#  Theorems of Chisini type

Vik.S. Kulikov  Steklov Mathematical Institute of Russian Academy of Sciences,
Moscow, Russia<EMAIL_ADDRESS>
###### Abstract.

A text of the talk given by the author on Conference on Algebra, Algebraic
Geometry and Number Theory dedicated to the 100th anniversary of I.R.
Shafarevich, Moscow, June 5 – 9, 2023.

##  0.1. Notations and definitions.

Denote by $S$ a smooth irreducible projective surface defined over the field
of complex numbers $\mathbb{C}$ and denote by

$B:=B_{f}=\\{y\in\mathbb{P}^{2}\,\mid\,\sharp f^{-1}(y)<n\\}$

the branch curve of a finite map $f:S\to\mathbb{P}^{2}$, $\deg f=n$.

A divisor $R_{f}$ given in affine neighborhoods $U\subset S$ by

$J(f):=\det\left(\begin{array}[]{cc}\frac{\partial u}{\partial
z}&\frac{\partial u}{\partial w}\\\ \frac{\partial v}{\partial
z}&\frac{\partial v}{\partial w}\end{array}\right)=0,$

is called the ramification divisor  of $f$, where $(z,w)$ are local
coordinates in neighborhoods $U\subset S$, $(u,v)$ are local coordinates in
$V=f(U)\subset\mathbb{P}^{2}$, and $f$ is given by $u=h_{1}(z,w)$,
$v=h_{2}(z,w).$

The curve ${R=R_{f,red}}\subset S$ is called the ramification curve  of $f$.
We have $f^{-1}(B)=R\cup C$, where $C$ is an additional curve to $R$ up to the
proper inverse image of the branch curve.

Note that $f:S\setminus f^{-1}(B)\to\mathbb{P}^{2}\setminus B$ is unramified
$n$-sheeted cover.

##  0.2. Generic covers.

######  Theorem 1.

(Enriques, Chisini, Ciliberto, Flamini, see, for example [3]). The restriction
$f:=\text{pr}_{\mid S}:S\to\mathbb{P}^{2}$ to $S$ of a linear projection
$\text{pr}:\mathbb{P}^{N}\to\mathbb{P}^{2}$ generic with respect to an
embedding $S$ in $\mathbb{P}^{N}$, has the following properties:

  * $(\text{i})$

$f$ is a finite cover, $\deg f=\deg S$,

  * $(\text{ii})$

the branch curve $B_{f}$ is irreducible and the points
$p\in\text{Sing}\,B_{f}$ are only ordinary nodes and ordinary cusps,

  * $(\text{iii})$

$f_{\mid R}:R\to B_{f}$ has degree $1$,

  * $(\text{iv})$

the ramification curve $R$ is smooth and reduced.

A finite map $f:S\to\mathbb{P}^{2}$ is called a generic cover if $f$ satisfies
conditions $(\text{i})$ – $(\text{iv})$.

Finite covers $f_{i}:S_{i}\to\mathbb{P}^{2}$, $i=1,2$, are equivalent
($f_{1}\sim f_{2}$)
a if there exists an isomorphism $h:S_{1}\to S_{2}$ such that
$f_{1}=f_{2}\circ h$,

$S_{1}$$S_{2}$$\mathbb{P}^{2}$$h$$f_{\text{\tiny 1}}$$f_{\text{\tiny 2}}$

O. Chisini in [2] (see also [1]) proved that two generic covers with the same
branch curve are equivalent ”under some suitable conditions on generality”.
Chisini Conjecture. Let $B\subset\mathbb{P}^{2}$ be the branch curve of
generic covers $f_{1}:S\to\mathbb{P}^{2}$ and $f_{2}:S\to\mathbb{P}^{2}$. If
$\max(\deg f_{1},\deg f_{2})\geqslant 5$, then $f_{1}\sim f_{2}$.

######  Example 1.

Let $B\subset\mathbb{P}^{2}$ be the dual curve to a smooth cubic
$\hat{B}\subset\mathbb{P}^{2}$. There exist four non-equivalent generic covers
$f_{i}:S_{i}\to\mathbb{P}^{2}$, $i=1,\dots,4$, branched in $B$, $\deg f_{1}=3$
and $\deg f_{i}=4$, $i\geqslant 2$.

Let $B$ be the branch curve of a generic cover $f:S\to\mathbb{P}^{2}$, then it
is easy to show that degree of $B$ is even, $\deg B=2d$. Denote by $g$ genus
of $B$ and $c$ the number of cusps of $B$.

Applying Hodge index Theorem, in [4] in 1999, it was proved

######  Theorem 2.

Let $B$ be the branch curve of two generic covers
$f_{1}:S_{1}\to\mathbb{P}^{2}$ and $f_{2}:S_{2}\to\mathbb{P}^{2}$. If
$\displaystyle\max(\deg f_{1},\deg f_{2})>\frac{4(3d+g-1)}{2(3d+g-1)-c}$, then
$f_{1}\sim f_{2}$.

Using Bogomolov – Miaoka – Yau inequality and Theorem 2, S. Nemirovski ([11])
proved the following

######  Theorem 3.

Let $B\subset\mathbb{P}^{2}$ be the branch curve of generic covers
$f_{1}:S\to\mathbb{P}^{2}$ and $f_{2}:S\to\mathbb{P}^{2}$. If $\max(\deg
f_{1},\deg f_{2})\geqslant 12$ then $f_{1}\sim f_{2}$.

Using Theorems 2 and 3, in [5], it was proved the following

######  Theorem 4.

Let $B\subset\mathbb{P}^{2}$ be the branch curve of generic projection
$f_{1}:S\to\mathbb{P}^{2}$ and generic cover $f_{2}:S\to\mathbb{P}^{2}$. If
$\deg f_{1}\geqslant 5$ then $f_{1}\sim f_{2}$.

##  0.3. Monodromy of finite covers.

An unramified $n$-sheeted cover

$f:X=S\setminus f^{-1}(B)\to Y=\mathbb{P}^{2}\setminus B$

defines a monodromy homomorphism $f_{*}:\pi_{1}(\mathbb{P}^{2}\setminus
B,q)\to\mathbb{S}_{n}$, where $\mathbb{S}_{n}$ is the symmetric group acting
on the fibre $f^{-1}(q)$. The group
$G_{f}:=\text{im}\,f_{*}\subset\mathbb{S}_{n}$ is called the monodromy group
of $f$.

Conversely, if a homomorphism $f_{*}:\pi_{1}(\mathbb{P}^{2}\setminus
B,q)\to\mathbb{S}_{n}$ is given, then it defines an unramified $n$-sheeted
cover $f:S\setminus f^{-1}(B)\to\mathbb{P}^{2}\setminus B$.

Theorem (Riemann - Stein, [12]). For an unramified finite cover
$\widetilde{f}:X\to Y=\mathbb{P}^{2}\setminus B,$ there exist an uniquely
defined extension $X\subset S$, where $S$ is a normal surface, and a finite
holomorphic map $f:S\to\mathbb{P}^{2}$ such that $f_{|X}=\widetilde{f}$. The
surface $S$ is irreducible if and only if the monodromy group $G_{f}$ is a
transitive subgroup of $\mathbb{S}_{n}$.

Denote by $V_{p}\subset\mathbb{P}^{2}$ a sufficiently small neighborhood of a
point $p\in B\subset\mathbb{P}^{2}$ bi-holomorphic to the ball
$\mathbb{B}_{\varepsilon}=\\{(u,v)\in\mathbb{C}^{2}\mid\sqrt{u\overline{u}+v\overline{v}}<\varepsilon^{2}\\}$,
$p=(0,0)$. If $\varepsilon\ll 1$, then $(B,p):=V_{p}\cap B$ is called a germ
of the curve $B$ at the point $p$.

Theorem–Definition. Let $V_{p}\simeq\mathbb{B}_{\varepsilon}$ be as above.
Then the group

$\pi_{1}^{loc}(B,p):=\pi_{1}(\mathbb{B}_{\varepsilon}\setminus B)$

does not depend on $\varepsilon$ if $\varepsilon\ll 1$. The group
$\pi_{1}^{loc}(B,p)$ is called the local fundamental group of the curve germ
$(B,p)$.

##  0.4. The local monodromy.

Let $B=B_{f}$ be the branch curve of a finite cover $f:S\to\mathbb{P}^{2}$,
$\deg f=n$, and let $U_{p}:=f^{-1}(V_{p})\subset S$. The restriction
$f_{p}=f_{\mid U_{p}}:U_{p}\to V_{p}$ of the cover $f$ to $U_{p}$ is called a
germ of the cover $f:S\to\mathbb{P}^{2}$ at the point $p$.

The embedding $\iota:V_{p}\hookrightarrow\mathbb{P}^{2}$ defines a
homomorphism

$\iota_{*}:\pi_{1}^{loc}(B,p)=\pi_{1}(V_{p}\setminus
B,q)\to\pi_{1}(\mathbb{P}^{2}\setminus B,q)$

and the local monodromy homomorphism

$f_{p*}:=f_{*}\circ\iota_{*}:\pi_{1}^{loc}(B,p)\to\mathbb{S}_{n}.$

The group $G_{f,p}$$:=\text{im}\,f_{p*}$ is called the local monodromy group
of $f:S\to\mathbb{P}^{2}$ at $p$.

The inverse image $U_{p}:=f^{-1}(V_{p})=\bigsqcup_{j=1}^{m_{p}}$
$U_{p,j}$$\subset S$ is a disjoint union of connected surfaces $U_{p,j}$,
$U_{p,j_{1}}\cap U_{p,j_{2}}=\emptyset$ for $j_{1}\neq j_{2}$. The restriction
$f_{p,j}:=f_{\mid U_{p,j}}:U_{p,j}\to V_{p}$, $\deg f_{p,j}=n_{j}$, of $f_{p}$
to $U_{p,j}$ is called a connected component of the cover germ $f_{p}:U_{p}\to
V_{p}$ of $f:S\to\mathbb{P}^{2}$, $n=\sum_{j=1}^{m_{p}}n_{j}$.

Let $(B_{j},p)$ be the branch curve germ of $f_{p,j}:U_{p,j}\to V_{p}$, $\deg
f_{p,j}>1$. We have $(B_{j},p)\subset(B,p)$ and $(B_{j},p)$ is called a
$f$-component of $(B,p)$. If $\deg f_{p,j}=n_{j}=1$ for some $j$, then we say
that $f_{p,j}$ is branched at $p$ and $B_{j}:=p$ is called also a
$f$-component of $(B,p)$ and, by definition, $\pi_{1}^{loc}(B_{j},p)=\text{\bf
1}$.

For the base point $q$ of the fundamental group
$\pi_{1}^{loc}(B,p)=\pi_{1}(V_{p}\setminus B,q)$, let us consider its inverse
image

$f^{-1}(q)=\\{q_{1},\dots,q_{n}\\}\subset
U_{p}=f^{-1}(V_{p})=\bigsqcup_{j=1}^{m_{p}}U_{p,j}\subset S.$

Denote by $I_{p,j}:=f^{-1}(q)\cap U_{p,j}$,
$f^{-1}(q)=\bigsqcup_{j=1}^{m_{p}}I_{p,j}$ and by
$\mathbb{S}_{n_{j}}:=\mathbb{S}(I_{p,j})\subset\mathbb{S}_{n}$ the symmetric
group acting on $I_{p,j}$ and considered as a subgroup of $\mathbb{S}_{n}$,

$\psi_{I}:\mathbb{S}_{n_{1}}\times\dots\times\mathbb{S}_{n_{m_{p}}}\,\hookrightarrow\,\mathbb{S}_{n}.$

The embedding $\iota_{j}:V_{p}\setminus B\hookrightarrow V_{p}\setminus B_{j}$
defines an epimorphism

$\iota_{j_{*}}:\pi_{1}^{loc}(B,p)=\pi_{1}(V_{p}\setminus
B)\twoheadrightarrow\pi_{1}(V_{p}\setminus B_{j})=\pi_{1}^{loc}(B_{j},p).$

Let $f_{p,j*}:\pi_{1}^{loc}(B_{j},p)\to\mathbb{S}_{n_{j}}$ be the monodromy
homomorphism of $f_{p,j}:U_{p,j}\to V_{p}$. Consider homomorphisms
$\varphi_{p,j}$$=f_{p,j*}\circ\iota_{j_{*}}:\pi_{1}^{loc}(B,p)\to\mathbb{S}_{n_{j}}$
and

$\varphi_{p}:=\varphi_{p,1}\times\dots\times\varphi_{p,m_{p}}:\pi_{1}^{loc}(B,p)\longrightarrow\mathbb{S}_{n_{1}}\times\dots\times\mathbb{S}_{n_{m_{p}}}.$

Then the local monodromy homomorphism $f_{p*}$ is

$f_{p*}=\psi_{I}\circ\varphi_{p}:\pi_{1}^{loc}(B,p)\to\mathbb{S}_{n}.$

Remind that singularities $(B_{1},p)$ and $(B_{2},p)$ are called deformation
equivalent if their weighted dual graphs of the minimal resolutions of
$(B_{1},p)$ and $(B_{2},p)$ up to divisors with normal crossings, are
isomorphic.

It is well-known ([13], [14]) that if $(B_{1},p)$ and $(B_{2},p)$ are
deformation equivalent, then their local fundamental groups are naturally
isomorphic, $\pi_{1}^{loc}(B_{1},p)\cong\pi_{1}^{loc}(B_{2},p)$ and if
$f_{1*}:\pi_{1}^{loc}(B_{1},p)\to\mathbb{S}_{n_{1}}$ is the monodromy
homomorphism of a finite cover $f_{1}:U_{1}\to V_{p}$, where $U_{1}$ is a
smooth surface, then ([8]) the monodromy homomorphism
$f_{2*}:\pi_{1}^{loc}(B_{2},p)\cong\pi_{1}^{loc}(B_{1},p)\stackrel{{\scriptstyle
f_{1*}}}{{\longrightarrow}}\mathbb{S}_{n_{1}}$ defines a finite cover
$f_{2}:U_{2}\to V_{p}$ of $V_{p}$ in which $U_{2}$ is a smooth surface.

By van Kampen – Zariski Theorem, the fundamental group
$\pi_{1}(\mathbb{P}^{2}\setminus B,q)$ is generated by $\deg B$, so called,
geometric generators, i.e elements $\gamma\in\pi_{1}(\mathbb{P}^{2}\setminus
B,q)$ represented by simple loops around the curve $B$. The local fundamental
group $\pi_{1}^{loc}(B,p)$ is also generated by $\mu_{p}(B)$ geometric
generators $\gamma\in\pi_{1}(V_{p}\setminus B,q)$, where $\mu_{p}(B)$ is the
multiplicity of the curve $B$ at the point $p\in B$.

##  0.5. The case when $B$ is an irreducible curve

Next, we assume that $B$ is an irreducible curve. In this case the geometric
generators $\gamma\in\pi_{1}(\mathbb{P}^{2}\setminus B,q)$ are conjugated in
$\pi_{1}(\mathbb{P}^{2}\setminus B,q)$ and if
$f_{*}:\pi_{1}(\mathbb{P}^{2}\setminus B,q)\to\mathbb{S}_{n}$ is the monodromy
homomorphism of a finite cover $f:S\to\mathbb{P}^{2}$, then
$f_{*}(\gamma)\in\mathbb{S}_{n}$ are conjugated in $\mathbb{S}_{n}$ for
geometric generators $\gamma\in\pi_{1}(\mathbb{P}^{2}\setminus B,q)$.

Denote by $\overline{r}_{f}=(r_{1},\dots,r_{k})$ the cyclical type of
permutation $f_{*}(\gamma)$, i.e., the collection of lengths of nontrivial
cycles included in the factorization of $f^{*}(\gamma)$ into the product of
disjoint cycles. We call $\overline{r}_{f}$ the monodromy cyclical type of the
cover $f:S\to\mathbb{P}^{2}$, the collection
$\mathcal{M}_{f}=\\{f_{p*}:\pi_{1}^{loc}(B_{f},p)\to\mathbb{S}_{n}\mid
p\in\text{Sing}\,B_{f}\\}$ is called the local monodromy data and a triple
$\text{pas}(f)=(B_{f},\overline{r}_{f},\mathcal{M}_{f})$ is called the
passport of the cover $f:S\to\mathbb{P}^{2}$.

##  0.6. Definition of the concept of a Chisini Theorem.

The next part of my talk is an overview of results related to answers to the
following

Question. For which finite covers $f:S\to\mathbb{P}^{2}$ do their passports
uniquely determine these covers up to equivalence?

Denote by $\mathcal{F}_{\overline{r},\mathcal{T}}$ a set of finite covers
$f:S\to\mathbb{P}^{2}$ such that $\overline{r}_{f}=\overline{r}$ and the
singularity types of the $f$-components $(B_{p},j)$ of $(B_{f},p)$,
$p\in\text{Sing}\,B_{f}$, belong to a set of singularity types $\mathcal{T}$
of plane curve germs. Let
$\mathcal{F}_{\overline{r}}=\displaystyle\bigcup_{\mathcal{T}}\mathcal{F}_{\overline{r},\mathcal{T}}$.

A statement is called Chisini Theorem with constant $\mathfrak{n}$ for the
covers belonging to a set $\mathcal{F}_{\overline{r},\mathcal{T}}$ if it
claims that there exists a constant
$\mathfrak{n}=\mathfrak{n}(\mathcal{F}_{\overline{r},\mathcal{T}})\in\mathbb{N}$
such that if

$\text{pas}(f_{1})=\text{pas}(f_{2})\,\,\,\text{and}\,\,\max(\deg f_{1},\deg
f_{2})\geqslant\mathfrak{n}$

for covers $f_{1}$ and $f_{2}\in\mathcal{F}_{\overline{r},\mathcal{T}}$, then
$f_{1}\sim f_{2}$.

The following Theorem is well-known.

######  Theorem 5.

For all $r\geqslant 2$, Chisini Theorem with constant $\mathfrak{n}=1$ holds
for finite covers belonging to the set $\mathcal{F}_{(r),\emptyset}$.

Let $\mathcal{G}en=\mathcal{F}_{(2),\\{A_{0},A_{2}\\}}$ be the set of generic
covers and
$\mathcal{G}en_{\text{pr}}\subset\mathcal{F}_{(2),\\{A_{0},A_{2}\\}}$ the set
of generic projections.

Theorems 2 and 3 are examples of Chisini Theorems with constant
$\mathfrak{n}=12$ for the covers belonging to the set $\mathcal{G}en$ and with
constant $\mathfrak{n}=5$ for the covers belonging to the set
$\mathcal{G}en_{\text{pr}}$.

######  Proposition 1.

For each $m>1$, there is no a constant $\mathfrak{n}\in\mathbb{N}$ for which
Chisini Theorem with constant $\mathfrak{n}$ would be fulfilled for the covers
belonging to $\mathcal{F}_{(\underbrace{2,\dots,2}_{m}),\\{A_{0},A_{2}\\}}$.

###### Proof.

Let $S_{o}$ be an Abelian surface whose endomorphism ring is $\mathbb{Z}$,
then, for example, for each prime number $m$, there are $\frac{m^{4}-1}{m-1}$
finite et$\hat{\text{a}}$le cyclic covers $f_{i}:S_{i}\to S_{o}$ of degree $m$
such that $S_{i_{1}}$ and $S_{i_{2}}$ are not isomorphic for $i_{1}\neq
i_{2}$. Therefore if $B$ is the branch curve of a generic projection
$f_{o}:S_{o}\to\mathbb{P}^{2}$, then $B$ is the branch curve of at least
$\frac{m^{4}-1}{m-1}$ covers $\\{\widetilde{f}_{i}=f_{o}\circ
f_{i}:S_{i}\to\mathbb{P}^{2}\\}\in\mathcal{F}_{(\underbrace{2,\dots,2}_{m}),\\{A_{0},A_{2}\\}}$
of degree $m\cdot\deg f_{o}$.

Note that for any $N\gg 1$ we can take a generic projection
$f_{o}:S_{o}\to\mathbb{P}^{2}$ of $\deg f_{o}>N$. ∎

Proposition 1 shows that there is no hope that Chisini Theorems hold for
monodromy cyclical types that do not consist of a single cycle. Note also that
by now almost nothing is known about theorems of Chisini type in the case when
the monodromy cyclical type is $\overline{r}_{f}=(r)$ with $r\geqslant 3$,
except, perhaps, Theorem 5. Therefore, further we will limit ourselves to
considering the case when $r=2$.

##  0.7.  The case of monodromy cyclical type $\overline{r}_{f}=(2)$.

In order to be able to formulate statements related to the answers to the
question formulated above, it is necessary to recall a few more definitions
related to the invariants of the singularities of plane curves.

Let $(B,p)=V_{p}\cap B$ be a germ of a curve $B\subset\mathbb{P}^{2}$ at a
point $p$. Denote by $\delta_{p}(B)$ the $\delta$-invariant of the singularity
$(B,p)$, and denote by $\mu_{p}(B)$ the multiplicity of the singularity
$(B,p)$ at $p$.

Let $(B,p)=(B_{1},p)\cup\dots\cup(B_{k},p)$ be the union of $k$ irreducible
germs $(B_{i},p)$ of $(B,p)$, The number

$c_{v,p}(B):=\sum_{i=1}^{k}(\mu_{p}(B_{i})-1)$

is called the number of virtual cusps of the curve germ $(B,p)$ and

$c_{v}(B):=\sum_{p\in\text{Sing}\,B}c_{v,p}$

is called the number of virtual cusps of the curve $B\subset\mathbb{P}^{2}$.
The number

$n_{v}(B):=\delta(B)-c_{v}(B)$

is called the number of virtual nodes of the curve $B$.

Note that the numbers of virtual cusps and nodes of a plane curve and its dual
curve play the same role as the numbers of ordinary casps and ordinary nods in
the case of cuspidal curves, that is, if we substitute the numbers of virtual
cusps and nods of a curve and its dual curve in the classical Plücker formulas
instead of the numbers of ordinary cusps and nods, then we get again equality
in these formulas (see [7]).

As it was mentioned above, if $f_{*}:\pi_{1}(\mathbb{P}^{2}\setminus
B,q)\to\mathbb{S}_{n}$ is the monodromy homomorphism of
$f:S\to\mathbb{P}^{2}$, then the images $f_{*}(\gamma)\in\mathbb{S}_{n}$ are
transpositions in $\mathbb{S}_{n}$ for geometric generators
$\gamma\in\pi_{1}(\mathbb{P}^{2}\setminus B,q)$. Consequently,
$G_{f}=\mathbb{S}_{n}$.

Let $(B,p)=V_{p}\cap B$ be a germ of $B$ at $p\in\text{Sing}\,B$,
$f_{p,j}:U_{p,j}\to V_{p}$, $\deg f_{p,j}=n_{j}$, a connected component of the
cover germ $f_{p}:U_{p}\to V_{p}$ of $f:S\to\mathbb{P}^{2}$, and
$(B_{j},p)\subset(B,p)$ is the $f$-component of $(B,p)$. Since fundamental
group $\pi_{1}^{loc}(B_{j},p)$ is generated by $\mu_{p}(B_{j})$ geometric
generators, then $G_{f,p}=\mathbb{S}_{n_{j}}$,
$n_{j}\leqslant\mu_{p}(B_{j})+1$. A connected component $f_{p,j}:U_{p,j}\to
V_{p}$ is called non-degenerate if $n_{j}=\mu_{p}(B_{j})+1$.

##  0.8. Extra property.

Consider two copies of a connected component $f_{p,j}:U_{p,j}\to V_{p}$ of a
cover germ $f_{p}:U_{p}\to V_{p}$ of $f:S\to\mathbb{P}^{2}$. Denote by

$f_{i}:=f_{p,j}:U_{i}:=U_{p,j}\to V_{p},$

$\deg f_{i}=n_{j}$, $f_{i}^{-1}(B_{j})=R_{i}\cup C_{i}\subset U_{i}$, $i=1,2$,
and consider the fibre product

$U_{1}\times_{V_{p}}U_{2}=\\{\,(x,y)\in U_{1}\times
U_{2}\,\,\mid\,\,f_{1}(x)=f_{2}(y)\,\,\\}.$

Let $\widetilde{X}$$=\widetilde{U_{1}\times_{V_{p}}U_{2}}$ be the
normalization of $U_{1}\times_{V_{p}}U_{2}$. Denote by

$g_{\text{\tiny 1}}:\widetilde{X}\to U_{1},\qquad g_{\text{\tiny
2}}:\widetilde{X}\to U_{2},\qquad g_{\text{\tiny 1,2}}:\widetilde{X}\to V_{p}$

the corresponding natural morphisms, $\deg g_{\text{\tiny 1}}=\deg
g_{\text{\tiny 2}}=n_{j}$, $\deg g_{\text{\tiny 1,2}}=n_{j}^{2}$.

Note that
$\widetilde{X}=\widetilde{X}^{\prime}\bigsqcup\widetilde{X}^{\prime\prime}$ is
a disjoint union of two connected components $\widetilde{X}^{\prime}$ and
$\widetilde{X}^{\prime\prime}$, $\deg g_{i\mid\widetilde{X}^{\prime}}=n_{j}-1$
and $\deg g_{i\mid\widetilde{X}^{\prime\prime}}=1$.

We have the following commutative diagram.

$p_{\text{\tiny 1}}\in U_{\text{\tiny 1}}$$U_{\text{\tiny 2}}\ni
p_{\text{\tiny 2}}$$V_{p}\ni p$$\widetilde{X}^{\prime}\ni p_{\text{\tiny
1,2}}$$g_{\text{\tiny 1}}$$g_{\text{\tiny 2}}$$f_{\text{\tiny
1}}$$f_{\text{\tiny 2}}$Fig. 1

Note that $p_{\text{\tiny
1,2}}=g_{1}^{-1}(f_{1}^{-1}(p))\cap\widetilde{X}^{\prime}$ is, possibly, the
only singular point of $\widetilde{X}^{\prime}$.

Let us add the inverse images of the curve germ $(B_{j},p)$ into the
commutative diagram depicted in Fig 1. We obtain a commutative diagram
depicted in Fig 2 in which

$\widetilde{R}\cup\widetilde{C}_{\text{\tiny}1}=g_{\text{\tiny}1}^{-1}(R_{\text{\tiny}1}),\qquad\widetilde{R}\cup\widetilde{C}_{\text{\tiny}2}=g_{\text{\tiny}2}^{-1}(R_{\text{\tiny}2}),$

and

$\widetilde{R}=g_{\text{\tiny 1}}^{-1}(R_{\text{\tiny 1}})\cap g_{\text{\tiny
2}}^{-1}(R_{\text{\tiny 2}}).$

$U_{\text{\tiny 1}}$$p_{\text{\tiny 1}}\in C_{\text{\tiny 1}}\cup
R_{\text{\tiny 1}}\subset$$U_{\text{\tiny 2}}$$\supset R_{\text{\tiny 2}}\cup
C_{\text{\tiny 2}}\supset p_{\text{\tiny 2}}$$V_{p}\ni p$$V_{p}\cap
B_{f}\supset(B_{j},p)\subset$$\widetilde{X}^{\prime}$$p_{\text{\tiny
1,2}}\in\widetilde{R}\cup\widetilde{C}_{\text{\tiny
1}}\subset$$\supset\widetilde{R}\cup\widetilde{C}_{\text{\tiny 2}}\ni
p_{\text{\tiny 1,2}}$$g_{\text{\tiny 1}}$$g_{\text{\tiny 2}}$$f_{\text{\tiny
1}}$$f_{\text{\tiny 2}}$Fig. 2

Denote by $\nu:X^{\prime}\to\widetilde{X}^{\prime}$ a resolution of
singularity of $\widetilde{X}^{\prime}$. We obtain the following commutative
diagram in which $\overline{R}=\nu^{-1}(\widetilde{R})$ and
$\overline{C}_{1}=\nu^{-1}(\widetilde{C}_{1})$ are the proper inverse images
of curve germs $\widetilde{R}$ and $\widetilde{C}_{1}$ and
$(g_{1}\circ\nu)^{*}(R_{1})=\overline{R}+\overline{C}_{1}+E$ is the inverse
image of divisor $R_{1}\in\text{Div}\,U_{1}$,
$\text{Supp}(E)\subset\nu^{-1}(p_{\text{\tiny 1,2}})$.

$U_{\text{\tiny 1}}$$p_{\text{\tiny 1}}\in R_{\text{\tiny
1}}\subset$$U_{\text{\tiny 2}}$$V_{p}\ni p$$V_{p}\cap
B_{f}\supset(B_{j},p)\subset$$\widetilde{X}^{\prime}\ni p_{\text{\tiny
1,2}}$$\widetilde{R}\cup\widetilde{C}_{\text{\tiny
1}}\,\,\subset$$g_{\text{\tiny 1}}$$X^{\prime}\ni\nu^{-1}(p_{\text{\tiny
1,2}})$$\nu$$g_{\text{\tiny 2}}$$\overline{R}+\overline{C}_{\text{\tiny
1}}+E\subset$$f_{\text{\tiny 1}}$$f_{\text{\tiny 2}}$Fig. 3

######  Definition 1.

([10]) A germ $f_{\text{\tiny 1}}=f_{p,j}:U_{1}=U_{p,j}\to V_{p}$ has extra
property over the point $p\in\text{Sing}\,B$ if either $\deg f_{\text{\rm\tiny
1}}\leqslant 2$, or if $\deg f_{\rm\text{\tiny 1}}>2$, then there is a
presentation of $E$ in the form $E=E_{\overline{R}}+E_{\overline{C}_{1}}$ such
that

$(\overline{R}+E_{\overline{R}},\overline{C}_{1}+E_{\overline{C}_{1}})_{X^{\prime}}\leqslant
2\delta_{p_{\rm\text{\tiny 1}}}(R_{1})+c_{v,p}(B_{j}).$

Denote by $\mathcal{T}_{Ch_{\text{\tiny 12}}}$ the set of singularity types of
$f$-components $(B_{j},p)$ of connected non-degenerate cover germs
$f_{p,j}:U_{p,j}\to V_{p}$ having extra property.

######  Theorem 6.

([9], [10]) Chisini Theorem with constant $\mathfrak{n}=12$ holds for covers
$f:S\to\mathbb{P}^{2}$ belonging to
$\mathcal{F}_{(2),\mathcal{T}_{Ch_{\text{\rm\tiny 12}}}}$, i.e., if
$pas(f_{1})=pas(f_{2})$ and $\max(\deg f_{1},\deg f_{2})\geqslant 12$ for
covers $f_{1}:S_{1}\to\mathbb{P}^{2}$ and $f_{2}:S_{2}\to\mathbb{P}^{2}$
belonging to $\mathcal{F}_{(2),\mathcal{T}_{Ch_{\text{\rm\tiny 12}}}}$, then
$f_{1}\sim f_{2}$.

######  Theorem 7.

(1) The set of $f$-components of branch curve germs of ADE-singularity
Theorem 6.aaaaa types is $\\{A_{0},E_{6},A_{3n+2},n\geqslant 0\\}$.
Theorem 6.aa (2) $\\{A_{0},E_{6},A_{3n+2},n\geqslant
0\\}\subset\mathcal{T}_{Ch_{\text{\rm\tiny 12}}}$.

######  Corollary 1.

Chisini Theorem with constant $\mathfrak{n}=12$ holds for finite covers
$f:S\to\mathbb{P}^{2}$ belonging to
$\mathcal{F}_{(2),\\{A_{0},E_{6},A_{3n+2},\,\,n\geqslant 0\\}}$.

##  0.9. Dualizing covers of the plane associated with curves immersed in the
plane.

Let $\iota:C\to{\mathbb{P}}^{2}$ be a morphism of smooth irreducible reduced
projective curve $C$ to the projective plane such that $\iota_{\mid
C}:C\to\iota(C)$ is a bi-rational morphism, $\deg\iota(C)=n\geqslant 2$ and
let $B\subset\hat{\mathbb{P}}^{2}$ be the dual curve to the curve $\iota(C)$
(the curve $B$ consists of lines $l_{p}\in\hat{\mathbb{P}}^{2}$, $p\in C$,
tangent to the curve $\iota(C)$). The correspondence graph between curves
$\iota(C)$ and $B$ is a curve $\check{C}$ in
${\mathbb{P}}^{2}\times\hat{\mathbb{P}}^{2}$ (the so-called Nash curve blowup
of $\iota(C)$), which lies in the variety of incident
$I=\\{(P,l)\in{\mathbb{P}}^{2}\times\hat{\mathbb{P}}^{2}\mid P\in l\\}$,

$\check{C}=\\{(\nu(p),l_{p})\in I\mid p\in C\,\,\text{and}\,\,l_{p}\,\text{is
a tangent line to}\,\,\iota(C)\,\,\text{at}\,\,\iota(p)\in C\\}.$

Let $\text{pr}_{1}:\mathbb{P}^{2}\times\hat{\mathbb{P}}^{2}\to\mathbb{P}^{2}$
and
$\text{pr}_{2}:\mathbb{P}^{2}\times\hat{\mathbb{P}}^{2}\to\hat{\mathbb{P}}^{2}$
be the projections to the factors,
$X^{\prime}=\text{pr}_{1}^{-1}(\iota(C))\cap I$, and
$h:X^{\prime}\to\hat{\mathbb{P}}^{2}$ the restriction of the projection
$\text{pr}_{2}$ to $X^{\prime}$. Obviously, $h^{-1}(l)$ consists of the points
$(P,l)\in\mathbb{P}^{2}\times\hat{\mathbb{P}}^{2}$ such that $P\in\iota(C)\cap
l$ and hence, $\deg h=\deg\iota(C)=n$.

In [6], the notion of the dualizing cover of the plane associated with the
curve $\iota(C)\subset\mathbb{P}^{2}$ was introduced. Namely, let $\nu:X\to
X^{\prime}$ be the normalization of the surface $X^{\prime}$, then the
morphism $f_{\iota(C)}=h\circ\nu:X\to\hat{\mathbb{P}}^{2}$ is called the
dualizing cover of the plane associated with the curve
$\iota(C)\subset\mathbb{P}^{2}$. We have $\deg f_{\iota(C)}=n$. It is easy to
see that $X$ is isomorphic to the fibre product $C\times_{\iota(C)}X^{\prime}$
of the morphism $\iota:C\to\iota(C)$ and the projection
$\text{pr}_{1}:X^{\prime}\to\iota(C)$. The projection
$\text{pr}_{1}:X^{\prime}\to\iota(C)$ defines a structure of ruled surface on
$X^{\prime}$ and this structure induces a ruled structure on $X$ over $C$.
Note that $\widetilde{C}=\nu^{-1}(\check{C})\subset X$ is a section of this
ruled structure, and the image $f_{\iota(C)}(F_{p})$ of a fibre $F_{p}$ is a
line $L_{\iota(p)}\subset\hat{\mathbb{P}}^{2}$ dual to the point
$\iota(p)\in\iota(C)\subset\mathbb{P}^{2}$.

######  Theorem 8.

Let $\iota:C\hookrightarrow\hat{\mathbb{P}}^{2}$ be an immersion of
irreducible projective curve $C$ of genus $g$ and
$f_{\iota(C)}:X_{\iota(C)}\to\mathbb{P}^{2}$ the dualizing cover associated
with $\iota(C)$, and let a cover $f:X\to\mathbb{P}^{2}$ belong to the set
$\mathcal{F}_{(2)}$. If $pas(f)=pas(f_{\iota(C)})$, then the covers $f$ and
$f_{\iota(C)}$ are equivalent if either $\deg\iota(C)\geqslant 8$ and
$g\geqslant 1$, or $\deg\iota(C)\geqslant 12$ and $g=0$.

Consider covers given by functions $u=z$, $v=w^{n}-nwz$. They are branched in
curves given by equation $v^{n-1}-(1-n)^{n-1}u^{n}=0$. Denote by $T_{n,1}$,
$n\geqslant 2$, the singularity type at the origin of function
$v^{n-1}-(1-n)^{n-1}u^{n}$.

######  Remark 1.

We have $T_{2,1}=A_{0}$, $T_{3,1}=A_{2}$, and $T_{4,1}=E_{6}$.

######  Theorem 9.

([9], [10]) $\\{T_{n,1},\,\,n\geqslant
2\\}\subset\mathcal{T}_{Ch_{\text{\rm\tiny 12}}}$, i.e., Chisini Theorem with
constant $\mathfrak{n}=12$ holds for finite covers $f:S\to\mathbb{P}^{2}$
belonging to $\mathcal{F}_{(2),\\{T_{n,1},\,\,n\geqslant 2\\}}$.

Denote by $T_{n,m}$ the singularity type of the branch curve germ $B_{n,m}$ of
a cover germ $f_{p,n,m,}:U_{p,n,m}\to V_{p}$ given by functions $u=z$,
$v=w^{n}-nwz^{m}$.

######  Conjecture 1.

$\\{T_{n,m},\,\,n\geqslant 2,m\geqslant
1\\}\subset\mathcal{T}_{Ch_{\text{\rm\tiny 12}}}$, i.e., Chisini Theorem with
constant $\mathfrak{n}=12$ holds for finite covers $f:S\to\mathbb{P}^{2}$
belonging to $\mathcal{F}_{(2),\\{T_{n,m},\,\,n\geqslant 2,m\geqslant 1\\}}$.

Question. Is it true that
$\mathcal{F}_{(2)}=\mathcal{F}_{(2),\mathcal{T}_{Ch_{\text{\tiny 12}}}}$?

## References

  * [1] F. Catanese: On a problem of Chisini, Duke Math. J., 53:1 (1986), 33 – 42.
  * [2] O. Chisini: Sulla identita birazionale delle funzioni algebriche di due variabili dot ate di una medesima curva di diramazione, Rend. 1st. Lombardo, 77 (1944), 339 – 356.
  * [3] C. Ciliberto, F. Flamini: On the branch curve of a generic projection of a surface to a plane, Trans. Amer. Math. Soc., 363:7 (2011), 3457 – 3471.
  * [4] Vik.S. Kulikov: On Chisini’s conjecture, Izv. Math., 63:6 (1999), 83 – 116.
  * [5] Vik.S. Kulikov: On Chisini’s conjecture. II, Izv. Math., 72:5 (2008), 63 – 76.
  * [6] Vik.S. Kulikov: Dualizing coverings of the plane, Izv. Math., 79:5 (2015), 1013 – 1042.
  * [7] Vik.S. Kulikov: A Remark on Classical Plücker’s formulae, Ann. Fac. Sci. Toulouse. Math., 25:5 (2016), 959 – 967.
  * [8] Vik.S. Kulikov: On germs of finite morphisms of smooth surfaces, Proceedings of the Steklov Institute of Mathematics, 307 (2019), 85 – 114.
  * [9] Vik.S. Kulikov: A Chisini Theorem for almost generic covers of the projective plane, Sb. Math., 213:3 (2022), 341 – 356.
  * [10] Vik.S. Kulikov: On quasi-generic covers of the projective plane, submitted to Sb. Math.
  * [11] S.Yu. Nemirovski: Kulikov’s theorem on the Chisini conjecture, Izv. Math., 65:1 (2001), 71 – 74.
  * [12] K. Stein: Analytische Zerlegungen komplexer Räume, Math. Ann. 132 (1956), 63 – 93.
  * [13] J.M. Wahl: Equisingular deformations of plane algebroid curves, Transactions of the AMS, 193 (174), 143 – 170.
  * [14] O. Zariski: Studies in singularity. I. Equivalent singularities of plane algebroid curves, Amer. J. Math., 87 (1965), 507 – 536.


#  $T\overline{T}$-Deformations of Holographic Warped CFTs

Rahul Poddar  Science Institute, University of Iceland,
Dunhaga 3, 107 Reykjavík, Iceland<EMAIL_ADDRESS>
###### Abstract

We explore $T\overline{T}$ deformations of Warped Conformal Field Theories
(WCFTs) in two dimensions as examples of $T\overline{T}$ deformed non-
relativistic quantum field theories. WCFTs are quantum field theories with a
Virasoro$\times$U(1) Kac-Moody symmetry. We compute the deformed symmetry
algebra of a $T\overline{T}$ deformed holographic WCFT, using the asymptotic
symmetries of AdS3 with $T\overline{T}$ deformed Compére, Song and Strominger
(CSS) boundary conditions. The U(1) Kac-Moody symmetry survives provided one
allows the boundary metric to transform under the asymptotic symmetry. The
Virasoro sector remains but is now deformed and no longer chiral.

##  I Introduction

A Warped Conformal Field Theory (WCFT) is a quantum field theory with an
SL(2,$\mathbb{R})\times$U(1) global symmetry in two dimensions, which breaks
Lorentz invariance. Such QFTs have translation invariance, but scaling
invariance is restricted to only one coordinate. Finite warped symmetry
transformations take the form [1, 2]

$z\to f(z),\quad{\bar{z}}\to{\bar{z}}+g(z).$ (1)

However, despite not being Lorentz invariant, this class of two-dimensional
quantum field theories still possesses an infinite-dimensional symmetry
algebra, namely a Virasoro$\times$U(1) Kac-Moody current algebra. WCFTs are
interesting as they appear in a number of holographic systems with an
SL(2,$\mathbb{R})\times$U(1) symmetry, such as Warped AdS3 [3], the near
horizon geometry of extremal rotating black holes [4, 5], and AdS3 with
Dirichlet-Neumann boundary conditions [6]. Holographic WCFTs have passed a
number of consistency checks, such as a Cardy formula [2], holographic
entanglement entropy [7, 8, 9, 10] and one-loop determinants [11].

Since WCFTs are non-relativistic, they do not couple to standard
(pseudo-)Riemannian manifolds. One approach is to couple WCFTs to “warped
geometries” [3], a variant of Newton-Cartan geometries. These geometries can
be found at the boundary of Warped AdS3 spacetimes. Unfortunately, these
geometries have certain pathologies, such as a degenerate metric, which make
some calculations untenable. Another way to couple WCFT to a background
manifold is to allow the manifold to transform with the warped symmetry
transformations. Holographically, this requires relaxing Dirichlet boundary
conditions of the bulk metric to boundary conditions which allow for
asymptotic symmetry transformations to transform the boundary metric under the
warped symmetry transformation of the boundary WCFT. The Dirichlet-Neumann
boundary conditions of Compére, Song and Strominger (CSS) [6] do exactly this.
This approach bypasses the need for a warped geometry with degenerate metrics,
and we can work with conventional techniques.

Two dimensional translationally invariant quantum field theories admit a class
of solvable irrelevant deformations built from conserved currents, most
notable of which is the $T\overline{T}$ deformation [12, 13]. The
$T\overline{T}$ operator is defined by the determinant of the energy-momentum
tensor of the quantum field theory, and the deformed action obeys the
following flow equation

$\begin{split}\partial_{\lambda}S_{\text{QFT}}(\lambda)&=-\frac{1}{2}\int
d^{2}x\sqrt{\gamma}\,\mathcal{O}_{T\overline{T}}^{(\lambda)},\\\
\mathcal{O}_{T\overline{T}}&=\det
T=\frac{1}{2}\epsilon^{\mu\rho}\epsilon^{\nu\sigma}T_{\mu\nu}T_{\rho\sigma},\end{split}$
(2)

where the deformation parameter $\lambda$ is the coupling to the
$T\overline{T}$ operator $\mathcal{O}_{T\overline{T}}$. This operator is
defined using point splitting, which in the coincident limit produces a well
defined local operator up to total derivatives. The expectation value of
$\mathcal{O}_{T\overline{T}}$ turns out to be a constant, and from this one
can derive the flow of energy eigenstates of the quantum field theory defined
on a cylinder of radius $R$,

$\partialderivative{E_{n}}{\lambda}=E_{n}\partialderivative{E_{n}}{R}+\frac{P_{n}^{2}}{R}.$
(3)

Even though the energy eigenvalues are changed, the Hilbert space remains
undeformed since there is a one-one correspondence between the states of the
original and deformed theory. Similarly, other observables can be calculated
in the deformed theory, for example the deformed Lagrangian, partition
function, two-two scattering matrices, correlation functions, etc [14, 15, 16,
17, 18, 19, 20, 21].

There are various ways to interpret how the $T\overline{T}$ deformation acts
on holographic CFTs. One proposal by [22] is to impose Dirichlet boundary
conditions for the bulk metric at a finite radius. Another proposal given by
[23] is to treat the $T\overline{T}$ deformation as a double trace
deformation, which will deform the asymptotic behaviour of the bulk fields
[24, 25]. This approach agrees with the cut-off AdS proposal when both are
valid, but has the advantage of working when there are bulk matter fields, and
also for either sign of the deformation parameter, which the former does not.
More recently there has also been the “Glue-on AdS holography” proposal [26]
which also agrees with [23] for the positive sign of the deformation parameter
in the absence of matter fields.

Using the mixed boundary conditions/Dirichlet boundary conditions at finite
radius, the asymptotic symmetry algebra of the bulk dual to $T\overline{T}$
deformed holographic CFT was calculated in [23, 27, 28]. Despite losing
conformal invariance, the asymptotic symmetry algebra turns out to still have
a Virasoro$\times$Virasoro structure. However either the central charge
becomes state dependent, or one loses the holomorphic factorization of the
symmetry algebra, which can also be expressed as a non-linear deformation of
the standard Virasoro algebra.

In this work, we explore $T\overline{T}$ deformations of WCFTs from a
holographic perspective. To establish what a $T\overline{T}$ deformation of a
WCFT is, one must first define what the energy-momentum tensor of a WCFT is.
Canonically, for WCFTs defined on a warped geometry, energy-momentum tensors
are not symmetric and a determinant is harder to define since the metric is
degenerate and non-invertible. The energy-momentum tensor turns out to be a
tensor with components being a chiral stress tensor and a U(1) current. For
WCFTs dual to Warped AdS3, it is also not possible to use the Fefferman-Graham
expansion to compute the energy momentum tensor for the same reason, the
boundary metric is not invertible. However, if we study WCFTs dual to AdS3
with CSS boundary conditions, for the price of a boundary metric which is not
invariant under warped transformations, we have an invertible metric and a
symmetric energy-momentum tensor, and a conventional definition for a
determinant. Given these considerations, it is possible to propose a
definition for a $T\overline{T}$ deformed WCFT which is dual to AdS3 with
$T\overline{T}$ deformed CSS boundary conditions.

This paper is organized as follows. We first briefly review the mixed boundary
conditions of [23] in Section II.1. Then in Section II.2 we review the CSS
boundary conditions and derive the Virasoro$\times$U(1) Kac-Moody algebra. In
Section III we derive the $T\overline{T}$ deformed CSS boundary conditions to
compute the deformed symmetry algebra of a $T\overline{T}$ deformed WCFT. We
will see that if one imposes deformed boundary conditions equivalent to
Dirichlet boundary conditions at the radial cut-off surface, we recover a
deformed Virasoro algebra, but we lose the U(1) Kac-Moody algebra. However, if
we allow the Dirichlet-Neumann boundary conditions to remain at the cut-off
surface, which is what the mixed boundary conditions suggests is the correct
approach, we recover an undeformed Kac-Moody symmetry. We then conclude and
discuss future directions in Section IV.

##  II Review

###  II.1 Mixed Boundary Conditions from $T\overline{T}$

We begin by briefly reviewing the mixed boundary conditions derived in [23]
from the variational principle. The variation of the boundary QFT action with
respect to the boundary metric sources the energy-momentum tensor of the QFT
and of the bulk dual. The flow of the variation of the QFT action is equal to
the variation of the deformation which generates the flow. So we have

$\begin{split}\partial_{\lambda}\delta S&=\delta\partial_{\lambda}S\\\
\partial_{\lambda}\left(\frac{1}{2}\int_{\partial\mathcal{M}}d^{2}x\sqrt{\gamma}T^{(\lambda)}_{ij}\delta\gamma^{(\lambda)ij}\right)&=\delta\int_{\partial\mathcal{M}}d^{2}x\sqrt{\gamma}\mathcal{O}^{(\lambda)}_{T\overline{T}}.\end{split}$
(4)

From this we can compute the flow equations for the boundary metric and the
energy-momentum tensor with respect to the deformation parameter $\lambda$.
Expressing the equations in terms of the trace reversed energy-momentum tensor
$\hat{T}_{ij}=T_{ij}-\gamma_{ij}T^{i}_{i}$, we have:

$\begin{split}\partial_{\lambda}&\gamma_{ij}=-2\hat{T}_{ij},\\\
\partial_{\lambda}&\hat{T}_{ij}=-\hat{T}_{il}\hat{T}_{j}^{~{}l},\\\
\partial_{\lambda}&(\hat{T}_{il}\hat{T}_{j}^{~{}l})=0.\end{split}$ (5)

Solving these equations, we can express the deformed metric and energy-
momentum tensor in terms of the undeformed metric and energy-momentum tensor,

$\begin{split}\gamma_{ij}(\lambda)&=\gamma_{ij}-2\lambda\hat{T}_{ij}+\lambda^{2}\hat{T}_{ik}\hat{T}_{jl}\gamma^{kl},\\\
\hat{T}_{ij}(\lambda)&=\hat{T}_{ij}-\lambda\hat{T}_{ik}\hat{T}_{jl}\gamma^{kl},\end{split}$
(6)

where everything on the right hand side are undeformed quantities. The new
deformed quantities are now the new boundary conditions for the bulk fields.
To see this, let us consider pure Einstein gravity.

For pure Einstein gravity in three dimensions, the Fefferman-Graham expansion
of the metric truncates at 2nd order in $1/r^{2}$ [29]

$\begin{split}ds^{2}&=l^{2}\frac{dr^{2}}{r^{2}}+g_{ab}dz^{a}dz^{b}\\\
&=l^{2}\frac{dr^{2}}{r^{2}}+l^{2}r^{2}\left(g_{ab}^{(0)}+\frac{g_{ab}^{(2)}}{r^{2}}+\frac{g_{ab}^{(4)}}{r^{4}}\right)dz^{a}dz^{b},\end{split}$
(7)

using which we can now express the boundary energy-momentum tensor in terms of
the Fefferman-Graham expansion

$\hat{T}_{ab}=\frac{k}{2\pi}g_{ab}^{(2)},$ (8)

where $k=\frac{l}{4G_{N}}$. For pure gravity, we also have

$g_{ab}^{(4)}=\frac{1}{4}g^{(2)}_{ac}g^{(2)}_{db}g_{(0)}^{cd}.$ (9)

Therefore we can express the deformed boundary metric and energy-momentum
tensor in terms of the Fefferman-Graham expansion

$\begin{split}\gamma_{ab}(\lambda)&=l^{2}\left(g_{ab}^{(0)}-\left(2\lambda\frac{k}{2\pi}\right)g_{ab}^{(2)}+\left(2\lambda\frac{k}{2\pi}\right)^{2}g_{ab}^{(4)}\right),\\\
\hat{T}_{ab}(\lambda)&=\frac{k}{2\pi}\left(g_{ab}^{(2)}-\left(2\lambda\frac{k}{2\pi}\right)g_{ab}^{(4)}\right).\end{split}$
(10)

Equating this to the Fefferman-Graham expansion (7), it is easy to see that
the deformed boundary metric can be thought of as being placed at a finite
radius $r_{c}=\sqrt{-\frac{\pi}{k\lambda}}$. Indeed, it turns out that the
Brown-York energy-momentum tensor (with the appropriate counterterm) evaluated
at this surface reproduces the deformed energy-momentum tensor derived here.
This makes it clear that in pure gravity, the mixed boundary conditions and
imposing Dirichlet boundary conditions at $r_{c}=\sqrt{-\frac{\pi}{k\lambda}}$
are equivalent111We will be absorbing the factor of $\pi$ into the
normalization of $\lambda$ and $\mathcal{O}_{T\overline{T}}$ from now on to
avoid clutter in the equations..

For a derivation of the mixed boundary conditions from the Chern-Simons
formulation of 3d gravity, see [31].

###  II.2 CSS boundary conditions

Examples of constructing a holographic bulk dual to a WCFT are either Warped
AdS3 or AdS3 with CSS boundary conditions. We shall use the CSS boundary
conditions since it is amenable to the mixed boundary conditions from the
$T\overline{T}$ deformation.

Expressing the metric in Fefferman-Graham gauge (7), we have the the following
Dirichlet-Neumann boundary conditions for the metric [6]

$g^{(0)}=\begin{pmatrix}P^{\prime}(z)&-\frac{1}{2}\\\
-\frac{1}{2}&0\end{pmatrix},\quad
g^{(2)}_{{\bar{z}}{\bar{z}}}=\frac{\Delta}{k},$ (11)

where $k=\frac{l}{4G_{N}}$, and $\Delta$ is a constant. These fall off
conditions are chiral, with $P(z)$ being an undetermined holomorphic function.
This is to accommodate (1), which shifts $P^{\prime}(z)$, and hence we must
leave it undetermined. Note that this is unlike the warped geometry in [3],
where the warped geometry metric is invariant under (1).

One can compute the full bulk metric with the CSS boundary conditions by
taking the Fefferman-Graham expansion (7) to be

$\begin{split}&\frac{ds^{2}}{l^{2}}=\frac{dr^{2}}{r^{2}}+\frac{\Delta}{k}d{\bar{z}}^{2}-\left(r^{2}+\frac{2\Delta
P^{\prime}(z)}{k}+\frac{\Delta L(z)}{k^{2}r^{2}}\right)dzd{\bar{z}}\\\
&+\left(r^{2}P^{\prime}(z)+\frac{(L(z)+\Delta
P^{\prime}(z))^{2}}{k}+\frac{\Delta
L(z)P^{\prime}(z)}{k^{2}r^{2}}\right)dz^{2}.\end{split}$ (12)

Here both $L(z)$ and $P(z)$ are undetermined holomorphic functions, and
parametrize the phase space of AdS3 with CSS boundary conditions. A special
case is the BTZ black hole when $P^{\prime}(z)=L^{\prime}(z)=0$ [32, 6].

The asymptotic symmetries of this metric are interesting as they differ from
the usual product of $SL(2,\mathbb{R})$ algebras despite being locally AdS3.
To compute asymptotic Killing vectors, we first require that they preserves
radial gauge,

$\mathcal{L}_{\xi}g_{r\mu}=0.$ (13)

This fixes the asymptotic Killing vector $\xi$ to have the form

$\xi=rf(z,{\bar{z}})\partial_{r}+\left(V^{a}(z,{\bar{z}})-\int\frac{g^{ab}}{r}\partial_{b}f(z,{\bar{z}})dr\right)\partial_{a}.$
(14)

Evaluating this for the CSS metric (12) we get,

$\begin{split}&\xi^{r}=rf(z,{\bar{z}}),\\\
&\xi^{z}=V^{z}(z,{\bar{z}})-\frac{k\big{(}\partial_{z}f(z,{\bar{z}})+(kr^{2}+\Delta
P^{\prime}(z))\partial_{\bar{z}}f(z,{\bar{z}})\big{)}}{k^{2}r^{4}-\Delta
L(z)},\\\ &\xi^{\bar{z}}=V^{{\bar{z}}}(z,{\bar{z}})-\frac{k}{k^{2}r^{4}-\Delta
L(z)}\Big{(}(kr^{2}+\Delta P^{\prime}(z))\partial_{z}f(z,{\bar{z}})\\\
&\qquad+((2kr^{2}+\Delta
P^{\prime}(z))P^{\prime}(z)+L(z))\partial_{\bar{z}}f(z,{\bar{z}})\Big{)}.\end{split}$
(15)

If we impose Dirichlet boundary conditions at infinity

$\lim_{r\to\infty}\mathcal{L}_{\xi}g_{\mu\nu}=0,$ (16)

we get conditions on the undetermined functions in $\xi$:

$\begin{split}&\partial_{\bar{z}}V^{a}(z,{\bar{z}})=0,\;f(z,{\bar{z}})=-\frac{1}{2}\partial_{z}V^{z}(z,{\bar{z}}),\\\
&V^{\bar{z}}(z,{\bar{z}})=P^{\prime}(z)V^{z}(z,{\bar{z}}),\end{split}$ (17)

and so we can write our asymptotic killing vector, (where $V(z)\equiv
V^{z}(z,{\bar{z}})$)

$\begin{split}\xi(V)&=-\frac{1}{2}V^{\prime}(z)\partial_{r}+\left(V(z)+\frac{k\Delta
V^{\prime\prime}(z)}{2(k^{2}r^{4}-\Delta L(z))}\right)\partial_{z}\\\
&\quad+\left(P^{\prime}(z)V(z)+\frac{k(kr^{2}+\Delta
P^{\prime}(z))}{2(k^{2}r^{4}-\Delta
L(z))}V^{\prime\prime}(z)\right)\partial_{\bar{z}}.\end{split}$ (18)

Asymptotic Killing vectors generate flow in the phase space, i.e.

$\mathcal{L}_{\xi}g_{\mu,\nu}=\partial_{L(z)}g_{\mu\nu}\delta_{\xi}L(z)+\partial_{P^{\prime}(z)}g_{\mu\nu}\delta_{\xi}P^{\prime}(z).$
(19)

From this, we can compute $\delta L$ and $\delta P$. It turns out that $\xi$
only transforms $L(z)$, and reproduces the infinitesimal Schwarzian
transformation.

$\delta_{\xi}L(z)=V(z)L^{\prime}(z)+2V^{\prime}(z)L(z)-\frac{k}{2}V^{\prime\prime\prime}(z),\;\delta_{\xi}P=0.$
(20)

To transform $P(z)$, we can not allow the asymptotic killing vector to satisfy
Dirichlet boundary conditions at infinity (16), since warped symmetry requires
changing the boundary metric. The “asymptotic Killing vector”222The quotes are
to indicate that since this vector does not satisfy Dirichlet boundary
conditions, it is technically not an asymptotic Killing vector, but since it
generates flows in the phase space it will continue to be referred to as such
later in this paper. which generates transformations in $P(z)$ is

$\eta(\sigma)=\sigma(z)\partial_{\bar{z}},$ (21)

and the transformations of the parametrizing functions are

$\delta_{\eta}L=0,\quad\delta_{\eta}P(z)=-\sigma(z).$ (22)

Note that $\eta$ also generates the warped symmetry transformation
${\bar{z}}\to{\bar{z}}+\sigma(z)$.

We can use the Fefferman-Graham expansion to compute the boundary energy-
momentum tensor

$\begin{split}T_{ab}&=\frac{k}{2\pi}\left(g^{(2)}_{ab}-g^{kl}_{(0)}g^{(2)}_{kl}g^{(0)}_{ab}\right)\\\
&=\frac{1}{2\pi}\begin{pmatrix}L(z)+\Delta P^{\prime}(z)^{2}&-\Delta
P^{\prime}(z)\\\\[5.69054pt] -\Delta
P^{\prime}(z)&\Delta\end{pmatrix}.\end{split}$ (23)

At this point it should be stated that this energy-momentum tensor is not the
canonical energy-momentum tensor for a warped CFT. For a warped CFT defined on
a manifold with warped geometry, the energy-momentum tensor is not symmetric,
since symmetry of the energy-momentum tensor is a result of Lorentz
invariance. However, a warped CFT dual to AdS3 with CSS boundary conditions is
not defined on a manifold with warped geometry. Rather, the manifold is not
invariant under warped transformations, but for that price we gain the
symmetry of the energy-momentum tensor.

The conserved charges corresponding to the asymptotic Killing vectors are

$\begin{split}Q_{\xi(f)}&=\frac{1}{2\pi}\int_{\partial\Sigma}d\phi\,n^{a}T_{ab}\xi^{b}=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}d\phi
f(z)L(z),\\\
Q_{\eta(f)}&=\frac{1}{2\pi}\int_{\partial\Sigma}d\phi\,n^{a}T_{ab}\eta^{b}=\frac{\Delta}{4\pi^{2}}\int_{0}^{2\pi}d\phi
f(z)(P^{\prime}(z)-1),\end{split}$ (24)

where $\partial\Sigma$ is at $r\to\infty$, $t=\frac{z+{\bar{z}}}{2}$ constant,
$\phi=\frac{z-{\bar{z}}}{2}\in(0,2\pi)$, and
$n=\partial_{t}=\partial_{z}+\partial_{\bar{z}}$.

We can now also compute the charge algebra, using the Dirac brackets of
Einstein gravity,

$\\{Q_{\zeta_{1}(f)},Q_{\zeta_{2}(g)}\\}=\delta_{\zeta_{1}(f)}Q_{\zeta_{2}(g)}.$
(25)

So we have

$\begin{split}&\\{Q_{\xi(f)},Q_{\xi(g)}\\}=\delta_{\xi(f)}Q_{\xi(g)}=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}d\phi\,g(z)\delta_{\xi(f)}L(z)\\\
&=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}g(z)\left(f(z)L^{\prime}(z)+2f^{\prime}(z)L(z)-\frac{k}{2}f^{\prime\prime\prime}(z)\right).\end{split}$
(26)

Expanding the functions in modes,

$\begin{split}f(z)&=\sum_{n}f_{n}e^{inz},\quad g(z)=\sum_{m}g_{m}e^{imz},\\\
L(z)&=\sum_{p}L_{p}e^{-ipz}.\end{split}$ (27)

Replacing Dirac brackets with commutators, we obtain the Virasoro algebra

$[L_{m},L_{n}]=(m-n)L_{m+n}-\frac{k}{2}n^{3}\delta_{m,-n}.$ (28)

Note that equating $\frac{k}{2}=\frac{c}{12}$ gives the familiar
$c=6k=\frac{3l}{2G_{N}}$. Similarly, we obtain a U(1) Kac-Moody algebra from
the commutator of the charges $Q_{\eta}$,

$[P_{m},P_{n}]=m\Delta\delta_{m,-n}.$ (29)

Note that the Virasoro and Kac-Moody algebra is factorized in this basis. This
is presented in this form in [34], which also gives the relation between this
and the algebra presented in [6].

##  III $T\overline{T}$ deformed CSS boundary conditions

To compute the $T\overline{T}$ deformed bulk metric corresponding to the
$T\overline{T}$ deformed boundary WCFT, we first compute the deformed boundary
metric using (10)

$\begin{split}\gamma_{ij}(\lambda)dz^{i}dz^{j}&=-\Big{(}d{\bar{z}}+(\lambda
L(z)-P^{\prime}(z))dz\Big{)}\times\\\
&\quad\Big{(}dz+\lambda\Delta(d{\bar{z}}-P^{\prime}(z)dz)\Big{)}.\end{split}$
(30)

This metric is flat, so we express it in explicitly flat coordinates with
indices $a,b$,

$\gamma_{ab}(\lambda)du^{a}du^{b}=-du\,dv.$ (31)

Equating the two, we can calculate the state dependent coordinate
transformation for a $T\overline{T}$ deformed WCFT, analogous to the ones
introduced in [35, 36]:

$\begin{split}du&=dz+\lambda\Delta\left(d{\bar{z}}-P^{\prime}(z)dz\right),\\\
dv&=d{\bar{z}}+dz\left(\lambda L(z)-P^{\prime}(z)\right),\\\
dz&=\frac{du-\lambda\Delta dv}{1-\lambda^{2}\Delta L(z)},\\\
d{\bar{z}}&=\frac{(P^{\prime}(z)-\lambda L(z))du+(\lambda\Delta
P^{\prime}(z)-1)dv}{1-\lambda^{2}\Delta L(z)}.\end{split}$ (32)

Furthermore, we can use the flow equations to compute the full bulk metric
dual to the $T\overline{T}$ deformed WCFT

$\begin{split}\frac{ds^{2}}{l^{2}}&=\frac{dr^{2}}{r^{2}}+\frac{\left(du\left(\lambda\Delta
L+kr^{2}\right)-\Delta dv\left(\lambda
kr^{2}+1\right)\right)}{k^{2}r^{2}\left(\lambda^{2}\Delta
L-1\right)^{2}}\times\\\ &\left(du\left(\lambda
kr^{2}+1\right)L-dv\left(\lambda\Delta L+kr^{2}\right)\right),\end{split}$
(33)

where $L\equiv L(u,v)=L(z)$. Note that on doing so, we lose the $P(z)$ degree
of freedom since this is equivalent to imposing Dirichlet boundary conditions
at the constant radial surface $r_{c}=\sqrt{-\frac{1}{k\lambda}}$. If we are
to impose Dirichlet-Neumann boundary conditions on this surface, we can
recover the U(1) degree of freedom. To do so, we have to perform the
transformation

$u\to u-\lambda\Delta P(u,v),\quad v\to v-P(u,v).$ (34)

This is the analogue of the warped symmetry transformation but now in the
state dependent coordinates. We will explore both types of boundary
conditions, starting with the simpler case of only imposing Dirichlet boundary
conditions.

###  III.1 Asymptotic Killing Vectors I: Dirichlet boundary conditions

We will first compute the $T\overline{T}$ deformed asymptotic symmetries which
preserve the deformed boundary conditions, which is equivalent to imposing
Dirichlet boundary conditions at the radial cutoff surface.

Preserving radial gauge (13), we see that the asymptotic Killing vector in the
deformed spacetime has the form

$\begin{split}\xi^{r}(\lambda)&=rf(u,v),\\\
\xi^{u}(\lambda)&=V^{u}(u,v)-\frac{k}{k^{2}r^{4}-\Delta
L}\Big{(}\Delta(2\lambda kr^{2}+\lambda^{2}\Delta L+1)\partial_{u}f\\\
&+(\lambda\Delta L(2+k\lambda r^{2})+kr^{2})\partial_{v}f\Big{)},\\\
\xi^{v}(\lambda)&=V^{v}(u,v)-\frac{k}{k^{2}r^{4}-\Delta L}\Big{(}L(2\lambda
kr^{2}+\lambda^{2}\Delta L+1)\partial_{v}f\\\ &+(\lambda\Delta L(2+k\lambda
r^{2})+kr^{2})\partial_{u}f\Big{)}.\\\ \end{split}$ (35)

It will be convenient to define

$\begin{split}W^{u}(u,v)&=V^{u}(u,v)+k\lambda\partial_{\bar{z}}f(u,v),\\\
W^{v}(u,v)&=V^{v}(u,v)+k\lambda\big{(}\partial_{z}f(u,v)+P^{\prime}(u,v)\partial_{\bar{z}}f(u,v)\big{)},\end{split}$
(36)

where, using (32), the derivatives in $z,{\bar{z}}$ are

$\partial_{\bar{z}}=\lambda\Delta\partial_{u}+\partial_{v},\quad\partial_{z}=\partial_{u}+\lambda
L(u,v)\partial_{v}-P^{\prime}(u,v)\partial_{\bar{z}}.$ (37)

In terms of $W^{a}$, the mixed boundary condition, or equivalently the
Dirichlet boundary condition at $r=r_{c}$

$\mathcal{L}_{\xi(\lambda)}g_{\mu\nu}(\lambda)|_{r=r_{c}}=0,$ (38)

constrains the functions in $\xi(\lambda)$ to obey

$\begin{split}f(u,v)&=-\frac{1}{2}\left(\frac{1-\lambda^{2}\Delta
L}{1+\lambda^{2}\Delta L}\right)(\partial_{u}W^{u}+\partial_{v}W^{v}),\\\
W^{u}&=-\left(\frac{\lambda\Delta}{1+\lambda^{2}\Delta
L}\right)(\partial_{u}W^{u}+\partial_{v}W^{v}),\\\ W^{v}&=-\left(\frac{\lambda
L}{1+\lambda^{2}\Delta
L}\right)(\partial_{u}W^{u}+\partial_{v}W^{v}).\end{split}$ (39)

It turns out that this is not enough to solve for $\delta L$. In the
undeformed case (18), the functions in the asymptotic Killing vector are all
holomorphic functions, so we apply the holomorphicity property in the deformed
case as well:

$\partial_{\bar{z}}W^{a}(u,v)=0,\quad\partial_{\bar{z}}L(u,v)=0.$ (40)

Combining the previous two equations, we get the conditions

$\begin{split}f(u,v)&=-\frac{1}{2}(1-\lambda^{2}\Delta
L(u,v))\partial_{u}W^{u}(u,v),\\\
\partial_{v}L(u,v)&=-\lambda\Delta\partial_{u}L(u,v),\\\
\partial_{v}W^{u}(u,v)&=-\lambda\Delta\partial_{u}W^{u}(u,v),\\\
\partial_{a}W^{v}(u,v)&=-\lambda L(u,v)\partial_{a}W^{u}(u,v).\end{split}$
(41)

We can use these equations to eliminate $v$ derivatives of all the functions,
and all derivatives of $W^{v}$.

Now we have enough information to be able to solve for $\delta L$. To do so,
we solve

$\mathcal{L}_{\xi(\lambda)}g_{\mu\nu}(\lambda)=\partial_{L(u,v)}g_{\mu\nu}\,\delta_{\xi}L(u,v).$
(42)

There are three equations but with the relations in (41), all three equations
become identical, and the $r$ dependence drops out. Solving for $\delta L$, we
get

$\begin{split}&\delta_{\xi}L(u,v)=(W^{u}-\lambda\Delta W^{v})L^{\prime}\\\
&+\frac{1}{2}\big{(}4L+\lambda^{2}k\Delta(1-\lambda^{2}\Delta
L)L^{\prime\prime}\big{)}(1-\lambda^{2}\Delta L){W^{u}}^{\prime}\\\
&+\lambda^{2}k\Delta(1-\lambda^{2}\Delta
L)^{2}L^{\prime}{W^{u}}^{\prime\prime}-\frac{k}{2}(1-\lambda^{2}\Delta
L)^{3}{W^{u}}^{\prime\prime\prime},\end{split}$ (43)

where ${}^{\prime}=\partial_{u}$. When $\lambda\to 0$, we recover (20). Note
that this depends on two arbitrary functions $W^{u}$ and $W^{v}$.

####  III.1.1 Deformed Charge Algebra

To compute the symmetry algebra of the $T\overline{T}$ deformed holographic
WCFT, we must compute the conserved charge algebra of the dual spacetime. We
first compute the deformed boundary energy-momentum tensor using the flow
equations, which also coincides with the AdS3 Brown-York energy-momentum
tensor evaluated on the constant radial surface $r=r_{c}$,

$T_{ij}^{(\lambda)}=-\frac{l}{2\pi}\begin{pmatrix}\frac{L}{1-\lambda^{2}\Delta
L}&\frac{1+\lambda k+\lambda^{2}\Delta L}{\lambda^{2}k(1-\lambda^{2}\Delta
L)}\\\\[5.69054pt] \frac{1+\lambda k+\lambda^{2}\Delta
L}{\lambda^{2}k(1-\lambda^{2}\Delta L)}&\frac{\Delta}{1-\lambda^{2}\Delta
L}\\\ \end{pmatrix}.$ (44)

Conserved charges are defined with respect to a constant time coordinate $t$,
which is defined in terms of $u,v$ by

$u=t+\phi,\quad v=t-\phi.$ (45)

Since $L$ is holomorphic in $z$, we can express the $t$ derivative in terms of
the $\phi$ derivative

$\partial_{t}L=\frac{1+\lambda\Delta}{1-\lambda\Delta}\partial_{\phi}L.$ (46)

So we can express the $u$ derivatives of holomorphic functions only in $\phi$
derivatives as well

$\partial_{u}=\frac{1}{2}(\partial_{t}+\partial_{\phi})=\frac{1}{1-\lambda\Delta}\partial_{\phi}.$
(47)

For constant $t$, we can now eliminate $W^{v}$ in (43), using equations (41)
and (47),

$\partial_{u}W^{v}=-\lambda
L(\phi)\partial_{u}W^{u}\implies\partial_{\phi}W^{v}=-\lambda
L(\phi)\partial_{\phi}W^{u}.$ (48)

Integrating over $\phi$, we have

$W^{v}(\phi)=\int^{\phi}d\phi^{\prime}\,L(\phi^{\prime})\partial_{\phi^{\prime}}W^{u}(\phi^{\prime}).$
(49)

Now we can label the variation of the conserved charges with only one
arbitrary function $W^{u}$. Using (24) but with the deformed energy-momentum
tensor, the conserved charge is

$Q_{f}=\frac{l}{4\pi^{2}}\int_{0}^{2\pi}d\phi
f(\phi)\frac{\Delta-L(\phi)}{1-\lambda^{2}\Delta L(\phi)}.$ (50)

We can now compute the charge algebra:

$\begin{split}&\\{Q_{W},Q_{f}\\}=\delta_{W}Q_{f}\\\
&=\frac{l}{4\pi^{2}}\int_{0}^{2\pi}d\phi\,f\left(\frac{-\delta_{W}L}{1-\lambda^{2}\Delta
L}+\frac{\Delta-L}{(1-\lambda^{2}\Delta
L)^{2}}(\lambda^{2}\Delta\delta_{W}L)\right).\end{split}$ (51)

Using (43) and (49), substituting $f(\phi)=e^{im\phi},\,W(\phi)=e^{in\phi}$,
and removing $\phi$ derivatives from $L$ using integration by parts, we have

$\begin{split}&\\{Q_{W},Q_{f}\\}=\delta_{W}Q_{f}=\frac{l(1+\lambda\Delta)}{8\pi^{2}(1-\lambda\Delta)^{2}}\int_{0}^{2\pi}d\phi\,\frac{1}{1-\lambda^{2}\Delta
L(\phi)}\Bigg{[}2in^{3}\,k(1-\lambda\Delta)^{3}e^{i(m+n)\phi}(1-\lambda^{2}\Delta
L(\phi))\\\
&\qquad+e^{im\phi}L(\phi)\Bigg{(}2mn\,\lambda\Delta(1-\lambda\Delta)^{2}\int^{\phi}e^{in\phi^{\prime}}L(\phi^{\prime})d\phi^{\prime}-ie^{in\phi}\big{(}n\,\lambda^{2}k\Delta(1-\lambda^{2}\Delta
L(\phi))\\\
&\times\big{(}n^{2}\,\lambda\Delta(3-\lambda\Delta(3-\lambda\Delta))-m^{2}\big{)}\big{)}-2(1-\lambda\Delta)^{2}\big{(}(m-n)-2n\,\lambda\Delta
L(\phi)\big{)}\Bigg{)}\Bigg{]}.\end{split}$ (52)

Since $L$ is not independent of $t$, only the zero modes are conserved in
time. In this choice of basis of functions and Fourier modes, the central
charge term is state dependent. This is similar to what was found in [23] for
a $T\overline{T}$ deformed CFT. It is straightforward to verify that on taking
the $\lambda\to 0$ limit and expressing $L$ in Fourier modes, one recovers the
Virasoro algebra.

###  III.2 Asymptotic Killing Vectors II: Dirichlet-Neumann boundary
conditions

If we want to impose the same boundary conditions at the radial cutoff in the
$T\overline{T}$ deformed metric as the undeformed Dirichlet-Neumann CSS
boundary conditions at infinity of the undeformed metric, we have to find a
global Killing vector which corresponds to translations on the boundary. It is
easy to verify that $\lambda\Delta\partial_{u}+\partial_{v}$ is such a global
Killing vector of (33). To generate transformations in the boundary metric, we
then promote this global Killing vector to an “asymptotic Killing vector”
analagous to (21),

$\eta(\lambda;\sigma)=-\sigma(u,v)(\lambda\Delta\partial_{u}+\partial_{v}).$
(53)

To introduce the $P(z)$ degree of freedom back into the metric (33), one can
make the coordinate transformation (34):

$u\to u-\lambda\Delta h(u,v),\quad v\to v-h(u,v),$ (54)

which is generated by the asymptotic Killing vector (53) as the analogue to
the warped transformation ${\bar{z}}\to{\bar{z}}-h(z)$. The state dependent
coordinate transformation is now

$\begin{split}&du-\lambda\Delta\differential(h(u,v))=dz+\lambda\Delta\differential({\bar{z}}-P(z)),\\\
&dv-\differential(h(u,v))=d{\bar{z}}+(\lambda L(z)-P^{\prime}(z))dz,\\\
&dz=\frac{du-\lambda\Delta dv}{1-\lambda^{2}\Delta L},\\\
&d{\bar{z}}=\frac{dv-\lambda Ldu+(du+\lambda\Delta
dv)P^{\prime}(z)}{1-\lambda^{2}\Delta L}-(\differential(h(u,v)),\end{split}$
(55)

where $\differential$ is the exterior derivative. Note that since both $h$ and
$P$ are arbitrary functions of $(u,v)$, we can choose the gauge where $h=P$.
The coordinate transformation now becomes much simpler,

$\begin{gathered}du=dz+\lambda\Delta{\bar{z}},\quad dv=d{\bar{z}}+\lambda
L(z)dz,\\\ dz=\frac{du-\lambda\Delta dv}{1-\lambda^{2}\Delta L},\quad
d{\bar{z}}=\frac{dv-\lambda Ldu}{1-\lambda^{2}\Delta L}.\end{gathered}$ (56)

The metric now reads

$\begin{split}ds^{2}&=l^{2}\frac{dr^{2}}{r^{2}}+\frac{l^{2}}{k^{2}r^{2}(1-\lambda^{2}\Delta
L)^{2}}\Big{(}\big{(}kr^{2}(\lambda^{2}\Delta L-1)\partial_{u}h-(1+\lambda
kr^{2})L\big{)}du+\big{(}kr^{2}\partial_{v}h\left(\lambda^{2}\Delta
L-1\right)+\lambda\Delta L+kr^{2}\big{)}dv\Big{)}\times\\\
&\Big{(}\big{(}\Delta\partial_{u}h\left(\lambda^{2}\Delta
L-1\right)-\lambda\Delta
L-kr^{2}\big{)}du+\Delta\big{(}\partial_{v}h\left(\lambda^{2}\Delta
L-1\right)+\lambda kr^{2}+1\big{)}dv\Big{)}.\end{split}$ (57)

This metric still has the asymptotic Killing vector (53), and when $\sigma=1$
it is a global Killing vector. Computing the flow in phase space generated by
(53), we have

$\mathcal{L}_{\eta(\lambda,\sigma)}g_{\mu\nu}(\lambda;L(u,v),\partial_{u}h(u,v))=\partial_{L}g_{\mu\nu}\delta
L+\partial_{h}g_{\mu\nu}\delta h,$ (58)

which on solving, we see that we recover the undeformed U(1) symmetry,

$\delta L=0,\quad\delta h=\sigma.$ (59)

Now we will see if on performing the warped transformation (54) we lose the
deformed Virasoro symmetry (43). The vector field which preserves radial gauge
is

$\begin{split}\xi^{r}&=rf(u,v),\\\ \xi^{u}&=V^{u}(u,v)+\frac{k}{(\Delta
L-k^{2}r^{4})(1-\lambda\Delta\partial_{u}h-\partial_{v}h)^{2}}\Big{[}\partial_{v}f\Big{(}\Delta\lambda
L\left(\Delta\lambda\partial_{u}h\left(-2\partial_{v}h+k\lambda
r^{2}+1\right)\right.\\\ &\quad\left.+\left(1-\lambda
kr^{2}\right)\partial_{v}h-\lambda
kr^{2}-2\right)+\Delta\partial_{u}h\left(\partial_{v}h-\lambda
kr^{2}-1\right)+kr^{2}\left(\partial_{v}h-1\right)+\lambda^{3}\Delta^{2}L^{2}\partial_{v}h\left(\lambda\Delta\partial_{u}h-1\right)\Big{)}\\\
&\quad-\Delta\partial_{u}f\left(2\left(\lambda
kr^{2}+1\right)\left(\lambda^{2}\Delta
L-1\right)\partial_{v}h+(\partial_{v}h)^{2}\left(\lambda^{2}\Delta
L-1\right)^{2}+\lambda^{2}\Delta L+2\lambda kr^{2}+1\right)\Big{]},\\\
\xi^{v}&=V^{v}(u,v)+\frac{k}{(\Delta
L-k^{2}r^{4})(1-\lambda\Delta\partial_{u}h-\partial_{v}h)^{2}}\Big{[}\partial_{u}f\Big{(}-\partial_{v}h\left(\Delta\lambda^{2}L-1\right)\left(\Delta\lambda
L+kr^{2}\right)\\\
&\quad+\Delta\partial_{u}h\left(\Delta\lambda^{2}L-1\right)\left(\partial_{v}h\left(\Delta\lambda^{2}L-1\right)+k\lambda
r^{2}+1\right)-\lambda\Delta L\left(\lambda kr^{2}+2\right)-kr^{2}\Big{)}\\\
&\quad+\partial_{v}f\Big{(}2\partial_{u}h\left(\lambda^{2}\Delta
L-1\right)\left(\Delta\lambda
L+kr^{2}\right)-\Delta(\partial_{u}h)^{2}\left(\Delta\lambda^{2}L-1\right)^{2}-L\left(\Delta\lambda^{2}L+2k\lambda
r^{2}+1\right)\Big{)}\Big{]}.\end{split}$ (60)

To have solutions which preserve the mixed boundary conditions, we are
required to impose holomorphicity of $h$ in $z,{\bar{z}}$ coordinates.

$\partial_{\bar{z}}h=0\implies\partial_{v}h=-\lambda\Delta\partial_{u}h.$ (61)

To simplify the equations, one can introduce the following definitions

$\begin{split}W^{u}&=V^{u}-\lambda^{3}k\Delta^{2}(1-\lambda^{2}\Delta
L)\partial_{u}f\partial_{u}h,\\\ W^{v}&=V^{v}+\lambda k(1-\lambda^{2}\Delta
L)\partial_{u}f(1-\lambda\Delta\partial_{u}h),\\\ X&=W^{u}-\lambda\Delta
W^{v}.\end{split}$ (62)

To compute the variation of the functions $L$ and $h$, it will be necessary to
impose holomorphicity in $z$ for the functions $W^{a},X,f$. We can now compute
the variation of the metric which preserves the mixed boundary conditions, or
equivalently impose Dirichlet boundary conditions at the constant radial
surface

$\mathcal{L}_{\xi(\lambda,\sigma)}g_{\mu\nu}(\lambda;L(u,v),\partial_{u}h(u,v))|_{r=r_{c}}=0.$
(63)

This is a set of three equations, however only two are linearly independent.
The conditions we get from solving the above equations are

$\begin{split}f(u,v)&=-\frac{1}{2}\left(\frac{1-\lambda^{2}\Delta
L}{1+\lambda^{2}\Delta L}\right)X^{\prime}(u,v),\\\
X(u,v)&=\frac{W^{v}(u,v)}{h^{\prime\prime}(u,v)}\\\
&\;-\frac{h^{\prime}(u,v)-\lambda L(1-\lambda\Delta
h^{\prime}(u,v))X^{\prime}(u,v)}{h^{\prime\prime}(u,v)(1+\lambda^{2}\Delta
L)},\end{split}$ (64)

where ${}^{\prime}\equiv\partial_{u}$.

We are now in a position to compute the flow of the metric in phase space
generated by this vector field subject to the above constraints,

$\mathcal{L}_{\xi(\lambda,\sigma)}g_{\mu\nu}(\lambda;L(u,v),\partial_{u}h(u,v))=\partial_{L}g_{\mu\nu}\delta
L_{\xi}+\partial_{h}g_{\mu\nu}\delta h_{\xi}.$ (65)

As before, this set of three equations, subject to the constraints reduce to
two equations, and remove any dependence on the radial coordinate $r$. The
variation of the functions $h$ and $L$ are

$\begin{split}\delta_{\xi}h&=0,\\\
\delta_{\xi}L&=\frac{1}{2\Theta^{2}h^{\prime\prime}}\Bigg{(}\Big{(}2\lambda
kL^{\prime}L_{m}^{2}L_{p}(2\lambda\Delta
h^{\prime}-1)h^{\prime\prime}+3kL_{m}^{3}L_{p}^{2}(h^{\prime\prime})^{2}\Big{)}W^{\prime\prime}-k\Theta
L_{m}^{3}L_{p}h^{\prime\prime}W^{\prime\prime\prime}\\\
&\qquad+X^{\prime}\Big{(}-2\lambda^{2}k(L^{\prime})^{2}L_{m}^{2}h^{\prime\prime}\big{(}1-3\lambda\Delta
h^{\prime}+2\lambda^{2}\Delta^{2}(h^{\prime})^{2}\big{)}-6kL_{m}^{3}L_{p}^{2}(h^{\prime\prime})^{3}+h^{\prime\prime}\big{(}-\Theta
L_{m}(\lambda kL^{\prime\prime}(1-2\lambda\Delta h^{\prime})\\\
&\qquad+4\lambda L^{2}(1-\lambda\Delta h^{\prime})+L(-\lambda^{3}k\Delta
L^{\prime\prime}-2(2-\lambda^{4}k\Delta^{2}L^{\prime\prime})h^{\prime}))+6k\Theta
L_{m}^{3}L_{p}h^{\prime\prime\prime}\big{)}-L^{\prime}\big{(}2\Theta^{3}\\\
&\qquad+\lambda kL_{m}^{2}(-7+8\lambda\Delta h^{\prime}-\lambda^{2}\Delta
L(1-8\lambda\Delta h^{\prime}))(h^{\prime\prime})^{2}+2\lambda k\Theta
L_{m}^{2}(1-2\lambda\Delta
h^{\prime})h^{\prime\prime\prime}\big{)}-k\Theta^{2}L_{m}^{3}h^{\prime\prime\prime\prime}\Big{)}\\\
&\qquad+W^{\prime}\Big{(}L^{\prime}\big{(}2\Theta^{2}L_{p}+2\lambda
kL_{m}^{2}L_{p}(1-2\lambda\Delta h^{\prime})h^{\prime\prime\prime}+k\Theta
L_{m}^{3}L_{p}h^{\prime\prime\prime\prime}\big{)}-3kL_{m}^{3}L_{p}^{2}h^{\prime\prime}h^{\prime\prime\prime}\Big{)}\Bigg{)},\end{split}$
(66)

where

$\begin{gathered}\Theta=\partial_{u}h-\lambda
L(1-\lambda\Delta\partial_{u}h),\quad W=W^{v},\\\ L_{m}=1-\lambda^{2}\Delta
L,\quad L_{p}=1+\lambda^{2}\Delta L.\end{gathered}$ (67)

We see that we still preserve a deformed Virasoro generator, and do not
generate a transformation in the U(1) generator. However, the algebra produced
the modes of the charges will not be closed as the variation of $L$ depends on
the U(1) generator $h$. We will not compute a charge algebra for this since it
is not illuminating, but in principle can be computed using the same procedure
outlined in the previous section.

##  IV Discussion

In this paper we computed the $T\overline{T}$ deformed generators of a Warped
CFT, using holographic techniques developed in [23, 27]. Previously,
holographic $T\overline{T}$ techniques have been used to compute
$T\overline{T}$ deformations of holographic CFTs. Since the $T\overline{T}$
deformation is a double trace deformation, the boundary conditions of the
holographic bulk dual are modified. For the $T\overline{T}$ deformation, this
can be interpreted as imposing Dirichlet boundary conditions at a finite
radial surface for the bulk metric. However, when considering a holographic
WCFT dual to AdS3, one has to employ the CSS boundary conditions [6], which
are Dirichlet-Neumann boundary conditions for the bulk metric.

We therefore computed the $T\overline{T}$ deformed CSS boundary conditions, by
imposing either only Dirchlet boundary conditions at the radial surface or
Dirichlet-Neumann boundary conditions at the same surface. Using this, we
computed the $T\overline{T}$ deformed asymptotic symmetry algebra for both
cases, and found that for a $T\overline{T}$ deformed holographic WCFT, the
U(1) Kac-Moody generators are not affected, but the Virasoro generators are
deformed in a non-linear way. In fact, when considering the Dirichlet-Neumann
boundary conditions at the finite radial surface, we see that the deformed
Virasoro generator will no longer create a closed algebra. This suggests that
the symmetry algebra of the $T\overline{T}$ deformed WCFT still contains the
U(1) Kac-Moody algebra, which follows from the fact that the $T\overline{T}$
deformation preserves translation invariance.

This result strengthens and extends the proposals of [22, 23, 27] to the case
of an example of bottom-up holography where the boundary theory is not a
conformal field theory, but instead a non-relativistic theory. It will be
interesting to explore how the holographic $T\overline{T}$ dictionary extends
to other examples of holography, and in particular, non-AdS holography.

There are many directions one can take from here. Another starting point for a
bulk dual to a non-relativistic QFT would be the a $J\overline{T}$ deformation
of a CFT dual to AdS3 with a U(1) Chern-Simons matter field, to generate CSS-
like boundary conditions [37]. Since Warped CFTs can also be formulated as
dual to modified gravity theories with a Warped AdS bulk, it would also be
interesting to use the Chern-Simons formalism of holographic $T\overline{T}$
[31] to compute the $T\overline{T}$ deformations of WCFT dual to Warped AdS3
as a solution to Lower Spin Gravity [3], or as a solution to Massive Gravity
[38, 39]. Stepping away from holography, it would be interesting to compute
the $T\overline{T}$ deformed WCFT partition function, and explore the
deformations of other non-relativistic QFTs such as the Quantum Lifshitz Model
in 2+1d, which will require understanding $T\overline{T}$ deformations in
higher dimensions.

###### Acknowledgements.

The author would like to thank Valentina Giangreco M. Puletti, Monica Guica,
and Lárus Thorlacius for their insights and helpful discussions. This work was
supported by the Icelandic Research Fund under the grant 228952-052 and by a
University of Iceland Doctoral Grant.

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