$v$$, there is an $$N$$ such that $$u_n v = 0$$ for $$n < N$$.
 * as a left-multiplication map $$V\rightarrow \mathrm{End}(V)z^{\pm 1}$$. This is the 'state-to-field' map of the so-called state-field correspondence. For each $$u\in V$$, the endomorphism-valued formal distribution $$Y(u,z)$$ is called a vertex operator or a field, and the coefficient of $$z^{-n-1}$$ is the operator $$u_{n}$$. In the context of vertex algebras, a field is more precisely an element of $$\mathrm{End}(V)z^{\pm 1}$$, which can be written $$A(z) = \sum_{n \in \mathbb{Z}}A_n z^n, A_n \in \mathrm{End}(V)$$ such that for any $$v \in V, A_n v = 0$$ for sufficiently small $$n$$ (which may depend on $$v$$). The standard notation for the multiplication is

Axioms
These data are required to satisfy the following axioms:

 * Identity. For any $$u\in V\,,\,Y(1,z)u=u$$ and $$\,Y(u,z)1\in u+zVz$$.
 * Translation. $$T(1)=0$$, and for any $$u,v\in V$$,
 * $$[T,Y(u,z)]v = TY(u,z)v - Y(u,z)Tv = \frac{d}{dz}Y(u,z)v$$

 * Locality (Jacobi identity, or Borcherds identity). For any $$u,v\in V$$, there exists a positive integer $N$ such that:
 * $$ (z-x)^N Y(u, z) Y(v, x) = (z-x)^N Y(v, x) Y(u, z).$$

Equivalent formulations of locality axiom
The Locality axiom has several equivalent formulations in the literature, e.g., Frenkel–Lepowsky–Meurman introduced the Jacobi identity:

 * $$\forall u,v, w \in V : \qquad z^{-1}\delta\left(\frac{y-x}{z}\right)Y(u,x)Y(v,y)w - z^{-1}\delta\left(\frac{-y+x}{z}\right)Y(v,y)Y(u,x)w = y^{-1}\delta\left(\frac{x+z}{y}\right)Y(Y(u,z)v,y)w,$$

where we define the formal delta series by:

 * $$\delta\left(\frac{y-x}{z}\right) := \sum_{s \geq 0, r \in \mathbf{Z}} \binom{r}{s} (-1)^s y^{r-s}x^s z^{-r}.$$

Borcherds initially used the following two identities: for any vectors u, v, and w, and integers m and n we have

 * $$(u_m (v))_n (w) = \sum_{i \geq 0} (-1)^i \binom{m}{i} \left (u_{m-i} (v_{n+i} (w)) - (-1)^m v_{m+n-i} (u_i (w)) \right)$$

and
 * $$ u_m v=\sum_{i\geq 0}(-1)^{m+i+1}\frac{T^{i}}{i!}v_{m+i}u $$.

He later gave a more expansive version that is equivalent but easier to use: for any vectors u, v, and w, and integers m, n, and q we have

 * $$\sum_{i \in \mathbf{Z}} \binom{m}{i} \left(u_{q+i} (v) \right )_{m+n-i} (w) = \sum_{i\in \mathbf{Z}} (-1)^i \binom{q}{i} \left (u_{m+q-i} \left(v_{n+i} (w) \right ) - (-1)^q v_{n+q-i} \left (u_{m+i} (w) \right ) \right)$$

Finally, there is a formal function version of locality: For any $$u,v,w\in V$$, there is an element

 * $$X(u,v,w;z,x) \in Vz,x \left[z^{-1}, x^{-1}, (z-x)^{-1} \right]$$

such that $$Y(u,z)Y(v,x)w$$ and $$Y(v,x)Y(u,z)w$$ are the corresponding expansions of $$X(u,v,w;z,x)$$ in $$V((z))((x))$$ and $$V((x))((z))$$.

Vertex operator algebra
A vertex operator algebra is a vertex algebra equipped with a conformal element $$\omega \in V$$, such that the vertex operator $$Y(\omega,z)$$ is the weight two Virasoro field $$L(z)$$:

 * $$Y(\omega, z) = \sum_{n\in\mathbf{Z}} \omega_{n} {z^{-n-1}} = L(z) = \sum_{n\in\mathbf{Z}} L_n z^{-n-2}$$

and satisfies the following properties:
 * $$[L_m,L_n]=(m-n)L_{m+n}+\frac{1}{12}\delta_{m+n,0}(m^3-m)c\,\mathrm{Id}_V$$, where $$c$$ is a constant called the central charge, or rank of $$V$$. In particular, the coefficients of this vertex operator endow $$V$$ with an action of the Virasoro algebra with central charge $$c$$.
 * $$L_0$$ acts semisimply on $$V$$ with integer eigenvalues that are bounded below.
 * Under the grading provided by the eigenvalues of $$L_0$$, the multiplication on $$V$$ is homogeneous in the sense that if $$u$$ and $$v$$ are homogeneous, then $$u_n v$$ is homogeneous of degree $$\mathrm{deg}(u)+\mathrm{deg}(v)-n-1$$.
 * The identity $$1$$ has degree 0, and the conformal element $$\omega$$ has degree 2.
 * $$L_{-1}=T$$.

A homomorphism of vertex algebras is a map of the underlying vector spaces that respects the additional identity, translation, and multiplication structure. Homomorphisms of vertex operator algebras have "weak" and "strong" forms, depending on whether they respect conformal vectors.

Commutative vertex algebras
A vertex algebra $$V$$ is commutative if all vertex operators $$Y(u,z)$$ commute with each other. This is equivalent to the property that all products $$Y(u,z)v$$ lie in $$Vz$$, or that $$Y(u, z) \in \operatorname{End}z$$. Thus, an alternative definition for a commutative vertex algebra is one in which all vertex operators $$Y(u,z)$$ are regular at $$z = 0$$.

Given a commutative vertex algebra, the constant terms of multiplication endow the vector space with a commutative and associative ring structure, the vacuum vector $$1$$ is a unit and $$T$$ is a derivation. Hence the commutative vertex algebra equips $$V$$ with the structure of a commutative unital algebra with derivation. Conversely, any commutative ring $$V$$ with derivation $$T$$ has a canonical vertex algebra structure, where we set $$Y(u,z)v=u_{-1}vz^0=uv$$, so that $$Y$$ restricts to a map $$Y:V \rightarrow \operatorname{End}(V)$$ which is the multiplication map $$u \mapsto u \cdot$$ with $$\cdot$$ the algebra product. If the derivation $$T$$ vanishes, we may set $$\omega=0$$ to obtain a vertex operator algebra concentrated in degree zero.

Any finite-dimensional vertex algebra is commutative. Thus even the smallest examples of noncommutative vertex algebras require significant introduction.

Basic properties
The translation operator $$T$$ in a vertex algebra induces infinitesimal symmetries on the product structure, and satisfies the following properties:

 * $$\,Y(u,z)1=e^{zT}u$$
 * $$\,Tu=u_{-2}1$$, so $$T$$ is determined by $$Y$$.
 * $$\,Y(Tu,z)=\frac{\mathrm{d}Y(u,z)}{\mathrm{d}z}$$
 * $$\,e^{xT}Y(u,z)e^{-xT}=Y(e^{xT}u,z)=Y(u,z+x)$$
 * (skew-symmetry) $$Y(u,z)v=e^{zT}Y(v,-z)u$$

For a vertex operator algebra, the other Virasoro operators satisfy similar properties:

 * $$\,x^{L_0}Y(u,z)x^{-L_0}=Y(x^{L_0}u,xz)$$
 * $$\,e^{xL_1}Y(u,z)e^{-xL_1}=Y(e^{x(1-xz)L_1}(1-xz)^{-2L_0}u,z(1-xz)^{-1})$$
 * (quasi-conformality) $$[L_m, Y(u,z)] = \sum_{k=0}^{m+1} \binom{m+1}{k} z^k Y(L_{m-k}u, z)$$ for all $$m\geq -1$$.
 * (Associativity, or Cousin property): For any $$u,v,w\in V$$, the element

 * $$X(u,v,w;z,x) \in Vz,x[z^{-1}, x^{-1}, (z-x)^{-1}]$$

given in the definition also expands to $$Y(Y(u,z-x)v,x)w$$ in $$V((x))((z-x))$$.

The associativity property of a vertex algebra follows from the fact that the commutator of $$Y(u,z)$$ and $$Y(v,z)$$ is annihilated by a finite power of $$z-x$$, i.e., one can expand it as a finite linear combination of derivatives of the formal delta function in $$(z-x)$$, with coefficients in $$\mathrm{End}(V)$$.

Reconstruction: Let $$V$$ be a vertex algebra, and let $$J_a$$ be a set of vectors, with corresponding fields $$J^a(z)\in \mathrm{End}(V)z^{\pm 1}$$. If $$V$$ is spanned by monomials in the positive weight coefficients of the fields (i.e., finite products of operators $$J^{a}_{n}$$ applied to $$1$$, where $$n$$ is negative), then we may write the operator product of such a monomial as a normally ordered product of divided power derivatives of fields (here, normal ordering means polar terms on the left are moved to the right). Specifically,

 * $$Y(J^{a_1}_{n_1+1}J^{a_2}_{n_2+1}...J^{a_k}_{n_k+1}1, z) = :\frac{\partial^{n_1}}{\partial z^{n_1}}\frac{J^{a_1}(z)}{n_1!}\frac{\partial^{n_2}}{\partial z^{n_2}}\frac{J^{a_2}(z)}{n_2!} \cdots \frac{\partial^{n_k}}{\partial z^{n_k}}\frac{J^{a_k}(z)}{n_k!}:$$

More generally, if one is given a vector space $$V$$ with an endomorphism $$T$$ and vector $$1$$, and one assigns to a set of vectors $$J^a$$ a set of fields $$J^a(z)\in \mathrm{End}(V)z^{\pm 1}$$ that are mutually local, whose positive weight coefficients generate $$V$$, and that satisfy the identity and translation conditions, then the previous formula describes a vertex algebra structure.

Operator product expansion
In vertex algebra theory, due to associativity, we can abuse notation to write, for $$A, B, C \in V,$$

This is the operator product expansion. Equivalently,

Since the normal ordered part is regular in $$z$$ and $$w$$, this can be written more in line with physics conventions as

where the equivalence relation $$\sim$$ denotes equivalence up to regular terms.

Commonly used OPEs
Here some OPEs frequently found in conformal field theory are recorded.

Examples from Lie algebras
The basic examples come from infinite-dimensional Lie algebras.

Heisenberg vertex operator algebra
A basic example of a noncommutative vertex algebra is the rank 1 free boson, also called the Heisenberg vertex operator algebra. It is "generated" by a single vector b, in the sense that by applying the coefficients of the field b(z) := Y(b,z) to the vector 1, we obtain a spanning set. The underlying vector space is the infinite-variable polynomial ring $$\mathbb{C}[b_{-1}, b_{-2}, \cdots]$$, where for positive $$n$$, $$b_{-n}$$ acts obviously by multiplication, and $$b_n$$ acts as $$n\partial_{b_{-n}}$$. The action of b0 is multiplication by zero, producing the "momentum zero" Fock representation V0 of the Heisenberg Lie algebra (generated by bn for integers n, with commutation relations [bn,bm]=n δn,–m), induced by the trivial representation of the subalgebra spanned by bn, n ≥ 0.

The Fock space V0 can be made into a vertex algebra by the following definition of the state-operator map on a basis $$b_{j_1}b_{j_2}...b_{j_k}$$ with each $$j_i < 0$$,

 * $$Y( b_{j_1}b_{j_2}...b_{j_k}, z) := \frac{1}{(-j_1 - 1)!(-j_2 - 1)!\cdots (-j_k - 1)!}:\partial^{-j_1 - 1}b(z)\partial^{-j_2 - 1}b(z)...\partial^{-j_k - 1}b(z):$$

where $$:\mathcal{O}:$$ denotes normal ordering of an operator $$\mathcal{O}$$. The vertex operators may also be written as a functional of a multivariable function f as:

 * $$ Y[f,z] \equiv :f\left(\frac{b(z)}{0!},\frac{b'(z)}{1!},\frac{b''(z)}{2!},...\right): $$

if we understand that each term in the expansion of f is normal ordered.

The rank n free boson is given by taking an n-fold tensor product of the rank 1 free boson. For any vector b in n-dimensional space, one has a field b(z) whose coefficients are elements of the rank n Heisenberg algebra, whose commutation relations have an extra inner product term: [bn,cm]=n (b,c) δn,–m.

The Heisenberg vertex operator algebra has a one-parameter family of conformal vectors with parameter $$\lambda \in \mathbb{C}$$ of conformal vectors $$\omega_\lambda$$ given by
 * $$\omega_\lambda = \frac{1}{2}b_{-1}^2 + \lambda b_{-2},$$

with central charge $$c_\lambda = 1 - 12\lambda^2$$.

When $$\lambda = 0$$, there is the following formula for the Virasoro character:

 * $$Tr_V q^{L_0} := \sum_{n \in \mathbf{Z}} \dim V_n q^n = \prod_{n \geq 1} (1-q^n)^{-1}$$

This is the generating function for partitions, and is also written as q1/24 times the weight −1/2 modular form 1/η (the reciprocal of the Dedekind eta function). The rank n free boson then has an n parameter family of Virasoro vectors, and when those parameters are zero, the character is qn/24 times the weight −n/2 modular form η−n.

Virasoro vertex operator algebra
Virasoro vertex operator algebras are important for two reasons: First, the conformal element in a vertex operator algebra canonically induces a homomorphism from a Virasoro vertex operator algebra, so they play a universal role in the theory. Second, they are intimately connected to the theory of unitary representations of the Virasoro algebra, and these play a major role in conformal field theory. In particular, the unitary Virasoro minimal models are simple quotients of these vertex algebras, and their tensor products provide a way to combinatorially construct more complicated vertex operator algebras.

The Virasoro vertex operator algebra is defined as an induced representation of the Virasoro algebra: If we choose a central charge c, there is a unique one-dimensional module for the subalgebra C[z]∂z + K for which K acts by cId, and C[z]∂z acts trivially, and the corresponding induced module is spanned by polynomials in L–n = –z−n–1∂z as n ranges over integers greater than 1. The module then has partition function

 * $$Tr_V q^{L_0} = \sum_{n \in \mathbf{R}} \dim V_n q^n = \prod_{n \geq 2} (1-q^n)^{-1}$$.

This space has a vertex operator algebra structure, where the vertex operators are defined by:

 * $$Y(L_{-n_1-2}L_{-n_2-2}...L_{-n_k-2}|0\rangle,z) \equiv \frac{1}{n_1!n_2!..n_k!}:\partial^{n_1}L(z)\partial^{n_2}L(z)...\partial^{n_k}L(z):$$

and $$\omega = L_{-2}|0\rangle$$. The fact that the Virasoro field L(z) is local with respect to itself can be deduced from the formula for its self-commutator:

$$[L(z),L(x)] =\left(\frac{\partial}{\partial x}L(x)\right)w^{-1}\delta \left(\frac{z}{x}\right)-2L(x)x^{-1}\frac{\partial}{\partial z}\delta \left(\frac{z}{x}\right)-\frac{1}{12}cx^{-1}\left(\frac{\partial}{\partial z}\right)^3\delta \left(\frac{z}{x}\right)$$

where c is the central charge.

Given a vertex algebra homomorphism from a Virasoro vertex algebra of central charge c to any other vertex algebra, the vertex operator attached to the image of ω automatically satisfies the Virasoro relations, i.e., the image of ω is a conformal vector. Conversely, any conformal vector in a vertex algebra induces a distinguished vertex algebra homomorphism from some Virasoro vertex operator algebra.

The Virasoro vertex operator algebras are simple, except when c has the form 1–6(p–q)2/pq for coprime integers p,q strictly greater than 1 – this follows from Kac's determinant formula. In these exceptional cases, one has a unique maximal ideal, and the corresponding quotient is called a minimal model. When p = q+1, the vertex algebras are unitary representations of Virasoro, and their modules are known as discrete series representations. They play an important role in conformal field theory in part because they are unusually tractable, and for small p, they correspond to well-known statistical mechanics systems at criticality, e.g., the Ising model, the tri-critical Ising model, the three-state Potts model, etc. By work of Weiqang Wang concerning fusion rules, we have a full description of the tensor categories of unitary minimal models. For example, when c=1/2 (Ising), there are three irreducible modules with lowest L0-weight 0, 1/2, and 1/16, and its fusion ring is Z[x,y]/(x2–1, y2–x–1, xy–y).

Affine vertex algebra
By replacing the Heisenberg Lie algebra with an untwisted affine Kac–Moody Lie algebra (i.e., the universal central extension of the loop algebra on a finite-dimensional simple Lie algebra), one may construct the vacuum representation in much the same way as the free boson vertex algebra is constructed. This algebra arises as the current algebra of the Wess–Zumino–Witten model, which produces the anomaly that is interpreted as the central extension.

Concretely, pulling back the central extension

 * $$0 \to \mathbb{C} \to \hat{\mathfrak{g}} \to \mathfrak{g}[t,t^{-1}] \to 0$$

along the inclusion $$\mathfrak{g}[t] \to \mathfrak{g}[t,t^{-1}]$$ yields a split extension, and the vacuum module is induced from the one-dimensional representation of the latter on which a central basis element acts by some chosen constant called the "level". Since central elements can be identified with invariant inner products on the finite type Lie algebra $$\mathfrak{g}$$, one typically normalizes the level so that the Killing form has level twice the dual Coxeter number. Equivalently, level one gives the inner product for which the longest root has norm 2. This matches the loop algebra convention, where levels are discretized by third cohomology of simply connected compact Lie groups.

By choosing a basis Ja of the finite type Lie algebra, one may form a basis of the affine Lie algebra using Jan = Ja tn together with a central element K. By reconstruction, we can describe the vertex operators by normal ordered products of derivatives of the fields

 * $$J^a(z) = \sum_{n=-\infty}^\infty J^a_n z^{-n-1} = \sum_{n=-\infty}^\infty (J^a t^n) z^{-n-1}.$$

When the level is non-critical, i.e., the inner product is not minus one half of the Killing form, the vacuum representation has a conformal element, given by the Sugawara construction. For any choice of dual bases Ja, Ja with respect to the level 1 inner product, the conformal element is

 * $$\omega = \frac{1}{2(k+h^\vee)} \sum_a J_{a,-1} J^a_{-1} 1$$

and yields a vertex operator algebra whose central charge is $$k \cdot \dim \mathfrak{g}/(k+h^\vee)$$. At critical level, the conformal structure is destroyed, since the denominator is zero, but one may produce operators Ln for n ≥ –1 by taking a limit as k approaches criticality.

Modules
Much like ordinary rings, vertex algebras admit a notion of module, or representation. Modules play an important role in conformal field theory, where they are often called sectors. A standard assumption in the physics literature is that the full Hilbert space of a conformal field theory decomposes into a sum of tensor products of left-moving and right-moving sectors:

 * $$\mathcal{H} \cong \bigoplus_{i \in I} M_i \otimes \overline{M_i}$$

That is, a conformal field theory has a vertex operator algebra of left-moving chiral symmetries, a vertex operator algebra of right-moving chiral symmetries, and the sectors moving in a given direction are modules for the corresponding vertex operator algebra.

Definition
Given a vertex algebra V with multiplication Y, a V-module is a vector space M equipped with an action YM: V ⊗ M → M((z)), satisfying the following conditions:

 * (Identity) YM(1,z) = IdM
 * (Associativity, or Jacobi identity) For any u, v ∈ V, w ∈ M, there is an element

 * $$X(u,v,w;z,x) \in Mz,x[z^{-1}, x^{-1}, (z-x)^{-1}]$$

such that YM(u,z)YM(v,x)w and YM(Y(u,z–x)v,x)w are the corresponding expansions of $$X(u,v,w;z,x)$$ in M((z))((x)) and M((x))((z–x)). Equivalently, the following "Jacobi identity" holds:

 * $$z^{-1}\delta\left(\frac{y-x}{z}\right)Y^M(u,x)Y^M(v,y)w - z^{-1}\delta\left(\frac{-y+x}{z}\right)Y^M(v,y)Y^M(u,x)w = y^{-1}\delta\left(\frac{x+z}{y}\right)Y^M(Y(u,z)v,y)w.$$

The modules of a vertex algebra form an abelian category. When working with vertex operator algebras, the previous definition is sometimes given the name weak $$V$$-module, and genuine V-modules must respect the conformal structure given by the conformal vector $$\omega$$. More precisely, they are required to satisfy the additional condition that L0 acts semisimply with finite-dimensional eigenspaces and eigenvalues bounded below in each coset of Z. Work of Huang, Lepowsky, Miyamoto, and Zhang has shown at various levels of generality that modules of a vertex operator algebra admit a fusion tensor product operation, and form a braided tensor category.

When the category of V-modules is semisimple with finitely many irreducible objects, the vertex operator algebra V is called rational. Rational vertex operator algebras satisfying an additional finiteness hypothesis (known as Zhu's C2-cofiniteness condition) are known to be particularly well-behaved, and are called regular. For example, Zhu's 1996 modular invariance theorem asserts that the characters of modules of a regular VOA form a vector-valued representation of $$\mathrm{SL}(2, \mathbb{Z})$$. In particular, if a VOA is holomorphic, that is, its representation category is equivalent to that of vector spaces, then its partition function is $$\mathrm{SL}(2, \mathbb{Z})$$-invariant up to a constant. Huang showed that the category of modules of a regular VOA is a modular tensor category, and its fusion rules satisfy the Verlinde formula.

Heisenberg algebra modules
Modules of the Heisenberg algebra can be constructed as Fock spaces $$\pi_\lambda$$ for $$\lambda \in \mathbb{C}$$ which are induced representations of the Heisenberg Lie algebra, given by a vacuum vector $$v_\lambda$$ satisfying $$b_nv_\lambda = 0$$ for $$n > 0$$, $$b_0v_\lambda = 0$$, and being acted on freely by the negative modes $$b_{-n}$$ for $$n>0$$. The space can be written as $$\mathbb{C}[b_{-1}, b_{-2}, \cdots]v_\lambda$$. Every irreducible, $$\mathbb{Z}$$-graded Heisenberg algebra module with gradation bounded below is of this form.

These are used to construct lattice vertex algebras, which as vector spaces are direct sums of Heisenberg modules, when the image of $$Y$$ is extended appropriately to module elements.

The module category is not semisimple, since one may induce a representation of the abelian Lie algebra where b0 acts by a nontrivial Jordan block. For the rank n free boson, one has an irreducible module Vλ for each vector λ in complex n-dimensional space. Each vector b ∈ Cn yields the operator b0, and the Fock space Vλ is distinguished by the property that each such b0 acts as scalar multiplication by the inner product (b, λ).

Twisted modules
Unlike ordinary rings, vertex algebras admit a notion of twisted module attached to an automorphism. For an automorphism σ of order N, the action has the form V ⊗ M → M((z1/N)), with the following monodromy condition: if u ∈ V satisfies σ u = exp(2πik/N)u, then un = 0 unless n satisfies n+k/N ∈ Z (there is some disagreement about signs among specialists). Geometrically, twisted modules can be attached to branch points on an algebraic curve with a ramified Galois cover. In the conformal field theory literature, twisted modules are called twisted sectors, and are intimately connected with string theory on orbifolds.

Vertex operator algebra defined by an even lattice
The lattice vertex algebra construction was the original motivation for defining vertex algebras. It is constructed by taking a sum of irreducible modules for the Heisenberg algebra corresponding to lattice vectors, and defining a multiplication operation by specifying intertwining operators between them. That is, if $Λ$ is an even lattice (if the lattice is not even, the structure obtained is instead a vertex superalgebra), the lattice vertex algebra $V_{Λ}$ decomposes into free bosonic modules as:

 * $$V_\Lambda \cong \bigoplus_{\lambda \in \Lambda} V_\lambda$$

Lattice vertex algebras are canonically attached to double covers of even integral lattices, rather than the lattices themselves. While each such lattice has a unique lattice vertex algebra up to isomorphism, the vertex algebra construction is not functorial, because lattice automorphisms have an ambiguity in lifting.

The double covers in question are uniquely determined up to isomorphism by the following rule: elements have the form $±e_{α}$ for lattice vectors $α ∈ Λ$ (i.e., there is a map to $Λ$ sending $e_{α}$ to α that forgets signs), and multiplication satisfies the relations eαeβ = (–1)(α,β)eβeα. Another way to describe this is that given an even lattice $Λ$, there is a unique (up to coboundary) normalised cocycle $ε(α, β)$ with values $±1$ such that $(−1)^{(α,β)} = ε(α, β) ε(β, α)$, where the normalization condition is that ε(α, 0) = ε(0, α) = 1 for all $α ∈ Λ$. This cocycle induces a central extension of $Λ$ by a group of order 2, and we obtain a twisted group ring $C_{ε}[Λ]$ with basis $e_{α} (α ∈ Λ)$, and multiplication rule $e_{α}e_{β} = ε(α, β)e_{α+β}$ – the cocycle condition on $ε$ ensures associativity of the ring.

The vertex operator attached to lowest weight vector $v_{λ}$ in the Fock space $V_{λ}$ is

 * $$Y(v_\lambda,z) = e_\lambda :\exp \int \lambda(z): = e_\lambda z^\lambda \exp \left (\sum_{n<0} \lambda_n \frac{z^{-n}}{n} \right )\exp \left (\sum_{n>0} \lambda_n \frac{z^{-n}}{n} \right ),$$

where $z^{λ}$ is a shorthand for the linear map that takes any element of the α-Fock space $V_{α}$ to the monomial $z^{(λ,α)}$. The vertex operators for other elements of the Fock space are then determined by reconstruction.

As in the case of the free boson, one has a choice of conformal vector, given by an element s of the vector space $Λ ⊗ C$, but the condition that the extra Fock spaces have integer L0 eigenvalues constrains the choice of s: for an orthonormal basis $x_{i}$, the vector 1/2 xi,12 + s2 must satisfy $(s, λ) ∈ Z$ for all λ ∈ Λ, i.e., s lies in the dual lattice.

If the even lattice $Λ$ is generated by its "root vectors" (those satisfying (α, α)=2), and any two root vectors are joined by a chain of root vectors with consecutive inner products non-zero then the vertex operator algebra is the unique simple quotient of the vacuum module of the affine Kac–Moody algebra of the corresponding simply laced simple Lie algebra at level one. This is known as the Frenkel–Kac (or Frenkel–Kac–Segal) construction, and is based on the earlier construction by Sergio Fubini and Gabriele Veneziano of the tachyonic vertex operator in the dual resonance model. Among other features, the zero modes of the vertex operators corresponding to root vectors give a construction of the underlying simple Lie algebra, related to a presentation originally due to Jacques Tits. In particular, one obtains a construction of all ADE type Lie groups directly from their root lattices. And this is commonly considered the simplest way to construct the 248-dimensional group E8.

Monster vertex algebra
The monster vertex algebra $$V^\natural$$ (also called the "moonshine module") is the key to Borcherds's proof of the Monstrous moonshine conjectures. It was constructed by Frenkel, Lepowsky, and Meurman in 1988. It is notable because its character is the j-invariant with no constant term, $$j(\tau) - 744$$, and its automorphism group is the monster group. It is constructed by orbifolding the lattice vertex algebra constructed from the Leech lattice by the order 2 automorphism induced by reflecting the Leech lattice in the origin. That is, one forms the direct sum of the Leech lattice VOA with the twisted module, and takes the fixed points under an induced involution. Frenkel, Lepowsky, and Meurman conjectured in 1988 that $$V^\natural$$ is the unique holomorphic vertex operator algebra with central charge 24, and partition function $$j(\tau) - 744$$. This conjecture is still open.

Chiral de Rham complex
Malikov, Schechtman, and Vaintrob showed that by a method of localization, one may canonically attach a bcβγ (boson–fermion superfield) system to a smooth complex manifold. This complex of sheaves has a distinguished differential, and the global cohomology is a vertex superalgebra. Ben-Zvi, Heluani, and Szczesny showed that a Riemannian metric on the manifold induces an N=1 superconformal structure, which is promoted to an N=2 structure if the metric is Kähler and Ricci-flat, and a hyperkähler structure induces an N=4 structure. Borisov and Libgober showed that one may obtain the two-variable elliptic genus of a compact complex manifold from the cohomology of the Chiral de Rham complex. If the manifold is Calabi–Yau, then this genus is a weak Jacobi form.

Vertex algebra associated to a surface defect
A vertex algebra can arise as a subsector of higher dimensional quantum field theory which localizes to a two real-dimensional submanifold of the space on which the higher dimensional theory is defined. A prototypical example is the construction of Beem, Leemos, Liendo, Peelaers, Rastelli, and van Rees which associates a vertex algebra to any 4d N=2 superconformal field theory. This vertex algebra has the property that its character coincides with the Schur index of the 4d superconformal theory. When the theory admits a weak coupling limit, the vertex algebra has an explicit description as a BRST reduction of a bcβγ system.

Vertex operator superalgebras
By allowing the underlying vector space to be a superspace (i.e., a Z/2Z-graded vector space $$ V=V_+\oplus V_-$$) one can define a vertex superalgebra by the same data as a vertex algebra, with 1 in V+ and T an even operator. The axioms are essentially the same, but one must incorporate suitable signs into the locality axiom, or one of the equivalent formulations. That is, if a and b are homogeneous, one compares Y(a,z)Y(b,w) with εY(b,w)Y(a,z), where ε is –1 if both a and b are odd and 1 otherwise. If in addition there is a Virasoro element ω in the even part of V2, and the usual grading restrictions are satisfied, then V is called a vertex operator superalgebra.

One of the simplest examples is the vertex operator superalgebra generated by a single free fermion ψ. As a Virasoro representation, it has central charge 1/2, and decomposes as a direct sum of Ising modules of lowest weight 0 and 1/2. One may also describe it as a spin representation of the Clifford algebra on the quadratic space t1/2C[t,t−1](dt)1/2 with residue pairing. The vertex operator superalgebra is holomorphic, in the sense that all modules are direct sums of itself, i.e., the module category is equivalent to the category of vector spaces.

The tensor square of the free fermion is called the free charged fermion, and by boson–fermion correspondence, it is isomorphic to the lattice vertex superalgebra attached to the odd lattice Z. This correspondence has been used by Date–Jimbo–Kashiwara-Miwa to construct soliton solutions to the KP hierarchy of nonlinear PDEs.

Superconformal structures
The Virasoro algebra has some supersymmetric extensions that naturally appear in superconformal field theory and superstring theory. The N=1, 2, and 4 superconformal algebras are of particular importance.

Infinitesimal holomorphic superconformal transformations of a supercurve (with one even local coordinate z and N odd local coordinates θ1,...,θN) are generated by the coefficients of a super-stress–energy tensor T(z, θ1, ..., θN).

When N=1, T has odd part given by a Virasoro field L(z), and even part given by a field

 * $$G(z) = \sum_n G_n z^{-n-3/2}$$

subject to commutation relations

 * $$[G_m,L_n] = (m-n/2)G_{m+n}$$
 * $$[G_m,G_n] = (m-n)L_{m+n} + \delta_{m,-n} \frac{4m^2+1}{12}c$$

By examining the symmetry of the operator products, one finds that there are two possibilities for the field G: the indices n are either all integers, yielding the Ramond algebra, or all half-integers, yielding the Neveu–Schwarz algebra. These algebras have unitary discrete series representations at central charge

 * $$\hat{c} = \frac{2}{3}c = 1-\frac{8}{m(m+2)} \quad m \geq 3$$

and unitary representations for all c greater than 3/2, with lowest weight h only constrained by h≥ 0 for Neveu–Schwarz and h ≥ c/24 for Ramond.

An N=1 superconformal vector in a vertex operator algebra V of central charge c is an odd element τ ∈ V of weight 3/2, such that
 * $$Y(\tau,z) = G(z) = \sum_{m \in \mathbb{Z}+1/2} G_n z^{-n-3/2},$$

G−1/2τ = ω, and the coefficients of G(z) yield an action of the N=1 Neveu–Schwarz algebra at central charge c.

For N=2 supersymmetry, one obtains even fields L(z) and J(z), and odd fields G+(z) and G−(z). The field J(z) generates an action of the Heisenberg algebras (described by physicists as a U(1) current). There are both Ramond and Neveu–Schwarz N=2 superconformal algebras, depending on whether the indexing on the G fields is integral or half-integral. However, the U(1) current gives rise to a one-parameter family of isomorphic superconformal algebras interpolating between Ramond and Neveu–Schwartz, and this deformation of structure is known as spectral flow. The unitary representations are given by discrete series with central charge c = 3-6/m for integers m at least 3, and a continuum of lowest weights for c > 3.

An N=2 superconformal structure on a vertex operator algebra is a pair of odd elements τ+, τ− of weight 3/2, and an even element μ of weight 1 such that τ± generate G±(z), and μ generates J(z).

For N=3 and 4, unitary representations only have central charges in a discrete family, with c=3k/2 and 6k, respectively, as k ranges over positive integers.

Additional constructions

 * Fixed point subalgebras: Given an action of a symmetry group on a vertex operator algebra, the subalgebra of fixed vectors is also a vertex operator algebra. In 2013, Miyamoto proved that two important finiteness properties, namely Zhu's condition C2 and regularity, are preserved when taking fixed points under finite solvable group actions.
 * Current extensions: Given a vertex operator algebra and some modules of integral conformal weight, one may under favorable circumstances describe a vertex operator algebra structure on the direct sum. Lattice vertex algebras are a standard example of this.  Another family of examples are framed VOAs, which start with tensor products of Ising models, and add modules that correspond to suitably even codes.
 * Orbifolds: Given a finite cyclic group acting on a holomorphic VOA, it is conjectured that one may construct a second holomorphic VOA by adjoining irreducible twisted modules and taking fixed points under an induced automorphism, as long as those twisted modules have suitable conformal weight. This is known to be true in special cases, e.g., groups of order at most 3 acting on lattice VOAs.
 * The coset construction (due to Goddard, Kent, and Olive): Given a vertex operator algebra V of central charge c and a set S of vectors, one may define the commutant C(V,S) to be the subspace of vectors v strictly commute with all fields coming from S, i.e., such that Y(s,z)v ∈ V z  for all s ∈ S.  This turns out to be a vertex subalgebra, with Y, T, and identity inherited from V.   and if S is a VOA of central charge cS, the commutant is a VOA of central charge c–cS.  For example, the embedding of SU(2) at level k+1 into the tensor product of two SU(2) algebras at levels k and 1 yields the Virasoro discrete series with p=k+2, q=k+3, and this was used to prove their existence in the 1980s.  Again with SU(2), the embedding of level k+2 into the tensor product of level k and level 2 yields the N=1 superconformal discrete series.
 * BRST reduction: For any degree 1 vector v satisfying v02=0, the cohomology of this operator has a graded vertex superalgebra structure. More generally, one may use any weight 1 field whose residue has square zero.  The usual method is to tensor with fermions, as one then has a canonical differential.  An important special case is quantum Drinfeld–Sokolov reduction applied to affine Kac–Moody algebras to obtain affine W-algebras as degree 0 cohomology.  These W algebras also admit constructions as vertex subalgebras of free bosons given by kernels of screening operators.

Related algebraic structures

 * If one considers only the singular part of the OPE in a vertex algebra, one arrives at the definition of a Lie conformal algebra. Since one is often only concerned with the singular part of the OPE, this makes Lie conformal algebras a natural object to study. There is a functor from vertex algebras to Lie conformal algebras that forgets the regular part of OPEs, and it has a left adjoint, called the "universal vertex algebra" functor. Vacuum modules of affine Kac–Moody algebras and Virasoro vertex algebras are universal vertex algebras, and in particular, they can be described very concisely once the background theory is developed.
 * There are several generalizations of the notion of vertex algebra in the literature. Some mild generalizations involve a weakening of the locality axiom to allow monodromy, e.g., the abelian intertwining algebras of Dong and Lepowsky. One may view these roughly as vertex algebra objects in a braided tensor category of graded vector spaces, in much the same way that a vertex superalgebra is such an object in the category of super vector spaces. More complicated generalizations relate to q-deformations and representations of quantum groups, such as in work of Frenkel–Reshetikhin, Etingof–Kazhdan, and Li.
 * Beilinson and Drinfeld introduced a sheaf-theoretic notion of chiral algebra that is closely related to the notion of vertex algebra, but is defined without using any visible power series. Given an algebraic curve X, a chiral algebra on X is a DX-module A equipped with a multiplication operation $$j_*j^*(A \boxtimes A) \to \Delta_* A$$ on X×X that satisfies an associativity condition. They also introduced an equivalent notion of factorization algebra that is a system of quasicoherent sheaves on all finite products of the curve, together with a compatibility condition involving pullbacks to the complement of various diagonals. Any translation-equivariant chiral algebra on the affine line can be identified with a vertex algebra by taking the fiber at a point, and there is a natural way to attach a chiral algebra on a smooth algebraic curve to any vertex operator algebra.

title: Vertex operator algebra