Mills theory are equal to certain form factors of full Yang-Mills theory, at
particular orders in the loop expansion and with particular helicity
configurations. For instance, the all $+$ form factor of the operator
$\operatorname{tr}(B^{n})$ at $n$ loops is a quantity that is the same in
self-dual Yang-Mills and in full Yang-Mills.

A full understanding of the chiral algebra would (in principle) allow a
computation of these form factors. This would be a major step forward, as
there are almost no form factor computations in QCD beyond low loop numbers.
As we see from the analysis of this paper, however, there is a lot of tedious
algebra to be done to generate form factors from chiral algebra OPEs.

In a different direction, one might ask if the methods discussed here give any
insight into the non-rational terms in the two-loop amplitudes and form
factors. Let us discuss this for the two-loop all $+$ amplitudes of QCD with
$N_{f}=N_{c}$. The chiral algebra method requires one to cancel the twistor
space anomaly by an axion field. With the axion, all two-loop all $+$
amplitudes are rational. Axion exchange does not contribute at leading order
in $N$, however, it does contribute at subleading order.

This suggests that we could calculate the transcendental terms in the two-loop
amplitude with $N_{f}=N_{c}$, by studying those Feynman diagrams in which an
axion is exchanged. Because the axion is introduced as part of a Green-Schwarz
mechanism, an axion exchange is effectively a one-loop process. This means
that the transcendental parts of the amplitude with $N_{f}=N_{c}$ can in
principle be computed using one-loop and tree-level Feynman diagrams which
contain either one or two axion exchanges.

### Acknowledgements

I would like to thank Roland Bittleston, Lance Dixon, Lionel Mason and Natalie
Paquette for helpful comments and conversations. I am particularly grateful to
Anthony Morales for pointing out a number of computational errors in the first
version of this paper, and for kindly cross-checking the colour algebra
computations in the appendix. Any remaining errors are, of course, my
responsibility.

I am grateful to the Krembil Foundation, the NSERC Discovery Grant program and
the Perimeter Institute for Theoretical Physics for support. Research at
Perimeter Institute is supported in part by the Government of Canada through
the Department of Innovation, Science and Economic Development Canada and by
the Province of Ontario through the Ministry of Colleges and Universities.

##  Appendix A Associativity of the OPE with matter included

The calculation in Costello:2022upu  goes _mutatis mutandis_ up to the point
where we find that the identity

$C\left(\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{b}{\sf{t}}_{c}{\sf{t}}_{d})+\operatorname{Tr}({\sf{t}}_{b}{\sf{t}}_{a}{\sf{t}}_{c}{\sf{t}}_{d})\right)-\tfrac{1}{2}D\tfrac{1}{2{\sf{h}}^{\vee}-D_{R}}\operatorname{Tr}\left([{\sf{t}}_{a},{\sf{t}}_{b}][{\sf{t}}_{c},{\sf{t}}_{\mathrm{d}}]\right)-D\tfrac{1}{2{\sf{h}}^{\vee}-D_{R}}\operatorname{Tr}\left([{\sf{t}}_{b},{\sf{t}}_{c}][{\sf{t}}_{a},{\sf{t}}_{d}]\right)\\\
=\widehat{\lambda}_{\mathfrak{g},R}^{2}\left(K_{ac}K_{bd}+K_{ab}K_{cd}+K_{ad}K_{bc}\right).$
(185)

needs to hold. Here $D_{R}$ is the Dynkin index of the representation $R$.

To verify that this holds, we need to show that the terms on the left hand
side of equation (185) that are not totally symmetric cancel. In what follows
we write $\operatorname{Tr}$ for trace in the adjoint minus trace in the
representation $R$. We have

$\begin{split}3\left(\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{b}{\sf{t}}_{c}{\sf{t}}_{d})+\operatorname{Tr}({\sf{t}}_{b}{\sf{t}}_{a}{\sf{t}}_{c}{\sf{t}}_{d})\right)&-2\left(\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{b}{\sf{t}}_{c}{\sf{t}}_{d})+\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{c}{\sf{t}}_{d}{\sf{t}}_{b})+\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{d}{\sf{t}}_{b}{\sf{t}}_{c})\right)\\\
&=\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{b}{\sf{t}}_{c}{\sf{t}}_{d})+\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{c}{\sf{t}}_{d}{\sf{t}}_{b})-2\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{d}{\sf{t}}_{b}{\sf{t}}_{c})\\\
&=\operatorname{Tr}([{\sf{t}}_{d},{\sf{t}}_{a}]{\sf{t}}_{b}{\sf{t}}_{c})+\operatorname{Tr}([{\sf{t}}_{a},{\sf{t}}_{c}]{\sf{t}}_{d}{\sf{t}}_{b}).\end{split}$
(186)

By dihedral symmetry we have

$\operatorname{Tr}([{\sf{t}}_{d},{\sf{t}}_{a}]{\sf{t}}_{b}{\sf{t}}_{c})=\tfrac{1}{2}\operatorname{Tr}([{\sf{t}}_{d},{\sf{t}}_{a}][{\sf{t}}_{b},{\sf{t}}_{c}])$
(187)

so that

$\begin{split}3\left(\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{b}{\sf{t}}_{c}{\sf{t}}_{d})+\operatorname{Tr}({\sf{t}}_{b}{\sf{t}}_{a}{\sf{t}}_{c}{\sf{t}}_{d})\right)&-2\left(\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{b}{\sf{t}}_{c}{\sf{t}}_{d})+\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{c}{\sf{t}}_{d}{\sf{t}}_{b})+\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{d}{\sf{t}}_{b}{\sf{t}}_{c})\right)\\\
&=\tfrac{1}{2}\operatorname{Tr}([{\sf{t}}_{d},{\sf{t}}_{a}][{\sf{t}}_{b},{\sf{t}}_{c}])+\tfrac{1}{2}\operatorname{Tr}([{\sf{t}}_{a},{\sf{t}}_{c}][{\sf{t}}_{d},{\sf{t}}_{b}])\\\
&=-\tfrac{1}{2}\operatorname{Tr}([{\sf{t}}_{a},{\sf{t}}_{d}][{\sf{t}}_{b},{\sf{t}}_{c}])+\tfrac{1}{2}\operatorname{Tr}([{\sf{t}}_{b},[{\sf{t}}_{a},{\sf{t}}_{c}]]{\sf{t}}_{d})\\\
&=-\tfrac{1}{2}\operatorname{Tr}([{\sf{t}}_{a},{\sf{t}}_{d}][{\sf{t}}_{b},{\sf{t}}_{c}])+\tfrac{1}{2}\operatorname{Tr}([[{\sf{t}}_{b},{\sf{t}}_{a}],{\sf{t}}_{c}]{\sf{t}}_{d})+\tfrac{1}{2}\operatorname{Tr}([{\sf{t}}_{a},[{\sf{t}}_{b},{\sf{t}}_{c}]]{\sf{t}}_{d})\\\
&=-\operatorname{Tr}([{\sf{t}}_{a},{\sf{t}}_{d}][{\sf{t}}_{b},{\sf{t}}_{c}])+\tfrac{1}{2}\operatorname{Tr}([[{\sf{t}}_{b},{\sf{t}}_{a}],{\sf{t}}_{c}]{\sf{t}}_{d})\\\
&=-\operatorname{Tr}([{\sf{t}}_{a},{\sf{t}}_{d}][{\sf{t}}_{b},{\sf{t}}_{c}])-\tfrac{1}{2}\operatorname{Tr}([{\sf{t}}_{b},{\sf{t}}_{a}][{\sf{t}}_{c},{\sf{t}}_{d}]).\end{split}$
(188)

Therefore we have

$\begin{split}\tfrac{3}{2}\left(\operatorname{Tr}({\sf{t}}_{a}{\sf{t}}_{b}{\sf{t}}_{c}{\sf{t}}_{d})+\operatorname{Tr}({\sf{t}}_{b}{\sf{t}}_{a}{\sf{t}}_{c}{\sf{t}}_{d})\right)&-\lambda_{\mathfrak{g},R}^{2}(K_{ac}K_{bd}+K_{ab}K_{cd}+K_{ad}K_{bc})\\\
&=-\frac{1}{2}\operatorname{Tr}([{\sf{t}}_{a},{\sf{t}}_{d}][{\sf{t}}_{b},{\sf{t}}_{c}])-\frac{1}{4}\operatorname{Tr}([{\sf{t}}_{b},{\sf{t}}_{a}][{\sf{t}}_{c},{\sf{t}}_{d}]).\end{split}$
(189)

Therefore, the Lie algebra identity (185) holds if we take

$\begin{split}C&=\frac{3}{2(2\pi i)^{2}12}\\\
D&=-\frac{2{\sf{h}}^{\vee}-D_{R}}{(2\pi i)^{3}24}\end{split}$ (190)

##  Appendix B Computing algebraic factors

We defined

$\begin{split}T_{ab}^{cd}&=f_{ea}^{c}f_{be}^{d}+f_{ae}^{d}f_{eb}^{c}\\\
S_{abj}^{i}&=g_{ka}^{i}g_{jb}^{k}+g_{ja}^{k}g_{kb}^{i}\end{split}$ (191)

where $g_{ka}^{i}$ is the matrix defining the action of a Lie algebra
generator ${\sf{t}}_{a}$ on the matter representation $R$. Thus, $S$ is the
matrix ${\sf{t}}_{a}{\sf{t}}_{b}+{\sf{t}}_{b}{\sf{t}}_{a}$, acting in the
matter representation.

We note that

$T_{a_{1}a_{2}}^{b_{1}b_{2}}{\sf{t}}_{b_{1}}{\sf{t}}_{b_{2}}=-[{\sf{t}}_{a_{1}},{\sf{t}}_{e}][{\sf{t}}_{a_{2}},{\sf{t}}_{e}]-[{\sf{t}}_{a_{2}},{\sf{t}}_{e}][{\sf{t}}_{a_{1}},{\sf{t}}_{e}]$
(192)

The pole in the two-loop $4$-point form factor at $z_{1}=z_{2}$, $z_{3}=z_{4}$
coming from the one loop corrected OPEs is

$\frac{[12][34]}{\left\langle 12\right\rangle\left\langle
34\right\rangle}\frac{1}{(2\pi i)^{4}2^{6}}$ (193)

multiplied by the rather complicated algebraic expression

$\begin{split}\left\langle\tfrac{1}{2}\operatorname{tr}(B^{2})\mid
T_{a_{1}a_{2}}^{b_{1}b_{2}}:J_{b_{1}}\widetilde{J}_{b_{2}}:(z_{1})T_{a_{3}a_{4}}^{b_{3}b_{4}}:J_{b_{3}}\widetilde{J}_{b_{4}}:(z_{3})\right\rangle\\\
+\left\langle\tfrac{1}{2}\operatorname{tr}(B^{2})\mid
T_{a_{1}a_{2}}^{b_{1}b_{2}}:J_{b_{1}}\widetilde{J}_{b_{2}}:(z_{1})S_{a_{3}a_{4}i_{4}}^{i_{3}}:M_{i_{3}}\widetilde{M}^{i_{4}}:(z_{3})\right\rangle\\\
+\left\langle\tfrac{1}{2}\operatorname{tr}(B^{2})\mid
S_{a_{1}a_{2}i_{2}}^{i_{1}}:M_{i_{1}}\widetilde{M}^{i_{2}}:(z_{1})T_{a_{3}a_{4}}^{b_{3}b_{4}}:J_{b_{3}}\widetilde{J}_{b_{4}}:(z_{3})\right\rangle\\\
+\left\langle\tfrac{1}{2}\operatorname{tr}(B^{2})\mid
S_{a_{1}a_{2}i_{2}}^{i_{1}}:M_{i_{1}}\widetilde{M}^{i_{2}}:(z_{1})S_{a_{3}a_{4}i_{4}}^{i_{3}}:M_{i_{3}}\widetilde{M}^{i_{4}}:(z_{3})\right\rangle\end{split}$
(194)

where

$\begin{split}\left\langle\tfrac{1}{2}\operatorname{tr}(B^{2})\mid:\widetilde{J}_{(a}J_{b)}:(z):\widetilde{J}_{(c}J_{d)}:(w)\right\rangle&=-2\operatorname{tr}({\sf{t}}_{(a}{\sf{t}}_{b)}{\sf{t}}_{(c}{\sf{t}}_{d)})+\operatorname{tr}({\sf{t}}_{a}{\sf{t}}_{c}{\sf{t}}_{b}{\sf{t}}_{d})+\operatorname{tr}({\sf{t}}_{a}{\sf{t}}_{d}{\sf{t}}_{b}{\sf{t}}_{c})\\\
\left\langle\tfrac{1}{2}\operatorname{tr}(B^{2})\mid:\widetilde{J}_{(a}J_{b)}:(z):\widetilde{M}_{(i}M_{j)}:(w)\right\rangle&=\tfrac{1}{2}(g^{j}_{ka}g^{k}_{ib}+g^{j}_{kb}g^{k}_{ia})\\\
\left\langle\tfrac{1}{2}\operatorname{tr}(B^{2})\mid:\widetilde{M}_{(i}M_{j)}:(z):\widetilde{M}_{(k}M_{l)}:(w)\right\rangle&=-\tfrac{1}{2}\left(g_{ila}g_{kja}+g_{jla}g_{kia}\right).\end{split}$
(195)

We thus need to compute the four terms

$\begin{split}T_{a_{1}a_{2}}^{b_{1}b_{2}}T_{a_{3}a_{4}}^{b_{3}b_{4}}\left(-2\operatorname{tr}({\sf{t}}_{(b_{1}}{\sf{t}}_{b_{2})}{\sf{t}}_{(b_{3}}{\sf{t}}_{b_{4})})+\operatorname{tr}({\sf{t}}_{b_{1}}{\sf{t}}_{b_{3}}{\sf{t}}_{b_{2}}{\sf{t}}_{b_{4}})+\operatorname{tr}({\sf{t}}_{b_{1}}{\sf{t}}_{b_{4}}{\sf{t}}_{b_{2}}{\sf{t}}_{b_{3}})\right)\\\
\tfrac{1}{2}T_{a_{1}a_{2}}^{b_{1}b_{2}}S_{a_{3}a_{4}j}^{i}\left(g^{j}_{kb_{1}}g^{k}_{ib_{2}}+g^{i}_{kb_{2}}g^{k}_{jb_{1}}\right)\\\
\tfrac{1}{2}T_{a_{3}a_{4}}^{b_{1}b_{2}}S_{a_{1}a_{2}j}^{i}\left(g^{j}_{kb_{1}}g^{k}_{ib_{2}}+g^{i}_{kb_{2}}g^{k}_{jb_{1}}\right)\\\
-\tfrac{1}{2}S_{a_{1}a_{2}i_{2}}^{i_{1}}S_{a_{3}a_{4}i_{4}}^{i_{3}}\left(g_{i_{1}i_{4}a}g_{i_{3}i_{2}a}+g_{i_{2}i_{4}a}g_{i_{3}i_{1}a}\right).\end{split}$
(196)

Our goal is to simplify these algebraic expressions.

We will do this for gauge group $\mathfrak{sl}_{N}$ with various choices of
matter. We will take $N_{F}$ copies of the fundamental plus the anti-
fundamental, $N_{\wedge^{2}F}$ copies of
$\wedge^{2}F\oplus\wedge^{2}F^{\vee}$, and $N_{a}$ copies of the adjoint.

We find that the total contribution is

$\begin{split}-2N^{2}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+8\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})-8\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
\
+(4N_{F}N+2N_{F}N^{-1})\operatorname{tr}_{F}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{4}})\\\
+N_{\wedge^{2}F}\left(2N^{2}-8N-8-48N^{-1}\right)\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+N_{\wedge^{2}F}(-16-16N^{-1})\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\\\
+2N_{a}N^{2}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+8N_{a}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-8N_{a}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\\\
+\text{ permutations }\end{split}$ (197)

where permutations are under the $\mathbb{Z}/2\times\mathbb{Z}/2$ inside
$S_{4}$ permuting $(12)$ and $(34)$.

As a consistency check, we note that, with $N=1$ supersymmetry, where there is
only adjoint matter, everything cancels. This is to be expected as the two
loop all $+$ amplitude vanishes in this case.

In the theory with fundamental matter and $N_{F}=N$, the planar (order
$N^{2}$) term is non-zero at two loops, although at one loop the planar term
vanishes with only external gluons. This is not a contradiction: at one loop,
the planar term does not vanish if two of the external fields are fermions.
These one-loop diagrams with two fermions contribute to the two-loop all $+$
amplitude.

###  B.1 First term

The first term is the only term which is independent of the matter content. We
should view this term as the contribution of pure glue. It is

$2T_{a_{1}a_{2}}^{b_{1}b_{2}}T_{a_{3}a_{4}}^{b_{3}b_{4}}\left(-\operatorname{tr}_{F}({\sf{t}}_{b_{1}}{\sf{t}}_{b_{2}}{\sf{t}}_{b_{3}}{\sf{t}}_{b_{4}})+\operatorname{tr}_{F}({\sf{t}}_{b_{1}}{\sf{t}}_{b_{3}}{\sf{t}}_{b_{2}}{\sf{t}}_{b_{4}})\right).$
(198)

(Here we have used the fact that $T_{a_{1}a_{2}}^{b_{1}b_{2}}$ is symmetric
under permutation of $b_{1},b_{2}$).

Using the definition of $T_{a_{1}a_{2}}^{b_{1}b_{2}}$, and working in an
orthonormal basis for the Lie algebra, this is

$\begin{split}-2\operatorname{tr}_{F}\left([{\sf{t}}_{a_{1}},{\sf{t}}_{e}][{\sf{t}}_{a_{2}},{\sf{t}}_{e}][{\sf{t}}_{a_{3}},{\sf{t}}_{f}][{\sf{t}}_{a_{4}},{\sf{t}}_{f}]\right)\\\
+2\operatorname{tr}_{F}\left([{\sf{t}}_{a_{1}},{\sf{t}}_{e}][{\sf{t}}_{a_{3}},{\sf{t}}_{f}][{\sf{t}}_{a_{2}},{\sf{t}}_{e}][{\sf{t}}_{a_{4}},{\sf{t}}_{f}]\right)\\\
+\text{ permutations }\end{split}$ (199)

where there are $8$ terms in total, which are the permutations of the two
terms written down under the $\mathbb{Z}/2\times\mathbb{Z}/2$ subgroup of
$S_{4}$ generated by $(12)$ and $(34)$.

We are only interested in the single-trace form factor. The quadratic Casimir
for $\mathfrak{sl}_{N}$ can be written as

$\sum{\sf{t}}_{e}\otimes{\sf{t}}_{e}=\sum{\sf{t}}_{i}^{j}\otimes{\sf{t}}_{j}^{i}-\frac{1}{N}\operatorname{Id}\otimes\operatorname{Id}.$
(200)

Since the quadratic Casimir only appears in commutators, we can use the
Casimir for $\mathfrak{gl}_{N}$ instead, dropping the term
$N^{-1}\operatorname{Id}\otimes\operatorname{Id}$.

To extract the single trace terms, we use the following rules. First, inside
any trace, we can replace an occurrence of ${\sf{t}}_{e}{\sf{t}}_{e}$ by $N$.
Next, an expression like

$\operatorname{tr}_{F}(\dots\sf{t}_{e}{\sf{t}}_{a_{1}}\dots\sf{t}_{a_{k}}{\sf{t}}_{e}\dots)$
(201)

where none the indices $a_{1},\dots,a_{k}$ nested between the two $t_{e}$’s
are contracted, is contracted, and they are not adjacent, is double trace.

Finally, an expression like

$\operatorname{tr}_{F}({\sf{t}}_{a}{\sf{t}}_{e}{\sf{t}}_{b}{\sf{t}}_{f}{\sf{t}}_{c}{\sf{t}}_{e}{\sf{t}}_{d}{\sf{t}}_{f})$
(202)

where two pairs of adjoint indices are contracted, but with a ${\sf{t}}_{f}$
occuring between the two ${\sf{t}}_{e}$’s, is single trace. This has the
permuted trace ordering
$\operatorname{tr}_{F}({\sf{t}}_{a}{\sf{t}}_{d}{\sf{t}}_{c}{\sf{t}}_{b})$.

Using these rules, the single-trace part of
$-2\operatorname{tr}_{F}\left([{\sf{t}}_{a_{1}},{\sf{t}}_{e}][{\sf{t}}_{a_{2}},{\sf{t}}_{e}][{\sf{t}}_{a_{3}},{\sf{t}}_{f}][{\sf{t}}_{a_{4}},{\sf{t}}_{f}]\right)$
is

$-2N^{2}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})$
(203)

The expression
$2\operatorname{tr}_{F}\left([{\sf{t}}_{a_{1}},{\sf{t}}_{e}][{\sf{t}}_{a_{3}},{\sf{t}}_{f}][{\sf{t}}_{a_{2}},{\sf{t}}_{e}][{\sf{t}}_{a_{4}},{\sf{t}}_{f}]\right)$
is entirely single trace. It has $16$ terms, each of which is a permutation of
$\operatorname{tr}_{F}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{4}})$ by some element
in $S_{4}$, with some sign. If one writes out all the $16$ terms, one finds
that $8$ terms cancel and we are left with

$\begin{split}2\operatorname{tr}_{F}\left([{\sf{t}}_{a_{1}},{\sf{t}}_{e}][{\sf{t}}_{a_{3}},{\sf{t}}_{f}][{\sf{t}}_{a_{2}},{\sf{t}}_{e}][{\sf{t}}_{a_{4}},{\sf{t}}_{f}]\right)+\text{
permutations }=\\\
8\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})-8\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+\text{
permutations }\end{split}$ (204)

The total contribution of the first term is

$-2N^{2}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+8\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})-8\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+\text{
permutations }$ (205)

###  B.2 Second and third terms

The second term is

$\tfrac{1}{2}T_{a_{1}a_{2}}^{b_{1}b_{2}}S_{a_{3}a_{4}i}^{j}\left(g^{j}_{kb_{1}}g^{k}_{ib_{2}}+g^{j}_{kb_{2}}g^{k}_{ib_{1}}\right).$
(206)

Here we are raising and lowering matter indices using the fact that the matter
representation has a symmetric invariant pairing.

Since $S_{a_{3}a_{4}i}^{j}$ is the matrix element of
${\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}+{\sf{t}}_{a_{4}}{\sf{t}}_{a_{3}}$, we can
write this as

$\tfrac{1}{2}T_{a_{1}a_{2}}^{b_{1}b_{2}}\operatorname{tr}_{R}\left(({\sf{t}}_{b_{1}}{\sf{t}}_{b_{2}}+{\sf{t}}_{b_{2}}{\sf{t}}_{b_{1}})({\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}+{\sf{t}}_{a_{4}}{\sf{t}}_{a_{3}})\right)$
(207)

where $R$ is the matter representation. Since

$T_{a_{1}a_{2}}^{b_{1}b_{2}}{\sf{t}}_{b_{1}}{\sf{t}}_{b_{2}}=-[{\sf{t}}_{a_{1}},{\sf{t}}_{e}][{\sf{t}}_{a_{2}},{\sf{t}}_{e}]-[{\sf{t}}_{a_{2}},{\sf{t}}_{e}][{\sf{t}}_{a_{1}},{\sf{t}}_{e}]$
(208)

we find that (206) is

$-\operatorname{tr}_{R}\left([{\sf{t}}_{a_{1}},{\sf{t}}_{e}][{\sf{t}}_{a_{2}},{\sf{t}}_{e}]{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)+\text{
permutations }$ (209)

where as before the permutations are under $\mathbb{Z}/2\times\mathbb{Z}/2$
swapping $(12)$ and $(34)$.

The sum of the second and third terms is then

$\begin{split}-\operatorname{tr}_{R}\left([{\sf{t}}_{a_{1}},{\sf{t}}_{e}][{\sf{t}}_{a_{2}},{\sf{t}}_{e}]{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)\\\
-\operatorname{tr}_{R}\left({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}[{\sf{t}}_{a_{3}},{\sf{t}}_{e}][{\sf{t}}_{a_{4}},{\sf{t}}_{e}]\right)\\\
+\text{ permutations }\end{split}$ (210)

###  B.3 The fourth term

The fourth term is

$-\tfrac{1}{2}S^{i_{1}}_{a_{1}a_{2}i_{2}}S^{i_{3}}_{a_{3}a_{4}i_{4}}\left(g_{i_{1}i_{4}a}g_{i_{3}i_{2}a}+g_{i_{2}i_{4}a}g_{i_{3}i_{1}ia}\right).$
(211)

This is

$-\operatorname{tr}_{R}\left({\sf{t}}_{e}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}+{\sf{t}}_{a_{2}}{\sf{t}}_{a_{1}}){\sf{t}}_{e}({\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}+{\sf{t}}_{a_{4}}{\sf{t}}_{a_{3}})\right)$
(212)

which is

$-\operatorname{tr}_{R}\left({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)+\text{
permutations }$ (213)

Putting this together with the second and third terms we get

$\begin{split}\operatorname{tr}_{R}\left({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)-\operatorname{tr}_{R}({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-\operatorname{tr}_{R}({\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
-\operatorname{tr}_{R}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}})-\operatorname{tr}_{R}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}}{\sf{t}}_{e})+\operatorname{tr}_{R}({\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+\operatorname{tr}_{R}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{e}{\sf{t}}_{a_{4}})\\\
+\text{ permutations }\end{split}$ (214)

This is the complete contribution of the matter terms.

###  B.4 Contribution of matter in the fundamental and anti-fundamental

Let us now take $R$ to be $N_{F}$ copies of the fundamental plus anti-
fundamental, and determine the contribution of $R$.

$\sum{\sf{t}}_{e}\otimes{\sf{t}}_{e}=\sum
t^{i}_{j}\otimes{\sf{t}}^{j}_{i}-\frac{1}{N}\operatorname{Id}\otimes\operatorname{Id}.$
(215)

Each term in equation (214) gives a single-trace expression coming from the
$-N^{-1}\operatorname{Id}\otimes\operatorname{Id}$. Together, these give

$2N_{F}N^{-1}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+\text{
permutations }.$ (216)

(The factor of $2$ is because we have $N_{F}$ fundamental and $N_{F}$ anti-
fundamental. Fundamental and anti-fundamental contribute the same, because the
contributions are related by a permutation in
$\mathbb{Z}/2\times\mathbb{Z}/2$).

There are also contributions from the $\mathfrak{gl}_{N}$ quadratic Casimir
$\sum{\sf{t}}_{i}^{j}\otimes{\sf{t}}_{j}^{i}$. These only occur when the two
copies of ${\sf{t}}_{e}$ are adjacent. We can replace
${\sf{t}}_{e}{\sf{t}}_{e}$ by $N$. These terms contribute

$4N_{F}N\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+\text{
permutations }.$ (217)

The total contribution of matter in the fundamental and anti-fundamental is

$(4N_{F}N+2N_{F}N^{-1})\operatorname{tr}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{4}})+\text{
permutations }.$ (218)

###  B.5 Contribution of adjoint matter

Next we will calculate what happens with adjoint matter. This will give us the
two-loop amplitudes for $N=1$ supersymmetric gauge theory. These we know to be
zero, so by showing that the matter contribution cancels with the pure gauge
theory contribution we will find a non-trivial cross-check.

We need to compute the single trace part of

$\begin{split}\operatorname{tr}_{F^{\vee}\otimes
F}\left({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)-\operatorname{tr}_{F^{\vee}\otimes
F}({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-\operatorname{tr}_{F^{\vee}\otimes
F}({\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
-\operatorname{tr}_{F^{\vee}\otimes
F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}})-\operatorname{tr}_{F^{\vee}\otimes
F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}}{\sf{t}}_{e})+\operatorname{tr}_{F^{\vee}\otimes
F}({\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+\operatorname{tr}_{F^{\vee}\otimes
F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{e}{\sf{t}}_{a_{4}})\\\
+\text{ permutations }.\end{split}$ (219)

We will use the fact that

$\operatorname{tr}_{F^{\vee}\otimes
F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})=\sum_{I\subset\\{1,2,3,4\\}}(-1)^{\left|I\right|}\operatorname{tr}_{F}({\sf{t}}_{\overline{I}})\operatorname{tr}_{F}({\sf{t}}_{I^{c}})$
(220)

where $\overline{I}$ is the set $I$ with opposite ordering.

Also, the identity in $\mathfrak{gl}_{N}$ acts by $0$ in the adjoint
representation, so that we can use the $\mathfrak{gl}_{N}$ Casimir instead of
the $\mathfrak{sl}_{N}$ Casimir.

We find that

$\begin{split}\operatorname{tr}_{F^{\vee}\otimes
F}\left({\sf{t}}_{j}^{i}{\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{i}^{j}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)&=(2N^{2}+8)\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-8\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\\\
+\text{ irrelevant terms }\\\ \operatorname{tr}_{F^{\vee}\otimes
F}({\sf{t}}_{j}^{i}{\sf{t}}_{a_{1}}{\sf{t}}_{i}^{j}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})&=2N^{2}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
+\text{ irrelevant terms }\\\ \operatorname{tr}_{F^{\vee}\otimes
F}({\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})&=4N^{2}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
+\text{ irrelevant terms }\end{split}$ (221)

where irrelevant terms are those that are either double trace, or average to
zero under the $\mathbb{Z}/2\times\mathbb{Z}/2$ action.

These equations tell us that (219) is

$\begin{split}2N^{2}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+8\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-8\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\\\
+\text{ permutations }\end{split}$ (222)

###  B.6 Contribution of matter in $\wedge^{2}F$

Matter in $\wedge^{2}F$ and $\wedge^{2}F^{\vee}$ is more challenging to deal
with. Let $P:F\otimes F\to F\otimes F$ be the permutation operator. Then, for
any elements ${\sf{t}}_{a_{i}}\in\mathfrak{sl}_{N}$, we have

$\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{n}})=\tfrac{1}{2}\operatorname{tr}_{F\otimes
F}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{n}}(1-P)).$ (223)

This is because $\wedge^{2}F$ is the image of the projector
$\tfrac{1}{2}(1-P)$.

We can write traces in $F\otimes F$ as follows. For any subset
$I\subset\\{1,\dots,n\\}$, let ${\sf{t}}_{I}$ be the ordered product of the
elements ${\sf{t}}_{a_{i}}$ for $i\in I$. Let $I^{c}$ be the complement of
$I$; similarly we have ${\sf{t}}_{I^{c}}$. Then,

$\operatorname{tr}_{F\otimes
F}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{n}})=\sum_{I\subset\\{1,\dots,n\\}}\operatorname{tr}_{F}({\sf{t}}_{I})\operatorname{tr}_{F}({\sf{t}}_{I^{c}}).$
(224)

This follows from the fact that the Lie algebra element ${\sf{t}}_{a}$ acts as
${\sf{t}}_{a}\otimes 1+1\otimes{\sf{t}}_{a}$ in the representation $F\otimes
F$.

Similarly,

$\operatorname{tr}_{F\otimes
F}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{n}}P)=\sum_{I\subset\\{1,\dots,n\\}}\operatorname{tr}_{F}({\sf{t}}_{I}{\sf{t}}_{I^{c}}).$
(225)

Therefore

$\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{n}})=\tfrac{1}{2}\sum_{I\subset\\{1,\dots,n\\}}\left(\operatorname{tr}_{F}({\sf{t}}_{I})\operatorname{tr}_{F}({\sf{t}}_{I^{c}})-\operatorname{tr}_{F}({\sf{t}}_{I}{\sf{t}}_{I^{c}})\right)$
(226)

Focusing on the contribution from $\wedge^{2}F$, we would like to extract the
single-trace part of

$\begin{split}\operatorname{tr}_{\wedge^{2}F}\left({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}})-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}}{\sf{t}}_{e})+\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{e}{\sf{t}}_{a_{4}})\\\
+\text{ permutations }.\end{split}$ (227)

The first thing to note is that the quadratic Casimir
${\sf{t}}_{e}{\sf{t}}_{e}$ acts as

$2N-2-4N^{-1}$ (228)

in $\wedge^{2}F$, so we can rewrite the previous expression as

$\begin{split}\operatorname{tr}_{\wedge^{2}F}\left({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}})-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}}{\sf{t}}_{e})+(4N-4-8N^{-1})\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
+\text{ permutations }.\end{split}$ (229)

The next step will be to expand the quadratic Casimir into the
$\mathfrak{gl}_{N}$ part ${\sf{t}}_{i}^{j}\otimes{\sf{t}}_{j}^{i}$ and the
identity contribution $-N^{-1}{\sf{t}}^{i}_{i}\otimes{\sf{t}}^{j}_{j}$.

The identity matrix ${\sf{t}}^{i}_{i}$ in $\mathfrak{gl}_{N}$ acts, in the
representation $\wedge^{2}F$, by $2$ (because by the Leibniz rule we must add
the actions on each tensor factor). The identity term in the Casimir is
$-N^{-1}{\sf{t}}^{i}_{i}\otimes{\sf{t}}^{j}_{j}$, with two copies of the
identity matrix. This acts by $-4N^{-1}$ in $\wedge^{2}F$. Therefore, each of
the first five terms in equation (232) contributes $-4N^{-1}$. The total
contribution is us
$12N^{-1}\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{4}})$.
This brings equation (227) to the form

$\begin{split}\operatorname{tr}_{\wedge^{2}F}\left({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}})-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}}{\sf{t}}_{e})+(4N-4+4N^{-1})\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
+\text{ permutations }.\end{split}$ (230)

where we are now using the $\mathfrak{gl}_{N}$ Casimir.

We need to write this in terms of single traces. We focus on the last term
first. We have

$\begin{split}\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})=(N-6)\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
-2\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})+\text{
irrelevant terms }\end{split}$ (231)

where irrelevant terms include double traces and expressions that vanish when
we average over the $\mathbb{Z}/2\times\mathbb{Z}/2$. This means we can
rewrite (230) as

$\begin{split}\operatorname{tr}_{\wedge^{2}F}\left({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}})-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}}{\sf{t}}_{e})\\\
+(4N-4+4N^{-1})\left((N-6)\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-2\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\right)+\text{
permutations }.\end{split}$ (232)

Now we need to determine the contribution of the five terms

$\begin{split}\operatorname{tr}_{\wedge^{2}F}\left({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{e}{\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{e}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{e}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}})-\operatorname{tr}_{\wedge^{2}F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{e}{\sf{t}}_{a_{4}}{\sf{t}}_{e})\end{split}$
(233)

where, as above, we are using the $\mathfrak{gl}_{N}$ Casimir. We will do this
using the expansion of trace in $\wedge^{2}F$ in terms of single and double
traces, given by (226) which I reproduce here for convenience:

$\operatorname{tr}_{\wedge^{2}F}({\sf{t}}^{i}_{j}{\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}^{j}_{i}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})=\tfrac{1}{2}\sum_{I\subset\\{\circ,1,2,\bullet,3,4\\}}\left(\operatorname{tr}_{F}({\sf{t}}_{I})\operatorname{tr}_{F}({\sf{t}}_{I^{c}})-\operatorname{tr}_{F}({\sf{t}}_{I}{\sf{t}}_{I^{c}})\right)$
(234)

Here the element $\circ$ in the set $\\{\circ,1,2,\bullet,3,4\\}$ indicates
${\sf{t}}^{i}_{j}$ and $\bullet$ indicates ${\sf{t}}^{j}_{i}$.

The following terms will be ultimately single trace:

  1. 1.

Double trace expressions where $t^{i}_{j}$ and $t_{j}^{i}$ are in different
traces.

  2. 2.

Single trace expressions where $t^{i}_{j}$ and $t_{j}^{i}$ are adjacent.

There are terms of order $N^{2}$, order $N$, and order $1$101010In the first
version of this article, I missed the order $N^{2}$ terms in this equation,
and made some further elementary errors in the colour algebra computation.
This led to an error in the formula for the two-loop amplitude which was
pointed out by Lance Dixon and Anthony Morales. I am very grateful to them for
pointing out this error, and to Anthony Morales for performing an independent
verification of the colour algebra in this section.. Let us discuss them in
turn, first studying the contribution of the expression

$\operatorname{tr}_{\wedge^{2}F}\left({\sf{t}}_{i}^{j}{\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{j}^{i}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right).$
(235)

The order $N^{2}$ term arises from those terms in (234) which are double
traces of the form

$\operatorname{tr}_{F}({\sf{t}}^{i}_{j}{\sf{t}}_{i}^{j})\operatorname{tr}_{F}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{4}}).$
(236)

This can happen in two ways: where the set $I$ in equation (234) consists of
$\\{\circ\bullet\\}$, or where the complement of $I$ consists of
$\\{\circ\bullet\\}$. But taking into account the factor of $\tfrac{1}{2}$, we
find the order $N^{2}$ term is
$N^{2}\operatorname{tr}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{4}})$.

The order $N$ term arises from those terms in (234) of the form
$-\tfrac{1}{2}\operatorname{tr}_{F}({\sf{t}}_{I}{\sf{t}}_{I^{c}})$ where
${\sf{t}}^{i}_{j}{\sf{t}}^{j}_{i}$ are adjacent. There several ways this can
happen. First, $I$ can contain both $\circ$ and $\bullet$. For
${\sf{t}}^{i}_{j}$ and ${\sf{t}}^{j}_{i}$ to be adjacent in the trace, $I$ can
not contain $1$ or $2$. This gives $4$ terms, all of which are related by a
permutation in $\mathbb{Z}/2\times\mathbb{Z}/2$ to
$-N\operatorname{tr}_{F}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{4}})$. There are
similar $4$ possibilities from the case when $I^{c}$ contains both $\circ$ and
$\bullet$. Together, taking account the factor of $\tfrac{1}{2}$, this
contributes $-4N\operatorname{tr}_{F}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{4}})$.

Or, $I$ contains exactly one of $\circ$ or $\bullet$. The two contributions
are symmetric, so we can assume that $I$ contains $\circ$; this absorbs the
factor of $\tfrac{1}{2}$. Then, for ${\sf{t}}^{i}_{j}$ and ${\sf{t}}^{j}_{i}$
to be adjacent in the trace, the final element of the set $I^{c}$ must be
$\bullet$. This happens if and only if $I$ also contains the elements $3$ and
$4$, together with an arbitrary subset of $\\{1,2\\}$. This gives $4$ terms,
all of which are related by a permutation in $\mathbb{Z}/2\times\mathbb{Z}/2$
to $-N\operatorname{tr}_{F}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{4}})$. Therefore
the complete order $N$ term is

$-8N\operatorname{tr}({\sf{t}}_{a_{1}}\dots\sf{t}_{a_{4}})+\text{ irrelevant
terms }$ (237)

where irrelevant terms are those which vanish when we average over the
$\mathbb{Z}/2\times\mathbb{Z}/2$ subgroup of $S_{4}$ permuting $(12)$ and
$(34)$.

Finally, there are $32$ order $1$ terms. These come from the terms
$\tfrac{1}{2}\operatorname{tr}({\sf{t}}_{I})\operatorname{tr}({\sf{t}}_{I^{c}})$
in the expansion (234) where ${\sf{t}}^{i}_{j}$ and ${\sf{t}}^{j}_{i}$ are in
different traces. Without loss of generality, we can assume that $\circ\in I$
and $\bullet\in I^{c}$; this absorbs the factor of $\tfrac{1}{2}$. Writing
$I=\\{\circ\\}\cup J$ for $J\subset\\{1,\dots,4\\}$, we find the order $1$
terms are

$\sum_{J\subset\\{1,\dots,4\\}}\operatorname{tr}({\sf{t}}_{J}{\sf{t}}_{\widetilde{J}^{c}})$
(238)

where $\widetilde{J}^{c}$ means the complement of the subset $J$, but ordered
according to the ordering as a subset of $\\{3,4,1,2\\}$.

Each of the $16$ order $1$ terms can be permuted, using an element of the
$\mathbb{Z}/2\times\mathbb{Z}/2$ inside $S_{4}$, to
$\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})$.
Thus, we can write the order $1$ terms as
$16\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})$
plus irrelevant terms.

In sum, we have

$\begin{split}\operatorname{tr}_{\wedge^{2}F}\left({\sf{t}}_{i}^{j}{\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{j}^{i}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)&=N^{2}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-8N\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+16\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})\\\
&+\text{ irrelevant terms }.\end{split}$ (239)

In a similar way, we can compute

$\operatorname{tr}_{\wedge^{2}F}\left({\sf{t}}_{i}^{j}{\sf{t}}_{a_{1}}{\sf{t}}^{j}_{i}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right).$
(240)

We follow the previous analysis. We use the formula

$\operatorname{tr}_{\wedge^{2}F}({\sf{t}}^{i}_{j}{\sf{t}}_{a_{1}}{\sf{t}}^{j}_{i}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})=\tfrac{1}{2}\sum_{I\subset\\{\circ,1,\bullet,2,3,4\\}}\left(\operatorname{tr}_{F}({\sf{t}}_{I})\operatorname{tr}_{F}({\sf{t}}_{I^{c}})-\operatorname{tr}_{F}({\sf{t}}_{I}{\sf{t}}_{I^{c}})\right)$
(241)

where $\circ$ corresponds to ${\sf{t}}^{i}_{j}$ and $\bullet$ corresponds to
${\sf{t}}^{j}_{i}$.

The order $N^{2}$ term is
$N^{2}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})$,
exactly as before.

To compute the order $N$ term, we argue as before. The order $N$ term comes
from the $-\tfrac{1}{2}\operatorname{tr}({\sf{t}}_{I}{\sf{t}}_{I^{c}})$ terms
where ${\sf{t}}^{i}_{j}$ and ${\sf{t}}^{j}_{i}$ are adjacent in the trace.

Suppose the the set $I$ contains exactly one of $\circ$ or $\bullet$. If $I$
contains $\circ$, then the final element of $I^{c}$ must be $\bullet$, in
order for ${\sf{t}}^{i}_{j}$ and ${\sf{t}}^{j}_{i}$ to be adjacent in the
trace. There are two ways this can happen, if $I^{c}=\\{\bullet\\}$ or
$I^{c}=\\{1,\bullet\\}$. Similarly, if $I$ contains $\bullet$, then this must
be the final element of the set $I$, so that either $I=\\{1,\bullet\\}$ or
$I=\\{\bullet\\}$. These $4$ terms (taking account the factor of
$\tfrac{1}{2}$) contribute

$-2N\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}).$
(242)

If $I$ contains both $\circ$ and $\bullet$, then it can not contain $1$,
leaving $8$ possibilities, and a similar $8$ possibilities from the case where
$I$ contains neither $\circ$ or $\bullet$. Taking into account the factor of
$\tfrac{1}{2}$, these contribute

$-6N\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-2N\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}).$
(243)

so that the total order $N$ term is

$-8N\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-2N\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}).$
(244)

There are $32$ order $1$ terms as before, coming from
$\tfrac{1}{2}\operatorname{tr}({\sf{t}}_{I})\operatorname{tr}({\sf{t}}_{I^{c}})$
where ${\sf{t}}^{i}_{j}$ and ${\sf{t}}^{j}_{i}$ are in different traces.
Taking account the factor of $\tfrac{1}{2}$, and considering traces related by
the $\mathbb{Z}/2\times\mathbb{Z}/2$ permutations to be equivalent, we find
the order $1$ term is

$12\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+4\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}}).$
(245)

Putting this together gives

$\begin{split}\operatorname{tr}_{\wedge^{2}F}\left({\sf{t}}_{i}^{j}{\sf{t}}_{a_{1}}{\sf{t}}^{j}_{i}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}}\right)&=N^{2}\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-8N\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-2N\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\\\
&+12\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+4\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\\\
&+\text{ irrelevant terms }.\end{split}$ (246)

Irrelevant terms are either double trace, or expressions that sum to zero when
we sum over permutations in $\mathbb{Z}/2\times\mathbb{Z}/2$.

Therefore the single trace contribution of the five terms in equation (233) is

$\begin{split}(-3N^{2}+24N-32)\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+(8N-16)\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\\\
+\text{ irrelevant terms }\end{split}$ (247)

Finally, our initial expression (227) becomes

$\begin{split}(-3N^{2}+24N-32)\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+(8N-16)\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\\\
+(4N-4+4N^{-1})\left((N-6)\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})-2\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\right)\\\
+\text{ irrelevant terms }.\end{split}$ (248)

This can be simplified to

$\begin{split}\left(N^{2}-4N-4-24N^{-1}\right)\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+(-8-8N^{-1})\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\\\
+\text{ irrelevant terms }.\end{split}$ (249)

If we include the contribution from the $\wedge^{2}F^{\vee}$ representation,
we double this expression to find

$\begin{split}\left(2N^{2}-8N-8-48N^{-1}\right)\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+(-16-16N^{-1})\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\\\
+\text{ irrelevant terms }.\end{split}$ (250)

Restoring the sum over permutations in $\mathbb{Z}/2\times\mathbb{Z}/2$, the
final answer is

$\begin{split}\left(2N^{2}-8N-8-48N^{-1}\right)\operatorname{tr}_{F}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{4}})+(-16-16N^{-1})\operatorname{tr}({\sf{t}}_{a_{1}}{\sf{t}}_{a_{3}}{\sf{t}}_{a_{2}}{\sf{t}}_{a_{4}})\\\
+\text{ permutations}.\end{split}$ (251)

## References

  * (1) K. Costello and N. M. Paquette, _Associativity of One-Loop Corrections to the Celestial Operator Product Expansion_ , _Phys. Rev. Lett._ 129 (2022) 231604, [2204.05301].
  * (2) N. Arkani-Hamed, L. J. Dixon, A. J. McLeod, M. Spradlin, J. Trnka and A. Volovich, _Solving Scattering in $N$ = 4 Super-Yang-Mills Theory_, 2207.10636.
  * (3) Z. Bern, L. J. Dixon and D. A. Kosower, _A Two loop four gluon helicity amplitude in QCD_ , _JHEP_ 01 (2000) 027, [hep-ph/0001001].
  * (4) S. Badger, H. Frellesvig and Y. Zhang, _A Two-Loop Five-Gluon Helicity Amplitude in QCD_ , _JHEP_ 12 (2013) 045, [1310.1051].
  * (5) S. Badger, G. Mogull, A. Ochirov and D. O’Connell, _A Complete Two-Loop, Five-Gluon Helicity Amplitude in Yang-Mills Theory_ , _JHEP_ 10 (2015) 064, [1507.08797].
  * (6) D. C. Dunbar and W. B. Perkins, _Two-loop five-point all plus helicity Yang-Mills amplitude_ , _Phys. Rev. D_ 93 (2016) 085029, [1603.07514].
  * (7) D. C. Dunbar, G. R. Jehu and W. B. Perkins, _Two-loop six gluon all plus helicity amplitude_ , _Phys. Rev. Lett._ 117 (2016) 061602, [1605.06351].
  * (8) D. C. Dunbar, J. H. Godwin, G. R. Jehu and W. B. Perkins, _Analytic all-plus-helicity gluon amplitudes in QCD_ , _Phys. Rev. D_ 96 (2017) 116013, [1710.10071].
  * (9) A. R. Dalgleish, D. C. Dunbar, W. B. Perkins and J. M. W. Strong, _Full color two-loop six-gluon all-plus helicity amplitude_ , _Phys. Rev. D_ 101 (2020) 076024, [2003.00897].
  * (10) D. C. Dunbar, G. R. Jehu and W. B. Perkins, _The two-loop n-point all-plus helicity amplitude_ , _Phys. Rev. D_ 93 (2016) 125006, [1604.06631].
  * (11) D. A. Kosower and S. Pögel, _A Unitarity Approach to Two-Loop All-Plus Rational Terms_ , 2206.14445.
  * (12) D. A. Kosower and S. Pögel, _Yang–Mills All-Plus: Two Loops for the Price of One_ , _PoS_ LL2022 (2022) 031, [2208.06209].
  * (13) L. F. Alday, D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, _An Operator Product Expansion for Polygonal null Wilson Loops_ , _JHEP_ 04 (2011) 088, [1006.2788].
  * (14) B. Basso, A. Sever and P. Vieira, _Spacetime and Flux Tube S-Matrices at Finite Coupling for N=4 Supersymmetric Yang-Mills Theory_ , _Phys. Rev. Lett._ 111 (2013) 091602, [1303.1396].
  * (15) S. Caron-Huot, L. J. Dixon, J. M. Drummond, F. Dulat, J. Foster, O. Gürdoğan et al., _The Steinmann Cluster Bootstrap for $N$ = Super Yang-Mills Amplitudes_, _PoS_ CORFU2019 (2020) 003, [2005.06735].
  * (16) K. Costello and N. M. Paquette, _Celestial holography meets twisted holography: 4d amplitudes from chiral correlators_ , 2201.02595.
  * (17) A. Guevara, E. Himwich, M. Pate and A. Strominger, _Holographic Symmetry Algebras for Gauge Theory and Gravity_ , 2103.03961.
  * (18) A. Strominger, _Lectures on the Infrared Structure of Gravity and Gauge Theory_ , 1703.05448.
  * (19) A. Ball, S. Narayanan, J. Salzer and A. Strominger, _Perturbatively Exact $w_{1+\infty}$ Asymptotic Symmetry of Quantum Self-Dual Gravity_, 2111.10392.
  * (20) K. J. Costello, _Quantizing local holomorphic field theories on twistor space_ , 2111.08879.
  * (21) F. Cachazo, P. Svrcek and E. Witten, _MHV vertices and tree amplitudes in gauge theory_ , _JHEP_ 09 (2004) 006, [hep-th/0403047].
  * (22) Z. Bern and D. A. Kosower, _The Computation of loop amplitudes in gauge theories_ , _Nucl. Phys. B_ 379 (1992) 451–561.
  * (23) G. Mahlon, _Multi - gluon helicity amplitudes involving a quark loop_ , _Phys. Rev. D_ 49 (1994) 4438–4453, [hep-ph/9312276].
  * (24) Z. Bern, G. Chalmers, L. J. Dixon and D. A. Kosower, _One loop N gluon amplitudes with maximal helicity violation via collinear limits_ , _Phys. Rev. Lett._ 72 (1994) 2134–2137, [hep-ph/9312333].
  * (25) S. J. Parke and T. R. Taylor, _An Amplitude for $n$ Gluon Scattering_, _Phys. Rev. Lett._ 56 (1986) 2459.
  * (26) Z. Bern, L. J. Dixon and D. A. Kosower, _One loop corrections to five gluon amplitudes_ , _Phys. Rev. Lett._ 70 (1993) 2677–2680, [hep-ph/9302280].
  * (27) Z. Bern, L. J. Dixon and D. A. Kosower, _The last of the finite loop amplitudes in QCD_ , _Phys. Rev. D_ 72 (2005) 125003, [hep-ph/0505055].
  * (28) L. J. Dixon, E. W. N. Glover and V. V. Khoze, _MHV rules for Higgs plus multi-gluon amplitudes_ , _JHEP_ 12 (2004) 015, [hep-th/0411092].
  * (29) R. Boels, L. J. Mason and D. Skinner, _From twistor actions to MHV diagrams_ , _Phys. Lett. B_ 648 (2007) 90–96, [hep-th/0702035].
  * (30) D. A. Kosower and P. Uwer, _One loop splitting amplitudes in gauge theory_ , _Nucl. Phys. B_ 563 (1999) 477–505, [hep-ph/9903515].
  * (31) S. Okubo, _Quartic Trace Identity for Exceptional Lie Algebras_ , _J. Math. Phys._ 20 (1979) 586.
  * (32) R. Britto, F. Cachazo, B. Feng and E. Witten, _Direct proof of tree-level recursion relation in Yang-Mills theory_ , _Phys. Rev. Lett._ 94 (2005) 181602, [hep-th/0501052].
  * (33) Z. Bern, L. J. Dixon and D. A. Kosower, _On-shell recurrence relations for one-loop qcd amplitudes_ , _Physical Review D_ 71 (2005) 105013.
  * (34) K. Zeng, _Twisted Holography and Celestial Holography from Boundary Chiral Algebra_ , 2302.06693.


The Economy’s Potential
Duality and Equilibrium
Jacob K. Goeree111AGORA Center for Market Design, UNSW. I gratefully
acknowledge funding from the Australian Research Council (DP190103888 and
DP220102893). I thank Brett Williams for useful comments.
August 28, 2024

Abstract

I introduce a concave function of allocations and prices – the economy’s
potential – which measures the difference between utilitarian social welfare
and its dual. I show that Walrasian equilibria correspond to roots of the
potential: allocations maximize weighted utility and prices minimize weighted
indirect utility. Walrasian prices are “utility clearing” in the sense that
the utilities consumers expect at Walrasian prices are just feasible. I
discuss the implications of this simple duality for equilibrium existence, the
welfare theorems, and the interpretation of Walrasian prices.

Keywords: Walrasian equilibrium, Pareto optimality, duality, potential, roots

##  1 Introduction

A landmark of economic theory concerns the determination of prices for all
goods in the economy. General equilibrium theory originated with Walras
“$\ldots$ whose system of equations, defining equilibria in a system of
interdependent quantities, is the Magna Carta of economic theory” (Schumpeter,
1954). The modern approach to general equilibrium theory is due to Arrow and
Debreu (1954) and McKenzie (1954) who established existence of Walrasian
equilibria. Duffie and Sonnenschein (1989) describe the history and importance
of this existence proof, which allowed general equilibrium theory to gain the
central role it now occupies in economics and finance.

Arrow and Debreu’s proof pertains to “abstract economies” or “generalized
games” and builds on Nash’s (1950) existence proof for normal-form games.
Their abstract approach establishes existence of a solution to a system of
equations but does not reveal if, or why, it has desirable properties. This is
the content of the welfare theorems, the modern versions of which are also due
to Arrow (1951) and Debreu (1951). The first welfare theorem – that the price
system results in a Pareto optimal allocation of resources – is perhaps the
central result in price theory. The second welfare theorem states the
converse, i.e. that every Pareto optimal allocation is Walrasian.

Despite their dual formulations, the proofs of the two welfare theorems are
very different in nature. The first welfare theorem requires only positive
marginal utility of income while the second welfare theorem hinges on the
assumption of convexity. Duffie and Sonnenschein (1989) note that as a result
of Arrow and Debreu’s work “$\ldots$ the separateness of the two welfare
theorems was brought into sharp focus.”

I demonstrate that the welfare theorems encapsulate a simple duality property
of Walrasian equilibria. To this end, I introduce the economy’s potential,
which is a weighted sum of consumers’ utilities of their allocations minus
their indirect utilities at given prices. I show that, for any welfare
weights, the potential is a non-positive and strictly concave function with a
unique root corresponding to the Walrasian equilibrium. The intuition is that
allocations solve the primal problem of maximizing weighted utilities and
prices solve the dual problem of minimizing weighted indirect utilities.
Walrasian prices are “utility clearing” in that the utilities consumers expect
are just feasible, i.e. the potential vanishes at the Walrasian equilibrium.

Usually, the economy is parameterized by initial endowments rather than
welfare weights. While the set of endowments is of higher dimension than the
set of welfare weights,111With $N>1$ consumers and $K>1$ goods the set of
endowments is $NK$ dimensional while the set of welfare weights is $N$
dimensional. no additional equilibria are introduced. The reason is that any
endowments and resulting Walrasian price imply some set of welfare weights.
Moreover, existence of Walrasian equilibrium for arbitrary endowments follows
if there is at least one vector of “equilibrium weights” that produce the
correct incomes. I show that existence of such equilibrium weights readily
follows from the Poincare–Hopf theorem.

The next section presents a graphical illustration of duality for a simple
exchange economy. Section 3 introduces the economy’s potential, generalizes
the duality result to any exchange economy, and shows that existence and the
welfare theorems are direct corollaries of duality. Furthermore, duality
provides a novel interpretation of the Walrasian price vector as the gradient
of utilitarian social welfare. Section 4 discusses possible applications of
the potential to non-convex economies. Proofs can be found in the Appendix.

##  2 An Example

Consider an exchange economy with two consumers with Cobb-Douglas preferences
$u_{1}(x,y)=2\log(x)+\log(y)$ and $u_{2}(x,y)=\log(x)+2\log(y)$. Suppose there
are three units of each good in the economy. Welfare maximization

$\max_{{0\,\leq\,x_{1}+x_{2}\,\leq\,3}\atop{0\,\leq\,y_{1}+y_{2}\,\leq\,3}}\,\alpha
u_{1}(x_{1},y_{1})+(1-\alpha)u_{2}(x_{2},y_{2})$ (1)

yields the Pareto-optimal allocations
$(x_{1}(\alpha),y_{1}(\alpha))=({\textstyle{6\alpha\over
1+\alpha}},{\textstyle{3\alpha\over 2-\alpha}})$ and
$(x_{2}(\alpha),y_{2}(\alpha))=({\textstyle{3-3\alpha\over
1+\alpha}},{\textstyle{6-6\alpha\over 2-\alpha}})$. The red curve in Figure 1
shows the resulting utility pairs $(u_{1}(\alpha),u_{2}(\alpha))$ for
$\alpha\in(0,1)$. The shaded area corresponds to the utility possibility set.

The dual of (1) entails minimizing a weighted sum of indirect utilities with
respect to prices. If prices are normalized to sum to one, i.e. the price
vector is $(p,1-p)$, then the indirect utility functions can be written as
$v_{1}(p,m_{1})=2\log({\textstyle{2m_{1}\over
3p}})+\log({\textstyle{m_{1}\over 3(1-p)}})$ and
$v_{2}(p,m_{2})=\log({\textstyle{m_{2}\over
3p}})+2\log({\textstyle{2m_{2}\over 3(1-p)}})$. The economy’s total income is
$3p+3(1-p)=3$ and a consumer’s welfare weight determines her share:222More
generally, a consumer’s welfare weight is equal to the inverse of her marginal
utility of income, see Theorem 1 below. For the economy studied here this
yields $\alpha=m_{1}/3$ and $1-\alpha=m_{2}/3$. $m_{1}=3\alpha$ and
$m_{2}=3(1-\alpha)$. The dual of (1) is thus

$\min_{{0\,\leq\,p\,\leq\,1}}\,\alpha
v_{1}(p,3\alpha)+(1-\alpha)v_{2}(p,3(1-\alpha))$ (2)

The blue curves show the indirect utility pairs
$(v_{1}(p,3\alpha),v_{2}(p,3(1-\alpha))$ for different values of $\alpha$ and
$p\in(0,1)$. For $\alpha={\textstyle{1\over 2}}$, the weighted sum of
utilities is constant on the dashed lines and increasing in the North-East
direction. The weighted sum of indirect utilities is constant on the dotted
lines and decreasing in the South-West direction. There is a unique point
where weighted utility is maximized and weighted indirect utility minimized
and their values are equal. This point corresponds to the Walrasian
equilibrium.

$u_{1}$$u_{2}$$\alpha={\textstyle{4\over 5}}$$\alpha={\textstyle{1\over
2}}$$\alpha={\textstyle{1\over 5}}$ Figure 1: Illustration of duality. The
shaded area is the utility possibility set and the red curve its frontier. The
blue curves depict indirect utilities for given welfare weights as functions
of prices. For $\alpha={\textstyle{1\over 2}}$, weighted utility is constant
on the dashed lines and increases to the North-East while weighted indirect
utility is constant on the dotted lines and decreases to the South-West. The
unique point where weighted utility and weighted indirect utility match
corresponds to the Walrasian equilibrium.

To see this, note that the solution to (2) is $p(\alpha)={\textstyle{1\over
3}}(1+\alpha)$ and it is readily verified that the price ratio

$\frac{p(\alpha)}{1-p(\alpha)}\,=\,\frac{1+\alpha}{2-\alpha}$

is equal to consumers’ marginal rates of substitution at
$(x_{1}(\alpha),y_{1}(\alpha))$ and $(x_{2}(\alpha),y_{2}(\alpha))$. Hence,
these allocations maximize consumers’ utilities given prices
$(p(\alpha),1-p(\alpha))$. In other words, the Pareto-optimal allocations that
follow from (1) form a Walrasian equilibrium together with the price that
follows from the dual program in (2).

##  3 The Economy’s Potential

Consider an exchange economy with $\mathcal{N}=\\{1,\ldots,N\\}$ consumers and
$\mathcal{K}=\\{1,\ldots,K\\}$ goods. For $i\in\mathcal{N}$, let
$u_{i}:\mathbb{R}^{K}_{\geq 0}\rightarrow\mathbb{R}$ denote consumer $i$’s
utility function and $\omega_{i}\in\mathbb{R}^{K}_{>0}$ consumer $i$’s
endowment. I assume the utility functions are strictly increasing, strictly
concave, and differentiable.333The usual assumption is that the utility
functions are quasi-concave, which can be made concave by a monotone
transformation. Intuitively, this construction implies that among utility
functions with the same indifference curves there is one with non-increasing
marginal utility. Concave functions are differentiable almost everywhere. For
ease of presentation, I assume differentiability everywhere. For
$k\in\mathcal{K}$, let $w_{k}=\sum_{i\in\mathcal{N}}\omega_{ik}$ denote the
total amount of good $k$ in the economy. The set of feasible allocations is

$F(w)\,=\,\\{x\in\mathbb{R}_{\geq
0}^{NK}\,|\,\sum_{i\,\in\,\mathcal{N}}x_{ik}\,\leq\,w_{k}\,\,\,\forall\,k\in\mathcal{K}\\}$

For vectors $v,v^{\prime}\in\mathbb{R}^{K}$ let $\langle
v|v^{\prime}\rangle=\sum_{k\in\mathcal{K}}v_{k}v^{\prime}_{k}$ denote the
usual inner product. The Fenchel dual of $u_{i}(x_{i})$ is defined as

$\overline{v}_{i}(p)\,=\,\max_{x_{i}\,\geq\,0}\,\,u_{i}(x_{i})-\langle
p|x_{i}\rangle$ (3)

which is a strictly convex function of prices. An envelope-theorem argument
establishes that the solution to (3) satisfies
$\overline{x}_{i}(p)=-\nabla_{p}\overline{v}_{i}(p)$, which is a simplified
version of Roy’s identity. The solution $\overline{x}_{i}(p)$ is strictly
decreasing in each price and further satisfies $\lim_{p_{k}\downarrow
0}\overline{x}_{ik}(p)=\infty$ and
$\lim_{p_{k}\rightarrow\infty}\overline{x}_{ik}(p)=0$.

The next lemma relates $\overline{v}_{i}(p)$ and $\overline{x}_{i}(p)$ to the
traditional indirect utility $v_{i}(p,m_{i})$ and the Marshallian demand
$x_{i}(p,m_{i})$ respectively.

###### Lemma 1

For $i\in\mathcal{N}$, let $\lambda_{i}(p,m_{i})$ solve $\langle
p|\overline{x}_{i}(\lambda_{i}p)\rangle=m_{i}$, then

$\displaystyle v_{i}(p,m_{i})$ $\displaystyle=$
$\displaystyle\lambda_{i}(p,m_{i})m_{i}+\overline{v}_{i}(\lambda_{i}(p,m_{i})p)$
(4) $\displaystyle x_{i}(p,m_{i})$ $\displaystyle=$
$\displaystyle\overline{x}_{i}(\lambda_{i}(p,m_{i})p)$ (5)

Moreover, $\lambda_{i}(p,m_{i})$ equals the marginal utility of income,
$\lambda_{i}(p,m_{i})=\partial v_{i}(p,m_{i})/\partial m_{i}$, and for any
price vector, $\lambda_{i}(p,m_{i})$ is strictly positive and strictly
decreasing in $m_{i}$ with $\lim_{m_{i}\downarrow
0}\lambda_{i}(p,m_{i})=\infty$ and
$\lim_{m_{i}\rightarrow\infty}\lambda_{i}(p,m_{i})=0$.

######  Example 1 (CES utility)

For $i\in\mathcal{N}$ and $\rho_{i}<1$, consider the CES utilities

$u_{i}(x_{i})\,=\,\frac{1}{\rho_{i}}\,\log(\sum_{k\,\in\,\mathcal{K}}a_{ik}x_{ik}^{\rho_{i}})$

with Fenchel duals

$\overline{v}_{i}(p)\,=\,\frac{1-\rho_{i}}{\rho_{i}}\,\log\bigl{(}\,\sum_{k\,\in\,\mathcal{K}}a_{ik}\bigl{(}p_{k}/a_{ik})^{\frac{\rho_{i}}{\rho_{i}-1}}\bigr{)}-1$

Roy’s identity yields

$\overline{x}_{ik}(p)\,=\,\frac{(p_{k}/a_{ik})^{\frac{1}{\rho_{i}-1}}}{\sum_{\ell\,\in\,\mathcal{K}}a_{i\ell}(p_{\ell}/a_{i\ell})^{\frac{\rho_{i}}{\rho_{i}-1}}}$

which satisfies $\overline{x}_{ik}(p/m_{i})=m_{i}\overline{x}_{ik}(p)$ and
$\langle p|\overline{x}_{i}(p)\rangle=1$ so
$\lambda_{i}(p,m_{i})={\textstyle{1\over m_{i}}}$. Marshallian demand is
$x_{i}(p,m_{i})=m_{i}\overline{x}_{i}(p)$ and indirect utility is
$v_{i}(p,m_{i})=1+\overline{v}_{i}(p)+\log(m_{i})$. $\hfill\blacksquare$

Let $U_{\alpha}(x)=\sum_{i\in\mathcal{N}}\alpha_{i}u_{i}(x_{i})$ denote
weighted aggregate utility for some positive weight vector
$\alpha\in\mathbb{R}^{N}_{>0}$. The welfare-maximization program

$W_{\alpha}(w)\,=\,\max_{x\,\in\,F(w)}\,U_{\alpha}(x)$ (6)

is equivalent to the saddle-point problem

$W_{\alpha}(w)\,=\,\min_{p\,\geq\,0}\,\,\max_{x\,\geq\,0}\,\sum_{i\,\in\,\mathcal{N}}\,\alpha_{i}u_{i}(x_{i})-\langle
p|x_{i}-\omega_{i}\rangle$

where the multiplier for the feasibility constraint
$\sum_{i\in\mathcal{N}}(x_{i}-\omega_{i})\leq 0$ is denoted $p$ on purpose.
The maximization over allocations is solved by
$\overline{x}_{i}(p/\alpha_{i})$ and using the Fenchel duals
$\overline{v}_{i}$ we can write the result as

$W_{\alpha}(w)\,=\,\min_{p\,\geq\,0}\,V_{\alpha}(p,w)$ (7)

where

$V_{\alpha}(p,w)\,=\,\langle
p|w\rangle+\sum_{i\,\in\,\mathcal{N}}\,\alpha_{i}\overline{v}_{i}(p/\alpha_{i})$
(8)

is strictly convex in prices.

The economy’s potential is defined as

$Y_{\alpha}(x,p,w)\,=\,U_{\alpha}(x)-V_{\alpha}(p,w)$ (9)

From (6) and (7) it follows that the potential is non-positive for all
feasible allocations and non-negative prices. Moreover, it is a strictly
concave function of allocations and prices since $U_{\alpha}(x)$ is strictly
concave and $V_{\alpha}(p,w)$ is strictly convex.

Let $\lambda_{i}^{(-1)}(p,\cdot\,)$ denote the inverse of
$\lambda_{i}(p,m_{i})$ with respect to the income $m_{i}$.

###### Theorem 1

If $(x,p)$ is a Walrasian equilibrium of the economy with incomes $m_{i}$ for
$i\in\mathcal{N}$ then $\,Y_{\alpha}(x,p,w)=0$ for
$\alpha_{i}=1/\lambda_{i}(p,m_{i})$ and $i\in\mathcal{N}$. Conversely, for any
$\alpha\in\mathbb{R}^{N}_{>0}$, $\,Y_{\alpha}(x,p,w)$ has a unique root
$(x(\alpha),p(\alpha))$, which is the Walrasian equilibrium of the economy
with incomes
$m_{i}(\alpha)=\lambda_{i}^{(-1)}(p(\alpha),\frac{1}{\alpha_{i}})$ for
$i\in\mathcal{N}$ and $\sum_{i\in\mathcal{N}}m_{i}(\alpha)=\langle
p(\alpha)|w\rangle$.

Evaluated at the equilibrium price, (8) can also be expressed in terms of the
standard indirect utilities

$V_{\alpha}(p(\alpha),w)\,=\,\sum_{i\,\in\,\mathcal{N}}\alpha_{i}v_{i}(p(\alpha),m_{i}(\alpha))$

which follows from (4) together with
$\lambda_{i}(p(\alpha),m_{i}(\alpha))={\textstyle{1\over\alpha_{i}}}$ and
$\sum_{i\in\mathcal{N}}m_{i}(\alpha)=\langle p(\alpha)|w\rangle$.

###  3.1 Welfare Theorems

A necessary condition for the pair $(x,p)$ to be a root of the potential is
that $x$ maximizes welfare, i.e. it is Pareto optimal. The first part of
Theorem 1 thus implies the first welfare theorem and the converse part implies
the second welfare theorem.

###### Corollary 1

Theorem 1 implies that any Walrasian allocation is Pareto-optimal and vice
versa.

$u_{1}$$u_{2}$$\alpha={\textstyle{9\over 10}}$$\alpha={\textstyle{1\over
2}}$$\alpha={\textstyle{1\over 10}}$ Figure 2: Illustration of duality for the
economy in Example 2.

For the economy in Section 2, the Walrasian equilibrium price and allocation
can be obtained from either the primal welfare-maximization program or the
dual program of minimizing weighted indirect utility. For instance, the primal
program yields the allocations and the price follows from consumers’ marginal
rates of substitution. Alternatively, the dual program yields the price and
the allocations follow from Roy’s identity. This is a consequence of the
strict concavity assumption. If we relax this assumption to concavity, the
tangency in Figure 1 does not necessarily occur. Nonetheless, Walrasian prices
are “utility clearing,” i.e. they force a zero potential.

######  Example 2 (Linear utility)

Consider an exchange economy with two consumers with linear preferences
$u_{1}(x,y)=\log(2x+y)$ and $u_{2}(x,y)=\log(x+2y)$. Suppose there is one unit
of each good in the economy. Welfare maximization yields the Pareto-optimal
allocations

$(x_{1}(\alpha),y_{1}(\alpha))\,=\,\left\\{\begin{array}[]{lll}(3\alpha,0)&\mbox{if}&\alpha\,\leq\,\frac{1}{3}\\\
(1,0)&\mbox{if}&\frac{1}{3}\,\leq\,\alpha\,\leq\,\frac{2}{3}\\\
(1,3\alpha-2)&\mbox{if}&\alpha\,\geq\,\frac{2}{3}\end{array}\right.$

and $(x_{2}(\alpha),y_{2}(\alpha))=(1-x_{1}(\alpha),1-y_{1}(\alpha))$. The
frontier of the utility-possibility set, depicted by the red curve in Figure
2, corresponds to the resulting utility pairs $(u_{1},u_{2})$. The Pareto
optimal allocations lie on the boundary of the Edgeworth box for any welfare
weight and the Walrasian price does not follow from consumers’ marginal rates
of substitution. Instead it follows from mininizing $V_{\alpha}(p,w)$, which
yields

$\frac{p(\alpha)}{1-p(\alpha)}\,=\,\left\\{\begin{array}[]{lll}\frac{1}{2}&\mbox{if}&\alpha\,\leq\,\frac{1}{3}\\\
\frac{\alpha}{1-\alpha}&\mbox{if}&\frac{1}{3}\,\leq\,\alpha\,\leq\,\frac{2}{3}\\\
2&\mbox{if}&\alpha\,\geq\,\frac{2}{3}\end{array}\right.$

This outcome is illustrated by the blue curves in Figure 2.
$\hfill\blacksquare$

###  3.2 The Greedy Invisible Hand

An envelope-theorem argument applied to (7) establishes the Walrasian price
vector as the gradient of utilitarian social welfare.

###### Corollary 2

If $(x,p)$ is a Walrasian equilibrium then

$p\,=\,\nabla_{w}W_{\alpha}(w)$ (10)

for $\alpha_{i}=1/\lambda_{i}(p,m_{i})$ and
$x_{i}=\overline{x}_{i}(\lambda_{i}(p,m_{i})p)$ for $i\in\mathcal{N}$.

The characterization of the Walrasian equilibrium price as the gradient of
utilitarian social welfare is surprising – Adam Smith’s “invisible hand”
steers market participants to a state of greatest happiness in a simple greedy
manner. Yet, the characterization is intuitive as it implies a balance of
individual and social incentives:

$\frac{\partial_{k}W_{\alpha}(w)}{\partial_{\ell}W_{\alpha}(w)}\,=\,\frac{\partial_{k}u_{i}(x_{i})}{\partial_{\ell}u_{i}(x_{i})}$
(11)

for $i\in\mathcal{N}$, $k,\ell\in\mathcal{K}$ and
$\alpha_{i}=1/\lambda_{i}(p,m_{i})$. In other words, individuals’ marginal
rates of substitution match the social marginal rate of substitution.
Corollary 2 provides a simple-yet-powerful way to analytically characterize
Walrasian equilibria.

######  Example 3 (Homogeneous CES)

Suppose there are $K$ goods and $N$ consumers with CES utilities

$u_{i}(x)\,=\,\frac{1}{\rho}\,\log\bigl{(}\,\sum_{k\,\in\,\mathcal{K}}\,a_{k}x_{k}^{\rho}\bigr{)}$

for $\rho<1$. Consumers’ endowments are $\omega_{i}$ and the total endowments
are $w=\sum_{i\in\mathcal{N}}\omega_{i}$. Aggregate utility
$U_{\alpha}(x)=\sum_{i\in\mathcal{N}}\alpha_{i}u_{i}(x_{i})$ is maximized at
$x_{i}=(\alpha_{i}/\sum_{j\in\mathcal{N}}\alpha_{j})w$. The social weights are
$\alpha_{i}=1/(\partial v_{i}/\partial m_{i})=m_{i}$, so the gradient of
utilitarian social welfare is

$\nabla_{w}W_{\alpha}(w)\,=\,\frac{\sum_{i\,\in\,\mathcal{N}}m_{i}}{\sum_{k\,\in\,\mathcal{K}}\,w_{k}^{\rho}}\,\,w^{\rho-1}\,=\,p$

resulting in price ratios $p_{k}/p_{\ell}=(w_{k}/w_{\ell})^{\rho}$. Using
$m_{i}=\langle p|\omega_{i}\rangle$ and
$x_{i}=(\alpha_{i}/\sum_{j\in\mathcal{N}}\alpha_{j})w$ yields the Walrasian
equilibrium allocations

$x_{i}\,=\,\frac{\sum_{k\,\in\,\mathcal{K}}\,\omega_{ik}w_{k}^{\rho-1}}{\sum_{k\,\in\,\mathcal{K}}\,w_{k}^{\rho}}\,w$

for $i\in\mathcal{N}$. Note that I did not need to solve any individual
consumer’s maximization problem to derive the Walrasian equilibrium price and
allocations. $\hfill\blacksquare$

###  3.3 Existence of Walrasian Equilibria

Theorem 1 shows there is a unique Walrasian equilibrium for any welfare
weights. No fixed-point arguments are needed. A simple duality result
establishes the Walrasian equilibrium as the unique maximum, and root, of a
strictly concave potential.

Usually, the economy is parameterized by endowments
$\omega_{i}\in\mathbb{R}_{>0}$ for $i\in\mathcal{N}$ rather than welfare
weights. Theorem 1 shows that if there is a Walrasian equilibrium $(x,p)$ then
it is a root of the potential for weights that equal the inverse of the
marginal utility of income: $\alpha_{i}=1/\lambda_{i}(p,\langle
p|\omega_{i}\rangle)$. Hence, despite the set of endowments being of higher
dimension $(NK)$ than the set of welfare weights $(N)$, no additional
Walrasian equilibria are added when parameterizing the economy by endowments.

Do Walrasian equilibria exist for any endowments? Arrow and Debreu (1954) have
answered this question affirmatively by extending Nash’s (1950) existence
proof. Here I present a simpler argument by showing that for any initial
endowments there are “equilibrium weights” that produces the correct incomes.
Since $Y_{\kappa\alpha}(x,\kappa p,w)=\kappa Y_{\alpha}(x,p,w)$ for any
$\kappa>0$ we can, without loss of generality, scale the weight vector
$\alpha$ so its entries sum to one, i.e. $\alpha$ belongs to the simplex
$\Sigma_{N}$. For $i\in\mathcal{N}$, consider the fixed-point equations

$\langle p(\alpha)|\omega_{i}\rangle-m_{i}(\alpha)\,=\,0$ (12)

By Theorem 1, $\sum_{i\in\mathcal{N}}m_{i}(\alpha)=\langle p|w\rangle$, so the
left side of (12) defines a vector field on $\Sigma_{N}$. Moreover,
$\lim_{\alpha_{i}\downarrow 0}m_{i}(\alpha)=0$ by Lemma 1, so the vector field
points inward on the boundary of $\Sigma_{N}$. By the Poincare–Hopf theorem
such a vector field has at least one zero in the interior of $\Sigma_{N}$.

###### Corollary 3

For any economy parametrized by endowments $\omega_{i}\in\mathbb{R}_{>0}$,
$i\in\mathcal{N}$ there exists a weight vector $\alpha\in\Sigma_{N}$ such that
the root of $Y_{\alpha}(x,p,w)$ is a Walrasian equilibrium.

The solution to (12) is not necessarily unique as the next example shows.

###### Example 4

Consider an exchange economy with two goods and two consumers with utilities
$u_{1}(x,y)={\textstyle{3\over 2}}x^{\frac{2}{3}}-{\textstyle{1\over
2}}y^{-2}$ and $u_{2}(x,y)={\textstyle{3\over
2}}y^{\frac{2}{3}}-{\textstyle{1\over 2}}x^{-2}$ and endowments
$\omega_{1}=({\textstyle{11\over 6}},{\textstyle{1\over 6}})$ and
$\omega_{2}=({\textstyle{1\over 6}},{\textstyle{11\over 6}})$. The Fenchel
duals are

$\overline{v}_{1}(p_{1},p_{2})\,=\,\frac{1}{2}\,\Bigl{(}\,\frac{1}{p_{1}^{2}}-3p_{2}^{\frac{2}{3}}\,\Bigr{)}$

and $\overline{v}_{2}(p_{1},p_{2})=\overline{v}_{1}(p_{2},p_{1})$. The prices
$(p_{1}(\alpha),p_{2}(\alpha))$ that minimize $V_{\alpha}(p,w)$ solve

$\Bigl{(}\frac{\alpha}{p_{1}(\alpha)}\Bigr{)}^{3}+\Bigl{(}\frac{1-\alpha}{p_{1}(\alpha)}\Bigr{)}^{\frac{1}{3}}\,=\,2$

and the equation for $p_{2}(\alpha)$ follows by interchanging $\alpha$ and
$1-\alpha$, i.e. $p_{2}(\alpha)=p_{1}(1-\alpha)$. The incomes are
$m_{1}(\alpha)=\alpha^{3}/p_{1}(\alpha)^{2}+\alpha^{\frac{1}{3}}p_{2}(\alpha)^{\frac{2}{3}}$
and $m_{2}(\alpha)=m_{1}(1-\alpha)$. The fixed-point condition (12) for the
equilibrium weight can be written as

$p_{1}(\alpha)\Bigl{(}\Bigl{(}\frac{1-\alpha}{p_{1}(\alpha)}\Bigr{)}^{\frac{1}{3}}-\frac{1}{6}\Bigr{)}\,=\,p_{1}(1-\alpha)\Bigl{(}\Bigl{(}\frac{\alpha}{p_{1}(1-\alpha)}\Bigr{)}^{\frac{1}{3}}-\frac{1}{6}\Bigr{)}$

An obvious solution is $\alpha={\textstyle{1\over 2}}$ and prices
$(p_{1},p_{2})=({\textstyle{1\over 2}},{\textstyle{1\over 2}})$. Two other
solutions are, approximately, $\alpha\approx 0.09$,
$(p_{1},p_{2})\approx(0.16,0.79)$ and $\alpha\approx 0.91$,
$(p_{1},p_{2})\approx(0.79,0.16)$. $\hfill\blacksquare$

To summarize, parameterizing the economy using welfare weights is economical
in two ways. First, there is exactly one Walrasian equilibrium for every
welfare weight. Second, this unique Walrasian equilibrium follows from duality
rather than from a system of fixed-point conditions. In contrast,
parameterizing the economy using endowments is uneconomical for three reasons.
First, if $(x,p)$ is a Walrasian equilibrium for the endowments $\omega_{i}$
then it is also a Walrasian equilibrium for any endowments
$\omega^{\prime}_{i}$ that satisfy $\langle p|\omega_{i}\rangle=\langle
p|\omega^{\prime}_{i}\rangle$ for $i\in\mathcal{N}$. Second, there may exist
multiple Walrasian equilibria for some endowments as Example 4 shows. Third,
computing Walrasian equilibria necessarily involves solving a system of fixed-
point conditions, see (12). It should be noted that, since the number of goods
$(K)$ is typically assumed to be far larger than the number of consumers
$(N)$, (12) provides a computationally efficient alternative to solving fixed-
point conditions for the equilibrium prices.

Importantly, even if the economy is parameterized by endowments there are no
additional Walrasian equilibria with possibly different features than the
potential’s roots. The Paretian–Walrasian duality derived above thus applies
to all Walrasian equilibria (even when computed using standard fixed-point
conditions).

##  4 Outlook

The potential provides a litmus test for equilibrium existence. Given
preferences and endowments it is a mechanical exercise to compute the
potential’s maximum value. A Walrasian equilibrium exists if and only if this
exercise returns nil.

If not, general equilibrium theory is quiet about the allocations and prices
that ensue even when there are obvious gains from trade. For instance, suppose
two consumers have $\max(x,y)$ preferences and endowments $\omega_{1}=(2,2)$
and $\omega_{2}=(1,1)$. At any price vector $(p,1-p)$ consumer 1 demands at
least four units of one of the goods, ruling out Walrasian equilibrium. Yet,
consumers can exchange one unit of either good for one unit of the other and
both be better off.

Non-existence of equilibrium usually stops any further analysis, but that does
not mean that gains from trade will not be seized – the economy continues to
operate after all. Goeree and Roger (2022) demonstrate that the outcome in
which one unit is exchanged corresponds to a maximum of the potential, albeit
not a root. They term these maxima “$Y$ equilibria” and show they exist in any
economy, including non-convex ones. As such, the potential complements general
equilibrium’s incomplete toolkit and provides a compass to navigate economics’
terra incognita of non-convex economies.

##  A Proofs

Proof of Lemma 1. The indirect utility $v_{i}(p,m_{i})$ follows from the
saddle-point problem

$v_{i}(p,m_{i})\,=\,\min_{\lambda_{i}\,\geq\,0}\max_{x_{i}\,\geq\,0}\,u_{i}(x_{i})-\lambda_{i}(\langle
p|x_{i}\rangle-m_{i})$ (A.1)

and is related to the Fenchel dual as follows

$v_{i}(p,m_{i})\,=\,\min_{\lambda_{i}\,\geq\,0}\,\lambda_{i}m_{i}+\overline{v}_{i}(\lambda_{i}p)$

Let $\lambda_{i}(p,m_{i})$ be the solution to the minimization problem then

$v_{i}(p,m_{i})\,=\,\lambda_{i}(p,m_{i})m_{i}+\overline{v}_{i}(\lambda_{i}(p,m_{i})p)$
(A.2)

where the $\lambda_{i}(p,m_{i})$ for $i\in\mathcal{N}$ are such that budget
constraints are binding:

$\langle p|\overline{x}_{i}(\lambda_{i}(p,m_{i})p)\rangle\,=\,m_{i}$ (A.3)

From (A.1), $\nabla_{p}v_{i}(p,m_{i})=\lambda_{i}(p,m_{i})(\omega_{i}-x_{i})$
since $m_{i}=\langle p|\omega_{i}\rangle$. In addition, from (A.2),
$\nabla_{p}v_{i}(p,m_{i})=\lambda_{i}(p,m_{i})(\omega_{i}+\nabla_{p}\overline{v}_{i}(\lambda_{i}(p,m_{i})p))$.
Combining these results yields a simplified version of Roy’s identity:
$x_{i}(p,m_{i})=\overline{x}_{i}(\lambda_{i}(p,m_{i})p)=-\nabla_{p}v_{i}(\lambda_{i}(p,m_{i})p)$.
From (A.2) it follows that $\partial v_{i}/\partial
m_{i}=\lambda_{i}(p,m_{i})$. The budget constraints (A.3) imply that
$\lambda_{i}(p,m_{i})>0$ and that $\overline{x}_{i}(\lambda_{i}(p,m_{i})p)$ is
strictly decreasing in $p$ and strictly increasing in $m_{i}$. The latter
implies that $\lambda_{i}(p,m_{i})$ is strictly decreasing in $m_{i}$. From
(A.3) it further follows that $\overline{x}_{i}(\lambda_{i}(p,m_{i})p)$
vanishes when $m_{i}=0$ and $\overline{x}_{i}(\lambda_{i}(p,m_{i})p)$ diverges
when $m_{i}=\infty$. Hence, $\lim_{m_{i}\downarrow
0}\lambda_{i}(p,m_{i})=\infty$ and
$\lim_{m_{i}\rightarrow\infty}\lambda_{i}(p,m_{i})=0$. $\hfill\blacksquare$

Proof of Theorem 1. If $(x,p)$ is a Walrasian equilibrium the Marshallian
demands satisfy $x_{i}=-\nabla\overline{v}_{i}(\lambda_{i}(p,m_{i})p)$, see
Lemma 1. This can be inverted to $\lambda_{i}(p,m_{i})p=\nabla u_{i}(x_{i})$,
see e.g. Rockafellar (1970, Th. 26.5).444Theorem 26.5 in Rockafellar (1970)
relates the gradient of a strictly convex function to the gradient of its
dual, which is also a strictly convex function. To apply the theorem use $-u$
and $\overline{v}$. When $\alpha_{i}=1/\lambda_{i}(p,m_{i})$ we thus have
$p=\alpha_{i}\nabla u_{i}(x_{i})$, so the $x_{i}$ satisfy the first-order
conditions for maximizing $U_{\alpha}$. Hence,
$x=\text{argmax}_{x^{\prime}}U_{\alpha}(x^{\prime})$ as $U_{\alpha}$ is
strictly concave. The Walrasian price $p$ is market clearing so
$0=\sum_{i\in\mathcal{N}}(\omega_{i}-x_{i}(p,m_{i}))=\nabla_{p}V_{\alpha}(p,w)$
when $\alpha_{i}=1/\lambda_{i}(p,m_{i})$ for $i\in\mathcal{N}$. So $p$
satisfies the first-order condition for minimizing $V_{\alpha}$ and
$p=\text{argmin}_{p^{\prime}}V_{\alpha}(p^{\prime},w)$ as $V_{\alpha}$ is
strictly convex. Since $\max_{x}U_{\alpha}(x)=\min_{p}V_{\alpha}(p,w)$ the
potential vanishes at $(x,p)$.

For the converse part, since $p(\alpha)$ minimizes $V_{\alpha}(p,w)$ it
follows that
$\nabla_{p}V_{\alpha}(p,w)=\sum_{i\in\mathcal{N}}(\omega_{i}-\overline{x}_{i}(p(\alpha)/\alpha_{i}))=0$,
i.e. the $\overline{x}_{i}(p(\alpha)/\alpha_{i})$ satisfy feasibility. The
unique solution $x(\alpha)$ to $\max_{x\in F(w)}\,U_{\alpha}(x)$ satisfies
$\alpha_{i}\nabla u_{i}(x_{i}(\alpha))=q(\alpha)$ for $i\in\mathcal{N}$ and
some price vector $q(\alpha)$. This can be inverted as
$x_{i}(\alpha)=-\nabla\overline{v}_{i}(q(\alpha)/\alpha_{i})=\overline{x}_{i}(q(\alpha)/\alpha_{i})$.
Since $x(\alpha)$ is feasible by construction, $q(\alpha)$ also minimizes
$V_{\alpha}(p,w)$. Strict convexity of $V_{\alpha}(p,w)$ implies
$q(\alpha)=p(\alpha)$. By Lemma 1 the $\overline{x}_{i}(p(\alpha)/\alpha_{i})$
are optimal Marshallian demands at $p(\alpha)$ and incomes $m_{i}$ if
$\alpha_{i}=1/\lambda_{i}(p(\alpha),m_{i})$, which can be inverted as
$m_{i}=\lambda_{i}^{(-1)}(p(\alpha),{\textstyle{1\over\alpha_{i}}})$. Taking
the inner product of the feasibility condition with the price vector yields
$\langle p(\alpha)|w\rangle=\sum_{i\in\mathcal{N}}\langle
p(\alpha)|\overline{x}_{i}(p(\alpha)/\alpha_{i})\rangle=\sum_{i\in\mathcal{N}}m_{i}(\alpha)$.
To summarize, $(x(\alpha),p(\alpha))$ is a Walrasian equilibrium of the
economy with incomes $m_{i}(\alpha)$. $\hfill\blacksquare$

## References

  * Arrow and Debreu (1954) Arrow, K. and G. Debreu (1954).  Existence of an Equilibrium for a Competitive Economy.  Econometrica 22, 265–290.
  * Arrow (1951) Arrow, K. J. (1951).  An extension of the basic theorems of classical welfare economics.  In Proceedings of the second Berkeley symposium on mathematical statistics and probability, Ed. Neyman, J., pp. 507–532. Berkeley: U. of California Press.
  * Debreu (1951) Debreu, G. (1951).  The coefficient of resource utilization.  Econometrica 19, 273–292.
  * Duffie and Sonnenschein (1989) Duffie, D. and H. Sonnenschein (1989).  Arrow and General Equilibrium Theory.  Journal of Economic Literature 27, 565–598.
  * Goeree and Roger (2022) Goeree, J. K. and G. Roger (2022).  $Y$ Equilibrium: A Theory For Non-Convex Markets.  AGORA working paper, in preparation.
  * McKenzie (1954) McKenzie, L. (1954).  On Equilibrium in Graham’s Model of World Trade and Other Competitive Systems.  Econometrica 22, 147–161.
  * Nash (1950) Nash, J. F. (1950).  Equilibrium points in $n$-person games.  Proceedings of the National Academy of Sciences 36(1), 48–49.
  * Rockafellar (1970) Rockafellar, R. T. (1970).  Convex Analysis.  Princeton University Press.
  * Schumpeter (1954) Schumpeter, J. (1954).  History of Economic Analysis.  New York, USA: Oxford University Press.


# Fiedler vector analysis for particular cases of connected graphs

Daniel Felisberto Traciná Filho  Daniel Felisberto Traciná Filho
Pós-graduação em Sistemas e Computação - Instituto Militar de Engenharia
<EMAIL_ADDRESS>Claudia Marcela Justel
Pós-graduação em Sistemas e Computação - Instituto Militar de Engenharia
<EMAIL_ADDRESS>
( )

###### Abstract

In this paper, some subclasses of block graphs are considered in order to
analyze Fiedler vector of its members. Two families of block graphs with
cliques of fixed size, the block-path and block-starlike graphs, are analyzed.
Cases A and B of classification for both families were considered, as well as
the behavior of the algebraic connectivity when some vertices and edges are
added for particular cases of block-path graphs.

Keyword: spectral graph theory, Fiedler vector, algebraic connectivity, block
graphs.

##  1 Introduction

We consider $G=(V,E)$ an undirected, unweighted and simple graph. The sizes of
its sets of vertices and edges are $|V|=n$, $|E|=m$. The Laplacian matrix of a
graph $G$, $L(G)$, is the symmetric and semidefinite positive matrix
$D(G)-A(G)$, where $D(G)$ is the diagonal matrix with the degrees of the
vertices of $G$ and $A(G)$ is the adjacency matrix of $G$. The eigenvalues of
$L(G)$, the roots of the characteristic polynomial of $L(G)$, are $n$ non-
negative real numbers, and zero is always an eigenvalue of $L(G)$. We denote
them as $0=\lambda_{1}\leq\lambda_{2}\leq....\leq\lambda_{n}$. The second
smallest eigenvalue of $L(G)$, $\lambda_{2}(G)$, is called the algebraic
connectivity of $G$. An eigenvector associated with the algebraic connectivity
is called a Fiedler vector. The algebraic connectivity gives some measure of
how connected the graph is. Fiedler ([3]), showed that $\lambda_{2}(G)>0$ if
and only if $G$ is connected. Moreover, the algebraic connectivity of $G$ does
not decrease if an edge is added to $G$ and the complete graph has
$\lambda_{2}(K_{n})=n$. Also, Fiedler proved that
$\lambda_{2}(G)\leq\kappa(G)$, where $\kappa(G)$ is the vertex connectivity of
$G$. For a connected graph $G$ and a Fiedler vector $\mathbf{y}$, the $i-th$
entry of $\mathbf{y}$, $y_{i}$, gives a valuation of vertex $i$. When the
graph is a tree (acyclic and connected graph), the entries of a Fiedler vector
of such tree provide extra information. A tree $T$ is called a Type 1 tree if
there is a vertex $z$ such that $y_{z}=0$ and is adjacent to a vertex with non
zero valuation. In this case $z$ is called characteristic vertex. When there
is no entry of $\mathbf{y}$ equal to zero, the tree has an edge $(u,w)$ with
$y_{u}>0$ and $y_{w}<0$, and in that case $T$ is called a Type 2 tree and the
edge $(u,w)$ the characteristic edge ([5]).

Fiedler vectors and algebraic connectivity of graphs have been studied by
several authors. For instance some results about trees are presented in [7]
and [10], others works such as [8] and [9] deal with connected graphs. The
surveys in [1] and [6] provide overviews of the literature on algebraic
connectivity.

Even for trees, to identify classes of connected graphs for which its elements
are of the same type is not an easy task. For instance, path graphs $P_{n}$
are classified by type according to their parity. Also, in [10] Patra shows
conditions for a broom tree by of Type 2 and claims that all the broom trees
are of Type 2, he also says that will be nice to get a proof of this claim.

This paper consider two families of connected graphs, both of them belonging
to the block graph class, the block-path and the block-starlike graphs. Some
properties of Fiedler vectors for graphs in those families are analyzed and
conditions for classifying by type such graphs are given.

The paper is organized as follows. In Section 2, we review some basic concepts
and results of the literature. Section 3 presents the families block-path and
block-starlike graphs, as well as some properties about them. Finally, the
conclusions of the work are presented in Section 4 and the references in the
last section.

##  2 Preliminaries

In this section, some concepts and results about Graph Theory and Spectral
Graph Theory are introduced.

First, let $G=(V,E)$ be a connected graph. The neighborhood of a vertex $v\in
V$ is denoted by $N(v)=\\{w\in V:(v,w)\in E\\}$ and its closed neighborhood by
$N[v]=N(v)\cup\\{v\\}$. Two vertices $u,v\in V$ are true twins if $N[u]=N[v]$.

The vertex connectivity of $G$, denoted $\kappa(G)$, is the minimum number of
vertices whose removal from $G$ leaves a disconnected or trivial graph.

A vertex $v\in V$ is a point of articulation or cutpoint if $G\backslash v$ is
disconnected, and in this case $\kappa(G)=1$. A maximal connected subgraph
without any points of articulation is called a block. The distance,
$d_{G}(u,v)$, in $G$ between two vertices is the length of a shortest path
between $u,v$ in $G$. The eccentricity, $e_{G}(v)$, of a vertex $v$ in $G$ is
its greatest distance from any other vertex. The center of $G$, denoted
$center(G)$, is the set of vertices with minimum eccentricity. For any
$S\subseteq V$, the subgraph of $G$ induced by $S$ is denoted $G[S]$. If
$G[S]$ is a complete subgraph then $S$ is a clique in $G$.

A tree is an acyclic and connected graph. We denote by $P_{j}$ the path on $j$
vertices and $K_{1,q}$ the star on $q+1$ vertices.

A tree $T$ is called starlike if $T$ is homeomorphic to a star $K_{1,m}$. For
$m\geq 3$, $T$ has a unique vertex $v$ of degree $m$ and $T\backslash v$ is a
union of $m$ paths ([11]).

A graph is a block graph if every block is a clique. A proper interval graph
is an interval graph that has an intersection model in which no interval
properly contains another. Proper interval graphs are also known as
indifference graphs. A graph is a block indifference graph when it belongs to
the intersection of both classes, indifference and block graphs [2].

Next, some concepts and properties on graph spectra are introduced. The
following results consider $G=(V,E)$ a weighted graph with $w(e)\geq 0$ for
all $e\in E$ (the weight of edge), and define the Laplacian matrix of $G$,
$L(G)$ whose entry $(i,j)$ equals $-w(i,j)$ if $(i,j)\in E$; zero if $i\not=j$
and $(i,j)\not\in E$; and the sum of weights of edges incident to vertex $i$,
if $i=j$.

In [4] Fiedler proved the following theorem which describes some structure of
any Fiedler vector of a connected graph.

###### Theorem 1

([4]) Let $G$ be a connected graph and $\mathbf{y}$ a Fiedler vector of $G$.
Then exactly one of the following cases occurs:

Case A: There is a single block $C$ in $G$ which contains both positively and
negatively valuated vertices. Each other block has either vertices with
positive valuation only, vertices with negative valuation only, or vertices
with zero valuation only. Every path $P$ which contains at most two points of
articulation in each block, which starts in $C$ and contains just one vertex
$k$ in $C$ has the property that the valuations at points of articulation
contained in $P$ form either an increasing, or decreasing or a zero sequence
among this path according to whether $y_{k}>0$, $y_{k}<0$ or $y_{k}=0$; in the
last case all vertices in $P$ have valuation zero.

Case B: No block of $G$ contains both positively and negatively valuated
vertices. There exists a unique vertex $z$ which has valuation zero and is
adjacent to a vertex with a non-zero valuation. This vertex $z$ is a point of
articulation. Each block contains either vertices with positive valuation
only, with negative valuation only, or with zero valuation only. Every path
containing both positively and negatively valuated vertices passes through
$z$.

Kirkland et al ([8]) presented another characterization of graphs for which
Case B of Theorem 1 holds. This characterization is based on the concept of
Perron components at points of articulation.
For a positive $n\times n$ matrix $M$, the Perron value of $M$, $\rho(M)$ is
defined as the maximum eigenvalue of the matrix. Let $G$ be a connected
weighted graph with Laplacian matrix $L(G)$. For a vertex $v$, we denote the
connected components of $G\backslash v$ by $C_{1},...,C_{p}$. If $p\geq 2$,
$v$ is a point of articulation. For each connected component $C_{i}$, $1\leq
i\leq p$ let $L(C_{i})$ be the principal submatrix of $L(G)$ corresponding to
the vertices of $C_{i}$. The Perron value of $C_{i}$ is the Perron value of
the positive matrix $L(C_{i})^{-1}$, denoted by $\rho(L(C_{i})^{-1})$. We say
that $C_{j}$ is the Perron component at $v$ if
$\rho(L(C_{j})^{-1})=\max_{1\leq i\leq p}\rho(L(C_{i})^{-1})$.

###### Theorem 2

([8]) Let $G=(V,E)$ be a weighted graph. Case B of Theorem 1 holds if and only
if there are two or more Perron components at $z$. Further, in that case, the
algebraic connectivity $\lambda_{2}(G)$ is given by
$\frac{1}{\rho(L(C)^{-1})}$ for any Perron component $C$ at $z$. If the Perron
components at $z$ are $C_{1},...,C_{m}$, for $1\leq i\leq m$, let the Perron
vector of $L(C_{i})^{-1}$ be $x_{i}$, normalized so that its entries sum to 1.
For each $2\leq i\leq m$, let $b_{i-1}$ be the Fiedler vector which valuates
the vertices of $C_{1}$ by $x_{1}$, the vertices of $C_{i}$ by $-x_{i}$ and
all other vertices zero; then $b_{1},...,b_{m-1}$ is a basis for the
eigenspace corresponding to $\lambda_{2}(G)$. In particular, if there are $m$
Perron components at $z$, then the multiplicity of $\lambda_{2}(G)$ is $m-1$,
and every Fiedler vector has zeros in the positions corresponding to $z$, and
to the vertices of the non-Perron components at $z$. Finally, for any vertex
$v\not=z$, the unique Perron component at $v$ is the component containing $z$.

In other words, Theorem 2 can be rewritten as in Corollary 2.1.

###### Corollary 2.1

([8]) Let $G=(V,E)$ be a weighted graph. Case A of Theorem 1 holds if and only
if there is a unique Perron component at every vertex of $G$. Case B of
Theorem 1 holds if and only if there is a unique vertex at which there are two
or more Perron components.

Observe that for unweighted graphs, we consider each of its edges weight
equals 1 in order to apply Theorem 2 and Corollary 2.1.

##  3 Block-path and block-starlike classes

The block-path class, denoted $\mathcal{B}$, is a subclass of block-
indifference graphs, with elements denoted $G_{k,p}$, that contain all cliques
of size $k\geq 2$ and $p\geq 1$ points of articulation. Moreover, the number
of vertices in $G_{k,p}$ is $n=k(p+1)-p$. For $p=0$, $G_{k,0}=K_{k}$. The
block-path graph $G_{4,3}$ is presented in Figure 1.

Figure 1: $G_{4,3}\in\mathcal{B}$

Let $G=S_{r,k,p_{1},...,p_{r}}\in\mathcal{A}$ with $r\geq 3$ and $v$ its
central vertex such that the connected components of $G\backslash v$ are
denoted by $G_{k,p_{1}}\backslash v$, $G_{k,p_{2}}\backslash v$, $\ldots$,
$G_{k,p_{r}}\backslash v$, with $p_{1}\geq p_{2}\geq\ldots\geq p_{r}$. $G$ is
a block-starlike. The class of block-starlike graphs is denoted $\mathcal{A}$.

Figure 2 shows $S_{3,4,1,2,3}\in\mathcal{A}$ with $v$ its central vertex and
$S_{3,4,1,2,3}\backslash v=G_{4,1}\backslash v\cup G_{4,2}\backslash v\cup
G_{4,3}\backslash v$. Observe that in this example, the central vertex $v$ do
not belong to $center(S_{3,4,1,2,3})$.

$v$ Figure 2: $S_{3,4,1,2,3}\in\mathcal{A}$

Next, we present some new results about classes $\mathcal{A}$ and
$\mathcal{B}$.

###### Lemma 3

Let $G$ be a block graph with a point of articulation. If $a,b$ are true twins
in $G$ and $\mathbf{y}$ is a Fiedler vector, then $y_{a}=y_{b}$.

Proof. Let $|V|=n$, $a,b\in V$ be true twin vertices and the Laplacian matrix
of $G$ denoted $L=(l_{i,j})$. As $\lambda_{2}(G)$ is a Laplacian eigenvalue of
$G$ associated with $\mathbf{y}$, $L\mathbf{y}=\lambda_{2}(G)\mathbf{y}$ and

$\lambda_{2}(G)(y_{a}-y_{b})=\sum_{s=1}^{n}l_{a,s}y_{s}-\sum_{s=1}^{n}l_{b,s}y_{s}$

Vertices $a$ and $b$ are true twins then, for all $s\in V\backslash\\{a,b\\}$
$l_{a,s}=l_{b,s}$. So,

$\lambda_{2}(G)(y_{a}-y_{b})=l_{a,a}y_{a}+l_{a,b}y_{b}-l_{b,a}y_{a}-l_{b,b}y_{b}$

$\lambda_{2}(G)(y_{a}-y_{b})=(l_{a,a}-l_{b,a})y_{a}+(l_{a,b}-l_{b,b})y_{b}$

Since $l_{a,a}=l_{b,b}$,

$\lambda_{2}(G)(y_{a}-y_{b})=\lambda_{2}(G)(y_{a}-y_{b})=q(y_{a}-y_{b})$, with
$q=l_{a,a}-l_{b,a}=deg_{G}(a)+1>1\in N$.

However, by hypothesis the graph $G$ has a point of articulation and then
$\lambda_{2}(G)\leq 1$. So it is not possible to be $y_{a}\not=y_{b}$. Then,
the result holds.

###### Theorem 4

Let $G_{k,p}\in\mathcal{B}$. Then $p$ is an odd number if and only if Case B
of Theorem 1 holds for $G_{k,p}$.

Proof. Label the vertices of $G_{k,p}$ from 1 to $n=k(p+1)-p$ in the following
way: from one block with only 1 point of articulation, all the points of
articulation are labelled by $k,2k-1,3k-2,...,(p+1)k-p$, and inside each
block, the true twin vertices are labeled in order by block with numbers
between the labels of the corresponding points of articulation of the block.
Then vertex $v*$ labeled $(\frac{p+1}{2}k-\frac{p+1}{2}+1)$ is the unique
vertex in $center(G_{k,p})$ with odd $p\geq 1$. So, by definition of the
block-path class, the graph $G_{k,p}-v*$ has two Perron components and by
Corollary 2.1 it holds if and only if Case B of Theorem 1 is true for
$G_{k,p}$.

###### Remark 1

By Theorem 4 we can conclude that $G_{k,p}\in\mathcal{B}$ with $p$ an even
number if and only if Case A of Theorem 1 holds for $G_{k,p}$.

###### Remark 2

If $G=S_{r,k,p_{1},...,p_{r}}\in\mathcal{A}$ with $r\geq 2$ and
$p_{1}=p_{2}=...=p_{r}$, then by Corollary 2.1 Case B of Theorem 1 holds for
$G$ and the center of the graph equals the central vertex. Figure 3 shows
$S_{3,4,1,1,1}\in\mathcal{A}$ with $v$ its central vertex.

$v$ Figure 3: $S_{3,4,1,1,1}\in\mathcal{A}$

###### Theorem 5

Let $G=S_{r,k,p_{1},...,p_{r}}\in\mathcal{A}$. If $p_{2}+p_{3}+1\geq p_{1}$
and $p_{1}>p_{2}$ then Case A of Theorem 1 holds for $G$.

Proof. We will prove that for every vertex in $G$ there is only one Perron
component. Then Corollary 2.1 holds for $G$ in the condition of the
hypothesis. Observe that is sufficient to consider points of articulations of
$G$. Let $L=L(G)$ and $v$ be the central vertex of $G$, then the connected
components of $G\backslash v$ are $G_{k,p_{1}}\backslash v$,
$G_{k,p_{2}}\backslash v$, …, $G_{k,p_{r}}\backslash v$ and for each $2\leq
i\leq r$, $G_{k,p_{i}}\backslash v$ is isomorphic to a proper subgraph of
$G_{k,p_{1}}\backslash v$, and $\rho(L(G_{k,p_{1}}\backslash
v)^{-1})>\rho(L(G_{k,p_{i}}\backslash v)^{-1})$. So,
$\rho(L_{v}^{-1})=\rho(L(G_{k,p_{1}}\backslash v)^{-1})$ and
$G_{k,p_{1}}\backslash v$ is the unique Perron component of $G\backslash v$.
Let $w\in V$, $w\not=v$. We consider two cases:
1\. $G\backslash w=K_{k-1}\cup G^{\prime}\backslash w$ with $G^{\prime}$ a
block starlike, and
2\. $G\backslash w=G_{k,a}\backslash w\cup H\backslash w$, where $H$ is a
block starlike containing as subgraph a block path graph with $p_{2}+p_{3}+1$
articulation points and exists $1\leq i\leq 3$ such that $1<a\leq p_{i}-1$.
When case 1 holds there is only one Perron component of $G$ in $w$.
On the other hand, if case 2 holds, consider the following situations:
If $i=1,p_{1}-1\geq a$. Since by hypothesis $p_{2}+p_{3}+1\geq p_{1}$, we can
conclude that $p_{2}+p_{3}\geq a$. Moreover, as $H$ contains at least 2
connected components of $G\backslash w$ and $p_{2}+p_{3}\geq a$ then $G_{k,a}$
is isomorphic to a subgraph of $H$.

If $i\not=1$, since $p_{i}-1\geq a$ then $a\leq p_{i}-1<p_{1}$ and then
$G_{k,a}$ is isomorphic to a subgraph of $H$.

Thus, by Corollary 2.1 the theorem is true.

Grone and Merris ([5]) showed that the algebraic connectivity of a Type 1 tree
$T$ does not change when a degree one vertex adjacent to the characteristic
vertex is added to $T$. The next theorem generalizes that result for graphs
$G_{k,p}\in\mathcal{B}$ with $p$ an odd number.

Observe that if $p$ is odd, then $center(G_{k,p})=u$ where $u$ is the point of
articulation labeled $(\frac{p+1}{2}k-\frac{p+1}{2}+1)$ as in proof of Theorem
4.

$v$$w$$u$ Figure 4: $G^{\prime}$, obtained from $G=G_{4,3}$ and $G_{4,0}$,
with $\lambda_{2}(G)=\lambda_{2}(G^{\prime})=0.32938$

###### Theorem 6

Let $G=G_{k,p}\in\mathcal{B}$ with $n$ vertices and $p$ an odd number. Let
$\mathbf{y}$ be a Fiedler vector of $G$ and $u\in V(G)$ the unique vertex in
$center(G)$. Let $G^{\prime}$ be the graph obtained as the coalescence between
vertex $u$ of $G$ with a vertex of $G_{k,0}$. Let $\mathbf{y}^{\prime}$ be a
$(n+k-1)$-dimensional vector such that $y^{\prime}_{v}=y_{v},\forall v\in
V(G)$ and $y^{\prime}_{v}=0$ otherwise. Then $G^{\prime}\in\mathcal{A}$,
$\mathbf{y}^{\prime}$ is a Fiedler vector of $G^{\prime}$ and
$\lambda_{2}(G^{\prime})=\lambda_{2}(G)$.

Proof. Let $v,w$ be a pair of vertices different from $u$ in the new block
$K_{k}$ of $G^{\prime}$ (as in Figure 4). Since $v,w$ are twin vertices, by
Lemma 3 $y_{v}=y_{w}$. Then

$\sum^{n}_{s=1}l_{v,s}y_{s}=(k+1)y_{v}-ky_{v}-y_{u}=y_{v}-y_{u}=\lambda_{2}(G^{\prime})y_{v}$

$y_{v}=\frac{1}{1-\lambda_{2}(G^{\prime})}y_{u}\quad(*)$

As $p\geq 1$, $G^{\prime}\backslash u$ has 3 connected components, 2 of them
are isomorphic and contain at least one block $K_{k}$ and the other one is
isomorphic to $K_{k-1}$. Then, by Theorem 2 case B holds for $G^{\prime}$ and
$y_{u}=0$. From equation $(*)$, we have that $y^{\prime}_{v}=0$ for all twin
vertices in the new block $K_{k}$ of $G^{\prime}$, and the result follows.

##  4 Conclusions

Two subclasses of block graphs were considered in this paper: block-path and
block-starlike graphs. First, we showed some properties of Fiedler vectors
associated with those graphs, such as identifying whether Cases A or B of
Theorem 1 holds for block-path graphs. Then we exhibited particular conditions
to have block-starlike graphs for which Case A of the same theorem holds.
Next, we obtained the necessary conditions to mantain the value of the
algebraic connectivity of a block-path graph when some vertices and edges are
added. We plan on exploring other subclasses of block graphs in the future.

## Acknowledgments

This work was supported by CAPES, Coordenação de Aperfeiçoamento de Pessoal do
Nível Superior - Código de Financiamento 001.

## References

  * [1] Abreu. N.M.M. “Old and new results on algebraic connectivity of graphs”. _Linear Algebra and its Applications_ , vol. 423, pp. 53-73, 2007. [Online] Available: https://doi.org/10.1016/j.laa.2006.08.017
  * [2] Brandstädt, A., Le, V. B., Spinrad, J. “Graph Classes - a Survey”, _SIAM Monographs in Discrete Mathematics and Applications_ , Philadelphia, PA, 1999.
  * [3] Fiedler, M.M. “Algebraic connectivity of graphs”. _Czechoslovak Math. J._ , vol. 23, pp. 298-305, 1973. [Online] Available: https://dml.cz/bitstream/handle/10338.dmlcz/101168/CzechMathJ_23-1973-2_11.pdf
  * [4] Fiedler, M.M. “A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory”. _Czechoslovak Math. J._ , vol. 25, pp. 607-618, 1975. [Online] Available: https://dml.cz/bitstream/handle/10338.dmlcz/101357/CzechMathJ_25-1975-4_12.pdf
  * [5] Grone, R., Merris, R. “Characteristic vertices of trees”. _Linear and Multilinear Algebra_ , vol. 22, pp. 115-131, 1987. [Online] Available: https://doi.org/10.1080/03081088708817827
  * [6] Kirkland, S., “Algebraic Connectivity, in Handbook of Linear Algebra”, L. Hogben, R. Brualdi, A. Greenbaum and R. Mathias, Eds., CRC Press, pp. 36-1 - 36-12, 2006.
  * [7] Kirkland, S., Neumann, M.S., Shader, B.L. “Characteristic vertices of weighted trees via Perron values”. _Linear and Multilinear Algebra_ , vol. 40 no. 4, pp. 311-325, 1996. [Online] Available: https://doi.org/10.1080/03081089608818448
  * [8] Kirkland, S., Fallat, S.“Perron components and algebraic connectivity for weighted graphs”. _Linear and Multilinear Algebra_ , vol. 44, no 2, pp. 131-148, 1998. [Online] Available: https://doi/abs/10.1080/03081089808818554
  * [9] Kirkland, S., Rocha, I., Trevisan, V. “Algebraic connectivity of k-connected graphs”. _Czechoslovak Mathematical Journal_ , vol. 65, pp. 219-236, 2015. [Online] Available: https://doi.org/10.1007/s10587-015-0170-9
  * [10] Patra, K.L. “Maximizing the distance between center, centroid and characteristic set of a tree”. _Linear and Multilinear Algebra_ , vol. 55, no. 4, pp. 381-397, 2007. [Online] Available: https://doi.org/10.1080/03081080701208512
  * [11] Watanabe, M., Schwenk, A.J. “Integral starlike trees”. _Journal Austral. Math. Soc. (Series A)_ , vol. 28, pp. 120-128, 1979. [Online] Available: https://doi.org/10.1017/S1446788700014981