# A Taylor Resolution Over Complete Intersections

Aleksandra Sobieska

###### Abstract.

The Taylor resolution is a fundamental object in the study of free resolutions
over the polynomial ring, due to its explicit formula, cellular/combinatorial
structure, and applicability to any and all monomial ideals. This paper
generalizes the Taylor resolution to complete intersection rings via the
Eisenbud–Shamash construction.

##  1\. Introduction

The Taylor resolution [Tay66] universally resolves any monomial ideal over the
polynomial ring; it generalizes the Koszul complex, has straightforward
cellular support, and is eminently computable. Indeed, it is often among the
first constructions one learns to build a free resolution. Its combinatorial
and explicit nature have influenced many advances in commutative algebra, some
of which can be found in [HH90, Lyu88, EK90, Eis95, BS98, BW02, MS05, Pee11]
and the references therein.

Resolutions over singular rings are typically infinite and more complicated.
Results for special types of rings – complete intersections [Gul74, Gul80,
Eis80, Avr89, AGP97, AB00, EP16], Golod rings [Gol62, HRW99, GPW00, BJ07],
Koszul algebras [Frö99, AE92, Roo95] – and/or special types of modules – the
ground field [Tat57, Pri70, ML63, Frö75, Ani82, RS98] – have been developed,
but still much remains to be understood. Some surveys on infinite free
resolutions can be found in [Avr98, MP15].

In this article, we generalize the Taylor resolution to complete intersection
rings. The fundamental idea behind this construction is to produce the
homotopies from the Eisenbud–Shamash construction entirely in terms of
combinatorial data (essentially ratios of lcms). In providing an explicit
formula for the system of homotopies, we show that no higher homotopies are
necessary to write the complete resolution. Because the entries are easily
expressed as ratios of $\operatorname{lcm}$s, this method generates a body of
examples of infinite free resolutions, and provides combinatorial recipes for
matrix factorizations. Finally, it is worth noting that, in addition to
generalizing the Taylor resolution of monomial ideals to complete
intersections, this construction recovers the well-known Tate resolution
[Tat57] as well as the resolution over monomial complete intersections in
[Iye97].

The main result can be found in Definition 3.1 and Theorem 3.2. We provide an
example of how this construction provides a direct way to resolve a monomial
ideal over a complete intersection.

######  Example 1.1.

Let $Q=\mathbbm{k}[x,y,z]$ and $I={\langle x^{2},y^{2},z^{2}\rangle}$, and
${\mathfrak{a}}={\langle x^{2}z+xy^{2}\rangle}$ and $R=Q/{\mathfrak{a}}$. The
Taylor resolution of $I$ over $R$ given by Theorem 3.2 is

${\mathbf{F}}:\cdots\xrightarrow{\varphi_{5}}\begin{array}[]{c}R(-6)\\\
\oplus\\\
R(-7)^{3}\end{array}\xrightarrow{\varphi_{4}}\begin{array}[]{c}R(-5)^{3}\\\
\oplus\\\ R(-6)\end{array}\xrightarrow{\varphi_{3}}\begin{array}[]{c}R(-3)\\\
\oplus\\\
R(-4)^{3}\end{array}\xrightarrow{\varphi_{2}}R(-2)^{3}\xrightarrow{\varphi_{1}}R\rightarrow
0$

where $F_{2i-1}=R(-3i+3)\oplus R(-3i+2)^{3}$ for $i\geq 2$ and
$F_{2i}=R(-3i+1)^{3}\oplus R(-3i)$, the initial maps are

$\varphi_{1}=\blockarray{*{4}{>{\scriptstyle}c}<{}}1&23\\\
\block{[*{3}r]>{\scriptstyle}c<{}}x^{2}y^{2}z^{2}\emptyset\\\ \text{ and
}\varphi_{2}=\blockarray{*{5}{>{\scriptstyle}c}<{}}\emptyset&121323\\\
\block{[r|*{3}r]>{\scriptstyle}c<{}}zy^{2}z^{2}01\\\ x-x^{2}0z^{2}2\\\
00-x^{2}-y^{2}3\\\ ,$

and the tail of the resolution is periodic with maps

$\varphi_{2i-1}=\blockarray{*{5}{>{\scriptstyle}c}<{}}1&23123\\\
\block{[*{3}r|r]>{\scriptstyle}c<{}}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{x^{2}}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{y^{2}}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{z^{2}}0\emptyset\\\
\BAhhline{----}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{x}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{-z}0\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{z^{2}}12\\\
00\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{-z}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{-y^{2}}13\\\
00\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{-x}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{x^{2}}23\\\
\text{ and
}\varphi_{2i}=\blockarray{*{5}{>{\scriptstyle}c}<{}}\emptyset&121323\\\
\block{[r|*{3}r]>{\scriptstyle}c<{}}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{z}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{y^{2}}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{z^{2}}01\\\
\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{x}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{-x^{2}}0\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{z^{2}}2\\\
00\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{-x^{2}}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{-y^{2}}3\\\
\BAhhline{----}00\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{-x}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{z}123\\\
$

for $i\geq 2$.

We can see all the maps from the classic Taylor resolution inside the
differentials of ${\mathbf{F}}$ (written in blue). As with the Taylor
resolution, rows and columns indexed with subsets of $\\{1,2,3\\}$
corresponding to subsets of the generators of $I$, and there is a non-zero
entry in the matrix factorization when the row label is a subset of the column
label with just one element missing.

The entries accounting for the singularity of $R$ are placed in prescribed
spots (written in red) via the expression $x^{2}z+y^{2}x=z\cdot x^{2}+x\cdot
y^{2}$, which gives the generator of ${\mathfrak{a}}$ as a combination of the
first and second generators of $I$. Therefore, there is a non-zero entry in
the matrix factorization when the row label is a union of the column label and
either $\\{1\\}$ or $\\{2\\}$. The formula for the entry is a ratio of lcm’s
of subsets of generators of $I$, similar to the Taylor resolution.

$123$$12$$13$$23$$1$$2$$3$$\emptyset$ (a) The Taylor resolution as a directed
graph

$123$$12$$13$$23$$1$$2$$3$$\emptyset$ (b) The matrix factorization as a
directed graph

Figure 1. A graphical description of the matrix factorization

Figure 1(a) shows the summands (represented by vertices) and non-zero
differential arrows (represented by blue arrows) from the Taylor resolution.
In Figure 1(b), the original directed graph is rearranged into a bipartite
graph with even-cardinality subsets on one side and odd-cardinality subsets on
the other. With the addition of the eight red arrows either from a vertex $S$
to $\\{1\\}\cup S$ or from $S$ to $\\{2\\}\cup S$, we can encapsulate the
induced matrix factorization with a edge-colored, directed bipartite graph,
where each edge represents a nonzero entry – the blue edges give a “Taylor”
entry and the red edges give a “lift” entry.

Acknowledgements: The author would like to thank Daniel Erman for many
illuminating conversations during all stages of this article’s development.
Most, if not all, computations were aided by Macaulay2 [GS].

##  2\. Background

######  Notation.

For a positive integer $r$, use $[r]$ to denote the set $\\{1,2,\ldots,r\\}$.
We will occasionally identify the singleton set $\\{s\\}$ and $s$ when
$s\in[r]$, especially when appearing in long and teeny subscripts.

###  2.1. The Taylor Resolution

Let $Q=\mathbbm{k}[x_{1},\ldots,x_{n}]$ be a polynomial ring over a field and
$I\subseteq Q$ be a monomial ideal $I=\langle m_{1},\ldots,m_{r}\rangle$. For
a set $S=\\{j_{1},\ldots,j_{k}\\}\subseteq[r]$ with $j_{1}<\ldots<j_{k}$, let
$m_{S}=\operatorname{lcm}(m_{j_{1}},\ldots,m_{j_{k}})$, and use $v_{S}$ to
denote $\operatorname{degree}m_{S}$.

######  Definition 2.1 ([Tay66]).

The Taylor resolution $Q/I$ for a monomial ideal $I=\ \langle
m_{1},\ldots,m_{r}\rangle$ is given by $(T_{k},\tau_{k})$, where

$T_{k}=\bigoplus\limits_{S\subseteq[r],\ |S|=k}Q(-v_{S})$

and

$\tau_{k}(\varepsilon_{S})=\sum\limits_{s_{i}\in
S}(-1)^{k-i}\frac{m_{S}}{m_{S-s_{i}}}\varepsilon_{S-s_{i}}\text{ for
}S=\\{s_{1},\ldots,s_{k}\\}.$

The resolution ${\mathbf{T}}$ is highly non-minimal, but is always a
resolution for any monomial ideal $I$.

######  Example 2.2.

Let $Q=\mathbbm{k}[x,y,z]$ and $I=\langle xy,xz,yz\rangle$. The Taylor
resolution of $Q/I$ over $Q$ is given by

$0\rightarrow
Q(-3)\xrightarrow{\tau_{3}}Q(-3)^{3}\xrightarrow{\tau_{2}}Q(-2)^{3}\xrightarrow{\tau_{1}}Q\rightarrow
0$

where we label and order the basis elements as follows:

  1. (1)

basis for $T_{0}=Q$: $\\{{\varepsilon}_{\emptyset}\\}$

  2. (2)

basis for $T_{1}=Q(-2)^{3}$:
$\\{{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}\\}$

  3. (3)

basis for $T_{2}=Q(-3)^{3}$:
$\\{{\varepsilon}_{12},{\varepsilon}_{13},{\varepsilon}_{23}\\}$

  4. (4)

basis for $T_{3}=Q(-3)$: $\\{{\varepsilon}_{123}\\}$.

The differentials for ${\mathbf{T}}$ are

$\tau_{1}=\blockarray{*{4}{>{\scriptstyle}c}<{}}1&23\\\
\block{[*{3}r]>{\scriptstyle}l<{}}xyxzyz\emptyset\\\
,\>\tau_{2}=\blockarray{*{4}{>{\scriptstyle}c}<{}}12&1323\\\
\block{[*{3}r]>{\scriptstyle}l<{}}zz01\\\ -y0y2\\\ 0-x-x3\\\ ,\text{ and
}\tau_{3}=\blockarray{*{2}{>{\scriptstyle}c}<{}}123\\\
\block{[r]>{\scriptstyle}l<{}}1&12\\\ -113\\\ 123\\\ .$

###  2.2. A System of Homotopies

Let ${\mathfrak{a}}={\langle a_{1},\ldots,a_{c}\rangle}$ be generated by a
regular sequence $a_{1},\ldots,a_{c}$ in $Q$ and $R=Q/{\mathfrak{a}}$. The
following definitions and constructions are due to Shamash [Sha69] in the
$c=1$ case, and to Eisenbud [Eis80, Eis77] when $c\geq 2$. We give a summary
of the necessary background with some finer details omitted to suit our
particular setting; a full treatment can be found in [Avr98, Chapter 3] [EP16,
Section 4.1].

######  Definition 2.3.

Let ${\mathbf{G}}$ be a complex of free $Q$-modules. Let
${\mathbf{u}}=(u_{1},\ldots,u_{c})$, where each $u_{i}\geq 0$ is an integer;
use ${\mathbf{e}_{t}}$ to denote the $t$th standard basis vector of length $c$
and ${\mathbf{0}}$ to denote the zero vector. Set
$|{\mathbf{u}}|=\sum_{i}u_{i}$. A system of higher homotopies $\sigma$ for
$a_{1},\ldots,a_{c}$ on ${\mathbf{G}}$ is a collection of maps

$\sigma_{\mathbf{u}}:{\mathbf{G}}\rightarrow{\mathbf{G}}[-2|{\mathbf{u}}|+1]$

on the modules in ${\mathbf{G}}$ where the three following conditions are
satisfied:

  1. (i)

$\sigma_{\mathbf{0}}$ is the differential on ${\mathbf{G}}$,

  2. (ii)

for each $j\in[c]$, the map
$\sigma_{\mathbf{0}}\sigma_{\mathbf{e}_{j}}+\sigma_{\mathbf{e}_{j}}\sigma_{\mathbf{0}}$
is multiplication by $a_{j}$ on ${\mathbf{G}}$, and

  3. (iii)

if $|{\mathbf{u}}|\geq 2$, then

$\sum\limits_{{\mathbf{b}}+{\mathbf{b}}^{\prime}={\mathbf{u}}}\sigma_{\mathbf{b}}\sigma_{{\mathbf{b}}^{\prime}}=0.$

######  Construction 2.4.

Consider the divided power algebra $Q\\{y_{1},\ldots,y_{c}\\}\cong\oplus
Qy_{1}^{(i_{1})}y_{2}^{(i_{2})}\cdots y_{c}^{(i_{c})}$ on variables
$y_{1},\ldots,y_{c}$ of degree $2$, and the polynomial ring
$Q[t_{1},\ldots,t_{c}]$, where we treat the monomials in
$Q\\{y_{1},\ldots,y_{c}\\}$ and in $Q[t_{1},\ldots,t_{c}]$ as dual to each
other. In this way, we can prescribe an action of $Q[t_{1},\ldots,t_{c}]$ on
$Q\\{y_{1},\ldots,y_{c}\\}$ by

$t_{j}y_{j}^{(i)}=y_{j}^{(i-1)}.$

For a system of homotopies $\sigma$ on ${\mathbf{G}}$, the graded free
$R$-module

$\Phi({\mathbf{G}})\coloneqq
Q\\{y_{1},\ldots,y_{c}\\}\otimes{\mathbf{G}}\otimes R$

with differential

$\varphi\coloneqq\sum t^{\mathbf{u}}\otimes\sigma_{\mathbf{u}}\otimes R$

is called the Shamash construction or Eisenbud–Shamash construction.

######  Theorem 2.5.

[Sha69, Eis80] If ${\mathbf{G}}$ is a $Q$-free resolution of a finitely
generated module $N$, then $\Phi({\mathbf{G}})$ is an $R$-free resolution of
$N$.

More information on divided power algebras can be found in [Eis95, Appendix
2]. For the reader who is satisfied to restrict to the characteristic zero
case, the divided power algebra and the symmetric algebra behave roughly
equivalently.

######  Example 2.6.

Let $Q=\mathbbm{k}[x_{1},x_{2}]$ and $R=Q/{\langle x_{1}^{5}\rangle}$. We will
use the Shamash construction to write a (minimal) resolution of $R/{\langle
x_{1}^{2},x_{2}^{2}\rangle}$. The Taylor resolution

${\mathbf{T}}:0\rightarrow
Q(-4)\xrightarrow{\tau_{2}}Q(-2)^{2}\xrightarrow{\tau_{1}}Q\rightarrow 0$

resolves $Q/{\langle x_{1}^{2},x_{2}^{2}\rangle}$ minimally. The maps

$\sigma_{1,0}:Q\xrightarrow{{\left[\begin{array}[]{c}x_{1}^{3}\\\
0\end{array}\right]}}Q(-2)^{2}\quad\text{ and
}\quad\sigma_{1,1}:Q(-2)^{2}\xrightarrow{{\left[\begin{array}[]{cc}0&-x_{1}^{3}\end{array}\right]}}Q(-4)$

form a system of homotopies on ${\mathbf{G}}$ for $x^{5}$, though from here on
out, we will drop the first index from the subscript to avoid notational
clutter. Identifying $Q(-d)\otimes R$ with $R(-d)$, we construct the Shamash
resolution of $Q/{\langle x_{1}^{2},x_{2}^{2}\rangle}$ as an $R$-module:

$\cdots\rightarrow\begin{array}[]{c}Qy^{(2)}\otimes R\\\ \bigoplus\\\
Qy^{(1)}\otimes
R(-4)\end{array}\xrightarrow{{\left[\begin{array}[]{c|c}\sigma_{0}&\tau_{2}\end{array}\right]}}Qy^{(1)}\otimes
R(-2)^{2}\xrightarrow{{\left[\begin{array}[]{c}\tau_{1}\\\
\hline\cr\sigma_{1}\end{array}\right]}}\begin{array}[]{c}Qy^{(1)}\otimes R\\\
\bigoplus\\\ Q\otimes
R(-4)\end{array}\xrightarrow{{\left[\begin{array}[]{c|c}\sigma_{0}&\tau_{2}\end{array}\right]}}Q\otimes
R(-2)^{2}\xrightarrow{{\left[\begin{array}[]{c}\tau_{1}\end{array}\right]}}Q\otimes
R\rightarrow 0$

Note that $Qy^{(d)}\otimes R(-k)$ can be identified with $R(-5d-k)$, so we
could in fact rewrite the constituent modules in the resolution above as
$R(-5i+1)\oplus R(-5i)$ in homological degrees $2i$ and $R(-5i+3)^{2}$ in
homological degrees $2i-1$, when $i>0$.

More generally, if $\deg(a_{j})=d_{j}$, then we can identify
$Qy_{1}^{(p_{1})}y_{2}^{(p_{2})}\cdots y_{c}^{(p_{c})}\otimes Q(-k)\otimes R$
with $R(-k-\sum\limits_{i=1}^{c}p_{i}d_{i})$ in the Shamash construction to
obtain the correct twist.

##  3\. Taylor Resolutions Over Complete Intersections

Let $I={\langle m_{1},\ldots,m_{r}\rangle}$ be a monomial ideal in $Q$ and
${\mathfrak{a}}={\langle a_{1},\ldots,a_{c}\rangle}$ be generated by a regular
sequence in $Q$ with ${\mathfrak{a}}\subseteq I$. This means that each
$a_{i}=f_{i,1}m_{1}+\cdots+f_{i,r}m_{r}$ for some $f_{1},\ldots,f_{r}\in Q$,
though such an expression is not inherently unique without first fixing a
monomial ordering on $Q$. Once these ${f_{i,j}}$ are fixed, the following
construction will produce a system $\sigma$ of higher homotopies for
$a_{1},\ldots,a_{c}$ on ${\mathbf{T}}$, where ${\mathbf{T}}$ is the Taylor
resolution of $Q/I$.

######  Definition 3.1.

Let ${\mathbf{u}}=(u_{1},\ldots,u_{c})$, where each $u_{i}\geq 0$ is an
integer; use ${\mathbf{e}_{i}}$ to denote the $i$th standard vector of length
$c$ and ${\mathbf{0}}$ to denote the zero vector. Set
$|{\mathbf{u}}|=\sum_{i}u_{i}$. Define a system of maps

$\sigma_{u}:{\mathbf{T}}\rightarrow{\mathbf{T}}[-2|{\mathbf{u}}|+1]$

in the following way:

  * •

for $S=\\{s_{1},\ldots,s_{k}\\}\subseteq[r]$,

$\sigma_{\mathbf{0}}({\varepsilon_{S}})=\sum\limits_{s_{i}\in
S}(-1)^{k-i}\frac{m_{S}}{m_{S-s_{i}}}\varepsilon_{S-s_{i}},$

that is, $\sigma_{\mathbf{0}}=\tau$,

  * •

for $S=\\{s_{1},\ldots,s_{k}\\}\subseteq[r]$ and ${\overline{S}}=[r]-S$,

$\sigma_{\mathbf{e}_{i}}({\varepsilon_{S}})=\sum\limits_{t\in{\overline{S}}}(-1)^{k-p_{t,S}-1}\frac{f_{i,t}m_{t}m_{S}}{m_{S\cup
t}}{\varepsilon_{S\cup t}}$

where $p_{t,S}$ is the position of $t$ in ${S\cup t}$, that is, ${S\cup
t}=\\{s_{1},\ldots,s_{p_{t,S}-1},t,s_{p_{t,S}},\ldots,s_{k}\\}$,

  * •

and

$\sigma_{\mathbf{u}}=0$

for all other ${\mathbf{u}}\in\mathbbm{N}^{c}$, i.e. if $|{\mathbf{u}}|\geq
2$.

######  Theorem 3.2.

The maps $\sigma$ given in Definition 3.1 form a system of higher homotopies
for $a_{1},\ldots,a_{c}$ on the Taylor complex ${\mathbf{T}}$. Therefore, when
the Shamash construction is applied to the system of higher homotopies for
$a_{1},\ldots,a_{c}$ on ${\mathbf{T}}$, these maps furnish a resolution $\Phi$
of $R/I$.

The existence of a system of higher homotopies is guaranteed by [Sha69, Eis80]
Eisenbud and Shamash, so the main content of the theorem is, first, that no
higher homotopies are necessary with the proposed $\sigma_{\mathbf{e}_{i}}$,
and, second, that the terms in the differentials are a simple combination of
the $f_{i,j}$ and, as in the Taylor resolution, a ratio of LCMs of the
$m_{i}$. Therefore, once expressions for the $a_{j}$ in terms of the $m_{i}$
are chosen, the resulting resolution is extremely computable and completely
prescribed – no further choices are necessary. In the hypersurface case, this
resolution automatically produces a matrix factorization that can be stated
purely combinatorially, just in terms of subsets and least common multiples.
Some overlap can be found in [Iye97] and [AB00], but the result here uses
different, less technical methods to provide more combinatorially-minded
results.

###### Proof.

The proof method is straightforward: the three criteria from Definition 2.3
must be checked.

(i) is satisfied by the definition of $\sigma_{\mathbf{0}}$.

To check (iii), first note that
$\sigma_{\mathbf{b}}\sigma_{{\mathbf{b}}^{\prime}}=0$ if
${\mathbf{b}}+{\mathbf{b}}^{\prime}>2$, since, in this case, at least one of
$|{\mathbf{b}}|,|{\mathbf{b}}^{\prime}|>2$. Therefore the only time that
$\sum\limits_{{\mathbf{b}}+{\mathbf{b}}^{\prime}={\mathbf{u}}}\sigma_{\mathbf{b}}\sigma_{{\mathbf{b}}^{\prime}}$
is not trivially zero is when $|{\mathbf{u}}|=2$. This occurs in two cases:
when ${\mathbf{u}}=2{\mathbf{e}_{i}}$ and when
${\mathbf{u}}={\mathbf{e}_{i}}+{\mathbf{e}_{j}}$.

In the case when ${\mathbf{u}}=2{\mathbf{e}_{i}}$, we have

$\sum\limits_{{\mathbf{b}}+{\mathbf{b}}^{\prime}=2{\mathbf{e}_{i}}}\sigma_{\mathbf{b}}\sigma_{{\mathbf{b}}^{\prime}}=\sigma_{2{\mathbf{e}_{i}}}\sigma_{\mathbf{0}}+\sigma_{\mathbf{e}_{i}}\sigma_{\mathbf{e}_{i}}+\sigma_{\mathbf{0}}\sigma_{2{\mathbf{e}_{i}}}=\sigma_{\mathbf{e}_{i}}\sigma_{\mathbf{e}_{i}},$

so we must check that $\sigma_{\mathbf{e}_{i}}\sigma_{\mathbf{e}_{i}}=0$.
Indeed, we see that

$\displaystyle\sigma_{\mathbf{e}_{i}}(\sigma_{\mathbf{e}_{i}}({\varepsilon_{S}}))$
$\displaystyle=\sigma_{\mathbf{e}_{i}}\left(\sum\limits_{t\in{\overline{S}}}(-1)^{k-{p_{t,S}}-1}\frac{{f_{i,t}}m_{t}m_{S}}{m_{S\cup
t}}{\varepsilon_{S\cup t}}\right)$
$\displaystyle=\sum\limits_{t\in{\overline{S}}}(-1)^{k-{p_{t,S}}-1}\frac{{f_{i,t}}m_{t}m_{S}}{m_{S\cup
t}}\sigma_{\mathbf{e}_{i}}({\varepsilon_{S\cup t}})$
$\displaystyle=\sum\limits_{t\in{\overline{S}}}(-1)^{k-{p_{t,S}}-1}\frac{{f_{i,t}}m_{t}m_{S}}{m_{S\cup
t}}\left(\sum\limits_{q\in{\overline{S\cup t}}}(-1)^{k+1-p_{q,{S\cup
t}}-1}\frac{{f_{i,q}}m_{S\cup t}m_{q}}{m_{S\cup t\cup q}}{\varepsilon}_{S\cup
t\cup q}\right)$
$\displaystyle=\sum\limits_{t\in{\overline{S}}}\sum\limits_{q\in{\overline{S\cup
t}}}(-1)^{-{p_{t,S}}-p_{q,{S\cup
t}}-1}\frac{{f_{i,t}}{f_{i,q}}m_{t}m_{q}m_{S}}{m_{S\cup t\cup
q}}{\varepsilon}_{S\cup t\cup q}$
$\displaystyle=\sum\limits_{\begin{subarray}{c}t<q\\\
t,q\in{\overline{S}}\end{subarray}}\left((-1)^{-{p_{t,S}}-p_{q,{S\cup
t}}-1}+(-1)^{-{p_{q,S}}-p_{t,{S\cup
q}}-1}\right)\frac{{f_{i,t}}{f_{i,q}}m_{t}m_{q}m_{S}}{m_{S\cup t\cup
q}}{\varepsilon}_{S\cup t\cup q}.$

Because $t<q$, we have that $p_{t,{S\cup q}}={p_{t,S}}$ but $p_{q,{S\cup
t}}={p_{q,S}}+1$, so we can adjust the final sum to

$\sum\limits_{\begin{subarray}{c}t<q\\\
t,q\in{\overline{S}}\end{subarray}}\left((-1)^{-{p_{t,S}}-(p_{q,S}+1)-1}+(-1)^{-{p_{q,S}}-p_{t,S}-1}\right)\frac{{f_{i,t}}{f_{i,q}}m_{t}m_{q}m_{S}}{m_{S\cup
t\cup q}}{\varepsilon}_{S\cup t\cup q},$

which then reduces to $0$.

In the case when ${\mathbf{u}}={\mathbf{e}_{i}}+{\mathbf{e}_{j}}$, we have

$\sum\limits_{{\mathbf{b}}+{\mathbf{b}}^{\prime}={\mathbf{e}_{i}}+{\mathbf{e}_{j}}}\sigma_{\mathbf{b}}\sigma_{{\mathbf{b}}^{\prime}}=\sigma_{{\mathbf{e}_{i}}+{\mathbf{e}_{j}}}\sigma_{\mathbf{0}}+\sigma_{\mathbf{e}_{i}}\sigma_{\mathbf{e}_{j}}+\sigma_{\mathbf{e}_{j}}\sigma_{\mathbf{e}_{i}}+\sigma_{\mathbf{0}}\sigma_{{\mathbf{e}_{i}}+{\mathbf{e}_{j}}}=\sigma_{\mathbf{e}_{i}}\sigma_{\mathbf{e}_{j}}+\sigma_{\mathbf{e}_{j}}\sigma_{\mathbf{e}_{i}},$

so we must check that
$\sigma_{\mathbf{e}_{i}}\sigma_{\mathbf{e}_{j}}+\sigma_{\mathbf{e}_{j}}\sigma_{\mathbf{e}_{i}}=0$.
First we see that

$\displaystyle\sigma_{\mathbf{e}_{i}}(\sigma_{\mathbf{e}_{j}}({\varepsilon_{S}}))$
$\displaystyle=\sigma_{\mathbf{e}_{i}}\left(\sum\limits_{t\in{\overline{S}}}(-1)^{k-{p_{t,S}}-1}\frac{{f_{j,t}}m_{t}m_{S}}{m_{S\cup
t}}{\varepsilon_{S\cup t}}\right)$
$\displaystyle=\sum\limits_{t\in{\overline{S}}}(-1)^{k-{p_{t,S}}-1}\frac{{f_{j,t}}m_{t}m_{S}}{m_{S\cup
t}}\sigma_{\mathbf{e}_{i}}({\varepsilon_{S\cup t}})$
$\displaystyle=\sum\limits_{t\in{\overline{S}}}(-1)^{k-{p_{t,S}}-1}\frac{{f_{j,t}}m_{t}m_{S}}{m_{S\cup
t}}\left(\sum\limits_{q\in{\overline{S\cup t}}}(-1)^{k+1-{p_{q,S\cup
t}}-1}\frac{{f_{i,q}}m_{q}m_{S\cup t}}{m_{S\cup t\cup q}}{\varepsilon}_{S\cup
t\cup q}\right)$
$\displaystyle=\sum\limits_{t\in{\overline{S}}}\sum\limits_{q\in{\overline{S\cup
t}}}(-1)^{-{p_{t,S}}-{p_{q,S\cup
t}}-1}\frac{{f_{j,t}}{f_{i,q}}m_{t}m_{q}m_{S}}{m_{S\cup t\cup
q}}{\varepsilon_{S\cup t\cup q}}$
$\displaystyle=\sum\limits_{\begin{subarray}{c}t<q\\\
t,q\in{\overline{S}}\end{subarray}}\left((-1)^{-{p_{t,S}}-{p_{q,S\cup
t}}-1}{f_{j,t}}{f_{i,q}}+(-1)^{-{p_{q,S}}-{p_{t,S\cup
q}}-1}{f_{j,q}}{f_{i,t}}\right)\frac{m_{t}m_{q}m_{S}}{m_{S\cup t\cup
q}}{\varepsilon_{S\cup t\cup q}}$
$\displaystyle=\sum\limits_{\begin{subarray}{c}t<q\\\
t,q\in{\overline{S}}\end{subarray}}\left((-1)^{-{p_{t,S}}-({p_{q,S}}+1)-1}{f_{j,t}}{f_{i,q}}+(-1)^{-{p_{q,S}}-{p_{t,S}}-1}{f_{j,q}}{f_{i,t}}\right)\frac{m_{t}m_{q}m_{S}}{m_{S\cup
t\cup q}}{\varepsilon_{S\cup t\cup q}}$
$\displaystyle=\sum\limits_{\begin{subarray}{c}t<q\\\
t,q\in{\overline{S}}\end{subarray}}(-1)^{-{p_{t,S}}-{p_{q,S}}}\left({f_{j,t}}{f_{i,q}}-{f_{j,q}}{f_{i,t}}\right)\frac{m_{t}m_{q}m_{S}}{m_{S\cup
t\cup q}}{\varepsilon_{S\cup t\cup q}}.$

A similar computation shows that

$\displaystyle\sigma_{\mathbf{e}_{j}}(\sigma_{\mathbf{e}_{i}}({\varepsilon_{S}}))$
$\displaystyle=\sum\limits_{\begin{subarray}{c}t<q\\\
t,q\in{\overline{S}}\end{subarray}}(-1)^{-{p_{t,S}}-{p_{q,S}}}\left({f_{i,t}}{f_{j,q}}-{f_{j,t}}{f_{i,q}}\right)\frac{m_{t}m_{q}m_{S}}{m_{S\cup
t\cup q}}{\varepsilon_{S\cup t\cup q}}.$

Combining the two computations, we see that
$\sigma_{\mathbf{e}_{i}}\sigma_{\mathbf{e}_{j}}+\sigma_{\mathbf{e}_{j}}\sigma_{\mathbf{e}_{i}}=0$.

It remains to check (ii). We compute

$\displaystyle\sigma_{\mathbf{0}}(\sigma_{\mathbf{e}_{j}}({\varepsilon_{S}}))$
$\displaystyle=\sigma_{\mathbf{0}}\left(\sum\limits_{t\in{\overline{S}}}(-1)^{k-{p_{t,S}}-1}\frac{{f_{j,t}}m_{t}m_{S}}{m_{S\cup
t}}{\varepsilon_{S\cup t}}\right)$
$\displaystyle=\sum\limits_{t\in{\overline{S}}}(-1)^{k-{p_{t,S}}-1}\frac{{f_{j,t}}m_{t}m_{S}}{m_{S\cup
t}}\sigma_{\mathbf{0}}\left({\varepsilon_{S\cup t}}\right)$
$\displaystyle=\sum\limits_{t\in{\overline{S}}}(-1)^{k-{p_{t,S}}-1}\frac{{f_{j,t}}m_{t}m_{S}}{m_{S\cup
t}}\left(\sum\limits_{\begin{subarray}{c}s_{i}<t\\\ s_{i}\in{S\cup
t}\end{subarray}}(-1)^{k+1-i}\frac{m_{S\cup t}}{m_{S\cup
t-s_{i}}}{\varepsilon}_{S\cup t-s_{i}}\right.$ $\displaystyle\hskip
90.3375pt\left.+(-1)^{k+1-{p_{t,S}}}\frac{m_{S\cup
t}}{m_{S}}{\varepsilon_{S}}+\sum\limits_{\begin{subarray}{c}s_{i}>t\\\
s_{i}\in{S\cup t}\end{subarray}}(-1)^{k+1-(i+1)}\frac{m_{S\cup t}}{m_{S\cup
t-s_{i}}}{\varepsilon}_{S\cup t-s_{i}}\right)$
$\displaystyle=\left(\sum\limits_{t\in{\overline{S}}}\sum\limits_{\begin{subarray}{c}s_{i}<t\\\
s_{i}\in
S\end{subarray}}(-1)^{-{p_{t,S}}-i}\frac{{f_{j,t}}m_{t}m_{S}}{m_{S\cup
t-s_{i}}}\right)+\sum\limits_{t\in{\overline{S}}}{f_{j,t}}m_{t}{\varepsilon_{S}}+$
$\displaystyle\hskip
90.3375pt+\left(\sum\limits_{t\in{\overline{S}}}\sum\limits_{\begin{subarray}{c}s_{i}>t\\\
s_{i}\in
S\end{subarray}}(-1)^{-{p_{t,S}}-i-1}\frac{{f_{j,t}}m_{t}m_{S}}{m_{S\cup
t-s_{i}}}{\varepsilon}_{S\cup t-s_{i}}\right)$

Similarly we can see that

$\displaystyle\sigma_{\mathbf{e}_{j}}(\sigma_{\mathbf{0}}({\varepsilon_{S}}))$
$\displaystyle=\left(\sum\limits_{s_{i}\in
S}\sum\limits_{\begin{subarray}{c}t<s_{i}\\\
t\in{\overline{S}}\end{subarray}}(-1)^{-i-{p_{t,S}}}\frac{{f_{j,t}}m_{t}m_{S}}{m_{S-s_{i}\cup
t}}{\varepsilon}_{S-s_{i}\cup t}\right)+\sum\limits_{s_{i}\in
S}f_{j,s_{i}}m_{s_{i}}{\varepsilon_{S}}$ $\displaystyle\hskip
90.3375pt+\left(\sum\limits_{s_{i}\in
S}\sum\limits_{\begin{subarray}{c}t>s_{i}\\\
t\in{\overline{S}}\end{subarray}}(-1)^{-i-{p_{t,S}}-1}\frac{{f_{j,t}}m_{t}m_{S}}{m_{S-s_{i}\cup
t}}{\varepsilon}_{S-s_{i}\cup t}\right).$

Adding the two expressions together, we see that the parenthesized double sums
cancel, and we are left with

$\sigma_{\mathbf{0}}(\sigma_{\mathbf{e}_{j}}({\varepsilon_{S}}))+\sigma_{\mathbf{e}_{j}}(\sigma_{\mathbf{0}}({\varepsilon_{S}}))=\sum\limits_{t\in{\overline{S}}}{f_{j,t}}m_{t}{\varepsilon_{S}}+\sum\limits_{s_{i}\in
S}f_{j,s_{i}}m_{s_{i}}{\varepsilon_{S}}=\sum\limits_{i=1}^{r}f_{j,i}m_{i}{\varepsilon_{S}}=a_{j}{\varepsilon_{S}}.$

∎

A few remarks should be made about the resolution ${\mathbf{F}}$. Due to its
origin in the Taylor resolution, ${\mathbf{F}}$ is universal for all monomial
ideals $I$ in $Q$ containing ${\mathfrak{a}}$. The same uniformity that
characterizes the Taylor resolution is also present in ${\mathbf{F}}$ – these
resolutions for two ideals generated by $r$ monomials will have the same ranks
in all their free modules. However, this provides a quick bound to the Betti
numbers of $I$. This bound could be calculated without explicit knowledge of
the homotopies (see [Avr98, Proposition 3.3.5]) but, for the sake of
completeness, we include the specific computation here.

######  Corollary 3.3.

The total Betti numbers $\beta_{i}$ of $R/I$ for a monomial ideal $I$ in a
complete intersection $R$ satisfy the inequality

$\beta_{2m}\leq\sum\limits_{j=0}^{m}\binom{r}{2j}\binom{c+m-j-1}{c-1}\quad\text{
and
}\quad\beta_{2m+1}\leq\sum\limits_{j=0}^{m}\binom{r}{2j+1}\binom{c+m-j-1}{c-1}.$

###### Proof.

This follows from a quick rank calculation of the modules in $F$ in the even
and odd cases. To wit,

$F_{2m}=\bigoplus\limits_{\begin{subarray}{c}j=0\\\
|{\mathbf{u}}|=m-j\end{subarray}}^{m}Q\mathbf{y}^{(\mathbf{u})}\otimes
T_{2j}\otimes R,$

where we use multinomial notation
$\mathbf{y}^{\mathbf{u}}=y_{1}^{(u_{1})}\cdots y_{c}^{(u_{c})}$. Since
$\operatorname{rank}T_{2j}=\binom{r}{2j}$ and there are $\binom{c+m-j-1}{c-1}$
monomials of degree $m-j$ in the $y$’s, we can see that

$\operatorname{rank}F_{2m}=\sum\limits_{j=0}^{m}\binom{r}{2j}\binom{c+m-j-1}{c-1},$

which immediately gives the bound for $\beta_{2m}$. The bound for
$\beta_{2m+1}$ can be found analogously. ∎

######  Remark 3.4.

Per [Avr98, Remark (p.33)], the minimality of ${\mathbf{F}}$ is closely linked
to the minimality of ${\mathbf{T}}$, with an extra caveat: ${\mathbf{F}}$ is
minimal if and only if ${\mathbf{T}}$ is minimal and
${\mathfrak{a}}\subseteq{\mathfrak{m}}I$. In our particular maps, the
necessity that ${f_{i,j}}\in{\mathfrak{m}}$ comes from

$\sigma_{\mathbf{e}_{i}}({\varepsilon}_{\emptyset})=\sum\limits_{t=1}^{r}(-1)^{0-1-1}\frac{{f_{i,t}}m_{t}m_{\emptyset}}{m_{t}}{\varepsilon}_{t}=\sum\limits_{t=1}^{r}{f_{i,t}}{\varepsilon}_{t}.$

Since the ${f_{i,t}}$ appear as entries in the maps of ${\mathbf{F}}$, they
must be in ${\mathfrak{m}}$ for ${\mathbf{F}}$ to be minimal. On the other
hand, the coefficient of ${\varepsilon_{S\cup t}}$ in
$\sigma_{\mathbf{e}_{i}}({\varepsilon_{S}})$ is always a multiple of
${f_{i,t}}$, so ${f_{i,t}}\in{\mathfrak{m}}$ is enough to ensure that
${\mathbf{F}}$ is minimal.

######  Example 3.5 (Resolution of the Ground Field).

When $I={\langle x_{1},\ldots,x_{n}\rangle}$, ${\mathbf{F}}$ recovers the Tate
complex [Tat57] resolving $\mathbbm{k}$ over $R$, since the Taylor resolution
coincides with the Koszul resolution of $\mathbbm{k}$, and the Tate resolution
can be obtained by applying the Shamash construction to the Koszul complex. By
Remark 3.4, the resolution is minimal, assuming $\deg a_{i}>1$ for
$a_{1},\ldots,a_{c}$.

######  Example 3.6 (Resolution of ${\mathfrak{m}}^{b}$).

If ${\mathfrak{a}}$ is a regular sequence of degree $d$, then ${\mathbf{F}}$
gives a resolution of ${\mathfrak{m}}^{b}$ for any power $1\leq b\leq d$. This
resolution is nonminimal when $b\geq 2$, since in this case the Taylor
resolution does not minimally resolve ${\mathfrak{m}}^{b}$ over $Q$.

The simplest case is when ${\mathfrak{a}}$ is generated by a single monomial
$a$. Here, at least one of the $m_{i}$ in $I$ must divide $a$, but it is more
interesting to examine a case when more than one $m_{i}$ divides $a$ to see
how the choice of expressing $a$ in terms of the $m_{i}$ affects the homotopy.

######  Example 3.7 (Codimension $1$ Monomial ${\mathfrak{a}}$ Case).

Let $Q=\mathbbm{k}[x,y,z]$, $a=xyz$, and $I=\langle xy,xz,yz\rangle$. The
Taylor resolution of $I$ over $Q$ is given in Example 2.2.

Writing $xyz=z\cdot xy+0\cdot xz+0\cdot yz$, the homotopy
$\sigma\coloneqq\sigma_{\mathbf{1}}$ has the following behavior:

$\sigma_{0}=\blockarray{*{2}{>{\scriptstyle}c}<{}}\emptyset\\\
\block{[r]>{\scriptstyle}l<{}}z&1\\\ 02\\\ 03\\\
,\>\sigma_{1}=\blockarray{*{4}{>{\scriptstyle}c}<{}}1&23\\\
\block{[*{3}r]>{\scriptstyle}l<{}}0-xz012\\\ 00-yz13\\\ 00023\\\ ,\text{ and
}\sigma_{2}=\blockarray{*{4}{>{\scriptstyle}c}<{}}12&1323\\\
\block{[*{3}r]>{\scriptstyle}l<{}}00xyz123\\\ $

Therefore a resolution of $I$ over $R\coloneqq Q/\langle a\rangle$ is

${\mathbf{F}}:\cdots\xrightarrow{\varphi_{5}}R(-6)^{4}\xrightarrow{\varphi_{4}}\begin{array}[]{c}R(-5)^{3}\\\
\oplus\\\
R(-3)\end{array}\xrightarrow{\varphi_{3}}R(-3)^{4}\xrightarrow{\varphi_{2}}R(-2)^{3}\xrightarrow{\varphi_{1}}R\rightarrow
0$

where $F_{2i}=R(-3i)^{4}$ and $F_{2i-1}=R(-3i+1)^{3}\oplus R(-3(i-1))$ for
$i\geq 2$, the initial maps are

$\varphi_{1}=\blockarray{*{4}{>{\scriptstyle}c}<{}}1&23\\\
\block{[*{3}r]>{\scriptstyle}c<{}}xyxzyz\emptyset\\\ \text{ and
}\varphi_{2}=\blockarray{*{5}{>{\scriptstyle}c}<{}}\emptyset&121323\\\
\block{[r|*{3}r]>{\scriptstyle}c<{}}zzz01\\\ 0-y0y2\\\ 00-x-x3\\\ ,$

and the (non-minimal) tail of the resolution is periodic with maps

$\varphi_{2i-1}=\blockarray{*{5}{>{\scriptstyle}c}<{}}1&23123\\\
\block{[*{3}r|r]>{\scriptstyle}c<{}}xyxzyz0\emptyset\\\
\BAhhline{----}0-xz0112\\\ 00-yz-113\\\ 000123\\\ \text{ and
}\varphi_{2i}=\blockarray{*{5}{>{\scriptstyle}c}<{}}\emptyset&121323\\\
\block{[r|*{3}r]>{\scriptstyle}c<{}}zzz01\\\ 0-y0y2\\\ 00-x-x3\\\
\BAhhline{----}000xyz123\\\ $

for $i\geq 2$.

The vertical lines in the maps are visual dividers indicating the origin of
the constituent blocks (whether they come from $\sigma_{\mathbf{0}}$ or
$\sigma_{\mathbf{1}}$). Note that the domains of $\varphi_{2i-1}$ and
$\varphi_{2i}$ are indexed by the odd and even subsets of $\\{1,2,3\\}$
respectively. This division of domains into even and odd subsets will always
appear in the tail of the resolution in the $c=1$ case.

Note that writing $xyz=0\cdot xy+y\cdot xz+0\cdot yz$, we get a different
homotopy and therefore a resolution with the same modules as ${\mathbf{F}}$
but different differentials:

$\varphi^{\prime}_{2}=\blockarray{*{5}{>{\scriptstyle}c}<{}}\emptyset&121323\\\
\block{[r|*{3}r]>{\scriptstyle}c<{}}0zz01\\\ y-y0y2\\\ 00-x-x3\\\
,\varphi^{\prime}_{2i-1}=\blockarray{*{5}{>{\scriptstyle}c}<{}}1&23123\\\
\block{[*{3}r|r]>{\scriptstyle}c<{}}xyxzyz0\emptyset\\\
\BAhhline{----}xy00112\\\ 000-113\\\ 00-yz123\\\ \text{ and
}\varphi^{\prime}_{2i}=\blockarray{*{5}{>{\scriptstyle}c}<{}}\emptyset&121323\\\
\block{[r|*{3}r]>{\scriptstyle}c<{}}0zz01\\\ y-y0y2\\\ 00-x-x3\\\
\BAhhline{----}00-xyz0123\\\ $

for $i\geq 2$.

Finally, writing $xyz=0\cdot xy+0\cdot xz+x\cdot yz$ to yields a third
resolution with maps

$\varphi^{\prime\prime}_{2}=\blockarray{*{5}{>{\scriptstyle}c}<{}}\emptyset&121323\\\
\block{[r|*{3}r]>{\scriptstyle}c<{}}0zz01\\\ 0-y0y2\\\ x0-x-x3\\\
,\varphi^{\prime\prime}_{2i-1}=\blockarray{*{5}{>{\scriptstyle}c}<{}}1&23123\\\
\block{[*{3}r|r]>{\scriptstyle}c<{}}xyxzyz0\emptyset\\\
\BAhhline{----}000112\\\ xy00-113\\\ 0xz0123\\\ \text{ and
}\varphi^{\prime\prime}_{2i}=\blockarray{*{5}{>{\scriptstyle}c}<{}}\emptyset&121323\\\
\block{[r|*{3}r]>{\scriptstyle}c<{}}0zz01\\\ 0-y0y2\\\ x0-x-x3\\\
\BAhhline{----}0xyz00123\\\ $

for $i\geq 2$.

Examples 3.7 shows that the universality of ${\mathbf{T}}$ does not quite
carry over; there are some choices to be made that change the form of the
resolution ${\mathbf{F}}$. However, characteristic of $\mathbbm{k}$
permitting, we can create a sort of “average homotopy”
$\frac{1}{3}(\sigma+\sigma^{\prime}+\sigma^{\prime\prime})$, which is also a
system of homotopies for $xyz$ on ${\mathbf{T}}$, and liberates us from making
a choice of how to express $xyz$ in terms of the generators of $I$. It is
straightforward to check that a convex combination of systems of higher
homotopies will also create a system of higher homotopies, so this averaging
scheme can be generalized. The idea of taking an average of all possibilities
to create a choice-free resolution also appears in [EMO19].

######  Example 3.8 (Codimension $1$ Polynomial ${\mathfrak{a}}$ Case).

Take $Q=\mathbbm{k}[x,y]$, $I={\langle x^{2},y^{2}\rangle}$, and
${\mathfrak{a}}={\langle x^{2}y+xy^{2}\rangle}$, so that $f_{1,1}=y$ and
$f_{1,2}=x$. Then the resolution of $I$ over $R=Q/{\mathfrak{a}}$ is

${\mathbf{F}}:\cdots\xrightarrow{\varphi_{5}}R(-6)\oplus
R(-7)\xrightarrow{\varphi_{4}}R(-5)^{2}\xrightarrow{\varphi_{3}}R(-3)\oplus
R(-4)\xrightarrow{\varphi_{2}}R(-2)^{2}\xrightarrow{\varphi_{1}}R\rightarrow
0$

where

$\varphi_{1}=\blockarray{*{3}{>{\scriptstyle}c}<{}}1&2\\\
\block{[*{2}r]>{\scriptstyle}c<{}}x^{2}y^{2}\emptyset\\\
,\quad\varphi_{2i}=\blockarray{*{3}{>{\scriptstyle}c}<{}}\emptyset&12\\\
\block{[r|r]>{\scriptstyle}c<{}}yy^{2}1\\\ x-x^{2}2\\\ ,\text{ and
}\varphi_{2i+1}=\blockarray{*{3}{>{\scriptstyle}c}<{}}1&2\\\
\block{[rr]>{\scriptstyle}c<{}}x^{2}y^{2}\emptyset\\\ \BAhhline{--}x-y12\\\ $

for $i\geq 1$.

######  Example 3.9 (Codimension $2$ Polynomial ${\mathfrak{a}}$ Case).

Take $Q=\mathbbm{k}[x,y,z,w]$, $I={\langle x^{2},y^{2},z^{2},w^{2}\rangle}$
and ${\mathfrak{a}}={\langle x^{3}+y^{3},z^{3}+w^{3}\rangle}$. The homotopies
for $x^{3}+y^{3}$ are

$\sigma_{{10},0}=\blockarray{*{2}{>{\scriptstyle}c}<{}}\emptyset\\\
\block{[r]>{\scriptstyle}l<{}}x&1\\\ y2\\\ 03\\\ 04\\\
,\sigma_{{10},1}=\blockarray{*{5}{>{\scriptstyle}c}<{}}1&234\\\
\block{[*{4}r]>{\scriptstyle}l<{}}y-x0012\\\ 00-x013\\\ 000-x14\\\ 00-y023\\\
000-y24\\\ 000034\\\
,\sigma_{{10},2}=\blockarray{*{7}{>{\scriptstyle}c}<{}}12&1314232434\\\
\block{[*{6}r]>{\scriptstyle}l<{}}0-y0x00123\\\ 00-y0x0124\\\ 00000x134\\\
00000y234\\\
,\sigma_{{10},3}=\blockarray{*{5}{>{\scriptstyle}c}<{}}123&124134234\\\
\block{[*{4}r]>{\scriptstyle}l<{}}00y-x1234\\\ ,$

while the homotopies for $z^{3}+w^{3}$ are

$\sigma_{{01},0}=\blockarray{*{2}{>{\scriptstyle}c}<{}}\emptyset\\\
\block{[r]>{\scriptstyle}l<{}}0&1\\\ 02\\\ z3\\\ w4\\\
,\sigma_{{01},1}=\blockarray{*{5}{>{\scriptstyle}c}<{}}1&234\\\
\block{[*{4}r]>{\scriptstyle}l<{}}000012\\\ z00013\\\ w00014\\\ 0z0023\\\
0w0024\\\ 00w-z34\\\
,\sigma_{{01},2}=\blockarray{*{7}{>{\scriptstyle}c}<{}}12&1314232434\\\
\block{[*{6}r]>{\scriptstyle}l<{}}z00000123\\\ w00000124\\\ 0w-z000134\\\
000w-z0234\\\
,\sigma_{{01},3}=\blockarray{*{5}{>{\scriptstyle}c}<{}}123&124134234\\\
\block{[*{4}r]>{\scriptstyle}l<{}}w-z001234\\\ .$

The modules in the resolution take the form

${\mathbf{F}}:\cdots\xrightarrow{\varphi_{6}}\begin{array}[]{c}R(-8)^{12}\\\
\bigoplus\\\
R(-9)^{8}\end{array}\xrightarrow{\varphi_{5}}\begin{array}[]{c}R(-6)^{3}\\\
\bigoplus\\\ R(-7)^{12}\\\ \bigoplus\\\
R(-8)\end{array}\xrightarrow{\varphi_{4}}\begin{array}[]{c}R(-5)^{8}\\\
\bigoplus\\\
R(-6)^{4}\end{array}\xrightarrow{\varphi_{3}}\begin{array}[]{c}R(-3)^{2}\\\
\bigoplus\\\
R(-4)^{6}\end{array}\xrightarrow{\varphi_{2}}R(-2)^{4}\xrightarrow{\varphi_{1}}R\rightarrow
0$

and a schematic for the differentials is illustrated below. To save some
notational hassle, we will shorten notation by using $\sigma$ and
$\widetilde{\sigma}$ for $\sigma_{10}$ and $\sigma_{01}$, respectively. To
visually distinguish the pieces, we color the blocks from $\tau$, $\sigma$,
and ${\widetilde{\sigma}}$ with blue, orange, and red, respectively.

$\tau_{1}$$\varphi_{1}=$$\sigma_{0}$${\widetilde{\sigma}}_{0}$$\tau_{2}$$\varphi_{2}=$

$\sigma_{0}$${\widetilde{\sigma}}_{0}$$\tau_{2}$$\sigma_{0}$${\widetilde{\sigma}}_{0}$$\tau_{2}$$\sigma_{2}$${\widetilde{\sigma}}_{2}$$\tau_{4}$$\varphi_{4}=$

$\tau_{1}$$\sigma_{1}$$\tau_{1}$${\widetilde{\sigma}}_{1}$$\tau_{3}$$\varphi_{3}=$

$\tau_{1}$$\tau_{1}$$\tau_{1}$$\sigma_{1}$${\widetilde{\sigma}}_{1}$$\tau_{3}$$\sigma_{1}$${\widetilde{\sigma}}_{1}$$\tau_{3}$$\sigma_{3}$${\widetilde{\sigma}}_{3}$$\varphi_{5}=$

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# More on spin-2 operators in holographic quantum mechanics

###### Abstract

We study the spectrum, unitarity bound and holographic central charge of
spin-2 operators in a warped
$\text{AdS}_{2}\times\text{S}^{2}\times\text{T}^{4}\times\mathcal{I}_{\psi}\times\mathcal{I}_{\rho}$
background in the Type IIB theory. We were able to identify a class of
solutions that is completely independent of the functions that define the
background solution. We comment on the relation of our results to the previous
ones in the literature.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 The supergravity solutions
    2. 2.2 The field theory interpretation
  3. 3 Holographic spin-2 fluctuations
  4. 4 Unitarity bound and universal solutions
  5. 5 The holographic central charge
  6. 6 Discussion and outlook

##  1 Introduction

The AdS/CFT correspondence [1, 2, 3] relates string theories defined in an
$\text{AdS}_{d+1}$ spacetime to superconformal field theories in
$d$-dimensions. Since its inception there has been a flurry of activities in
the systematic study of conformal and/or supersymmetric field theories using
both field theory methods as well as supergravity techniques. Tremendous
efforts have been put towards the classification of supersymmetric
$\text{AdS}_{d+1}$ backgrounds in (massive) type IIA/B and M-theory. In the
special case that supersymmetry is half-maximal and $d\geq 2$ there has been
significant progress in the supergravity classification and the mapping of
these solutions to specific quantum field theories [4, 5, 6, 7, 8, 9, 10, 11,
12, 13].

The AdS2/CFT1 case is of particular interest both for the microscopical
description and understanding of extremal black holes since it appears as the
near-horizon limit [14, 15], and also in the context of studying holographic
defects. Despite its interesting status much less is known due to subtle
points that arise in AdS2/CFT1 and are not present in the higher-dimensional
examples [16, 17, 18, 19]. To begin with, when one considers the backreaction
effect of finite-energy excitations, one ends up with a divergence at one of
the asymptotic boundaries. In addition to that, global $\text{AdS}_{2}$ has
two boundaries that are disconnected. Therefore, naively, it appears that the
proper holographic description is in terms of two copies of the same
1-dimensional field theory. This, however, is in contradiction with the
calculations of black hole entropy in string theory. Furthermore, gravity in
AdS2 is non-dynamical111This argument is true in the case of pure
$\text{AdS}_{2}$. More specifically, adding matter and/or any other warp
factors bypasses this hindrance, see e.g. [20]..

An interesting case study was presented in [21]. The author presented a
general class of regular geometries that asymptote to
$\text{AdS}_{2}\times\text{S}^{2}$. These are solutions that resemble
wormholes and the presence of the two boundaries appears as a feature of the
construction instead of a problematic description. Expanding these regular
solutions results to metric perturbations very close to the ones we will
consider in this work, although in a completely different string theory setup.

Recently we have a lot and exciting examples of new supergravity solutions
with an $\text{AdS}_{2}$ factor [20, 22, 23, 24, 25, 26, 27, 28, 29, 30]. From
these recent works, it was understood that the additional fluxes and warp
factors lead to an $\text{AdS}_{2}$ theory where gravity is no longer non-
dynamical. These solutions are generalizations of previous studies with
constant fluxes [31, 32, 33, 34, 35]. In some of the recently derived
$\text{AdS}_{2}$ solutions, the dual field theories have been identified and
several checks have been performed [20, 22, 23, 24, 25, 26, 27, 28].

Based on the principles of holography the spectrum of field operators is
derived by considering the linearized fluctuations of the fields in the bulk
for a specific supergravity solution. However, computing the complete set of
linearized fluctuations for the full supergravity background is a formidable
task. To this day, we have a handful of explicit examples at our disposal [36,
37, 38].

In spite of this complication, the work of [39] provided a proof that if we
focus only on the spin-2 operators and their spectrum the situation is
dramatically simplified. In particular, the authors proved that the spin-2
fluctuations, which result as perturbations of the Minkowski part of the
geometry, satisfy an equation depending solely on the geometrical data of the
background supergravity configuration. While the authors of that paper focused
on an $\text{AdS}_{4}$ solution, a generalization of their logic to arbitrary
space-time dimensions is straightforward. This laid the foundations for many
new and exciting results on the spectrum of holographic spin-2 operators
across all dimensions [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51].

It is worthwhile mentioning that for theories with 8 Poincaré supersymmetries
in AdS4,5,6,7, the spin-$2$ fluctuations have been put into a universal
treatment [52]. We note, however, that only dynamics were described, and the
equations were not solved except for the case of the AdS7 backgrounds.

This paper is organized as the following: In section 2 we briefly show the
background we are going to focus on and it was derived in [20]. In section 3
we study the linear fluctuations and solve the equation of motion. We find
that it becomes a Sturm-Liouville problem. In section 4 we calculate the
unitarity bound and so-called “minimal universal class of solutions”. Here
“universal” means that the solutions do not depend on the warp factors of the
backgrounds and “minima” is in the sense that we get these solutions under the
condition that we saturate the unitarity bound. In the section 5 we compute
the holographic central charge. We show our conclusions and outlooks in
section 6.

##  2 Preliminaries

The purpose of this section is to present and discuss the supergravity
backgrounds that we will examine in this work and their holographic field
theory descriptions. In doing so, we also set up our conventions. These
$\text{AdS}_{2}$ solutions were obtained in the work [20], which we will
closely follow in the forthcoming presentation.

###  2.1 The supergravity solutions

Let us begin by reviewing the main characteristics of the $\text{AdS}_{2}$
solutions that will be the focus of this work. We will be following [20], as
we have already mentioned, in our presentation where this class of backgrounds
was originally developed. The family of supergravity backgrounds has the
schematic form
$\text{AdS}_{2}\times\text{S}^{2}\times\text{CY}_{2}\times\mathcal{I}_{\psi}\times\mathcal{I}_{\rho}$.
For the purposes of our analysis we make a concrete choice for the Calabi-Yau
$2$-fold and we choose it to be the $4$-dimensional torus, which means
$\text{CY}_{2}\equiv\text{T}^{4}$. This solution was obtained after a
T-duality transformation in an AdS-coordinate from the mother massive type IIA
backgrounds that assume the form
$\text{AdS}_{3}\times\text{S}^{2}\times\text{T}^{4}\times\mathcal{I}_{\rho}$
and were originally developed in [10]. In the $\text{AdS}_{2}$ picture, $\psi$
is the T-dual coordinate and has a $\text{U}(1)$ isometry. We note that the
interval $\mathcal{I}_{\rho}$ is, also, bounded as required for the
$8$-dimensional manifold to be compact [20]. The NS-NS sector of the
$\text{AdS}_{2}$ solutions that we examine in this work is given by [20]:

$\displaystyle\text{d}s^{2}$
$\displaystyle=\frac{u}{\sqrt{h_{4}h_{8}}}\left(\frac{1}{4}\text{d}s^{2}_{\text{AdS}_{2}}+\frac{h_{4}h_{8}}{4h_{4}h_{8}+(u^{\prime})^{2}}\text{d}s^{2}_{\text{S}^{2}}\right)+\sqrt{\frac{h_{4}}{h_{8}}}\text{d}s^{2}_{\text{T}^{4}}+\frac{\sqrt{h_{4}h_{8}}}{u}(\text{d}\rho^{2}+\text{d}\psi^{2}),$
(2.1) $\displaystyle e^{-2\Phi}$
$\displaystyle=h_{8}^{2}+\frac{h_{8}(u^{\prime})^{2}}{4h_{4}},$ $\displaystyle
H_{3}$
$\displaystyle=\frac{1}{2}\text{vol}_{\text{S}^{2}}\wedge\text{d}\left(\rho-\frac{uu^{\prime}}{4h_{4}h_{8}+(u^{\prime})^{2}}\right)+\frac{1}{2}\text{vol}_{\text{AdS}_{2}}\wedge\text{d}\psi,$

where we wrote the above in the string frame. We have used $\Phi$ for the
dilaton, $H=\text{d}B_{2}$ is the NS-NS flux 3-form and the functions $h_{4}$,
$h_{8}$ and $u$ are functions of the $\rho$-coordinate222Note that in [20] the
authors have used both $h_{4}$ and $\hat{h}_{4}$ and in that context there is
a subtle difference between the two. We will be using $h_{4}$ without hat here
for convenience with the understanding that
$h^{\texttt{here}}_{4}=\hat{h}^{\texttt{there}}_{4}$.333Note that there is
another possibility where we have $h_{4}=h_{4}(\rho,z_{\text{CY}_{2}})$ and
$h_{8}=h_{8}(\rho,z_{\text{CY}_{2}})$[20]. We do not consider this kind of
solutions in this paper.. We have used $u^{\prime}$ to represent
$\partial_{\rho}u$, and the same is true for $h_{4}$ and $h_{8}$. The $\psi$
is the T-dual-coordinate taking values within $[0,2\pi]$ as we have already
explained. The torus is given by:

$\text{d}s^{2}_{\text{T}^{4}}=\text{d}\theta_{1}^{2}+\text{d}\theta_{2}^{2}+\text{d}\theta_{3}^{2}+\text{d}\theta_{4}^{2}.$
(2.2)

While it will not be necessary in our computations, for completeness we
present the R-R sector of the theory. It is given by :

$\displaystyle F_{1}$ $\displaystyle=h^{\prime}_{8}\text{d}\psi,$ (2.3)
$\displaystyle F_{3}$
$\displaystyle=-\frac{1}{2}\left(h_{8}-\frac{h^{\prime}_{8}u^{\prime}u}{4h_{8}h_{4}+(u^{\prime})^{2}}\right)\text{vol}_{\text{S}^{2}}\wedge\text{d}\psi+\frac{1}{4}\left(\text{d}\left(\frac{u^{\prime}u}{2h_{4}}\right)+2h_{8}\text{d}\rho\right)\wedge\text{vol}_{\text{AdS}_{2}},$
$\displaystyle F_{5}$
$\displaystyle=-(1+\ast)h^{\prime}_{4}\text{vol}_{\text{T}^{4}}\wedge\text{d}\psi=-h^{\prime}_{4}\text{vol}_{\text{T}^{4}}\wedge\text{d}\psi+\frac{h^{\prime}_{4}h_{8}u^{2}}{4h_{4}(4h_{8}h_{4}+(u^{\prime})^{2})}\text{vol}_{\text{AdS}_{2}}\wedge\text{vol}_{\text{S}^{2}}\wedge\text{d}\rho,$
$\displaystyle F_{7}$
$\displaystyle=\frac{4h_{4}^{2}h_{8}-uu^{\prime}h^{\prime}_{4}+h_{4}(u^{\prime})^{2}}{8h_{4}h_{8}+2(u^{\prime})^{2}}\text{vol}_{\text{T}^{4}}\wedge\text{vol}_{\text{S}^{2}}\wedge\text{d}\psi-\frac{4h_{4}h_{8}^{2}-uu^{\prime}h^{\prime}_{8}+h_{8}(u^{\prime})^{2}}{8h_{8}^{2}}\text{vol}_{\text{AdS}_{2}}\wedge\text{vol}_{\text{T}^{4}}\wedge\text{d}\rho,$
$\displaystyle F_{9}$
$\displaystyle=-\frac{h_{4}h^{\prime}_{8}u^{2}}{4h_{8}(4h_{8}h_{4}+(u^{\prime})^{2})}\text{vol}_{\text{AdS}_{2}}\wedge\text{vol}_{\text{T}^{4}}\wedge\text{vol}_{\text{S}^{2}}\wedge\text{d}\rho.$

It is obvious from equations 2.1 and 2.3 that the functions $u,h_{4}$ and
$h_{8}$ determine completely the supergravity class of solutions. These
functions are constrained such that the BPS equations and the Bianchi
identities are satisfied. One can explicitly check that satisfying the type
IIB equations of motion implies:

$h^{\prime\prime}_{4}(\rho)=0,\qquad h^{\prime\prime}_{8}(\rho)=0\,,$ (2.4)

and

$u^{\prime\prime}(\rho)=0.$ (2.5)

In the above, the first set of equations is a consequence of the Bianchi
identities, while the condition on the function $u(\rho)$ is the BPS equation.
In passing and for completeness, we mention that, in principle, we can have a
violation of the Bianchi identities at some specific points where explicit
brane sources are located. In this work we consider that the Bianchi
identities always hold.

Various cases for the definitions of $h_{4}$ and $h_{8}$ can be considered in
relevance to the corresponding ones in the mother $\text{AdS}_{3}$ theories
[53, 54, 55]. In this work, we focus on an infinite class of solutions defined
in a such a way that the these characteristic functions are piecewise
continuous. In particular they are given by:

$h_{4}(\rho)=\Upsilon\begin{cases}\beta_{0}\frac{\rho}{2\pi}&\quad
0\leq\rho\leq 2\pi\\\
\sum\limits_{i=0}\limits^{k-1}\beta_{i}+\beta_{k}\left(\frac{\rho}{2\pi}-k\right)&\quad
2\pi k<\rho\leq 2\pi(k+1),\quad k=1,\cdots,P-1\\\
\alpha_{P}+\alpha_{P}\left(-\frac{\rho}{2\pi}+P\right)&\quad 2\pi P<\rho\leq
2\pi(P+1),\end{cases}$ (2.6)
$h_{8}(\rho)=\begin{cases}\nu_{0}\frac{\rho}{2\pi}&\quad 0\leq\rho\leq 2\pi\\\
\sum\limits_{i=0}\limits^{k-1}\nu_{i}+\nu_{k}\left(\frac{\rho}{2\pi}-k\right)&\quad
2\pi k<\rho\leq 2\pi(k+1),\quad k=1,\cdots,P-1\\\
\mu_{P}+\mu_{P}\left(-\frac{\rho}{2\pi}+P\right)&\quad 2\pi P<\rho\leq
2\pi(P+1).\end{cases}$ (2.7)

In the above $\Upsilon$ is an overall normalisation444The precise relation is
$\hat{h}_{4}=\Upsilon h_{4}$, see e.g. [53].. The variables
$\alpha_{P},\beta_{k},\mu_{P},\nu_{k}$ with $(k=0,\cdots,P-1)$ are integration
constants. Demanding that the functions $h_{4}(\rho)$ and $h_{8}(\rho)$ are
continuous we have:

$\alpha_{P}=\sum_{i=0}^{k-1}\beta_{i},\qquad\mu_{P}=\sum_{i=0}^{k-1}\nu_{i}.$
(2.8)

We are interested in such a case that the $\rho$-coordinate defines a finite
interval [20] denoted by $\mathcal{I}_{\rho}$ within the range
$[0,\rho^{\ast}]$. It is convenient to set the value $\rho^{\ast}$ to be equal
to $\rho^{\ast}=2\pi(P+1)$, where $P$ is a large integer [20]. In the class of
solutions relevant to our studies, the functions $h_{4}(\rho)$ and
$h_{8}(\rho)$ should vanish at the endpoints of the $\mathcal{I}_{\rho}$,
namely $h_{4}(0)=h_{8}(0)=h_{4}(\rho^{\ast})=h_{4}(\rho^{\ast})=0$555We would
like to stress that there are more exotic choices that could be made for the
characteristic functions $h_{4}$ and $h_{8}$ related to the more exotic
behaviour of the corresponding functions in the mother $\text{AdS}_{3}$
theories [56]. We will not examine this kind of solutions here.. We show an
example of this behavior in figure 1:

Figure 1: An example of the characteristic functions $h_{4}(\rho),h_{8}(\rho)$
and $u$. Here we show a background for which the functions
$h_{4}(\rho),h_{8}(\rho)$ vanish at the endpoints of the interval
$\mathcal{I}_{\rho}$ and $u$ is being given by a constant.

It is clear from the BPS condition that the solution for the characteristic
function $u(\rho)$ can be at most linear in $\rho$. Hence, we distinguish
between two different behaviours that are equally well-defined:

  * •

$u(\rho)=u_{1}\rho$,

  * •

$u(\rho)=u_{0}$,

where in the above $u_{0,1}$ are just constants. With an eye towards the KK-
spectrum, we mention that the case $u(\rho)=u_{1}\rho$ has been analyzed in
[40], and thus in this work we focus on $u(\rho)=u_{0}$. To avoid cluttering
our equations, we use the shortcut $u_{0}=u$ in the following. This result to
the following simplification of the NS-NS sector:

$\displaystyle\text{d}s^{2}=\frac{u}{4\sqrt{h_{4}h_{8}}}(\text{d}s^{2}_{\text{AdS}_{2}}+\text{d}s^{2}_{\text{S}^{2}})+\sqrt{\frac{h_{4}}{h_{8}}}\text{d}s^{2}_{\text{T}^{4}}+\frac{\sqrt{h_{4}h_{8}}}{u}(\text{d}\rho^{2}+\text{d}\psi^{2}),$
(2.9) $\displaystyle e^{-2\Phi}=h_{8}^{2},$ $\displaystyle
H_{3}=\frac{1}{2}(\text{vol}_{\text{S}^{2}}\wedge\text{d}\rho+\text{vol}_{\text{AdS}_{2}}\wedge\text{d}\psi),$

and the relevant simplification of the R-R sectors readily follows, and we
refrain from presenting the simplified R-R sectors as it is not relevant in
our computations.

The supergravity background presented above has the following brane
realization, see table 1.

| $x^{0}$ | $x^{1}$ | $x^{2}$ | $x^{3}$ | $x^{4}$ | $x^{5}$ | $x^{6}$ | $x^{7}$ | $x^{8}$ | $x^{9}$
---|---|---|---|---|---|---|---|---|---|---
D1 | — | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | — | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
D3 | — | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | — | — | — | $\bullet$
D5 | — | — | — | — | — | — | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
D7 | — | — | — | — | — | $\bullet$ | — | — | — | $\bullet$
NS5 | — | — | — | — | — | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | —
F1 | — | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | —
Table 1: A visualization of the supergravity solution given by equations 2.1
and 2.3 in terms of branes. We have used (—) to denote that the brane extends
along that direction. A $\bullet$ represents a coordinate that is transverse
to the brane.

In table 1, $x^{0}$ is the direction associated with time in the ten-
dimensional spacetime. $x^{1},x^{2},x^{3},x^{4}$ are the directions of
$\text{T}^{4}$. $x^{5}$ is the $\rho$-dimension, and $x^{6},x^{7},x^{8}$ are
the three coordinates of the SO(3) symmetry. Finally, $x^{9}$ is the
$\psi$-internal.

###  2.2 The field theory interpretation

In this section we will discuss the basic features regarding the CFT which is
dual to the AdS2 background that we described in equations 2.1 and 2.3.
However, we find it more convenient to start by spelling out the basic
characteristics of the $2$-dimensional seed CFT theories.

As we have already mentioned, the AdS2 backgrounds that we examine in this
work are related, by means of a T-duality transformation, to the seed AdS3
solution of [54]. In [53] it was shown that these AdS3 solutions are the duals
of the IR fixed-point limit of the quiver theory presented in figure 2. In
particular, the quiver diagram depicted in figure 2 flows to an IR fixed-point
solution with dynamics that is captured by the AdS3 background solution of
[54].

Figure 2: The general 2d quiver field theory that is the seed of the CFT1
considered in this work. The black line denotes a (4,4) hypermultiplet, the
red line represents a (4,0) hypermultiplet and the blue line a (2,0)
multiplet. Each gauge node has $\mathcal{N}=(4,4)$ vector multiplets degree of
freedom.

Let us explain the basic building blocks of the quiver diagram depicted in
figure 2. We have used a circle to denote a gauge node in the quiver with the
associated group being SU. Hence, the $\beta_{0}$ in the circle means
$\text{SU}(\beta_{0})$ and likewise for the rest. We have, also, flavor groups
which are denoted by squares and hence a square with $F_{0}$ is the flavor
group $\text{SU}(F_{0})$. The black lines connecting the different gauge nodes
represent $\mathcal{N}=(4,4)$ hypermultiplets, and similarly, the red lines in
here denote $\mathcal{N}=(4,0)$ hypermultiplets. We have used blue lines to
denote $\mathcal{N}=(2,0)$ Fermi multiplets between two gauge nodes. Each
gauge node is equipped with $\mathcal{N}=(4,4)$ vector multiplets degrees of
freedom.

Note that there are two rows of flavour groups, and the same is true for the
color nodes. Crucially, the $F$’s and $\tilde{F}$’s are not independent of the
other characteristic numbers of the quiver theory. The reasoning, as was
explicitly observed and resolved in [53], is that the field theory is chiral
and hence we have to make sure to cancel gauge anomalies. This cancellation
was carried out explicitly in [53], leading to:

$F_{0}=\nu_{0}-\nu_{1}\,,\qquad\tilde{F}_{0}=\beta_{0}-\beta_{1}\,.$ (2.10)

We note that the above is an application to cancel out the anomalies at the
first note and that we have to insist on anomaly cancellation at at each gauge
node of the quiver. Similar relations can be derived straightforwardly, see
[53] for further details. It is worthwhile to note that the numbers in
equation 2.10 correspond to the number of color and flavor branes in the bulk
description of the field theory.

In [20] the authors argued that the type IIB solutions they derived are dual
to the IR limit of a $1$-dimensional quiver. This $1$-dimensional quiver
inherits its shape directly from the seed $2$-dimensional theory that is
depicted in figure 2.

The way to think about the dynamics and matter content of the superconformal
quantum mechanics theory is to regard it as the result of a dimensional
reduction along the spatial direction of the $2$-dimensional
$\mathcal{N}=(4,0)$ quivers that were previously derived in [53] and reviewed
above. There are some additional twisted multiplets obtained after the
dimensional reduction and we refer the reader to [20, Appendix B] for a
detailed exposition.

##  3 Holographic spin-2 fluctuations

We start from the perturbations of the metric. We find it most convenient for
the purposes of our analysis to re-write the metric in Einstein frame. To do
so, we multiply by a factor of the dilaton, more precisely
$e^{-\frac{\Phi}{2}}$. We will use the shorthand definitions for simplicity,

$\displaystyle\hat{g}_{ab}\text{d}z^{a}\text{d}z^{b}$
$\displaystyle=e^{-\frac{\Phi}{2}}\left(\frac{u}{4\sqrt{h_{4}h_{8}}}\text{d}s^{2}_{\text{S}^{2}}+\sqrt{\frac{h_{4}}{h_{8}}}\text{d}s^{2}_{\text{T}^{4}}+\frac{\sqrt{h_{4}h_{8}}}{u}(\text{d}\rho^{2}+\text{d}\psi^{2})\right)\,,$
(3.1) $\displaystyle f_{1}$ $\displaystyle=\frac{u}{4\sqrt{h_{4}h_{8}}}\,,$

such that the metric in the Einstein frame becomes:

$\text{d}s^{2}=e^{-\frac{\Phi}{2}}f_{1}\text{d}s^{2}_{\text{AdS}_{2}}+\hat{g}_{ab}\text{d}z^{a}\text{d}z^{b}.$
(3.2)

We consider fluctuations of the metric, which we will denote by $h$, along the
directions of the $\text{AdS}_{2}$ part of the $10$-dimensional background. In
particular:

$\text{d}s^{2}=e^{-\frac{\Phi}{2}}f_{1}(\text{d}s^{2}_{\text{AdS}_{2}}+h_{\mu\nu}(x,z)\text{d}x^{\mu}\text{d}x^{\nu})+\hat{g}_{ab}\text{d}z^{a}\text{d}z^{b}.$
(3.3)

We decompose $h_{\mu\nu}(x,z)$ into the transverse-traceless part in the bulk
and a scalar in the internal part like:

$h_{\mu\nu}(x,z)=h_{\mu\nu}^{(tt)}(x)\psi(z).$ (3.4)

Here the transverse-traceless tensor $h_{\mu\nu}^{(tt)}(x)$ satisfies:

$\Box_{\text{AdS}_{2}}^{(2)}h_{\mu\nu}^{(tt)}(x)=(M^{2}-2)h_{\mu\nu}^{(tt)}(x)=(\Box_{\text{AdS}_{2}}-2)h_{\mu\nu}^{(tt)}(x).$
(3.5)

The $\Box_{\text{AdS}_{2}}^{(2)}$ here is the Laplacian in $\text{AdS}_{2}$
acting on a rank-2 tensor and $\Box_{\text{AdS}_{2}}$ is the scalar Laplacian
in $\text{AdS}_{2}$ [57].

In order to make further progress, we recall that in [39] the authors showed
that the linearized Einstein equations can be reduced to the ten dimensional
Laplace equation for the $h_{\mu\nu}$ given by:

$\frac{1}{\sqrt{-g}}\partial_{M}\sqrt{-g}g^{MN}\partial_{N}h_{\mu\nu}=0.$
(3.6)

The above is the starting point of our analysis for the dynamics in the setup
we consider here. In the above, the capital indices ${M,N}$ denote directions
in the ten-dimensional background, and schematically are split into
$M=\\{\mu,a\\}$; that is in an index taking values in the non-compact space,
$\mu$, and another one along the compact directions. After separating equation
3.6 into the two different parts as described above and a straightforward
computation, we obtain:

$\hat{\Box}\psi(z)=-M^{2}\psi(z)\,,$ (3.7)

where we have defined the operator $\hat{\Box}$ to be given by:

$\hat{\Box}\equiv\frac{1}{\sqrt{\hat{g}}}\partial_{a}(f_{1}e^{-\frac{\Phi}{2}}\sqrt{\hat{g}}\hat{g}^{ab}\partial_{b}),$
(3.8)

for notational convenience and we have also used equation 3.5 in deriving the
above. As we see, by using the eigenvalues for the mode in the 2-dimensional
AdS subspace, we have, now, simplified the 10-dimensional equations of motion
to an 8-dimensional dynamical problem with a mass term. Upon evaluating the
contributions from the different parts of the internal space, we arrive at:

$\left(\nabla_{\text{S}^{2}}+\frac{u}{4h_{4}}\sum_{i=1}^{n}\partial_{\theta_{i}}^{2}+\frac{u^{2}}{4h_{4}h_{8}}(\partial_{\psi}^{2}+\partial_{\rho}^{2})+M^{2}\right)\psi=0.$
(3.9)

In order to solve the differential equation obtained above we decompose the
internal part of the metric perturbation $\psi$ into the appropriate
eigenstates along the different parts of the internal space in the following
manner:

$\psi=\sum_{\ell mnp}\Psi_{\ell mnp}Y_{\ell
m}e^{in\cdot\theta}e^{ip\cdot\psi}\,,$ (3.10)

where in the above
$n\cdot\theta=n_{1}\theta_{1}+n_{2}\theta_{2}+n_{3}\theta_{3}+n_{4}\theta_{4}$.
We proceed by inserting the decomposition given by equation 3.10 into equation
3.9 and we obtain:

$\frac{\text{d}^{2}}{\text{d}\rho^{2}}\Psi-\left(\frac{h_{8}n^{2}}{u}+p^{2}\right)\Psi+4\frac{M^{2}-\ell(\ell+1)}{u^{2}}h_{4}h_{8}\Psi=0\,,$
(3.11)

where in the above, to avoid cluttering our notation, we have suppressed the
quantum numbers as subscripts; $\Psi_{\ell mnp}=\Psi$.

This ordinary differential equation is a Sturm-Liouville problem. We can write
it in a more suggestive way, namely in the standard way of expressing any
Sturm-Liouville problem in the following way:

$\frac{\text{d}}{\text{d}\rho}\left(L(\rho)\frac{\text{d}}{\text{d}\rho}\right)\Psi+Q(\rho)\Psi=-\lambda
W(\rho)\Psi\,,$ (3.12)

with the characteristic quantities being given by:

$\displaystyle L(\rho)$ $\displaystyle=1\,,$ (3.13) $\displaystyle Q(\rho)$
$\displaystyle=-\left(\frac{h_{8}n^{2}}{u}+p^{2}\right)\,,$ $\displaystyle
W(\rho)$ $\displaystyle=h_{4}h_{8}\,,$ $\displaystyle\lambda$
$\displaystyle=4\frac{M^{2}-\ell(\ell+1)}{u^{2}}\,,$

and, of course, $\lambda$ being the eigenvalues of the Sturm-Liouville
problem. The coordinate $\rho$ is valued in the range of $[0,2\pi(P+1)]$,
where we remind the reader that $P$ is a large integer. We have already
mentioned that the defining functions vanish at the endpoints of
$\mathcal{I}_{\rho}$. This defines what is known in the mathematical
literature as a singular Sturm-Liouville problem. We discuss the boundary
conditions necessary to fully define the Sturm-Liouville problem in the next
section, where we also proceed to derive its solutions.

##  4 Unitarity bound and universal solutions

The next step we wish to make in this work is to obtain a bound for that mass
eigenvalue $M^{2}$ from the Sturm-Liouville problem we derived in the previous
section, see equation 3.11. From this bound and by using the basic holographic
relation between the mass and the conformal dimension, we will derive the
unitarity bound. To obtain an inequality for $M^{2}$ from the equation of
motion, we focus on a particular kind of solutions that is known in the
existing literature as “minimal universal class of solution”. Said class of
solutions is universal in the sense that it does not depend on the exact form
of $h_{4}$ and $h_{8}$.

Let us begin by deriving the mass inequality that will eventually lead to the
unitarity bound. To that end, we multiply the Sturm-Liouville problem, defined
in equation 3.11, by $\Psi$ and then integrate over the $\mathcal{I}_{\rho}$.
This yields:

$\int_{\mathcal{I}_{\rho}}\Psi\frac{\text{d}^{2}}{\text{d}\rho^{2}}\Psi\text{d}\rho-\int_{\mathcal{I}_{\rho}}\left(\frac{h_{8}n^{2}}{u}+p^{2}\right)\Psi^{2}\text{d}\rho+4\frac{M^{2}-\ell(\ell+1)}{u^{2}}\int_{\mathcal{I}_{\rho}}h_{4}h_{8}\Psi^{2}\text{d}\rho=0,$
(4.1)

where we remind the reader that the interval $\mathcal{I}_{\rho}$ is bounded
as $[0,\rho^{\ast}]$ and $\rho^{\ast}>0$. Firstly, we focus on the term that
contains the second derivative of $\Psi$. We perform integration by part, such
that we obtain:

$4\frac{M^{2}-\ell(\ell+1)}{u^{2}}\int_{0}^{\rho^{\ast}}h_{4}h_{8}\Psi^{2}\text{d}\rho+\Psi\Psi^{\prime}\big{|}_{0}^{\rho^{\ast}}=\int_{0}^{\rho^{\ast}}\left((\Psi^{\prime})^{2}+\left(\frac{h_{8}n^{2}}{u}+p^{2}\right)\Psi^{2}\right)\text{d}\rho.$
(4.2)

In the above, $\Psi^{\prime}$ is $\Psi$’s derivative of $\rho$.

Although the interval $\mathcal{I}_{\rho}$ does not possess an exact U(1)
isometry, we have seen that the $\rho^{\ast}$ endpoint is an integer multiple
of $2\pi$. In other words, it is topologically, but not metrically, equivalent
to $\text{S}^{1}$. Demanding single-valuedness for the wave-function leads to
the choice of the boundary conditions
$\Psi\big{|}_{\rho=0}=\Psi\big{|}_{\rho=\rho^{\ast}}$ and
$\Psi^{\prime}\big{|}_{\rho=0}=\Psi^{\prime}\big{|}_{\rho=\rho^{\ast}}$.
Focusing on the Hilbert space of such functions, it is clear that the term
$\Psi\Psi^{\prime}\big{|}_{0}^{\rho^{\ast}}$ vanishes.

Because $h_{4},h_{8},u,n,p$ are non-negative, we obtain straightforwardly:

$M^{2}\geq\ell(\ell+1).$ (4.3)

The next step is to obtain the “minimal universal class of solutions”. We have
already explained what “universal” is in this context. The “minimal” refers to
this fact that we wish to saturate the bound we obtained earlier for $M^{2}$,
see equation 4.3. Upon setting $M^{2}=\ell(\ell+1)$ the Sturm-Liouville
problem given by equations 3.12 and 3.13 reduces to:

$\frac{\text{d}^{2}}{\text{d}\rho^{2}}\Psi=0\,,$ (4.4)

that has the obvious solution

$\Psi=a\rho+\Psi_{0}\,,$ (4.5)

where in the above $a$ and $\Psi_{0}$ are constants of integration. Notice,
however, that we need to examine the compatibility of the general solution
with the boundary conditions imposed earlier. A simple inspection yields:

$\Psi=\Psi_{0}\,,$ (4.6)

namely the “minimal universal class of solutions” that describe these spin-2
fluctuations is given by just a constant $\Psi_{0}$. Note that this situation
is strikingly different compared to the solutions for a linear function
$u(\rho)$ obtained in [40]. Some commentary is in order. The modern intuition
regarding these spin-2 operators in AdS2 holography has been that the
additional fluxes in the NS-NS and R-R sectors of the theory as well as the
non-trivial warp factors in the geometry are sufficient to provide us with a
richer spectrum in the CFT [20, 23, 40, 58, 27, 59]. Our findings suggest,
however, that a more important role is played by the constraints arising from
the BPS conditions of the background supergravity theory. Our reasoning for
suggesting this comes from carefully observing the background for
$u=\text{constant}$ given by equation 2.9, for which we obviously have non-
trivial fluxes and warp factors and the only different with [40] is the
solution to the BPS equations for $u$.

The supergravity mode solutions that correspond to the metric perturbations we
studied in this work are dual to operators in the field theory that carry
conformal dimension $\Delta$. We can relate the mass, $M^{2}$, of the
supergravity states to the conformal dimension of the operators using the
standard AdS/CFT formula

$M^{2}=\Delta(\Delta-1).$ (4.7)

From the inequality we derived earlier, see equation 4.3, we can conclude for
the scaling weight of the operators a lower bound given by

$\Delta\geq\Delta_{\text{min}},\qquad\text{with}\qquad\Delta_{\text{min}}=\ell+1.$
(4.8)

We note that we can, in principle, study non-minimal solution, namely
solutions for which either $M^{2}>\ell(\ell+1)$ or there are non-trivial
excitations along T4 or $\mathcal{I}_{\rho}$. However, in order to study this
latter kind of solutions we need to make very specific choices for $h_{4}$ and
$h_{8}$ and we do not examine these solutions here.

##  5 The holographic central charge

The analysis we performed in the previous sections has provided us with all
the necessary information, in order to proceed to the computation of the
central charge for the class of solutions described in equation 2.9. This is
the purpose of this section.

Before we proceed with the computational part of this section, however, we
feel that some comments are in order. Defining the notion of the central
charge in a one-dimensional conformal field theory is a delicate and subtle
issue. The reasoning is quite straightforward. Let us consider a general one-
dimensional conformal field theory. The energy-momentum tensor, denoted by
$T_{\mu\nu}$, has only one component. Let us denote this component by
$T_{00}$. Then by virtue of the tracelessness property, this component
vanishes, $T_{00}=0$.

Taking the above into consideration, we wish to clarify that the quantity we
call the “holographic central charge” has the interpretation as the number of
vacuum states in the dual conformal quantum mechanics theory. Our reasoning
for choosing to name said quantity as the central charge is to remain
consistent with past literature.

In order to be concrete on how we make progress with this computation, we
borrow intuition from higher-dimensional examples where this methodology has
been successfully applied see, e.g. [46, 45, 41]. In these examples, using the
unitarity bound and the minimal universal class of solutions, the analogous
expressions to the ones we have obtained in the previous sections, the task at
hand was to compute the normalization of the two-point functions of the
operators $T_{\mu\nu}$. This is the operator that is sourced by the the
graviton fluctuations. For the case of the minimal solution with no excitation
in the internal space the supergravity mode is that of the massless graviton
fluctuation, which is dual to the the energy-momentum tensor. Using
holography, the normalization of the two-point function of the energy-momentum
tensor can be determined from the effective action for the lower-dimensional
graviton666By lower-dimensional graviton we mean that in an $\text{AdS}_{5}$
theory we have the five-dimensional graviton and likewise for other examples..
While some might argue that this is a flawed analogy, recent studies suggest
the same principles also apply to the $\text{AdS}_{2}$ case since the
computation of the holographic central charge agrees with the entanglement
entropy as they should [20, 40]. This gives us confidence on how to proceed
and indeed we will see that even when the function $u$ is simply given by a
constant the final result is consistent with the expression of the
entanglement entropy.

Our starting point is the Einstein-frame Type IIB action which we can write
schematically in the following form:

$S_{\text{IIB}}=\frac{1}{2\kappa_{10}^{2}}\int\sqrt{-g}(R+\cdots)\text{d}^{10}x$
(5.1)

Note that in the above $\cdots$ represent terms in the action that are
irrelevant to our computation.

Expanding the action given by equation 5.1 to quadratic order in the metric
fluctuations yields:

$\delta^{2}S=\frac{1}{\kappa_{10}^{2}}\int
h^{\mu\nu}\partial_{M}\sqrt{-g}g^{MN}\partial_{N}h_{\mu\nu}\text{d}^{10}x+\ldots$
(5.2)

where in the above $\ldots$ denote a boundary term. Expanding out the action
given by equation 5.2 explicitly and dropping the boundary term leads to:

$\delta^{2}S[h_{\mu\nu}]=\frac{1}{\kappa_{10}^{2}}\int\sqrt{-g_{\text{AdS}_{2}}}\sqrt{\hat{g}}h^{\mu\nu}\left(\Box_{\text{AdS}_{2}}^{(2)}+2+\hat{\Box}\right)h_{\mu\nu}\text{d}^{10}x$
(5.3)

We remind the reader that the operator $\hat{\Box}$ in the above relation is
defined in equation 3.8.

Expanding the $h_{\mu\nu}$ mode according to equation 3.10

$h_{\mu\nu}=\sum_{\ell mnp}(h_{\ell mnp}^{(tt)})_{\mu\nu}Y_{\ell m}\Psi_{\ell
mnp}e^{in\cdot\theta}e^{ip\cdot\psi}$ (5.4)

leads to

$\delta^{2}S=\sum_{\ell mnp}C_{\ell
mnp}\int_{\text{AdS}_{2}}\sqrt{-g_{\text{AdS}_{2}}}(h_{\ell
mnp}^{(tt)})^{\mu\nu}\left(\Box_{\text{AdS}_{2}}^{(2)}+2-\ell(\ell+1)\right)(h_{\ell
mnp}^{(tt)})_{\mu\nu}\text{d}^{2}x$ (5.5)

with spherical harmonics begin normalized as:

$\int Y_{\ell m}Y_{\ell^{\prime}m^{\prime}}\text{d}^{2}x=1$ (5.6)

and the coefficients $C_{lmnp}$ being given by:

$C_{\ell
mnp}=\frac{1}{\pi}\int_{\mathcal{I}_{\rho}}\sqrt{\hat{g}}(e^{-\tfrac{\Phi}{2}}f_{1})^{\tfrac{1}{2}}|\Psi_{\ell
mnp}|^{2}$ (5.7)

The integral in equation 5.7 is finite for the fluctuations we have considered
in this work, since they remain finite at every point of the background
spacetime. We specialize to the “minimal universal solution” obtained in
section 4. Unlike the situation in [20], in our case $\Psi$ is just a
constant. Setting all the quantum numbers to be equal to zero, we can set
$|\Psi_{0000}|^{2}=1$, such that we obtain:

$C_{0000}=\frac{1}{2\pi}\int_{0}^{\rho^{\ast}}h_{4}h_{8}\text{d}\rho,$ (5.8)

where in deriving the above we have used the standard results:

$\text{vol}_{\text{S}^{2}}=4\pi,\quad\text{vol}_{\text{T}^{4}}=(2\pi)^{4},\quad\text{vol}_{\mathcal{I}_{\psi}}=2\pi,\quad
2\kappa_{10}^{2}=(2\pi)^{7}.$ (5.9)

The above result, equation 5.8, agrees with the computation of holographic
entanglement entropy derived in [20], up to an irrelevant numerical factor, as
it should.

##  6 Discussion and outlook

In this paper we considered aspects of spin-2 perturbations around a warped
solution in the Type IIB theory of the schematic form
$\text{AdS}_{2}\times\text{S}^{2}\times\text{T}^{4}\times\mathcal{I}_{\psi}\times\mathcal{I}_{\rho}$.
The solution comes equipped with fluxes in the NS-NS and R-R sectors. They are
derived after a T-duality transformation from a seed
$\text{AdS}_{3}\times\text{S}^{2}\times\text{T}^{4}\times\mathcal{I}_{\rho}$
family of solutions in massive Type IIA supergravity. We have derived the
equations of motion governing the dynamics of these spin-2 states following
the blueprint of [39]. As expected from earlier studies, we can distinguish
them into two broad classes; the universal and non-universal. We note that the
universal class of solutions are independent of the defining functions of the
background, and thus are present in all 1-dimensional field theories dual to
the backgrounds examined here.

In this work we have explicitly computed the so-called “minimal universal
class of solutions”. This is a class of solutions that saturates the bound on
the mass. These supergravity states source operators in the dual field theory
of classical dimension $\Delta=\ell+1$, where $\ell$ is the quantum number on
the $\text{S}^{2}$; the holographic realization of the $SU(2)_{R}$ R-symmetry.

Using this class of solutions and the expansion of supergravity action we were
able to compute the expression for the holographic central charge. We checked
that it agrees with the computation of the holographic entanglement entropy
that was performed in [20] as it should.

We mention that our work is complementary to [40]. The difference in these two
works is the choice of the function $u(\rho)$. In particular, we chose it to
be a constant, while [40] examined a linear solution. The difference between
the different choices of $u(\rho)$ is reflected on the solutions of the metric
perturbations $h_{\mu\nu}(x,z)$. More specifically, we found that
$u(\rho)=\text{constant}$ leads to a solution given by a constant, while the
non-trivial functional dependence of $h_{\mu\nu}(x,z)$ is achieved only for
$u(\rho)$ being a linear function. Of course the expressions for the unitarity
bound and central charge agree in the two pictures.

Our results lead us to the following interpretation. In backgrounds with pure
$\text{AdS}_{2}$ factors, it was well-understood that the modes examined here
are non-trivial if and only if the mass is zero, see e.g [60, 18, 61, 62]. The
more recent explorations of $\text{AdS}_{2}$ supergravity solutions have been
suggesting that non-trivial warp factors and fluxes lead to a richer and more
exotic description [20, 40, 59]. However, our choice of $u$ led to trivial
spin-2 states without making the warp factors and fluxes equal to some
constants. Therefore, we conclude that a more pivotal role is played by the
supersymmetry conditions.

Before we finish this section we would like to discuss some interesting future
topics.

Based on [41, 40], we know that we can extract concrete information for the
seed $\text{AdS}_{3}$ theories from the daughter $\text{AdS}_{2}$ and vice
versa. More precisely, it was found that the expressions for the “minimal
universal class of solutions” and central charge of the theories were the
same. It would be interesting to examine if this connections persists for the
choice $u=\text{constant}$ by a direct computation. This would be strong
indicative suggestions on the importance of the BPS conditions in this type of
quiver theories.

Another interesting avenue of future work is to delineate more aspects of the
dual superconformal quantum mechanics following the recent progress of [56] in
the mother 2-dimensional theories. In that work the author found that in some
cases the dual quiver theories were anomalous and additional matter had to be
added, in order to cancel gauge anomalies. The way that this mechanism was
realized in supergravity was via bound states of D-branes. Interestingly,
these anomalous quiver arose for choices of both $u(\rho)$ given by a linear
function or a constant. Owing to the connections of the 2-dimensional CFT to
the conformal quantum mechanics, it would be very desirable to understand
these phenomena in the $\text{AdS}_{2}$ solutions.

Furthermore, another interesting question is the status of (non)integrability
and chaos in these theories. This is a rich subject, with well-developed
analytical and numerical methods. We cannot do justice to the literature, so
we mention as representative examples [63, 64, 65, 66, 67, 68, 69, 70, 71].
The gist of the approach is to carefully consider the embedding of a
fundamental string in the background and examine the resulting dynamics.

Finally, there has been recent activity in the construction of warped
$\text{AdS}_{2}$ solutions that have a holographic realization as defect
within higher-dimensional solutions [22, 23, 72, 27, 25]. We find it
interesting to address the question of the spectrum in those theories as well.

We hope to report on some of the above in the near future.

## Acknowledgments

We thank Carlos Nunez and Xinan Zhou for feedback on the draft and Kostas
Rigatos for suggesting this problem to us. We are, also, grateful to Xinan
Zhou and Kostas Rigatos for help with the English language in writing this
paper. The work of Shuo Zhang is supported by funds from the Kavli Institute
for Theoretical Sciences (KITS).

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