# How to Use the IEEEtran LaTeX Templates

IEEE Publication Technology Department Manuscript created October, 2020; This
work was developed by the IEEE Publication Technology Department. This work is
distributed under the LaTeX Project Public License (LPPL) ( http://www.latex-
project.org/ ) version 1.3. A copy of the LPPL, version 1.3, is included in
the base LaTeX documentation of all distributions of LaTeX released 2003/12/01
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###### Abstract

This document describes the most common article elements and how to use the
IEEEtran class with LaTeX to produce files that are suitable for submission to
the Institute of Electrical and Electronics Engineers (IEEE). IEEEtran can
produce conference, journal and technical note (correspondence) papers with a
suitable choice of class options.

###### Index Terms:

Class, IEEEtran, LaTeX, paper, style, template, typesetting.

##  I Introduction

Welcome to the updated and simplified documentation to using the IEEEtran
LaTeX class file. The IEEE has examined hundreds of author submissions using
this package to help formulate this easy to follow guide. We will cover the
most commonly used elements of a journal article. For less common elements we
will refer back to the “IEEEtran_HOWTO.pdf”.

This document applies to version 1.8b of IEEEtran.

The IEEEtran template package contains the following example files:

  * bare_jrnl.tex

  * bare_conf.tex

  * bare_jrnl_compsoc.tex

  * bare_conf_compsoc.tex

  * bare_jrnl_comsoc.tex

These are “bare bones” templates to quickly understand the document structure.

It is assumed that the reader has a basic working knowledge of LaTeX. Those
who are new to LaTeX are encouraged to read Tobias Oetiker’s “The Not So Short
Introduction to LaTeX”, available at:
http://tug.ctan.org/info/lshort/english/lshort.pdf which provides an overview
of working with LaTeX.

##  II The Design, Intent and
Limitations of the Templates

The templates are intended to approximate the final look and page length of
the articles/papers. Therefore, they are NOT intended to be the final produced
work that is displayed in print or on IEEEXplore®. They will help to give the
authors an approximation of the number of pages that will be in the final
version. The structure of the LaTeXfiles, as designed, enable easy conversion
to XML for the composition systems used by the IEEE’s outsource vendors. The
XML files are used to produce the final print/IEEEXplore® pdf and then
converted to HTML for IEEEXplore®. Have you looked at your article/paper in
the HTML version?

##  III LaTeX Distributions: Where to Get Them

IEEE recommends using the distribution from the TeXUser Group at
http://www.tug.org. You can join TUG and obtain a DVD distribution or download
for free from the links provided on their website:
http://www.tug.org/texlive/. The DVD includes distributions for Windows, Mac
OS X and Linux operating systems.

##  IV Where to get the IEEEtran Templates

The IEEE Template Selector will always have the most up-to-date versions of
the LaTeX and MSWord templates. Please see: https://template-
selector.ieee.org/ and follow the steps to find the correct template for your
intended publication. Many publications use the IEEETran LaTeX templates,
however, some publications have their own special templates. Many of these are
based on IEEEtran, but may have special instructions that vary slightly from
those in this document.

##  V Where to get LaTeX help - user groups

The following on-line groups are very helpful to beginning and experienced
LaTeX users. A search through their archives can provide many answers to
common questions.

  * http://www.latex-community.org/

  * https://tex.stackexchange.com/

##  VI Document Class Options in IEEEtran

At the beginning of your LaTeX file you will need to establish what type of
publication style you intend to use. The following list shows appropriate
documentclass options for each of the types covered by IEEEtran.

  * Regular Journal Article

  * $\backslash$documentclass[journal]IEEEtran

  * Conference Paper

  * $\backslash$documentclass[conference]IEEEtran

  * Computer Society Journal Article

  * $\backslash$documentclass[10pt,journal,compsoc]IEEEtran

  * Computer Society Conference Paper

  * $\backslash$documentclass[conference,compsoc]IEEEtran

  * Communications Society Journal Article

  * $\backslash$documentclass[journal,comsoc]IEEEtran

  * Brief, Correspondence or Technote

  * $\backslash$documentclass[9pt,technote]IEEEtran

There are other options available for each of these when submitting for peer
review or other special requirements. IEEE recommends to compose your article
in the base 2-column format to make sure all your equations, tables and
graphics will fit the final 2-column format. Please refer to the document
“IEEEtran_HOWTO.pdf” for more information on settings for peer review
submission if required by your EIC.

##  VII How to Create Common Front Matter

The following sections describe general coding for these common elements.
Computer Society publications and Conferences may have their own special
variations and will be noted below.

###  VII-A Paper Title

The title of your paper is coded as:

    \title{The Title of Your Paper}

Please try to avoid the use of math or chemical formulas in your title if
possible.

###  VII-B Author Names and Affiliations

The author section should be coded as follows:

    \author{Masahito Hayashi
    \IEEEmembership{Fellow, IEEE}, Masaki Owari
    \thanks{M. Hayashi is with Graduate School
    of Mathematics, Nagoya University, Nagoya,
    Japan}
    \thanks{M. Owari is with the Faculty of
    Informatics, Shizuoka University,
    Hamamatsu, Shizuoka, Japan.}
    }

Be sure to use the $\backslash$IEEEmembership command to identify IEEE
membership status. Please see the “IEEEtran_HOWTO.pdf” for specific
information on coding authors for Conferences and Computer Society
publications. Note that the closing curly brace for the author group comes at
the end of the thanks group. This will prevent you from creating a blank first
page.

###  VII-C Running Heads

The running heads are declared by using the $\backslash$markboth command.
There are two arguments to this command: the first contains the journal name
information and the second contains the author names and paper title.

    \markboth{Journal of Quantum Electronics,
    Vol. 1, No. 1, January 2021}
    {Author1, Author2,
    \MakeLowercase{\textit{(et al.)}:
    Paper Title}

###  VII-D Copyright Line

For Transactions and Journals papers, this is not necessary to use at the
submission stage of your paper. The IEEE production process will add the
appropriate copyright line. If you are writing a conference paper, please see
the “IEEEtran_HOWTO.pdf” for specific information on how to code ”Publication
ID Marks”.

###  VII-E Abstracts

The abstract is the first element of a paper after the $\backslash$maketitle
macro is invoked. The coding is simply:

    \begin{abstract}
    Text of your abstract.
    \end{abstract}

Please try to avoid mathematical and chemical formulas in the abstract.

###  VII-F Index Terms

The index terms are used to help other researchers discover your paper. Each
society may have it’s own keyword set. Contact the EIC of your intended
publication for this list.

    \begin{IEEEkeywords}
    Broad band networks, quality of service
    \end{IEEEkeywords}

##  VIII How to Create Common Body Elements

The following sections describe common body text elements and how to code
them.

###  VIII-A Initial Drop Cap Letter

The first text paragraph uses a “drop cap” followed by the first word in ALL
CAPS. This is accomplished by using the $\backslash$IEEEPARstart command as
follows:

    \IEEEPARstart{T}{his} is the first paragraph
    of your paper. . .

###  VIII-B Sections and Subsections

Section headings use standard LaTeX commands: $\backslash$section,
$\backslash$subsection and $\backslash$subsubsection. Numbering is handled
automatically for you and varies according to type of publication. It is
common to not indent the first paragraph following a section head by using
$\backslash$noindent as follows:

    \section{Section Head}
    \noindent The text of your paragraph . . .

###  VIII-C Citations to the Bibliography

The coding for the citations are made with the LaTeX $\backslash$cite command.
This will produce individual bracketed reference numbers in the IEEE style. At
the top of your LaTeX file you should include:

    \usepackage{cite}

For a single citation code as follows:

    see \cite{ams}

This will display as: see [1]

For multiple citations code as follows:

    \cite{ams,oxford,lacomp}

This will display as [1, 2, 3]

###  VIII-D Figures

Figures are coded with the standard LaTeX commands as follows:

    \begin{figure}[!t]
    \centering
    \includegraphics[width=2.5in]{fig1}
    \caption{This is the caption for one fig.}
    \label{fig1}
    \end{figure}

The [!t] argument enables floats to the top of the page to follow IEEE style.
Make sure you include:

    \usepackage{graphicx}

at the top of your LaTeXfile with the other package declarations.

To cross-reference your figures in the text use the following code example:

    See figure \ref{fig1} ...

This will produce:
See figure 1 . . .

Figure 1: This is the caption for one fig.

###  VIII-E Tables

Tables should be coded with the standard LaTeX coding. The following example
shows a simple table.

    \begin{table}
    \begin{center}
    \caption{Filter design equations  ...}
    \label{tab1}
    \begin{tabular}{| c | c | c |}
    \hline
    Order & Arbitrary coefficients &
    coefficients\\
    of filter & $e_m$ &   $b_{ij}$ \\
    \hline
    1& $b_{ij}=\hat{e}.\hat{\beta_{ij}}$,
    & $b_{00}=0$\\
    \hline
    2&$\beta_{22}=(~1,-1,-1,~~1,~~1,~~1)$ &\\
    \hline
    3& $b_{ij}=\hat{e}.\hat{\beta_{ij}}$,
    & $b_{00}=0$,\\
    \hline
    \end{tabular}
    \end{center}
    \end{table}

To reference the table in the text, code as follows:

    Table~\ref{tab1} lists the closed-form...

to produce:

Table I lists the closed-form . . .

TABLE I: A Simple Table Example. Order | Arbitrary coefficients | coefficients
---|---|---
of filter | $e_{m}$ | $b_{ij}$
1 |  $b_{ij}=\hat{e}.\hat{\beta_{ij}}$, | $b_{00}=0$
2 | $\beta_{22}=(~{}1,-1,-1,~{}~{}1,~{}~{}1,~{}~{}1)$ |
3 |  $b_{ij}=\hat{e}.\hat{\beta_{ij}}$, |  $b_{00}=0$,

###  VIII-F Lists

In this section, we will consider three types of lists: simple unnumbered,
numbered and bulleted. There have been numerous options added to IEEEtran to
enhance the creation of lists. If your lists are more complex than those shown
below, please refer to the “IEEEtran_HOWTO.pdf” for additional options.

A plain unnumbered list

  * bare_jrnl.tex

  * bare_conf.tex

  * bare_jrnl_compsoc.tex

  * bare_conf_compsoc.tex

  * bare_jrnl_comsoc.tex

coded as:

    \begin{list}{}{}
    \item{bare\_jrnl.tex}
    \item{bare\_conf.tex}
    \item{bare\_jrnl\_compsoc.tex}
    \item{bare\_conf\_compsoc.tex}
    \item{bare\_jrnl\_comsoc.tex}
    \end{list}

A simple numbered list

  1. 1.

bare_jrnl.tex

  2. 2.

bare_conf.tex

  3. 3.

bare_jrnl_compsoc.tex

  4. 4.

bare_conf_compsoc.tex

  5. 5.

bare_jrnl_comsoc.tex

coded as:

    \begin{enumerate}
    \item{bare\_jrnl.tex}
    \item{bare\_conf.tex}
    \item{bare\_jrnl\_compsoc.tex}
    \item{bare\_conf\_compsoc.tex}
    \item{bare\_jrnl\_comsoc.tex}
    \end{enumerate}

A simple bulleted list

  * •

bare_jrnl.tex

  * •

bare_conf.tex

  * •

bare_jrnl_compsoc.tex

  * •

bare_conf_compsoc.tex

  * •

bare_jrnl_comsoc.tex

coded as:

    \begin{itemize}
    \item{bare\_jrnl.tex}
    \item{bare\_conf.tex}
    \item{bare\_jrnl\_compsoc.tex}
    \item{bare\_conf\_compsoc.tex}
    \item{bare\_jrnl\_comsoc.tex}
    \end{itemize}

###  VIII-G Other Elements

For other less common elements such as Algorithms, Theorems and Proofs, and
Floating Structures such as page-wide tables, figures or equations, please
refer to the “IEEEtran_HOWTO.pdf” section on “Double Column Floats.”

##  IX How to Create Common Back Matter Elements

The following sections demonstrate common back matter elements such as
Acknowledgments, Bibliographies, Appendicies and Author Biographies.

###  IX-A Acknowledgments

This should be a simple paragraph before the bibliography to thank those
individuals and institutions who have supported your work on this article.

    \section{Acknowledgments}
    \noindent Text describing those who
    supported your paper.

###  IX-B Bibliographies

References Simplified: A simple way of composing references is to use the
$\backslash$bibitem macro to define the beginning of a reference as in the
following examples:

[6] H. Sira-Ramirez. “On the sliding mode control of nonlinear systems,”
Systems & Control Letters, vol. 19, pp. 303–312, 1992.

coded as:

    \bibitem{Sira3}
    H. Sira-Ramirez. ‘‘On the sliding mode
    control of nonlinear systems,’’
    \textit{Systems \& Control Letters},
    vol. 19, pp. 303--312, 1992.

[7] A. Levant.“Exact differentiation of signals with unbounded higher
derivatives,” in Proceedings of the 45th IEEE Conference on Decision and
Control, San Diego, California, USA, pp. 5585–5590, 2006.

coded as:

    \bibitem{Levant}
    A. Levant. ‘‘Exact differentiation of
    signals with unbounded higher
    derivatives,’’  in \textit{Proceedings
    of the 45th IEEE Conference on
    Decision and Control}, San Diego,
    California, USA, pp. 5585--5590, 2006.

[8] M. Fliess, C. Join, and H. Sira-Ramirez. “Non-linear estimation is easy,”
International Journal of Modelling, Identification and Control, vol. 4, no. 1,
pp. 12–27, 2008.

coded as:

    \bibitem{Cedric}
    M. Fliess, C. Join, and H. Sira-Ramirez.
    ‘‘Non-linear estimation is easy,’’
    \textit{International Journal of Modelling,
    Identification and Control}, vol. 4,
    no. 1, pp. 12--27, 2008.

[9] R. Ortega, A. Astolfi, G. Bastin, and H. Rodriguez. “Stabilization of
food-chain systems using a port-controlled Hamiltonian description,” in
Proceedings of the American Control Conference, Chicago, Illinois, USA, pp.
2245–2249, 2000.

coded as:

    \bibitem{Ortega}
    R. Ortega, A. Astolfi, G. Bastin, and H.
    Rodriguez. ‘‘Stabilization of food-chain
    systems using a port-controlled Hamiltonian
    description,’’ in \textit{Proceedings of the
    American Control Conference}, Chicago,
    Illinois, USA, pp. 2245--2249, 2000.

###  IX-C Accented Characters in References

When using accented characters in references, please use the standard LaTeX
coding for accents. Do not use math coding for character accents. For example:

    \’e, \"o, \‘a, \~e

will produce: é, ö, à, ẽ

###  IX-D Use of BibTeX

If you wish to use BibTeX, please see the documentation that accompanies the
IEEEtran Bibliography package.

###  IX-E Biographies and Author Photos

Authors may have options to include their photo or not. Photos should be a
bit-map graphic (.tif or .jpg) and sized to fit in the space allowed. Please
see the coding samples below:

    \begin{IEEEbiographynophoto}{Jane Doe}
    Biography text here without a photo.
    \end{IEEEbiographynophoto}

or a biography with a photo

    \begin{IEEEbiography}[{\includegraphics
    [width=1in,height=1.25in,clip,
    keepaspectratio]{fig1.png}}]
    {IEEE Publications Technology Team}
    In this paragraph you can place
    your educational, professional background
    and research and other interests.
    \end{IEEEbiography}

Please see the end of this document to see the output of these coding
examples.

##  X Mathematical Typography
and Why It Matters

Typographical conventions for mathematical formulas have been developed to
provide uniformity and clarity of presentation across mathematical texts. This
enables the readers of those texts to both understand the author’s ideas and
to grasp new concepts quickly. While software such as LaTeX and MathType® can
produce aesthetically pleasing math when used properly, it is also very easy
to misuse the software, potentially resulting in incorrect math display.

IEEE aims to provide authors with the proper guidance on mathematical
typesetting style and assist them in writing the best possible article.

As such, IEEE has assembled a set of examples of good and bad mathematical
typesetting. You will see how various issues are dealt with. The following
publications have been referenced in preparing this material:

  * _Mathematics into Type_ , published by the American Mathematical Society

  * _The Printing of Mathematics_ , published by Oxford University Press

  * _The LaTeXCompanion_, by F. Mittelbach and M. Goossens

  * _More Math into LaTeX_ , by G. Grätzer

  * AMS-StyleGuide-online.pdf, published by the American Mathematical Society

Further examples can be seen at http://journals.ieeeauthorcenter.ieee.org/wp-
content/uploads/sites/7/IEEE-Math-Typesetting-Guide.pdf

###  X-A Display Equations

A simple display equation example shown below uses the “equation” environment.
To number the equations, use the $\backslash$label macro to create an
identifier for the equation. LaTeX will automatically number the equation for
you.

$x=\sum_{i=0}^{n}2{i}Q.$ (1)

is coded as follows:

    \begin{equation}
    \label{deqn_ex1}
    x = \sum_{i=0}^{n} 2{i} Q.
    \end{equation}

To reference this equation in the text use the $\backslash$ref macro. Please
see (1)
is coded as follows:

    Please see (\ref{deqn_ex1})

###  X-B Equation Numbering

Consecutive Numbering: Equations within an article are numbered consecutively
from the beginning of the article to the end, i.e., (1), (2), (3), (4), (5),
etc. Do not use roman numerals or section numbers for equation numbering.

Appendix Equations: The continuation of consecutively numbered equations is
best in the Appendix, but numbering as (A1), (A2), etc., is permissible.

Hyphens and Periods: Hyphens and periods should not be used in equation
numbers, i.e., use (1a) rather than (1-a) and (2a) rather than (2.a) for sub-
equations. This should be consistent throughout the article.

###  X-C Multi-line equations and alignment

Here we show several examples of multi-line equations and proper alignments.

A single equation that must break over multiple lines due to length with no
specific alignment.

$\text{The first line of this example}\\\ \text{The second line of this
example}\\\ \text{The third line of this example}$ (2)

is coded as:

    \begin{multline}
    \text{The first line of this example}\\
    \text{The second line of this example}\\
    \text{The third line of this example}
    \end{multline}

A single equation with multiple lines aligned at the = signs

$\displaystyle a$ $\displaystyle=c+d$ (3) $\displaystyle b$
$\displaystyle=e+f$ (4)

is coded as:

    \begin{align}
    a &= c+d \\
    b &= e+f
    \end{align}

The align environment can align on multiple points as shown in the following
example:

$\displaystyle x$ $\displaystyle=y$ $\displaystyle X$ $\displaystyle=Y$
$\displaystyle a$ $\displaystyle=bc$ (5) $\displaystyle x^{\prime}$
$\displaystyle=y^{\prime}$ $\displaystyle X^{\prime}$
$\displaystyle=Y^{\prime}$ $\displaystyle a^{\prime}$ $\displaystyle=bz$ (6)

is coded as:

    \begin{align}
    x &= y & X & =Y & a &=bc\\
    x’ &= y’ & X’ &=Y’ &a’ &=bz
    \end{align}

###  X-D Subnumbering

The amsmath package provides a subequations environment to facilitate
subnumbering. An example:

$\displaystyle f$ $\displaystyle=g$ (7a) $\displaystyle f^{\prime}$
$\displaystyle=g^{\prime}$ (7b) $\displaystyle\mathcal{L}f$
$\displaystyle=\mathcal{L}g$ (7c)

is coded as:

    \begin{subequations}\label{eq:2}
    \begin{align}
    f&=g \label{eq:2A}\\
    f’ &=g’ \label{eq:2B}\\
    \mathcal{L}f &= \mathcal{L}g \label{eq:2c}
    \end{align}
    \end{subequations}

###  X-E Matrices

There are several useful matrix environments that can save you some
keystrokes. See the example coding below and the output.

A simple matrix:

$\begin{matrix}0&1\\\ 1&0\end{matrix}$ (8)

is coded as:

    \begin{equation}
    \begin{matrix}  0 &  1 \\
    1 &  0 \end{matrix}
    \end{equation}

A matrix with parenthesis

$\begin{pmatrix}0&-i\\\ i&0\end{pmatrix}$ (9)

is coded as:

    \begin{equation}
    \begin{pmatrix} 0 & -i \\
     i &  0 \end{pmatrix}
    \end{equation}

A matrix with square brackets

$\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}$ (10)

is coded as:

    \begin{equation}
    \begin{bmatrix} 0 & -1 \\
    1 &  0 \end{bmatrix}
    \end{equation}

A matrix with curly braces

$\begin{Bmatrix}1&0\\\ 0&-1\end{Bmatrix}$ (11)

is coded as:

    \begin{equation}
    \begin{Bmatrix} 1 &  0 \\
    0 & -1 \end{Bmatrix}
    \end{equation}

A matrix with single verticals

$\begin{vmatrix}a&b\\\ c&d\end{vmatrix}$ (12)

is coded as:

    \begin{equation}
    \begin{vmatrix} a &  b \\
    c &  d \end{vmatrix}
    \end{equation}

A matrix with double verticals

$\begin{Vmatrix}i&0\\\ 0&-i\end{Vmatrix}$ (13)

is coded as:

    \begin{equation}
    \begin{Vmatrix} i &  0 \\
    0 & -i \end{Vmatrix}
    \end{equation}

###  X-F Arrays

The array environment allows you some options for matrix-like equations. You
will have to manually key the fences, but you’ll have options for alignment of
the columns and for setting horizontal and vertical rules. The argument to
array controls alignment and placement of vertical rules.

A simple array

$\left(\begin{array}[]{cccc}a+b+c&uv&x-y&27\\\
a+b&u+v&z&134\end{array}\right)$ (14)

is coded as:

    \begin{equation}
    \left(
    \begin{array}{cccc}
    a+b+c & uv & x-y & 27\\
    a+b & u+v & z & 134
    \end{array} \right)
    \end{equation}

A slight variation on this to better align the numbers in the last column

$\left(\begin{array}[]{cccr}a+b+c&uv&x-y&27\\\
a+b&u+v&z&134\end{array}\right)$ (15)

is coded as:

    \begin{equation}
    \left(
    \begin{array}{cccr}
    a+b+c & uv & x-y & 27\\
    a+b & u+v & z & 134
    \end{array} \right)
    \end{equation}

An array with vertical and horizontal rules

$\left(\begin{array}[]{c|c|c|r}a+b+c&uv&x-y&27\\\ \hline\cr
a+b&u+v&z&134\end{array}\right)$ (16)

is coded as:

    \begin{equation}
    \left(
    \begin{array}{c|c|c|r}
    a+b+c & uv & x-y & 27\\
    a+b & u+v & z & 134
    \end{array} \right)
    \end{equation}

Note the argument now has the pipe ”$|$” included to indicate the placement of
the vertical rules.

###  X-G Cases Structures

Many times we find cases coded using the wrong environment, i.e., array. Using
the cases environment will save keystrokes (from not having to type the
$\backslash$left$\backslash$lbrace) and automatically provide the correct
column alignment.

${z_{m}(t)}=\begin{cases}1,&{\text{if}}\ {\beta}_{m}(t)\\\
{0,}&{\text{otherwise.}}\end{cases}$

is coded as follows:

    \begin{equation*}
    {z_m(t)} =
    \begin{cases}
    1,&{\text{if}}\ {\beta }_m(t),\\
    {0,}&{\text{otherwise.}}
    \end{cases}
    \end{equation*}

Note that the “&” is used to mark the tabular alignment. This is important to
get proper column alignment. Do not use $\backslash$quad or other fixed spaces
to try and align the columns. Also, note the use of the $\backslash$text macro
for text elements such as “if” and “otherwise”.

###  X-H Function Formatting in Equations

In many cases there is an easy way to properly format most common functions.
Use of the $\backslash$ in front of the function name will in most cases,
provide the correct formatting. When this does not work, the following example
provides a solution using the $\backslash$text macro.

$d_{R}^{KM}=\underset{d_{l}^{KM}}{\text{arg
min}}\\{d_{1}^{KM},\ldots,d_{6}^{KM}\\}.$

is coded as follows:

    \begin{equation*}
     d_{R}^{KM} = \underset {d_{l}^{KM}}
     {\text{arg min}} \{ d_{1}^{KM},
     \ldots,d_{6}^{KM}\}.
    \end{equation*}

###  X-I Text Acronyms inside equations

This example shows where the acronym “MSE” is coded using $\backslash$text{}
to match how it appears in the text.

$\text{MSE}=\frac{1}{n}\sum_{i=1}^{n}(Y_{i}-\hat{Y_{i}})^{2}$

    \begin{equation*}
     \text{MSE} = \frac {1}{n}\sum _{i=1}^{n}
    (Y_{i} - \hat {Y_{i}})^{2}
    \end{equation*}

###  X-J Obsolete Coding

Avoid the use of outdated environments, such as eqnarray and $$ math
delimiters, for display equations. The $$ display math delimiters are left
over from PlainTeX and should not be used in LaTeX, ever. Poor vertical
spacing will result.

###  X-K Use Appropriate Delimiters for Display Equations

Some improper mathematical coding advice has been given in various YouTubeTM
videos on how to write scholarly articles, so please follow these good
examples:

For single-line unnumbered display equations, please use the following
delimiters:

    \[ . . . \] or

    \begin{equation*} . . . \end{equation*}

Note that the * in the environment name turns off equation numbering.

For multiline unnumbered display equations that have alignment requirements,
please use the following delimiters:

    \begin{align*} . . . \end{align*}

For single-line numbered display equations, please use the following
delimiters:

    \begin{equation} . . . \end{equation}

For multiline numbered display equations, please use the following delimiters:

    \begin{align} . . . \end{align}

##  XI LaTeX Package Suggestions

Immediately after your documenttype declaration at the top of your LaTeX file
is the place where you should declare any packages that are being used. The
following packages were used in the production of this document.

    \usepackage{amsmath,amsfonts}
    \usepackage{algorithmic}
    \usepackage{array}
    \usepackage[caption=false,font=normalsize,
      labelfont=sf,textfont=sf]{subfig}
    \u00sepackage{textcomp}
    \usepackage{stfloats}
    \usepackage{url}
    \usepackage{verbatim}
    \usepackage{graphicx}
    \usepackage{balance}

##  XII Additional Advice

Please use “soft” (e.g., `\eqref{Eq}`) or `(\ref{Eq})` cross references
instead of “hard” references (e.g., `(1)`). That will make it possible to
combine sections, add equations, or change the order of figures or citations
without having to go through the file line by line.

Please note that the `{subequations}` environment in LaTeX will increment the
main equation counter even when there are no equation numbers displayed. If
you forget that, you might write an article in which the equation numbers skip
from (17) to (20), causing the copy editors to wonder if you’ve discovered a
new method of counting.

BibTEX does not work by magic. It doesn’t get the bibliographic data from thin
air but from .bib files. If you use BibTEX to produce a bibliography you must
send the .bib files.

LaTeX can’t read your mind. If you assign the same label to a subsubsection
and a table, you might find that Table I has been cross referenced as Table
IV-B3.

LaTeX does not have precognitive abilities. If you put a `\label` command
before the command that updates the counter it’s supposed to be using, the
label will pick up the last counter to be cross referenced instead. In
particular, a `\label` command should not go before the caption of a figure or
a table.

Please do not use `\nonumber` or `\notag` inside the `{array}` environment. It
will not stop equation numbers inside `{array}` (there won’t be any anyway)
and it might stop a wanted equation number in the surrounding equation.

##  XIII A Final Checklist

  1. 1.

Make sure that your equations are numbered sequentially and there are no
equation numbers missing or duplicated. Avoid hyphens and periods in your
equation numbering. Stay with IEEE style, i.e., (1), (2), (3) or for sub-
equations (1a), (1b). For equations in the appendix (A1), (A2), etc..

  2. 2.

Are your equations properly formatted? Text, functions, alignment points in
cases and arrays, etc.

  3. 3.

Make sure all graphics are included.

  4. 4.

Make sure your references are included either in your main LaTeX file or a
separate .bib file if calling the external file.

## References

  * [1] Mathematics into Type, American Mathematical Society. Online available:
  * [2] T.W. Chaundy, P.R. Barrett and C. Batey, The Printing of Mathematics, Oxford University Press. London, 1954.
  * [3] The LaTeXCompanion, by F. Mittelbach and M. Goossens
  * [4] More Math into LaTeX, by G. Grätzer
  * [5] AMS-StyleGuide-online.pdf, published by the American Mathematical Society
  * [6] H. Sira-Ramirez. “On the sliding mode control of nonlinear systems,” Systems & Control Letters, vol. 19, pp. 303–312, 1992.
  * [7] A. Levant. “Exact differentiation of signals with unbounded higher derivatives,” in Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, California, USA, pp. 5585–5590, 2006.
  * [8] M. Fliess, C. Join, and H. Sira-Ramirez. “Non-linear estimation is easy,” International Journal of Modelling, Identification and Control, vol. 4, no. 1, pp. 12–27, 2008.
  * [9] R. Ortega, A. Astolfi, G. Bastin, and H. Rodriguez. “Stabilization of food-chain systems using a port-controlled Hamiltonian description,” in Proceedings of the American Control Conference, Chicago, Illinois, USA, pp. 2245–2249, 2000.

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spelled out, appropriately relaxed, correspond to deformations of a symmetric
product orbifold CFT by a “single-trace” $T\bar{T}$ and $J\bar{T}$
deformation, i.e. an operator of the form

$\sum_{i=1}^{N}T_{i}\bar{T}_{i}\;\;\;\;\;\;\;\mbox{or}\;\;\;\;\;\;\;\sum_{i=1}^{N}J_{i}\bar{T}_{i}$
(4.8)

where the index $i$ runs over the $N$ copies of the CFT. For specific choices
of the seed CFT, such theories have been proposed to be holographically dual
to the background obtained from the NS5 decoupling limit of the NS5-F1 system
[32] and, respectively, to a decoupling limit that yields a warped AdS3
spacetime [23, 16].

We concentrate on the single-trace $T\bar{T}$ deformation (the $J\bar{T}$ case
is entirely analogous). We will moreover focus on the untwisted sector only,
where the effect of the single-trace $T\bar{T}$ deformation on the spectrum is
clearly understood, as it acts independently in each copy. In this case, one
can build a Virasoro algebra $L_{m}^{\mu,i}$ associated to each of the copies,
and expect that the corresponding $L_{0}^{\mu,i}$ is related via (4.4) to the
Hamiltonian $H_{i}$ associated to that copy. The total Hamiltonian can then be
written as

$H=\sum_{i=1}^{N}\frac{\sqrt{1+4\mu\left(L_{0}^{\mu,i}+\bar{L}_{0}^{\mu,i}\right)+4\mu^{2}\left(L_{0}^{\mu,i}-\bar{L}_{0}^{\mu,i}\right)^{2}}-1}{2\mu}$
(4.9)

The total Virasoro generators can be built as
$L_{m}^{\mu}=\sum_{i}L_{m}^{\mu,i}$ and are manifestly symmetric under the
interchange of the copies. Since the various copies commute with each other,
we find

$[L_{m}^{\mu},H]=\sum_{i}\alpha_{m}(H^{i},P^{i})L_{m}^{\mu,i}$ (4.10)

Given this commutator, it is straightforward to construct a generator whose
eigenvalue is conserved

$L^{\mu}_{m,S}(t)=\sum_{i}e^{i\alpha_{m}(H^{i},P_{i})t}L_{m,S}^{\mu,i}=\sum_{i}L^{\mu,i}_{m,S}(t)$
(4.11)

Since the states in the symmetric product orbifold are symmetrized, the
expectation values of $H_{i}$ across the copies should be equal, thus
producing an overall time-dependent phase factor, which would be interesting
to reproduce from a gravitational calculation. For such a symmetric state, in
which $\langle H_{i}\rangle=\langle H\rangle/N$, it is interesting to note
that the departure of $\alpha_{m}$ from its usual CFT value is

$\langle\alpha_{m}\rangle=m\hbar-\frac{2m\mu\hbar}{N}\frac{\langle
H\rangle-\langle P\rangle}{1+2\mu m\hbar}+\mathcal{O}(1/N^{2})$ (4.12)

suggesting that even if the non-locality scale of the theory is set by $\mu$,
the departure of the non-local Virasoro algebra from a standard one, as
measured by the commutator $\alpha_{m}$, is suppressed by powers of $1/N$, and
thus possibly hard to detect in the dual gravity picture. If the corrections
associated to the inclusion of twisted sectors do not significantly modify
this picture, this ‘large N’ supression mechanism of the non-local commutator
could provide an explanation for the ubiquity of Virasoro asymptotic symmetry
groups in three-dimensional spacetimes that are not asymptotically AdS3.

_More general deformations_

As noted in the introduction, there exist many examples of string backgrounds
that are obtained through non-trivial decoupling limits, and which could be
dual to non-local, UV complete two-dimensional QFTs, which appear to fulfill
at least the first condition on having a Virasoro algebra.

To understand whether a direct relation between $L_{0}^{\lambda}$ and $H$ can
be established, as required by our second condition, one can try to analyse
the spectrum of the deformed theory (representing the $H$ eigenvalues) as a
function of the undeformed one (representing the eigenvalues of
$L_{0}^{\lambda}$). In $T\bar{T}$ and $J\bar{T}$ \- deformed CFTs, the two are
related in a universal manner that does not depend at all on the details (or
particular spectrum) of the seed CFT. For this reason, the universal relation
between the deformed and undeformed spectra in these theories can be lifted to
a general relation between the operators $H$ and $L_{0}^{\lambda}$.

For the more general deformations we are interested in, the situation is more
nuanced242424We thank Gregory Korchemsky for discussions of this point.. On
the one hand, reducing the existence of the Virasoro symmetry to a question
about the spectrum of the theory is an enormous simplification, because the
spectrum is an observable that can be computed not just in the field theory -
where tracking the irrelevant deformation is hard, if not impossible - but
also in the dual supergravity or string theory picture, where the description
of the deformation may significantly simplify252525For example, in the case of
TsT transformations, their field-theoretical description is usually
intractable, whereas their effect in the dual string theory description is to
simply change the boundary conditions of the worldsheet fields [33].. On the
other hand, it is harder to argue that knowledge of the deformed spectrum will
immediately imply a relation between $H$ and $L^{\lambda}_{0}$, because: i)
the deformations are less universal, and therefore they will only exist in
specific seed CFTs, whose spectra may not be generic and ii) even if one
succeeds in computing the spectrum, this will usually be for a subsector of
the theory (e.g., described by supergravity excitations, or string solutions),
and thus one will not in general have access to the relation between
$L_{0}^{\lambda}$ and $H$ for all possible eigenvalues that they can take.

This being said, it does appear to be true that in many string-theoretical
examples of continuous deformations, either exactly marginal [34], or of the
more general kind that change the asymptotics [35], the deformed dimensions
are (especially at strong coupling) smooth, universal functions of the
undeformed ones for entire subsectors of the theory. One may therefore be able
to write at least an approximate relation between $H$ and $L_{0}^{\lambda}$ in
these subsectors, which may allow one to show, via an argument that parallels
the one given for the $J\bar{T}$ and $T\bar{T}$ deformations, that the
respective subsector is acted upon by a Virasoro symmetry. Such an aproximate
relation could be sufficient to explain why various asymptotic symmetry group
analyses in the dual gravity picture were uncovering Virasoro symmetries. Note
the latter may even act locally, perhaps through a ‘large N’ supression
mechanism of the non-local part of the commutator with $H$, similar to the one
we hinted at in the case of single-trace $T\bar{T}$ deformations. Whether the
full theory has Virasoro symmetry would however depend on the exact expression
for $L_{0}^{\lambda}$ in terms of $H$ and other operators in the theory, which
would presumably need to be rather special, and likely difficult to establish.

##  5 Discussion

In this article, we have analysed the symmetries of $J\bar{T}$ \- deformed
CFTs, at both the classical and the quantum level and showed that in a certain
basis, they consist of two commuting copies of the Virasoro-Kac-Moody algebra,
with the same central extension as that of the undeformed CFT. While the
possibility of having Virasoro symmetries in a non-local theory may seem
surprising, we provided a concrete mechanism for reconciling the non-locality
of the model with these symmetries: the zero mode of the Virasoro algebra is
not identified with the Hamiltonian, as in usual CFTs; rather, it is a non-
linear function of it. We showed such a condition was still compatible with
the existence of an infinite set of Virasoro conserved charges and discussed
possible extensions of this mechanism to more general non-local, UV complete
QFTs.

There are many interesting future directions to explore. First, our
construction of the symmetry generators is somewhat abstract, even in the
classical case, where we did provide entirely explicit formulae for the
conserved charges. It would be worth having a better physical picture of the
action of these non-local symmetries on the fields, as well as a better
understanding of the physical implications of an operator-dependent time
dependence in the generators. One may be able to address these questions by
focussing on a simple example, such as the $J\bar{T}$ \- deformed free boson.

Still on the technical side, it would be interesting to develop an algorithmic
procedure for explicitly computing quantum corrections to the $J\bar{T}$
operator, to all orders in the coupling. As explained in the main text, this
would involve reconstructing the right-moving translation current $T_{\alpha
V}$ from knowledge of the right-moving conserved charges. While this exercise
is clearly possible in principle, it is less clear how to do it in practice,
due to the non-linear nature of the charges. Perhaps reconstructing the right-
moving spectral-flow-invariant current, whose relation to the associated
conserved charges is simpler (2.49), may be a good place to start.

Another interesting direction would be to understand the Virasoro symmetries
discussed in this article from a dual gravitational point of view. The
holographic description of $J\bar{T}$ \- deformed CFTs is given by AdS3
gravity coupled to a Chern-Simons gauge field, with mixed boundary conditions
connecting the two [36]. While the field-dependent Virasoro symmetries of
$J\bar{T}$ \- deformed CFTs were first uncovered in the asymptotic symmetry
group analysis of these boundary conditions, that analysis did not take into
account the subtleties associated with finite size which, as we have recently
learned, play an essential role in rendering the charges well-defined in
compact space. It would thus be very interesting to understand how the fully
non-local piece of the transformation is implemented in the gravitational
description, and in particular whether the current formalisms for computing
asymptotic conserved charges already incorporate such structures, or not.
Since field-dependent asymptotic symmetries are not uncommon in space-times
with non-trivial asymptotics, this analysis could be rather useful in future
applications of asymptotic symmetry group methods to non-AdS holography.

So far, we have only described the action of the non-local Virasoro generators
on the spectrum of the theory. An obvious next step is to study the
consequences of these symmetries on other observables, such as correlation
functions. For this, one should find a basis of operators that transform
nicely under them, analogously to primary operators in usual CFTs. Should one
be able to fix the form of correlation functions using symmetry arguments
alone, one may attempt to give an axiomatic definition of non-local CFTs that
does not specifically refer to their construction in terms of an irrelevant
deformation, which does not appear very natural from a renormalisation group
point of view.

All the questions that we listed above also apply to the $T\bar{T}$
deformation. In this case, it would be interesting to first achieve an
explicit classical realisation of the symmetry generators, as we already have
for $J\bar{T}$ \- deformed CFTs. One can also ask whether an analogue of the
unflowed generators exists in these theories, given that the quantum argument
we provided only predicts the existence of the analogues of the flowed ones.
After finding the classical symmetries, it would be interesting to understand
how they match to the asymptotic symmetry group analysis in the dual AdS3 with
mixed boundary conditions [37].

Finally, a rather interesting avenue to explore is to find more general
examples of non-local CFTs, along the lines of our discussion in section 4.
Possibly the simplest such generalizations are the single-trace analogues of
the $T\bar{T}$ and $J\bar{T}$ deformations that we have already mentioned,
whose advantage is to have a rather concrete definition. In this setting, it
appears worthwhile to better understand the effects of these single trace
deformations on the twisted sectors. By doing so, one would not only be able
to test the holographic duality proposed in [32] at a more refined level, but
also make new field-theoretical predictions that should be reproduced by the
gravitational side of the correspondence. One can also consider deformations
of this duality and understand what is the most general structure of the
deformed CFT that still allows for the existence of a Virasoro symmetry, as
well as whether, in the holographic duals to the string backgrounds discussed
in the introduction, this symmetry is exact or, rather, emergent in the large
$N$, large gap limit.

#### Acknowledgements

The author would like to thank Zohar Komargodsky, Gregory Korchemsky, Mark
Mezei, Nikita Nekrasov, Anton Pribytok, Sylvain Ribault, Samson Shatashvili
and especially Blagoje Oblak for interesting discussions. This research was
supported in part by the ERC Starting Grant 679278 Emergent-BH.

##  Appendix A Properties of $\alpha_{m}^{l,r}$

The quantities $\alpha_{m}^{l,r}$ are operator-valued functions, defined via
the commutation relations of the flowed generators with the right-moving
Hamiltonian

$[\widetilde{\bar{L}}{}^{\lambda}_{m},H_{R}]=\widetilde{\bar{L}}{}^{\lambda}_{m}\alpha_{m}^{r}=\alpha_{m}^{l}\widetilde{\bar{L}}{}^{\lambda}_{m}$
(A.1)

The $\alpha_{m}^{l,r}$ need not commute with
$\widetilde{\bar{L}}{}^{\lambda}_{n}$, reason for which they can in principle
be different. They do commute with $H_{R}$, being a function of it. Their
explicit form to all orders in $\hbar$ can be found using (A.1) and the
following relations

$[\widetilde{\bar{L}}{}{}_{m}^{\lambda},\widetilde{\bar{L}}{}_{0}^{\lambda}]=m\hbar\widetilde{\bar{L}}{}^{\lambda}_{m}\;,\;\;\;\;\;\;\widetilde{\bar{L}}{}^{\lambda}_{0}=R_{v}H_{R}-\lambda
H_{R}\bar{J}_{0}-\frac{k\lambda^{2}}{4}H_{R}^{2}$ (A.2)

both of which are assumed to be exact in $\hbar$. Commuting the second
equation with $\widetilde{\bar{L}}{}_{m}^{\lambda}$, we find

$\frac{k\lambda^{2}}{4}(\alpha^{r}_{m})^{2}+(R-\lambda
Q_{K})\alpha^{r}_{m}-m\hbar=0$ (A.3)

$\frac{k\lambda^{2}}{4}(\alpha^{l}_{m})^{2}-(R-\lambda
Q_{K})\alpha^{l}_{m}+m\hbar=0$ (A.4)

where, as before, $Q_{K}=J_{0}+\frac{\lambda k}{2}H_{R}$. The solutions are
given by

$\alpha_{m}^{r}=-2\frac{R-\lambda Q_{K}-\sqrt{(R-\lambda Q_{K})^{2}+\hbar
km\lambda^{2}}}{k\lambda^{2}}\approx\frac{m\hbar}{R-\lambda
Q_{K}}-\frac{km^{2}\lambda^{2}\hbar^{2}}{4(R-\lambda
Q_{K})^{3}}+\mathcal{O}(\hbar^{3})$ (A.5)

$\alpha^{l}_{m}(E_{R})=2\frac{R-\lambda Q_{K}-\sqrt{(R-\lambda
Q_{K})^{2}-\hbar km\lambda^{2}}}{k\lambda^{2}}\approx\frac{m\hbar}{R-\lambda
Q_{K}}+\frac{km^{2}\lambda^{2}\hbar^{2}}{4(R-\lambda
Q_{K})^{3}}+\mathcal{O}(\hbar^{3})$ (A.6)

This reproduces the classical result (2.27) to leading order in $\hbar$. It
may be sometimes useful to rewrite this using
$\widetilde{\bar{L}}{}^{\lambda}_{0}$ instead of $Q_{K}$, using the identity

$(R-\lambda Q_{K})^{2}=(R-\lambda
J_{0})^{2}-\lambda^{2}k\widetilde{\bar{L}}{}^{\lambda}_{0}$ (A.7)

Note that, to the extent that $Q_{K}$ has real eigenvalues, $\alpha_{m}^{r}$
is well defined for arbitrarily large positive $m$, while $\alpha_{m}^{l}$ is
well defined for large negative $m$. One can consequently choose the best way
to write (A.1), dpending on the value of $m$.

Using the fact that the commutator
$[\widetilde{\bar{K}}{}_{m}^{\lambda},\widetilde{\bar{L}}{}_{0}^{\lambda}]=m\hbar\widetilde{\bar{K}}{}_{m}^{\lambda}$,
one can also show that

$[\widetilde{\bar{K}}{}^{\lambda}_{m},H_{R}]=\widetilde{\bar{K}}{}^{\lambda}_{m}\alpha_{m}^{r}=\alpha_{m}^{l}\widetilde{\bar{K}}{}^{\lambda}_{m}$
(A.8)

for the same values of $\alpha_{m}^{l,r}$.

It is interesting to work out the commutator of
$\widetilde{\bar{L}}{}^{\lambda}_{m}$ with an arbitrary function of $H_{R}$.
We find

$[\widetilde{\bar{L}}_{m},f(H_{R})]=\widetilde{\bar{L}}_{m}(f(H_{R})-f(H_{R}-\alpha_{m}^{r}))=(f(H_{R}+\alpha^{l}_{m})-f(H_{R}))\widetilde{\bar{L}}_{m}$
(A.9)

and similarly for $\widetilde{\bar{K}}{}_{m}^{\lambda}$. One can use this to
show in particular that the difference between $\alpha_{m}^{l}$ and
$\alpha_{m}^{r}$ is consistent with (A.1), because

$\alpha_{n}^{r}(H_{R})=\alpha_{n}^{l}(H_{R}-\alpha_{n}^{r})\;,\;\;\;\;\;\alpha_{n}^{l}(H_{R})=\alpha_{n}^{r}(H_{R}+\alpha_{n}^{l})$
(A.10)

Other useful relations are

$\alpha_{n}^{r}(H_{R}-\alpha^{r}_{m})=\alpha^{r}_{m+n}(H_{R})-\alpha^{r}_{m}(H_{R})\;,\;\;\;\;\;\;\alpha_{n}^{l}(H_{R}+\alpha_{m}^{l})=\alpha_{m+n}^{l}-\alpha_{m}^{l}$
(A.11)

which can be used to show that

$[\widetilde{\bar{L}}{}^{\lambda}_{m},\alpha_{n}^{r}]=\widetilde{\bar{L}}{}^{\lambda}_{m}(\alpha_{m}^{r}+\alpha_{n}^{r}-\alpha^{r}_{m+n})\;,\;\;\;\;\;\;[\widetilde{\bar{L}}{}^{\lambda}_{m},\alpha_{n}^{l}]=(\alpha^{l}_{m+n}-\alpha^{l}_{m}-\alpha^{l}_{n})\widetilde{\bar{L}}{}^{\lambda}_{m}$
(A.12)

and similarly for $\widetilde{\bar{K}}{}^{\lambda}_{m}$. These relations,
together with $\alpha^{l,r}_{0}=0$, can be used to show that any combination
of the flowed right-moving generators and $H_{R}$ satisfies the the Jacobi
identities.

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