and therefore resemble conserved currents.

To compare with the common conventions for supersymmetric quantum mechanics,
which use a complex Grassmann coordinate $\theta,\bar{\theta}$ rather than
$\theta^{\pm}$, we will now translate (81) into this new notation. The details
of this change of variables are described in appendix 9.B; we summarize the
results. In complex coordinates, the operator (77) is on-shell proportional to

$\displaystyle\int\,dt\,d^{2}\theta\,\frac{\mathcal{Q}\bar{\mathcal{Q}}}{\frac{1}{2}-2\lambda\bar{D}\mathcal{Q}}\,.$
(82)

Similarly, we call this combination
$f(\mathcal{Q},\bar{\mathcal{Q}})=\frac{\mathcal{Q}\bar{\mathcal{Q}}}{\frac{1}{2}-2\lambda\bar{D}\mathcal{Q}}$.
The new supercurrents $(\mathcal{Q},\bar{\mathcal{Q}})$ satisfy the
conservation equation $\bar{D}\mathcal{Q}+D\bar{\mathcal{Q}}=0$ as shown in
appendix 9.B. Since the complex supercovariant derivatives obey the algebra
$D^{2}=\bar{D}^{2}=0$, one also has $D\bar{D}\mathcal{Q}=0$ and
$\bar{D}D\bar{\mathcal{Q}}=0$. This presentation makes it more transparent
that no additional terms are generated when the overall $D_{+}$ and $D_{-}$
derivatives act on the denominator of
(LABEL:scsquare_reduction_intermediate_four). In complex notation, this is
simply the statement that:

$\displaystyle
D\bar{D}\left(\frac{1}{4\lambda\bar{D}\mathcal{Q}-1}\right)=\bar{D}D\left(\frac{1}{4\lambda\bar{D}\mathcal{Q}-1}\right)=0\,,$
(83)

which is clear since $\bar{D}^{2}\mathcal{Q}=D\bar{D}\mathcal{Q}=0$ as we have
pointed out.

Because $\mathcal{Q},\bar{\mathcal{Q}}$ contain the single component $T$ of
the stress tensor in their component expansion, the combination (81) can be
viewed as a manifestly supersymmetric extension of the deforming operator (1),
which is now written directly in superspace. This gives a prescription for
deforming any theory of supersymmetric quantum mechanics that descends via
dimensional reduction from a $2d$ superconformal field theory (we have assumed
that the seed theory in $2d$ is conformal to use the trace flow equation).

One shortcoming of this presentation is that we have not provided any
operational method for computing the objects $\mathcal{Q},\bar{\mathcal{Q}}$
within a given theory of supersymmetric quantum mechanics. To construct these
objects using the procedure described in this section, one would need to lift
such a $(0+1)$-dimensional theory to theory in $(1+1)$-dimensions, construct
the supercurrents of this parent theory, and then assemble the appropriate
combination of the supercurrents which appear upon reducing back down to
quantum mechanics. It is, of course, desirable to have a complementary view of
$\mathcal{Q},\bar{\mathcal{Q}}$, which facilitates direct computation of these
conserved superfields in a $(0+1)$-dimensional theory, ideally via some
Noether procedure that provides an interpretation of these objects as
conserved charges associated with time translations. We turn to this issue
next.

###  5 Direct definition of deformation in $1d$

In this section, we will define a manifestly supersymmetric deformation
directly in the superspace of a $(0+1)$-dimensional quantum mechanics theory.
We first develop some formalism for defining a conserved Noether “current,”
denoted $\mathcal{Q}$, associated with time translation invariance. Because
the theory has no spatial dimensions, this conserved quantity is a charge
rather than a current; however, we will still use the term “supercurrent”
rather than “supercharge” for this object to avoid confusion with the
supercharges $Q_{i}$, which represents the action of the SUSY algebra on
superfields.

####  1 Noether currents in $\mathcal{N}=2$ theories

We will follow the strategy of using a superspace Noether procedure, which
closely parallels the discussion of section 3; we note that a similar analysis
for supersymmetric quantum mechanics appeared in [205] for a different class
of superspace Lagrangians.

Begin by considering a theory for a collection of real scalars $X^{i}$
described by the action:

$\displaystyle
I=\int\,dt\,d\theta\,d\bar{\theta}\,\mathcal{A}\left(X^{i},DX^{i},\bar{D}X^{i},D\bar{D}X^{i},\bar{D}DX^{i}\right)\,,$
(84)

Although we have not allowed the superspace Lagrangian $\mathcal{A}$ to
explicitly depend on the time derivatives $\dot{X}^{i}=\partial_{t}X^{i}$,
such dependence is implicitly allowed since

$\displaystyle D\bar{D}X^{i}+\bar{D}DX^{i}=-2i\dot{X}^{i}$ (85)

according to our conventions for the supersymmetry algebra which are described
in appendix 9.A. Therefore, since $\mathcal{A}$ depends on both
$D\bar{D}X^{i}$ and $\bar{D}DX^{i}$, arbitrary dependence on $\dot{X}^{i}$ can
also be accommodated.

For the moment, we will make no additional assumptions about the superfields
$X^{i}$ besides the reality condition $\left(X^{i}\right)^{\ast}=X^{i}$. We
first consider an arbitrary variation of the superspace Lagrangian under a
field fluctuation $\delta X^{i}$:

$\displaystyle\delta\mathcal{A}$ $\displaystyle=\delta
X^{i}\,\frac{\delta\mathcal{A}}{\delta
X^{i}}+\delta(DX^{i})\,\frac{\delta\mathcal{A}}{\delta(DX^{i})}+\delta(\bar{D}X^{i})\,\frac{\delta\mathcal{A}}{\delta(\bar{D}X^{i})}$
(86)
$\displaystyle+\delta(D\bar{D}X^{i})\,\frac{\delta\mathcal{A}}{\delta(D\bar{D}X^{i})}+\delta(\bar{D}DX^{i})\,\frac{\delta\mathcal{A}}{\delta(\bar{D}DX^{i})}\,.$

It will be convenient to re-express (86) by writing each term as the
derivative of a product minus an appropriate correction. For instance,

$\displaystyle\delta(DX^{i})\,\frac{\delta\mathcal{A}}{\delta(DX^{i})}$
$\displaystyle=D\left(\delta
X^{i}\,\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)-\left(\delta
X^{i}\right)D\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)\,,$ (87)
$\displaystyle\delta\left(D\bar{D}X^{i}\right)\frac{\delta\mathcal{A}}{\delta(D\bar{D}X^{i})}$
$\displaystyle=D\left(\frac{\delta\mathcal{A}}{\delta(D\bar{D}X^{i})}\bar{D}\delta
X^{i}\right)+\bar{D}\left(\delta
X^{i}D\frac{\delta\mathcal{A}}{\delta(D\bar{D}X^{i})}\right)$
$\displaystyle-(\delta
X^{i})\bar{D}D\left(\frac{\delta\mathcal{A}}{\delta(D\bar{D}X^{i})}\right)\,.$

This gives

$\displaystyle\delta\mathcal{A}=D\left(\delta
X^{i}\,\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)+\bar{D}\left(\delta
X^{i}\,\frac{\delta\mathcal{A}}{\delta(\bar{D}X^{i})}\right)$ (88)
$\displaystyle+D\left(\frac{\delta\mathcal{A}}{\delta(D\bar{D}X^{i})}\bar{D}\delta
X^{i}\right)+\bar{D}\left(\delta
X^{i}D\frac{\delta\mathcal{A}}{\delta(D\bar{D}X^{i})}\right)$
$\displaystyle+\bar{D}\left(\frac{\delta\mathcal{A}}{\delta(\bar{D}DX^{i})}D\delta
X^{i}\right)+D\left(\delta
X^{i}\bar{D}\frac{\delta\mathcal{A}}{\delta(\bar{D}DX^{i})}\right)$
$\displaystyle-\delta X^{i}\bigg{(}-\frac{\delta\mathcal{A}}{\delta
X^{i}}+D\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)+\bar{D}\left(\frac{\delta\mathcal{A}}{\delta(\bar{D}X^{i})}\right)$
$\displaystyle+D\bar{D}\left(\frac{\delta\mathcal{A}}{\delta(D\bar{D}X^{i})}\right)+\bar{D}D\left(\frac{\delta\mathcal{A}}{\delta(\bar{D}DX^{i})}\right)\bigg{)}\,.$

One advantage of the form (LABEL:susy_variation_ibp) is that we can
immediately read off the superspace equations of motion. Suppose we consider a
linearized fluctuation $\delta X^{i}$ around a configuration $X^{i}$ which
satisfies the equations of motion, and demand that $\delta
S=\int\,dt\,d\theta\,d\bar{\theta}\,\delta\mathcal{A}=0$. Since the terms in
the first two lines are total superspace derivatives, and the final line must
vanish for any $\delta X^{i}$, we obtain the on-shell condition:

$\displaystyle\frac{\delta\mathcal{A}}{\delta
X^{i}}=D\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)+\bar{D}\left(\frac{\delta\mathcal{A}}{\delta(\bar{D}X^{i})}\right)+D\bar{D}\left(\frac{\delta\mathcal{A}}{\delta(D\bar{D}X^{i})}\right)+\bar{D}D\left(\frac{\delta\mathcal{A}}{\delta(\bar{D}DX^{i})}\right)\,.$
(89)

Next, we study the conserved charge associated with time translations $t\to
t^{\prime}=t+\delta t$. Under such a transformation, the superspace Lagrangian
varies as

$\displaystyle\delta\mathcal{A}=\left(\delta
t\right)\partial_{t}\mathcal{A}=\frac{i}{2}\left(\delta
t\right)\left(D\bar{D}\mathcal{A}+\bar{D}D\mathcal{A}\right)\,,$ (90)

where we have again used the algebra $\\{D,\bar{D}\\}=-2i\partial_{t}$.
Meanwhile, each superfield $X^{i}$ also transforms as

$\displaystyle\delta X^{i}=(\delta t)\dot{X}^{i}=\frac{i}{2}\left(\delta
t\right)\left(D\bar{D}X^{i}+\bar{D}DX^{i}\right)\,.$ (91)

We use these expressions in (LABEL:susy_variation_ibp) and also restrict to
the case of on-shell variations, which means that the equations of motion are
satisfied, and we can discard the term proportional to $\delta X^{i}$ in the
final line. This gives

$\displaystyle 0$ $\displaystyle=-\frac{i}{2}\left(\delta
t\right)\left(D\bar{D}\mathcal{A}+\bar{D}D\mathcal{A}\right)+\left(\delta
t\right)D\left(\frac{i}{2}\left(D\bar{D}X^{i}+\bar{D}DX^{i}\right)\,\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)$
(92) $\displaystyle+\frac{i}{2}\left(\delta
t\right)\bar{D}\left(\left(D\bar{D}X^{i}+\bar{D}DX^{i}\right)\,\frac{\delta\mathcal{A}}{\delta(\bar{D}X^{i})}\right)+\frac{i}{2}\left(\delta
t\right)D\left(\frac{\delta\mathcal{A}}{\delta(D\bar{D}X^{i})}\bar{D}\left(D\bar{D}X^{i}\right)\right)$
$\displaystyle+\frac{i}{2}(\delta
t)\bar{D}\left(\frac{\delta\mathcal{A}}{\delta(\bar{D}DX^{i})}D(\bar{D}DX^{i})\right)+\frac{i}{2}(\delta
t)D\left(\left(D\bar{D}X^{i}+\bar{D}DX^{i}\right)\bar{D}\frac{\delta\mathcal{A}}{\delta(\bar{D}DX^{i})}\right)$
$\displaystyle+\frac{i}{2}(\delta
t)\bar{D}\left(\left(D\bar{D}X^{i}+\bar{D}DX^{i}\right)D\frac{\delta\mathcal{A}}{\delta(D\bar{D}X^{i})}\right)\,.$

To ease notation, we define $\eta^{i}=D\bar{D}X^{i}$,
$\tilde{\eta}^{i}=\bar{D}DX^{i}$. After simplifying and collecting terms, the
resulting equation can be written as

$\displaystyle 0$ $\displaystyle=\frac{i}{2}(\delta
t)\,D\left[(\eta^{i}+\tilde{\eta}^{i})\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}+\bar{D}\left(\frac{\delta\mathcal{A}}{\delta\tilde{\eta}^{i}}\right)\right)+\frac{\delta\mathcal{A}}{\delta\eta^{i}}\bar{D}\eta^{i}-\bar{D}\mathcal{A}\right]$
$\displaystyle\qquad+\frac{i}{2}(\delta
t)\,\bar{D}\left[(\eta^{i}+\tilde{\eta}^{i})\left(\frac{\delta\mathcal{A}}{\delta(\bar{D}X^{i})}+D\left(\frac{\delta\mathcal{A}}{\delta\eta^{i}}\right)\right)+\frac{\delta\mathcal{A}}{\delta\tilde{\eta}^{i}}D\tilde{\eta}^{i}-D\mathcal{A}\right]\,.$
(93)

This can be interpreted as a superspace conservation equation of the form:

$\displaystyle D\bar{\mathcal{Q}}+\bar{D}\mathcal{Q}=0\,,$ (94)

where

$\displaystyle\mathcal{Q}$
$\displaystyle=(\eta^{i}+\tilde{\eta}^{i})\left(\frac{\delta\mathcal{A}}{\delta(\bar{D}X^{i})}+D\left(\frac{\delta\mathcal{A}}{\delta\eta^{i}}\right)\right)+\frac{\delta\mathcal{A}}{\delta\tilde{\eta}^{i}}D\tilde{\eta}^{i}-D\mathcal{A}\,,$
$\displaystyle\bar{\mathcal{Q}}$
$\displaystyle=(\eta^{i}+\tilde{\eta}^{i})\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}+\bar{D}\left(\frac{\delta\mathcal{A}}{\delta\tilde{\eta}^{i}}\right)\right)+\frac{\delta\mathcal{A}}{\delta\eta^{i}}\bar{D}\eta^{i}-\bar{D}\mathcal{A}\,.$
(95)

We note that there is an overall factor of $i$ multiplying each term in (1),
but we have chosen to strip off this factor in defining the charges (1). From
the perspective of conservation properties, this is, of course, irrelevant
because any scalar multiple of the combination
$D\bar{\mathcal{Q}}+\bar{D}\mathcal{Q}$ still vanishes. Thus, we are free to
rescale $\mathcal{Q}$ and $\bar{\mathcal{Q}}$ by any constant. However, this
means that there will be a relative factor of $i$ when comparing
$\mathcal{Q}$, $\bar{\mathcal{Q}}$ to $\mathcal{Q}_{+}$, $\mathcal{Q}_{-}$, in
which case there was no factor of $i$ naturally appearing in the conservation
equation. We will account for this rescaling when converting between
conventions in appendix 9.B.

In summary, the Noether procedure leading to (1) provides a direct definition
of the objects $\mathcal{Q},\bar{\mathcal{Q}}$ obtained in (81) without
relying upon dimensional reduction. Since $D^{2}=\bar{D}^{2}=0$, the
superspace conservation (94) also implies that
$D\bar{D}\mathcal{Q}=\bar{D}D\bar{\mathcal{Q}}=0$. As mentioned above, we note
that the supercurrents $\mathcal{Q},\bar{\mathcal{Q}}$ are not to be confused
with the supercharges $Q,\bar{Q}$ defined by

$\displaystyle
Q=\frac{\partial}{\partial\theta}+i\bar{\theta}\frac{\partial}{\partial
t}\,,\qquad\bar{Q}=\frac{\partial}{\partial\bar{\theta}}+i\theta\frac{\partial}{\partial
t}\,,$ (96)

which represents the action of the supersymmetry algebra on superfields.

####  2 Definition of $f(\mathcal{Q},\bar{\mathcal{Q}})$ deformation

To acquire some intuition for the objects $\mathcal{Q},\bar{\mathcal{Q}}$, it
is useful to consider a simple example. The theory of a single real scalar is
described by

$\displaystyle L=\frac{m}{2}\int\,d\theta\,d\bar{\theta}\,DX\,\bar{D}X\,.$
(97)

Setting $m=2$ for simplicity, the supercurrent $\mathcal{Q}$ and its conjugate
are given by

$\displaystyle\mathcal{Q}$
$\displaystyle=-\left(\eta+\tilde{\eta}\right)DX-D(DX\,\bar{D}X)$
$\displaystyle=-\tilde{\eta}DX\,,$ $\displaystyle\bar{\mathcal{Q}}$
$\displaystyle=\left(\eta+\tilde{\eta}\right)\bar{D}X-\bar{D}(DX\,\bar{D}X)$
$\displaystyle=\eta\bar{D}X\,,$ (98)

where we used $\bar{D}DX=-2i\dot{X}-D\bar{D}X$. The component expressions for
these charges are

$\displaystyle\mathcal{Q}$
$\displaystyle=\psi(i\dot{x}-F)-2i\theta\psi\dot{\psi}+\bar{\theta}\left(\dot{x}^{2}+2iF\dot{x}-F^{2}\right)+\theta\bar{\theta}\left(i\psi\dot{F}+\psi\ddot{x}-3\dot{x}\dot{\psi}-3iF\dot{\psi}\right)\,,$
$\displaystyle\bar{\mathcal{Q}}$
$\displaystyle=\bar{\psi}(F+i\dot{x})+2i\bar{\theta}\bar{\psi}\dot{\bar{\psi}}+\theta\left(F^{2}+2iF\dot{x}-\dot{x}^{2}\right)+\theta\bar{\theta}\left(i\bar{\psi}\dot{F}-\bar{\psi}\ddot{x}+3\dot{x}\dot{\bar{\psi}}-3iF\dot{\bar{\psi}}\right)\,.$
(99)

We can also compute the highest component of the product
$\mathcal{Q}\bar{\mathcal{Q}}$. For simplicity, we will set the auxiliary
field to zero using its equation of motion. Then

$\displaystyle\mathcal{Q}\bar{\mathcal{Q}}\Big{|}_{\theta^{2},F=0}=-4\psi\bar{\psi}\dot{\psi}\dot{\bar{\psi}}+3i\left(\psi\dot{\bar{\psi}}+\bar{\psi}\dot{\psi}\right)\dot{x}^{2}+\dot{x}^{4}\,.$
(100)

We compare this with the Lagrangian of the theory written in components,

$\displaystyle
L=\dot{x}^{2}+i\left(\bar{\psi}\dot{\psi}-\dot{\bar{\psi}}\psi\right)+F^{2}\,.$
(101)

To further develop our intuition, we focus on the bosonic sector. Setting
$F=0$ and $\dot{\psi}=\dot{\bar{\psi}}=0$, we have the relation

$\displaystyle L^{2}=\mathcal{Q}\bar{\mathcal{Q}}\,.$ (102)

Interpreting the Euclidean Lagrangian as the Hamiltonian, we see that
deforming the bosonic sector by the product $\mathcal{Q}\bar{\mathcal{Q}}$ is
equivalent to a deformation by $H^{2}$. Next, the lowest component of
$\bar{D}\mathcal{Q}$ is given by

$\displaystyle\bar{D}\mathcal{Q}\Big{|}_{\theta=\bar{\theta}=0}=-F^{2}+2iF\dot{x}+\dot{x}^{2}\,.$
(103)

When the auxiliary field equation of motion is satisfied, the lowest component
of $\bar{D}\mathcal{Q}$ is, therefore, $\dot{x}^{2}$, which is the Hamiltonian
for the bosonic degree of freedom.

Using the intuition that $\mathcal{Q}\bar{\mathcal{Q}}$ has $H^{2}$ as its top
component and $\bar{D}\mathcal{Q}$ has $H$ as its bottom component, a natural
guess for a combination of superfields which has the deforming operator (1) as
its top component is
$\frac{\mathcal{Q}\bar{\mathcal{Q}}}{\frac{1}{2}-2\lambda\bar{D}\mathcal{Q}}$,
which suggests the flow equation

$\displaystyle\frac{\partial\mathcal{A}}{\partial\lambda}=\frac{\mathcal{Q}\bar{\mathcal{Q}}}{\frac{1}{2}-2\lambda\bar{D}\mathcal{Q}}\,.$
(104)

This is exactly the form of the deformation (81), which we obtained by
dimensionally reducing the supercurrent-squared operator in
$(1+1)$-dimensions.

####  3 Solution for one scalar

We now provide evidence that this is on-shell equivalent to the deformation
(1). In particular, we will check that the flow (104) generates the expected
superspace Lagrangian on-shell for the case of a single real scalar field $X$.
From the expressions (2) for the conserved charges in the free theory, we see
that the leading deformation is $-\lambda\eta\tilde{\eta}DX\bar{D}X$.
Motivated by this, we will make an ansatz for the finite-$\lambda$ solution of
the form:

$\displaystyle\mathcal{A}=f(\lambda\eta\tilde{\eta})\,DX\,\bar{D}X\,,$ (105)

with $f(y)\to 1-y+\mathcal{O}(y^{2})$ as $y\to 0$. Using the definition (1) we
can compute the conserved superspace charges associated with a Lagrangian of
this form, which gives

$\displaystyle\mathcal{Q}$
$\displaystyle=-(\eta+\tilde{\eta})\left(fDX+\lambda\eta\tilde{\eta}f^{\prime}DX+\lambda^{2}\tilde{\eta}f^{\prime\prime}D(\eta\tilde{\eta})DX\bar{D}X\right)+\lambda\eta
f^{\prime}DX\,\bar{D}X\,D\tilde{\eta}$ $\displaystyle\qquad+f\eta DX-\lambda
f^{\prime}D(\eta\tilde{\eta})DX\,\bar{D}X\,,$ $\displaystyle\bar{\mathcal{Q}}$
$\displaystyle=(\eta+\tilde{\eta})\left(f\bar{D}X+\lambda\eta\tilde{\eta}f^{\prime}\,\bar{D}X+\lambda^{2}\eta
f^{\prime\prime}D(\eta\tilde{\eta})DX\bar{D}X\right)+\lambda\tilde{\eta}f^{\prime}DX\,\bar{D}X\,\bar{D}\eta$
$\displaystyle\qquad-f\tilde{\eta}\bar{D}X-\lambda
f^{\prime}\bar{D}(\eta\tilde{\eta})DX\,\bar{D}X\,.$ (106)

The product of these is therefore

$\displaystyle\mathcal{Q}\bar{\mathcal{Q}}=-\left(\eta\tilde{\eta}\left(f+f^{\prime}\eta\lambda(\eta+\tilde{\eta})\right)\left(f+f^{\prime}\lambda\tilde{\eta}(\eta+\tilde{\eta}\right)\right)DX\,\bar{D}X\,.$
(107)

We now pause to investigate an implication of the superspace equations of
motion, which will allow us to simplify this expression for
$\mathcal{Q}\bar{\mathcal{Q}}$ and, therefore, the flow equation. This will be
the analog of (32), which we used to make a similar simplification in the
field theory setting. The equation of motion (89) can be written as

$\displaystyle
0=D\left(\frac{\delta\mathcal{A}}{\delta(DX)}\right)+\bar{D}\left(\frac{\delta\mathcal{A}}{\delta(\bar{D}X)}\right)+D\bar{D}\left(\frac{\delta\mathcal{A}}{\delta\eta}\right)+\bar{D}D\left(\frac{\delta\mathcal{A}}{\delta\tilde{\eta}}\right)\,,$
(108)

which for our ansatz (105) is

$\displaystyle
0=D\left(f\bar{D}X\right)-\bar{D}\left(fDX\right)+D\bar{D}\left(\tilde{\eta}\lambda
f^{\prime}DX\bar{D}X\right)+\bar{D}D\left(\eta\lambda
f^{\prime}DX\,\bar{D}X\right)\,.$ (109)

Suppose we multiply both sides of (109) by $DX\bar{D}X$. Since
$(DX)^{2}=(\bar{D}X)^{2}=0$, several terms vanish by nilpotency, and the
surviving contributions are

$\displaystyle f\eta
DX\bar{D}X-f\tilde{\eta}DX\bar{D}X+\lambda\tilde{\eta}^{2}\eta
f^{\prime}DX\bar{D}X-\lambda\eta^{2}\tilde{\eta}f^{\prime}DX\bar{D}X=0\,.$
(110)

It follows that

$\displaystyle\left(f-\lambda
f^{\prime}\eta\tilde{\eta}\right)\left(\eta-\tilde{\eta}\right)DX\bar{D}X=0\,.$
(111)

Therefore, either $\eta-\tilde{\eta}$ or $f-\lambda
f^{\prime}\eta\tilde{\eta}$ vanishes when multiplying $DX\bar{D}X$ . The
latter cannot hold identically unless $f(y)=\frac{c}{y}$, which is not
consistent with the boundary condition $f(0)=1$. We conclude that

$\displaystyle\left(\eta-\tilde{\eta}\right)DX\,\bar{D}X=0\,.$ (112)

In particular, this means that, on-shell, we can replace $\eta$ with
$\tilde{\eta}$ or vice-versa when either is multiplying $DX\,\bar{D}X$. Making
this replacement in the expression (107) for the bilinear
$\mathcal{Q}\bar{\mathcal{Q}}$ gives

$\displaystyle\mathcal{Q}\bar{\mathcal{Q}}=-\eta\tilde{\eta}\left(f+2f^{\prime}\lambda\eta\tilde{\eta}\right)^{2}DX\,\bar{D}X\,.$
(113)

Next, to construct our deforming operator (104), we consider the combinations
$D\bar{\mathcal{Q}}$ and $\bar{D}\mathcal{Q}$ (which are of course related by
the conservation equation). Any term appearing in these combinations which is
proportional to $DX$ or $\bar{D}X$ will not contribute to the deformation,
since the function of $\bar{D}\mathcal{Q}$ appearing in (104) comes
multiplying $\mathcal{Q}\bar{\mathcal{Q}}$, which is already proportional to
$DX\,\bar{D}X$. The only terms which we need to retain are therefore

$\displaystyle\bar{D}\mathcal{Q}$
$\displaystyle\sim-\eta\tilde{\eta}f-2\lambda\eta^{2}\tilde{\eta}^{2}f^{\prime}\,,$
$\displaystyle D\bar{\mathcal{Q}}$
$\displaystyle\sim\eta\tilde{\eta}f+2\lambda\eta^{2}\tilde{\eta}^{2}f^{\prime}\,,$
(114)

where by “$\sim$” we mean equivalence up to terms proportional to $DX$ or
$\bar{D}X$, which vanish when multiplying $DX\,\bar{D}X$ by nilpotency. We
have thus found that, on-shell, the combination of superfields which drives
our flow equation can be written as

$\displaystyle\frac{\mathcal{Q}\bar{\mathcal{Q}}}{\frac{1}{2}-2\lambda\bar{D}Q}=-\frac{\eta\tilde{\eta}(f+2\lambda\eta\tilde{\eta}f^{\prime})^{2}}{\frac{1}{2}+2\lambda\eta\tilde{\eta}(f+2\lambda\eta\eta
f^{\prime})}\,DX\,\bar{D}X\,.$ (115)

In terms of the dimensionless variable $y=\lambda\eta\tilde{\eta}$, the flow
equation then reduces to an ordinary differential equation for $f(y)$,

$\displaystyle
f^{\prime}(y)=\frac{2\left(f(y)+2yf^{\prime}(y)\right)^{2}}{1+4y\left(f(y)+2yf^{\prime}(y)\right)}\,,$
(116)

whose solution is

$\displaystyle f(y)=\frac{1}{4y}\left(\sqrt{1+8y}-1\right)\,.$ (117)

We, therefore, conclude that the all-orders solution to the flow equation
(104) is on-shell equivalent to the expression:

$\displaystyle\mathcal{A}(\lambda)=\frac{1}{4\lambda\eta\tilde{\eta}}\left(\sqrt{1+8\lambda\eta\tilde{\eta}}-1\right)\,DX\,\bar{D}X\,.$
(118)

To facilitate comparison with our earlier analysis, we re-scale
$\lambda\to\frac{\lambda}{2}$ and replace $\eta,\tilde{\eta}$ with their
explicit expressions. The resulting deformed quantum mechanics theory is

$\displaystyle
I=\int\,dt\,d\theta\,d\bar{\theta}\frac{1}{2\lambda(D\bar{D}X)(\bar{D}DX)}\left(-1+\sqrt{1+4\lambda(D\bar{D}X)(\bar{D}DX)}\right)\,DX\,\bar{D}X\,.$
(119)

We see that this matches (48) on the nose after identifying $X$ with $\Phi$
and setting the metric to $g(\Phi)=1$. The case with a non-trivial metric for
the $(0+1)$-dimensional theory can be treated similarly.

One could also consider flows driven by other operators constructed from
$\mathcal{Q}$ and $\bar{\mathcal{Q}}$. These are supersymmetric versions of
the $f(H)$ deformations considered in [115, 116]. From the perspective of the
quantum mechanics theory, there is no distinguished choice of $f(H)$ since any
such function drives a qualitatively similar flow where all energy eigenstates
remain eigenstates, and their energy eigenvalues change in a prescribed way.
The only reason for treating the particular $f(H)$ corresponding to
$T\overline{T}$ is due to the connections to interesting deformations of
higher dimensional theories.

As an example of a different supersymmetric $f(H)$ deformation, one could
instead study the flow:

$\displaystyle\frac{\partial\mathcal{A}}{\partial\lambda}=\mathcal{Q}\bar{\mathcal{Q}}\,,$
(120)

which is analogous to the deformation $\frac{\partial
L}{\partial\lambda}=H^{2}$. If we again restrict to the case of a single real
scalar considered above and use the result (113), which is equivalent to
$\mathcal{Q}\bar{\mathcal{Q}}$ on-shell, this leads to a differential equation

$\displaystyle f^{\prime}(y)=\left(f(y)+2yf^{\prime}(y)\right)^{2}\,.$ (121)

for the function $f(y)$ appearing in the ansatz (105). This is a quadratic
equation that can be solved for $f^{\prime}(y)$ as

$\displaystyle f^{\prime}(y)=\frac{1-4yf(y)-\sqrt{1-8yf(y)}}{8y^{2}}\,.$ (122)

However, we note that this deformation, and flows driven by other functions of
$\mathcal{Q}$ and $\bar{\mathcal{Q}}$, will not be related to the usual two-
dimensional $T\overline{T}$ deformation by dimensional reduction. Only the
operator appearing in (104) has this property, and even in that case, the
relationship only holds for deformations of conformal $2d$ seed theories since
the derivation relies on the trace flow equation.

###  6 Theories with $\mathcal{N}=1$ supersymmetry

Thus far, we have focused on theories with two real supercharges, such as $2d$
field theories with $\mathcal{N}=(1,1)$ supersymmetry or quantum mechanical
theories with $\mathcal{N}=2$ supersymmetry. However, one could carry out an
analogous study of theories with only a single real supercharge. This would be
relevant for theories with either $\mathcal{N}=(0,1)$ or $\mathcal{N}=(1,0)$
theories in two dimensions, which then reduce to theories with $\mathcal{N}=1$
SUSY in $(0+1)$-dimensions.

We will not carry out an extensive analysis of the three different methods for
constructing a supersymmetric $T\overline{T}$ deformation in the
$\mathcal{N}=1$ case, as we did in sections 3 \- 5 for $\mathcal{N}=2$.
However, in this section, we will briefly outline some of the ingredients that
would go into such an analysis and argue that similar results hold.

####  1 Noether currents in $\mathcal{N}=1$ theories

Consider a theory of a collection of $\mathcal{N}=1$ superfields $X^{i}$ in
$(0+1)$-dimensions. The $\mathcal{N}=1$ superspace has a single anticommuting
coordinate $\theta$, so the superfields $X^{i}$ can be expanded in components
as

$\displaystyle X^{i}=x^{i}+i\theta\psi^{i}\,.$ (123)

The supercovariant derivative associated with $\theta$ is

$\displaystyle
D=\frac{\partial}{\partial\theta}-i\theta\frac{\partial}{\partial t}\,,$ (124)

which satisfies the algebra

$\displaystyle\\{D,D\\}=-2i\partial_{t}\,.$ (125)

As a simple example, the free superspace Lagrangian for such a collection of
$\mathcal{N}=1$ superfields is written:

$\displaystyle\mathcal{A}=\frac{i}{2}\dot{X}^{i}DX^{i}\,.$ (126)

We now carry out a version of the Noether procedure, which was used to obtain
expressions for the supercurrents $\mathcal{Q},\bar{\mathcal{Q}}$ in the
$\mathcal{N}=2$ case. Consider a superspace Lagrangian that depends on the
$X^{i}$, their superspace derivatives $DX^{i}$, and their time derivatives
$\dot{X}^{i}$:

$\displaystyle I=\int\,dt\,d\theta\,\mathcal{A}(X^{i},DX^{i},\dot{X}^{i})\,.$
(127)

We note that, unlike in the $\mathcal{N}=2$ case, the superspace Lagrangian
$\mathcal{A}$ in (127) must be fermionic so that the action $S$ itself is
bosonic. The variation of the superspace Lagrangian is given by

$\displaystyle\delta\mathcal{A}=\delta X^{i}\frac{\delta\mathcal{A}}{\delta
X^{i}}+\delta(DX^{i})\frac{\delta\mathcal{A}}{\delta(DX^{i})}+\delta\dot{X}^{i}\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}\,.$
(128)

The only difference in our Noether procedure is that rather than specializing
to the case of time translations $\delta X^{i}=(\delta t)\dot{X}^{i}$ as we
did for $\mathcal{N}=2$, we will now consider both translations along the
Grassmann coordinates $\theta$ and along time. The reason for this is that we
would like to construct current bilinears, which require the presence of two
current-like objects, such as $\mathcal{Q}$ and $\bar{\mathcal{Q}}$. However,
for the $\mathcal{N}=1$ case, there is only a single current associated with
time translations, and (as we will see shortly) its square does not have
$T\overline{T}$ as its top component. Similarly, there is a single current
associated with superspace translations, but because this current is
fermionic, we cannot square it to construct bilinears since the result would
vanish by nilpotency.444Another way to see that we need two separate currents
is that the superspace Lagrangian for $\mathcal{N}=1$ is itself fermionic.
Thus, we could not have constructed a fermionic current bilinear out of a
single conserved current since the square of such a current is necessarily
bosonic.

With this motivation, we will again re-express (128) using the product rule as
before. Now, we must be careful because $\delta$ does not commute with $D$
since we are allowing translations along the $\theta$ direction as well. One
finds

$\displaystyle\delta\mathcal{A}$ $\displaystyle=\delta
X^{i}\frac{\delta\mathcal{A}}{\delta X^{i}}+D\left(\delta
X^{i}\right)\frac{\delta\mathcal{A}}{\delta(DX^{i})}+\left(\big{[}\delta,D\big{]}X^{i}\right)\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)+\delta\dot{X}^{i}\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}$
$\displaystyle=D\left(\delta
X^{i}\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)+\partial_{t}\left(\delta
X^{i}\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}\right)$
$\displaystyle\qquad-\delta X^{i}\left(-\frac{\delta\mathcal{A}}{\delta
X^{i}}+D\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)+\partial_{t}\left(\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}\right)\right)+\left(\big{[}\delta,D\big{]}X^{i}\right)\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)\,.$
(129)

Exactly, as before, in (89), we can read off the superspace equation of
motion:

$\displaystyle\frac{\delta\mathcal{A}}{\delta
X^{i}}=D\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)+\partial_{t}\left(\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}\right)\,.$
(130)

Now consider a combined superspace translation of the form $t\to t+\delta t$,
$\theta\to\theta+\delta\theta$ for a commuting constant $\delta t$ and a
Grassmann constant $\delta\theta$. The resulting change in the fields is

$\displaystyle\delta
X^{i}=(\delta\theta)\frac{\partial}{\partial\theta}X^{i}+(\delta
t)\dot{X}^{i}\,.$ (131)

Using the definition
$DX^{i}=\frac{\partial}{\partial\theta}X^{i}-i\theta\dot{X}^{i}$, we can
rewrite

$\displaystyle\frac{\partial}{\partial\theta}X^{i}=DX^{i}+i\theta\dot{X}^{i}\,,$
(132)

and therefore repackage the variation (131) as

$\displaystyle\delta X^{i}$ $\displaystyle=(\delta\theta)DX^{i}+(\delta
t+i(\delta\theta)\theta)\dot{X}^{i}$
$\displaystyle=(\delta\theta)DX^{i}+(\delta\tilde{t})\dot{X}^{i}\,,$ (133)

where in the last step we have defined $\delta\tilde{t}\equiv\delta
t+i(\delta\theta)\theta$. Likewise, the variation $\delta\mathcal{A}$ of the
superspace Lagrangian can be written in the same way:

$\displaystyle\delta\mathcal{A}=(\delta\theta)D\mathcal{A}+(\delta\tilde{t})\partial_{t}\mathcal{A}\,.$
(134)

We can also compute the commutator:

$\displaystyle\big{[}\delta,D\big{]}X^{i}$
$\displaystyle=\delta(DX^{i})-D(\delta X^{i})$
$\displaystyle=\left((\delta\theta)DDX^{i}+(\delta\tilde{t})D\dot{X}^{i}\right)-D\left((\delta\theta)DX^{i}+(\delta\tilde{t})\dot{X}^{i}\right)$
$\displaystyle=-i(\delta\theta)\dot{X}^{i}\,.$ (135)

Substituting these variations into (1) and going on-shell so that we can
discard the equation of motion term gives

$\displaystyle(\delta\theta)D\mathcal{A}+i(\delta\tilde{t})DD\mathcal{A}$
$\displaystyle=D\left((\delta\theta)(DX^{i})\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)+DD\left((\delta\theta)(DX^{i})\left(\frac{\delta\mathcal{A}}{\delta
DDX^{i}}\right)\right)$
$\displaystyle\quad+D\left(i(\delta\tilde{t})(DDX^{i})\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)+DD\left(i(\delta\tilde{t})(DDX^{i})\left(\frac{\delta\mathcal{A}}{\delta
DDX^{i}}\right)\right)$
$\displaystyle\quad-i(\delta\theta)\dot{X}^{i}\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)\,.$
(136)

where we have rewritten time derivatives in terms of $D$ using
$\partial_{t}=iD^{2}$. Collecting terms then gives

$\displaystyle 0$
$\displaystyle=-(\delta\theta)\Bigg{[}D\left((DX^{i})\frac{\delta\mathcal{A}}{\delta(DX^{i})}-D\left((DX^{i})\frac{\delta\mathcal{A}}{\delta(DDX^{i})}\right)+\mathcal{A}\right)+i\dot{X}^{i}\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)\Bigg{]}\,$
$\displaystyle\quad+iD\left[(\delta\tilde{t})(DDX^{i})\frac{\delta\mathcal{A}}{\delta(DX^{i})}+D\left((\delta\tilde{t})(DDX^{i})\frac{\delta\mathcal{A}}{\delta(DDX^{i})}\right)\right]-i(\delta\tilde{t})DD\mathcal{A}\,.$
(137)

It is now tempting to commute the $\delta\tilde{t}$ past various instances of
$D$ in the second line of (1) and define two charge-like objects corresponding
to the quantities in brackets, namely

$\displaystyle\mathcal{Q}_{\theta}$
$\displaystyle=(DX^{i})\frac{\delta\mathcal{A}}{\delta(DX^{i})}-iD\left((DX^{i})\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}\right)+\mathcal{A}\,,$
$\displaystyle\mathcal{Q}_{t}$
$\displaystyle=-i\dot{X}^{i}\frac{\delta\mathcal{A}}{\delta(DX^{i})}+D\left(\dot{X}^{i}\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}\right)-D\mathcal{A}\,.$
(138)

One might then conclude that $D\mathcal{Q}_{t}$ must vanish and
$D\mathcal{Q}_{\theta}$ must be related to the remaining term
$i\dot{X}^{i}\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)$, giving us
one conserved charge and one object, which is not conserved, but which has a
known source. However, this manipulation is not valid because
$\delta\tilde{t}$ itself depends on $\theta$ and therefore does not commute
with $D$. If we first set $\delta\theta=0$ and consider only a finite $\delta
t$, then $\delta\tilde{t}=\delta t$ does commute with the $D$ operator so we
can write

$\displaystyle 0$ $\displaystyle=(\delta
t)D\left[\left((DDX^{i})\frac{\delta\mathcal{A}}{\delta(DX^{i})}+D\left((DDX^{i})\frac{\delta\mathcal{A}}{\delta(DDX^{i})}\right)-D\mathcal{A}\right)\right]\,,$
(139)

which is interpreted as a conservation equation of the form

$\displaystyle D\mathcal{Q}_{t}=0\,,$ (140)

with $\mathcal{Q}_{t}$ defined in (1). Next, let us set $\delta t=0$ and look
at a fermionic translation $\delta\theta$. First, we must account for the
additional terms introduced when commuting
$\delta\tilde{t}=i(\delta\theta)\theta$ past the $D$ operators. Note that

$\displaystyle-i(\delta\tilde{t})DD\mathcal{A}$
$\displaystyle=(\delta\theta)\theta DD\mathcal{A}$
$\displaystyle=(\delta\theta)\left(-D\left(\theta
D\mathcal{A}\right)+D\mathcal{A}\right)$
$\displaystyle=(\delta\theta)D\left(\mathcal{A}-\theta D\mathcal{A}\right)\,.$
(141)

Then one finds

$\displaystyle(\delta\theta)D\mathcal{Q}_{\theta}$
$\displaystyle=-D\left[(\delta\theta)\theta(DDX^{i})\frac{\delta\mathcal{A}}{\delta(DX^{i})}+D\left((\delta\theta)\theta\dot{X}^{i}\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}\right)\right]+(\delta\theta)D\left(\mathcal{A}-\theta
D\mathcal{A}\right)\,$
$\displaystyle\qquad-i(\delta\theta)\dot{X}^{i}\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)$
$\displaystyle=(\delta\theta)D\left[\theta(DDX^{i})\frac{\delta\mathcal{A}}{\delta(DX^{i})}-D\left(\theta\dot{X}^{i}\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}\right)+\mathcal{A}-\theta
D\mathcal{A}\right]$
$\displaystyle-i(\delta\theta)\dot{X}^{i}\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)$
$\displaystyle=(\delta\theta)D\left[\theta\mathcal{Q}_{t}-\dot{X}^{i}\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}+\mathcal{A}\right]-i(\delta\theta)\dot{X}^{i}\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)$
$\displaystyle=(\delta\theta)\left(\mathcal{Q}_{t}-\theta
D\mathcal{Q}_{t}-D\left(\dot{X}^{i}\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}\right)+D\mathcal{A}-i\dot{X}^{i}\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)\right)\,.$
(142)

Using the conservation equation $D\mathcal{Q}_{t}=0$, we then have

$\displaystyle
D\mathcal{Q}_{\theta}=\mathcal{Q}_{t}-D\left(\dot{X}^{i}\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}\right)+D\mathcal{A}-i\dot{X}^{i}\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)\,.$
(143)

Thus, we see that the “charge” $\mathcal{Q}_{\theta}$ is not an independent
quantity but is related to the time translation charge $\mathcal{Q}_{t}$, as
one might expect from the intuition that the supersymmetry algebra relates
successive superspace translations to time translations through
$D^{2}=-i\partial_{t}$. In particular, $\mathcal{Q}_{\theta}$ itself is not
conserved in general. We can quantify this non-conservation by acting again on
(143) with $D$ and using $D\mathcal{Q}_{t}=0$ to write

$\displaystyle\partial_{t}\mathcal{Q}_{\theta}=\partial_{t}\left(\mathcal{A}-\dot{X}^{i}\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}\right)-iD\left(\dot{X}^{i}\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)\right)\,.$
(144)

Therefore, one could define a modified charge $\widetilde{Q}_{\theta}$ and a
correction term $\mathcal{Q}_{c}$ by

$\displaystyle\widetilde{\mathcal{Q}}_{\theta}=\mathcal{Q}_{\theta}+\dot{X}^{i}\frac{\delta\mathcal{A}}{\delta\dot{X}^{i}}-\mathcal{A}\,,\qquad\mathcal{Q}_{c}=i\dot{X}^{i}\left(\frac{\delta\mathcal{A}}{\delta(DX^{i})}\right)\,,$
(145)

with the property that

$\displaystyle\partial_{t}\widetilde{\mathcal{Q}}_{\theta}+D\mathcal{Q}_{c}=0\,.$
(146)

####  2 Definition of $\mathcal{Q}_{\theta}\mathcal{Q}_{t}$ deformation and
solution for one scalar

To get some intuition for these objects constructed in the preceding
subsection, we compute them for the free theory (126):

$\displaystyle\mathcal{Q}_{\theta}$ $\displaystyle=iDX^{i}\dot{X}^{i}\,,$
$\displaystyle\widetilde{\mathcal{Q}}_{\theta}$
$\displaystyle=iDX^{i}\dot{X}^{i}\,,$ $\displaystyle\mathcal{Q}_{c}$
$\displaystyle=-\frac{1}{2}\dot{X}^{i}\dot{X}^{i}\,,$
$\displaystyle\mathcal{Q}_{t}$
$\displaystyle=\frac{1}{2}\dot{X}^{i}\dot{X}^{i}\,.$ (147)

Note that $\mathcal{Q}_{\theta}$, $\widetilde{\mathcal{Q}}_{\theta}$ are
fermionic and $\mathcal{Q}_{t}$ is bosonic, as expected for Noether currents
associated with Grassmann translations and time translations respectively.
Therefore, the product $\mathcal{Q}_{\theta}\mathcal{Q}_{t}$ is a fermion and
thus an appropriate quantity to add to the Lagrangian as a deformation. In
particular, for the free theory, we note that the top component of
$\mathcal{Q}_{\theta}\mathcal{Q}_{t}$ is proportional to
$(\dot{x}^{i}\dot{x}^{i})^{2}$, which is the square of the Hamiltonian. Using
this intuition, we propose an $\mathcal{N}=1$ version of the SUSY-QM
deformation as

$\displaystyle\frac{\partial\mathcal{A}}{\partial\lambda}=\frac{1}{2}\mathcal{Q}_{\theta}\mathcal{Q}_{t}\,.$
(148)

In this proposal we do _not_ divide by the combination
$\frac{1}{2}-2\lambda\mathcal{Q}_{t}$, as one might expect from the analogous
$f(\mathcal{Q},\bar{\mathcal{Q}})$ expression in the $\mathcal{N}=2$ case.
This may seem strange because the form of this deformation is very different
than in the preceding cases that we have considered. However, we will later
see that there is an equivalent rewriting of this flow equation as

$\displaystyle\frac{\partial\mathcal{A}}{\partial\lambda}=\frac{\tilde{\mathcal{Q}}_{\theta}\mathcal{Q}_{t}}{1+2\lambda\mathcal{Q}_{t}}\,.$
(149)

For the class of Lagrangians that we focus on in this work, the solution to
the flow equation (149) is identical to the solution of (148). We will explore
the reason for this in section 3, where we see that there is a simpler way to
understand this equivalence by studying an analogous pair of deformations in
the non-supersymmetric setting. For the moment, however, we will work with the
first deformation (148).

First, we argue that this is on-shell equivalent to the dimensional reduction
of the supercurrent-squared flow for theories with $\mathcal{N}=(0,1)$
supersymmetry. In particular, we will solve the flow equation (148) for the
seed theory of a single free boson and verify that it matches the dimensional
reduction of the corresponding $2d$ flow. We make an ansatz for the
finite-$\lambda$ deformed superspace Lagrangian of the form:

$\displaystyle\mathcal{A}^{(\lambda)}=\frac{i}{2}f(\lambda\dot{X}^{2})\,\dot{X}DX\,.$
(150)

Next, we compute the supercurrents. To ease notation, we define the
dimensionless combination $\xi=\lambda\dot{X}^{2}$. Then using (1) one finds

$\displaystyle\mathcal{Q}_{\theta}$ $\displaystyle=if(\xi)\dot{X}DX\,,$
$\displaystyle\mathcal{Q}_{t}$
$\displaystyle=\frac{1}{2}\dot{X}^{2}f(\xi)+\frac{i}{2}D\Big{(}\dot{X}\left(f(\xi)+2\xi
f^{\prime}(\xi)\right)DX\Big{)}-\frac{i}{2}D\left(f(\xi)\dot{X}DX\right)\,.$
(151)

Here we have used that $DX$ is fermionic so $(DX)^{2}=0$. Furthermore, since
$\mathcal{Q}_{\theta}$ is proportional to $DX$, when we construct the
combination $\mathcal{Q}_{\theta}\mathcal{Q}_{t}$, any terms proportional to
$DX$ in $\mathcal{Q}_{t}$ will drop out by nilpotency. Therefore, we can write

$\displaystyle\mathcal{Q}_{t}\sim\frac{1}{2}\left(f(\xi)+2\xi
f^{\prime}(\xi)\right)\dot{X}^{2}\,,$ (152)

where “$\sim$” means equality up to terms which will not contribute to the
product $\mathcal{Q}_{\theta}\mathcal{Q}_{t}$. The flow equation (148) becomes

$\displaystyle\frac{i}{2}f^{\prime}(\xi)\dot{X}^{3}DX=\frac{i}{2}\cdot\left(f(\xi)\left(f(\xi)+2\xi
f^{\prime}(\xi)\right)\right)\,\dot{X}^{3}\,DX\,,$ (153)

whose solution is

$\displaystyle f(\xi)=\frac{1}{2\xi}\left(1-\sqrt{1-4\xi}\right)\,.$ (154)

Thus, the full solution for the deformed superspace Lagrangian is

$\displaystyle\mathcal{A}^{(\lambda)}=\frac{i}{4\lambda\dot{X}^{2}}\left(1-\sqrt{1-4\lambda\dot{X}^{2}}\right)\,\dot{X}DX\,.$
(155)

As we mentioned around (149), the same flow can be acquired by deforming with
another operator

$\frac{\widetilde{\mathcal{Q}}_{\theta}\mathcal{Q}_{t}}{1+2\lambda\mathcal{Q}_{t}}\,,$
(156)

similar to the irrelevant operators used in the previous sections. To see that
(156) yields the same flow, we first compute

$\displaystyle\widetilde{\mathcal{Q}}_{\theta}$
$\displaystyle=\mathcal{Q}_{\theta}+\dot{X}\frac{\delta\mathcal{A}}{\delta\dot{X}}-\mathcal{A}$
(157)
$\displaystyle=if(\xi)\dot{X}DX+\frac{i}{2}\left(f(\xi)+2f^{\prime}(\xi)\xi\right)\dot{X}DX-\frac{i}{2}f(\xi)\dot{X}DX$
$\displaystyle=i(f(\xi)+\xi f^{\prime}(\xi))\dot{X}DX\,.$

Then using the expression for $\mathcal{Q}_{t}$ from (152), we have

$\frac{\widetilde{\mathcal{Q}}_{\theta}\mathcal{Q}_{t}}{1+2\lambda\mathcal{Q}_{t}}=\frac{\left(f(\xi)+\xi
f^{\prime}(\xi)\right)\left(f(\xi)+2\xi
f^{\prime}(\xi)\right)}{1+\xi\left(f(\xi)+2\xi
f^{\prime}(\xi)\right)}\frac{i\dot{X}^{3}DX}{2}\,.$ (158)

The flow equation
$\frac{\partial\mathcal{A}}{\partial\lambda}=\frac{\widetilde{\mathcal{Q}}_{\theta}\mathcal{Q}_{t}}{1+2\lambda\mathcal{Q}_{t}}$
leads to the following differential equation:

$f^{\prime}(\xi)=\frac{(f(\xi)+2\xi f^{\prime}(\xi))(f(\xi)+\xi
f^{\prime}(\xi))}{1+\xi(f(\xi)+2\xi f^{\prime}(\xi))}\,,$ (159)

which has the same solution as in (154) from the flow triggered by the
operator $\mathcal{Q}_{\theta}\mathcal{Q}_{t}$,

$f(\xi)=\frac{1-\sqrt{1-4\xi}}{2\xi}\,.$ (160)

Therefore, the operator
$\frac{\tilde{\mathcal{Q}}_{\theta}\mathcal{Q}_{t}}{1+2\lambda\mathcal{Q}_{t}}$
also triggers the same $T\overline{T}$-like flow. Notice that from
$\mathcal{Q}_{t}=D\widetilde{\mathcal{Q}}_{\theta}+\mathcal{Q}_{c}$, we can
express the operator
$\frac{\widetilde{\mathcal{Q}}_{\theta}\mathcal{Q}_{t}}{1+2\lambda\mathcal{Q}_{t}}$
in terms of the conserved currents $\mathcal{Q}_{c}$ and
$\widetilde{\mathcal{Q}}_{\theta}$, satisfying the conservation equation

$\partial_{t}\widetilde{\mathcal{Q}}_{\theta}+D\mathcal{Q}_{c}=0\,.$ (161)

We now argue that this result is on-shell equivalent to the dimensional
reduction of the solution to the supercurrent-squared flow for the
corresponding $\mathcal{N}=(0,1)$ theory in $2d$. The dimensional lift of this
theory can be written as

$\displaystyle I=\int\,d^{2}x\,d\theta\,D_{+}\Phi\partial_{++}\Phi\,.$ (162)

A superspace Noether procedure analogous to the one that we have used in the
$\mathcal{N}=(1,1)$ analysis of section 3 can also be applied here. The input
of this process is a superspace Lagrangian
$\mathcal{A}(D_{+}\Phi,\partial_{\pm\pm}\Phi)$. The output is a conservation
equation

$\displaystyle\begin{split}&\partial_{--}\mathcal{S}_{+++}+D_{+}\mathcal{T}_{++--}=0\,,\\\
&\partial_{--}\mathcal{S}_{--+}+D_{+}\mathcal{T}_{----}=0\,,\end{split}$ (163)

where $\mathcal{S}_{\pm\pm+}$ and $\mathcal{T}_{\pm\pm--}$ are superfields
given by:

$\displaystyle\begin{split}&\mathcal{S}_{+++}=\frac{\delta\mathcal{A}}{\delta(\partial_{--}\Phi)}\partial_{++}\Phi\,,\\\
&\mathcal{S}_{--+}=\frac{\delta\mathcal{A}}{\delta(\partial_{--}\Phi)}\partial_{--}\Phi-\mathcal{A}\,,\\\
&\mathcal{T}_{++--}=\frac{\delta\mathcal{A}}{\delta(D_{+}\Phi)}\partial_{++}\Phi+D_{+}\left(\frac{\delta\mathcal{A}}{\delta(\partial_{++}\Phi)}\partial_{++}\Phi\right)-D_{+}\mathcal{A}\,,\\\
&\mathcal{T}_{----}=\frac{\delta\mathcal{A}}{\delta(D_{+}\Phi)}\partial_{--}\Phi+D_{+}\left(\frac{\delta\mathcal{A}}{\delta(\partial_{++}\Phi)}\partial_{--}\Phi\right)\,.\end{split}$
(164)

The $\mathcal{N}=(0,1)$ supercurrent-squared flow is defined by

$\displaystyle\frac{\partial}{\partial\lambda}\mathcal{A}^{(\lambda)}=\mathcal{S}_{+++}\mathcal{T}_{----}-\mathcal{S}_{--+}\mathcal{T}_{++--}\,.$
(165)

Beginning from the seed superspace Lagrangian (162), we make an ansatz for the
finite-$\lambda$ solution:

$\displaystyle\mathcal{A}^{(\lambda)}=f(\lambda\partial_{++}\Phi\partial_{--}\Phi)D_{+}\Phi\partial_{--}\Phi\,.$
(166)

After evaluating the supercurrents, computing the combination of bilinears
(165), and simplifying the differential equation, one finds

$\displaystyle x\frac{\partial f}{\partial x}=-xf^{2}-2x^{2}f\frac{\partial
f}{\partial x}\,,$ (167)

where $x=\lambda\partial_{++}\Phi\partial_{--}\Phi$. The solution is

$\displaystyle f(x)=\frac{\sqrt{1+4x}-1}{2x}.$ (168)

Thus, the fully deformed superspace Lagrangian is

$\displaystyle\mathcal{A}^{(\lambda)}=\frac{1}{2\lambda\partial_{++}\Phi\partial_{--}\Phi}\left(\sqrt{1+4\lambda\partial_{++}\Phi\partial_{--}\Phi}-1\right)D_{+}\Phi\partial_{++}\Phi$
(169)

giving the $\mathcal{N}=(1,0)$ superspace action

$\displaystyle S=\int
dtd\theta\frac{\left(\sqrt{1+4\lambda\partial_{++}\Phi\partial_{--}\Phi}-1\right)D_{+}\Phi\partial_{++}\Phi}{2\lambda\partial_{++}\Phi\partial_{--}\Phi}\,.$
(170)

Upon dimensional reduction, we identify the superfield $\Phi$ with $X$, and
all partial derivatives $\partial_{\pm\pm}\Phi$ are proportional to $\dot{X}$.
Doing this, we see that – up to various constant factors that can be absorbed
into rescalings – the solution (169) exactly matches (155).

We will not perform the analog of the analysis in section 4, where we
dimensionally reduced the supercurrent-squared operator itself using the trace
flow equation, in this $\mathcal{N}=1$ case. However, such a procedure should
certainly be possible. One would identify a superfield analog of the trace
flow equation which relates $\mathcal{S}_{\pm\pm+}$ and
$\mathcal{T}_{\pm\pm--}$, and then use this to eliminate the appropriate
linear combinations of these superfields that correspond to the $x$
directions. One might even expect the process to be simpler, in this case,
since the dimensionally reduced deformation should be bilinear of the form
$\mathcal{Q}_{\theta}\mathcal{Q}_{t}$ rather than a rational function of
supercurrents. In particular, it appears $\mathcal{T}_{++--}$ is structurally
similar to $\mathcal{Q}_{t}$, so one might believe that the correct
dimensionally reduced deformation would be some product of
$\mathcal{T}_{++--}$ with another superfield that plays the role of
$\mathcal{Q}_{\theta}$.

We conclude this subsection with a few comments about the relationship between
$\mathcal{N}=1$ and $\mathcal{N}=2$ theories.

  1. 1.

Every SUSY-QM theory with $\mathcal{N}=2$ SUSY can be viewed as a special case
of a theory with $\mathcal{N}=1$ supersymmetry. Therefore, one can always
write the $f(\mathcal{Q},\bar{\mathcal{Q}})$ deformation for a theory with
$\mathcal{N}=2$ supersymmetry and integrate out one of the fermionic
directions to obtain a deformation in $\mathcal{N}=1$ superspace. The
resulting $\mathcal{N}=1$ deformation should be on-shell equivalent to the
combination $\mathcal{Q}_{\theta}\mathcal{Q}_{t}$ which we described in this
section since this generates the appropriate supercurrent-squared flows for
$\mathcal{N}=1$ theories. Evidence for the on-shell equivalence of these two
flows in the case of $2d$ field theory was given in [98]; the SUSY-QM case
should be similar.

  2. 2.

As pointed out in [206], quantum mechanical theories with $\mathcal{N}=1$
supersymmetry are often equivalent to $\mathcal{N}=2$ theories because they
have a hidden second supersymmetry. In particular, this will be true for any
$\mathcal{N}=1$ theory with a fermion number symmetry. A second hidden
supersymmetry of this form was not present in the case of a single
$\mathcal{N}=1$ superfield, which we considered in this section, but it would
be present in other cases (such as those with an even number of
$\mathcal{N}=1$ superfields). For those theories, one should be able to
present the supercurrent-squared deformation of the theory in either
$\mathcal{N}=1$ or $\mathcal{N}=2$ language, and we expect the results to be
equivalent on-shell.

####  3 The $HT$ form of the deformation

We now turn to the question of why our deforming operator in the case of
$\mathcal{N}=1$ supersymmetry could be written either as a bilinear
$\mathcal{Q}_{\theta}\mathcal{Q}_{t}$ or a rational function of the form

$\displaystyle\frac{\widetilde{\mathcal{Q}}_{\theta}\mathcal{Q}_{t}}{1+2\lambda\mathcal{Q}_{t}}\,,$
(171)

although these two expressions appear quite different. This equivalence is
related to an exact correspondence between two expressions involving the
Hamiltonian, which holds for $T\overline{T}$-like deformations of any kinetic
seed theory. It will be simplest to discuss this correspondence in the purely
bosonic context first, without any supersymmetry.

To begin, we first point out that there are two natural notions of energy in
the theory of quantum mechanics. The first is the Hamiltonian $H$ of the
system. Since we work in Euclidean signature and interpret the Euclidean
Lagrangian as the Hamiltonian, $H$ is the object that sits under the integral
sign in the action:

$\displaystyle I=\int\,dt\,H\,.$ (172)

The second notion of energy is the (Euclidean) Hilbert stress tensor
$T^{(\text{Hilb})}$. In $(0+1)$ dimensions, there is only a single component
of the stress tensor. It is defined by coupling the theory to a worldline
metric $g_{tt}$, or equivalently an einbein $e_{t}$, and computing:

$\displaystyle T^{(\text{Hilb})}=-\frac{2}{\sqrt{g^{tt}}}\frac{\delta
I}{\delta g^{tt}}=H-2\frac{\partial H(g^{tt})}{\partial
g^{tt}}\Big{|}_{g^{tt}=1}\,.$ (173)

Here, by $H(g^{tt})$, we mean the expression obtained by minimally coupling
$H$ to a worldline metric. Since generically $\frac{\partial
H(g^{tt})}{\partial g^{tt}}\neq 0$, the two notions of energy differ. Thus
far, we have been somewhat sloppy and used the symbols $H$ and $T$
interchangeably, for instance, in the deformation (1). Although this
deformation is written in terms of $T$, it is more properly a flow equation
for the object $H$ appearing under the integral in the Euclidean action.
Therefore, we will be more careful and write:

$\displaystyle\frac{\partial
H}{\partial\lambda}=\frac{H^{2}}{\frac{1}{2}-2\lambda H}\,,$ (174)

whose solution is

$\displaystyle H(\lambda)=\frac{1}{4\lambda}\left(1-\sqrt{1-8\lambda
H_{0}}\right)\,.$ (175)

We recall that (174) was derived using the trace flow equation, which means
that it is valid only for theories that descend from CFTs. In particular, it
does not hold for theories with potential. We now restrict to a particular
class of theories for which (174) is valid, which we will refer to as “kinetic
seed theories.” Explicitly, we assume that the undeformed Hamiltonian does not
depend on any dimensionful scale but depends linearly on the inverse metric
$g^{tt}$ when coupled to worldline gravity. For instance, the free scalar
Hamiltonian

$\displaystyle H_{0}(g^{tt})=\dot{x}^{2}=g^{tt}\partial_{t}x\partial_{t}x$
(176)

belongs to this class of theories. Since $H_{0}(g^{tt})$ depends linearly on
the metric, which is a scalar, the Hilbert stress tensor (173) associated with
$H(\lambda)$ is

$\displaystyle T^{(\text{Hilb})}$ $\displaystyle=H(\lambda)-2\frac{\partial
H}{\partial H_{0}}\frac{\partial H_{0}(g^{tt})}{\partial
g^{tt}}\Big{|}_{g^{tt}=1}$ $\displaystyle=H(\lambda)-2H_{0}\frac{\partial
H}{\partial H_{0}}\,.$ (177)

We now ask whether we can express the right side of the flow equation (174)
more simply in terms of $H$ and $T^{(\text{Hilb})}$, rather than $H$. One can
verify by explicit calculation that, for a Hamiltonian of the form (175), the
operator appearing in the flow is

$\displaystyle\frac{H^{2}}{\frac{1}{2}-2\lambda
H}=\frac{\left(\sqrt{1-8\lambda
H_{0}}-1\right)^{2}}{8\lambda^{2}\sqrt{1-8\lambda H_{0}}}\,.$ (178)

On the other hand, using the expression (3) for the Hilbert stress tensor, one
can also compute the combination

$\displaystyle HT^{(\text{Hilb})}$
$\displaystyle=H\left(H-2H_{0}\frac{\partial H}{\partial H_{0}}\right)$
$\displaystyle=-\frac{\left(\sqrt{1-8\lambda
H_{0}}-1\right)^{2}}{16\lambda^{2}\sqrt{1-8\lambda H_{0}}}\,.$ (179)

Therefore, for this class of theories, we conclude

$\displaystyle
HT^{(\text{Hilb})}=-\frac{1}{2}\left(\frac{H^{2}}{\frac{1}{2}-2\lambda
H}\right)\,.$ (180)

The upshot of this discussion is that, up to an overall constant that can be
absorbed into the scaling of $\lambda$, we are free to deform _either_ by the
combination $\frac{H^{2}}{\frac{1}{2}-2\lambda H}$ or by the combination
$HT^{(\text{Hilb})}$. We refer to this latter expression as the $HT$ form of
the flow equation (dropping the superscript (Hilb) for simplicity).

This $HT$ deformation has a simple interpretation.555In the holographic
context, we interpret the addition of the double-trace $HT$ operator as a
change in boundary conditions for the dual BF gauge theory fields. This is
discussed in section 2. From the form (173) of the Hilbert stress tensor, one
has

$\displaystyle\frac{\partial H}{\partial\lambda}=H^{2}-2H\frac{\partial
H}{\partial g^{tt}}\,.$ (181)

This is reminiscent of the form of the inviscid Burgers’ equation (39) for the
cylinder energy levels of a $T\overline{T}$-deformed CFT in two dimensions,
which we repeat:

$\displaystyle\frac{\partial E_{n}}{\partial\lambda}=E_{n}\frac{\partial
E_{n}}{\partial R}+\frac{1}{R}P_{n}^{2}\,.$ (182)

In the zero-momentum sector, the inviscid Burgers’ equation (39) admits an
implicit solution:

$\displaystyle E_{n}(R,\lambda)=E_{n}(R+\lambda E_{n}(R,\lambda),0)\,.$ (183)

This has the interpretation that the theory has effectively been put on a
cylinder with an “energy-dependent radius.” That is, energy eigenstates with
different energy eigenvalues see different effective geometries.

There is no straightforward analog of the limit $P_{n}=0$ in the quantum
mechanical case, but if we restrict to the case of small energies so that
$H^{2}$ is negligible compared to $H$, the (181) becomes

$\displaystyle\frac{\partial H}{\partial\lambda}\approx-2H\frac{\partial
H}{\partial g^{tt}}\,,$ (184)

which likewise has the implicit solution

$\displaystyle H(g^{tt},\lambda)=H(g^{tt}-2\lambda H(g^{tt},\lambda),0)\,.$
(185)

This has a similar interpretation that different states in the deformed
quantum mechanics theory see different effective energy-dependent metrics.
Note that the relative factor of $-2$ between the rescalings in (183) and
(185) is because the relation (180) between $HT$ and the dimensionally reduced
$T\overline{T}$ operator required us to re-scale $\lambda$ by a factor of
$-\frac{1}{2}$.

We now see why there were also two equivalent ways of writing the deformation
in the $\mathcal{N}=1$ case. On-shell, the quantity $\mathcal{Q}_{\theta}$ is
always proportional to the superspace Lagrangian $\mathcal{A}$ for
deformations of a free seed theory, whereas the time translation current
$\mathcal{Q}_{t}$ contains the Hilbert stress tensor. Therefore, the top
component of their product is proportional to

$\displaystyle\mathcal{Q}_{\theta}\mathcal{Q}_{t}\Big{|}_{\theta}\sim
HT^{(\text{Hilb})}\,.$ (186)

That is, the bilinear $\mathcal{Q}_{\theta}\mathcal{Q}_{t}$ is the superspace
analog of the $HT$ deformation. On the other hand, the second form of the
deformation

$\displaystyle\frac{\widetilde{Q}_{\theta}\mathcal{Q}_{t}}{1+2\lambda\mathcal{Q}_{t}}\,,$
(187)

is the $\mathcal{N}=1$ superspace analog of the
$\frac{H^{2}}{\frac{1}{2}-2\lambda H}$ form of the deformation. The fact that
we obtained square root solutions to the two flows driven by
$\mathcal{Q}_{\theta}\mathcal{Q}_{t}$ and
$\frac{\widetilde{Q}_{\theta}\mathcal{Q}_{t}}{1+2\lambda\mathcal{Q}_{t}}$ is
therefore expected since this is related to the statement that we likewise
obtain square roots solutions in the bosonic sector using either $HT$ or
$\frac{H^{2}}{\frac{1}{2}-2\lambda H}$.

###  7 Conclusion

In this chapter, we have proposed a manifestly supersymmetric deformation of
the superspace Lagrangian for a theory of $\mathcal{N}=2$ quantum mechanics:

$\displaystyle\frac{\partial\mathcal{A}}{\partial\lambda}=f(\mathcal{Q},\bar{\mathcal{Q}})\equiv\frac{\mathcal{Q}\bar{\mathcal{Q}}}{\frac{1}{2}-2\lambda\bar{D}\mathcal{Q}}\,.$
(188)

The conserved superfields $\mathcal{Q},\bar{\mathcal{Q}}$ are computed using a
Noether prescription, for which we have given explicit formulas that apply to
a class of theories involving scalar superfields $X^{i}$. We have also
performed several non-trivial checks that this superspace deformation is on-
shell equivalent to the dimensional reduction of the $\mathcal{N}=(1,1)$
supercurrent-squared deformation of two-dimensional field theories, at least
for conformally invariant seed theories. For such conformal seeds, this
deformation is, therefore, a natural candidate for the appropriate
supersymmetric version of $T\overline{T}$ for $(0+1)$-dimensional theories.

Additionally, we proposed two manifestly supersymmetric deformations for an
$\mathcal{N}=1$ quantum mechanics theory

$\displaystyle\frac{\partial\mathcal{A}}{\partial\lambda}=\frac{1}{2}\mathcal{Q}_{\theta}\mathcal{Q}_{t}\quad\text{
and
}\quad\frac{\partial\mathcal{A}}{\partial\lambda}=\frac{\widetilde{\mathcal{Q}_{\theta}}\mathcal{Q}_{t}}{1+2\lambda\mathcal{Q}_{t}}\,.$
(189)

Although the forms of this deformation appear different, we showed that they
produce the same flow equation when applied to the seed theory of a single
free scalar. This flow equation matches the dimensional reduction of the
$\mathcal{N}=(0,1)$ supercurrent-squared operator. We also interpreted the
equivalence of these deformations by pointing out an analogous rewriting that
holds for deformations of the bosonic sector of kinetic seed theories, namely
$-\frac{1}{2}HT$ and $\frac{H^{2}}{\frac{1}{2}-2\lambda H}$.

We close this chapter with future research.

_More Supersymmetry_

Perhaps the most obvious follow-up to this work is to exhibit a version of our
superspace deformation with differing amounts of supersymmetry. For instance,
it should be possible to define deformations of SUSY-QM theories, which are
related by dimensional reduction to the supercurrent-squared deformations of
theories with $(0,2)$ or $(2,2)$ supersymmetry [100, 99]. The case of an
$\mathcal{N}=4$ SUSY-QM theory, which descends from a $\mathcal{N}=(2,2)$
field theory is perhaps more interesting since such field theories are
especially well-studied.

It may be that such an analysis is more amenable to a different technique for
obtaining the supercurrents than the one we have used here. In the $2d$ case,
such supercurrents for theories with $\mathcal{N}=(1,1)$ supersymmetry were
straightforward to compute using either a Noether procedure [98] or via
coupling to supergravity [97]. However, in the case of
$2d\,,\,\mathcal{N}=(2,2)$ theories, it was more convenient to couple to the
appropriate supergravity rather than employing a Noether approach [100]. From
this intuition, one might expect that the computation of supercharges for
deformations of $\mathcal{N}=4$ SUSY-QM theories might likewise be easier to
perform by coupling to worldline supergravity.

It would also be interesting to understand $T\overline{T}$-type deformations
in theories with even more supersymmetry, like $\mathcal{N}=8$ or maximal
SUSY.666Other deformations of QM theories with more supersymmetry, albeit not
related to $T\overline{T}$, have been considered in [207, 208, 209, 210]. See
also [211, 212, 213] for discussions of the super-Schwarzian with more SUSY.
Such an endeavor is complicated by the absence of a conventional superspace,
which makes all of the supersymmetries manifest. One could, of course, work
with a reduced superspace like $\mathcal{N}=2$ or $\mathcal{N}=4$, which
geometrizes a subset of the supersymmetry transformations, but the action of
the non-manifest SUSY generators will then be corrected order-by-order in
$\lambda$ after turning on a $T\overline{T}$-like deformation.

_Connections to Supersymmetric BF Gauge Theory_

Another direction concerns the holographic interpretation of these results. We
have emphasized that part of the motivation for considering deformations of
$(0+1)$-dimensional theories whose Lagrangians take the purely kinetic form:

$\displaystyle S=\int\,dt\,g_{ij}(X)\dot{X}^{i}\dot{X}^{j}\,,$ (190)

where the $X^{i}$ are coordinates on a Lie group $G$, is that such theories
are dual to BF gauge theories with gauge group $G$. This relationship holds
with or without supersymmetry; in the SUSY case, the dual is a SUSY-BF theory,
and the quantum mechanics theory admits an interpretation as a particle moving
on a supergroup. In the special case that the gauge group is an extension of
$\mathrm{SL}(2,\mathbb{R})$, the dual BF theory is also related to JT gravity
[72] and to other interesting theories such as SYK.

There have been some interpretations offered for the holographic
interpretations of the $T\overline{T}$-like deformation of quantum mechanics
in these various dual theories, at least in the non-manifestly-supersymmetric
context. For instance, connections to cutoff JT gravity and the Schwarzian
have been discussed in [115, 116, 117, 131, 214], related analyses of the dual
matrix models have been carried out in [215, 5], and a connection to modified
boundary conditions in BF gauge theory is discussed in chapter 2.

It would be very interesting to extend these holographic interpretations to
the case with manifest supersymmetry. In the undeformed case, the
correspondence between the quantum mechanical theory of a particle moving on
an $\mathrm{SL}(2,\mathbb{R})$ group manifold and the BF gauge theory with
gauge group $\mathrm{SL}(2,\mathbb{R})$ is lifted to the supersymmetric
setting by promoting the gauge group to either $\mathrm{OSp}(1\,|\,2)$ for
$\mathcal{N}=1$ SUSY or $\mathrm{OSp}(2\,|\,2)$ for $\mathcal{N}=2$ SUSY, as
is nicely reviewed in Section 4.2 of [166]. Here $\mathrm{OSp}(N\,|\,2p)$ is
the orthosymplectic supergroup, a particular sub-supergroup of
$\mathrm{GL}(N\,|\,2p)$, which is the supergroup version of the general linear
group $\mathrm{GL}(N)$.777In particular, $\mathrm{OSp}(N\,|\,2p)$ is the sub-
supergroup of $\mathrm{GL}(N\,|\,2p)$ which preserves a symmetric bilinear
form on the bosonic elements (analogous to the orthogonal group) and preserves
a symplectic form on the fermionic elements (analogous to the symplectic
group); hence the name “orthosymplectic.” We focus on the $\mathcal{N}=2$ case
which was the main focus of this thesis.888 Various aspects of the
$\mathcal{N}=1$ version of this theory, including its relationship to the
super-Schwarzian and the properties of boundary-anchored Wilson lines, have
been studied in [216]. This $\mathcal{N}=2$ supersymmetric BF theory was
analyzed in [217, 218], and its action can be written as

$\displaystyle S_{\text{BF}}^{\mathcal{N}=2}=\int_{M}\mathrm{STr}\left(\Phi
F\right)-\frac{1}{2}\oint_{\partial M}\mathrm{STr}(\Phi\mathscr{A}_{t})\,,$
(191)

where $\mathrm{STr}$ is the supertrace, $\Phi$ is the supersymmetric analogue
of the scalar $\phi$ appearing in the usual BF Lagrangian
$\mathcal{L}_{\text{BF}}=\mathrm{tr}(\phi F)$, and
$F=d\mathscr{A}+\mathscr{A}\wedge\mathscr{A}$ is the field strength of a
supersymmetric gauge connection $\mathscr{A}$. In this $\mathcal{N}=2$ case,
each of $\mathscr{A}$ and $\Phi$ admit an expansion in terms of the $8$
generators of the $\mathfrak{osp}(2\,|\,2)$ Lie superalgebra; these consist of
the usual $3$ generators of $\mathfrak{sl}(2,\mathbb{R})$, along with four
fermionic generators, and one additional bosonic $\mathfrak{u}(1)$ generator
required by supersymmetry.

One would like to understand what modification of the bulk super-BF theory
corresponds to turning on the $f(\mathcal{Q},\bar{\mathcal{Q}})$ operator in
the boundary SUSY-QM theory. We can view this question as the dimensional
reduction of a related query: what has happened to a bulk $\mathrm{AdS}_{3}$
supergravity theory, written in Chern-Simons variables when the dual
supersymmetric field theory is deformed by supercurrent-squared? The standard
intuition from $\mathrm{AdS}/\mathrm{CFT}$ is that the addition of a double-
trace operator in the field theory corresponds to a modification of the
boundary conditions for the bulk fields [162, 163], although it is not clear
that this intuition should generically apply for irrelevant double-trace
deformations as opposed to relevant (or marginal) operators. In the non-
supersymmetric context, it has been argued that this expectation is indeed
correct and that activating $T\overline{T}$ in a $2d$ CFT corresponds to a
rotation of the sources and expectation values in the dual
$\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ Chern-Simons theory
[110]. A similar rotation of boundary conditions appears in the non-
supersymmetric setting of a $2d$ BF gauge theory, which is dual to a boundary
$(0+1)$-dimensional theory in chapter 2. It would be interesting to see
whether the deformed super-BF theory, dual to a quantum mechanics theory
deformed by $f(\mathcal{Q},\bar{\mathcal{Q}})$ likewise admits such an
interpretation, perhaps involving a linear mixing of the coefficient functions
multiplying the $8$ generators of $\mathfrak{osp}(2\,|\,2)$ in the expansions
of $\Phi$ and $\mathcal{A}$.

_Deformations of Multiple Scalars; Target Space Geometry_

Another avenue for investigation is seeking solutions to the
$f(\mathcal{Q},\bar{\mathcal{Q}})$ flow equations for theories with multiple
scalars. In this work, we have only managed to find a closed-form result (119)
for the deformed theory in the case of a single scalar, and even then, we have
only found an expression that is on-shell equivalent to the full solution
since we have imposed one implication of the superspace equations of motion.
But of course, the most interesting examples are the deformed theories of a
particle moving on a higher-dimensional manifold, such as the $3$-dimensional
$\mathrm{SL}(2,\mathbb{R})$ group manifold relevant for the Schwarzian theory.
We have already mentioned in the analysis of the corresponding question for
$2d$ field theories around equation (26) that solving the flow in this context
is much more difficult because one expects a system of coupled PDEs for the
functions multiplying the various two-fermion, four-fermion, etc. terms in the
superspace Lagrangian. However, if one could find a partial or approximate
solution with multiple scalars – perhaps after going partly on-shell, as we
have done here – the result could be interesting.

For example, given such a solution, we could ask whether the resulting
deformed theory still admits an interpretation as a point particle moving on
some deformed target-space geometry. One might think not since our intuition
is that the ordinary $T\overline{T}$ flow in $2d$ generates theories that are
no longer local QFTs. Analogously, one might expect that
$f(\mathcal{Q},\bar{\mathcal{Q}})$ deformed SUSY-QM theories exhibit some
signature of non-locality. For instance, the particle whose position is
described by the $X^{i}$ in the undeformed theory could become delocalized
into a “fuzzy particle” over a length scale controlled by $\lambda$. It would
be interesting to ask whether other properties of the target manifold can be
probed in this case or if the target manifold itself is changed.

On the other hand, in the undeformed theory, the Witten index of the theory is
controlled by the Euler characteristic of the target space. Since our
$f(\mathcal{Q},\bar{\mathcal{Q}})$ flow is the supersymmetric extension of an
$f(H)$ deformation – which does not affect the energy eigenstates but merely
modifies their energy eigenvalues – it seems that this index remains unchanged
under our deformation, which suggests that the target space topology is also
unmodified.

There is some evidence that other indices cannot flow under
$T\overline{T}$-like deformations. For instance, related indices like the
elliptic genus have been shown not to flow under the usual $T\overline{T}$ in
two dimensions if the seed theory is conformal [84], and the same conclusion
seems likely to hold if the undeformed theory is integrable but not conformal
[9]. Nonetheless, it would be worthwhile to make this intuition precise in the
SUSY-QM case and perhaps look for other index-like quantities that do flow
under $f(\mathcal{Q},\bar{\mathcal{Q}})$ and which may admit an interpretation
via target space geometry.

_Relation to Supersymmetric SYK_

We mention one final future direction, along the lines of the previously
mentioned question about the relationship of this deformation with super-BF
theory, but which is also related to the issue of defining our
$f(\mathcal{Q},\bar{\mathcal{Q}})$ deformation in quantum mechanics with a
potential.

It is well-known that the Schwarzian or particle-on-a-group theory is also
related to the SYK model of Majorana fermions with random all-to-all
interactions [219, 220]. The SYK model has a supersymmetric extension [221,
222, 223]; for an (incomplete) collection of related works on the SYK model
and supersymmetry, see [224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234,
235, 236, 237, 238] and references therein.

The application of $T\overline{T}$-like deformations in quantum mechanics to
the non-supersymmetric SYK model was carried out in [116] (see also [239]). In
that case, after shifting the ground state energy of the model by a constant
$E_{0}$, it was pointed out that there are two choices for how to perform the
deformation:

  1. 1.

First, perform the average over disorder in the undeformed model and then
deform the Hamiltonian by the desired $T\overline{T}$ or $f(H)$ operator.

  2. 2.

Begin by deforming the Hamiltonian by some $f(H)$ operator and then perform
the disorder average in the deformed theory.

The authors of [116] point out that it is easier to do the former since if one
first deforms the Hamiltonian then this procedure will introduce higher powers
of the disorder, which makes the resulting disorder average difficult.
Although the latter provides a microscopic picture of physics.

It would be interesting to carry out a version of this analysis in the
supersymmetric setting using the techniques developed in the present work. To
do this, one should use a presentation of the supersymmetric SYK action which
is written directly in $\mathcal{N}=1$ or $\mathcal{N}=2$ superspace, such as
those developed in [221, 240]. For concreteness, let us focus on the
$\mathcal{N}=2$ case. The degrees of freedom for the $\mathcal{N}=2$ super-SYK
model are packaged into chiral superfields $\Psi,\bar{\Psi}$ which obey the
constraints

$\displaystyle\bar{D}\Psi=D\bar{\Psi}=0\,.$ (192)

The Lagrangian is a sum of a kinetic $F$-term plus a holomorphic
superpotential:

$\displaystyle L=\int\,d\bar{\theta}\,\mathcal{A}_{\text{kin}}+\left(\int
d\theta\,\mathcal{A}_{\text{potential}}+\text{c.c}\right)\,,$
$\displaystyle\mathcal{A}_{\text{kin}}=\bar{\Psi}D_{i}\Psi\,\qquad\mathcal{A}_{\text{potential}}=C_{i_{1}\cdots
i_{k}}\Psi_{i_{1}}\cdots\Psi_{i_{k}}\,.$ (193)

To study the appropriate $T\overline{T}$-type deformation of such a superspace
Lagrangian, one would, therefore, need to generalize the analysis presented in
this work to allow for fermionic superfields and potentials. One could also
attempt to understand deformations of SYK via the dimensional reductions of
appropriate two-dimensional field theories [241, 222], or to investigate
$T\overline{T}$-like deformations in related disordered supersymmetric models
[242, 243, 244].

##  Chapter 4 $T\overline{T}$-Deformed Free Energy of the Airy Model

Sharpening the correspondence of JT gravity and its matrix model description
under the $T\overline{T}$ deformation is of interest. To proceed, we simplify
the problem by considering the Airy model and deform Airy correlators in the
same way as in $T\overline{T}$-deformed JT gravity. We use those correlators
to numerically compute the annealed and quenched free energies for $\lambda>0$
and $\lambda<0$ from an integral representation of the replica trick. At the
leading order in $\lambda$ and low temperatures, we confirm that the genus-
zero quenched free energy monotonically decreases as a function of temperature
when perturbation theory is valid. We then study the all-genus quenched free
energy at low temperatures, where we discover and discuss subtleties due to
non-perturbative effects in the Airy model, as well as the contributions from
the non-perturbative branch under the $T\overline{T}$ deformation.

###  1 Introduction

An application of the $T\overline{T}$ deformation relevant to this chapter is
probing the low-temperature limit of JT gravity and its Airy model
description.111Hereafter, we will refer to the Airy limit of Gaussian matrix
models as the “Airy model.” To further motivate the Airy model, we list a few
reasons why it is interesting and useful to study. Firstly, the genus
expansion can be summed via [245] allowing one to make definite statements on
non-perturbative corrections. Secondly, JT gravity at low temperatures, or
more precisely the ’t Hooft limit:

$\hbar\rightarrow
0\,,\quad\beta\rightarrow\infty\,,\quad\operatorname{for~{}fixed}~{}\hbar\beta\,,$
(1)

is dual to the Airy model at all genera known from [246]. The authors of [246]
showed all the non-trivial information of the spectral curve of JT gravity is
still preserved under the ’t Hooft limit (1). Additionally, there is an
important caveat as [79] finds: the exact eigenvalue density has an
exponential leakage when $E<0$, which is denoted as the “classically
forbidden” region causing the system unstable. This non-perturbative
instability is not special for the Airy model as the same phenomena occur for
the matrix model dual of JT gravity.

However, a recent proposal by [247] improves the non-perturbative behavior of
JT gravity by removing the non-perturbative instabilities. Therefore, one
should only expect this relation between JT gravity and the Airy model to
confidently hold perturbatively. There are subtleties for the Airy model’s
quenched free energy not monotonically decreasing as a function of temperature
using directly the replica trick as done by [248].

Fortunately, Okuyama [249] showed this failure of monotonicity arises due to
analytical continuation issues in the correlators $\langle Z^{n}\rangle$ to
$\langle Z^{n=0}\rangle$ and proposed an alternative formulation of the
replica trick to correctly give a monotonically decreasing quenched free
energy in the low-temperature limit after summing over all genus. With these
motivations of the Airy model, we wish to investigate how some of these
features change under the $T\overline{T}$ deformation with a holographic
picture in mind to understand JT gravity and its matrix model dual.

Through the lens of the $T\overline{T}$ deformation, it is interesting to
probe observables between JT gravity and its double-scaled matrix model dual
description at a finite cutoff governed by $\lambda$, as done in [115, 116,
117, 131, 215, 214]. The purpose of this chapter is to further sharpen the
correspondence between JT gravity at low temperatures and its Airy model
description by computing the correlators and quenched free energy. We now
comment on the deformed energy spectrum in JT gravity.

Throughout several instances in this thesis, we established and used the
following differential equation for the one-dimensional stress scalar
$T^{\tau}_{\tau}=f^{\pm}_{\lambda}(E)$:

$\frac{\partial
f^{\pm}_{\lambda}(E)}{\partial\lambda}=\frac{f^{\pm}_{\lambda}(E)^{2}}{\frac{1}{2}-2\lambda
f^{\pm}_{\lambda}(E)}$ (2)

with two solutions

$f^{\pm}_{\lambda}(E)=\frac{1\pm\sqrt{1-8\lambda E}}{4\lambda}\,.$ (3)

As can be seen in the deformed energy spectrum (3) and figure 1, the sign of
$\lambda>0$ violates unitarity when the undeformed energy is
$E>\frac{1}{8\lambda}$. The authors of [117] carefully dealt with this
violation of unitarity and restored it in their rigorous non-perturbative
treatment for the deformed partition function from a Wheeler-de Witt
wavefunction perspective. In short, one writes a linear combination of the
wavefunction of the two branches (3) such that the density of states stays
real for all energies.

(a) $|\lambda|=1$

(b) $|\lambda|=0.01$

Figure 1: We plot the entire real deformed energy spectrum as a function of
the undeformed energy $E$ when $|\lambda|=1$ and $|\lambda|=0.01$. Here
$f^{+}_{\lambda>0}(E)$ (blue curve) and $f^{+}_{\lambda<0}(E)$ (green curve)
describe the non-perturbative branch. While $f^{-}_{\lambda>0}(E)$ (orange
curve) and $f^{-}_{\lambda<0}(E)$ (red curve) describe the perturbative
branch.

Other treatments of the deformation parameter $\lambda$’s sign are addressed
by [215] from a double-scaled matrix model perspective of JT gravity with an
attempt to define the dual deformed matrix model description at a finite
cutoff. Unfortunately, the analysis of [215] was unable to match the
$T\overline{T}$-deformed correlators between JT gravity and the matrix model
description, but did provide several alternative methods on how one could
properly make this correspondence well-defined. We will discuss the importance
of [117] and [215] more throughout various places of this chapter.222A recent
proposal in [250] successfully matched the $T\overline{T}$-deformed partition
functions of $\mathcal{N}=1$ type 0A and type 0B JT supergravity with the
associated matrix models. Additional evidence of the duality from [250] was
calculating the deformed $T\overline{T}$-deformed matrix model correlators via
topological recursion relations.

It is helpful to point out in this chapter, that we will think about JT
gravity and its $T\overline{T}$ deformation from the point of view of the
matrix model description. As shown in [79], higher topology contributions in
JT gravity are captured entirely by a double-scaled matrix model. The
partition function of JT gravity at any genus and with any number of
boundaries can be computed from the correlators in its matrix model
description. From the point of view of boundary theory, we only know that JT
gravity on a disk is dual to Schwarzian theory on the boundary. Also, the
result of [79] showed the connected $n$-point function in JT gravity – namely,
the partition function on a connected surface with $n$ boundaries – does not
factorize. The boundary theory is an ensemble of theories, not a single
theory. Since the replica trick is essential to compute quenched free energy
and requires knowledge of $n$-point correlators, we will think from the point
of view of the matrix model and its correlators rather than directly from the
boundary theory.

In section 1, we first explain how the $T\overline{T}$-deformed partition
function of various topologies in JT gravity is found through an integral
transformation of the undeformed partition function. Additionally, in section
1, we then extend this integral transformation to find a relation between the
deformed and undeformed correlators via:

$\langle Z(\beta_{1})\cdots
Z(\beta_{n})\rangle_{\lambda}=\int_{-\infty}^{\infty}dE_{1}\cdots
dE_{n}\,\rho(E_{1},E_{2},\cdots,E_{n})e^{-\beta_{1}f^{-}_{\lambda}(E_{1})}\cdots
e^{-\beta_{n}f^{-}_{\lambda}(E_{n})}\,,$ (4)

where $\rho(E_{1},\cdots,E_{n})$ is defined by ensuring

$\langle Z(\beta_{1})\cdots
Z(\beta_{n})\rangle_{0}=\int_{-\infty}^{\infty}dE_{1}\cdots
dE_{n}\,\rho(E_{1},E_{2},\cdots,E_{n})e^{-\beta_{1}E_{1}}\cdots
e^{-\beta_{n}E_{n}}\,.$ (5)

Only when $\lambda<0$, we can use the integration kernel
$K(\beta,\beta^{\prime})=\frac{\beta}{\sqrt{-8\pi\lambda}\beta^{\prime\frac{3}{2}}}e^{\frac{(\beta-\beta^{\prime})^{2}}{8\lambda\beta^{\prime}}}$
to be reviewed in section 1 to conveniently compute the deformed correlators:

$\langle Z(\beta_{1})\cdots
Z(\beta_{n})\rangle_{\lambda}=\int^{\infty}_{0}d\beta_{1}^{\prime}K(\beta_{1},\beta_{1}^{\prime})\cdots\int^{\infty}_{0}d\beta_{n}^{\prime}K(\beta_{n},\beta_{n}^{\prime})\langle
Z(\beta_{1}^{\prime})\cdots Z(\beta_{n}^{\prime})\rangle_{0}\,,$ (6)

where

$\int_{0}^{\infty}d\beta^{\prime}K(\beta,\beta^{\prime})e^{-\beta^{\prime}E}=e^{-\beta
f_{\lambda}^{-}(E)}\,.$ (7)

This is the case when we look at any particular order in the genus expansion.
However, when we look at the exact solution of the Airy model, non-
perturbative effects there extend the integration range to
$E_{i}\in(-\infty,\infty)$, reflecting the translational invariance of the
underlying effective Hamiltonian [247]. As we later encounter the non-
perturbative effect of the $T\overline{T}$ deformation, hereafter, we refer to
this as the non-perturbative instability to distinguish the two.

With the density of states for the Airy model, the undeformed one-point
function is given by

$\langle
Z(\beta)\rangle_{\text{Airy}}=\int_{-\infty}^{\infty}dE\,\rho_{\text{Airy}}(E)e^{-\beta
E}\,,$ (8)

where, from [251, 252], the density of states is

$\rho_{\text{Airy}}(E)=\text{Ai}^{\prime}(-E)^{2}+E\,\text{Ai}(-E)^{2}\,,$ (9)

and $\text{Ai}(x)$ is the Airy function of the first kind.

In this Airy model, we should use the following

$\langle
Z(\beta)\rangle_{\text{Airy},\lambda}=\int_{-\infty}^{\infty}dE\,\rho_{\text{Airy}}(E)e^{-\beta
f^{-}_{\lambda}(E)}\,,$ (10)

instead of (6), which is invalid because the integral transformation (7) of
$e^{-\beta E}$ diverges when $E<\frac{1}{8\lambda}$.

From determining the deformed correlators in section 1, an immediate
application is to calculate the annealed and quenched free energies in JT
gravity at low temperatures where certain approximations are feasible. In
section 2, we review a new way of computing the quenched free energy from an
integral representation of replica trick due to Okuyama [249]:

$\lim_{n\rightarrow 0}\frac{\left\langle Z^{n}\right\rangle-1}{n}=\ln\langle
Z\rangle-\int_{0}^{\infty}\frac{dx}{x}\left[\left\langle
e^{-Zx}\right\rangle-e^{-\langle Z\rangle x}\right]\,,$ (11)

where $\left\langle e^{-Zx}\right\rangle$ is fully determined by connected
correlators $\langle Z^{n}\rangle_{c}$. We elaborate later in section 2 why we
use this integral representation (11) instead of directly using the replica
trick in the low-temperature regime, and we comment more on the non-
perturbative contributions from the $T\overline{T}$ deformation in section 3.

However, it is generally difficult to evaluate the correlators of JT gravity
under the $T\overline{T}$ deformation let alone sum over all the connected
correlators to use Okuyama’s formula (11). Since JT gravity is known to have a
matrix model description, we will simplify the problem by studying a simpler
matrix model, the Airy model, to make progress. Here, it is important to
notice that although the $T\overline{T}$ deformation of random matrix models
has been studied in [215], the correlators there do not match those of
$T\overline{T}$-deformed JT gravity studied in [117].

The goal is to understand JT gravity and its matrix model description at a
finite cutoff. We will not follow the $T\overline{T}$ deformation of the
matrix model defined in [215]. Instead, we require the correlators of the Airy
model to transform in the same way as the correlators in JT gravity do under
the $T\overline{T}$ deformation, and we will take this as a working definition
for our version of $T\overline{T}$ deformation applied to the Airy model. This
chapter will not attempt to completely explore this version of deformation for
matrix models.

It is also important to point out one caveat of our simplification. Naïvely,
the double-scaled matrix model dual to JT gravity shares the same non-
perturbative instability as the Airy model we considered; for instance, see
[79, 247]. However, there have been studies on how to improve the non-
perturbative behavior of JT gravity and remove this undesired feature [247].

In section 3, we use (11) to numerically evaluate the quenched free energy in
the Airy model for both $\lambda>0$ and $\lambda<0$. As a warm-up, we first
compute the quenched free energy $F_{q,\lambda}(T)$ at genus-zero at the
leading order of $\lambda$ in perturbation theory. We confirm that
$F_{q,\lambda}(T)$ is a monotonic function in $T$ at low temperature when a
leading order approximation is valid. Additionally, we find that $\lambda<0$
deformation decreases the quenched free energy while the $\lambda>0$ increases
it. Intuitively, this sign of the deformation parameter $\lambda>0$
corresponds to JT gravity in a finite box with Dirichlet boundary conditions
at $r_{c}=\frac{\pi\lambda}{4G}$. The increase of the quenched free energy for
the $\lambda>0$ theory is related to the fact that the $T\overline{T}$
deformation cuts off the spectrum.

On the gravity side, one can think of this phenomenon as a gravitational
redshift. For an object in a gravitational potential, the energy measured at
the conformal boundary (which is the undeformed case) is reduced compared to
the energy measured at the particle location. For example, in the extreme case
where a particle is sent towards a black hole’s horizon, the energy at
infinity vanishes and is negative when the particle is inside the event
horizon. Here, in the cutoff gravitational theory, we are measuring this local
energy at the finite cutoff boundary, which is closer to the particle compared
to the conformal boundary. Thus, the amount of redshift decreases, and the
energy we measure increases.

Next, we use the low-temperature approximation $\langle
Z(\beta)^{n}\rangle_{c}\simeq\langle Z(n\beta)\rangle$ of the Airy model to
compute the quenched free energy for finite $\lambda$. Notice that this
computation includes contributions from all genera as well as non-perturbative
effects in the Airy model, which makes $\rho_{\text{Airy}}(E<0)\neq 0$.

For the $\lambda>0$, as studied in [117, 214], there are contributions from
the non-perturbative branch. We compute the quenched free energy with this
branch included and excluded. In both cases, we find the quenched free energy
to be divergent.

For $\lambda<0$, the issue of the complex-valued energy always arises
regardless of the value of $\lambda$ since $\rho_{\text{Airy}}(E)$ has support
on the entire real axis. Although the issue of complex energy for $\lambda<0$
has been noticed before in [80] for $2d$ CFT since the ground state energy is
$-\frac{c}{12}$, we emphasize this is different from our case, as the energy
spectrum is still bounded below by $-\frac{c}{12}$.

However, for the Airy model, the issue of complex energy will always arise
regardless of the value of $\lambda$. This has not been noticed before for
$\lambda<0$. A simple solution is imposing a hard cutoff in the energy $E$,
namely, we simply remove these states with complex-valued energy. We will see
in section 4 that this option would lead to a violation of the $T\overline{T}$
flow equation by a boundary term in the integral.

Another option would be to include those states with complex-valued energy to
demand the partition function to be real-valued. One must include the other
branch as well. By a careful choice of coefficient, one can make sure that the
boundary terms cancel each other properly so the flow equation is satisfied.
We then compute the quenched free energy for both cases: If we exclude the
non-perturbative contribution, we find the quenched free energy to be finite
and monotonic at low temperatures and decrease under the deformation. If we
include the non-perturbative contribution, we find the quenched free energy
diverges.

###  2 $T\overline{T}$ deformation in JT gravity and matrix models

####  1 $T\overline{T}$ deformed correlation functions in JT gravity

We review how one computes the deformed partition function in JT gravity via
an integral transformation of the partition function in the undeformed theory.
As already alluded to in the introduction, the deformed energy spectrum is
given by (3), and given the density of states, one can immediately compute the
deformed partition function systematically as

$Z_{\lambda}(\beta)=\int^{\infty}_{-\infty}dE~{}\rho(E)e^{-\beta
f^{-}_{\lambda}(E)}\,.$ (12)

For the moment, we will only consider the contribution from the perturbative
branch $f_{\lambda}^{-}(E)$. We will return to the potential contribution from
the non-perturbative branch $f_{\lambda}^{+}(E)$ later in section3.

The partition function $Z_{\lambda}(\beta)$ satisfies a flow equation derived
by [117]:

$\left[4\lambda\partial_{\lambda}\partial_{\beta}+2\beta\partial^{2}_{\beta}+\left(1-\frac{4\lambda}{\beta}\right)\partial_{\lambda}\right]Z_{\lambda}(\beta)=0\,,$
(13)

which is closely related to the usual inviscid Burgers’ equation from the $2d$
$T\overline{T}$-deformed energy spectra (2) (e.g. see [253]). The deformed
partition function (12) may be written in terms of an integral transformation
involving a kernel and the undeformed partition function [115] when
$\rho(E<\frac{1}{8\lambda})=0$

$\displaystyle Z_{\lambda}(\beta)$ $\displaystyle=\frac{1}{2\pi
i}\int^{\infty}_{0}d\beta^{\prime}Z_{0}(\beta^{\prime})\int^{i\infty}_{-i\infty}dEe^{-\beta
f^{-}_{\lambda}(E)+\beta^{\prime}E}$ (14)
$\displaystyle=\int^{\infty}_{0}d\beta^{\prime}K(\beta,\beta^{\prime})Z_{0}(\beta^{\prime})\,,$

where $K(\beta,\beta^{\prime})$ is a kernel determined from the deformed
energy spectrum (3):

$\displaystyle K(\beta,\beta^{\prime})$ $\displaystyle=\frac{1}{2\pi
i}\int^{i\infty}_{-i\infty}dEe^{-\beta
f^{-}_{\lambda}(E)+\beta^{\prime}E}=\frac{\beta}{\sqrt{-8\pi\lambda}\beta^{\prime\frac{3}{2}}}e^{\frac{(\beta-\beta^{\prime})^{2}}{8\lambda\beta^{\prime}}}\,,$
(15)

namely, the inverse Laplace transform of the Boltzmann factor after
deformation for $\lambda<0$.

With the integral transform (14) and its kernel (15) at hand, one can proceed
to compute the partition function of the deformed JT gravity on disk, trumpet,
and other topologies for $\lambda<0$. For example, the map between the
undeformed and deformed disk and trumpet partition functions, respectively,
are

$\displaystyle
Z_{0}(\beta^{\prime})_{\text{D}}=\frac{e^{\frac{\pi^{2}}{\beta^{\prime}}}}{4\sqrt{\pi}\beta^{\prime\frac{3}{2}}}\implies
Z_{\lambda}(\beta)_{\text{D}}=\frac{\beta}{\sqrt{-8\lambda}\pi}\frac{e^{-\frac{\beta}{4\lambda}}}{\beta^{2}+8\pi^{2}\lambda}K_{2}\left(-\frac{\sqrt{\beta^{2}+8\pi^{2}\lambda}}{4\lambda}\right)\,,$
(16) $\displaystyle
Z_{0}(b,\beta)_{\text{T}}=\frac{e^{-\frac{b^{2}}{4\beta}}}{2\sqrt{\pi\beta}}\implies
Z_{\lambda}(b,\beta)_{\text{T}}=\frac{\beta}{2\pi\sqrt{-2\lambda}}\frac{e^{-\frac{\beta}{4\lambda}}}{\sqrt{\beta^{2}-2b^{2}\lambda}}K_{1}\left(-\frac{\sqrt{\beta^{2}-2b^{2}\lambda}}{4\lambda}\right)\,,$

where we have used an identity for the modified Bessel functions of the second
kind

$\int^{\infty}_{0}d\beta^{\prime}(\beta^{\prime})^{-m-\frac{3}{2}}e^{\frac{a}{\beta^{\prime}}}e^{\frac{(\beta-\beta^{\prime})^{2}}{8\beta^{\prime}\lambda}}=\frac{2e^{-\frac{\beta}{4\lambda}}}{(\beta^{2}+8a\lambda)^{\frac{2m+1}{4}}}K_{\frac{2m+1}{2}}\left(-\frac{\sqrt{\beta^{2}+8a\lambda}}{4\lambda}\right)\,,\quad
m\in\mathbb{R}\,,$ (17)

when $\lambda<0$ and $8a\lambda+\beta^{2}>0$.

As in [117], knowing $Z_{\lambda}(\beta)_{\text{D}}$ and
$Z_{\lambda}(b,\beta)_{\text{T}}$ allows us to build general correlation
functions in the deformed JT gravity. The connected correlators on a
hyperbolic Riemann surface with $n$ boundary components and genus $g$ are

$\left\langle
Z(\beta)^{n}\right\rangle_{\mathrm{conn},\lambda}=\sum_{g=0}^{\infty}e^{-S_{0}(2g+n-2)}Z_{g,n;\lambda}(\beta)\,,$
(18)

where $S_{0}$ is the two-dimensional Einstein-Hilbert action, and
$Z_{g,n;\lambda}(\beta)$ are defined from [214] as follows:

$\displaystyle Z_{0,1;\lambda}(\beta)$
$\displaystyle=Z_{\lambda}(\beta)_{\mathrm{D}}\,,$ (19) $\displaystyle
Z_{0,2;\lambda}(\beta_{1},\beta_{2})$
$\displaystyle=\int_{0}^{\infty}db~{}bZ_{\lambda}(b,\beta_{1})_{\text{T}}Z_{\lambda}(b,\beta_{2})_{\text{T}}\,,$
$\displaystyle Z_{g,n;\lambda}(\beta_{1},\cdots,\beta_{n})$
$\displaystyle=\int_{0}^{\infty}\Bigg{(}\prod_{j=1}^{n}db_{j}~{}b_{j}Z_{\lambda}\left(b_{j},\beta_{i}\right)_{\text{T}}\Bigg{)}V_{g,n}\left(b_{1},\ldots,b_{n}\right)\,,$

with the Weil-Petersson volume $V_{g,n}\left(b_{1},\ldots,b_{n}\right)$ of a
Riemann surface $\Sigma_{g,n}$ (i.e. with genus $g$ and $n$ distinct marked
points $p_{i}$) defined in [254] as 333A few of $Z_{g,n;\lambda}(\beta)$ has
already been computed in [214]. Also, see [255, 79] for a review of Weil-
Petersson volumes in $2d$ topological gravity and matrix models.

$V_{g,n}\left(b_{1},\ldots,b_{n}\right)=\frac{1}{(2\pi^{2})^{3g-3+n}}\int_{\overline{\mathcal{M}}_{g,n}}\exp\left(\omega+\frac{1}{2}\sum_{i=1}^{n}\psi_{i}b_{i}^{2}\right)\,.$
(20)

Here $\overline{\mathcal{M}}_{g,n}$ is the Deligne-Mumford compactification of
the moduli space $\mathcal{M}_{g,n}$ of $\Sigma_{g,n}$ of complex dimension
$(3g-3+n)$, $\psi_{i}\equiv c_{1}(\mathcal{L}_{i})$ is the first Chern class
of the tautological line bundle $\mathcal{L}_{i}$ over
$\overline{\mathcal{M}}_{g,n}$ whose fiber at the point
$(x_{1},\cdots,x_{n})\in\overline{\mathcal{M}}_{g,n}$ is the cotangent line to
the curve $C$ at $x_{i}$, $\omega$ is the Weil-Petersson symplectic form on
$\overline{\mathcal{M}}_{g,n}$, and $b_{i}$ is the length of the $i$th
geodesic boundary component of $\Sigma_{g,n}$.

From the definition in (19), one might wonder if the Weil-Petersson volumes
should flow under the $T\overline{T}$ deformation. One possible way to see why
Weil-Petersson volumes do not flow under the deformation444However, in the
context of topological recursion, both the resolvent $R_{g,n;\lambda}$ and
function $W_{g,n;\lambda}$ is deformed [214], while their relation to each
other $W_{g,n;\lambda}(z_{1},\cdots,z_{n})\equiv(-2)^{n}z_{1}\cdots
z_{n}R_{g,n;\lambda}(-z_{1}^{2},\cdots,-z_{n}^{2})$ is intact. So in the
deformed theory, $V_{g,n}$ is no longer the Laplace transform of
$W_{g,n;\lambda}$. The topological recursion formula in terms of
$W_{g,n;\lambda}$ [256] is covariant under the deformation, and retains the
same form. is that the flow equation (13) should be satisfied on each
asymptotic boundary component with proper legnth $\beta_{i}$. But by
definition, the flow equation only contains derivatives with respect to
$\lambda$ and $\beta$, not $b_{i}$, the length of geodesic boundary component
to be glued together. This fact was also adopted in [215, 214].

Then for generic $Z_{g,n;\lambda}(\beta)$, we write the deformed partition
functions as

$Z_{g,n;\lambda}(\beta_{1},\cdots,\beta_{n})=\int_{0}^{\infty}dE_{1}\cdots
dE_{n}\,\rho_{g,n}(E_{1},\cdots,E_{n})e^{-\beta_{1}f_{\lambda}^{-}(E_{1})}\cdots
e^{-\beta_{n}f_{\lambda}^{-}(E_{n})}\,,$ (21)

where

$\rho_{g,n}(E_{1},\cdots,E_{n})=\int_{0}^{\infty}\bigg{(}\prod_{j=1}^{n}db_{j}\,b_{j}\,\rho_{\text{T}}(b_{j},E_{j})\bigg{)}V_{g,n}(b_{1},\cdots,b_{n})\,,$
(22)

and

$\rho_{\text{T}}(b,E)=\frac{\cos(b\sqrt{E})}{2\pi\sqrt{E}}\quad\text{such
that}\quad\int_{0}^{\infty}dE\,\rho_{\text{T}}(b,E)e^{-\beta
E}=Z_{0}(b,\beta)_{\text{T}}\,.$ (23)

To conclude this section, we recall the differential operator presentation of
the $T\overline{T}$ deformation introduced in (194), which is sufficient for
computing perturbative expansions in $\lambda$ and will be used in section1,
by rewriting the exponential

$\displaystyle e^{-\beta f^{-}_{\lambda}(E)}$
$\displaystyle=e^{-\beta\sum^{\infty}_{m=1}c_{m}\lambda^{m}E^{m+1}}e^{-\beta
E}$ (24) $\displaystyle=\mathscr{D}_{y;\lambda}|_{y=\beta}\,e^{-yE}\,.$

It is then straightforward to compute partition functions of generic
topologies using this differential operator. However, for multiple boundary
components, one starts from the undeformed partition function with different
inverse temperatures $\beta_{i}$ on each boundary component and then applies
the differential operator $\mathscr{D}_{y_{i};\lambda}|_{y_{i}=\beta_{i}}$ to
each boundary component separately

$\langle Z(\beta_{1})\cdots
Z(\beta_{n})\rangle_{\lambda}=\left(\prod_{i=1}^{n}\mathscr{D}_{y_{i};\lambda}|_{y_{i}=\beta_{i}}\right)\langle
Z(y_{1})\cdots Z(y_{n})\rangle_{0}\,,$ (25)

which, in particular, yields

$\langle
Z(\beta)^{n}\rangle_{\lambda}=\left(\prod_{i=1}^{n}\mathscr{D}_{y_{i};\lambda}|_{y_{i}=\beta}\right)\langle
Z(y_{1})\cdots Z(y_{n})\rangle_{0}\,.$ (26)

We will use this differential operator presentation to perform perturbation
calculation in the leading order of $\lambda$ in section 1.

####  2 Quenched free energy

We briefly review a method of performing the replica trick

$\langle\ln Z\rangle=\lim_{n\rightarrow 0}\frac{\left\langle
Z^{n}\right\rangle-1}{n}\,,$ (27)

following [249]. The replica trick (27) has been shown in [249] to be written
as a rather convenient integral representation

$\langle\ln Z\rangle=\ln\langle
Z\rangle-\int_{0}^{\infty}\frac{dx}{x}\left[\left\langle
e^{-Zx}\right\rangle-e^{-\langle Z\rangle x}\right]$ (28)

such that the analytical continuation from $\langle Z^{n}\rangle$ to $\langle
Z^{n=0}\rangle$ remains unambiguous. From (28), the first term is the annealed
free energy while the second term encodes the contribution from Euclidean
replica wormholes with the interpretation that the operator $e^{-Zx}$ creates
spacetime boundary components, so-called “spacetime D-brane” or “SD-brane”
introduced in the context of baby universes [257]. Additionally, the term
$\langle e^{-Zx}\rangle$ can be rewritten as $e^{-\mathcal{Z}(x)}$ in terms of
the following generating function of connected correlators

$\mathcal{Z}(x)=\sum^{\infty}_{n=1}\frac{(-x)^{n+1}}{n!}\left\langle
Z^{n}\right\rangle_{c}\,.$ (29)

This will turn out to be an important quantity when we compute the
$T\overline{T}$-deformed annealed and quenched free energies of the Airy model
in the upcoming sections.

A motivation to study a simple observable, such as free energy, is to see how
Euclidean replica wormholes contribute to the Euclidean gravitational path
integral

$\langle Z(B)\rangle=\int_{B}~{}[dg]~{}e^{-I[g]}$ (30)

with spacetime boundary $B$, metric measure $[dg]$ and Euclidean JT gravity
action $I[g]$. An obvious way to determine the presence of Euclidean wormholes
is to see whether correlation functions of the partition function cease to
factorize

$\langle Z(B)^{n}\rangle\overset{?}{=}\langle Z(B)\rangle^{n}$ (31)

among $n$ boundary components.

It turns out not to be the case due to the factorization failure as shown in
(28), and this fact can be confirmed by directly computing the annealed and
quenched free energies555In [258], the quenched and annealed free energies in
JT gravity with conical deficit angles were computed following the formalism
developed by [259, 176, 260, 261] and observed the same pathology of
monotonicity failing at low temperatures as [248] finds. as done in [248]:

$\displaystyle F_{a}(\beta)=-\beta^{-1}\ln\langle Z\rangle\,,\quad
F_{q}(\beta)=-\beta^{-1}\langle\ln Z\rangle\,,$ (32)

at inverse temperature $\beta$. They are shown not to be the same indeed,
clearly indicating the factorization failure already hinted at by (28).

Unfortunately, the authors of [248] computed $F_{q}(\beta)$ with direct usage
of the replica trick (27), and their analysis at low temperature found that
$F_{q}(\beta)$ is _not_ monotonically decreasing as a function of temperature.
This is fundamentally due to the non-uniqueness of analytically continuing
$\langle Z^{n}\rangle$ to $\langle Z^{n=0}\rangle$. Given this conundrum at
low temperature, the correct analytical continuation was performed by Okuyama
[249] without directly using the replica trick (27), and there $F_{q}(\beta)$
was shown to be a monotonically decreasing function of temperature.

We now review how this is done following the proof in [249]. The correlator
$\langle Z^{n}\rangle$ can be expanded in terms of connected correlators
$\langle Z^{k}\rangle_{c}$ as [262]:

$\langle Z^{n}\rangle=\langle
Z\rangle^{n}\left[1+\frac{1}{2}n(n-1)\frac{\langle Z^{2}\rangle_{c}}{\langle
Z\rangle^{2}}+\cdots\right],$ (33)

so now the analytic continuation of $\langle Z^{n}\rangle$ is unambiguous due
to being rewritten as a polynomial in $n$ up to an overall $\langle
Z\rangle^{n}$. To further find the integral representation (28), we generalize
the above expansion to

$\langle Z^{n}\rangle=\langle Z\rangle^{n}\sum_{j_{i}\geq
0}\frac{n!}{\left(n-\sum_{\ell\geq 2}\ell j_{\ell}\right)!}\prod_{k\geq
2}\frac{1}{j_{k}!}\left(\frac{1}{k!}\frac{\langle Z^{k}\rangle_{c}}{\langle
Z\rangle^{k}}\right)^{j_{k}}\,,$ (34)

where $i\geq 2$ and the $j_{k}$’s constitute an integer partition

$\sum_{k=1}^{n}kj_{k}=n\,.$ (35)

Now, using the standard prescription for analytical continuation

$\lim_{n\rightarrow 0}\frac{1}{n}\frac{n!}{(n-m)!}=(-1)^{m-1}(m-1)!$ (36)

and then the identity

$\int^{\infty}_{0}dy~{}y^{k-1}e^{-y}=(k-1)!,$ (37)

the quenched free energy $F_{q}(\beta)$ can now be written as the integral
representation (28), whose integration range inherits that of (37).

####  3 Comments on non-perturbative contributions

We clarify more of the _non-perturbative_ features arising from the
$T\overline{T}$ deformation appearing in this chapter.666Another non-
perturbative effect we will encounter is in the Airy model, whose density of
states $\rho_{\text{Airy}}(E<0)\neq 0$. This is already present in the
undeformed theory and is unrelated to [214]. Yet another non-perturbative
effect takes place upon summing over all genus. We believe the non-
perturbative effects in summing over genus results in non-perturbative
instability. We will often refer to this effect as the non-perturbative
instability to distinguish the two. But based on context, the readers should
have no trouble in distinguishing the two. The perturbative branch is denoted
by the negative branch in the energy spectrum (3) since
$\lim_{\lambda\rightarrow 0}f^{-}_{\lambda}(E)=E$. In contrast, the
$\lambda\rightarrow 0$ limit of the positive branch for (3) diverges as
expected. Most papers omit this branch in their perturbative analysis and only
consider the negative branch. Unfortunately, when $\lambda>0$, the spectrum
along the flow becomes complex-valued for large enough energies. To resolve
this issue, as explicitly shown in [117], one is forced to include the non-
perturbative contribution such that the partition function

$Z_{\lambda>0}(\beta)=\int^{\infty}_{-\infty}dE\,\rho_{+}(E)e^{-\beta
f^{+}_{\lambda}(E)}+\int^{\infty}_{0}dE\,\rho_{-}(E)e^{-\beta
f^{-}_{\lambda}(E)}$ (38)

is real with appropriate constraints on the density of states $\rho_{\pm}(E)$
for JT gravity and is a solution to the flow equation (13). The appropriate
constraints on $\rho_{\pm}(E)$ are explained more in-depth by [117]. An
alternative approach to naturally incorporate non-perturbative effects is
through a resurgent analysis. In [214], the disk and trumpet deformed JT
partition functions written as power series in $\lambda$ are Borel resummed to
obtain non-perturbative results, which are used to study how the partition
functions summed over topologies (i.e. topological recursion) and spectral
form factor are modified under the $T\overline{T}$ deformation.

###  3 Quenched deformed free energy for Airy model

In this section, we compute the $T\overline{T}$-deformed annealed and quenched
free energies in the Airy model. It is important to point out that our
deformation of the double-scaled matrix model is different from the ones
considered in [215], whose $T\overline{T}$ deformation of the double-scaled
matrix model dual to JT gravity does not completely match the correlators with
the $T\overline{T}$-deformed JT gravity.

Eventually, our ultimate goal is to understand the quenched free energy in
$T\overline{T}$-deformed JT gravity. However, this is a rather difficult
problem since even for the one-point function $\langle
Z(\beta)\rangle_{\operatorname{JT}}$ of the undeformed theory, there is no
analytical expression that includes all-genus contributions, let alone non-
perturbative effects. In contrast, there has been much progress with numerical
calculations [247, 263, 264, 265, 266]. Therefore, we simplify and study the
Airy model instead, which is known to be the low-energy approximation of JT
gravity. Since it is a double-scaled matrix model, one could apply the
$T\overline{T}$ deformation defined in [215]. We already know that the
$T\overline{T}$ deformation defined and studied there does not provide an
exact match between JT gravity and its matrix model dual in terms of general
correlators. This is not a good choice if the goal is to understand the
quenched free energy in $T\overline{T}$-deformed JT gravity.

To investigate the Airy model with the hope of retaining some essential
features of the $T\overline{T}$-deformed JT gravity, we will have to work with
a different deformation from the one considered in [215]. Since one can
construct $T\overline{T}$-deformed JT correlators with any number of boundary
components and genera from basic observables like the deformed disk and
trumpet partition functions by gluing together pants decomposition [117], we
can recast the $T\overline{T}$ deformation of the correlators of JT gravity in
various ways (e.g. using the differential operator
$\mathscr{D}_{y;\lambda}|_{y=\beta}$) as reviewed in section 1. To match the
$T\overline{T}$-deformed matrix model dual to JT gravity, correlators on both
sides must deform the same way. We can then apply the same recipe of deforming
the matrix model correlators to the Airy model instead of using the one
defined in [215]. This is the deformation we will adopt and study in this
chapter.

To be more specific, we will take the deformed $n$-point functions in the Airy
model as

$\langle Z(\beta_{1})\cdots Z(\beta_{n})\rangle_{\text{Airy},\lambda}=\int
dE_{1}\cdots
dE_{n}\,\rho_{\text{Airy},n}(E_{1},\cdots,E_{n})e^{-\beta_{1}f_{\lambda}^{-}(E_{1})}\cdots
e^{-\beta_{n}f_{\lambda}^{-}(E_{n})}$ (39)

where $\rho_{\text{Airy},n}(E_{1},\cdots,E_{n})$ is such that

$\langle Z(\beta_{1})\cdots Z(\beta_{n})\rangle_{\text{Airy},0}=\int
dE_{1}\cdots
dE_{n}\,\rho_{\text{Airy},n}(E_{1},\cdots,E_{n})e^{-\beta_{1}E_{1}}\cdots
e^{-\beta_{n}E_{n}}\,.$ (40)

It is important to notice that if we are interested in computing the deformed
correlators at any given genus, there will not be non-perturbative effects
(from the Airy model itself before $T\overline{T}$ deforming) so the
integration range of $E_{i}$ is $[0,\infty)$, and we can safely apply the
integral transformation (14) on (40) to obtain (39) for convenience. However,
if we are interested in the exact result, for instance, $\langle
Z(\beta)\rangle_{\text{Airy},\lambda}$, then we cannot use the integral
transformation, no matter if $\lambda$ is positive or negative, because even
for $\lambda<0$, $\rho_{\text{Airy}}$ has support over the entire real axis of
$E$.

There are a few other important caveats to mention.

Firstly, carrying over the $T\overline{T}$ deformation of JT gravity
correlators to matrix model correlators determines the perturbative branch of
the $T\overline{T}$ deformation. There will be non-perturbative contributions
of the $T\overline{T}$ deformation if we are interested in results with
_finite_ $\lambda$. We will analyze those non-perturbative effects case-by-
case as we encounter them by matching the deformed Airy correlators with
either the genus-zero result or the flow equation. A possible systematical
treatment would be adapting the resurgent analysis performed in [214].
However, in our study, eventually, we will have to sum over not only all
orders in $\lambda$ but all genus $g$ as well. As hinted in this chapter’s
introduction, it is well-known that there is a non-perturbative instability in
the exact spectral density in the Airy model that is related to the genus
expansion.

Secondly, related to the non-perturbative instability of the Airy model, the
same non-perturbative instability appears in the naïve matrix model dual of
the JT gravity as well [79, 247]. There have been works to improve the non-
perturbative feature of the JT gravity and remove this undesired feature [79,
247, 267]. There are possibilities that the non-perturbative instability can
qualitatively affect the behavior of the quenched free energy in the deformed
theory and is important to understand if this is the case. A possible proposal
is to extend our study to the JT gravity at general temperatures with the non-
perturbative instability taken care of.

Thirdly, it should be emphasized that we do not provide a complete description
of our version of the $T\overline{T}$ deformation for the matrix model. We
require that in the deformed theory, every correlator has to transform to
match the gravity side. One can take this as the working definition of our
version of the $T\overline{T}$ deformation for matrix models. To further
illustrate the difference between our version and the one in [215], let us
begin with the standard Hermitian matrix integral

$Z=\int[dM]e^{-\text{Tr}V(M)}=\int d^{N}x\prod_{1\leq i<j\leq
N}(x_{i}-x_{j})^{2}e^{-\sum^{N}_{i=1}V(x_{i})}\,,$ (41)

where $x_{i}$ are the eigenvalues of the $N\times N$ Hermitian matrix $M$ and
the Vandermonde determinant $\prod_{1\leq i<j\leq N}(x_{i}-x_{j})^{2}$ appears
from diagonalizing $M$.

The undeformed correlation functions are determined by integrating operators
against (41):

$\langle O_{1}(M)\cdots
O_{n}(M)\rangle_{0}=\int[dM]e^{-\text{Tr}V(M)}O_{1}(M)\cdots O_{n}(M)\,.$ (42)

In [215], it is assumed that the deformed matrix model still takes the form of
(41), and the only physical quantity that changes is the matrix model
potential which shifts from $V(M)$ to
$V_{\lambda}(M)=c_{\lambda}V\left(M-2\lambda M^{2}\right)$. As a somewhat
unfortunate consequence of this assumption, the $n$-point functions in the
deformed matrix model still do not match the $n$-point functions in the
deformed JT gravity. In our case, we start by requiring that the $n$-point
functions in the deformed matrix model match the results in deformed JT
gravity leading to a different deformation on the matrix model side. In
particular, naïvely the integration measure $[dM]$ of matrices will receive
corrections from the $T\overline{T}$ deformation as well if we want to keep
the potential $\operatorname{Tr}V(M)$ as a single trace operator as implicitly
assumed in [215]. To see this, consider the correlators

$\displaystyle\left\langle\text{Tr}\,e^{-\beta_{1}M}\cdots\text{Tr}\,e^{-\beta_{n}M}\right\rangle_{0}$
(43) $\displaystyle\quad\quad\quad\quad\quad\quad=\int d^{N}x\prod_{1\leq
i\leq j\leq
N}(x_{i}-x_{j})^{2}e^{-\sum^{N}_{i=1}V(x_{i})}\bigg{(}\sum^{N}_{i_{1}=1}e^{-\beta_{1}x_{i_{1}}}\bigg{)}\cdots\bigg{(}\sum^{N}_{i_{n}=1}e^{-\beta_{n}x_{i_{n}}}\bigg{)}$

in the undeformed matrix model.

To match the gravitational result, its deformation should take the following
form:

$\displaystyle\left\langle\text{Tr}\,e^{-\beta_{1}f_{\lambda}^{-}(M)}\cdots\text{Tr}\,e^{-\beta_{n}f_{\lambda}^{-}(M)}\right\rangle_{\lambda}$
(44) $\displaystyle\quad\quad\,=\int d^{N}x\prod_{1\leq i<j\leq
N}(x_{i}-x_{j})^{2}e^{-\sum^{N}_{i=1}V(x_{i})}\bigg{(}\sum^{N}_{i_{1}=1}e^{-\beta_{1}f^{-}_{\lambda}(x_{i_{1}})}\bigg{)}\cdots\bigg{(}\sum^{N}_{i_{n}=1}e^{-\beta_{n}f^{-}_{\lambda}(x_{i_{n}})}\bigg{)}\,.$

To determine the deformed matrix integral, we consider a change of variables
for $x_{i}$, such that $f^{-}_{\lambda}(x_{i})=x_{i}^{\prime}$, so that we are
computing the same correlators

$\left\langle\text{Tr}\,e^{-\beta_{1}M^{\prime}}\cdots\text{Tr}\,e^{-\beta_{n}M^{\prime}}\right\rangle_{\lambda}$
(45)

in terms of the new variable $M^{\prime}$. This will not only change the
potential $V(M)$, but the integration measures $[dM]$ as well if we want
$e^{-\text{Tr}V(M)}$ to contain only single trace operators.

To further illustrate this fact, we consider the change of variables
$y_{i}=f_{\lambda}^{-}(x_{i})$ as well as neglect for the moment issues with
the branch cut from the square root in $f^{-}_{\lambda}(x_{i})$ and potential
non-perturbative subtleties related to the change of variable for $M$.777This
resembles the “half-diffeomorphism,” which appears in the formulation of
$T\overline{T}$ deformation by coupling the 2D field theory to topological
gravity in [101] where either the metric or coordinates change under the
deformation, but not both. Then $x_{i}=y_{i}(1-2\lambda y_{i})$ and this
implies

$\displaystyle\left[dM\right]e^{-\sum_{i=1}^{N}V(x_{i})}=(d^{N}y)\prod_{1\leq
i<j\leq N}(y_{i}-y_{j})^{2}e^{-\operatorname{Tr}V(y_{i}(1-2\lambda
y_{i}))}\prod_{i,j=1,\cdots,N}(1-2\lambda(y_{i}+y_{j}))$ (46)

Since only the product of the matrix measure $[dM]$ and exponential
$e^{-\operatorname{Tr}V(M)}$ are unambiguously defined, one could consider
absorbing the extra piece

$\prod_{i,j=1,\cdots,N}(1-2\lambda(y_{i}+y_{j}))$ (47)

into the definition of $e^{-\operatorname{Tr}V(M)}$ to retain the form of the
measure $[dM]$. However, this will lead to an infinite sum over double-trace
operators as follows:

$\displaystyle\prod_{i,j=1,\cdots,N}(1-2\lambda(y_{i}+y_{j}))$
$\displaystyle=\exp\Bigg{(}\sum_{i,j=1,\cdots,N}\log(1-2\lambda(y_{i}+y_{j}))\Bigg{)}$
$\displaystyle=\exp\Bigg{(}\sum_{i,j=1,\cdots,N}\sum_{m=1}^{\infty}\frac{(2\lambda)^{m}}{m}\sum_{p=0}^{m}C_{m}^{p}y_{i}^{p}y_{j}^{m-p}\Bigg{)}$
$\displaystyle=\exp\Bigg{(}\sum_{m=1}^{\infty}\frac{(2\lambda)^{m}}{m}\sum_{p=0}^{m}\operatorname{Tr}(M^{p})\operatorname{Tr}(M^{m-p})\Bigg{)}\,.$

Either way, the deformation presented here violates the implicit assumptions
in [215] and would be the starting point on the matrix model side. However, it
is unclear whether one should first $T\overline{T}$ deform and take the
double-scaling limit or vice versa. Further analysis is beyond the scope of
this thesis.

####  1 Genus-zero quenched free energy at leading order of $\lambda$

We begin our analysis on the deformed quenched free energy $F_{q,\lambda}(T)$
at genus-zero perturbatively. To determine $F_{q,\lambda}(T)$, we first
compute the deformed genus-zero multi-boundary connected correlators $\langle
Z(\beta_{1})\cdots Z(\beta_{n})\rangle_{c,\lambda}^{g=0}$. The undeformed
connected correlators at genus-zero have been computed in [246] using the
genus-zero Korteweg–De Vries (KdV) flow:

$\langle Z(\beta_{1})\cdots
Z(\beta_{n})\rangle_{c,0}^{g=0}=g_{s}^{n-2}\bigg{(}\sum^{n}_{i=1}\beta_{i}\bigg{)}^{n-3}\prod^{n}_{i=1}\left(\frac{\beta_{i}}{2\pi}\right)^{\frac{1}{2}}\,,$
(48)

where $g_{s}\equiv\sqrt{2}$ is the genus-counting parameter.888Here in fact
$g_{s}=\sqrt{2}\hbar$ and we will set $\hbar=1$. Adopting the conventions by
[249], $\hbar$ is the genus-counting parameter in the Airy limit of matrix
models while $g_{s}$ is the natural genus-counting parameter in 2D topological
gravity. Also refer to earlier discussions in [268, 246].

The deformed correlators can be computed directly using the integral
transformation (14) and, by construction, solve the flow equation (13). For
instance, for $\lambda<0$, the deformed one-point, two-point, and $n$-point
correlators are given by

$\displaystyle\langle Z(\beta)\rangle^{g=0}_{\lambda}$
$\displaystyle=g_{s}^{-1}\frac{e^{-\frac{\beta}{4\lambda}}}{2\pi\beta\sqrt{-\lambda}}K_{2}\bigg{(}-\frac{\beta}{4\lambda}\bigg{)}\,,$
(49) $\displaystyle\langle Z(\beta)^{2}\rangle^{g=0}_{c,\lambda}$
$\displaystyle=-\frac{\beta_{1}\beta_{2}}{16\pi^{2}\lambda}\int^{\infty}_{0}d\beta_{1}^{\prime}\int^{\infty}_{0}d\beta_{2}^{\prime}\,\frac{\sqrt{\beta_{1}^{\prime}\beta_{2}^{\prime}}}{\beta_{1}^{\prime}+\beta_{2}^{\prime}}\,\frac{1}{(\beta_{1}^{\prime}\beta_{2}^{\prime})^{\frac{3}{2}}}e^{\frac{(\beta_{1}-\beta_{1}^{\prime})^{2}}{8\lambda\beta_{1}^{\prime}}+\frac{(\beta_{2}-\beta_{2}^{\prime})^{2}}{8\lambda\beta_{2}^{\prime}}}\,,$
$\displaystyle\langle Z(\beta)^{n}\rangle^{g=0}_{c,\lambda}$
$\displaystyle=g_{s}^{n-2}\frac{e^{-\frac{n\beta}{4\lambda}}}{\beta^{3}}\bigg{(}\frac{\beta^{2}}{2\pi\sqrt{-\lambda}}\bigg{)}^{n}\,\sum_{i_{j}\geq
0}^{\sum_{j}i_{j}=n-3}\frac{(n-3)!}{i_{1}!\cdots
i_{n}!}\prod_{j=1}^{n}K_{i_{j}}\left(-\frac{\beta}{4\lambda}\right)\,,\quad
n\geq 3\,.$

Figure 2: As an example, we have drawn a genus-three Riemann surface with
three boundary components, each of which is a circle of circumference
$\beta_{i}$.

Generally, how one evaluates the above sums of products of modified Bessel
functions remains unclear, let alone compute

$\mathcal{Z}^{g=0}_{\lambda}(x)=\sum_{n=1}^{\infty}\frac{(-x)^{n}}{n!}\langle
Z(\beta)^{n}\rangle_{c,\lambda}^{g=0}$ (50)

in (29) as in [249].

Therefore, instead, we consider working with perturbation theory and keep the
leading order in $\lambda$. For this purpose, it is convenient (in comparison
with our “working definition” introduced at the beginning of section 3) to
express the $T\overline{T}$ deformation in terms of differential operators and
only keep the leading order term in $\lambda$ so that the deformation acts as

$\langle Z(\beta_{1})\cdots
Z(\beta_{n})\rangle_{\lambda}\rightarrow\bigg{(}1-2\lambda\sum^{n}_{i=1}\beta_{i}\partial_{\beta_{i}}^{2}\bigg{)}\langle
Z(\beta_{1})\cdots Z(\beta_{n})\rangle_{0}+O(\lambda^{2})\,.$ (51)

Then, we find

$\langle Z(\beta)^{n}\rangle_{c,\lambda}^{g=0}=\langle
Z(\beta)^{n}\rangle_{c}^{g=0}-\frac{\lambda}{\beta^{4}}n^{n-4}\left(\frac{\beta^{3}}{\pi}\right)^{\frac{n}{2}}\left(\frac{7}{4}n^{2}-10n+12\right)+O(\lambda^{2})\,.$
(52)

On the other hand,

$\mathcal{Z}_{\lambda}^{g=0}(x)=-\sum_{n=1}^{\infty}\frac{(-x)^{n}}{n!}\langle
Z(\beta)^{n}\rangle_{c,\lambda}^{g=0}$ (53)

can be evaluated using a similar trick as in [249]. Specifically, focusing on
the leading order piece in $\lambda$, we have the following sum:

$\mathcal{Z}^{g=0}_{\lambda}(x)-\mathcal{Z}^{g=0}_{0}(x)=-\frac{\lambda}{\beta^{4}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}n^{n-4}}{n!}\left(x\sqrt{\frac{\beta^{3}}{\pi}}\right)^{n}\left(\frac{7}{4}n^{2}-10n+12\right)\,.$
(54)

Let $z=x\sqrt{\frac{\beta^{3}}{\pi}}$, and the sum can be decomposed into
three separate pieces:

$\displaystyle
A(W(z))=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}n^{n-2}}{n!}z^{n}\,,\quad
B(W(z))=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}n^{n-3}}{n!}z^{n}\,,$ (55)
$\displaystyle
C(W(z))=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}n^{n-4}}{n!}z^{n}\,,$

satisfying

$(z\partial_{z})A(W(z))=W(z),\quad(z\partial_{z})^{2}B(W(z))=W(z),\quad(z\partial_{z})^{3}C(W(z))=W(z)\,,$
(56)

where $W(z)$ is the Lambert function defined by the following Taylor series
expansion:

$W(z)\equiv\sum_{n=1}^{\infty}\frac{(-1)^{n-1}n^{n-1}}{n!}z^{n}\,.$ (57)

The above differential equations (55) can be solved by making the ansatz

$A=A_{2}W^{2}+A_{1}W\,,\quad B=B_{3}W^{3}+B_{2}W^{2}+B_{1}W\,,\quad
C=C_{4}W^{4}+C_{3}W^{3}+C_{2}W^{2}+C_{1}W\,,$ (58)

and using the property of the Lambert function

$z\partial_{z}W(z)=\frac{W(z)}{1+W(z)}\,.$ (59)

We then find

$A=\frac{1}{2}W^{2}+W,\quad B=\frac{1}{6}W^{3}+\frac{3}{4}W^{2}+W,\quad
C=\frac{1}{24}W^{4}+\frac{11}{36}W^{3}+\frac{7}{8}W^{2}+W$ (60)

and

$\mathcal{Z}_{\lambda}^{g=0}(z)=\frac{B(W(z))}{2}+\frac{c_{1}\lambda}{2\beta}\left(\frac{7}{4}A(W(z))-10B(W(z))+12C(W(z))\right)\,.$
(61)

Then using (28), we compute the $\mathcal{O}(\lambda)$ corrections

$\displaystyle\langle\log Z\rangle^{g=0}_{\lambda}$ (62) $\displaystyle=$
$\displaystyle\log\langle
Z\rangle^{g=0}_{\lambda}-\int_{0}^{\infty}dx\frac{e^{-\beta^{-3}\mathcal{Z}^{g=0}_{\lambda}(x)}-e^{-x\langle
Z\rangle^{g=0}_{\lambda}}}{x}$ $\displaystyle=$
$\displaystyle\log\left[\frac{1}{\sqrt{4\pi\beta^{3}}}\bigg{(}1-\frac{15\lambda}{2\beta}\bigg{)}\right]$
$\displaystyle-\int_{0}^{\infty}dW\,\frac{1+W}{W}\left[e^{-\frac{1}{\beta^{3}}\left(\frac{B(W)}{2}-\frac{\lambda}{\beta}\left(\frac{7}{4}A(W)-10B(W)+12C(W)\right)\right)}-e^{-\frac{1}{2\beta^{3}}We^{W}\big{(}1-\frac{15\lambda}{2\beta}\big{)}}\right]$
$\displaystyle+\mathcal{O}(\lambda^{2})$ $\displaystyle=$
$\displaystyle\log\bigg{(}\frac{1}{\sqrt{4\pi\beta^{3}}}\bigg{)}-\frac{15\lambda}{2\beta}-\int_{0}^{\infty}\frac{dW}{W}(1+W)\bigg{(}e^{-\frac{B(W)}{2\beta^{3}}}-e^{-\frac{1}{2\beta^{3}}We^{W}}\bigg{)}$
$\displaystyle-\frac{\lambda}{8\beta^{4}}\int_{0}^{\infty}dW(1+W)\bigg{(}e^{-\frac{1}{24\beta^{3}}W(12+9W+2W^{2})}(30+31W+16W^{2}+4W^{3})$
$\displaystyle-30e^{W\left(1-\frac{1}{2\beta^{3}}e^{W}\right)}\bigg{)}+\mathcal{O}(\lambda^{2})\,,$

where we have changed the integration variable from $z$ to $W(z)$ and used

$\langle
Z(\beta)\rangle_{\lambda}^{g=0}=\frac{1}{\sqrt{4\pi\beta^{3}}}\bigg{(}1-\frac{15\lambda}{2\beta}+\mathcal{O}(\lambda^{2})\bigg{)}\,.$
(63)

Notice that in the last line of (LABEL:intg0), we expanded in $\lambda$ for
the logarithm and exponential since our result is only valid in the leading
order of $\lambda$.

This integral (LABEL:intg0) can be evaluated numerically, and, below we plot
the quenched free energy $F_{q}(\beta)$ at genus-zero with $T\overline{T}$
deformation coupling $\lambda=-\frac{1}{20}$ against temperature $T$.

Figure 3: The annealed and quenched free energies for the deformed Airy model
as a function of temperature $T$ are plotted. The blue and orange curves are
quenched and annealed free energies, respectively, for the deformed theory at
$\lambda=-\frac{1}{20}$ and the green curve is the quenched free energy for
the undeformed Airy model. Here $\lambda<0$ lowers the quenched free energy.

We also plot the genus-zero quenched free energy for various signs of
$\lambda$, again using differential operators as in (51), see figure 4. Notice
in the perturbative expansion of $\lambda$, that $\lambda$ also always appears
as $\lambda T$. Hence, for the leading order approximation to hold, $\lambda
T$ must be small. In the numerical calculation, we find $F_{q,\lambda}(T)$
monotonically decreases as a function of $T$ for $\lambda>0$. However,
monotonicity can break down when $\lambda T$ is too large for $\lambda>0$ and
this is likely due to $\lambda T$ exceeding the range of validity of leading
order approximation in $\lambda$.

Figure 4: The quenched free energy $F_{q,\lambda}(T)$ for the deformed Airy
model as a function of temperature $T$ is plotted. We notice that a
deformation with $\lambda<0$ lowers the quenched free energy while the one
with $\lambda>0$ increases $F_{q}(T)$. For both signs of $\lambda$,
$F_{q,\lambda}(T)$ monotonically decreases as $T$ increases for $T<1$;
however, for $\lambda<0$, this breaks down for $\lambda=\frac{1}{20}$ and at
around $T=1$. Notice that the perturbative expansion of $\lambda$ always
appears as $\lambda T$. For the first-order approximation to hold, $\lambda T$
should be small. Therefore, this breakdown of monotonicity when $T$ becomes
large is likely because we go beyond the validity of perturbation theory.

####  2 All genus quenched free energy in low temperature limit

Now, using the low-temperature approximation as in [249], we compute the
quenched free energy starting from the relation in the undeformed theory [262,
249]:

$\langle Z(\beta)^{n}\rangle_{c}\simeq\langle Z(n\beta)\rangle,\quad T\lesssim
1\,.$ (64)

Under the low-temperature approximation, the deformation of $\langle
Z(\beta)^{n}\rangle_{c}\simeq\langle Z(n\beta)\rangle$ is easily computable.
This is seen from expressing the undeformed correlator for $n$ boundary
components $\left\langle Z(\sum_{i=1}^{n}\beta_{i})\right\rangle_{0}$ as

$\left\langle Z\left(\sum_{i=1}^{n}\beta_{i}\right)\right\rangle_{0}=\int
dE\,\rho(E)e^{-\sum_{i=1}^{n}\beta_{i}E}\,,$ (65)

the term dependent on $\beta$ essentially factorizes. Thus, the deformed
$n$-point function in low temperatures is

$\left\langle
Z(n\beta)\right\rangle_{\lambda}=\int_{-\infty}^{\infty}dE\,\rho(E)e^{-n\beta
f^{-}_{\lambda}(E)}\,.$ (66)

However, we must show that the change from the $T\overline{T}$ deformation for
$\langle Z(n\beta)\rangle_{c,\lambda}-\langle Z(n\beta)\rangle_{c,0}$ is of
lower order compared to the correction $\langle
Z(\beta)^{n}\rangle_{c}-\langle Z(n\beta)\rangle$ with (64). This is the case
because the correction to the approximation $\langle Z(\beta)^{n}\rangle\simeq
Z(n\beta)\rangle$ is exponentially suppressed as $e^{-c_{0}\beta^{3}}$ in the
low temperature limit $\beta\rightarrow\infty$, where $c_{0}$ is some positive
constant. One can explicitly check this approximation for $n=2,3$, where the
exact expression of the partition functions can be conveniently found in
[245]. For instance, with $n=2$, we have

$\langle Z(\beta)^{2}\rangle_{c,0}=\langle
Z(2\beta)\rangle_{0}\,\text{erf}\left(\sqrt{\frac{\beta^{3}}{2}}\right)\,.$
(67)

Using the asymptotic expansion of the error function:

$\text{erf}(x)=1-\frac{e^{-x^{2}}}{x\sqrt{\pi}}\sum_{n=0}^{\infty}(-1)^{n}\frac{(2n-1)!!}{(2x^{2})^{n}},\quad\text{as\,\,}x\rightarrow\infty\,,$
(68)

it is clear that the correction is suppressed by $e^{-\frac{\beta^{3}}{2}}$ in
the low temperature limit for (67).

Similarly, for $n=3$:

$\langle Z(\beta)^{3}\rangle_{c,0}=\langle
Z(3\beta)\rangle_{0}\left(1-12T\left(\sqrt{3\beta^{3}},\frac{1}{\sqrt{3}}\right)\right)\,.$
(69)

From the definition of Owen’s $T$ function:

$T(h,a)\equiv\frac{1}{2\pi}\int_{0}^{a}\frac{e^{-\frac{1}{2}h^{2}(1+x^{2})}}{1+x^{2}}dx\,,$
(70)

one can see in the large $h$ limit (i.e. large $\beta$ limit), it is indeed
suppressed as $e^{-\frac{1}{2}h^{2}}=e^{-\frac{3}{2}\beta^{3}}$.

The connected correlator has a well-known integral representation given by
[245] and is known to be a closed form only for $n=1,2,3$. We can easily check
that $\langle Z(\beta)^{n}\rangle_{c}\simeq\langle Z(n\beta)\rangle$ when
$n=1,2,3$ for small temperatures, but proving this for $n>3$ is numerically
difficult. We will content ourselves and assume $\langle
Z(\beta)^{n}\rangle_{c}\simeq\langle Z(n\beta)\rangle$ is true for all $n$.

Meanwhile, even at the leading order of $\lambda$, the $T\overline{T}$
deformation will give polynomial corrections in $\beta$ for low temperatures.
Hence, the corrections in the approximation $\langle
Z(\beta)^{n}\rangle_{0}\simeq\langle Z(n\beta)\rangle_{0}$ can still be
neglected even when we consider the $T\overline{T}$ deformation.

There is a caveat: the all-genus density of states does not vanish for $E<0$.
Instead, it is exponentially suppressed when $E<0$:

$\rho_{\text{Airy}}(E)=\text{Ai}^{\prime}\left(-E\right)^{2}+E\,\text{Ai}\left(-E\right)^{2}$
(71)

and is plotted in figure 5.

Figure 5: We plot the Airy density of states (71). As one can see, the density
of states is supported along the entire real line and exponentially suppressed
when $E<0$.

This means for either sign of $\lambda$, there will be states with complex-
valued energy. We discuss the two cases separately, and in each case, there
will be a plausible non-perturbative contribution from the
$f^{+}_{\lambda}(E)$ branch defined in (2).

The deformed quenched free energy is defined and computed as

$F_{q,\lambda}(\beta)=-T\langle\log
Z\rangle_{\lambda}=T\int_{0}^{\infty}\frac{dx}{x}\left[e^{-\mathcal{Z}_{\lambda}(x)}-e^{-x\langle
Z\rangle_{\lambda}}\right]-T\log\langle Z\rangle_{\lambda}\,.$ (72)

####  3 Quenched free energy for $\lambda>0$

In this subsection, we study the quenched free energy of the deformed Airy
model with $\lambda>0$. In the deformed theory’s spectrum, as explained in
this chapter’s introduction, the deformed energy
$f^{-}_{\lambda}(E>\frac{1}{8\lambda})$ is complex-valued. Therefore, a cutoff
in $E$ at $\frac{1}{8\lambda}$ is needed to remove the complex-valued energy
states. Furthermore, there could be contributions arising from non-
perturbative states with energy given by the other branch
$f^{+}_{\lambda}(E)$. This kind of non-perturbative effects have been
rigorously studied in [214, 117]. Their analyses lead one to conjecture what
the non-perturbative contribution could be in the deformed Airy theory for
$\lambda>0$.

We study the quenched free energy with and without the non-perturbative
contributions. In both cases, we find the quenched free energy