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For a long time, English was the most widely used language in the digital world. Even today, it's estimated that 55 percent of all web content is in English. But that's changing—more and more users expect to get products in their native languages. In fact, more than 70 percent of consumers want to get information in their own language, making the potential of localization in product design clear.
But offering content in several languages can add many new layers of complexity to design. In this article, I want to describe some practical tips that will help you localize your products. But before we get into that, it's important to define what exactly location is.
What is localization?
Localization is the adaptation of a product to a specific market. It includes changes that are necessary to make a product culturally appropriate without breaking its functionality.
"Localization is the adaptation of a product to a specific market. It includes changes that are necessary to make a product culturally appropriate without breaking its functionality."
Localization is a lot more than just a translation. Product creators need to not only translate the content, but also consider the cultural specifics of the new region. The ultimate goal of localization is to adapt a product so that new users believe that it was designed especially for their language and culture.
Do not put off supporting localization
This is by far the most important rule of localization. It's vital to keep localization in mind from the very beginning of the product design process. All too often, product teams put off localization until late in the development cycle. This decision has a negative impact on the product—product designers create a UX dept and increase the overall cost of localization.
As a product designer, your responsibility is to build an awareness of localization. Make sure the aspects of localization are reflected in the specification, and that mocks and prototypes are designed to support localization.
You might also like: The Future of Commerce is Global: Why and How to Take Your Business International with Shopify.
Common technical issues of localization
When localizing a digital product, there are some technical issues that should be considered. Below are five things to consider.
1. Don't use hardcoded strings
Many localization issues stem from legacy code. Hardcoded strings are one of the most common problems that make it hard to extract content for translation.
All elements in your user interface such as titles, menu items, toolbar labels should be defined as variables, not hardcoded constants.
2. Choose your fonts wisely
Make sure you choose a font that's easy to read in all languages you support and has all the required characters. Use a Unicode-based encoding such as UTF-8, because it supports many languages and can accommodate pages in any mixture of those languages. Services like Google Fonts and Type Kit all allow you to test a typeface before using it in your design.
Quick tip: Note that some fonts may need a different line-height. For example, Chinese characters are visually more complex than Latin characters, meaning they might need more space for good readability.
3. Create a flexible layout
Expansion of the length of text strings is quite common when translating them into other languages—certain languages are more 'wordy' and require more space. For example, a label that says 'Submit' might be translated in French to 'Soumettre'. The English version consists of 6 characters, the French version 9, taking up 30 percent more space.
Below are a few tips to follow to allow easier text expansion:
Make containers responsive. Ideally, you shouldn't have containers with fixed dimensions.
Place labels above fields in forms. In comparison with left-aligned labels, labels above the fields give you more space and make it much easier to accommodate the lengthy character strings.
- Design for 'Left to Right' and 'Right to Left.' The script the language is written has a direction. Most languages use scripts which are written and read from left to right, but some scripts, such as Arabic, are written right to left. If you need to support right to left, the layout you design should support mirroring. You should be able to mirror the layout of the whole page—text, images, buttons, and navigation—should all be mirrored.
4. Consider local formats
Avoid hardcoding units in the user interface. You need to adjust the following units based on region and user preference:
Dates. Date format varies between countries, so it's important to be aware of the forms of all-numeric calendar dates used in every particular country.
Measurements. Be aware of the measurement system each country uses.Users in the United States typically expect to view measurement data in the imperial system (pounds, inches, and feet), while users in Europe expect to get the information in the metric system (grams and meters).
Phone numbers. Country codes and the number of digits in a phone number vary depending on the country.
Currency. Having the right currency is especially important for ecommerce apps and websites. When users view an item, they expect to see prices in their native currency. Every time users see a foreign currency, they have to spend extra time converting it to their own.
- Social media buttons. People around the world use different social media services.
5. Ask users to select a language
One of the things that differentiates good products from bad is the user's ability to complete the task with the least possible amount of effort. Common sense tells us that when it comes to choosing a language for our users, we should strive to auto-detect user location and set the language automatically.
But there are a few cases when this solution doesn't work properly—for example, the user might travel to different locations, so if the app or website hides the language option, they won't be able to use it—it'll be in an unfamiliar language.
If you design a website, you should give the user the ability to change languages. Present the language selector at the top of the page (the area where the user will search for it).
If you design a mobile app, it's safer to ask users to confirm the language selection during onboarding.
Quick tip: Make sure you correctly label languages. Many designers use flags for this purpose, but this isn't ideal, since flags represent countries, not languages. And one country can have two or more official languages.
Adapt the language of your product
Solving technical issues is just half of the battle. To create a great multilingual product, you need to look beyond code and focus on the language you use. Below are eight tips to help make translation and localization easier.
1. Research your target audience
Chances you already know who your English speaking users are. But do you spend enough time researching your target audience in the new region? To adapt your product for that audience, you need to know what language they speak and how they expect to be spoken too. And you only get that kind of information by researching your demographics.
Quick tip: Be sensitive to cultural customs and social taboos. Countries have differing ethical views, and humor and symbolism does not always cross borders. For example, the thumb up gestures that means 'great' for U.S. users, but in the Middle East it means 'Up yours!'
2. Avoid long sentences
Long sentences are hard to understand, both for your target audience and translators. It's always better to use short and catchy sentences. Ideally, each sentence should be up to 25 words.
3. Use simple language in your products
People interact with app and web user interfaces through language. Every time you introduce slang or idiomatic phrases into your copy, you sacrifice one important property of your UI: clarity. Users who are not able to decode the message easily become confused. But you also make the work of your translator much harder—not only do they need to be familiar with the slang, but they also have to translate it to a different language without losing the original meaning, which can be very difficult.
By choosing to use simple language, you create translation-friendly content.
Tip: If you have to use specific terms or phrases in your app that require a specific translation, it's best to add them into a separate document. This document should contain the terms, their translations, and an explanation or definition. By providing contextual information, you make it much easier for translators to understand the meaning.
4. Don't rely solely on machine translation
It might be tempting to use machine translation such as Google Translate to increase the speed of translation, but it better to resist that temptation. Tech giants such as Google have made exciting advances in auto-translation, but the tool they've created still can't compete with humans—the accuracy of the translated text varies significantly. In many cases, machine translation will be inaccurate, unnatural, and error-prone. A poorly translated product can easily lead to frustration.
For this reason, it's a good idea to get a human translator, preferably an in-country native speaker, since native speakers have a better knowledge of the subtle nuances of language.
5. Adapt visual elements
The language you use in your product should be adapted to account for cultural nuances and differences. Words are only one part of the translation equation; visual elements like colors and icons should also be adapted.
The meaning of icons and colors can vary depending on the culture. It's important to follow a few simple rules to be sure your UI works for your users:
Try to use icons with universal meaning. Many icons are now universally recognizable, such as those for home, play. or printer, which do not require text labels to convey their meanings. Test your icons to make sure they have the same meaning in all of your supported languages.
- Research color meaning. Colors have different meanings in different cultures. For example, red means different things for users in China and users in the Western world. While Westerners often associate red with danger, users in China associate red with good luck.
6. Strive for consistency
The language you use in your app should be consistent. Terms, icons, and colors should all be used consistently throughout your product.
This requirement is especially important when a few visual designers and content managers work on the product—even when each designer or content manager has their own style, the final product should sound like the same person created the entire design and text.
7. Create editorial guidelines
The English version of the product likely has a distinct voice, and you probably want your content to have that same voice and tone in any language. To make that happen, you need to create editorial guidelines for translators. People who work on localizing your product should follow the guidelines to keep your branding consistent.
8. Always test your design
'Test early, test often' is a fundamental requirement for good product design. When it comes to testing your localization, you need to hire people who represent the target audience. Ideally, those people should interact with the team regularly, so the product team can get regular feedback from them during the localization process.
You might also like: How To Use Translation Keys To Create Internationalized Themes.
Building for global markets
As our world becomes more and more connected, failing to adapt your product to people from different regions undermines your long-term ability to stay competitive. I hope that the tips mentioned in this article will help you create better multilingual products.
What tips do you follow to create digital products for international markets? Share your experience in the comments below!


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Many thanks to Tristan Morris for creating a beautiful illustrated hardcover print edition of the site |
(Sorry, this page has not yet been translated into the requested language.) At precisely 7:07 on the first clear morning of the first week of spring, the old scribe Qi awoke to a knocking on his door. An abbess waited outside, flanked by two heavily-armed guards in black and white robes. Said the abbess: "We have read your account of the burning of Zjing's bridge, and we judge it to be a deliberate lie from beginning to end. The annals are sacrosanct: by falsifying them, you have forfeited your own head." The scribe replied: "What was written, was, and is." Said the abbess: "Yet the bridge stands undamaged, and all parties refute your account. As proof of your willful deceit I offer these printouts of the various versions of the page in question, obtained from your own repository." The scribe said nothing. Said the abbess: "The first version contains only the text of an email. The sender suggests that you author a case wherein a stubborn monk is corrected by finding himself halfway across an impassible bridge." The scribe nodded and replied: "One point oh." Said the abbess: "The next version gives an elaborate account of master Suku and the monk Landhwa. Here, Suku corrects the monk by unfastening the planks on her side of bridge, much as the email suggests." The scribe nodded again and replied: "One point one." Said the abbess: "A later version changes the monk to Wangohan. Here, Suku releases a basket of trained cobras on the bridge to slither towards the hapless monk. This one is my favorite, for its elegant use of snakes throughout." The scribe bowed and replied: "One point four." Said the abbess: "The final version has master Suku reduce the bridge to ashes. A most obvious falsehood." The scribe replied: "One point six." Said the abbess: "Can you speak a word which will convince me to let you keep your head?" The scribe said nothing, but calmly sat at his desk and took up his keyboard. Behind him, the guards unsheathed their swords. At precisely 7:07 on the first clear morning of the first week of spring, the old scribe Qi awoke to the chattering of a magpie in the tree outside his window. Through the branches, the cold, charred remains of Zjing's bridge could be seen. Said the scribe to the magpie: "One point six point one." An excerpt from The Codeless Code, by Qi (email@example.com). Provided under the Creative Commons Attribution-NonCommercial 3.0 Unported License. |

subset: commoncrawl


text: As you may know, I'm just now coming out of several weeks of horrible pregnancy nausea/vomiting. Food has been my arch-nemesis. I felt sick if I didn't eat. I felt sick if I did eat. The thought of food was just unbearable. Now that my cookbooks aren't vomit triggers and I can enjoy them again, I took an afternoon getting to know the beautiful new edition of my friend Stephanie Cornais's From … [Read more...]


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If I get a protection order, will it show up in an internet search?
According to federal law, which applies to all states, territories, and tribal lands, the courts are not supposed to make available publicly on the internet any information that would be likely to reveal your identity or location. This applies to all of these documents:
- the petition you file;
- the protection order, restraining order, or injunction that was issued by the court; or
- the registration of an order in a different state.1
1 18 U.S.C. § 2265(d)(3)


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Previous Lesson
Complete and Continue
You're Not Crazy. You're Not Alone.
You're Not Alone. You're Not Crazy. Betrayal Trauma from a Bonding Science Perspective
Journal Prompts
Intro (6:27)
What is Betrayal Trauma? (5:59)
It's Not Your Fault (7:54)
Bonding Science (7:03)
PTSD Responses (10:54)
Reactive Interactive Pattern (10:10)
How to Build Back Trust (11:34)
How Do I Know When to Be Done? (9:04)
For Those That Want to Leave (9:07)
For Those That Want to Stay (7:16)
How Emotion is Processed (18:15)
What Can I Expect Long Term? (7:39)
Negative Interactive Pattern (9:34)
Loss of Identity (6:52)
When a Good Memory Turns Bad (5:31)
Nightmares (5:47)
What to Tell Kids? (5:13)
Who to Tell and What to Share? (7:45)
What About Sex? (6:00)
Notice Your Body (7:50)
To Not Exist in Your Partner's Eyes (3:48)
There's No Lightswitch for Trust (6:19)
What If It's Me? (5:23)
The Range of Emotion After Discovery (4:18)
Finding Out vs Being Told (4:49)
Disclosure in a Process Way (8:09)
Checking Behavior (4:45)
The Search For Answers (4:35)
When Your Guard is Down (7:43)
Anniversaries (4:30)
Villianizing Your Partner (6:20)
Grounding Techniques (10:32)
Grieving the Relationship You Thought You Had (3:54)
What to Expect in Couples Therapy? (9:15)
How to Choose A Couples Therapist (8:03)
What Does Healing Actually Look Like? (5:39)
Closing Video (2:00)
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From IPT:
The head of the Council on American-Islamic Relations (CAIR) vows to complain to U.S. Department of Homeland Security officials after they blocked a radical Saudi cleric from entering the country this week to attend a national Islamist conference in Chicago.
Sheikh Ayed al-Qarni was scheduled to speak twice during the Muslim American-Society (MAS)/Islamic Circle of North America (ICNA) national convention Dec. 22-25. But a statement released during the convention expressed "the unpleasant and saddening news" that al-Qarni had been removed from his flight from Saudi Arabia despite having a visa from the U.S. embassy, and that he appears to be on the U.S. "no-fly list." Al-Qarni is described as "one of our great speakers" and as someone known "for his logical discourse and balanced views, he promotes understanding and collaboration between all people, regardless of their faith, background, or language."
Al-Qarni has advocated jihad in the past and his preaching on the subject has been described as influential among al-Qaida followers.
CAIR Executive Director Nihad Awad told an Arabic news outlet that he would protest al-Qarni's exclusion with DHS and State Department officials. "We defend all Muslims who are subject to arbitrary measure, and by this logic, we will act but not formally plead, unless we obtain authorization from him."' [...]
Through Awad, also a listed speaker at the convention, CAIR is fighting to bring a Saudi cleric into the United States who has argued the exact opposite message. During a 2005 sermon flagged by the Middle East Media Research Institute (MEMRI), al-Qarni called the jihad against American forces in Fallujah "a source of pride … downing their planes, destroying equipment, slaughtering them, taking them hostage, and proclaiming 'Allah Akhbar' from the mosques, and the worshippers and the preacher cursing them in their prayers, and then come others begging for forgiveness, and requesting a dialogue and a ceasefire and negotiations. Who can say even one word against this true Jihad against these colonialist occupiers?"
He belittled Muslims who failed to take action, including "harming the Jews." He invoked Israel's targeted killings of Hamas leaders Ahmed Yassin and Abdel-Aziz Rantisi, saying he prayed that Allah "will destroy the Jews and their helpers from among the Christians and the Communists, and that He will turn them into the Muslims' spoils. I praise the Jihad, the sacrifice, and the resistance against the occupiers in Iraq. We curse them all of them every night and pray that Allah will annihilate them, tear them apart, and grant us victory over them…"
Their ideas are evil. They are anti-American. It is time for them to go.


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And now, what we've all been waiting for: the Ten-Tec Argonaut VI has finally hit the market. Manufacturer Ten-Tec has already begun shipping the new units–I hear they've already sold out the first production run. For the past two months, I have had the pleasure of beta-testing this newest QRP transceiver, and I'm ready to share my findings. [Do please note that, other than beta-testing, I have no relationship with Ten-Tec.]
I authored a post about the Model 539 when Ten-Tec first disclosed it at their 2011 Hamfest. The reactions and questions from readers came flooding in–so many, in fact, that I invited readers to send in those questions to share with the engineers at Ten-Tec. I presented these to the company, and posted Ten-Tec's helpful responses.
In truth, I don't think that Ten-Tec was quite prepared for all of the interest in their modest QRP transceiver. But it was no surprise to me: I've always been a fan of Ten-Tec, and although I've not been as excited by the QRP offerings since the early Argonauts, I knew I wasn't alone in my appreciation of this US-based radio company's quality products.
The following review is not a test-bench review–it is, rather, a consideration of the usability, ergonomics, design, basic performance and, well, fun factor of the new Argonaut VI. It's only fair to note that I don't review transceivers often; rather, I focus primarily on receiver reviews at my alternate radio blog. But I could not resist the opportunity to investigate the newest in this venerable line of transceivers.
You will note that I compare the Argonaut VI to the Elecraft K2 a number of times. Why? In my opinion, the K2 is the Argonaut VI's closest competitor. It, too, is a front-panel QRP transceiver not for general coverage. While there are a number of differences, of course, I nonetheless feel the K2 is a closer match than the new Elecraft KX3, the Yaesu FT-817, or the Icom IC-703. Plus, I have a K2 that I already know and love here in my shack, so by default it has been my point of comparison throughout the beta-testing period.
Argonaut VI: first impressions
The Argonaut VI is an attractive, simple, sturdy little radio. It reminds me a great deal of the Ten-Tec Scout outfitted with its simple front panel. The front features two knobs: one controls the AF gain, while its outer ring controls RF gain; the other controls the bandwidth, while its outer ring controls the pass band. There is also an appropriately-sized display panel, quality tuning knob (see below), four multi-function buttons, and a three-position toggle switch.
A toggle switch? I can't think of a recent front-panel radio in production that has had a proper mechanical toggle switch. On the Argo VI, this makes for a simple method to give the four function buttons a total of three one-push functions, each, for a total of twelve functions. Ten-Tec refers to this switch as the "TMB" (i.e., "Top-Middle-Bottom") switch.
Size-wise, the Argonaut VI is smaller in every dimension than the K2 (see photos). Its physical dimensions are 2.25″H x 6.5″W x 7.6″D, less the knobs and connectors. It weighs a mere 3.6 lbs, and feels very light in my hands. The Argo VI has a sturdy Ten-Tec bail that snaps into the perfect position for tabletop operations. The display is crisp and clear, and actually contains a lot of information:
- Frequency
- 2nd VFO frequency
- Mode
- AGC speed
- S Meter/SWR
- Output power
- Pre-amp
The display can be switched (via an internal setting) to blue (default), green, or red. One nice touch: the dot in the Ten-Tec logo is actually a red LED that lights up on transmit and ALC peaks.
Tuning knob
Perhaps I place more emphasis on a tuning knob that other hams. I liken it to shutting the door on a quality car: you want the door to shut solidly and feel substantial. But it may be more like a car's steering wheel–after all, the tuning knob is how one interacts with the radio. To me, the tuning knob is often a measure of a radio's overall quality, in my humble opinion. As for the Argonaut VI? Here's the answer: I was so impressed with the tuning knob on the Argonaut VI that I actually confirmed with Ten-Tec that the beta-unit's tuning knob would also be used on production models. In short, the Argo VI's tuning knob is heavy, perfectly-sized, has a light tactile grip, and is silky-smooth to operate. There is no play whatsoever in the action. I like the adaptive tuning, too–when you tune slowly, you're changing the frequency by hundredths of a kHz; spin the knob quickly and you've just shot across the band. After tuning the Argo VI for a bit, other small radios' tuning knobs begin to feel cheap.
When I first played with the Argo VI at the Ten-Tec hamfest, I was impressed with the simplicity of the front panel. This is an important factor because I simply won't use a radio that isn't pleasant to use/control, and I find that too many front buttons and general visual fussiness can be a distraction. To illustrate my point, when the Yaesu FT-817 hit the market over a decade ago, I was among the first to purchase one. I liked the idea of a small transceiver that I could tuck in my carry-on and take with me as I traveled. But I ended up selling the FT-817, however, because I hated the ergonomics and multi-function buttons. Button spacing was too tight for my larger hands, important multi-functions seemed to overlap, and menus were buried too deep for convenient operation. The FT-817 had a profound impact on all other buying decisions I've made since, and taught me that too much can simply be…too much.
Happily, on the Argo VI the most often used transceiver functions have dedicated buttons/knobs, and the display includes everything I need. Clean, clear, straightforward–this rig provides a pleasant operating experience.
For basic operations like rag-chewing, scanning the band, switching modes, switching bands, adjusting RF/AF/BW and PBT, you'll be pleased, too. None of these operations require calling multi-functions or toggling the TMB switch (assuming you're already in the "M" position).
But after spending some time on the air, I realized that there is a bit of a learning curve you'll have to overcome before front-panel operations become entirely fluid and intuitive. To change RIT, you need to toggle the TMB switch to "B", then press the RIT (BAN) button to toggle RIT on and off. If you hear DX working split, you'll need to move the TMB switch to the "T" position, then set the A/B and SPL buttons; if you need to change modes or turn on the pre-amp, then you'll have to move TMB back to "M."
In the first few hours of testing the Argo VI, I found it easy to forget that I had the TMB switch set to a certain position when pressing a multi-function button, thus I was sometimes not receiving the response I expected. Several times I intended to change the band, but had the TMB set to "T" and resultingly opened the output power setting, or pressed the MOD button only to find that TMB was set to "T" as I toggled A/B VFOs.
While this was distracting at first, I soon became accustomed to changing the setting, then moving the TMB back to the "M" setting afterward. Now I find I very rarely make a mistake.
On the Argo VI, all of the buttons and knobs are adequately spaced. You could operate this rig outside with lightweight insulated gloves on, should the need arise.
All in all, the ergonomics are excellent on the Argonaut VI.
Before I begin talking about this little transceiver's performance, I want to point out its two most obvious shortcomings:
- The Argonaut VI lacks 12 and 60 meters (ouch!)
- There is no internal ATU (auto antenna tuner), nor is there an option for one
If those two negatives are deal-breakers for you, I could certainly understand. You might want to consider a basic KX3 with ATU ($1070 unassembled, $1170 assembled) or a K2/10 with ATU ($1280 unassembled), instead.
But if, like me, you use neither 12 nor 60 meters very often, you may not miss them. Admittedly, the lack of 12 meters is unfortunate because it's such an ideal band–when conditions are right–for easy QRP field operation. However, I am very pleased the Argo VI has 160 meters.
An internal ATU–or the option to have one installed later–is certainly a negative for those of us who like a simple Field Day radio set-up. From my point of view, other than my Elecraft KX1, I've never had a radio with an internal ATU; I have two portable tuners (the LDG Z11 Pro and Elecraft T1) that work wonders. The way my shack is designed, I have a remote auto-tuner outside at the feed point of my antennas and thus have no tuner in my shack–so if I had an internal ATU, I'd have to turn it off 95% of the time. If your shack is set up similarly, you might not mind not having an ATU.
If you're concerned about the performance of the Argonaut VI, let me assure you now: you will not be disappointed. Indeed, the receiver in the Argonaut VI must be one of the best I've ever heard in any radio—especially in this price class ($1000). It is truly remarkable. I'm eager to learn how Rob Sherwood rates the Argonaut VI, but I suspect it will rank among the top few.
What is most impressive in the Argo VI is its incredibly effective variable DSP filtering. I experimented with the variable bandwidth and pass band during crowded CW conditions, and found that each and every time I could zero in on one QSO and block everything else. It's also highly effective when used with SSB. Based on the reviews I've read of the Eagle, this is obviously derived from its DSP architecture and has similar performance characteristics. [Future Argo VI owners, I eagerly welcome A/B comparisons of the Eagle and Argo VI–please comment!]
You'll be happy with both the Argo's sensitivity and its ability to reject adjacent signals. Compared with my Elecraft K2/10, the Argo VI's sensitivity had an edge in every band I tested, and to my ear, the noise floor is lower on the Argo VI as well. Most noticeably, however, is the Argonaut's audio fidelity, which is far superior to that of the K2. Whether using headphones or using the built-in top mounted speaker, you will be pleased. The speaker delivers an impressive sound for its size. I tend to hook up external speakers to my smaller transceivers, but in this case I never felt I needed to. With headphones, the audio is even more impressive. I do wish the headphone jack was on the front panel instead of the back, though.
Though I haven't spent enough time with the KX3 to compare audio fidelity, I imagine the KX3 and Argonaut VI would be a fair contest.
I can say that audio fidelity is the primary reason I continue to turn to Ten-Tec for transceivers and receivers. In my opinion, like Kenwood, Ten-Tec invests more resources into insuring superb audio fidelity–even at the cost (in this case) of uber-low current drain numbers (although the 550 mA drain on receive must be the lowest Ten-Tec has ever produced in a digital transceiver). The Argo VI's audio is rich, inviting enjoyable listening for hours on end. It would certainly be a great pick for long-haul events like Field Day or 24/48-hour contests.
But how does she sound on the other end? Immediately after unpacking the Argonaut VI, I caught band openings on 10, 15, 17 and 20 meters. Though I was only running 7 watts at the time (production units run a full 10W) I received great audio reports on SSB and was even heard through a pile-up on 17M. Though I believe the default settings would have worked well, setting up the mic gain on this rig is also very easy and straightforward.
As for CW, reports have also been very positive. CW ops will be happy to note that the Argonaut VI has Ten-Tec's silky-smooth QSK. Frankly, I expected nothing less.
Since I don't operate digital modes often, I did not test the Argo VI in this capacity. I imagine reviews will emerge soon, but I expect them to be positive as several beta testers were impressed.
When I begin a radio review, I keep a checklist of pros and cons as I discover them to remind myself of my initial discoveries. Here's my list from the Argonaut VI:
- Excellent top-of-the-line audio fidelity
- Extremely effective DSP variable bandwidth
- Silky smooth QSK
- All mode with optional AM
- Easily accessible primary controls
- Uncompromised ham band performance (see con)
- Quality tuning knob and adaptive tuning rate make for easy band scanning
- Simple front face and comfortably spaced buttons/knobs (see con)
- Undoubtedly the best receiver of any QRP rig produced by Ten-Tec
- Comprehensive and detailed owner's manual
- Made in USA
- Ten-Tec's US-based customer support, a major plus over many foreign manufacturers
- No 12 nor 60 meters
- No internal ATU, nor option for one
- No internal battery nor option
- Headphone jack on rear panel
- Not general coverage (see pro)
- Learning curve when using multi-functions (see pro)
- Price of $995 is a little steep
- RX current drain is high when compared with Elecraft K1/KX1/K2/K3 or KX3, or the Yaesu FT-817
To buy, or not to buy…
I really think the Argonaut VI is a streamlined Ten-Tec Eagle, and a very good rig. Its DSP architecture is based on the Eagle's (in beta, we even used the Eagle software for the frequent firmware updates).
When I ask myself, "Who will buy the Argonaut VI?" I believe the answer is anyone who wants a QRP radio with the performance and interface we've come to expect from Ten-Tec. If you can live without 12 and 60 meters, then you will be buying a rig that does not compromise on performance. The Argonaut VI is not a QRP radio designed for backpacking like the KX1 or KX3, but it would hang with the best in a QRP contest or on Field Day; operators would experience little to no listening fatigue with this smooth rig.
The Argo VI comes factory-assembled, warrantied, and ready to go, right out of the box. There's nothing to put together nor configure.
On a side note, I should mention that this is the first time I've beta-tested with Ten-Tec; this provided insight into the process of rolling out a new product and just how responsive and open the company is to both frank criticism and design requests. I believe all of the beta-testers would agree. Every concern I reported to Ten-Tec was eventually addressed by version 1.0 of the firmware. That's responsiveness I can really appreciate. Ten-Tec's Software Engineer, John Henry, is nothing short of amazing. I'm not sure when he finds the time to sleep.
I'm happy to have had this little radio in my shack for the past two months. I'll have to bid it farewell, though: Ten-Tec has already sold out of their first production run, so my unit will be returned, upgraded with a new board, tested, inspected, and sold as a demo. Alas, it's hard to say good-bye.
I guess the proof is in the Christmas pudding: I have to admit that I now prefer the Argonaut VI even over my trusty Elecraft K2. So, kind readers, if one of you is in the market for a K2/10 with SSB, and 160M, I may be in the market for an Argonaut VI.
It would bring tidings of great joy, indeed. Happy holidays!


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To me, all yeast doughs are "magical," but there's no real magic to sourdough. Just wild yeast and lactobacilli doing their thing in their own sweet time, it's more art than science on the baker's part. As they say, it isn't rocket surgery!
I've received so many questions about sourdough that I thought it was time to write another post about it. Time to dispel the myths! Really, nearly everything I've ever read on the subject makes it seem way more complicated than it is.
If you are an experienced sourdough bread baker, you might be having wonderful success with entirely different methods than mine. If so, just keep on doing what you're doing. But if you're new to sourdough bread (true sourdough bread made with a wild yeast starter and no added commercial yeast), this post might be just what you need.
First of all, here's How to Make Your Own Sourdough Starter. And it doesn't require you to invest in a fancy crock or imported flour. You don't have to wait until grapes are in season or until you remember to pick up some distilled water.
Just do it!
You don't have to wait until you buy a digital kitchen scale (though it is a wonderful thing to have for many things). People sending me questions have quoted experts who give extremely precise directions for making a starter.
Not necessary!
And you don't have to discard huge amounts of your starter. People sending me questions have pointed me to experts who would have them regularly discarding starter--both while they're starting the starter and when they're using the starter.
Not necessary!
Now then, after a week or two, you have a nice bubbly starter. How do you keep it going? Do you have to get a babysitter for it when you go on vacation?
Not necessary!
Since I like to bake sourdough bread twice a week, I could probably just leave my starter on the counter; but I keep it in the refrigerator. If I'm not baking bread for a while, I have had the habit of taking it out and feeding it once a week as mentioned in my How to Make Your Own Sourdough Starter post. But guess what?
Not necessary!
Once every three weeks would be good, but we had an incredibly long spell of heat and humidity that killed my interest in anything to do with the oven. And I totally ignored my little half-pint jar of starter for months as it gradually migrated to a back corner of the refrigerator. When I finally revived myself and remembered the starter, I was sure it would be past reviving.
Not true!
Sure, it had separated into a couple layers and looked a little down on its luck, but I thought I should at least give it a try before I tossed it and started over. Hey, if I could be revived, anything could! I just took out a couple teaspoons of it and put it into a 2-cup glass measure and started feeding it. When I'm baking regularly, it just takes two feedings; but I fed this next-to-dead starter twice a day for a few days (small amounts, with NO discarding) until it was bubbly.
It was just fine!
You'll find your own rhythm!
So I guess the moral of the story is: Just do it! Sourdough seems to have a mystique about it that some might enjoy but that holds people back from giving it a try.
Don't be discouraged if your first bakes aren't completely successful. It takes a while to know your starter and your dough.
Just do it! And let me know if you have any questions.
PS Here's my recipe for Sourdough English Muffins.


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Into the Field: Andrew Zawacki
I meet Andrew Zawacki in this episode of Into the Field, recorded on the campus of the University of Georgia in Athens. Zawacki teaches in the creative writing program at UGA, and holds degrees from the College of William and Mary, Oxford, the University of St. Andrews in Scotland, and the University of Chicago. His books of poetry are By Reason of Breakings (University of Georgia Press, 2002), Anabranch (Wesleyan University Press, 2004), and Petals of Zero Petals of One (Talisman House, 2009). Zawacki's writing has appeared in The New Yorker, The Nation, and The New Republic, as well as in many anthologies and journals. We talk about his ambivalence toward his role as a "professional" poet, and discuss what he's learned from his students over the years. The show begins with a reading from his long poem "Georgia," which explores his sense of cultural alienation after moving to Athens in 2005.
You can find more of Andrew Zawacki's poetry and criticism on his website.


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Presentation Days for TAE40116
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A two day block either face-to-face or online, where eligible students enrolled in the Cert IV TAE (TAE40116) complete their final presentations and assessments.
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Presentation days are held over a two day block (approximately 8 hours per day), with each student delivering 2 x 40 minute presentations. In addition, students must also conduct assessments throughout the presentation day blocks.
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The presentation day block is included in the course fee for Certificate IV in Training and Assessment (TAE40116.
The need for additional training or assessment arising from students who are deemed not yet competent or who fail to attend their scheduled session may incur additional charges.
This is the final assessment piece for your Certificate IV in Training and Assessment (TAE40116).
Irrespective of your chosen delivery format, your presentation must demonstrate competency against the following criteria:
- Organised resources and set up training environment ready for delivery
- Conducted delivery of training program (minimum of 2 x 40 minutes per session)
- Used effective communication skills, group facilitation skills, presentation skills and interpersonal skills to convey learning outcomes, information, instruction, demonstrate skills and make learners comfortable and engaged in learning
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If the answer to any of the above questions is "no", we recommend attending one of our face-to-face sessions. Students who elect to undertake their presentations via Virtual Classroom must agree with all terms and conditions pertaining to this mode of delivery.
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I completed a Cert 4 TAE40116 with Pinnacle. The trainer Mary-Ann and training material were excellent. It was a really intensive course and even though we were all mentally exhausted at the end of the 10 days I can genuinely say that I came away with a high degree of knowledge and confidence in moving forward and applying all I learned in this course. Once again two thumbs up to Mary-Ann!


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Mary Pope Osborne
The #1 bestselling chapter book series of all time celebrates 25 years with new covers and a new, easy-to-use numbering system! Jack and Annie are headed to a land of fierce samurai and great beauty, the capital city of Edo (now the city of Tokyo), in ancient Japan in the 1600s. They bring only a research book to guide them and a magic wand with three special rules. Formerly numbered as Magic Tree House #37, the title of this book is now Magic Tree House Merlin Mission #9: Dragon of the Red Dawn.Did you know that there's a Magic Tree House book for every kid? Magic Tree House: Adventures with Jack and Annie, perfect for readers who are just beginning chapter books Merlin Missions: More challenging adventures for the experienced reader Super Edition: A longer and more dangerous adventure Fact Trackers: Nonfiction companions to your favorite Magic Tree House adventures


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superalgebra and (synthetic ) supergeometry
The Green-Schwarz action functional is an action functional for a sigma-model that describes the propagation of a fundamental super -brane in a super spacetime supermanifold.
For this is the Green-Schwarz superparticle.
For this is the Green-Schwarz superstring (Green-Schwarz 84)
For this is the Green-Schwarz supermembrane (Bergshoeff-Sezgin-Townsend 87)
The Green-Schwarz model of the superstring is in contrast to the NSR-string model (the original spinning string), which has manifest worldsheet supersymmetry but no manifest spacetime supersymmetry. It is a non-trivial theorem that the spectrum of the NSR-string enjoys spacetime supersymmetry (after GSO projection) and may hence be identified with perturbative excitations of a supergravity background. The construction of the Green-Schwarz functional was motivated by the desire to find an equivalent alternative formulation in which spacetime supersymmetry is manifest (see e.g. Schwarz 16, slides 24-25).
manifest supersymmetry for brane sigma-models:
manifest worldvolume supersymmetry | manifest target+worldvolume supersymmetry | manifest target space supersymmetry |
NSR action functional | superembedding approach | Green-Schwarz action functional |
graphics grabbed from FSS19c
For more discussion see also at geometry of physics – fundamental super p-branes.
Perturbativestring theory on geometric backgrounds is defined by the Neveu-Schwarz-Ramond model, namely by sigma-model 2d super conformal field theories (of central charge 15) on worldsheets that are super Riemann surfaces, with target spaces that are ordinary (i.e. "bosonic") spacetime manifolds.
These worldsheet field theories are induced from action functionals, namely from variants of the standard energy functional (Polyakov action) on the mapping space of smooth functions
from the worldsheet to target spacetime .
The central theorem of perturbative superstring theory (the no ghost theorem with GSO projection) says that the excitation spectrum of such a 2d SCFT are the quanta of the perturbations of a higher dimensional effective supergravity field theory on target spacetime, hence transforms under supersymmetry on target spacetime.
This is the fundamental prediction of the assumption of fundamental strings:
assuming that the fundamental particles that run in Feynman diagrams are fundamentally (at high energy) the ground state modes of a fundamental string,
demanding that there are fermionic particles among these,
that the string must be the spinning string (have fermions in its worldsheet theory), which in turn implies…
that it is the superstring (worldsheet supersymmetry mixes the worldsheet bosons and fermions), precisely: the Neveu-Schwarz-Ramond superstring, which then in addition implies…
that its target space effective field theory is a supergravity theory, hence that also the effective target space fields exhibit local supersymmetry (i.e. "high energy supersymmetry", different from "low energy supersymmetry" that the LHC was looking for).
main theorem of perturbative super-string theory |
The first step in this implication (identifying the spinning string as the superstring) is fairly straightforward (in fact this is how the concept of supersymmetry was discovered in "the west", in the first place), but the second step (that the superstring excitations necessarily are quanta of a spacetime supergravity theory) appears as a miracle from the point of view of the Neveu-Schwarz-Ramond superstring. It comes out this way by non-trivial computation, but is not manifest in the theory.
In order to improve on this situation, Michael Green and John Schwarz searched for and found (Green-Schwarz 81, Green-Schwarz 82 Green-Schwarz 84, for the history see Schwarz 16, slides 24-25) a suitably equivalent string action functional that would manifestly exhibit spacetime supersymmetry. Acordingly, this is now called the Green-Schwarz action functional.
action functional for superstring | manifest supersymmetry |
Neveu-Ramond-Schwarz super-string | on worldsheet |
Green-Schwarz super-string | on target spacetime |
The basic idea is to pass to the evident supergeometric analogue of the bosonic string action:
Let be a closed manifold of dimension 2 – representing the abstract worldsheet of a string. Let be a pseudo-Riemannian manifold – representing a purely gravitational spacetime background. Then the action functional governing the bosonic string propagating in this spacetime is the functional
on the smooth mapping space (of smooth functions ), that simply assigns the proper relativistic volume of the image of the worldsheet in spacetime:
(This is the Nambu-Goto action. It is classically equivalent to the Polyakov action which is the genuine starting point for the quantum Neveu-Ramond-Schwarz super-string. However, since, as we discuss below, the Green-Schwarz action naturally generalizes to that of other p-branes it is more natural to consider the Nambu-Goto form of the action here.)
When here is generalized to a superspacetime supermanifold with orthogonal structure encoded by a super-vielbein , then the same form of the action functional still makes sense and produces a functional on the supergeometric mapping space . Moreover, by construction this action functional now is invariant under the superisometry group of , hence under global spacetime supersymmetry.
However, Green and Schwarz noticed that this kinetic action functional does not quite yield dynamics that is equivalent to that of the Neveu-Schwarz-Ramond super-string: when the equations of motion hold ("on shell") it has more fermionic degrees of freedom than present in the Neveu-Ramond-Schwarz super-string. The key insight of Green and Schwarz was that one may add an extra summand (whose notation we explain in a moment) to the plain super-Nambu-Goto action , such that the resulting action functional enjoys a further 1-parameter symmetry, called kappa-symmetry. This is the Green-Schwarz action functional:
Moreover, they showed that restricting the dynamics of the Green-Schwarz superstring to the -symmetric states, then it does become equivalent, classically to that of the Neveu-Ramond-Schwarz super-string.
Finally they showed that when gauge fixing the Green-Schwarz action functional to light-cone gauge (which is possible whenever target spacetime admits two lightlike Killing vector) then the Green-Schwarz string may be quantized by a standard procedure and the resulting quantum dynamics is equivalent to that of the Neveu-Schwarz-Ramond super-string. This provides the desired conceptual proof for the observed local target spacetime supersymmetry of super-string effective field theory, at least for backgrounds that admit two lightlike Killing vectors. (The quantization of the Green-Schwarz superstring away from light cone gauge remains an open problem.)
While Green-Schwarz's extra kappa-symmetry term this serves a clear purpose as a means to an end, originally its geometric meaning was mysterious. However, in (Henneaux-Mezincescu 85) it was observed (expanded on in (Rabin 87, Azcarraga-Townsend 89, Azcarraga-Izquierdo 95,chapter 8)), that the Green-Schwarz action functional describing the super-string in -dimensions does have a neat geometrical interpretation: it is simply the (parameterized) Wess-Zumino-Witten model for
target space being locally super Minkowski spacetime regarded as the coset supergroup
for a real spin representation (the "number of supersymmetries"), the corresponding super Poincaré group and its Lorentz-signature Spin subgroup;
WZW-term being a local potential for the unique (up to rescaling, if it exists) -invariant super Lie algebra 3-cocycle on the super Poincaré Lie algebra , with components locally given by the Gamma-matrices of the given Clifford algebra representation; in terms of the super vielbein :
and so in components the bi-fermionic component of is
and all other components vanish.
More in detail, just as ordinary Minkowski spacetime may be identified with the translation group along itself, with canonical linear basis of left invariant 1-forms given by the canonical vielbein field
where are the canonical coordinates on , so super Minkowski spacetime for some real spin representation is characterized as the supergroup whose left invariant 1-forms constitute the -bigraded differential with generators the super-vielbein
where are the canonical coordinates on , with the odd-graded elements spanning the given real Spin(d-1,1)-representation with Clifford algebra generators .
Now while ordinary Minkowski spacetime is an abelian group, reflected by the fact that its left-invariant 1-forms are all closed
the key phenomenon of supersymmetry (that two fermions pair to a bosons) means that is slightly non-abelian, reflected by the fact that the super-vielbein is not closed
This elementary effect is the source of all the rich structure seen in the Green-Schwarz super-string and generally in all super p-brane theory. (The above differential is equivalently that in the Chevalley-Eilenberg algebra of super Minkowski spacetime, hence its cohomology is the super-Lie algebra cohomology of super Minkowski spacetime. In parts of the physics literature this is referred to a "tau cohomology".)
In particular, for special combinations of spacetime dimension and number of supersymmetries (i.e. real spin representation ) then the 3-form
is a non-trivial super Lie algebra cocycle on , in that and so that there is no left invariant differential form with (beware here the left-invariance condition: there are of course non-left-invariant potentials for , and in fact these are exactly the possible Lagrangian densities for the WZW action functional ).
This happens notably for
and (heterotic string)
and (type IIB superstring)
and (type IIA superstring).
(It also happens in some lower dimensions, where however the corresponding Neveu-Schwarz-Ramond string develops a conformal anomaly after quantization ("non-critical strings"). This classification of cocycles is part of what has come to be known as the brane scan in superstring theory, see below.)
In this equivalent formulation, the Green-Schwarz action functional for the superstring has the following simple form:
Let be a superspacetime, hence a supermanifold equipped with a super-vielbein (super-orthogonal structure) which is locally modeled on (technically: a torsion-free super-Cartan geometry modeled on ). Write for the super differential form on which is the induced definite globalization of the cocycle over . For any contractible subspace, then the restriction of of to is exact, and hence admits a potential , i.e. such that .
Then for a 2-dimensional closed manifold, the Green-Schwarz action functional
is the function on the super-smooth mapping space of morphisms of supermanifolds which factor through , given by
In order to get rid of the restriction to some chart one needs to add global data. The need for this is at least mentioned briefly in (Witten 86, p. 261 (17 of 20)), but seems to have otherwise been ignored in the physics literature. The general solution is to promote the local potentials to the connection on a super gerbe (FSS 13). This is a choice of higher prequantization
Writing for the volume holonomy of a circle 2-bundle with connection , then the globally defined Green-Schwarz sigma model
is given by
This form of the Green-Schwarz action functional for the string has evident generalization to other p-branes. Whenever there is a Spin(d-1,1)-invariant -cocycle on , then one may ask for a higher gerbe (higher prequantum line bundle) with curvature and consider the analogous functional.
The triples (spacetime dimension, number of supersymmetries, dimension of brane) such that
is a nontrivial cocycle, hence for which there is such a Green-Schwarz action functional for -branes on may be classified and form what is called the brane scan (Achúcarro-Evans-TownsendWiltshire 87, Brandt 12-13):
The graphics on the left is from (Duff 87). The diagonal lines indicate double dimensional reduction, taking a -brane in dimensions to a -brane in -dimensions.
For instance for one finds a cocycle, and the corresponding GS-action functional is that of the fundamental M2-brane.
This was a striking confluence of brane physics and classification of super Lie algebra cohomology. But just as striking as the matching, was what it lacked to match: the D-branes and the M5-brane (, ) are lacking from the old brane scan. Incidentally, these lacking branes are precisely those branes on which the branes that do appear on the brane scan may end, equivalently those branes that have higher gauge fields on their worldvolume (tensor multiplets).
An action functional for the M5-brane analogous to a Green-Schwarz action functional was found in (BLNPST 97, APPS 97). It is again the sum of a kinetic term and a WZW-like term, but the WZW-like term does not come from a cocycle on a (super-)group.
In order to deal with this, it was suggested in (CAIB 99, Sakaguchi 00, Azcarraga-Izquierdo 01) that there is an algebraic structure called "extended super-Minkowski spacetimes" that generalizes super Minkowski spacetime and serves to unify the Green-Schwarz-like models for the D-branes and the M5-brane with the original Green-Schwarz models for the string and the M2-brane.
These extended super-Minkowski spacetimes carry algebraic analogs of super Lie algebra cocycles, such that the relevant terms for the D-branes and the M5-brane do appear after all, hence such that all the branes in string theory/M-theory are unified. In fact these "extended super-Minkowski spacetimes" are precisely the "FDA"s that have been introduced before in the D'Auria-Fré formulation of supergravity and what became identified as the 7-cocycle for the M5-brane this way had earlier been recognized algebraically as an stepping stone for an elegant re-derivation of 11-dimensional supergravity (D'Auria-Fré 82).
The (higher) geometric meaning of these constructions was found in (Fiorenza-Sati-Schreiber 13): these algebraic structures of "extended super-Minkowski spacetimes"/FDAs are precisely the Chevalley-Eilenberg algebras of super Lie n-algebra-extensions of super-Minkowski spacetime which are classified by the cocycles that serve as the GS-WZW terms of the -branes that may end on those -branes whose cocycles are carried by the extended super-Minkowski spacetime.
Hence the missing -branes in the old brane scan (classifying just cocycles on super Lie algebras) do appear as one generalizes (super) Lie algebras to (super) strong homotopy Lie algebras = L-infinity algebras. Moreover, each brane intersection law (one brane species may end on another) is now matched to a super -algebra extension and so the old brane scan is generalized to a tree of branes The brane bouquet:
Each item in this bouquet denotes a super L-infinity algebra and each arrow denotes an L-infinity extension classified by a cocycle which encodes the GS-WZW term of the brane named by the domain of the arrow. Moreover, arrows pass exactly from one brane species to the brane species that may end on the former.
In (Fiorenza-Sati-Schreiber 13) it is shown that all these super L-infinity algebras Lie integrate to smooth super-n-groups, and all the cocycles Lie integrate to super-gerbes on these, such that the induced volume holonomy is the relevant generalized GS-WZW term. For detailed exposition see at Structure Theory for Higher WZW Terms.
With this generalized perspective, now the Green-Schwarz-type action functionals describe all the p-branes in string theory/M-theory.
Again, in order to make this generally true one needs to apply a higher prequantization – a choice of line (p+1)-bundle with connection – in order to globalize the WZW-terms (Fiorenza-Sati-Schreiber 13)
Hence is the actual background field that the -brane couples to. There is considerably more information in than in its curvature . For instance for the M2-brane one may find the local super moduli space for local choices of for the given on KK-compactifications to . It turns out that the bosonic body of this moduli space is the exceptional tangent bundle on which the U-duality group E7 has a canonical action (see at From higher to exceptional geometry).
This highlights that Green-Schwarz functionals capture fundamental ("microscopic") aspects of -branes. In contrast, often -branes are discussed in their solitonic incarnation as black branes. These solitonic branes sit at asymptotic boundaries of anti-de Sitter spacetime and carry conformal field theories, related to the ambient supergravity by AdS-CFT duality.
This phenomenon is indeed a consequence of the fundamental Green-Schwarz branes:
Consider a 1/2-BPS state solution of type II supergravity or 11-dimensional supergravity, respectively. These solutions locally happen to have the same classification as the Green-Schwarz branes. Hence we may consider a configuration of the corresponding fundamental -brane which embeds into the asymptotic AdS boundary of the given 1/2 BPS spacetime . Then it turns out that restricting the Green-Schwarz action functional to small fluctuations around this configuration, and applying a diffeomorphism gauge fixing, then the resulting action functional is that of a supersymmetric conformal field theory on as in the AdS-CFT dictionary:
fundamental -brane | -fluctuations about asymptotic AdS configuration | solitonic -brane |
Green-Schwarz action functional | super-conformal field theory |
(Claus-Kallosh-Proeyen 97, Claus-Kallosh-Kumar-Townsend 98, AFFFTT 98 Pasti-Sorokin-Tonin 99)
In fact the BPS-state condition itself is neatly encoded in the Green-Schwarz action functionals: by construction they are invariant under the spacetime superisometry group. Hence the Noether theorem implies that there are corresponding conserved currents, whose Dickey bracket forms a super-Lie algebra extension of the Lie algebra of supersymmetries.
Here the "" filling the triangles is the non-trivial gauge transformation by which the WZW term (as any WZW term) is preserved under the symmetries (instead of being fixed identically). It is the information in this transformations which makes the currents form an extension of the symmetries.
Here this yields the famous brane charge extensions of the super-isometry super Lie algebra of the schematic form
(for a Killing spinor and its corresponding Killing vector) known as the type II supersymmetry algebra and the M-theory supersymmetry algebra, respectively (Azcárraga-Gauntlett-Izquierdo-Townsend 89). In fact it yields super-Lie n-algebra extensions of which the familiar super Lie algebra extensions are the 0-truncation (Sati-Schreiber 15, Khavkine-Schreiber 16).
In summary, the nature and classification of Green-Schwarz action functionals captures in a mathematically precise way a good deal of the core structure of string/M-theory.
In fact, the super Lie-n algebraic perspective on the Green-Schwarz functionals via the brane bouquet also solves the following open problem on M-branes:
it is famously known from Freed-Witten anomaly-cancellation that the D-brane charges are not in fact just in de Rham cohomology in every second degree, but are in twisted K-theory, hence rationally in twisted de Rham cohomology, with the twist being the F1-brane charge (from the fundamental). It is an open problem to determine what becomes of these twisted K-theory charge groups as one lifts F1/D-branes in string theory to M2/M5-branes in M-theory.
intersecting branes | charges in generalized cohomology theory | |
string theory | F1/Dp-branes | twisted K-theory |
M-theory | M2/M5-branes | ??? |
Notice that there are "microscopic degrees of freedom" of the theory encoded by the choice of generalized cohomology theory here, generalizing the extra degrees of freedom in the choice of a WZW-term already mentioned above. In general for a cohomology theory and its Chern character map (for instance from topological K-theory to ordinary cohomology in every second degree), then a choice of genuine charges is the extra information encoded in a lift
But rationally The brane bouquet allows to derive this from first principles:
Above we saw that the naive cocycles of the D-branes and of the M5-brane are not defined on the actual spacetime, but on some "extended" spacetime, which is really a smooth super infinity-groupoid extension of spacetime. Hence we should ask if these cocycles descend to the actual super-spacetime while picking up some twists.
One may prove that:
the F1/D-brane GS-WZW cocycles descend to 10d type II superspacetime to form a single cocycle in rational twisted K-theory, just as the traditional lore reqires (Fiorenza-Sati-Schreiber 16);
the M2/M5 GS-WZW cocycles descent to 11d superspacetime to form a single cocycle with values in the rational 4-sphere (Fiorenza-Sati-Schreiber 16).
This has implications on some open conjectures regarding M-theory, for more on this see Equivariant cohomology of M2/M5-branes.
The Green-Schwarz action functionals are of the standard sigma-model form for target spaces that are super-homogeneous spaces for a Lie supergroup and a sub-super-group, and for background gauge fields that are super-WZW-circle n-bundles with connection/bundle gerbes on .
These action functionals were first considered in (Green-Schwarz 84) for superstrings in various dimensions. The full interpretation of the action functional as an higher Wess-Zumino-Witten theory-type action controled by the Lie algebra cohomology of the super Poincaré Lie algebra (or rather of the super translation Lie algebra inside it) is due to (Azcárraga-Townsend89).
We briefly review some basics of the canonical coordinates and the super Lie algebra cohomology of the super Poincaré Lie algebra and super Minkowski space, which are referred to below (see for instance Azcárraga-Townsend 89, and see at super Cartesian space and at signs in supergeometry.).
By the general discussion at Chevalley-Eilenberg algebra, we may characterize the super Poincaré Lie algebra by its CE-algebra "of left-invariant 1-forms" on its group manifold.
The Chevalley-Eilenberg algebra is generated on
elements and of degree
and elements of degree
with the differential defined by
Removing the terms involving here this is the super translation algebra.
In this way the super-Poincaré Lie algebra and its extensions is usefully discussed for instance in (D'Auria-Fré 82) and in (Azcárraga-Townsend 89, CAIB 99). In much of the literature instead the following equivalent notation is popular, which more explicitly involves the coordinates on super Minkowski space.
The abstract generators in def. are identified with left invariant 1-forms on the super-translation group (= super Minkowski space) as follows.
Let be the canonical coordinates on the supermanifold underlying the super translation group. Then the identification is
Notice that this then gives the above formula for the differential of the super-vielbein in def. as
The term is sometimes called the supertorsion of the super-vielbein , because the defining equation
may be read as saying that is torsion-free except for that term. Notice that this term is the only one that appears when the differential is applied to "Lorentz scalars", hence to object in which have "all indices contracted". (See also at torsion constraints in supergravity.)
Notably we have
This remaining operation "" of the differential acting on Loretz scalars is sometimes denoted "", e.g. in (Bossard-Howe-Stelle 09, equation (8)).
This relation is what govers all of the exceptional super Lie algebra cocycles that appear as WZW terms for the Green-Schwarz action below: for some combinations of a Fierz identity implies that the term
vanishes identically, and hence in these dimensions the term
is a cocycle. See also the brane scan table below.
Let be the standard generators of the Chevalley-Eilenberg algebra of the super Poincaré Lie algebra, as discussed there.
The part of the Lie algebra cohomology of the super translation Lie algebra that is invariant under the Lorentz transformations is spanned by closed elements of the form
These exist (are closed) only for certain combinations of and . The possible values are listed below.
For a bosonic WZW model the background gauge field induced by such a cocycle would be the corresponding Lie integration to a circle n-bundle with connection. Here, since the super translation group is contractible, a Poincaré lemma applies and these circle -connections are simply given by globally defined connection form satisfying
The WZW part of the GS action is then
The Green-Schwarz action has an extra fermionic symmetry, on top of the genuine supersymmetry, first observed in (Siegel 83) for the superparticle and in (Siegel 84) for the super 1-brane in 3d, and finally in (GreenSchwarz 84) for the critical superstring in 10-dimensions. This is also called kappa-symmetry. It has a natural interpretation in terms of the super-Cartan geometry of target space (McArthur, GKW). Discussion from the point of view of the D'Auria-Fré formulation of supergravity is in (AFFFTT 98, section 3, Fré-Grassi 07, section 2.2).
The Green-Schwarz action functional of a -brane propagating on an -dimensional target spacetimes makes sense only for special combinations of , for which there are suitanble super Lie algebra cocycles on the super translation Lie algebra (see above).
The corresponding table has been called the brane scan in the literature, now often called the "old brane scan", since it has meanwhile been further completed (see below). In (Duff 87) the "old brane scan" is displayed as follows.
In the -row we see the critical superstring of string theory and its magnetic dual, the NS5-brane. The top row shows the M2-brane in 11-dimensional supergravity.
Moving down and left the diagonals corresponds to double dimensional reduction.
The first non-empty column of the table is a reflection of the exceptional isomorphisms of the spin group in low dimensions and the normed division algebras:
exceptional spinors and real normed division algebras
What is missing in the "old brane scan" are the D-branes in and the M5-brane in (See also BPST). The reason is that the M5 corresponds to a 7-cocycle not on the ordinary super Poincaré Lie algebra, but on its L-infinity algebra extension, the supergravity Lie 3-algebra. The completion in super L-infinity algebra theory is discussed in (FSS 13), as The brane bouquet.
So (with notation as above) we have the following.
The brane scan.
The Green-Schwarz type super -brane sigma-models (see at table of branes for further links and see at The brane bouquet for the full classification):
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
9 | * | |||||||||
8 | * | |||||||||
5 | * | |||||||||
4 | * | * | ||||||||
3 | * |
(The first columns follow the exceptional spinors table.)
The corresponding exceptional super L-∞ algebra cocycles (schematically, without prefactors):
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
11 | on sIso(10,1) | on m2brane | ||||||||
10 | on sIso(9,1) | on StringIIA | on StringIIB | on StringIIA | on sIso(9,1) | on StringIIA | on StringIIB | in StringIIA | on StringIIB | |
9 | on sIso(8,1) | |||||||||
8 | on sIso(7,1) | |||||||||
7 | on sIso(6,1) | |||||||||
6 | on sIso(5,1) | on sIso(5,1) | ||||||||
5 | on sIso(4,1) | |||||||||
4 | on sIso(3,1) | on sIso(3,1) | ||||||||
3 | on sIso(2,1) |
Furthermore, there exists a more general classification of possible supermembranes in spacetime with spatial dimensions and time dimensions, appearing in (Blencowe-Duff 88). In this sense, the brane scan is but the branch of the brane molecule. The objects appearing here are expected to be related to other generalizations of string theory. See D=12 supergravity and bosonic M-theory.
Miles Blencowe, Mike Duff, Supermembranes and the Signature of Space-time, Nucl. Phys. B310 (1988) 387-404 (spire:262142, 10.1016/0550-3213(88)90155-1, pdf)
In the first order formulation of gravity a field configuration on a spacetime manifold is a Cartan connection
hence a principal connection for the super Poincaré group such such that at each point it identifies the tangent space with
Hence given a Lie algebra cocycle
as for the Green-Schwarz superstring we can pull it back along this Cartan connection to a differential 3-form on spacetime.
In general this 3-form is no longer closed. If it is closed, then the Green-Schwarz superstring is again well defined on as a WZW model.
The claim now is that requiring this 3-form still to be closed is, as a condition on the field of gravity , precisely the equations of motion of supergravity (the super-Einstein equations).
This is due to (Nilsson 81, Bergshoeff-Sezgin-Townsend 86) and others, see the references below.
For the membrane(M2-brane) in a background of 11-dimensional supergravity (Bergshoeff-Sezgin-Townsend 87) find that consistency requires that (in a given coordinate chart with super-vielbein field ) the 4-form flux is of the form
where the first summand is the super-Lie algebra cocycle that classifies the supergravity Lie 3-algebra and the second is the field strength of the supergravity C-field proper (hence a purely bosonic differential form). In the second line we have rewritten this more manifestly in terms of the super-vielbein , this way the expression is directly analogous to that of definite 3-forms in the theory of G₂-manifolds (see this example for details).
Moreover the torsion tensor is to have its -component equal to , see at torsion constraints in supergravity.
In addition the Bianchi identities have to hold:
All this is implied by the equations of motion of 11-dimensional supergravity.
Notice that in view of the above analogy to G₂-structure, the covariant constancy condition is precisely the analog of G₂-manifold structure.
Discussion of this in the somewhat more streamlined D'Auria-Fré formulation of supergravity is in (AFFFTT 98, section 3.1).
Discussion that for the GS-version of the heterotic string consistency of the background is equivalent to the equations of motion of heterotic supergravity is in (Shapiro-Taylor 87).
Discussion with the hetetoric gauge field included is in (Atick-Dhar-Ratra 86).
Discussion for the GS-version of the type II superstring in type II supergravity-backgrounds is in (GHMNT 85), and for the D-branes in type II in (CGNSW 97).
The super-WZW term of the GS action functionals is invariant under supersymmetry only up to a divergence. Hence the Noether theorem in its generality for "weak" symmetries applies and gives that the conserved currents receive an extra contribution from this divergence term. The resulting algebra is a central extension of the given super translation Lie algebra, extending to the famous polyvector extensions "by brane charges" of the super Poincaré Lie algebra (AGIT 89).
By the above discussion, Green-Schwarz super -branes are consistent on superspacetimes that satisfy the respective higher supergravity equations of motion. These turn out to have solutions which exhibit black branes in essentially just the combinations of dimensions and supersymmetries that the original Green-Schwarz sigma-models exist in, hence they look precisely like the non-perturbative avatars of whatever these sigma models give the perturbation theory of by second quantization. (See at black holes in string theory for more on this correspondence between branes in string perturbation theory and black branes in supergravity.)
Moreover, the near-horizon geometries of these black branes are always anti de Sitter spacetime times orthogonal directions.
Therefore it is natural to consider the perturbation of the Green-Schwarz sigma-models around their asymptotic embeddings into AdS spaces, hence effectively the perturbation theory of the degrees of freedom at those naked singularity at which the corresponding black brane sits.
After diffeomorphism gauge fixing one finds that the resulting field theories now on the -brane worldvolumes are precisely the superconformal field theories for all the allowed superconformal supersymmetries (see also at singleton representation):
superconformal super Lie algebra | R-symmetry | brane worldvolume theory | ||
3 | osp | |||
3 | osp | M2-brane | ||
4 | D3-brane | |||
5 | 1 | |||
6 | osp | M5-brane |
This is effectively the AdS-CFT correspondence.
Detailed discussion of the above steps is in (AFFFTT 98, Pasti-Sorokin-Tonin 99). The resulting "super-conformal brane scan" is discussed in Blencowe & Duff 1988 with further developments in Duff 2009, Duff 2021.
The quantization of the Green-Schwarz super -brane sigma models is discussed in the literature in terms of light-cone gauge quantization.
This is actually how the Green-Schwarz superstring was first introduced in (Green-Schwarz 81, Green-Schwarz 82) before its generally covariant formulation was found in (Green-Schwarz 84). A textbook account of this is in (Green-Schwarz-Witten, section 5).
While, by the brane scan discussed above, the action functional for the Green-Schwarz superstring exists for target super Minkowski spacetimes of dimension , 4, 6, and 10, its light-cone gauge quantization produces a quantum anomaly for the spacetime Lorentz group symmetry in dimension and . For the anomaly disappears and the thus quantized Green-Schwarz string becomes equivalent to the quantum NSR string, hence to "the" critical string (of heterotic string theory, type II string theory).
Curiously, the light-cone gauge quantization of the GS-string also does wor however for , see at super 1-brane in 3d for more on this.
A precursor to the actual Green-Schwarz action functional is
Michael Green, John Schwarz, Supersymmetrical Dual String Theory, Nucl. Phys. B 181 (1981) 502;
Michael Green, John Schwarz, Supersymmetrical Dual String Theory. 2. Vertices and Trees, Nucl. Phys. B 198 (1982) 252.
which presented a light-cone gauge quantization of superstring with manifest target spacetime supersymmetry.
The observation that this has a generally covariant formulation lead to what is now called the Green-Schwarz action functional proper, for the superstring:
Michael Green, John Schwarz, Covariant description of superstrings, Phys. Lett. B136 (1984), 367–370 (spire:193596, arXiv;10.1016/0370-2693(84)92021-5)
Michael Green, John Schwarz, Properties of the Covariant Formulation of Superstring Theories, Nucl. Phys. B 243 (1984) 285 (spire:196623, doi:10.1016/0550-3213(84)90030-0)
See also the historical comments in
A standard textbook reference for the GS superstring is
and a brief paragraph in Volume II, section 10.2, page 983 of
Eric D'Hoker, String theory – lecture 10: Supersymmetry and supergravity , in part 3 of
Pierre Deligne, Pavel Etingof, Dan Freed, L. Jeffrey,
David Kazhdan, John Morgan, D.R. Morrison and Edward Witten, eds. Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
Textbook discussion of the Green-Schwarz version of the heterotic string is in
Comprehensive review:
Quantization of the Green-Schwarz string using pure spinors is discussed in
Pietro A. Grassi, Giuseppe Policastro, Peter van Nieuwenhuizen, An Introduction to the Covariant Quantization of Superstrings, Class. Quant. Grav. 20 (2003) 395-410 [arXiv:hep-th/0302147]
Pietro A. Grassi, Y. Oz, Non-Critical Covariant Superstrings [arXiv:hep-th/0507168]
Review in the superembedding approach
The observation that the Green-Schwarz action functional is an example of a WZW-model on super-Minkowski spacetime is due to
For more references on this WZW perspective see below.
For references on curved backgrounds see below.
The Green-Schwarz sigma-model-type formulation of the super-membrane in 11d (as in the brane scan and in contrast to the black brane-solutions of 11d supergravity) first appears in:
Via the superembedding approach the equations of motion were obtained in
and the Lagrangian density in:
The double dimensional reduction of the M2-brane to the Green-Schwarz superstring was observed in
Michael Duff, Paul Howe, T. Inami, Kellogg Stelle, Superstrings in from Supermembranes in , Phys. Lett. B 191 (1987) 70 [doi:10.1016/0370-2693(87)91323-2]
also in: Michael Duff (ed.): The World in Eleven Dimensions 205-206 (1987) [spire:245249]
Paul Townsend, The eleven-dimensional supermembrane revisited, Phys. Lett. B 350 (1995) 184-187 [arXiv:hep-th/9501068, doi:10.1016/0370-2693(95)00397-4]
around the time when M-theory became accepted due to
See also
Igor Bandos, Paul Townsend, SDiff Gauge Theory and the M2 Condensate (arXiv:0808.1583)
Maria P. Garcia del Moral, C. Las Heras, P. Leon, J. M. Pena, Alvaro Restuccia, Fluxes, Twisted tori, Monodromy and Supermembranes, J. High Energ. Phys. 2020 97 (2020) [arXiv:2005.06397, doi:10.1007/JHEP09(2020)097]
Discussion from the point of view of Green-Schwarz action functional-∞-Wess-Zumino-Witten theory:
On possible structures in M2-brane dynamics and M2-M5-brane bound states which could be M-theory-lifts of the familiar integrability of the Green-Schwarz superstring on :
Green-Schwarz action functionals for the D-branes (including the DBI action):
Martin Cederwall, Alexander von Gussich, Bengt Nilsson, Per Sundell, Anders Westerberg, The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity, Nucl.Phys. B490 (1997) 179-201 (arXiv:hep-th/9611159)
Mina Aganagic, Jaemo Park, Costin Popescu, John Schwarz, Dual D-Brane Actions, Nucl. Phys. B496 (1997) 215-230 (arXiv:hep-th/9702133)
Discussion of T-duality for the Green-Schwarz string is in
Mirjam Cvetic, H. Lu, Christopher Pope, Kellogg Stelle, T-Duality in the Green-Schwarz Formalism, and the Massless/Massive IIA Duality Map, Nucl.Phys.B573:149-176,2000 (arXiv:hep-th/9907202)
Bogdan Kulik, Radu Roiban, T-duality of the Green-Schwarz superstring, JHEP 0209 (2002) 007 (arXiv:hep-th/0012010)
Igor Bandos, Bernard Julia, Superfield T-duality rules, JHEP 0308 (2003) 032 (arXiv:hep-th/0303075)
reviewed in Superfield T-duality rules in ten dimensions with one isometry (arXiv:hep-th/0312299)
The WZW nature of the second term in the GS action, recognized in (Henneaux-Mezincescu 85) is discussed in
Jeffrey Rabin, Supermanifold Cohomology and the Wess-Zumino Term of the Covariant Superstring Action, Commun Math. Phys. 108, 375-389 (1987) (Euclid:1104116532)
B. Milewski, Superstrings from sigma models, Classical and Quantum Gravity, Volume 4, Number 3 (1987)
A.P. Isaev, E.A. Ivanov, On Sigma Model Formulation of Green-Schwarz Superstring, Mod.Phys.Lett. A4 (1989) 351-359 (spire:266793, doi:10.1142/S0217732389000423)
José de Azcárraga, Paul Townsend, Superspace geometry and the classification of supersymmetric extended objects, Physical Review Letters Volume 62, Number 22 (1989) (spire)
an with its Lie theoretic meaning made fully explicit (in "FDA" language) in
The original "brane scan" classification of GS action functionals by WZW terms:
A complete rigorous classification of all the relevant cocycles on the super Poincaré Lie algebra was given in
Friedemann Brandt, Supersymmetry algebra cohomology
I: Definition and general structure J. Math. Phys.51:122302, 2010, (arXiv:0911.2118)
II: Primitive elements in 2 and 3 dimensions, J. Math. Phys. 51 (2010) 112303 (arXiv:1004.2978)
III: Primitive elements in four and five dimensions, J. Math. Phys. 52:052301, 2011 (arXiv:1005.2102)
IV: Primitive elements in all dimensions from to , J. Math. Phys. 54, 052302 (2013) (arXiv:1303.6211)
For the relevant super Lie algebra cocycles have also been discussed (but not related to the Green-Schwarz action functional) in
A review is in
from which the above table is taken.
Systematic review and discussion of the 3- and 4-cocycles in the old brane scan via the relation between division algebras and supersymmetry is in
See also
More along these lines is in
The Green-Schwarz-type action for the M5-brane was found in
Igor Bandos, Kurt Lechner, Alexei Nurmagambetov, Paolo Pasti, Dmitri Sorokin, Mario Tonin, Covariant Action for the Super-Five-Brane of M-Theory (arXiv:hep-th/9701149)
Mina Aganagic, Jaemo Park, Costin Popescu, John Schwarz, World-Volume Action of the M Theory Five-Brane (arXiv:hep-th/9701166)
The 7-cocycle on the supergravity Lie 3-algebra which gives the supergravity Lie 6-algebra appears in these articles (somewhat secretly) in equation (BLNPST, equation (9)).
See also
The 7-cocycle for the M5-brane on the supergravity Lie 3-algebra is equation (8.8) there.
The interpretation of the cocycles for the D-branes and for the M5-brane as cocycles on "extended super-Minkowski spacetime" is due to
C. Chryssomalakos, José de Azcárraga, José M. Izquierdo, C. Pérez Bueno, The geometry of branes and extended superspaces, Nucl. Phys. B 567 (2000) 293-330 [arXiv:hep-th/9904137, doi:10.1016/S0550-3213(99)00512-X]
Makoto Sakaguchi, IIB-Branes and New Spacetime Superalgebras, JHEP 0004 (2000) 019 (arXiv:hep-th/9909143)
José de Azcárraga, J. M. Izquierdo, Superalgebra cohomology, the geometry of extended superspaces and superbranes (arXiv:hep-th/0105125)
See also division algebras and supersymmetry.
A corresponding refinement of the brane scan to a "brane bouquet" of super L-∞ algebra extensions (hence in infinity-Lie theory via ∞-Wess-Zumino-Witten theory) is discussed in
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, International Journal of Geometric Methods in Modern Physics 12:02 (2015) 1550018 (arXiv:1308.5264)
Domenico Fiorenza, Hisham Sati, Urs Schreiber The WZW term of the M5-brane and differential cohomotopy, J. Math. Phys. 56, 102301 (2015) (arXiv:1506.07557)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Rational sphere valued supercocycles in M-theory and type IIA string theory (arXiv:1606.03206)
Vincent Braunack-Mayer, Hisham Sati, Urs Schreiber, Gauge enhancement of Super M-Branes (arXiv:1806.01115)
These cohomological arguments also appear in what is called the "ectoplasm" method for invariants in super Yang-Mills theory in
G. Bossard, Paul Howe, K.S. Stelle, A note on the UV behaviour of maximally supersymmetric Yang-Mills theories, Phys. Lett. B682:137-142 (2009) (arXiv:0908.3883)
Paul Howe, T. G. Pugh, K. S. Stelle, C. Strickland-Constable, Ectoplasm with an Edge, JHEP 1108:081,2011 (arXiv:1104.4387)
G. Bossard, Paul Howe, Ulf Lindström, Kellogg Stelle, L. Wulff, Integral invariants in maximally supersymmetric Yang-Mills theories (arXiv:1012.3142)
The connection is made in
The other brane scan, listing consistent asymptotic AdS/CFT spacetimes/superconformal algebras is due to:
with further developments discussed in
Michael Duff, Near-horizon brane-scan revived, Nucl. Phys. B 810:193-209, 2009 (arXiv:0804.3675)
Michael Duff, The conformal brane-scan: an update (arXiv:2112.13784)
Discussion of Green-Schwarz strings on super anti de Sitter spacetimes includes the following.
Machiko Hatsuda, Makoto Sakaguchi, Wess-Zumino term for AdS superstring, Phys.Rev. D66 (2002) 045020 (arXiv:hep-th/0205092, doi:10.1103/PhysRevD.66.045020)
Machiko Hatsuda, Makoto Sakaguchi, Wess-Zumino term for the AdS superstring and generalized Inonu-Wigner contraction, Prog.Theor.Phys. 109 (2003) 853-867 (arXiv:hep-th/0106114)
The super 3-cocycle for the Green-Schwarz superstring on the super anti de Sitter spacetime (i.e. on ) is originally due to
However, a supersymmetric trivialization of this cocycle seems to have been obtained in
see Hatsuda-Sakaguchi 02, around (1.2) and (2.6), Hatsuda-Sakaguchi 01, around (1.2)
(amplified in arxiv:1808.04470, p. 5 and equation (5.5)).
See also
Gleb Arutyunov, Sergey Frolov, Foundations of the Superstring. Part I, J. Phys. A 42 (2009) (arXiv:0901.4937)
John Schwarz, The Superstring (arXiv:2004.09661)
For the superstring:
Gleb Arutyunov, Sergey Frolov, Superstrings on as a Coset Sigma-model, JHEP 0809:129,2008 (arXiv:0806.4940)
Pietro Fré, Pietro Antonio Grassi, Pure Spinor Formalism for backgrounds (arXiv:0807.0044)
Riccardo D'Auria, Pietro Fré, Pietro Antonio Grassi, Mario Trigiante, Superstrings on from Supergravity (arXiv:0808.1282)
D. V. Uvarov, superstring in the light-cone gauge, Nucl.Phys.B826:294-312,2010 (arXiv:0906.4699)
For the M2-brane:
Bernard de Wit, Kasper Peeters, Jan Plefka, Alexander Sevrin, The M-Theory Two-Brane in and , Phys.Lett. B443 (1998) 153-158 (arXiv:hep-th/9808052)
Makoto Sakaguchi, Hyeonjoon Shin, Kentaroh Yoshida, Semiclassical Analysis of M2-brane in , JHEP 1012:012,2010 (arXiv:1007.3354)
For the M2-brane and the M5-brane:
Piet Claus, Super M-brane actions in and , Phys. Rev. D 59 (1999) 066003 [arXiv:hep-th/9809045, doi:10.1103/PhysRevD.59.066003]
Makoto Sakaguchi, Kentaroh Yoshida, Open M-branes on Revisited, Nucl.Phys. B714 (2005) 51-66 (arXiv:hep-th/0405109)
Discussion of the self-dual string in 6d as a Green-Schwarz-type sigma model includes
The consistency of the Green-Schwarz action functional for the superstring in a supergravity background should be equivalent to the background satiyfying the supergravity equations of motion. For the superstring this is due to
and for the supermembrane due to
with a quick re-derivation using that the torsion constraint in 11d supergravity already imples the sugra equations of motion is in
These authors amplify the role of closed -forms in super -brane backgrounds (p. 3) and clearly state the consistency conditions for the M2-brane in a curved backround in terms of the Bianchi identities on p. 7-8, amounting to the statment that the 4-form field strength has to be the pullback of the cocycle plus the supergravity C-field curvature and has to be closed.
That the heterotic supergravity equations of motion are sufficient for the 3-form super field strength to be closed was first argued in
and the computation there was highlighted and a little simplified around p. 17 of
A more comprehensive result arguing that the heterotic supergravity equations of motion of the background are not just sufficient but also necessary for (and hence equivalent to) the heterotic GS-string on that background being consistent was then claimed in
Discussion of this with the heterotic gauge-field included (hence including the Green-Schwarz anomaly cancellation) is in
Similar arguments for the type II string in type II supergravity appeared in
and for GS sigma-model D-branes in
That the M2-brane sigma-model is consistent on backgrounds of 11-dimensional supergravity that satisfy their equations of motion is discussed in (Bergshoeff-Sezgin-Townsend 87).
The role of the 4-form here is also amplified around (2.29) in
and in section 2.2 of
See also
All this is actually subsumed by imposing the Bianchi identities of the corresponding supergravity Lie 3-algebra etc. in "rheonomic parameterization", of the D'Auria-Fré formulation of supergravity, this is discussed in (AFFFTT 98, section 3.1, Fré-Grassi 07).
Discussion including also the RR-field background includes
Over a decade before the modern formulation of the AdS-CFT correspondence, a candicate "microscopic" explanation was observed:
Immersing the worldvolume of a sigma-model super -brane (hence a "light" brane without backreaction) along the near horizon geometry (an AdS supergravity solution) of its own black brane incarnation (hence the "heavy" incarnation of the same brane, causing backreaction), its worldvolume fluctuations (after super-diffeomorphism gauge fixing) are described by the corresponding superconformal field theory (exhibited by superconformal multiplets such as "supersingletons").
The original observation for the M2-brane:
Eric Bergshoeff, Michael Duff, Christopher Pope, Ergin Sezgin: Supersymmetric Supermembrane Vacua and Singletons, Phys. Lett. B 199 (1987) 69-74 [doi:10.1016/0370-2693(87)91465-1, InSpire:250244]
Miles P. Blencowe, Mike Duff, Supersingletons, Physics letters B, 203 3 (1988) 229-236 [doi:10.1016/0370-2693(88)90544-8, cds:184143]
Gianguido Dall'Agata, Davide Fabbri, Christophe Fraser, Pietro Fré, Piet Termonia, Mario Trigiante, The singleton action from the supermembrane, Nucl. Phys. B 542 (1999) 157-194 [doi:10.1016/S0550-3213(98)00765-2, arXiv:hep-th/9807115]
with popular exposition in:
Further discussion including also M5-branes and D-branes:
Piet Claus, Renata Kallosh, Antoine Van Proeyen, M 5-brane and superconformal tensor multiplet in 6 dimensions, Nucl. Phys. B 518 (1998) 117-150 [doi:10.1016/S0550-3213(98)00137-0, arXiv:hep-th/9711161]
Piet Claus, Renata Kallosh, J. Kumar, Paul K. Townsend, Antoine Van Proeyen, Conformal Theory of M2, D3, M5 and "D1+D5" Branes, JHEP 9806 (1998) 004 [doi:10.1088/1126-6708/1998/06/004, arXiv:hep-th/9801206]
Paolo Pasti, Dmitri Sorokin, Mario Tonin, Branes in Super-AdS Backgrounds and Superconformal Theories, Proceedings, International Workshop on Supersymmetries and Quantum Symmetries (SQS'99), Moscow (July 27-31, 1999) [arXiv:hep-th/9912076, inspire:511348]
C. Grojean, J. Mourad, Super fivebranes near the boundary of , Nuclear Physics B 567 1–2 (2000) 133-150 [doi:10.1016/S0550-3213(99)00335-1, arXiv:hep-th/9903164]
Alexei Nurmagambetov, I. Y. Park On the M5 and the correspondence, Physics Letters B
524 1–2 (2002) 185-191 [doi:10.1016/S0370-2693(01)01375-2, arXiv:hep-th/0110192]
Mike Duff, Anti-de Sitter space, branes, singletons, superconformal field theories and all that, Int. J. Mod. Phys. A 14 (1999) 815-844 [doi:10.1142/S0217751X99000403, arXiv:hep-th/9808100]
Mike Duff, TASI Lectures on Branes, Black Holes and Anti-de Sitter Space, lectures at 9th CRM Summer School: Theoretical Physics at the End of the 20th Century at TASI 99 (1999) [arXiv:hep-th/9912164, inspire:511822]
The resulting super-conformal brane scan:
Michael Duff, Near-horizon brane-scan revived, Nucl. Phys. B 810 (2009) 193-209 [doi:10.1016/j.nuclphysb.2008.11.001, arXiv:0804.3675]
Michael Duff, The conformal brane-scan: an update, J. High Energ. Phys. 2022 54 (2022) [doi:10.1007/JHEP06(2022)054, arXiv:2112.13784]
Analogous discussion for embeddings with less supersymmetry, corresponding to defects
Nadav Drukker, Simone Giombi, Arkady A. Tseytlin, Xinan Zhou, Defect CFT in the 6d theory from M2 brane dynamics in , J. High Energ. Phys. 2020 101 (2020) [doi:10.1007/JHEP07(2020)101, arXiv:2004.04562]
Varun Gupta, Holographic M5 branes in , J. High Energ. Phys. 2021 32 (2021) [doi:10.1007/JHEP12(2021)032, arXiv:2109.08551]
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That higher WZW functionals and hence Green-Schwarz super -brane action functionals have conserved current BPS charge algebras which are polyvector extensions of the supersymmetry algebras was observed in
reviewed in
and generalized to super-Lie n-algebras of BPS charges in
This is for branes in the old brane scan (strings, membranes, NS5-branes), excluding D-branes and M5-brane.
The generalization oft this perspective to the M5-brane is discussed in
and the generalizatin to D-branes is discussed in
The existence of kappa-symmetry was first noticed around
Warren Siegel, Hidden Local Supersymmetry In The Supersymmetric Particle Action Phys. Lett. B 128, 397 (1983)
Warren Siegel, Light Cone Analysis Of Covariant Superstring , Nucl. Phys. B 236, 311 (1984).
Michael Green, John Schwarz, Covariant Description Of Superstrings , Phys. Lett. B 136, 367 (1984) (web)
The meaning of -symmetry in terms of the super-Cartan geometry of super-target space is discussed in
Discussion from the point of view of D'Auria-Fré formulation of supergravity is in
Gianguido Dall'Agata, Davide Fabbri, Christophe Fraser, Pietro Fré, Piet Termonia, Mario Trigiante, The singleton action from the supermembrane, Nucl. Phys. B542:157-194,1999, (arXiv:hep-th/9807115)
Pietro Fré, Pietro Antonio Grassi, Pure Spinors, Free Differential Algebras, and the Supermembrane, Nucl. Phys. B763:1-34,2007 (arXiv:hep-th/0606171)
Discussion of the Green-Schwarz action for the open M2-brane ending on the M5-brane is in
C. S. Chu, Ergin Sezgin, M-Fivebrane from the Open Supermembrane, JHEP 9712 (1997) 001 (arXiv:hep-th/9710223)
Ph. Brax, J. Mourad, Open Supermembranes Coupled to M-Theory Five-Branes, Phys. Lett. B416 (1998) 295-302 (arXiv:hep-th/9707246)
Last revised on July 18, 2024 at 11:06:15. See the history of this page for a list of all contributions to it.