# Root-${{T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}}$
Deformations in Two-Dimensional Quantum Field Theories

Christian Ferko<EMAIL_ADDRESS>Center for Quantum Mathematics and
Physics (QMAP), Department of Physics & Astronomy, University of California,
Davis, CA 95616, USA.  Alessandro Sfondrini<EMAIL_ADDRESS>IBM-Einstein Fellow Dipartimento di Fisica e Astronomia, Università degli
Studi di Padova, & Istituto Nazionale di Fisica Nucleare, Sezione di Padova,
via Marzolo 8, 35131 Padova, Italy.
Institute for Advanced Study
Einstein Drive, Princeton, New Jersey, 08540 USA  Liam Smith
<EMAIL_ADDRESS>Gabriele Tartaglino-Mazzucchelli  g.tartaglino-
<EMAIL_ADDRESS>School of Mathematics and Physics, University of
Queensland, St Lucia, Brisbane, Queensland 4072, Australia.

###### Abstract

In this letter we introduce a one-parameter deformation of two-dimensional
quantum field theories generated by a non-analytic operator which we call
Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$. For a
conformal field theory, the operator coincides with the square-root of the
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ operator. More
generally, the operator is defined so that classically it is marginal and
generates a flow which commutes with the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-flow. Intriguingly, the
Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow is
closely related to the ModMax theory recently constructed by Bandos, Lechner,
Sorokin and Townsend.

##  I Introduction and Summary

The so-called $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
operator is a universal composite operator defined from the stress-energy
tensor $T^{\mu\nu}$ of any two-dimensional quantum field theory (QFT) as [1]

$\displaystyle{\cal{O}}$ $\displaystyle=$
$\displaystyle\int\text{d}^{2}z\,O(z,\bar{z})\,,$ (1a) $\displaystyle
O(z,\bar{z})$ $\displaystyle=$
$\displaystyle-\text{Det}\left[{T^{\mu\nu}}\right]=\frac{1}{2}\left(T^{\mu\nu}T_{\mu\nu}-T_{\mu}{}^{\mu}T_{\nu}{}^{\nu}\right)\,.~{}~{}~{}~{}~{}~{}$
(1b)

In a conformal field theory (CFT) the trace of the stress-energy tensor
vanishes and we have, in light-cone coordinates,
$O(z,\bar{z})=T(z)\overline{T}(\bar{z})$. This is however not the case for a
generic QFT, making “$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$” somewhat of a misnomer. Given a two-dimensional QFT defined by a
“seed” action ${\cal S}_{0}$, the operator ${\cal O}$ can be used to define a
one-parameter family of theories ${\cal S}_{\lambda}$ [2, 3]

$\partial_{\lambda}{\cal S}_{\lambda}={\cal{O}}_{\lambda}\,,\qquad{\cal
S}_{\lambda=0}={\cal S}_{0}\,.$ (2)

Because $O$ has scaling dimension four, the resulting flow is irrelevant in
the sense of the renormalization group (RG). This is counter-intuitive in the
usual RG picture, where one typically considers relevant or marginal (i.e.
conformal) perturbations. Nonetheless, the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow has many
remarkable properties. Applied to a CFT, it breaks scale-invariance but
preserves infinitely-many commuting charges, and it results in a family of
integrable QFTs (IQFTs). If ${\cal S}_{0}$ is not conformal, but integrable,
the flow still preserves integrability. In fact, the flow is so well-behaved
that it is possible to express the finite-volume spectrum of the deformed
theory ${\cal S}_{\lambda}$ in closed form starting from the spectrum of the
seed theory ${\cal S}_{0}$. These properties allowed for a detailed study of
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-deformed
theories which has revealed and is revealing many surprising connections to
string theory [3, 4, 5], holography [6, 7, 8], two-dimensional gravity [9],
geometry [10], random geometry [11], and more — see [12, 13] for reviews and
references on $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$.

In this letter we propose a new “square-root”, $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-like deformation. By
this we mean a marginal deformation defined by $\partial_{\gamma}{\cal
S}_{\gamma}={\cal{R}}_{\gamma}$ which, when applied to a two-dimensional
conformal field theory (CFT2) ${\cal S}_{0}$, amounts to deforming by a non-
analytic operator

$\displaystyle{\cal{R}}$ $\displaystyle=$
$\displaystyle\int\text{d}^{2}z\,R(z,\bar{z})\,,$ (3a) $\displaystyle
R(z,\bar{z})$ $\displaystyle=$
$\displaystyle\sqrt{T(z)\overline{T}(\bar{z})}\,,\qquad\text{(for a CFT).}$
(3b)

Despite the square-root, this promises to be a well-behaved expression in a
CFT2 because the operator-product expansion between a chiral and an anti-
chiral field (non-analytic as they may be) is regular. This deformation was
recently considered in [14, 15] to study the BMS3 symmetry of ultra/non-
relativistic limits of a CFT2. It is however not obvious how such an
expression should be generalised when ${\cal S}_{0}$ is not conformal. We do
this by requiring that the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow commutes with the
ordinary $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow,
i.e. that we can construct a two-parameter action ${\cal
S}_{(\lambda,\gamma)}$ so that the diagram in Figure 1 commutes. As we shall
explain in the conclusions, this choice will make it easier to study a vaster
class of theories.

Figure 1: We require that first deforming ${\cal S}_{0}$ by $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ with
${\cal{O}}_{\lambda}$ and then deforming the result by Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ with
${\cal{R}}_{\gamma}$ is the same as deforming ${\cal S}_{0}$ by
${\cal{R}}_{\gamma}$ and then by ${\cal{O}}_{\lambda}$.

We will show that, classically, this can be obtained by setting

$R(z,\bar{z})=\sqrt{\frac{1}{2}T^{\mu\nu}T_{\mu\nu}-\frac{1}{4}T_{\mu}{}^{\mu}T_{\nu}{}^{\nu}}=\sqrt{-\text{Det}\big{[}t^{\mu\nu}\big{]}}\,,$
(4)

where $t^{\mu\nu}=T^{\mu\nu}-\frac{1}{2}g^{\mu\nu}T_{\rho}{}^{\rho}$ is the
traceless part of the stress-energy tensor $T^{\mu\nu}$. We will construct the
explicit two-parameter action ${\cal S}_{(\lambda,\gamma)}$ for a theory of
$N$ scalar fields. This turns out to be related to the action for the unique
conformal- and Electro-Magnetic duality-invariant deformation of Maxwell
theory, the ModMax theory [16, 17] (see also [18, 19]). ModMax is a four-
dimensional model, and it depends on a single marginal parameter $\gamma$

$\mathcal{S}^{\text{MM}}_{\gamma}=\int\text{d}^{4}x\left[-\frac{\text{ch}\gamma}{4}F^{2}+\frac{\text{sh}\gamma}{4}\sqrt{(F^{2}){}^{2}+(F\widetilde{F}){}^{2}}\right],$
(5)

where $F$ indicates the electromagnetic field strength and $\tilde{F}$ its
dual. Eq. (5) admits a $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-like deformation in
terms of a Born-Infeld action with parameter $\lambda$. The resulting two
parameter, four-dimensional action ${\cal S}_{(\lambda,\gamma)}$ is the
solution of two commuting flow-equations [20, 21, 22]. The flows we introduce
here precisely generate, and extend, the reduction to two dimensions of the
ModMax-Born-Infeld action recently obtained in Ref. [23]. After discussing the
generalisation of our construction to actions involving Fermions, we will
comment on the implications of this relation and on several immediate
directions of study.

### Note added:

On June $22^{\text{nd}}$ 2022, the day this work appeared for the first time
on the arXiv, the Authors of [23] released a second version of their
manuscript where they independently showed that the operator (4) generates the
$\gamma$-flow of the Lagrangian (20) in the particular case of $N$ Bosons with
seed Lagrangian (6) when $G_{ij}(\phi)=\delta_{ij}$ and $B_{ij}(\phi)=0$. On
June $25^{\text{th}}$, Ref. [24] appeared and presented an exploration of the
relation of Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ and ModMax theories.

##  II Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
Flow for $N$ Scalars

We start by considering a rather general Lagrangian density for $N$ real
scalars $\phi^{i}$, of the form

$L=\frac{1}{2}\Big{(}G_{ij}(\phi)g^{\mu\nu}+B_{ij}(\phi)\varepsilon^{\mu\nu}\Big{)}\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{j}-V(\phi)\,,$
(6)

where $g^{\mu\nu}$ is the two-dimensional inverse metric (which we take to be
Minkowski), $\varepsilon^{\mu\nu}$ is the Levi-Civita tensor, and
$i=1,\cdots,N$. This ansatz encompasses many QFTs of interest including vector
models, non-linear sigma models (where $G_{ij}$ and $B_{ij}$ represent the
target-space metric and Kalb-Ramond fields, respectively), and Wess-Zumino-
Witten models, but it is not closed under Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations. In fact,
terms with a higher number of derivatives will appear along the flow, as it
can be seen recalling that the (Hilbert) stress-energy tensor is

$T_{\mu\nu}=-2\frac{\partial L}{\partial g^{\mu\nu}}+g_{\mu\nu}L\,.$ (7)

To write the general ansatz for a flow driven by the operator $R$ (4), we list
the tensor structures that may arise from the stress-energy tensor associated
to (6) and from its powers. As $g^{\mu\nu}$ only appears in the first term of
(6), we consider

$(X_{1})_{\mu}{}^{\nu}=G_{ij}\partial_{\mu}\phi^{i}\partial^{\nu}\phi^{j},\quad(X_{2})_{\mu}{}^{\nu}=(X_{1})_{\mu}{}^{\rho}(X_{1})_{\rho}{}^{\nu}.$
(8)

In principle, we could define $(X_{3})_{\mu}{}^{\nu}$,
$(X_{4})_{\mu}{}^{\nu}$, etc., in an obvious way. However, in the Lagrangian
only the trace of these tensors may appear, and for any $2\times 2$ matrix the
only independent trace-invariants are $x_{1}=(X_{1})_{\mu}{}^{\mu}$ and
$x_{2}=(X_{2})_{\mu}{}^{\mu}$, due to identities such as

$2x_{3}=3x_{1}x_{2}-x_{1}^{3},\quad
2x_{4}=2x_{1}^{2}x_{2}-x_{1}^{4}+x_{2}^{2}\,,$ (9)

with $x_{n}=(X_{n})_{\mu}{}^{\mu}$. Hence the most general Lagrangian will
depend on $x_{1}$, $x_{2}$, and possibly on a term $x_{0}$ which does not
couple to $g^{\mu\nu}$, $L_{\gamma}=L_{\gamma}(x_{0},x_{1},x_{2})$. For such a
Lagrangian

$T_{\mu\nu}=-2(X_{1})_{\mu\nu}\frac{\partial L_{\gamma}}{\partial
x_{1}}-4(X_{2})_{\mu\nu}\frac{\partial L_{\gamma}}{\partial
x_{2}}+g_{\mu\nu}L_{\gamma}\,.$ (10)

Computing the contractions $T_{\mu\nu}T^{\mu\nu}$ and $(T_{\mu}{}^{\mu})^{2}$,
and simplifying the resulting expression with the help of the trace identities
(9), we can evaluate the square of the operator $R$ of Eq. (4) as

$R^{2}=\left(2x_{2}-x_{1}^{2}\right)\left(\frac{\partial L_{\gamma}}{\partial
x_{1}}+2x_{1}\frac{\partial L_{\gamma}}{\partial x_{2}}\right)^{2}.$ (11)

This expression is surprisingly nice in more than one way. Firstly, the
partial derivatives appear in a perfect square so that, away from
$\partial_{x_{1}}L_{\gamma}=-2x_{1}\partial_{x_{2}}L_{\gamma}$, the flow
equation for the Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ operator (4) will be linear,

$\frac{\partial}{\partial\gamma}L_{\gamma}=R_{\gamma}=\pm\sqrt{2x_{2}-x_{1}^{2}}\left(\frac{\partial
L_{\gamma}}{\partial x_{1}}+2x_{1}\frac{\partial L_{\gamma}}{\partial
x_{2}}\right)\,.$ (12)

Both choices of the branch can be recovered by flipping the sign of $\gamma$.
As $2x_{2}=x_{1}^{2}$ gives a fixed point of (12), the branch cannot change
along the flow. Additionally, the flow equation will not depend explicitly on
$L$, so that the dependence on $x_{0}$ will affect the flow only through its
initial conditions. Finally, (12) can be integrated,

$L_{\gamma}\left(x_{0},x_{1},x_{2}\right)=L_{\gamma=0}\left(x_{0},x_{1}^{(\gamma)},x_{2}^{(\gamma)}\right)\,,$
(13)

where

$\displaystyle x_{1}^{(\gamma)}$ $\displaystyle=$
$\displaystyle\text{ch}(\gamma)\,x_{1}\pm\text{sh}(\gamma)\sqrt{2x_{2}-x_{1}^{2}}\,,$
(14a) $\displaystyle x_{2}^{(\gamma)}$ $\displaystyle=$
$\displaystyle\text{ch}(2\gamma)\,x_{2}\pm\text{sh}(2\gamma)\,x_{1}\sqrt{2x_{2}-x_{1}^{2}}\,,$
(14b)

and $x_{0}$ is not deformed. Note that the combination

$\big{(}x_{1}^{(\gamma)}\big{)}^{2}-x_{2}^{(\gamma)}=x_{1}^{2}-x_{2}\,,$ (15)

is constant along the flow. In fact, an action which depends on only this
combination and through $x_{0}$ is a fixed-point of this flow. We also note
that (14) has the property

$\displaystyle x_{1}\frac{\partial L_{\gamma}}{\partial
x_{1}}+2x_{2}\frac{\partial L_{\gamma}}{\partial
x_{2}}=x_{1}^{(\gamma)}\frac{\partial L_{\gamma}}{\partial
x_{1}^{(\gamma)}}+2x_{2}^{(\gamma)}\frac{\partial L_{\gamma}}{\partial
x_{2}^{(\gamma)}}\,,$ (16)

and that the trace of the stress-energy tensor for $L_{\gamma}$ (10) vanishes
if and only if

$\frac{1}{2}T_{\mu}{}^{\mu}=L_{\gamma}-\left(x_{1}\frac{\partial
L_{\gamma}}{\partial x_{1}}+2x_{2}\frac{\partial L_{\gamma}}{\partial
x_{2}}\right)=0\,.$ (17)

Eq. (16) therefore implies that, if the stress-energy tensor for a seed theory
$L_{0}$ is traceless, then the same is true for the deformed theory
$L_{\gamma}$. Importantly, this confirms that the operator $R_{\gamma}$
defines a classically marginal deformation.

Note that, in the special case of $N$ free Bosons, it holds

$L_{\gamma}=\frac{\text{ch}\gamma}{2}\Phi_{\mu}{}^{\mu}\pm\frac{\text{sh}\gamma}{2}\,\sqrt{2\Phi_{\mu}{}^{\nu}\Phi_{\nu}{}^{\mu}-(\Phi_{\mu}{}^{\mu})^{2}}\,,$
(18)

with $\Phi_{\mu\nu}=\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{i}$. This is the
result found in Ref. [23] from dimensional reduction of ModMax, equation (5).
Notice also that this flow is non-trivial only for $N\geq 2$, for which
$x_{1}$ and $x_{2}$ are algebraically independent, otherwise, it is merely a
rescaling of the Lagrangian, $L_{\gamma}=e^{\gamma}L_{0}$.

##  III Two-Parameter Flow

Having determined the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow, we will now
verify that it commutes with the ordinary $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow. The latter is
sourced by the operator $O$ of Eq. (1). We can express $O$ in terms of the
Lagrangian and its derivatives following what we did for $R$ around Eq. (11),
and we get the (rather cumbersome) formula

$O=-L^{2}-2\left(\left(\frac{\partial L}{\partial
x_{1}}\right)^{2}+2\frac{\partial L}{\partial x_{1}}\frac{\partial L}{\partial
x_{2}}x_{1}+2\left(\frac{\partial L}{\partial
x_{2}}\right)^{2}(x_{1}^{2}-x_{2})\right)(x_{1}^{2}-x_{2})+2L\left(\frac{\partial
L}{\partial x_{1}}x_{1}+2\frac{\partial L}{\partial x_{2}}x_{2}\right).$ (19)

Now, consider a $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ deformation of $L_{\gamma}$ (13), where the dependence on $x_{1}$ and
$x_{2}$ appears exclusively through $x_{1}^{(\gamma)}$ and $x_{2}^{(\gamma)}$.
A change of variables and some algebra, along with the observations (15–16),
allows us to reformulate the operator $O$ in terms of $L$ and its partial
derivatives with respect to $x_{1}^{(\gamma)}$ and $x_{2}^{(\gamma)}$. In
fact, we find that $O$ is given precisely by the same expression (19) up to
promoting $x_{1}$ and $x_{2}$ to $x_{1}^{(\gamma)}$ and $x_{2}^{(\gamma)}$,
respectively.

Therefore, to take a theory defined by $L_{\gamma}$ and $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-deform (obtaining
$L_{(\lambda,\gamma)}$), we may as well first $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-deform $L_{0}$ (thereby
obtaining $L_{\lambda}$), and then replace everywhere $x_{1}$ and $x_{2}$ with
$x_{1}^{(\gamma)}$ and $x_{2}^{(\gamma)}$. This last step is precisely
tantamount to a Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ deformation, which goes to show that the diagram of Fig. 1 indeed
commutes.

The two-parameter Lagrangian can then be found by taking any $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-deformed Lagrangian,
such as the ones in [25], and replacing $x_{1}$, $x_{2}$ with the expressions
in Eq. (14). For the special case of $N$ free scalars we find

$L_{(\lambda,\gamma)}=\frac{\sqrt{1+2\lambda
x_{1}^{(\gamma)}+2\lambda^{2}\big{[}\big{(}x_{1}^{(\gamma)}\big{)}^{2}-x_{2}^{(\gamma)}\big{]}}-1}{2\lambda}\,,$
(20)

which is precisely the result obtained in Ref. [23] from the dimensional
reduction of the four-dimensional ModMax-Born-Infeld theory [16, 17, 21, 22].

##  IV Fermions

It is natural to wonder whether our discussion may be extended to models with
Fermions. It is unclear how to treat a non-analytic function of Grassmann
variables only, such as the one that would arise in a Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformed model
involving only Fermions. It is more natural to first look at a theory of
Bosons and Fermions, which admits a regular expansion in Grassman variables
(with non-analytic coefficients in the Boson fields). Let us consider a free
model of $N_{B}$ Bosons and $N_{F}$ Dirac Fermions,

$L=\frac{1}{2}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{i}+\frac{{\rm
i}}{2}\mkern 1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern
1.5mu^{I}\gamma^{\mu}\partial_{\mu}\psi^{I}-\frac{{\rm
i}}{2}\left(\partial_{\mu}\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern
1.5mu^{I}\right)\gamma^{\mu}\psi^{I}\,,$ (21)

where $i=1,\dots N_{B}$, and $I=1,\dots N_{F}$.

To build an ansatz for the $\gamma$-flowed action we need to consider the
stress-energy tensor of this model. Here we will consider the Hilbert, rather
than Noether, stress-energy tensor. Much like for $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flows [26], we expect
that the deformed Lagrangian differs in the two choices by terms which vanish
on-shell. It is possible to show that, for theories that depend on the
Fermions only via the symmetric tensor $\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern
1.5mu\gamma_{(\mu}\partial_{\nu)}\psi-(\partial_{(\mu}\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern 1.5mu)\gamma_{\nu)}\psi$,
the Hilbert tensor is the symmetrized Noether tensor. This ensures that the
deformed Lagrangian will be a function of

$(X_{1})_{\mu\nu}=\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{i}+{\rm i}\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern
1.5mu^{I}\gamma_{(\mu}\partial_{\nu)}\psi^{I}-{\rm
i}\big{(}\partial_{(\mu}\mkern
1.5mu\overline{\mkern-1.5mu\psi\mkern-1.5mu}\mkern
1.5mu^{I}\big{)}\gamma_{\nu)}\psi^{I},$ (22)

and of $(X_{2})_{\mu\nu}=(X_{1})_{\mu\rho}(X_{1})^{\rho}{}_{\nu}$ — more
precisely of their traces $x_{1}$ and $x_{2}$. Given such a dependence, the
flow equation can be determined by truncating all expressions to the Bosons,
which immediately reproduces Eq. (12).

Hence, given an action which depends on $x_{1}$ and $x_{2}$ as in Eq. (22),
the Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow
is found by replacing the $\gamma$-dependent expressions (14). The same
argument of the previous section confirms this flow too commutes with the
usual $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow.
Note that in (21) we started from a system of free Dirac Fermions. However,
the same argument concerning the integration of the flows holds if chirality
conditions are imposed. Interestingly, in the presence of at least one real
chiral Fermion the deformed Lagrangian is non-trivial even with $N_{B}=1$.

##  V Conclusions and Outlook

We have discussed Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations, which
commute with the $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ flow and are defined in terms of a classically marginal non-analytic
operator. One might have expected the deformed theories to be pathological and
unnatural. Instead, they are appealing, satisfying a simple flow equation, and
encompass the models arising from dimensional reduction of ModMax [16, 17].
ModMax is also a non-analytic classically conformal theory, with many
intriguing properties: it admits various physical solutions such as plane
waves [16] and dyons [27], its Hamiltonian is bounded from below [28], it can
be supersymmetrized [29, 30, 31], and it can be related to Maxwell theory
coupled to an axion-dilaton [27] — see also the recent review [19]. While many
questions about ModMax remain open, it is certainly a physically interesting
theory and it is inspiring that the two-dimensional Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation falls in
the same class of models.

The main open question is how to quantize such non-analytic theories. This
will be subtle, but may be possible for Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flows of CFTs, by
demanding the deformed theory to be conformal and bootstrapping its
properties. If successful, one could construct new marginal deformations of
generic CFTs. In general, the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ operator will not be
part of spectrum of local operators of the theory (with the intriguing
exception of the single free Boson). Hence the deformation should result in
one-parameter families of theories that are quite different from the familiar
local unitary two-dimensional CFTs — and it would be very interesting to
understand such differences. A first step is understanding how $N\geq 2$ free
Bosons get coupled by the deformation and extending the analysis to WZW models
and minimal models where algebraic techniques are very well developed. The
quantization of Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$-deformed QFTs (as opposed to CFTs) is likely to be harder. By imposing
that $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ and
Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
deformations commute, one can first perform the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation (of a CFT)
and only later break conformal invariance by $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$. This will clarify the
general quantum properties of Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$.

Again with quantization in mind, it would also be useful to study
Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ in
Hamiltonian terms [14, 15], as was recently done [28] for ModMax, and to
investigate whether the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation acts in a
simple manner on the S-matrix of (integrable and non-integrable) theories,
like $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ does [32,
9]. (See [33] for recent work on the integrability of Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$.) In turn,
understanding the quantum properties of Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ would also extend our
understanding of ModMax, where quantization remains a major stumbling block.
This is expected for a $D=4$ model for which no perturbative quantization
scheme is available. Approaching this issue from two dimensions, where CFTs
are under especially good control, may shed new light on the issue.

Recalling that $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
is only one of many current-current deformations [2], all obeying a flow
equation [34], it would be interesting to see if the higher-spin analogues of
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ also commute
with Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$.
Moreover, one could also consider higher roots of higher-spin $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-like operators. They
would also be naïvely marginal (non-analytic) operators. Their study would
probably be difficult, as is the case for higher-spin $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations [34], but
might be instructive already in the case of simple theories, see [35] for
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$. By allowing
for the breaking of Lorentz symmetry, more general scale-invariant roots of
current-current deformations could also be considered — for example a
Root-$J\overline{T}$ deformation based on the one of [36].

In this letter we have merely initiated the study of models involving
Fermions, considering a relatively small class of models. It would be
especially interesting to perform a systematic study of the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations of
superconformal theories. It is well-established that a supersymmetric model
remains supersymmetric under $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flows [26, 37, 38, 39,
40]. The fact that ModMax may be supersymmetrized is also encouraging.
However, it is not clear that the deformation of a free superconformal action
of the type (21) (for example the ${\cal N}=(1,1)$ supersymmetric system when
$N_{B}=N_{F}$) gives a superconformal model — this certainly requires the
supersymmetry transformations to be modified. In the case of the standard
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation it
was shown that the operator ${\cal O}$ of Eq. (1) is (on-shell) equivalent to
a manifestly supersymmetric supercurrent-squared operator defined in
superspace [26, 37]. It is straightforward to propose manifestly
supersymmetric extensions of the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ operator, Eq. (3), for
superconformal field theories. In the simplest case of ${\cal N}=(0,1)$
supersymmetry the following superspace Lagrangian is an example

$\int d\theta^{+}\frac{{\cal S}_{+++}{\cal T}_{----}}{\sqrt{{\cal
T}_{++++}{\cal T}_{----}}}=\sqrt{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}+\text{Fermions\,}.$
(23)

Here ${\cal S}_{+++}$ and ${\cal T}_{----}$ define the ${\cal N}=(0,1)$
supercurrents, and ${\cal T}_{++++}=D_{+}{\cal S}_{+++}$, see Refs. [26, 37].
For a single ${\cal N}=(0,1)$ real scalar multiplet, the superspace Lagrangian
${\cal A}_{-}(\gamma)=e^{\gamma}(D_{+}\Phi)(\partial_{--}\Phi)$ satisfies a
classical flow driven by the operator (23). This is a direct analogue of the
bosonic Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
deformation of a single real free scalar field discussed above. Similar
superspace Root-$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ operators can be defined for models with more supersymmetry. Aside from
supersymmetry, it would also be interesting to consider the bosonization and
fermionization of simple two-dimensional CFTs and their Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flows.

Finally, a major application of $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations has been
to holography, mapping them to gravitational theories in three-dimensional
anti-De Sitter space-time [6, 7], see also [41, 42, 43]. The holographic
interpretation of such a marginal but non-analytic deformation of the boundary
CFT2 should be understood more clearly. An immediate application of this
letter is to recast the Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation in terms of
the boundary conditions of the AdS3 metric much like in [44], which will
provide a holographic formulation of the deformation and yield a new class of
holographic backgrounds [45].

Holography provided a way to define new irrelevant deformations in $D>2$
dimensions [46, 47], in particular in $D=4$ for Maxwell [20], its
supersymmetrization [48] and ModMax theories [21, 22]. Both our $D=2$ operator
$R$ (4), and the $D=4$ operator generating the $\gamma$-flow of ModMax-Born-
Infeld [21] can be written as

$\sqrt{-t^{\mu\nu}t_{\mu\nu}}\,,\qquad
t_{\mu\nu}=T_{\mu\nu}-\frac{1}{D}g_{\mu\nu}T_{\rho}{}^{\rho}\,,$ (24)

where $t_{\mu\nu}$ is the traceless part of $T_{\mu\nu}$. It is natural to
wonder whether this operator plays a universal role in generating ModMax-like
deformations in any dimension, and to investigate generic Root-$T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-type deformations in
$D>2$ models, including holographic ones. One piece of evidence that non-
analytic deformations may be relevant for holography in $D>2$ is that
operators involving square roots of field gradients have been studied in the
context of the fluid-gravity correspondence [49] in general dimension $D$.

A different, but very interesting, direction is the “generalized holographic
principle” of [50], which also involves stress-tensor deformations in $D>2$;
one might wonder whether $\sqrt{TT}$-type operators may be needed to UV-
complete the deformations used in that proposal. Furthermore, the interplay
between ultra/non-relativistic limits of a CFT2 and the BMS3 symmetry found in
[14, 15] also suggests possible relations between operators like (24) and
deformations of space-time symmetries in dimensions higher than two. Finally,
see also [51] for an analogue of the Root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation in $(0+1)$
spacetime dimensions.

We believe that addressing all these questions may result in exciting new
insights on the structure of conformal and non-conformal quantum field
theories and of their holographic duals, and we plan to return to some of
these in the near future.

###### Acknowledgements.

We thank Gleb Arutyunov, Grisha Korchemsky, Dima Sorokin, Marika Taylor,
Roberto Volpato, and Aron Wall for helpful discussions, and Dima Sorokin for
useful comments on a draft of this letter. C. F. is supported by U.S.
Department of Energy grant DE-SC0009999 and by funds from the University of
California. A. S. acknowledges support from the European Union –
NextGenerationEU, and from the program STARS@UNIPD, under project “Exact-
Holography”, A new exact approach to holography: harnessing the power of
string theory, conformal field theory, and integrable models. L. S. is
supported by a postgraduate scholarship at the University of Queensland. The
work of G. T.-M. is supported by the Australian Research Council (ARC) Future
Fellowship FT180100353, and by the Capacity Building Package of the University
of Queensland. L. S. and G. T.-M. also thank the mathematical research
institute MATRIX in Australia where part of this research was performed.

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