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# Is it possible that elements in the StreamingAssets folder are not read in build?

I'm making a 2D game. In my StreamingAssets folder I have a JSON file with some info about inventory items (id, description, rarity, name, is stackable or not) and a png file which contains a pixel map (128x32) of the level I want to load after a first scene (this is from Quill18's tutorial: https://www.youtube.com/watch?v=5RzziXSwSsg , basically a specific color corresponds to a specific prefab. An object reads through all the pixels and then spawns the level). The fact is that I have issues with both these elements when the game is built (everything works perfectly in the game window in Unity): the inventory items don't collide with the player and the level is not loaded.

This is the code that gets the image name

public string levelFileName;
string filePath = Application.dataPath + "/StreamingAssets/" + levelFileName;
Texture2D levelMap = new Texture2D (2, 2);

And the pickable objects basically have a collision detection script and have a specific ID, so that when they collide with the player, the inventory adds in a free slot an image with the specs correspondant to that ID (i.e. ID=1 then it's an health potion, is stackable, it's not rare and so on). Everything else works fine in the game. Do you think that the issue is linked to the StreamingAssets folder (first time using it) or is it just a coincidence? And also could you help me solve the problem? Thank you in advance for any kind of help.

Edit: my Player logs:

To me it looks like part of the code was simply not executed due to an error. You should get the logs for your build and check if you do not have any error.

You should not assume that because you do not have errors in the editor you will not have any in the build.

You can find the info here : unity logs

EDIT :

The streaming asset path might change depending on the platform you are building to or on the version of Unity you are Using you should use

Application.streamingAssetsPath

instead. All info here : Streaming assets path

• Hi, thank you for your answer. It looks like you put me on the right track. I updated the question adding the screenshot. I named the build UntitledDelete just to test things out. If I'm reading this right, it looks like it can't find the files I'm referencing. Can you help me further? – Jules Oct 30 '16 at 7:42
• What platform are building for? Linux? – Géry Arduino Oct 30 '16 at 8:15
• In the build settings I selected "PC, Mac & Linux Standalone", but I'm playing and developing it on the Mac – Jules Oct 30 '16 at 8:19
• I edited my answer to include what i think is the actual solution. – Géry Arduino Oct 30 '16 at 8:43
• Thank you so much for your help, you solved a problem I had been working on for a week now! – Jules Oct 30 '16 at 10:06


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Alan Wright

2023-02-18

How to simplify $|-23|$?

Abbigail Joyce

The straight lined brackets surrounding the -23 represents "absolute value". The absolute value of a number is it's distance from 0 (so for example the absolute value of 1 is 1, the absolute value of -1 is also 1).
Now, the absolute value of a negative number is always positive. Therefore, the absolute value of -23 is 23.

Do you have a similar question?

format: plain


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Computing $K$ - theory groups of certain $C^{\ast}$- algebras

I am trying to show that the $K$-theory groups of the following $C^*$-algebra $A$ vanish:

Let $\mathcal{H}$ be a separable Hilbert space. Now consider the subalgebra of $\mathcal{B}(\mathcal{H}\oplus\mathcal{H})$ given by those matrices

\begin{pmatrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{pmatrix}

where $T_{11}, T_{12}$ and $T_{21}$ are compact and $T_{22}$ is arbitrary but bounded. Schematically this is the algebra

$$A = \begin{pmatrix} \mathcal{K} & \mathcal{K} \\ \mathcal{K} & \mathcal{B} \end{pmatrix}$$

where $\mathcal{K}$ denotes the compact operators of $\mathcal{H}$.

As is said I want to show $K_0(A) = 0 = K_1(A)$.

Unfortunately I do not really know what tools to use here. Any suggestions are appreciated.

-
+1 for a question on k-theory. There are not that many. –  Raskolnikov Apr 25 '12 at 17:20
to bad there aren't. but thanks anyway :) –  mland Apr 25 '12 at 18:24

1 Answer

Hints: Consider the ideal $$I = \begin{pmatrix} \mathcal{K} & \mathcal{K} \\ \mathcal{K} & \mathcal{K} \end{pmatrix}=\mathcal{K}\otimes\mathbb M_2\mathbb C\cong\mathcal{K}$$ in $A$. What is $A/I$? Write down the six-term exact sequence in K-theory for the extension $$0\to I\to A\to A/I\to 0.$$ Plug in what you know about the K-theory of $I$ and $A/I$. The desired result follows.

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of course. thanks. –  mland Apr 25 '12 at 18:23


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# Ambiguous [fisher] tag should be disambiguated

We have an ambiguous tag that says right in the tag excerpt that it's ambiguous and should better be avoided. @gung moved most of the questions out of it several years ago, but did not empty it completely so it got lots of new threads in the meantime (75 currently). I think this should still be done.

More specific tags are , , , , , and -- all fine tags with clear scopes and excerpts.

Update: We now also have the following synonyms:

I am posting it here hoping that maybe somebody will want to do this task or join the effort.

• @gung I retagged some Qs about [discriminant-analysis] but it might make sense to create a synonym [fisher-linear-discriminant]->[discriminant-analysis] for the future. Nov 5 '17 at 13:50
• Similarly, I retagged some Qs to [combining-p-values] but it might make sense to create synonyms [fisher-method] and [stouffer-method] (or fishers- and stouffers-?) pointing to [combining-p-values]. Nov 5 '17 at 13:54
• I've done 6 now of the ones since I last did it (July '13). I'm thinking we should remove the tag from everything, though. It's best for the tag to disappear, this isn't something people are going to be able to use correctly. Nov 5 '17 at 14:33
• @gung Yes, I agree; that is what I meant. Nov 5 '17 at 16:12
• @gung What do you think of the synonym suggestions above? Nov 5 '17 at 16:14
• They're fine w/ me, @amoeba. I don't really know these topics well enough. It seems there were several FLA threads. I don't know FLA vs L / Q DA. Is it worth its own tag? We could also make it a synonym, but maybe there's enough interest on its own. Nov 5 '17 at 20:36
• @gung Nah, it's the same thing. "FLA" is LDA for two classes. We definitely don't need a separate tag for it (but the synonym would be handy). I have some authority as the highest-rep user in discriminant-analysis tag :-) Nov 5 '17 at 21:01
• I made a synonym. Nov 6 '17 at 1:55
• Great, thanks. I still think that en.wikipedia.org/wiki/Fisher%27s_method deserves a tag (mapped to combining-p-values) as well (see my 2nd comment above). Nov 6 '17 at 8:34
• There are of course other methods for combining p values which have eponyms so we may be starting down a slippery path if we create tags for any more of them although I suppose if they are synonyms that does not matter too much. Nov 6 '17 at 9:36
• @mdewey I think it's a question of what eponyms are popular enough to warrant a tag synonym. This is an empirical question that should be easy to answer using our website: a search for "Fisher's method" yields 77 results, for "Stoufer's method" - 28 results. Anything else with >10 mentions? Nov 6 '17 at 9:53
• @amoeba, [fishers-method] is a synonym here. Nov 6 '17 at 12:43
• Are we trying to eliminate the tag? There are four left now for none of which I can find a good replacement. Nov 8 '17 at 17:05
• I noticed that a lot of the [fisher] questions were about fishers-lsd, which is legitimate, so I created that & retagged them, @amoeba. I'll post an answer. LSD is mentioned in various threads on CV, but I found 9 that were sufficiently about LSD that I added the tag. Nov 8 '17 at 19:59
• @gung It looks like almost all of these threads are not really about Fisher's scoring. I noticed one that clearly is and put the tag on it now, but overall I don't think we are missing a lot of questions. Nov 8 '17 at 20:17

Years ago I reorganized the [fisher] tag. I created several new tags (e.g., [f-e], [f-i], [f-t], and [r-f]). But I left the tag on ~25 threads that seemed to merit some connection to "fisher", but where idiosyncratic. I edited the excerpt to put a rather specific message that the tag should only rarely be used, and the other tags should be preferred. I think now that that strategy is unworkable and was a mistake. Not surprisingly, perhaps, many people over the intervening years ignored that usage guidance, necessitating this thread.
Over the past couple days, @mdewey, @amoeba (thanks, guys), and I have cleaned up the remaining [fisher] tagged threads. In doing so, we have created and . I also created two synonyms (noted in the body of the question above). After 24 hours, [fisher] should disappear. Let us hope it won't be recreated.


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# secondary chord calculator

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## Deskripsi secondary chord calculator

A Bag Of Marbles Book Pdf, Kohler Toilet Model Number Lookup, Another way to progress a chart is by using solar arc directions, which is an extremely valuable tool as well. Rust Skin Ids, You may also be interested in our Voltage Divider Calculator or Motor FLA Calculator Note that E sharp and G sharp require accidentals. Major Scale: 5 patterns. Lorne Michaels Maine House, Ok, now letâs breakdown what a secondary dominant chord is. Chord diagrams can be labelled with letters or scale degrees. The chord identifier knows most types of chords: major, minor, augmented, diminished, 7th chords (7, maj7, m7, m(maj7), dim7, 7b5, 7#5, m7b5), ninth chords, eleventh chords, 13th chords, sixth chords and suspended chords. material science. There could be more than one solution to a given set of inputs. A secondary dominant chord is an added dominant chord to a key that is not originally in the key. Bass Quiz - Secondary, or applied, dominants are a special use of a dominant 7th chord leading to chords other than the tonic. Ice Cream Freezer Temperature Celsius, Maybe go slower?1) Secondary dominant is a dominant chord leading to any tone of the scale other than the tonic (the root, the 1).2) In this method, it involves âtonicizingâ the other tones of the scale (that is, thinking of them as being the key youâre in temporarily). WSE Calculator. Intervals Part 2. Chord Progressions I - IV - V7 - I Primary Chords - Harmony and Theory: A Comprehensive Source for All Musicians (Musicians Institute) by Carl Schroeder and Keith Wyatt. It's the tritone that most wants to resolve into the Target Chord. Length of Arc = No. Chord and Arc Calculator. Secondary progressions offer insight into the unfolding of our unique personalities over time. Half-diminished seventh chord. They are by far the most common sort of altered chord in tonal music. Judgment Of The Nephilim Pdf, With this method, the chart is progressed by finding the âarcâ ... â Ascendant Calculator (e.g. 13 Inch Quilt Block Patterns, The half diminished seventh chord is the first of the two 7th chords which give a feeling of tension. Here is an example of a deceptive secondary dominant: V/vi to IV . minor scale: 5 patterns. For example, the secondary chords to the supersonic chord in C major come from the key of d minor. The Best Years Of Our Lives Trick Shots, à¤²à¤¾à¤à¤° / à¤¡à¥à¤²à¥à¤«à¥ / à¤ªà¥à¤°à¥à¤à¥à¤°à¤¾à¤®à¤¿à¤à¤ à¤à¥à¤¯à¥à¤à¥à¤°à¤¿à¤¯à¤². Womens Gym Workout Plan Pdf, A C#7 chord has the pitches C sharp, E sharp, G sharp, and B. Code to add this calci to your website . How To Write Email With Attachment Sample Pdf, Filed Under. Chord database navigation: List of 7th chords in the database . I assume you know and understand the diatonic chords (chords in a key) and can work them out easily. Intervals Part 2. for a chord based on the options you specify. The chord calculator automatically determines every mathematically possible fingering for a chord based on the options you specify. radius r . This page enables you to analyze chord progressions in order to identify which are the possible underlying musical scales.There are many scales. This online calculator computes the last nth term of arithmetic progression and the sum of the members. Area of an arch given height and chord Calculator . Diagrams & Notation » Theory » Chord Progressions with Borrowed Chords & Secondary Dominants. A secondary dominant is any chord that has the dominant function over another chord that is not the tonic of the song.. For example, in the key of C major, the dominant chord is G7. Secondary dominants are one of the many inspiring songwriting tools you can learn from music theory. Newest. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. A secondary dominant chord is when you turn a minor or major 7th chord into a dominant chord, in order to make another chord the tonic chord. Small Log Cabin Plans With Loft, If the fifth of the chord is the lowest note, the chord is in second inversion. It may take a bit of practice with key signatures and chord building to get comfortable with them, but once you do you’ll have access to a powerful sound in your chord progressions. Home / Mathematics / Area; Calculates the area, circular arc, center angle and radius of an arch given the chord and height. Chord Function. Lucky Bamboo In Turtle Tank, The harmonic functions of the C major/A minor scale will be called as follows: that is: I use the Greek letter \sigmaÏ(sigma) to denote the diminished subdominant, as there is not a third variant of the letter âSâ in the Latin alphabet. Chord Identifier (Reverse Chord Finder) The chord identifier provides a way to find chords that have a specific set of notes, and other related chords. Just as we can tonicize non-tonic chords by borrowing their dominant chords, we can also tonicize non-tonic chords by borrowing the leading-tone chord (vii o) from that same key.A secondary leading-tone chord follows all of the same voice-leading rules as if it were written in the borrowed key. Find out what makes a secondary dominant and how best to use them. Timothy Patrick Murphy National Enquirer, I am going to show how easy chord inversions are to understand and give you a few examples of when you should try to use them in your songs/pieces.. Understanding Chord Inversions. Sumerset Houseboat Owners Manual, chord c: height h: area S . "root" and "chord" pulldowns. The corresponding chord symbol shows up above the fretboard and the spots of the chord shape will display the according notes or intervals. The Oval Portrait Comprehension Questions And Answers, Since G7 is the V of C, we would call this secondary dominant the âV of IVâ (âfive of fourâ) and traditionally you would write it V/IV. This is then followed by a secondary dominant tonicising the FMaj7 chord and we go straight into another backdoor progression. It could be in one of two keys.But, a dominant 7th chord only exists in one key. Bob Gunton Net Worth, Nba Youngboy Album, Chord Function. Ok, now letâs breakdown what a secondary dominant chord is. Then divide the result by the radius squared (make sure that the units are the same) to get the central angle in radians. For example in an E7 chord the 3rd and 7th of the chord make up a tritone (G# and D). Please be guided by the angle subtended by the arc. ... Chord Shapes Calculator for Guitar 3. Let's say that we want to figure out the secondary dominant for the Target Chord Am. If, in that tonality, the A7 chord appeared, that chord would be a âsecondary dominantâ, since it is a dominant that resolves in D, not in C (our tonic in this case). Is Maria Quiban Filipino, The same way the dominant seventh chord of the 5th degree resolves to chord 1, there are other dominant seventh triads that resolve to other degrees of the scale â the second, third, fourth, fifth, sixth, and seventh degrees. Key Signatures part2: Flat keys. General Tso Vs Teriyaki, Chord Scale Finder for Music Composers. Scales-chords Article Directory 5. In A minor the secondary chords are B diminished (II), C major (III) and F major (VI). Susan Cameron Net Worth, Fortunately, the voice leading considerations for SECONDARY LEADING TONE chords are the same as they would be for any chord progression, when holding proper COUNTERPOINT in mind. Melissa Babish Bio, You could also argue that bars 15 – 16 are both in the key of C Major and the D7 is just a passing chord between Am7 and Dm7. Key Signatures part2: Flat keys. a half diminished seventh chord on C contains the notes C-Eb-Gb-Bb. Secondary Dominants are an important harmonic function that will help you understand modulations and some chords that 'don't seem to belong' in the key. Symbiotic Relationship Between Cattle And Cattle Egret, How To Soak A Sulcata Tortoise, â¦ The $\times$ means “look for the next chord; this will be your tonic”. For a comprehensive, concise reference about primary chords, secondary chords & chord progressions, this book is a "MUST HAVE". For angles in circles formed from tangents, secants, radii and chords click here. We Are The Ship, . Ultimate Chord â¦ This chord is said to have a dominant function, which means that it creates instability and typically leads to the tonic for resolution. Likewise, the triad built on the dominant note is called the dominant chord. Click here for the formulas used in this calculator. So, whatâs the corresponding dominant in the key of A minor?Considering that the A minor triad is chord 6 in the key of C major, its dominant (E dominant seventh) is called Chord 2 in the key of C is the D minor triad. Atlas » Learn more about the world with our collection of regional and country maps. Again, these contrast really well with the minor primary chords. Secondary dominants most commonly occur in the approach of the V (dominant) chord in a piece. A secondary chord is an analytical label for a specific harmonic device that is prevalent in the tonal idiom of Western music beginning in the common practice period: the use of diatonic functions for tonicization.. Bulletproof 2 Parents Guide, e.g. How Long Do Diving Boards Last, The chord calculator automatically determines every mathematically possible fingering for a chord based on the options you specify. Weapon, Secondary, Emblem Calculator. Tweet Follow @teoriaEng. It could be in one of two keys.But, a dominant 7th chord only exists in one key. Required fields are marked *. The upper sign stated that this chord is a dominant of the lower sign. Diagrams & Notation » Theory » Chord Progressions with Borrowed Chords & Secondary Dominants. This calculator calculates for the radius, length, width or chord, height or sagitta, apothem, angle, and area of an arc or circle segment given any two inputs. 14b Examples - Secondary Leading-tone Chords Secondary leading-tone chords. This chord however cannot function as a real dominant since it is not a major chord. Search • Write to us. Applied chords. If we make 7th chords out of all the diatonic chords above, we only have one dominant chord â G7, the âV7â chord. â¢ The root of the secondary chord resolves to the root of the next chord, as expected. Bending is a manufacturing process that produces a V-shape, U-shape, or channel shape along a straight axis in ductile materials, m It is called the dominant because it is next in importance after the first scale degree, the tonic. The chord identifier provides a way to find chords that have a specific set of notes, and other related chords. Your email address will not be published. Sun Dolphin Jon Boat Modifications, Valchromat Near Me, . In the functional notation, secondary dominants will be denoted just like normal dominants, just with a little $\times$ in the subscript. Letâs say weâre in D Major. Yandel Net Worth 2020, Tips for Playing Rock Bass 4. Secondary Dominants are an important harmonic function that will help you understand modulations and some chords that 'don't seem to belong' in the key. Carbs In 1 Single French Fry, Non-Reboot. Tim Considine Wyoming, A chord in second inversion may also be called a six-four chord, because the intervals in it are a sixth and a fourth. Secondary Chords Secondary chords relate to the key of the chord which follows (key of the moment) but do not effect a modulation. Full chord database with images and chord formation explanation. Chord inversions are really easy to understand! Best Power Supply For GTX 1080 in 2020 that you must SEE! Caloric content of … If you were, for example, a secondary dominant preparing for the fourth degree, we would write V7/ IV.This sequence is one preparation after another, which was resolved only at the end in C. Because of that trait, the way to find a secondary dominant is to look at the chord it’s approaching. Bass Quiz - Secondary, or applied, dominants are a special use of a dominant 7th chord leading to chords other than the tonic. Newest. The apparent root of the CT°7 resolves to the third of the next chord. Raze 2 Game, Value City Plush Sectional Review, Be careful So in the key of C the Target Chord of a G is C. Since a Secondary Dominant is the dominant of a chord other than the tonic, it is the dominant (V chord) from the Target Chord's key. Secondary Chords in Every Key. Sue Narramore Biography, Andrew Aronow Net Worth, person_outlineTimurschedule 2010-03-28 16:20:52. So bars 9 â 12 are all in the key of C Major, even though the contain some non-diatonic chords. Secondary Chords Construction . Tonicization = Changing key for a short period of time by using a dominant chord Itâs useful to make a similar differentiation between Modulation and Modal Interchange: 1. Bass: Chromatic secondary leading-tone E sharp resolves up to F sharp (Fi-Sol mimicing Ti-Do). Bass: Chromatic secondary leading-tone E sharp resolves up to F sharp (Fi-Sol mimicing Ti-Do). Tonicization is the process of momentarily emphasizing a non-tonic chord by using chords borrowed from the key in which that chord is tonic. This pattern is repeated for all 7 notes in the scale, resulting in 7 seventh chords. 14b Examples - Secondary Leading-tone Chords Secondary leading-tone chords. A secondary chord contains at least one chromatically altered note; the chromatically altered note is never doubled. Note that E sharp and G sharp require accidentals. The defaults for the other options should suffice Please enter any two values and leave the values to be calculated blank. I assume you know and understand the diatonic chords (chords in a key) and can work them out easily. What this type of chord does: A secondary chord is a dominant function chord that is not the dominant chord in the key of the piece, but is the dominant of one of the other major or minor triads in that key. The secondary dominant resolves to its tonic, taking special care with the active tones: leading tones want to be resolved up to their tonic note, and sevenths, if present, should resolve down by step. By now you should know the power of the bass.For example, the band may play G â G â C while the bass player plays G â G7 â C. Or, the progression may go Am â Am â Dm and the bass player can imply an A7 chord (Am â A7 â Dm) without breaking anything.As a bassist, how do you imply or emphasize the secondary dominant? The defaults for the other options should suffice for most situations. Modulation = Changing key for a long period of time 2. an inversion.) Take On Me Chords, In the case of deceptive secondary dominant chords, they will connect to chords related to the expected chord a fourth above. Los Zetas Net Worth, A secondary dominant is any chord that has the dominant function over another chord that is not the tonic of the song.. For example, in the key of C major, the dominant chord is G7. when adjusting the advanced options as they may result in chords that are very difficult to play. For example, in the previous case, the natural dominant of the song was G7, so we could play another dominant before it to prepare going into G. Observe:Gâs dominant is D7. The chord calculator automatically determines every mathematically possible fingering In most cases, you will only need to adjust the "root" and "chord" pulldowns. So really this is a tritone substitution of a secondary dominant chord. center angle θ . In a dominant 7th chord the 3rd and 7th notes of the chord create an interval of a tritone, which is the interval of greatest unrest. With the button inversions and slash chords you can navigate through the list of the different chord interpretations. Remember that the secondary dominant is in first inversion and that with inversions both chords are complete. A secondary dominant chord is an added dominant chord to a key that is not originally in the key. Estimate how many calories you need a day based on your total daily energy expenditure. There are two such notes in a dominant seventh chord, the leading tone and the seventh.Secondary chords also follow all other rules that we learned when using them diatonically, such as which notes may be omitted or doubled.Although there are seven triads in the scale, there are only five that may have secondary dominants.Either a V triad or a V7 chord can be used to tonicize, with one exception: the triad that would be the V of IV.When spelling secondary dominants, it is important to be accurate when locating the root of the secondary chord.Since it is not directly obvious where the root is located from the analysis symbol, this makes it more difficult than usual to see if you have the correct notes.Another thing to notice about secondary chords in major keys is that, other than the V7 of IV described above, which requires its seventh to be lowered one half step, all the rest of the secondary dominant chords require only Observe the type and placement of the accidentals in the secondary dominants in the chart below.Because we usually need raised notes to build these chords, you will find it sometimes requires a double-sharp accidental when you are working in a key with several sharps in the key signature.Secondary chords in minor work the same way as in major: we donât tonicize the tonic or any chords that are not major or minor triads. It is created by taking a diminished triad and adding a note a minor seventh (10 semitones) above the root. ... By clicking 'I Understand', you are confirming that you understand that the Cost of Second Level Education Calculator is an educational rather than advisory tool and is not a substitute for professional advice from your professional Financial Broker or Advisor. Secondary-dominants Author: Dudu-Yzhaki Subject: Chord theory Keywords: chord construction, three chord songs, chord theory, three chords and â¦ This calculator was created by one of my subscribers (thank you Shirley), and it uses the Sim’s interests to calculate their secondary aspiration. Secondary full-load current, I 2 = (50 × 1000 / 400) = 125 A. b) Turns Ratio = N 1 / N 2 = V 1 / V 2 = (4000 / 400) = 10. radian = degree; circular arc L . After entering or changing a chord shape in the fretboard, the tone on the lowest string is considered as the root. Filed Under. Just as we can tonicize non-tonic chords by borrowing their dominant chords, we can also tonicize non-tonic chords by borrowing the leading-tone chord (vii o) from that same key.A secondary leading-tone chord follows all of the same voice-leading rules as if it were written in the borrowed key. The power of the resolution is so strong it works even if the other instruments ignore it. Actually, they are complementary, even though some astrologers deem that the transits technique is the most reliable one. Articles about Playing Guitar 6. Chords/Keys/Relative minor/Major. Online calculator to calculate the chord length of a circle with the radius and distance of circle. You would expect the chord B7 to resolve to an E or Em chord. The chord identifier knows most types of chords: major, minor, augmented, diminished, 7th chords (7, maj7, m7, m(maj7), dim7, 7b5, 7#5, m7b5), ninth chords, eleventh chords, 13th chords, sixth chords and suspended chords. The defaults for the other options should suffice for most situations. Section Modulus Equations and Calculators Common Shapes. the 1st, 3rd, 5th and 7th positions relative to that 2nd root note.. Bernese Mountain Dog South Dakota, When we build a triad on this scale degree, we caâ¦ Old Town Canoe Serial Number Decoder, The calculator could not be displayed because JavaScript is disabled. Part write as follows. Introduction To Topology Gamelin Pdf, So, B7 is not a iii chord, but a V/vi (âfive of sixâ) chord.B7 is the V of Em7. Then in bar 15 we have a ii-V in G Major, which is the relative major of E minor. Beano Boost Clone, Best Bicycles For Seniors 2020 [RANKED] | Buyer’s Guide. There is a link to this calculator deep in my gameplay rules page. I can't tell you exactly what's the theory behind this, but I can tell you a bit about how they're used, not so much for composition but for arrangements: Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. The chord that is tonicized is typically a chord that belongs to the present key. If you were, for example, a secondary dominant preparing for the fourth degree, we would write V7/ IV.This sequence is one preparation after another, which was resolved only at the end in C. It could be in one of two keys.But, a dominant 7th chord only exists in one key. You no longer need to guess how much should you eat or how much food you need per day - use this TDEE-based calorie calculator to scientifically estimate your recommended caloric intake. For instance, the progression I – V$^7$/ii – ii – V$^7$ – I would be: If you were, for example, a secondary dominant preparing for the fourth degree, we would write V7/ IV.This sequence is one preparation after another, which was resolved only at the end in C. First, E7 prepared for A, but A was in the seventh, preparing for D, and so on until ending in C. This type of progression is widely used in jazz.As we have already seen, these are âextended dominantsâ, as they form a circle of fifths (or of fourths, depending on which side you are looking at). 90 degree bend calculation. Part write as follows. How Old Is Chris Sullivan's Wife, It is used to change keys momentarily â for the duration the chord is played. Secondary chords are a type of altered or borrowed chord, chords which are not part of the key the piece is in. Enter note names to discover which chord they form - Piano Chord Finder on 8notes.com Click for details Reboot. Chinese Rural Dog For Sale, An easy to use online calculator to calculate the arc length s , the length d of the Chord and the area A of a sector given its radius and its central angle t. Formulas for arc Length, chord and area of a sector Figure 1. formulas for arc Length, chord and area of a sector In the above formulas t is in radians. My All Mp3 Songs Download, This V chord existing in the key is called the If you simply say âthe dominantâ you will be understood as saying the V chord in the key.No one ever calls the standard V chord in the key the âprimary dominant,â but that's what it isâit is the main dominant chord. Threshold Cube Organizer Replacement Parts, A secondary dominant is a way of adding a touch of harmonic colour to a piece of music. It is widely believed that Secondary Progressions reflect the evolution of the personality, whereas the other techniques are more linked to more concrete events (Symbolic, Solar Arcs, and Primary Directions, particularly). Whole Foods Prices Vs Kroger, In most cases, you will only need to adjust the It does not matter how far the higher notes are from the lowest note, or how many of each note there are (at different octaves or on different instruments); all that matters is which note is lowest. How Much Weight Can A Barn Owl Carry, So, we grouped the scales to keep results focused. The calculator could not be displayed because JavaScript is disabled. Now the thing is A7 has C# in it. And we finish with a ii-V back in our original key of C Major. for most situations. In most cases, you will only need to adjust the "root" and "chord" pulldowns. First of all, secondary dominant chords are dominant chords, and dominant chords are 7th chords (major triad with a minor 7th on top). This G#ø7 chord is said to be a SECONDARY LEADING TONE chord because it resolves to the IV, rather than the tonic. Circle Calculator. I feel it’s much more realistic than randomly rolling, which was my previous method. Chords/Keys/Relative minor/Major. Chord Progression Style This section is the most fun! Other geometric properties used in design include area for tension, radius of gyration for compression, and moment of inertia for stiffness. Strength of Materials | Beam Deflection and Stress. Crosby Off Road Park, Liz In September Soundtrack, A C#7 chord has the pitches C sharp, E sharp, G sharp, and B. Best Recliner For Short Heavy Person, 16 Ft Skiff For Sale, In fact, it tonicizes the IV chord. That minor 7th chord could exist in 3 different major keys. Cherry Pie Mylar Bags, Yamaha Gp1800 Vs Kawasaki 310, This tool is designed to optimize your WSE and to also compare against various WSE options Step 1. There are too many to explain each one in depth, but the first one will generate a linear, straight chord progression while the rest begin to get a little crazy! A7 is the V of D7 (âfive of fiveâ). Queen Of Elegance Rose, When you'll look above the secondary dominant of each chord (in the second half of the bar) you'll see two signs and a slash. Bass Quiz - Secondary, or applied, dominants are a special use of a dominant 7th chord leading to chords other than the tonic. This V$^7$/ii (A$^7$) chord is called a secondary dominant. Vectra Vs Scotchgard, Looking at the image below, the fifth scale degree (the dominant) is an A. Theory . Why Are Barnacles On Crabs Bad, Unlike modulation, there is no cadence in a new key, only a short progression of chords borrowed from another key.. Theory . Calculate school start dates Enter the date of birth, to calculate the start of school date: For example in the progression C - Am - D7 - G7 â C, Am is the II chord of the dominant key. Are Cheetos Made From Potatoes, B7 â¦ 14b Examples - Secondary Leading-tone Chords Secondary leading-tone chords. 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Chord b7 to resolve into the Target chord Am dominant tonicising the FMaj7 chord and finish! They are by far the most reliable one key that is not originally in the of..., this book is a way of adding a touch of harmonic colour a. Remember that the secondary dominant is to look at an example in a key ) and can them. E sharp resolves up to F sharp ( Fi-Sol mimicing Ti-Do ) tool well... Tension, radius of gyration for compression, and moment of inertia stiffness... Is played 7th of the secondary chords are a type of altered chord second. The 1st, 3rd, 5th and 7th positions relative to that 2nd note. The same way, in that they typically precede a specific chord & secondary Dominants are one of two,... Why are secondary Dominants i distinguished between: 1 chord ; this will your...

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10 – M3L611 Trading Lists SC V1

In this notebook, we will take a look at trading lists and trading trajectories. In particular, you will see how trading lists vary depending on your initial trading parameters. You will also see how to implement a trade list in a simulated trading environment. In this first section, we will see how trading lists vary according to the traders risk aversion, the number of trades, and the liquidation time. The code outputs the plot of the trading lists and the corresponding trading trajectory. It also prints the total number of shares sold, and the trading list telling us how many shares to sell at each trading step. In this example, since we want it to sell our shares in 60 trades, this is exactly what we see. We can change the traders risk aversion to see how the plots change. If I simulate a trader that wants to take a lot of risk, we can see that the plots now look very different. A trader that wants to take a lot of risk would rather sell his shares evenly across the liquidation period. This is exactly what we see by looking at the trade list. We can also simulate a trader that doesn’t like to take risk and see how the blocks change. We can see that the plots now are very different. A trader that doesn’t like to take risks would rather sell his shares quickly rather than to sell them out evenly over the entire liquidation period. This is exactly what we see in the trading list. In the second section, we will see how to implement a trading list. Once we have a trading list, we would like to implement it. What this means is that we would like to sell our shares in the stock market according to the trading list and see how much money we made or lost. To do this, we’re going to simulate the stock market with a very simple trading environment. The details of the simulated trading environment will be covered in another notebook. In this notebook, we will just take a look at some of its parameters. For example, we can see that we’re going to use an annual volatility of 12 percent. Below, we can see the parameters we’re going to use to calculate the trading list according to the Almgren and Chriss model, such as the total number of shares to sell, and the starting price per share. This code will implement a trade list in our simulated trading environment. Again, you can set the values for the traders risk aversion, the number of trades, and the liquidation time. The code will output the value of the implementation shortfall, which tells you how much money you made or lost by selling your shares according to the trade list. The code also outputs the plot of the impacted stock price. This impacted stock price reflects the effects of both the permanent and temporary market impacts. Each dot in the plot represents when a trade took place and the corresponding price per share at which the stocks were sold. We can change the parameters in the code to see how the implementation shortfall varies. We can see that for this level of traders risk aversion, we have an implementation shortfall of around $580,000, and the plot shows many trades. Now, we can simulate a trader that doesn’t like to take risk. We can see that now the implementation shortfall is around two million dollars. We can also see that the plot only shows a few trades. This is consistent with the fact that in this case, since the trader doesn’t like to take any risk, he wants to sell his shares as quickly as possible. Notice that as a consequence of selling his shares quickly, the stock price of the first trade is greatly impacted from its initial value of$50 because this is when most of the stocks are sold, as can be seen from the trading list that shows that most of the stocks were sold in the first trade. The last thing to notice is that implementing the same trading list for different random realizations of the simulated stock price will yield different values for the implementation shortfall. For example, given these trading parameters and this random seed, we get that the implementation shortfall is around $580,000. Now, if we leave all the trading parameters the same, but only change the random seed, I will get the same trading list. However, we can see that now the implementation shortfall is around minus$788,000. The negative sign indicates that this time we actually made money. We can see why this happened from the plot. Remember, the initial stock price was \$50. As we can see from the plot, most of the trade took place when the stock price was above its initial value. Since we sold most of our shares at a higher price, that means that we actually made money. Notice that this happened simply because of random fluctuations in the stock price. So, even if we use the same trade list, we’re not always going to get the same implementation shortfall. This is why we had to re-frame the problem of finding the optimal strategy, not in terms of a single realization, but in terms of the average or the expected value of the shortfall.


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Circle compactification and ’t Hooft anomaly

# Circle compactification and ’t Hooft anomaly

## Abstract

Anomaly matching constrains low-energy physics of strongly-coupled field theories, but it is not useful at finite temperature due to contamination from high-energy states. The known exception is an ’t Hooft anomaly involving one-form symmetries as in pure Yang-Mills theory at . Recent development about large- volume independence, however, gives us a circumstantial evidence that ’t Hooft anomalies can also remain under circle compactifications in some theories without one-form symmetries. We develop a systematic procedure for deriving an ’t Hooft anomaly of the circle-compactified theory starting from the anomaly of the original uncompactified theory without one-form symmetries, where the twisted boundary condition for the compactified direction plays a pivotal role. As an application, we consider -twisted sigma model and massless -QCD, and compute their anomalies explicitly.

## 1 Introduction

Quantum field theory (QFT) provides us a universal description about collective quantum phenomena that appear in huge varieties of physical systems from particle and nuclear physics to condensed matter physics. However, we often encounter the situation where things we observe at low energies look completely different from microscopic degrees of freedom describing QFT. That happens when QFTs of our interest is strongly coupled, and nonperturbative aspects of QFTs are still in big mystery.

One of the possible approaches to tackle this situation is to find a rigorous nature of QFTs, especially related to symmetries and topologies. Symmetry has always played a key role in the development of QFTs; for instance, the idea of spontaneous symmetry breaking classifies traditional phases of matter following Landau’s characterization landau1937theory (); ginzburg1950theory (). In order to refine the data of QFTs related to symmetry, one can try to promote global symmetry to local gauge symmetry, but sometimes topology related to the symmetry gives an obstruction. Such obstruction is called an ’t Hooft anomaly, which is of great importance because of its preservation under the renormalization group flow tHooft:1979rat (); Frishman:1980dq (); Coleman:1982yg (): An ’t Hooft anomaly computed by the low-energy effective theory must be equal to that of microscopic degrees of freedom. It provides us an important consistency check to determine the structure of vacuum and its low-energy excitations and we can use it irrespective of QFTs of our interest being strongly coupled or not. Originally, anomaly matching was proposed for studying chiral symmetries of gauge theories with massless fermions tHooft:1979rat (); Frishman:1980dq (); Coleman:1982yg (). Recent development on topological phases of matter pushes that notion further Vishwanath:2012tq (); Wang:2014pma (); Kapustin:2014lwa (); Kapustin:2014zva (); Cho:2014jfa () and it is now applicable also for systems with discrete symmetries, higher-form symmetries, and so forth, to derive nontrivial consequences on vacuum structures Witten:2015aba (); Seiberg:2016rsg (); Witten:2016cio (); Tachikawa:2016nmo (); Gaiotto:2017yup (); Wang:2017txt (); Tanizaki:2017bam (); Komargodski:2017dmc (); Komargodski:2017smk (); Cho:2017fgz (); Shimizu:2017asf (); Wang:2017loc (); Metlitski:2017fmd (); Kikuchi:2017pcp (); Gaiotto:2017tne ().

Although anomaly matching is a powerful technique to study nonperturbative physics, it cannot uncover details of dynamical aspects of QFTs and just provides us a consistency condition. For example, Yang–Mills theory is believed to exhibit confinement in four-dimensional spacetime, and an ’t Hooft anomaly can tell us about some additional information on vacuum assuming confinement, but it does not show how confinement can happen. Analytic computation of confinement in four dimensions is currently impossible because of its strong coupling nature. Still, it is found that the confinement of Yang-Mills theory is realizable on by adding several massive adjoint fermions or deformations of the action itself, and this confinement is calculable with reliable semiclassical computations Unsal:2007vu (); Kovtun:2007py (); Unsal:2007jx (); Unsal:2008ch (); Shifman:2008ja (); Shifman:2009tp (). What is more interesting is that this semiclassical confinement is argued to be adiabatically connected to the confinement in the strongly-coupled regime by decompactifying the circle especially in the large- limit. This finding motivated many studies of various asymptotically-free field theories by compactifying one direction to a circle with an appropriate boundary condition, and it is expected to map the strongly-coupled dynamics into the semiclassical regime without losing its essential information Cossu:2009sq (); Cossu:2013ora (); Argyres:2012ka (); Argyres:2012vv (); Dunne:2012ae (); Dunne:2012zk (); Poppitz:2012sw (); Anber:2013doa (); Basar:2013sza (); Cherman:2013yfa (); Cherman:2014ofa (); Misumi:2014raa (); Misumi:2014jua (); Misumi:2014bsa (); Dunne:2016nmc (); Misumi:2016fno (); Cherman:2016hcd (); Fujimori:2016ljw (); Sulejmanpasic:2016llc (); Yamazaki:2017ulc (); Buividovich:2017jea (); Aitken:2017ayq ().

In this paper, we would like to make a connection between these two recent developments of nonperturbative QFTs; adiabatic circle compactification and ’t Hooft anomaly matching. If the vacuum structures of the original and circle-compactified theories are really adiabatically connected, it is natural to think that both vacuum structures reproduce the same ’t Hooft anomaly matching condition. However, there is the following difficulty in this idea: Anomaly is renormalization group invariant, and it is matched by the vacuum or its low-energy excitations. Since other high-energy states do not produce the anomaly, the effect of anomaly would disappear once those high-energy states give dominant contributions at finite temperature. How can this observation be consistent with the story about adiabatic continuity? In order to understand the situation better, we consider two quick examples.

Let us consider a three-dimensional free Dirac fermion, which has a symmetry and time-reversal symmetry . Consider the partition function under the gauge field , then

 Z[A]=|Z[A]|exp(iη[A]/2). (1)

Here, is the eta invariant Witten:2015aba (), which is roughly the level- Chern–Simons action but is gauge-invariant modulo , and itself has no ’t Hooft anomaly. With the background gauge field, the time-reversal symmetry is broken because

That is, and has a mixed ’t Hooft anomaly and it is characterized by the Chern-Simons action. Now, we consider the three-dimensional manifold of the form , and let be small enough. We want to check the fate of above anomaly in the two-dimensional effective theory, whose symmetry is and . To gauge the symmetry of this two-dimensional theory, we set with -independent , which is a connection on . The anomaly vanishes with this gauge field , since . As we can see in this example, the anomaly is characterized by a topological invariant of the background gauge field , and if we make be independent of one compactified direction the topological invariant vanishes identically. Physical interpretation is that thermal fluctuation appears after circle compactification and information of topology is lost because of it. This observation seems to be generic and shows the fundamental difficulty in making a connection between vacuum structures of the original theory and the circle-compactified theory from the viewpoint of ’t Hooft anomaly matching.

What is recently found is that if the ’t Hooft anomaly involves a one-form symmetry then it does survive even at finite temperatures Gaiotto:2017yup (). The known example is the Yang–Mills theory at . Yang–Mills theory has the one-form symmetry that acts on Wilon lines. At the theory also has the time-reversal symmetry . If we consider the partition function with the background two-form gauge field for the center symmetry, the time-reversal symmetry is broken:

 Zθ=π[T⋅B]=Zθ=π[B]exp(iN4π∫B∧B). (3)

Therefore, there is a mixed ’t Hooft anomaly between the one-form symmetry and time-reversal symmetry. At , either of them must be broken spontaneously if we assume the mass gap at .

Now, we compactify one direction and set the four-dimensional manifold as , where the size of is sufficiently small, i.e., the temperature is sufficiently high. In this case, in addition to one-form symmetry, there exists zero-form symmetry that acts on Polyakov loop . In order to gauge these symmetries, we introduce the two-form gauge field and also the one-form gauge field . In the four-dimensional language, these gauge fields for three-dimensional effective theory can be regarded as

 B=B(2)+B(1)∧L−1dx4. (4)

Substituting this form into the anomaly relation, we obtain

 Zθ=π[T⋅B]=Zθ=π[B]exp(iN2π∫B(2)∧B(1)). (5)

This suggests that there is a mixed ’t Hooft anomaly among zero-form, one-form, and time-reversal symmetries. Even at finite temperatures, one of these three symmetries must be spontaneously broken. The intuitive difference between anomalies involving only ordinary symmetry and containing one-form symmetries is the following: If the anomaly involves one-form symmetry, the line operator wrapping around is affected by the compactified direction even if that direction is small, and information of topology survives in the circle-compactified theory. This provides positive support to the idea of adiabatic continuity for Yang-Mills theory with adjoint matters Unsal:2007vu (); Kovtun:2007py (); Unsal:2007jx (); Unsal:2008ch (); Shifman:2008ja (); Shifman:2009tp () because we can claim that vacuum structures of the original and circle-compactified theories are controlled by the same ’t Hooft anomaly.

What happens if the gauge theory contains some matter fields not in the adjoint representation like quantum chromodynamics (QCD)? Typically, such theories do not have one-form symmetries, but still its vacuum property is sometimes constrained by ’t Hooft anomalies. The same situation occurs in some two-dimensional nonlinear sigma models, such as the model, which has no one-form symmetries and an ’t Hooft anomaly exists. There is a circumstantial evidence that ’t Hooft anomalies should survive under circle compactifications even for these cases from the viewpoint of adiabatic continuity: Adiabatic continuity of two-dimensional sigma models seems to be valid under specific boundary conditions on Dunne:2012ae (); Dunne:2012zk (); Cherman:2013yfa (); Misumi:2014jua (); Misumi:2014bsa (), and it is rigorously proven for or sigma models in large- limit Sulejmanpasic:2016llc (). It is elucidated in the large- limit that the twisted boundary condition eliminates most of contributions to the partition function from high-energy states Basar:2013sza (); Sulejmanpasic:2016llc () as seen in Witten index of supersymmetric theories Witten:1982im (), and the property of the vacuum is correctly captured at any size of circle compactification.

In this paper, we positively answer the question whether there is any example in which an ’t Hooft anomaly only of ordinary symmetries survives after circle compactification. We develop a systematic procedure generating the anomaly in the circle compactification starting from the anomaly of the original uncompactified theory, where the twisted boundary condition plays a pivotal role in our construction of anomaly. Although no one-form symmetry exists, the above calculation for the case with one-form symmetry Gaiotto:2017yup () gives us a strong motivation of our construction. Theories covered by our procedure contain sigma model with twisted boundary condition, massless QCD with with twisted boundary condition (-QCD) Kouno:2012zz (); Sakai:2012ika (); Kouno:2013zr (); Kouno:2013mma (); Poppitz:2013zqa (); Iritani:2015ara (), and so on. Our systematic procedure is valid even away from large- limit so long as an ’t Hooft anomaly exists.

The paper is organized as follows. In Section 2, we establish a systematic procedure to compute the ’t Hooft anomaly of the circle compactified theory when the original one has no one-form symmetry. We there give a concrete construction of the anomaly, and the importance of appropriately twisted boundary condition is clarified. In Section 3, we demonstrate our method in two-dimensional model. Starting from two-dimensional ’t Hooft anomaly of model at , we derive the anomaly of -twisted model on . In Section 4, we discuss an anomaly of massless -QCD and derive it starting from four-dimensional massless QCD. We compare the ’t Hooft anomaly with results of previous studies, and discuss the application of anomaly matching to the phase diagram. Section 5 is devoted to conclusion and discussion.

## 2 Formalism

In this section, we develop a systematic procedure for deriving an ’t Hooft anomaly of circle-compactified theories starting from the ’t Hooft anomaly of an original theory. Only when there exists one-form symmetry, it has been already well-understood that ’t Hooft anomaly survives even at finite temperatures Gaiotto:2017yup (). Our procedure given below derives the anomaly of circle-compactified theories when the original theory has no one-form symmetries. It turns out that the boundary condition twisted by global symmetry plays an important role for nonvanishing anomaly.

### 2.1 Systematic procedure for anomaly with circle compactification

We consider a -dimensional quantum field theory (QFT), and assume that the QFT has two symmetries and acting on its physical Hilbert space faithfully. We call as a flavor symmetry, and consider the case when with and . It is straightforward to extend the discussion for the case and (center of ). However, it makes notations for the following discussion complicated, and we try to make our explanation as simple as possible.

We further assume that and have a mixed ’t Hooft anomaly but and have no anomaly. In order to see the anomaly, we introduce the background gauge field for the flavor symmetry . -gauge field consists of two ingredients Kapustin:2014gua (); Gaiotto:2014kfa (); Aharony:2013hda ():

• -gauge field that is locally a one-form in dimensions.

• -gauge field that is locally a two-form in dimensions.

The physical interpretation is that the electric one-form symmetry appears after gauging and its charged object is the Wilson line

 (6)

Gauging this one-form symmetry by , the Wilson line is no longer a genuine line operator and we obtain -gauge theory instead of -gauge theory. We denote the partition function of this QFT under the background -gauge field as . The above assumption on anomaly implies that

 Z[h⋅(A,B)]=Z[(A,B)]exp(iAh[B]), (7)

where , is the -transformation of -gauge field , and is a -dimensional topological -gauge theory1 determined by . For some , the anomaly cannot be canceled by variations of local counterterms. Under this setup, we will derive the ’t Hooft anomaly of -dimensional effective theory when one of the direction is compactified to a small circle.

In order to consider the circle compactification, we set -dimensional manifolds as , and the size of its -dimensional part is much larger than the circle of size . At this stage, all the fields of QFT obey the periodic boundary condition along . To describe the theory at first, we turn off and -dimensional components of , i.e., . In this process, we can still fix the Polyakov-loop matrix of along to a nontrivial one, and denote it as

 Ω=Pexp(i∫L0Acl). (8)

For each that is uniform on , we obtain a -dimensional QFT on , and denote its partition function as . This is equivalent to imposing a twisted boundary condition on fields of QFT along by performing a boundary-condition-changing -gauge transformation, but we keep the periodic boundary condition with nontrivial holonomy during our explanation of the general strategy. Before introducing the two-form gauge field , we originally have -dimensional one-form symmetry , and it induces -dimensional zero-form and one-form symmetries after circle compactification when is dynamical. The zero-form symmetry acts as with some , and we thus identify2 its action on itself as . However, since we define the theory by fixing the holonomy , the above transformation maps one theory to another theory : It is not the symmetry of .

In order to have a nontrivial anomaly on , we need to have a symmetry involving the above zero-form transformation, . We specify the holonomy such that there exists satisfying

 SΩS−1=ωΩ. (9)

Since is not an element of the center of by definition, . Recall that acts faithfully on the physical Hilbert space, and it means that generates a faithful symmetry of QFT, which we call a “shift symmetry”. When the shift symmetry acts on fields, the holonomy matrix is changed to . The requirement (9) states that the symmetry generated by is intertwined with the zero-form symmetry , , in order to maintain the holonomy , and the symmetry of is obtained: We denote this zero-form symmetry generated by as in order to distinguish it from the original one, . Because of (9), cannot be proportional to the identity matrix: Typical example of and satisfying (9) is ()

 Ω=ω−(N−1)/2⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝100⋯00ω0⋯000ω2⋯0⋮⋮⋮⋮000⋯ωN−1⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,S=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝010⋯0001⋯0⋮⋮⋮⋮000⋯1100⋯0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (10)

Since the flavor symmetry of -dimensional theory must commute with , flavor symmetry might be explicitly broken to a maximal Abelian subgroup as . Symmetry with the faithful representation is again given by the quotient . Let us assume that is not explicitly broken by fixing , then the -dimensional effective theory has three symmetries; shift symmetry , flavor symmetry , and .

Let us try to introduce the background gauge fields for and . We denote the -gauge field as that is locally a one-form on . The -gauge field consists of two ingredients:

• -gauge field that is locally a one-form on .

• -gauge field that is locally a two-form on .

When is gauged, the -dimensional one-form symmetry emerges, so we can introduce which is a two-form on . Using these gauge fields, we define the -gauge field on as

 A=AK+B(1)+Acl,B=B(2)+B(1)∧L−1dxD+1. (11)

The field in the expression (11) may require some explanations: When it appears in , it is regarded as a gauge field for the subgroup of the flavor symmetry, while it should be regarded as a gauge field for the zero-form symmetry induced from the one-form symmetry when it appears inside . We use this slightly abused notation in order to emphasize that these two transformations are intertwined as a symmetry of . Let be the partition function with the background gauge fields for and , then the given construction of the -dimensional theory shows that

 ZΩ[(AK,B(1),B(2))]=Z[(A,B)], (12)

where and are given in (11).

We can now derive the mixed -dimensional anomaly among , , and as follows: Using the -dimensional expression of the -dimensional theory, (11) and (12), we obtain

 ZΩ[h⋅(AK,B(1),B(2))] (13) = Z[h⋅(A,B)] = Z[(A,B)]exp(iAh[B]) = ZΩ[(AK,B(1),B(2))]exp(iAh[B(2)+B(1)∧L−1dxD+1]).

Here, we use the anomaly relation (7) in dimensions. Since and have no dependence on , the compactified direction of can be integrated out in a trivial manner. As a result, becomes the -dimensional topological -gauge theory of and , and this defines the anomaly of the -dimensional QFT . That is, the -dimensional QFT has a mixed ’t Hooft anomaly among , , and , and the trivial gapped state is forbidden.

### 2.2 Comments on choice of the background holonomy

One of the most important part in our construction of -dimensional anomaly via circle compactification is the choice of nontrivial holonomy background . In order to clarify the importance of the condition (9), let us take the trivial one as a “bad” example.

Whether or not (9) is satisfied, the zero-form transformation acts on holonomy as . However, any elements commutes with , and then this zero-form transformation on the Polyakov loop has no connection with the symmetry of -dimensional system. Since it is not a faithful symmetry of the system, we must set in the trivial choice . Instead, the flavor symmetry is unbroken, and the corresponding gauge field on is denoted as . As a result, the anomaly relation (13) becomes

 Z1[h⋅(AG,0,B(2))]=Z1[(AG,0,B(2))]exp(iAh[B(2)]). (14)

Since is a gauge-field on while is a -dimensional topological theory, we obtain . This means that the trivial boundary condition eliminates the anomaly of dimensions. Two-form gauge fields on are not enough for anomaly.

In our construction, the zero-form transformation on Polyakov-loop is translated into the faithful symmetry generated by on fields of QFT via (9). The appearance of the faithful zero-form symmetry allows us to introduce the -gauge field . Since and is intertwined in dimensions, it is built into the -dimensional two-form gauge field as a form . Therefore, the ’t Hooft anomaly can survive even if the original theory has no one-form symmetry.

## 3 ZN-twisted CPN−1 sigma model

As a demonstration of the systematic procedure in Sec. 2, we calculate the anomaly of -twisted model starting from the two-dimensional ’t Hooft anomaly. Two-dimensional sigma model can be realized as a gauged linear sigma model,

 S=∫d2x[12|(∂μ+iaμ)→z|2+λ4(|→z|2−μ2)2]−iθ2π∫da, (15)

where is an -component complex vector-valued fields, and is the -gauge field3. We study theta-dependence of this theory from the viewpoint of anomaly. We start with the two-dimensional discussion Komargodski:2017dmc () first, and move to the circle compactification with -twisted boundary condition. In order to understand the formalism better, we follow each of the steps explained in Sec. 2.1 in detail.

### 3.1 ’t Hooft anomaly and global inconsistency in two dimensions

The symmetry of this theory (15) consists of the following Komargodski:2017dmc ():

• Flavor symmetry, , which is given by with .

• Time-reversal symmetry at .

Since the symmetry is gauged, the center elements of cannot act faithfully on gauge-invariant operators. The flavor symmetry with the faithful representation is thus given by . In the notation used in Sec. 2, we have the following correspondence: , , , and .

We introduce the background gauge field for flavor symmetry. Such a background gauge field consists of two ingredients: gauge field and two-form gauge field . To explain it, let us first gauge the symmetry, then the action obtained by the minimal coupling becomes

 Sgauged=∫d2x[12|(∂μ+iaμ−iAμ)→z|2+V(|→z|2)]−iθ2π∫da, (16)

where . Flavor symmetry is gauged and no longer a global symmetry, but the theory acquires the one-form symmetry: Considering the and Wilson lines,

 (17)

then the theory has a symmetry under the simultaneous rotation,

 WU(1)(C)↦e2πi/NWU(1)(C),WSU(N)(C)↦e2πi/NWSU(N)(C). (18)

The Wilson lines charged under this one-form symmetry must be dropped from the spectrum of genuine line operators if we appropriately gauge the flavor symmetry Kapustin:2014gua (); Gaiotto:2014kfa (). For this purpose, we introduce the two-form gauge field , and then we obtain

At , we consider the time-reversal transformation under the background flavor gauge field , and we obtain Komargodski:2017dmc ()

 Zπ[T⋅(A,B)]=Zπ[(A,B)]e−i∫B. (20)

We should check whether this anomaly is genuine or fake, so we consider whether it can be canceled by local counter terms of . The topological two-form gauge theory is given by with some integer modulo , and it is a candidate for the counterterm. The transformation after adding this counterterm behaves as

 Zπ[T⋅(A,B)]exp(−ik∫T⋅B)=Zπ[(A,B)]exp(−ik∫B)ei(2k−1)∫B. (21)

Thus, anomaly is fake if and only if modulo . For even , this is impossible and we find the ’t Hooft anomaly between the flavor and time-reversal symmetries.

For odd , we can eliminate the anomaly by choosing modulo , and no ’t Hooft anomaly exists. If we do the same computation at , the time-reversal symmetry is respected by choosing , and thus there is no common counterterm that respects both at . This is the global inconsistency condition, and we can derive a nontrivial consequence although it is slightly weaker than ’t Hooft anomaly matching (see Ref. Kikuchi:2017pcp () for global inconsistency condition).

In two dimensions, Coleman–Mermin–Wagner theorem Coleman:1973ci (); mermin1966absence () tells us that flavor symmetry with a continuous parameter cannot be broken. This naturally gives us the nonperturbative data that time-reversal symmetry at is spontaneously broken for two-dimensional model.

### 3.2 ZN-twisted CPN−1 model and its anomaly

We consider the circle compactification from to , where the circumference of the circle , i.e. , is regarded to be small. We impose the -twisted boundary condition,

 →z(x1,x2+L)=Ω→z(x1,x2), (22)

where ()

 Ω=diag(1,ω,…,ωN−1). (23)

We call this as -twisted sigma model, and we denote its partition function at as .

For our purpose, it is better to regard this twisting matrix as a holonomy of gauge field along compactified direction. Indeed, the boundary-condition-changing gauge transformation makes the boundary condition of periodic (up to gauge symmetry) and the price to be paid is the background holonomy . When is gauged, there is a one-form symmetry and it induces the zero-form symmetry, . What is special for this choice of the nontrivial holonomy is that the above transformation induces the symmetry given by

 →z=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝z1z2⋮zN−1zN⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠↦S→z=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝z2z3⋮zNz1⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,S=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝010⋯0001⋯0⋮⋮⋮⋮000⋯1100⋯0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (24)

We call this a shift symmetry, and it is a faithful transformation on the physical spectrum. As an example, a gauge-invariant operator is mapped to another gauge-invariant operator . When the transformation is performed as , the boundary condition for the transformed field becomes

 →z′(x1,x2+L)=SΩS−1→z′(x1,x2)=ωΩ→z′(x1,x2). (25)

That is, the zero-form symmetry on the Polyakov loop is intertwined with the shift symmetry generated by in order to maintain the boundary condition (22), and we call it . The symmetry acts on local operators on as

 →z↦S→z,exp(i∫S1a)↦ω−1exp(i∫S1a). (26)

We give a short summary of the situation: The compactified theory obtained here has a zero-form symmetry, and it is induced by the one-form symmetry in two dimensions when is gauged. Continuous part of the flavor symmetry is explicitly broken to , but it is not relevant for the following discussion and we do not introduce gauge fields for it.

As a result, the ’t Hooft anomaly (or global inconsistency) in two dimensions has the same meaning in the -twisted model on . The -twisted model has the shift symmetry and the time-reversal symmetry at . We introduce the one-form gauge field for gauging , which is independent of . Since is intertwined with the zero-form symmetry acting on the and Polyakov loops, we can embed it into the two-form gauge field in the two-dimensional language by setting . The two-dimensional anomaly (20) tells us that

 Zπ,Ω[T⋅B(1)] = Zπ,Ω[B(1)]exp(−i∫B(1)∫L0L−1dx2) (27) = Zπ,Ω[B(1)]exp(−i∫B(1)).

We find that and at has an ’t Hooft anomaly (or global inconsistency depending on even or odd ), and either of them must be spontaneously broken.

For usual periodic boundary condition, it means that we put and thus there is no room to introduce . Therefore, we cannot obtain ’t Hooft anomaly in such cases. The emergence of symmetry by the twisted boundary condition is essential for a deep connection with two-dimensional anomaly.

### 3.3 Comparison with previous studies and Discussion

The -angle dependence of model in two dimensions is studied in large- limit Witten:1978bc (); Affleck:1979gy (), and the ground state energy should behave as

 E(θ)∝mink∈Z1N(θ+2πk)2. (28)

This behavior matches the ’t Hooft anomaly (20), because the time-reversal symmetry is spontaneously broken at . Our derivation of the anomaly (27) for -twisted model claims that the same multi-branch structure would naturally appear under adiabatic circle compactification.

Indeed, -dependence of -twisted model is studied in Refs. Dunne:2012ae (); Dunne:2012zk (). Under the twisted boundary condition, there are types of fractional instantons which has the topological charge . As a result, the quasi-ground states are composed of states and the -th ground-state energy behaves as

 Ek(θ)∝−Ncos(θ+2πkN). (29)

The ground state energy is thus given by minimum of these,

 E(θ)=mink=1,…,NEk(θ). (30)

We can see that the time-reversal symmetry is spontaneously broken at , which satisfies matching of ’t Hooft anomaly or global inconsistency (27). What we have shown in this paper is that these two behaviors (28) and (29) are both consistent with anomalies, and those anomalies have essentially the same origin.

We argue that this observation gives a positive support for the adiabatic continuity. For -twisted model, it is rigorously shown that expectation values of any invariant operators does not depend on in the large- limit Sulejmanpasic:2016llc (), and our consideration on anomaly gives a complementary and consistent analysis.

## 4 Massless ZN-Qcd

As another demonstration, we consider a four-dimensional example: Yang-Mills theory with massless Dirac fermions in the fundamental representation, i.e. massless QCD with . The action of this theory is given by

 S=12g2∫tr(Gc∧∗Gc)+∫d4xtr{¯¯¯¯ΨγμDμ(a)Ψ}, (31)

where is the color gauge field, is the covariant derivative, is the -gauge field strength, and are matrix-valued Dirac fermions. The color group acts on from left, and the flavor group acts on from right, i.e., for ; the quark field is in the bifundamental representation of the color and flavor groups.

This theory possesses various symmetries, but we pay attention only to the vector-like flavor symmetry, , and the anomaly-free discrete subgroup of the axial symmetry, , in our demonstration. The complete analysis involving other symmetries will be discussed at future opportunity. We first compute the ’t Hooft anomaly of the above symmetries in four dimensions. Using the four-dimensional computation, we derive the anomaly of the circle-compactified theory with the -twisted boundary condition, -QCD.

Our discussion can be generalized to the case when color and flavor have different numbers so long as they have a nontrivial common divisor, . For notational simplicity, we only consider the case in this paper.

### 4.1 Four-dimensional ’t Hooft anomaly of massless N-flavor QCD

Here, we start with explanation on the vector-like flavor symmetry and . In the notation used in Sec. 2, we have the following correspondence: , , , and .

Quark field is in the bifundamental representation, for , and thus the subgroup generated by does not act on faithfully (). Therefore, the group acting faithfully on is . Since the color group symmetry is gauged, the flavor symmetry with the faithful representation on the physical Hilbert space is given by .

Since the quark field is massless, there is a symmetry , , at the Lagrangian level, but the fermion integration measure generates the additional term due to the index theorem. Therefore, it is a symmetry only when is quantized to , and is explicitly broken to by quantum anomaly.

We shall derive the mixed ’t Hooft anomaly between and , and we introduce the background gauge field of for that purpose. We first introduce the flavor gauge field , then the minimal-coupling procedure4 changes the action (31) as

 Sgauged=12g2∫tr(Gc∧∗Gc)+∫d4xtr{¯¯¯¯ΨγμDμ(a,A)Ψ}, (32)

where the covariant derivative is replaced by

 Dμ(a,A)Ψ=∂μΨ+iaμΨ−iΨAμ. (33)

The theory (32) has one-form symmetry that does not exist in the original massless QCD. It acts on the color and flavor Wilson lines, and , as the simultaneous rotation of phase,

 W(C)color↦ωW(C)color,W(C)flavor↦ωW(C)flavor. (34)

This symmetry does not arise in the gauge theories, and we have to introduce two-form gauge field  Kapustin:2014gua (); Gaiotto:2014kfa (), which is a two-form gauge field satisfying

 NB+dC=0, (35)

with a certain one-form gauge field . This constraint respects the one-form gauge invariance under and , and respecting this gauge invariance prevents us from adding extra degrees of freedom to the theory. We introduce the gauge fields made of gauge fields , , and a gauge field , as

 ˜a=a+1NC,˜A=A+1NC, (36)

and define their gauge field strengths as

 Gc=d˜a+i˜a∧˜a,Gf=d˜A+i˜A∧˜A. (37)

These field strengths transform under the one-form gauge transformation as and , and we obtain the gauge invariant combinations, and . This tells us that the introduction of two-form gauge field changes the action (32) as

 Sgauged=12g2∫tr{(Gc+B)∧∗(Gc+B)}+∫d4xtr{¯¯¯¯ΨγμDμ(˜a,˜A)Ψ}. (38)

We can shortly summarize the set of above procedures as follows: Introducing the flavor gauge field , the theory becomes bifundamental QCD, which has one-form symmetry, so we again introduce the two-form gauge field  Tanizaki:2017bam (); Shimizu:2017asf ().

Let us perform the axial rotation, then the Lagrangian is invariant again, but the action acquires the additional topological term due to the fermion measure:

 ΔS=i4π∫tr{(Gc+B)∧(Gc+B)}+i4π∫tr{(Gf+B)∧(Gf+B)}=−i2N4π∫B∧B. (39)

The last equality holds modulo 5. This means that the symmetry under the background gauge field is broken as

 Z[(A,B)]↦Z[(A,B)]exp(−2iN4π∫B∧B). (40)

For , this additional phase is nontrivial, and the ’t Hooft anomaly exists between the flavor symmetry and the discrete axial symmetry . Four-dimensional QCD is believed to break the chiral symmetry spontaneously, which also breaks axial symmetry to spontaneously ( is the fermion number operator), and the ’t Hooft anomaly is matched.

### 4.2 Massless ZN-QCD and its anomaly

We compactify one-direction, and derive the associated three-dimensional effective theory. We fix the holonomy as6

 Ω=eiϕdiag[1,ω,ω2,…,ωN−1]. (41)

Equivalently, we introduce the boundary condition on the quark field as

 Ψ(x,x4+L)=Ψ(x,x4)Ω. (42)

The extended gauge transformation eliminates the holonomy, but the quark field obeys the -twisted boundary condition. This is called -QCD, and we denote its partition function as .

Circle compactification induces zero-form transformation, , from the one-form symmetry (34), but it changes the boundary condition and maps a theory to another theory . We should intertwine it with the flavor rotation , where is defined in (10), in order to maintain the boundary condition. This generates the zero-form symmetry of , and we call this as the shift symmetry, , which acts on local operators on as

 Ψ↦ΨS,tr[Pexpi∫S1a]↦ωtr[Pexpi∫S1a]. (43)

To obtain the three-dimensional anomaly, we gauge the shift symmetry and denote the corresponding gauge field as . Because of the holonomy , the explicit breaking of the flavor symmetry occurs , and the faithful flavor symmetry is . We introduce the background gauge field and three-dimensional two-form gauge field . The -twisted partition function under these backgrounds are given by

 ZΩ[(AK,B(1),B(2))]=Z[(AK+B(1)+Acl,B(2)+B(1)∧L−1dx4)]. (44)

We thus obtain under the transformation as

 ZΩ[(AK,B(1),B(2))]↦ZΩ[(AK,B(1),B(2))]exp


text:
# Characteristic function as limit of integrals.

I've seen the following claim in a book without proof and don't know why it holds. Let $a<b \in\mathbb{R}$. Then the integral $$\frac{1}{2\pi i}\int_a^b\left(\frac{1}{t-i\epsilon-\lambda}-\frac{1}{t+i\epsilon-\lambda}\right)dt$$ converges to $0$ if $\lambda\not\in(a,b)$, $\frac{1}{2}$ if $\lambda=a$ or $b$ and $1$ if $\lambda\in(a,b)$ as $\epsilon\searrow0$. The $2\pi i$ made me think that this could be an application of the residue theorem but I don't know which integration path to choose.

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Hint: just split the integral up in real and imaginary part similar this proof.

Spoiler below:

Expand: $$\frac{1}{t-\lambda \pm i \epsilon} = \frac{1}{t-\lambda \pm i \epsilon} \frac{t-\lambda \mp i \epsilon}{t-\lambda \mp i \epsilon} = \frac{t-\lambda \mp i \epsilon}{(t-\lambda)^2 + \epsilon^2}.$$ Thus $$\frac{1}{t-\lambda-i\epsilon}-\frac{1}{t-\lambda+i\epsilon} = \frac{2i \epsilon}{(t-\lambda)^2 + \epsilon^2} .$$ and $$\frac{1}{2\pi i}\int_a^b\left(\frac{1}{t-i\epsilon-\lambda}-\frac{1}{t+i\epsilon-\lambda}\right)dt = \frac{\epsilon}{\pi} \int_a^b \frac{dt}{(t-\lambda) + \epsilon^2} = \frac{1}{\pi}\left[\arctan\left(\frac{b-\lambda}{\epsilon}\right) - \arctan\left(\frac{a-\lambda}{\epsilon}\right)\right].$$ Now, we know that \begin{align} \lim_{x\to \pm \infty} \arctan(x) &=\pm\frac\pi2, & \arctan(0)&=0.\end{align} from which the result stated follows immediately.

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Thank you! Thought to much about complex integration! – Julian Jan 11 '13 at 10:56
@Julien: yes, I know this. One is tempted to use complex analysis methods but they make the problem just more difficult. – Fabian Jan 11 '13 at 15:36