$U\rightarrow e^{h(\Phi)}U\,,\quad g^{a}(\Phi)\rightarrow
e^{-h(\Phi)}g^{a}(\Phi)\,.$ (D.19)

Such a residual symmetry may be used to re-absorb a Kähler transformation
(D.7)-(D.8).

The bosonic components of the Lagrangian (D.16) acquire a very simple form

$\begin{split}e^{-1}{\cal L}_{\rm bos}&=-\frac{1}{6}{\cal K}\,R-{\cal
K}_{a\bar{b}}D_{\mu}z^{a}\bar{D}^{\mu}\bar{z}^{b}+{\cal
K}_{a\bar{b}}f^{a}\bar{f}^{b}+\left(\hat{\cal W}_{a}f^{a}+{\rm
c.c.}\right)\,,\end{split}$ (D.20)

which is quite similar to the rigid case, as for example (2.41). However, here
we have redefined $f^{a}\equiv\bar{M}z^{a}-F^{a}_{Z}$ and introduced the
$U(1)$-covariant derivatives

$D_{\mu}z^{a}=\partial_{\mu}z^{a}+{\rm i}A_{\mu}z^{a}\,\qquad{\rm with}\quad
A_{\mu}=\frac{3{\rm i}}{2{\cal K}}({\cal K}_{a}\partial_{\mu}z^{b}-{\cal
K}_{\bar{b}}\partial_{\mu}\bar{z}^{b})\;.$ (D.21)

In order to pass to the Einstein frame, we isolate the compensator $U$ and
gauge-fix the super-Weyl invariance by setting [96]

$|U|^{\frac{2}{3}}=e^{\frac{{K(\Phi,\bar{\Phi})}}{3}}\,.$ (D.22)

In order to arrive at the bosonic components (D.11), one has to isolate the
compensator $u$ and split the index $a=(0,i)$, with $0$ associated to the
compensator $u$ and $i$ to the physical fields $\varphi^{i}$, namely we may
set $z^{a}=(u,u\varphi^{i})$. Then, the kinetic matrix ${\cal K}_{a\bar{b}}$
splits as

${\cal
K}_{a\bar{b}}=e^{-K}\begin{pmatrix}-\frac{1}{3}&\frac{K_{\bar{\jmath}}}{3}\\\
\frac{K_{i}}{3}&K_{i\bar{\jmath}}-\frac{1}{3}K_{i}K_{\bar{\jmath}}\end{pmatrix}\;.$
(D.23)

The Cremmer et al. potential instead stems after integrating out the auxiliary
fields $f^{a}$. This requires the inverse of ${\cal K}_{\bar{b}a}$, which is

${\cal
K}^{\bar{b}a}=e^{K}\begin{pmatrix}-3+K^{\bar{m}n}K_{n}K_{\bar{m}}&K^{\bar{m}i}K_{\bar{m}}\\\
K^{\bar{\jmath}n}K_{n}&K^{\bar{\jmath}i}\end{pmatrix}\;,$ (D.24)

From the discussion above it is clear that we may assign a couple of _super-
Weyl weights_ $(w_{1},\bar{w}_{2})$ to each physical quantity according to how
they transform under (D.13) and its anti-holomorphic counterpart. In Table
D.1, we have collected the super-Weyl weights of all the relevant quantities
which enter the supergravity actions considered in this Appendix and in
Chapter 4. Super-Weyl invariant physical quantities are built simply by
identifying combinations of objects in Table D.1 so that the sum of their
super-Weyl weights is zero.

| Supergravity quantity | super-Weyl weights
---|---|---
$U$ | chiral compensator | $(1,0)$
$Z^{a}$ | chiral superfields | $(1,0)$
$L^{\Lambda}$ | linear multiplets | $(\frac{1}{3},\frac{1}{3})$
${\cal K}(Z,\bar{Z},T)$ | kinetic function | $(\frac{1}{3},\frac{1}{3})$
${\cal F}(Z,\bar{Z},L)$ | Legendre transform | $(\frac{1}{3},\frac{1}{3})$
$E$ | Berezinian super-determinant | $(-\frac{1}{3},-\frac{1}{3})$
${\rm{d}}^{2}\Theta\,2{\cal E}$ | chiral superspace measure | $(-1,0)$
Table D.1: Super-Weyl weights $(w_{1},\bar{w}_{2})$ of the relevant quantities
which enter the Supergravity Lagrangian. The super-Weyl weight $w_{1}$
($\bar{w}_{2}$) refers to how the sections transform in the holomorphic (anti-
holomorphic) line bundle. Quantities which only depend on the physical fields
$\Phi^{i}$ and $\ell^{\Lambda}$ carry zero super-Weyl weights.

####  D.3 Super-Weyl invariant Lagrangians with chiral and linear multiplets

Let us now couple linear multiplets to the Lagrangians considered in the
previous section. The linear multiplets are endowed with the components

$L^{\Lambda}=\\{l^{\Lambda},{\cal
H}_{3}^{\Lambda}\\}\,,\quad\text{with}\quad\Lambda=1,\ldots,M\,,$ (D.25)

with $l^{\Lambda}$ real scalar fields and $H_{3}^{\Lambda}$ real field
strengths of gauge two-forms $B_{2}^{\Lambda}$. These transform under the
super-Weyl transformations as $L^{\Lambda}\rightarrow
e^{-2\Upsilon-2\bar{\Upsilon}}L^{\Lambda}$. The most general Lagrangian that
we can build out of the chiral multiplets (D.14) and linear multiplets (D.25)
is

$\begin{split}{\cal L}&=\int{\rm{d}}^{4}\theta\,E\,{\cal
F}(Z,\bar{Z};L)+\left(\int{\rm{d}}^{2}\theta\,2{\cal E}\,{\cal W}(Z)+{\rm
c.c.}\right)\,.\end{split}$ (D.26)

As in (4.107), the kinetic function ${\cal F}(Z,\bar{Z};L)$ can be seen as
Legendre transform of a kinetic function ${\cal K}(Z,\bar{Z};{\rm Im\hskip
1.00006pt}T)$

${\cal K}(Z,\bar{Z};{\rm Im\hskip 1.00006pt}T)={\cal F}(Z,\bar{Z};L)-{\cal
F}_{\Lambda}L^{\Lambda}\,,$ (D.27)

with ${\rm Im\hskip 1.00006pt}T_{\Lambda}$ dual to $L^{\Lambda}$ as in
(4.106). We require that ${\cal F}$ satisfies the same homogeneity condition
as ${\cal K}$ in (D.18), namely

${\cal F}(\lambda
Z,\bar{\lambda}\bar{Z};|\lambda|^{\frac{2}{3}}L)=|\lambda|^{\frac{2}{3}}{\cal
F}(Z,\bar{Z};L)$ (D.28)

The bosonic components of (D.26) are

$\begin{split}e^{-1}{\cal L}_{\rm bos}&=-\frac{1}{6}\tilde{\cal F}\,R-{\cal
F}_{a\bar{b}}D_{\mu}z^{a}\bar{D}^{\mu}\bar{z}^{b}+\frac{1}{4}{\cal
F}_{\Lambda\Sigma}\partial_{\mu}l^{\Lambda}\partial^{\mu}l^{\Sigma}+\frac{1}{4\cdot
3!}{\cal F}_{\Lambda\Sigma}{\cal H}_{\mu\nu\rho}^{\Lambda}{\cal
H}^{\Sigma\mu\nu\rho}\\\ &\quad\,+\left(\frac{{\rm i}}{2\cdot 3!}{\cal
F}_{\bar{a}\Sigma}\varepsilon^{\mu\nu\rho\sigma}{\cal
H}^{\Sigma}_{\nu\rho\sigma}D_{\mu}\bar{z}^{a}+{\rm c.c.}\right)+{\cal
F}_{a\bar{b}}f^{a}\bar{f}^{b}+\left({\cal W}_{a}f^{a}+{\rm
c.c.}\right)\end{split}$ (D.29)

where we have defined $\tilde{\cal F}={\cal F}-l^{\Lambda}{\cal F}_{\Lambda}$
and introduced the $U(1)$–covariant derivative

$\begin{split}D_{\mu}z^{a}=\partial_{\mu}z^{a}+{\rm
i}A_{\mu}z^{a}\;,\end{split}$ (D.30)

with

$\begin{split}A_{\mu}&=\frac{3}{2(\tilde{\cal F}-\tilde{\cal
F}_{\Lambda}l^{\Lambda})}\left[{\rm i}(\tilde{\cal
F}_{a}\partial_{\mu}z^{a}-\bar{\tilde{\cal
F}}_{\bar{a}}\partial_{\mu}\bar{z}^{a})+\frac{1}{3!}\tilde{\cal
F}_{\Lambda}\varepsilon_{\mu\nu\rho\sigma}{\cal
H}^{\Sigma\nu\rho\sigma}\right]\;.\end{split}$ (D.31)

In order to gauge fix the super-Weyl invariance, we need to isolate the
compensator from the chiral multiplets $Z^{a}$ and linear multiplets
$L^{\Lambda}$ as

$Z^{a}=Ug^{a}(\Phi)\,,\qquad L^{\Lambda}=\tilde{\cal F}\tilde{L}^{\Lambda}$
(D.32)

with $\tilde{\cal F}={\cal F}-{\cal F}_{\Lambda}L^{\Lambda}$ and, afterwards,
set

$|U|^{\frac{2}{3}}=e^{\frac{{\tilde{F}(\Phi,\bar{\Phi};L)}}{3}}\,,$ (D.33)

where $\tilde{F}=F-\ell^{\Lambda}F_{\Lambda}$, with $F(\Phi,\bar{\Phi};L)$ the
Legendre transform of the Kähler potential $K(\Phi,\bar{\Phi},{\rm Im\hskip
1.00006pt}T)$.

We now explain, with further details, the gauge-fixing procedure adopted in
Section 4.4.1 in order to pass from (4.110) to (4.115). In principle,
implementing the gauge-fixing in (D.33) allows one to compute the Lagrangian
in the Einstein frame. The gauge-fixing relations (D.32) may suggest that the
proper linear multiplets to consider, after the gauge-fixing, are the
$\tilde{L}^{\Lambda}$ obtained ‘extracting’ the compensator form
$L^{\Lambda}$. This is a necessary first step, but this is not the end of the
story. In order to get kinetic terms in a compact form, it is then convenient
to redefine the lowest components $\tilde{l}^{\Lambda}$ of
$\tilde{L}^{\Lambda}$ as
$\tilde{l}^{\Lambda}=e^{-\frac{1}{3}\tilde{F}}\ell^{\Lambda}$. Comparing this
change of variables with (D.32)-(D.33), it is clear that we are somehow
‘undoing’ the super-Weyl rescaling for the lowest component of $L$. Restoring
the Planck mass, passing from our starting $l^{\Lambda}$ to $\ell^{\Lambda}$
simply results in (4.113). In order to express the kinetic matrices in terms
of the Legendre transform of the Kähler potential (4.111), we preliminary
notice that

$K(\varphi,\bar{\varphi};{\rm Im\hskip
1.00006pt}t)=F(\varphi,\bar{\varphi};\ell)-\ell^{\Lambda}F_{\Lambda}(\varphi,\bar{\varphi};\ell)\equiv\tilde{F}(\varphi,\bar{\varphi};\ell)\;.$
(D.34)

We then get the following identifications between the derivatives of the
Kähler potential and its Legendre transform111We notice that partial
derivatives of $K$ with respect to $\varphi^{i}$ are computed for constant
${\rm Im\hskip 1.00006pt}t_{\Lambda}$, while those of $F$ for constant
$\ell^{\Lambda}$.

$\displaystyle K_{i}=\frac{\partial K}{\partial\varphi^{i}}\Big{|}_{{\rm
Im\hskip 0.70004pt}t_{\Lambda}}$ $\displaystyle=\frac{\partial
F}{\partial\varphi^{i}}\Big{|}_{\ell^{\Lambda}}=F_{i}\;,$ (D.35)
$\displaystyle
K_{i\bar{\jmath}}+F^{\Lambda\Sigma}F_{\Lambda_{i}}F_{\Sigma\bar{\jmath}}$
$\displaystyle=\frac{\partial^{2}F}{\partial\varphi^{i}\partial\bar{\varphi}^{\bar{\jmath}}}\Big{|}_{\ell^{\Lambda}}=F_{i\bar{\jmath}}\;.$

Now, from

$\tilde{\cal F}\equiv{\cal F}-{\cal
F}_{\Lambda}l^{\Lambda}=-3e^{-\frac{\tilde{F}}{3}}\equiv-3e^{-\frac{F-F_{\Lambda}\ell^{\Lambda}}{3}}$
(D.36)

we get the following identities

$\displaystyle-\frac{3}{\tilde{\cal F}}\left({\cal
F}_{a\bar{b}}-\frac{\tilde{\cal F}_{a}\tilde{\cal F}_{\bar{b}}}{\tilde{\cal
F}-\tilde{\cal F}_{\Pi}l^{\Pi}}\right)$ $\displaystyle=F_{a\bar{b}}\;,$ (D.37)
$\displaystyle-\frac{\tilde{\cal F}}{3}\left({\cal
F}_{\Lambda\Sigma}-\frac{\tilde{\cal F}_{\Lambda}\tilde{\cal
F}_{\Sigma}}{\tilde{\cal F}-\tilde{\cal F}_{\Pi}l^{\Pi}}\right)$
$\displaystyle=F_{\Lambda\Sigma}\;,$ $\displaystyle{\cal
F}_{a\Sigma}-\frac{{\tilde{\cal F}}_{a}\tilde{\cal F}_{\Sigma}}{\tilde{\cal
F}-\tilde{\cal F}_{\Pi}l^{\Pi}}$ $\displaystyle=F_{a\Lambda}\;.$

which relate the super-Weyl invariant quantities with the gauge-fixed ones.
These relations can be rewritten so as to get, more explicitly

${\cal F}_{\Lambda\Sigma}=-\frac{3}{\tilde{\cal
F}}\left(F_{\Lambda\Sigma}-\frac{\tilde{F}_{\Lambda}\tilde{F}_{\Sigma}}{3-\ell^{\Pi}\tilde{F}_{\Pi}}\right)$
(D.38)

whence we can compute

${\cal F}^{\Lambda\Sigma}=-\frac{\tilde{\cal
F}}{3}\left(F^{\Lambda\Sigma}+\frac{1}{3}\ell^{\Lambda}\ell^{\Sigma}\right)$
(D.39)

and

$\displaystyle{\cal F}_{\Lambda i}$ $\displaystyle=\frac{1}{u}F_{\Lambda
i}-\frac{1}{u}\frac{\tilde{F}_{\Lambda}\tilde{F}_{i}}{3-\ell^{\Pi}\tilde{F}_{\Pi}}\;,$
(D.40) $\displaystyle{\cal F}_{\Lambda 0}$
$\displaystyle=\frac{1}{u}\frac{\tilde{F}_{\Lambda}}{3-\ell^{\Pi}\tilde{F}_{\Pi}}\;,$

where the $0$th-index is associated to the compensator $u$, while the $i$s to
the physical fields $\varphi^{i}$. Moreover, from (D.27), we may further
relate the derivatives of ${\cal K}$ with those of the Legendre transform
${\cal F}$ as

$\displaystyle{\cal F}_{a}|_{l^{\Lambda}}$ $\displaystyle={\cal K}_{a}|_{{\rm
Im\hskip 0.70004pt}t_{\Lambda}}\,,$ (D.41) $\displaystyle{\cal
F}_{a\bar{b}}|_{l^{\Lambda}}$ $\displaystyle={\cal K}_{a\bar{b}}|_{{\rm
Im\hskip 0.70004pt}t_{\Lambda}}-{\cal K}^{\Lambda\Sigma}\frac{\partial{\rm
Im\hskip 1.00006pt}t_{\Lambda}}{\partial z^{a}}\frac{\partial{\rm Im\hskip
1.00006pt}t_{\Sigma}}{\partial\bar{z}^{b}}={\cal K}_{a\bar{b}}|_{{\rm Im\hskip
0.70004pt}t_{\Lambda}}+{\cal F}^{\Lambda\Sigma}{\cal F}_{\Lambda a}{\cal
F}_{\Sigma\bar{b}}$

where we have used

$\displaystyle{\cal F}_{\Lambda\Sigma}{\cal K}^{\Sigma\Pi}$
$\displaystyle=-4\delta_{\Lambda}{}^{\Pi}\quad\Rightarrow\quad{\cal
K}^{\Lambda\Sigma}|_{l^{\Pi}}=-4{\cal F}^{\Lambda\Sigma}|_{{\rm Im\hskip
0.70004pt}t_{\Pi}}$ (D.42) $\displaystyle{\cal K}^{\Lambda}_{a}$
$\displaystyle=-{\cal K}^{\Lambda\Sigma}\frac{\partial{\rm Im\hskip
1.00006pt}t_{\Sigma}}{\partial z^{a}}$ $\displaystyle{\cal F}_{\Lambda a}$
$\displaystyle=-2\frac{\partial{\rm Im\hskip 1.00006pt}t_{\Lambda}}{\partial
z^{a}}$

These relations, combined with (D.35), allow us to rewrite

${\cal
F}_{i\bar{\jmath}}=\frac{e^{-\frac{\tilde{F}}{3}}}{|u|^{\frac{4}{3}}}\left(F_{i\bar{\jmath}}-\frac{\tilde{F}_{i}\tilde{F}_{\bar{\jmath}}}{3-\ell^{\Pi}\tilde{F}_{\Pi}}\right)$
(D.43)

whence

${\cal
F}_{a\bar{b}}=\frac{e^{-\frac{\tilde{F}}{3}}}{|u|^{\frac{4}{3}}}\begin{pmatrix}-\frac{1}{3-\ell^{\Pi}\tilde{F}_{\Pi}}&\frac{\tilde{F}_{\bar{\jmath}}}{3-\ell^{\Pi}\tilde{F}_{\Pi}}\\\
\frac{\tilde{F}_{i}}{3-\ell^{\Pi}\tilde{F}_{\Pi}}&F_{i\bar{\jmath}}-\frac{\tilde{F}_{i}\tilde{F}_{\bar{\jmath}}}{3-\ell^{\Pi}\tilde{F}_{\Pi}}\end{pmatrix}$
(D.44)

We can now get the Lagrangian in the gauge-fixed Einstein frame. Integrating
out the fields $f^{a}$, extracting the compensator $u$ and using (D.44),
(D.40) and (D.38), allows us to finally rewrite the Lagrangian (D.29) as

$\begin{split}e^{-1}{\cal L}_{\rm
bos}&=\frac{1}{2}R-F_{i\bar{\jmath}}\partial\varphi^{i}\bar{\partial}\bar{\varphi}^{\bar{\jmath}}+\frac{1}{4}F_{\Lambda\Sigma}\left(\partial_{\mu}\ell^{\Lambda}\partial^{\mu}\ell^{\Sigma}+\frac{1}{3!}{\cal
H}_{\mu\nu\rho}^{\Lambda}{\cal H}^{\Sigma\mu\nu\rho}\right)\\\
&\quad\,+\left\\{\frac{{\rm i}}{2\cdot
3!}F_{\bar{\imath}\Sigma}\varepsilon^{\mu\nu\rho\sigma}{\cal
H}^{\Sigma}_{\nu\rho\sigma}\partial_{\mu}\bar{\varphi}^{\bar{\imath}}+{\rm
c.c.}\right\\}\\\
&\quad\,-e^{\tilde{F}}\left[F^{\bar{\jmath}i}D_{i}W\bar{D}_{\bar{\jmath}}\bar{W}-(3-\ell^{\Lambda}\tilde{F}_{\Lambda})|W|^{2}\right]\end{split}$
(D.45)

where the Kähler covariant derivatives are now given by

$D_{i}=\partial_{i}+\tilde{F}_{i}\;.$ (D.46)

####  D.4 Super-Weyl invariant Lagrangians with three-form and gauged linear
multiplets

Finally, let us consider the more general case where some chiral superfields
are endowed with gauge three-forms, namely they are constrained as in (4.30),
and also linear multiplets are present. The linear multiplets can also be
gauged by the three-form potentials as in (4.125). The most general Lagrangian
including these ingredients (plus other hidden multiplets) is given by
(4.128). However, in order to obtain its expression in bosonic components, it
is convenient to start from the master Lagrangian (4.118). After integrating
out $T_{\Sigma}$ from (4.118), we arrive at a master Lagrangian of the form

$\begin{split}{\cal L}&=\int{\rm{d}}^{4}\theta\,E{\cal
F}(Z,\bar{Z},\hat{L})+\left(\int{\rm{d}}^{2}\Theta\,2{\cal E}\,\hat{\cal
W}(Z)+\text{c.c.}\right)\\\ &\quad\,+\left[\int{\rm{d}}^{2}\Theta\,2{\cal
E}\,X_{A}{\cal V}^{A}(Z)+\frac{{\rm i}}{8}\int{\rm{d}}^{2}\Theta\,2{\cal
E}\,(\bar{\cal D}^{2}-8{\cal
R})(X_{A}-\bar{X}_{A})P^{A}+\text{c.c.}\right]\,.\end{split}$ (D.47)

We recall that $X_{A}$ is a chiral superfield, with bosonic components
$\\{x_{A},F^{(X)}_{A}\\}$ and ${\cal V}^{A}$ are homogeneous of degree one as
in (4.21). The homogeneity properties for ${\cal F}$ and $\hat{W}$ are the
same as in (D.18)-(D.28).

The bosonic components of the Lagrangian (D.47) are

$\small{\begin{split}e^{-1}{\cal L}_{\rm bos}&=-\frac{1}{6}\tilde{\cal
F}\,R-{\cal F}_{a\bar{b}}Dz^{a}\bar{D}\bar{z}^{b}+\frac{1}{4}{\cal
F}_{\Lambda\Sigma}\partial_{\mu}l^{\Lambda}\partial^{\mu}l^{\Sigma}+\frac{1}{4\cdot
3!}{\cal F}_{\Lambda\Sigma}\hat{\cal H}_{\mu\nu\rho}^{\Lambda}\hat{\cal
H}^{\Sigma\mu\nu\rho}\\\ &\quad\,+\left(\frac{{\rm i}}{2\cdot 3!}{\cal
F}_{\bar{a}\Sigma}\varepsilon^{\mu\nu\rho\sigma}\hat{\cal
H}^{\Sigma}_{\nu\rho\sigma}D_{\mu}\bar{z}^{\bar{a}}+{\rm c.c.}\right)+\\\
&\quad\,+{\cal F}_{a\bar{b}}f^{a}\bar{f}^{b}+{\cal
F}_{\Lambda\Sigma}c_{A}^{\Lambda}c_{B}^{\Sigma}{\cal V}^{A}\bar{\cal V}^{B}\\\
&\quad\,+\left\\{-\frac{{\rm i}}{2}{\cal F}_{\Lambda}c_{A}^{\Lambda}{\cal
V}^{A}_{a}f^{a}-{\rm i}{\cal F}_{\Lambda b}f^{b}c_{A}^{\Lambda}{\cal
V}^{A}+\hat{\cal W}_{a}f^{a}+{\rm c.c.}\right\\}\\\
&\quad\,+\Big{[}\left(F^{(X)}_{A}-\bar{M}x_{A}\right)\left(-s^{A}+{\cal
V}^{A}_{a}z^{a}\right)\\\ &\quad\quad\,+x_{A}{\cal
V}^{A}_{a}F^{a}_{Z}-\frac{{\rm i}}{2}x_{A}d^{A}-\frac{1}{2\cdot
3!e}\varepsilon^{\mu\nu\rho\sigma}\partial_{\mu}x_{A}A^{A}_{\nu\rho\sigma}-x_{A}{\rm
Re\hskip 0.90005pt}(\bar{M}s^{A})+{\rm c.c.}\Big{]}\,,\end{split}}$ (D.48)

where now $\hat{\cal H}^{\Lambda}_{\mu\nu\rho}={\cal
H}^{\Lambda}_{\mu\nu\rho}+c^{\Lambda}_{A}A_{\mu\nu\rho}^{A}$ and with the same
covariant derivative as (D.30), modulo the replacement ${\cal
H}^{\Lambda}\to\hat{\cal H}^{\Lambda}$.

We now proceed as follows. First, we integrate out the real auxiliary fields
$d^{A}$, which constrain $x_{A}$ to be real, and $F^{(X)}_{A}$, identifying
${\cal V}^{A}(z)=s^{A}$ as in (4.31a). Then, (D.48) becomes

$\small{\begin{split}e^{-1}{\cal L}_{\rm bos}&=-\frac{1}{6}\tilde{\cal
F}\,R-{\cal F}_{a\bar{b}}Dz^{a}\bar{D}\bar{z}^{b}+\frac{1}{4}{\cal
F}_{\Lambda\Sigma}\partial_{\mu}l^{\Lambda}\partial^{\mu}l^{\Sigma}+\frac{1}{4\cdot
3!}{\cal F}_{\Lambda\Sigma}\hat{\cal H}_{\mu\nu\rho}^{\Lambda}\hat{\cal
H}^{\Sigma\mu\nu\rho}\\\ &\quad\,+\left(\frac{{\rm i}}{2\cdot 3!}{\cal
F}_{\bar{a}\Sigma}\varepsilon^{\mu\nu\rho\sigma}\hat{\cal
H}^{\Sigma}_{\nu\rho\sigma}D_{\mu}\bar{z}^{\bar{a}}+{\rm c.c.}\right)\\\
&\quad\,+{\cal F}_{a\bar{b}}f^{a}\bar{f}^{b}+{\cal
F}_{\Lambda\Sigma}c_{A}^{\Lambda}c_{B}^{\Sigma}{\cal V}^{A}\bar{\cal V}^{B}\\\
&\quad\,+\left\\{-\frac{{\rm i}}{2}{\cal F}_{\Lambda}c_{A}^{\Lambda}{\cal
V}^{A}_{a}f^{a}-{\rm i}{\cal F}_{\Lambda b}f^{b}c_{A}^{\Lambda}{\cal
V}^{A}+\hat{\cal W}_{a}f^{a}+{\rm c.c.}\right\\}\\\ &\quad\,+\Big{[}x_{A}{\cal
V}^{A}_{a}f^{a}-\frac{1}{2\cdot
3!e}\varepsilon^{\mu\nu\rho\sigma}\partial_{\mu}x_{A}\,A^{A}_{\nu\rho\sigma}+{\rm
c.c.}\Big{]}\,.\end{split}}$ (D.49)

Then, we further integrate out the auxiliary fields $f^{a}$ of the chiral
multiplets $Z^{a}$ via

$\bar{f}^{b}=-{\cal F}^{a\bar{b}}\left(x_{A}{\cal V}^{A}_{a}-\frac{{\rm
i}}{2}{\cal F}_{\Lambda}c_{A}^{\Lambda}{\cal V}^{A}_{a}-{\rm i}{\cal
F}_{a\Lambda}c_{E}^{\Lambda}{\cal V}^{E}+\hat{\cal W}_{a}\right)$ (D.50)

and, subsequently, the real Lagrange multipliers $x_{A}$. We then arrive at
the super-Weyl invariant Lagrangian

$\begin{split}e^{-1}{\cal L}_{\rm bos}&=-\frac{1}{6}\tilde{\cal F}\,R-{\cal
F}_{a\bar{b}}Dz^{a}\bar{D}\bar{z}^{b}+\frac{1}{4}{\cal
F}_{\Lambda\Sigma}\partial_{\mu}l^{\Lambda}\partial^{\mu}l^{\Sigma}+\frac{1}{4\cdot
3!}{\cal F}_{\Lambda\Sigma}\hat{\cal H}_{\mu\nu\rho}^{\Lambda}\hat{\cal
H}^{\Sigma\mu\nu\rho}\\\ &\quad\,+\left(\frac{{\rm i}}{2\cdot 3!}{\cal
F}_{\bar{a}\Sigma}\varepsilon^{\mu\nu\rho\sigma}\hat{\cal
H}^{\Sigma}_{\nu\rho\sigma}D_{\mu}\bar{z}^{\bar{a}}+{\rm
c.c.}\right)+e^{-1}{\cal L}_{\text{3-forms}}\,,\end{split}$ (D.51)

where the three-form Lagrangian ${\cal L}_{\text{three-forms}}$ can be recast
as in (4.44), with

$\displaystyle T^{AB}(z,\bar{z})$ $\displaystyle\equiv 2\,{\rm Re\hskip
1.00006pt}\left({\cal F}^{\bar{b}a}\,{\cal V}_{a}^{A}\bar{\cal
V}_{\bar{b}}^{B}\right)\,,$ (D.52a) $\displaystyle h^{A}(z,\bar{z})$
$\displaystyle\equiv 2{\rm Re\hskip 1.00006pt}\left[{\cal
F}^{\bar{b}a}\left(\bar{\hat{\cal W}}_{\bar{b}}+\frac{{\rm i}}{2}{\cal
F}_{\Lambda}c_{B}^{\Lambda}\bar{\cal V}^{B}_{\bar{b}}+{\rm i}{\cal
F}_{\Lambda\bar{b}}c_{F}^{\Lambda}\bar{\cal V}^{\bar{F}}\right){\cal
V}_{a}^{A}\right]\,,$ (D.52b)
$\displaystyle\begin{split}\hat{V}(z,\bar{z})&\equiv{\cal
F}^{\bar{b}a}\left(\hat{\cal W}_{a}-\frac{{\rm i}}{2}{\cal
F}_{\Lambda}c_{A}^{\Lambda}{\cal V}^{A}_{a}-{\rm i}{\cal F}_{\Lambda
a}c_{E}^{\Lambda}{\cal V}^{E}\right)\\\ &\quad\quad\times\left(\bar{\hat{\cal
W}}_{\bar{b}}+\frac{{\rm i}}{2}{\cal F}_{\Lambda}c_{B}^{\Lambda}\bar{\cal
V}^{B}_{\bar{b}}+{\rm i}{\cal F}_{\Lambda\bar{b}}c_{F}^{\Lambda}\bar{\cal
V}^{\bar{F}}\right)\\\ &\quad\,-{\cal
F}_{\Lambda\Sigma}c_{A}^{\Lambda}c_{B}^{\Sigma}{\cal V}^{A}\bar{\cal
V}^{B}\,.\end{split}$ (D.52c)

We can now proceed to gauge-fixing the super-Weyl invariance by following the
same procedure described in the previous subsection. After having imposed the
Einstein frame condition (D.33), we arrive at

$\begin{split}e^{-1}{\cal L}_{\rm
bos}&=\frac{1}{2}R-F_{i\bar{\jmath}}\partial\phi^{i}\bar{\partial}\bar{\phi}^{\bar{\jmath}}+\frac{1}{4}F_{\Lambda\Sigma}\left(\partial_{\mu}\ell^{\Lambda}\partial^{\mu}\ell^{\Sigma}+\frac{1}{3!}\hat{\cal
H}_{\mu\nu\rho}^{\Lambda}\hat{\cal H}^{\Sigma\mu\nu\rho}\right)\\\
&\quad\,+\left\\{\frac{{\rm i}}{2\cdot
3!}F_{\bar{\imath}\Sigma}\varepsilon^{\mu\nu\rho\sigma}\hat{\cal
H}^{\Sigma}_{\nu\rho\sigma}\partial_{\mu}\bar{\phi}^{\bar{\imath}}+{\rm
c.c.}\right\\}+e^{-1}{\cal L}_{\text{3-forms}}\,,\end{split}$ (D.53)

where the three-form Lagrangian has the same form as (4.44) with

$\displaystyle T^{AB}$ $\displaystyle\equiv 2e^{\tilde{F}}\,{\rm Re\hskip
0.90005pt}\left({F}^{i\bar{\jmath}}\,D_{i}\Pi^{A}\bar{D}_{\bar{\jmath}}\bar{\Pi}^{B}-(3-\ell^{\Lambda}\tilde{F}_{\Lambda})\Pi^{A}\bar{\Pi}^{B}\right)\,,$
(D.54a) $\displaystyle\begin{split}h^{A}&\equiv 2e^{\tilde{F}}{\rm Re\hskip
0.90005pt}\Bigg{[}{F}^{i\bar{\jmath}}\,\left(D_{\bar{\jmath}}\bar{\hat{W}}+\frac{{\rm
i}}{2}F_{\Lambda}c_{B}^{\Lambda}D_{\bar{\jmath}}\bar{\Pi}^{B}+{\rm
i}F_{\Lambda\bar{\jmath}}c_{F}^{\Lambda}\bar{\Pi}^{\bar{F}}\right)D_{i}\Pi^{A}\\\
&\qquad\qquad\,-(3-\ell^{\Lambda}\tilde{F}_{\Lambda})\left(\overline{\hat{W}}+\frac{{\rm
i}}{2}F_{\Lambda}c_{D}^{\Lambda}\bar{\Pi}^{D}\right)\Pi^{A}-{\rm
i}\tilde{F}_{\Lambda}c_{F}^{\Lambda}\bar{\Pi}^{\bar{F}}\Pi^{A}\Bigg{]}\,,\end{split}$
(D.54b) $\displaystyle\begin{split}\hat{V}&\equiv
e^{\tilde{F}}{F}^{\bar{\jmath}i}\left(D_{i}{\hat{W}}-\frac{{\rm
i}}{2}F_{\Lambda}c_{A}^{\Lambda}D_{i}\Pi^{A}-{\rm i}F_{\Lambda
i}c_{F}^{\Lambda}\Pi^{F}\right)\left(D_{\bar{\jmath}}\bar{\hat{W}}+\frac{{\rm
i}}{2}F_{\Lambda}c_{B}^{\Lambda}D_{\bar{\jmath}}\bar{\Pi}^{B}+{\rm
i}F_{\Lambda\bar{\jmath}}c_{F}^{\Lambda}\bar{\Pi}^{\bar{F}}\right)\\\
&\quad\,-(3-\ell^{\Lambda}\tilde{F}_{\Lambda})e^{\tilde{F}}\left|\hat{W}-\frac{{\rm
i}}{2}F_{\Lambda}c_{A}^{\Lambda}\Pi^{A}\right|^{2}-e^{\tilde{F}}F_{\Lambda\Sigma}c_{A}^{\Lambda}c_{B}^{\Sigma}\Pi^{A}\bar{\Pi}^{B}\\\
&\quad\,+e^{\tilde{F}}\left[{\rm
i}c_{F}^{\Lambda}\tilde{F}_{\Lambda}\Pi^{F}\left(\bar{\hat{W}}+\frac{{\rm
i}}{2}F_{\Lambda}c_{A}^{\Lambda}\bar{\Pi}^{A}\right)+{\rm
c.c.}\right].\end{split}$ (D.54c)

In addition to the various identities among the kinetic matrices enlisted in
the previous section, we have also employed the following relations

$\displaystyle{\cal F}^{\Lambda\Sigma}{\cal F}_{\Lambda 0}{\cal
F}_{\Sigma\bar{0}}$
$\displaystyle=-\frac{e^{-\frac{\tilde{F}}{3}}}{3|u|^{\frac{4}{3}}}\frac{\tilde{F}_{\Lambda}\ell^{\Lambda}}{3-\ell^{\Pi}\tilde{F}_{\Pi}}\;,$
(D.55) $\displaystyle{\cal F}^{\Lambda\Sigma}{\cal F}_{\Lambda i}{\cal
F}_{\Sigma\bar{0}}$
$\displaystyle=-\frac{e^{-\frac{\tilde{F}}{3}}}{3|u|^{\frac{4}{3}}}\frac{3F_{\Sigma
i}\ell^{\Sigma}-(\ell^{\Pi}\tilde{F}_{\Pi})\tilde{F}_{i}}{3-\ell^{\Pi}\tilde{F}_{\Pi}}\;,$
$\displaystyle{\cal F}^{\Lambda\Sigma}{\cal F}_{\Lambda i}{\cal
F}_{\Sigma\bar{\jmath}}$
$\displaystyle=\frac{e^{-\frac{\tilde{F}}{3}}}{|u|^{\frac{4}{3}}}\left(F^{\Lambda\Sigma}F_{i\Lambda}F_{\bar{\jmath}\Sigma}+\frac{1}{3}F_{i}F_{\bar{\jmath}}-\frac{\tilde{F}_{i}\tilde{F}_{\bar{\jmath}}}{3-\ell^{\Pi}\tilde{F}_{\Pi}}\right)\;,$

which can be obtained from (D.39) and (D.40).

###  Appendix E Freed-Witten anomalies and axion monodromies

The configurations that we introduced in Sections 1.4.1 and 1.5.2, namely of
membranes ending on strings or of 3-branes ending on membranes are not unusual
in string theory constructions. In fact, ‘attaching’ 3-branes or membranes to
a string provides a mechanism to cure the anomalies of the gauge theories
defined over the worldvoumes of such objects. This effect, known as Freed-
Witten anomaly cancellation mechanism [55, 56], has a higher-dimensional
origin in string theory contexts. The descriptions that we gave in Sections
1.4.1 and 1.5.2, as well as in supergravity settings in Sections 5.1.3 and
5.1.5, provides effective four-dimensional descriptions of such effects
inherited from higher dimensions.

####  E.1 Freed-Witten anomaly cancellation and Hanany-Witten brane creation
effects

From the ten-dimensional string theory description, configurations of branes
with nontrivial background fluxes may lead to anomalies in the gauge theories
living on their worldvolumes. The Freed-Witten anomaly cancellation mechanisms
are as follows:

  * •

consider a D$p$-brane whose worldvolume $S_{p+1}$ is threaded by a nontrivial
background NS-NS $\overline{H}_{3}$ flux. Then, the D$p$-brane has to emit a
D$(p-2)$-brane spanning the directions orthogonal to $\overline{H}_{3}$ [56];

  * •

consider an NS5-brane whose worldvolume $S_{6}$ is threaded by nontrivial
background R-R $\overline{F}_{p}$ fluxes. Then, the NS5-brane has to emit a
D$(6-p)$-brane spanning the directions orthogonal to $\overline{F}_{p}$;

  * •

consider a D$p$-brane whose worldvolume $S_{p+1}$ is threaded by a nontrivial
background $\overline{F}_{p}$. Then, an F1-string is emitted spanning the
direction orthogonal to $\overline{F}_{p}$.

The Freed-Witten anomaly cancellation conditions, as explained in [110], are
in one-to-one correspondence with the Hanany-Witten brane creation effects
[184]. To exemplify the discussion, let us consider two examples:

  1. 1.

consider an NS5-brane stretching along all the external directions
$x^{0},\ldots,x^{3}$ and the internal $x^{4}$ and $x^{5}$ and a D8-brane
covering all the directions but $x^{9}$, as summarized in Table E.1. The
NS5-brane may transverse the D8-brane along the $x^{9}$-direction and this
results in creating a D6-brane covering the directions enlisted in Table E.1.

| $x^{0,\ldots,3}$ | $x^{4}$ | $x^{5}$ | $x^{6}$ | $x^{7}$ | $x^{8}$ | $x^{9}$
---|---|---|---|---|---|---|---
NS5 | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $-$ | $-$
D8 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $-$
$[\overline{H}_{3}]$ | $-$ | $-$ | $-$ | $\bullet$ | $\bullet$ | $\bullet$ | $-$
D6 | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $-$ | $\times$
Table E.1: The brane/flux-configuration considered in the first example of
Hanany-Witten effect, where $\times$ denotes the spanned direction, while $-$
a direction orthogonal to the brane. Analogously, $\bullet$ denotes the cycle
spannes, in homology, by the background flux.

In terms of Freed-Witten anomaly cancellation mechanism, the NS5-brane may be
understood as a source for an $H_{3}$ flux over the D8-brane worldvolume. This
anomalous configuration is cured by emitting a D6-brane in the directions
orthogonal to $H_{3}$.

  2. 2.

consider a D0-brane and a D8-brane covering all the directions but $x^{9}$, as
in Table E.2. The D0-brane may transverse the D8-brane along the
$x^{9}$-direction and this results in creating an F1-string stretching along
$x^{0}$ and $x^{9}$.

| $x^{0}$ | $x^{1}$ | $x^{2}$ | $x^{3}$ | $x^{4}$ | $x^{5}$ | $x^{6}$ | $x^{7}$ | $x^{8}$ | $x^{9}$
---|---|---|---|---|---|---|---|---|---|---
D0 | $\times$ | $-$ | $-$ | $-$ | $\times$ | $\times$ | $-$ | $-$ | $-$ | $-$
D8 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $-$
$[\overline{F}_{8}]$ | $-$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $-$
F1 | $\times$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $\times$
Table E.2: The brane/flux-configuration considered in the second example of
Hanany-Witten effect.

Again, this effect may be understood in terms of Freed-Witten anomaly
cancellations, as the D0-brane generate an $F_{8}$ fluxes over the D8-brane
worldvolume. According the third point enlisted above, an F1-string cures such
an anomalous configuration.

####  E.2 Freed-Witten anomalies and four-dimensional EFTs

Four-dimensional effective theories inevitably inherit such higher-dimensional
phenomena: what in higher dimensions were branes emitting other branes, in
four dimensions we get extended objects, whose boundary is nontrivial, being
constituted by another object of one dimension less. To exemplify the
discussion, let us again consider a simple example. An NS5-brane wrapping a
special Lagrangian four cycle in the internal space gives rise to a
supersymmetric string in four dimensions. Assume that over the NS5-brane a
nontrivial zero-form flux $\overline{F}_{0}=p$ is turned on. Now, according to
the Freed-Witten anomaly cancellation mechanism, such a configuration is cured
by attaching $p$ D6-branes to the string, whose number is counted by the zero-
form flux, as depicted in Fig. E.1. These D6-branes look like membranes in
four dimensions.

Figure E.1: An example of Freed-Witten anomaly cancellation mechanism.

A D6-brane generates a jump for the background two-form flux
$\overline{F}_{2}$, from $n$ units on the left to $n+p$ units on the right. At
the level of four-dimensional theories, the superpotential induced by such a
flux, schematically ${\cal W}_{\text{R-R}}=nZ$ (see (6.79) for the complete
expression), is different on the two sides of the membrane. However, also a
string is present, which couples to a gauge two-form in such a way that, after
transversing the string, the dual axion is subjected to a monodromy
transformation that changes $a\to a+p$. The superpotential associated to such
a configuration, as explained in Section 4.4.2, is of the form ${\cal
W}_{\text{string}}=-TZ$, with $a={\rm Re\hskip 1.00006pt}T$. It is then clear
that the dull superpotential ${\cal W}={\cal W}_{\text{R-R}}+{\cal
W}_{\text{string}}$ is unaltered if both a monodromy transformation and a flux
jump are performed.

This simple example can be further generalized to more complicated monodromy
transformations as in [110, 185, 121]. In particular, it can be also
generalized to the case where the four-dimensional ‘anomalous’ object is a
membrane, whose anomaly is cured by attaching 3-branes to it, in a similar
fashion that we did for a string.

In other words, the Freed-Witten anomaly cancellation mechanism is inherited
in the four-dimensional theory with the appearance of three elements, although
connected among each other: the composite extended objects that constitute the
spectrum of the theory; the gauging of forms that are coupled to the objects;
the symmetries of the superpotential and, thus, the potential.

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# Math Multiple Choice Question Generation via Human-Large Language Model
Collaboration

Jaewook Lee1 Digory Smith2 Simon Woodhead2 Andrew Lan1
University of Massachusetts Amherst1

Eedi2<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Multiple choice questions (MCQs) are a popular method for evaluating students’
knowledge due to their efficiency in administration and grading. Crafting
high-quality math MCQs is a labor-intensive process that requires educators to
formulate precise stems and plausible distractors. Recent advances in large
language models (LLMs) have sparked interest in automating MCQ creation, but
challenges persist in ensuring mathematical accuracy and addressing student
errors. This paper introduces a prototype tool designed to facilitate
collaboration between LLMs and educators for streamlining the math MCQ
generation process. We conduct a pilot study involving math educators to
investigate how the tool can help them simplify the process of crafting high-
quality math MCQs. We found that while LLMs can generate well-formulated
question stems, their ability to generate distractors that capture common
student errors and misconceptions is limited. Nevertheless, a human-AI
collaboration has the potential to enhance the efficiency and effectiveness of
MCQ generation.

###### keywords:

Multiple Choice Question, Large Language Models, Human-in-the-loop.

##  1 Introduction

Multiple choice questions (MCQs) are widely used to evaluate students’
knowledge since they enable quick and accurate administration and grading [2,
6, 9]. MCQs are constructed in a specific format. The stem refers to the
statement on the problem setup and context, followed by a question that needs
to be answered. Among the options, the correct one can be referred to as the
key, while incorrect ones can be referred to as distractors. As the name
implies, distractors in MCQs are typically formulated to align with common
errors among students. These distractors are chosen because students either i)
lack the necessary comprehension of the knowledge components (KCs) or
concepts/skills tested in the question to accurately identify the key as the
correct answer or ii) exhibit misconceptions that make them think a specific
distractor is correct.

While MCQs offer many advantages in student knowledge assessment, manually
crafting high-quality MCQs, especially in math-related domains, is a demanding
and labor-intensive process [5]. There are three main tasks in this process:
First, educators need to formulate a question stem that effectively
encapsulates the KCs they aim to test. Second, educators need to anticipate
common errors and/or misconceptions among students and create corresponding
distractors. Third, educators need to provide feedback to students who select
distractors that can help them identify their errors and lead them to the
correct answer, to expedite their learning process.

The emergence of large language models (LLMs) has raised hopes for making MCQ
creation more scalable by automating the process. Specifically, few-shot, in-
context learning is promising for generating math MCQs since LLMs can follow
instructions based on contextual information conveyed by a few examples. While
automated question generation for open-ended questions has shown notable
success, generating plausible distractors within MCQs presents a different
challenge: distractors should be based on anticipated student
errors/misconceptions [12], whereas LLMs have not necessarily learned this
information during training. Moreover, math MCQs are challenging since they
require mathematical reasoning, which means that distractors cannot be
generated using a knowledge graph [13] or paraphrasing tool [8]. Consequently,
math educators need to take an important role in guiding LLMs in math MCQ
generation: LLMs are responsible for scaling up the process while humans use
their expertise efficiently. Therefore, we raise following are two core
research questions (RQs) that help identify opportunities to generate math
MCQs through collaboration between LLMs and human educators: 1) RQ1: Can LLMs
generate valid MCQs, especially distractors and feedback corresponding to
common student errors/misconceptions? 2) RQ2: What are the key design elements
in a system where human math educators and LLMs collaborate on MCQ generation?

###  1.1 Contributions

In this paper, we introduce a prototype tool called the Human Enhanced
Distractor Generation Engine(HEDGE) for math MCQ creation, which leverages the
expertise of educators by asking them to edit LLM-generated MCQs in a two-step
process. In the first step, we prompt the LLM to generate stem, key, and
explanation in an MCQ, and ask educators to evaluate and edit the output to
make sure it is mathematically correct and relevant to the intended KC. In the
second step, we prompt the LLM to generate a set of possible
errors/misconceptions and the corresponding distractors and feedback, and ask
educators to evaluate and edit the output to make sure they correspond to
valid distractors to the generated question stem. In a pilot study, we recruit
four former/current math teachers to evaluate our tool on generating math MCQs
related to five pre-defined KCs. Results show that educators considered 70% of
the generated stem, key, and explanation generated by GPT-4 as valid. However,
they only considered 37% of the generated misconception, distractor, and
feedback valid, which reveals significant limitations of LLMs in capturing
anticipated common errors/misconceptions among real students. This observation
underscores the necessity of involving humans in the process of generating
math MCQs and leveraging real math educators’ expertise on common errors among
students.

##  2 Human Enhanced Distractor Generation Engine

###  2.1 Overview

Figure 1: HEDGE Overview: the human-AI collaboration setting for generating
math MCQs for a given KC. Strikethrough text represents edits made to LLM-
generated content while boldface text indicates misconceptions that correspond
to distractors.

HEDGE is our prototype for math MCQ generation that generates math MCQ for a
given mathematical KC, as illustrated in Figure 1. These KCs are categorized
into three levels of granularity: coarse, medium, and fine-grained. For
instance, KCs can cover either a broad topic such as “basic arithmetic” or a
specific topic like “Identify that a problem needs to be solved using
addition.” HEDGE is designed to utilize LLMs within OpenAI. The provided
example is generated using ChatGPT. We take a two-step approach for MCQ
generation: 1) generate the question step and answer key, and an explanation,
and 2) generate a list of possible misconceptions, corresponding distractors,
and feedback messages. We implement both steps using by prompting LLMs with an
in-context example of these tasks.

Table 1: The in-context example used for prompting LLMs for math MCQ
generation.

KC |  Coarse - Ratio, Medium - Writing ratios, Fine - Convert ratios to fractions
---|---
Stem |  Kate and Isaac share yogurt in a $2:5$ ratio. Kate has $\square$ of the total. Identify the fraction.
Key |  $\frac{2}{7}$
Explanation |  The total ratio is 7 parts. Kate’s share of $\frac{2}{7}$ is derived by dividing her 2 parts by the total.
Misconceptions |  1\. Misinterpreting the ratio as a fraction.
|  2\. Confusing the difference in ratio parts as relevant.
|  3\. Calculating Isaac’s share instead of Kate’s.
Distractors |  1\. $\frac{2}{5}$ 2\. $\frac{3}{7}$ 3\. $\frac{5}{7}$
Feedback |  1\. The ratio $2:5$ means 7 parts total, not $\frac{2}{5}$.
|  2\. The ratio splits the total, not the difference between parts.
|  3\. Ensure you are calculating Kate’s share, not Isaac’s.

The in-context example shows the KC converting ratios to fractions, employing
a real-life scenario in which Kate and Isaac share yogurt in a $2:5$ ratio.
The objective is to calculate the fraction representing Kate’s share,
$\frac{2}{7}$. In this context, we list three common misconceptions. First, a
student mistakenly thinks that the ratio $2:5$ could be directly converted
into the fraction $\frac{2}{5}$. Second, a student mistakenly calculates the
difference between Kate’s and Issac’s share. Third, a student mistakenly think
the goal is to calculate Issac’s share. These misconceptions, along with the
corresponding feedback on how to resolve them, are included as part of the in-
context example.

Now, we explore a scenario where an educator creates MCQs using our tool based
on the concept of basic arithmetic, specifically focusing on mental addition.
In the first step, given the target KC, along with an in-context example
consisting of the concept, stem, key, and explanation, the LLM generates the
following stem: “Sally has 5 apples. She gives 2 apples to her friend. How
many apples does Sally have left?” However, this stem mistakenly embodies the
KC of subtraction rather than addition. Therefore, the educator edits the
generated results to align it with the intended KC of addition. In the second
step, using the adjusted stem, key, and explanation, as well as incorporating
in-context examples with distractors, misconceptions, and feedback, the LLM
generates distractors along with corresponding misconceptions and feedback.
Figure 1 illustrates option $B$, which contains a misconception related to
subtraction instead of addition, accompanied by feedback designed to correct
this error. Additionally, the educator has the option to edit option $D$ to
address any misconceptions associated with multiplication.

###  2.2 User Interface

We develop HEDGE interface, as illustrated in Figure 2. This interface is
built using React and employs Firestore as its database for data storage. The
interface comprises three components: a Sidebar, a Preview, and a Generation.

The educator generates MCQs using the Generation component as discussed in
Section 2.1. Here, after prompting LLMs using the edited stem, key, and
explanation, we add a rating step to assess the overall quality of
misconceptions, distractors, and feedback that the educator rates based on a
5-point Likert scale.

Once the educator completes the distractor editing process, the Preview
component displays a fully structured MCQ, with the answer options randomized.
We store any metadata that isn’t visually represented within the image.
Following the completion of distractor editing, the Sidebar component is
refreshed. The educator can click on the stem to view the generated image
along with the answer sheet or create a new MCQ.

Figure 2: HEDGE Interface: what human participants use to generating an MCQ by
editing LLM output.

##  3 Pilot Study

Table 2: Pre-defined math KCs used in the pilot study.

Coarse-grained | Medium-grained | Fine-grained
---|---|---
Factors, Multiples and Primes | Factors and Highest Common Factor | Identify factors of a number
Fractions | Equivalent Fractions | Identify equivalent fractions when presented as numbers
Indices, Powers and Roots | Squares, Cubes, etc | Understand the notation for powers
Percentages | Repeated Percentages and Compound Interest | Understand the elements of the formula for compound percentage decrease
Surds | Simplifying Surds | Write a simplified surd in a non-simplified form

###  3.1 Experimental Setup

We perform a pilot study to assess the usability of HEDGE in generating MCQs.
In this study, we select pre-defined KCs and instruct participants to utilize
these KCs to simulate a scenario where an educator is crafting MCQs. We select
the KCs and the in-context example from a large education company’s content
repository, categorized under the label “Number,” encompasses various
subtopics, such as “Basic Arithmetic,” “Fractions,” and “Rounding and
Estimating.” We choose five KCs, as shown in Table 2, from the KCs that
incorporate mathematical expressions, such as fractions, powers, and surds. We
utilize GPT-4 as LLM for the study and set the parameters to temperature
$=0.7$ and top_p value $=0.9$ to balance creativity and consistency of the
generated MCQs. After completing the study, participants are asked to complete
an exit survey. The survey includes open-ended questions and ratings on their
satisfaction with the quality of LLM-generated responses and the usability of
the tool using a 5-point Likert scale.

###  3.2 Participants

We recruit four participants for the study, comprising one male and three
females, all recruited through Upwork [14]. Among them, two currently work as
middle/high school math teachers, while the other two currently work as
tutors, with prior experience as former math teachers. All participants are
selected based on their qualifications and expertise in mathematics education.
Each participant was tasked with creating five MCQs using the HEDGE, employing
the five KCs specified in Table 2.

Table 3: Question stems generated using HEDGE and the corresponding KCs.

Fine-grained KC |  Stem
---|---
$1$. Identify factors of a number
Which of these numbers is not a factor of 9? |  $a$. What are all the factors of the number 12?
$b$. What are the factors of $18$?
$c$. Which of the following is a factor of 18?
$d$. Which of the following numbers is a factor of 36?
$2$. Identify equivalent fractions when presented as numbers
Which fraction is equivalent to $\frac{9}{13}$? |  $a$. Sue has a fraction of $\frac{4}{8}$. What fraction is equivalent to the fraction she has?
$b$. The fraction $\frac{6}{18}$ is equivalent to which of the following
fractions?
$c$. Which of the following fractions is equivalent to $\frac{3}{9}$?
$d$. Which of the following fractions is equivalent to $\frac{2}{4}$?
$3$. Understand the notation for powers
To calculate ${53}^{2}$ you need to do … |  $a$. The number $3^{2}$ is equal to $\square$. What number completes the sentence?
$b$. The number $3^{4}$ represents $\square$. What number completes the
sentence?
$c$. If $a^{3}$ is read as "a cubed", how is $a^{4}$ read?
$d$. What is the value of $2^{3}$?
$4$. Understand the elements of the formula for compound percentage decrease
A car depreciates in value by 10% each year. If a car was bought for $4500, what calculation would find the value of the car after 3 years? |  $a$. A car that costs $5000 loses 12% of its value each year. After one year, the car is worth $\square$. What completes the sentence?
$b$. A new car loses 20% of its value each year. If the car was originally
priced at $15,000, what will be its value after 2 years?
$c$. The price of a car is reduced by 5% each year. If the car was originally
priced at $5000, what will be the price of the car after two years?
$d$. A car depreciates in value by 10% each year. If the car is initially
worth $35000, what is the formula to calculate the car’s value after n years?
$5$. Write a simplified surd in a non-simplified form
$5\sqrt{13}=\sqrt{n}$ What is the value of $n$? |  $a$. If $2\sqrt{5}$ is a simplified surd, what is its non-simplified form?
$b$. The square root of 18 is written in simplified surd form as $3\sqrt{2}$.
How can it be rewritten in a non-simplified form?
$c$. Simplify the surd $\sqrt{45}$.
$d$. A non-simplified surd is $\sqrt{8}$. How can it be represented in
simplified form?

##  4 Results

###  4.1 Stem, Key, and Explanation

Table 3 shows the stems produced by participants utilizing HEDGE. In the
“Fine-grained KC” column, the original stem is indicated in italics, while the
stems modified by each participant denoted as $a$, $b$, $c$, and $d$,
respectively. In what follows, we label each MCQ in the format of $1a$, where
$1$ denotes the index of the fine-grained KC and $a$ denotes index of the
participant.

Out of 20 sets of stem, key, and explanation generated by the LLM,
participants deemed 14 sets of them as valid. Among these valid sets, two
added more details in their explanations, while the remaining sets were
adopted without any need for edits. For example, italicized details were added
in the explanation for $2c$: “The fraction $\frac{3}{9}$ simplifies to
$\frac{1}{3}$ because both the numerator and the denominator can be divided by
a common factor of 3\. 3 divided by 3 is 1, and 9 divided by 3 is 3. Hence,
$\frac{1}{3}$ is an equivalent fraction to $\frac{3}{9}$.” The other case was
to make the question setting more realistic: In $4d$, the educator edited the
initial price of the car worth $5000 to $35000. This adjustment reveals the
limitations of LLMs in accurately representing real-life problem scenario. We
now analyze the cases that participants deemed invalid.

##### Grammar error

In $2a$, educator corrected grammar error of “she have” to “she has.” No other
grammar errors occurred in the study besides this one, underscoring the
capability of LLMs to consistently produce grammatically correct sentences.

##### Not mastering KC

Regarding 5th KC, GPT-4 shows a lack of knowledge on the distinction between
simplified and non-simplified surd. The followings are invalid stems generated
by GPT-4: 1) $5a$. If $\sqrt{20}$ is a simplified surd, what is its non-
simplified form? 2) $5c$. Express the simplified surd $\sqrt{45}$ in a non-
simplified form. 3) $5d$. A simplified surd is $\sqrt{8}$. How can it be
represented in non-simplified form?

This invalid stem has misled a participant to edit a stem to convey KC as
simplifying surd, which is the opposite of non-simplifying surd ($5c$).

##### Calculation error

In $4c$, GPT-4 generated a key of $4750, erroneously calculating the car price
after one year instead of two years. However, in the other three cases within
the same KC, GPT-4 calculated correctly, showing its math problem-solving
skills.

###  4.2 Distractor, Misconception, and Feedback

Table 4 shows a breakdown of 60 distractors (comprising three distractors for
20 stems), categorized based on the validity of misconceptions, distractors,
and feedback.

Table 4: Breakdown of the 60 generated distractors and their quality ratings.
(: valid, : invalid)

Case | Misconception | Distractor | Feedback | Ratio | Rating
---|---|---|---|---|---
1 |  |  |  | 37% | 4.8
2 |  |  |  | 8% | 2.8
3 |  |  |  | - | -
4 |  |  |  | 18% | 2.1
5 |  |  |  | 12% | 3.4
6 |  |  |  | 5% | 3.0
7 |  |  |  | - | -
8 |  |  |  | 20% | 2.3

##### Adopt All Responses (Case 1, 37%)

Among 60 distractors, educators identified 22 responses as valid, including
two cases that are actually invalid.

##### Edit Feedback Only (Case 2, 8%)

These cases have valid misconception and distractor and educators has made
adjustments to the feedback to enhance its clarity. For example, one of the
distractors for $2d$ is $\frac{2}{3}$. The feedback generated by GPT-4 is as
follows: “You seem to have compared only the numerators of the fractions.
However, when checking for equivalent fractions, both the numerator and
denominator need to be considered. The fraction $\frac{2}{3}$ is not
equivalent to $\frac{2}{4}$.” The educator removed the redundant final
sentence and introduced “Remember, equivalent fractions require both the
numerator and denominator to be proportional.”, which helps students better
understand the importance of considering both the numerator and denominator
when comparing fractions for equivalence. This adjustment emphasizes that the
equivalence between fractions relies on maintaining proportionality between
the numerator and denominator. While GPT-4 provides valid explanations, it
sometimes fail to include critical insights that are necessary for students’
improvement.

##### Adopt Misconception Only (Case 4, 18%)

These cases are often due to a mismatch between the misconception and the
distractor. In $4c$, the misconception “The student mistakenly believed that
the car depreciates by a constant amount each year, not a percentage.” did not
match the distractor $35000-0.10n$. Additionally, there are cases when, even
if the distractor is valid, it may not effectively encapsulate student
misconceptions. In $1a$, the educator updated the distractor from
$1,2,3,4,6,12,24$ to $12,24,36,48,60$, making it a more attractive distractor
for those who confuse factors for multiples.

##### Edit Misconception Only (Case 5, 12%)

As in Case 4, invalid cases are often due to a mismatch between the
misconception and the distractor. In $5d$, the misconception “The student may
believe that all square roots are in their simplest form.” did not match the
distractor “$\sqrt{2}$.” The educator updated the misconception as “The
student may have confused square roots with cube roots.” providing a more
accurate misconception for the distractor. Additionally, there are cases when,
even if the misconception is valid, it may not likely be the misconception why
the student selects the distractor. In $1c$, the educator updated the
misconception of distractor “$4$” from “The student might think that only the
numbers less than 18 can be the factors of 18.” to “The student might think
that any even number can be a factor of an even number.”, making it more
accurate for addressing the student’s misconception.

##### Adopt Distractor Only (Case 6, 5%)

These cases were when educators adopted distractors and edited wrong
misconceptions and feedback. For example, in the case of $5a$, $\sqrt{10}$ is
a valid distractor as the student could simply multiply $2$ and $5$. However,
the misconception and feedback generated by GPT-4 did not align with the
distractor; therefore the educator had to edit it accordingly.

In Cases 4, 5, and 6, LLMs revealed inconsistent mathematical reasoning when
analyzing misconceptions, distractors, and feedback for a given stem. The
inconsistency underscores a necessity for human educators to manually align
distractors and their underlying misconceptions and corresponding feedback in
many cases.

##### Reject All Responses (Case 8, 20%)

These cases were when misconceptions had poor quality or were wrong, resulting
in inadequate distractors and feedback. Two of the distractors generated for
$2b$ by GPT-4 shows both poor quality and wrong misconceptions. While the
misconception in the first distractor is valid, stating that “The student may
not divide both the numerator and denominator by the same number,” the
distractor itself, represented by $\frac{3}{9}$, and its associated feedback
lack coherence and fail to align with this misconception. Meanwhile, the
misconception in the second distractor ($\frac{8}{24}$) lacks coherence, as
expressed in the following manner: “The student may confuse the concept of
equivalent fractions with simplifying fractions.” These results reveal that
LLMs often fail to anticipate valid misconceptions and errors that are common
among students, making human educators’ involvement crucial in the creation of
math MCQs.

###  4.3 Takeaways from the Survey

After the study, participants were asked to fill out the survey, asking the
experience using HEDGE. We categorize result into two: Quality of LLM-
generated responses and Tool Usability.

####  4.3.1 Quality of LLM-generated responses.

##### Stem, Key, and Explanation

On a 5-point Likert scale, the participants gave an average rating of 4. This
rating aligns with the open-ended responses regarding most of the generated
stem, key, and explanation valid. However, two participants addressed the
tool’s limitation in terms of the level of question difficulty. One
participant points out that the questions appear to be at a low Bloom’s
Taxonomy level. For example, “If $a^{3}$ is read as ‘a cubed’, how is $a^{4}$
read?” While it’s important for students to grasp the verbal representation of
these terms, educators often place greater emphasis on whether students
understand the equivalent expressions and concepts associated with them. The
other participant points out that the Depth of Knowledge (DOK) levels
predominantly focused on Level 1 (Recall) and Level 2 (Skill or Concept). We
can prompt LLMs to generate questions at various Bloom’s or DOK levels to
enhance the question difficulty and promote deeper understanding [3].
Moreover, we can invite educators to craft in-context examples with higher
Bloom’s or DOK levels.

##### Distractor, Misconception, and Feedback

On a 5-point Likert scale, the participants gave an average rating of 2.5.
This rating aligns with the open-ended responses regarding most of the
generated misconceptions, distractors, and feedback that do not reflect what
students typically make in the classroom based on the participant’s teaching
experience. The responses again point to the observation that LLMs do not
understand errors that student are likely to make. One participant suggest
providing a “bank” of misconceptions that educators could refer to. We can
prompt LLMs to generate multiple misconceptions and engage educators in
ranking these misconceptions based on their alignment with actual student
errors.

####  4.3.2 Tool Usability

##### User Interface

On a 5-point Likert scale, the participants gave an average rating of 4 for
comfort level with generating MCQs using HEDGE while giving an average rating
of 3.25 for the effectiveness of generating high-quality MCQs. Participants
are enthusiastic about the tool’s potential for simplifying the process of
generating MCQs but are nevertheless skeptical about LLMs’ capability to
generate valid distractors. We will need to enhance the tool by making
improvements in the quality of generated distractors to align more closely
with educators’ expectations.

##  5 Conclusions and Future Work

In this paper, we conducted a pilot study using a prototype tool HEDGE to
explore the opportunity for collaboration between LLMs and humans in
generating math MCQs. We identified that while LLMs can generate valid stems,
keys, and explanations, they are currently limited in capturing anticipated
student errors, which is reflected in invalid misconceptions, distractors, and
feedback.

This study opens up many avenues for future work. First, we can extend the
prompt with more in-context examples. Currently, we use only one in-context
example; using multiple in-context examples can help guide LLMs to help
capture valid misconceptions for the target stem. As mentioned in our survey
takeaways, we can also add in-context examples with different Bloom’s taxonomy
and difficulty levels to enhance the diversity of the generated questions. We
could also use techniques for optimally selecting these in-context examples
[10]. Second, we can change the interface to choose generated distractors from
a bank that contains more than three distractors. When building the bank, we
can employ a $k$-nearest neighbor approach that gauges question similarity and
leverage LLMs to generate distractors [4]. Educators will have less burden on
thinking of anticipated misconceptions and even benefit from discovering
misconceptions they might have overlooked. Third, improving this tool as a
platform so that educators can share their misconceptions will result in a
constantly-expanding error bank, which will benefit future MCQ generation.
Fourth, we can provide educators choice to create personalized questions by
adding a module to customize the named entity or topic (e.g., sports, popular
culture) in an MCQ to stimulate student interest and make the question more
culturally relevant [15]. Lastly, we can extend the area of Human-LLM
collaboration to other domains beyond math MCQ generation, such as feedback
generation [11], question generation [7], and programming education [1].

##  6 Acknowledgements

We thank Schmidt Futures and the NSF (under grants IIS-2118706 and
IIS-2237676) for partially supporting this work.

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