$\displaystyle=\frac{1}{2}\sum_{\begin{subarray}{c}i,j=1\\\ j\neq
i\end{subarray}}^{n}\langle u_{\gamma}^{*}\otimes u_{\delta}^{*},v_{i}\otimes
v_{j}\rangle\mathcal{S}_{n}(u_{\alpha}\otimes u_{\beta}\otimes
v_{\setminus\\{i,j\\}})$ $\displaystyle=\sum_{1\leqslant i<j\leqslant
n}\langle u_{\gamma}^{*}\otimes u_{\delta}^{*},\mathcal{S}_{2}(v_{i}\otimes
v_{j})\rangle\mathcal{S}_{n}(u_{\alpha}\otimes u_{\beta}\otimes
v_{\setminus\\{i,j\\}})$ $\displaystyle=\sum_{1\leqslant i<j\leqslant
n}\langle\mathcal{S}_{2}(u_{\gamma}^{*}\otimes
u_{\delta}^{*}),\mathcal{S}_{2}(v_{i}\otimes
v_{j}))\rangle\mathcal{S}_{n}(u_{\alpha}\otimes u_{\beta}\otimes
v_{\setminus\\{i,j\\}}).$ (162)

where eq. 30 was used in the last line. Inserting this equality into section
B.3, the operator fulfills

$\mathcal{B}v=\frac{1}{2}\sum_{\begin{subarray}{c}\alpha\leqslant\beta\\\
\gamma\leqslant\delta\end{subarray}}\frac{1}{s_{\alpha\beta}s_{\gamma\delta}}\langle\mathcal{S}_{2}(u_{\alpha}^{*}\otimes
u_{\beta}^{*}),B(u_{\gamma}\otimes
u_{\delta})\rangle\,\,a^{+}\\{u_{\alpha}\\}a^{+}\\{u_{\beta}\\}a^{-}\\{u_{\gamma}^{*}\\}a^{-}\\{u_{\delta}^{*}\\}\,v.$
(163)

By means of eq. 35, and using the symmetry of the summands, we finally obtain

$\mathcal{B}v=\frac{1}{2}\sum_{\alpha,\beta,\gamma,\delta}\langle
u_{\alpha}^{*}\otimes u_{\beta}^{*},B(u_{\gamma}\otimes
u_{\delta})\rangle\,\,a^{+}\\{u_{\alpha}\\}a^{+}\\{u_{\beta}\\}a^{-}\\{u_{\gamma}^{*}\\}a^{-}\\{u_{\delta}^{*}\\}\,v.$
(164)

Note that the need of the factor $1/2$ becomes clear as a consequence of
applying the annihilation operator twice; removing first particle $i$ and then
$j$ is the same as removing particle $j$ and then $i$. If an operator
expansion consists of $n$ annihilation operators, it will require a factor
$1/n!$ to account for all the equivalent orderings.

One can verify the equation above directly by expanding the creation and
annihilation operators and by contracting the expansions in the non-
symmetrized basis:

$\displaystyle\frac{1}{2}$
$\displaystyle\sum_{\alpha,\beta,\gamma,\delta}\langle u_{\alpha}^{*}\otimes
u_{\beta}^{*},B(u_{\gamma}\otimes
u_{\delta})\rangle\,\,a^{+}\\{u_{\alpha}\\}a^{+}\\{u_{\beta}\\}a^{-}\\{u_{\gamma}^{*}\\}a^{-}\\{u_{\delta}^{*}\\}v$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:apapamam_gen}}}{{=}}\frac{1}{2}\sum_{\alpha,\beta,\gamma,\delta}\langle
u_{\alpha}^{*}\otimes u_{\beta}^{*},B(u_{\gamma}\otimes
u_{\delta})\rangle\,\,\mathcal{S}_{n}\left(\sum_{\begin{subarray}{c}i,j=1\\\
j\neq i\end{subarray}}^{n}\langle u_{\delta}^{*},v_{i}\rangle\langle
u_{\gamma}^{*},v_{j}\rangle(u_{\alpha}\otimes u_{\beta}\otimes
v_{\setminus\\{i,j\\}})\right)$
$\displaystyle=\frac{1}{2}\mathcal{S}_{n}\sum_{\begin{subarray}{c}i,j=1\\\
j\neq i\end{subarray}}^{n}\sum_{\alpha,\beta,\gamma,\delta}\langle
u_{\alpha}^{*}\otimes u_{\beta}^{*},B(u_{\gamma}\otimes
u_{\delta})\rangle\,\,\langle u_{\gamma}^{*}\otimes
u_{\delta}^{*},v_{i}\otimes v_{j}\rangle(u_{\alpha}\otimes u_{\beta}\otimes
v_{\setminus\\{i,j\\}})$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:resunity}}}{{=}}\frac{1}{2}\mathcal{S}_{n}\sum_{\begin{subarray}{c}i,j=1\\\
j\neq i\end{subarray}}^{n}B(v_{i}\otimes v_{j})\otimes v_{\setminus\\{i,j\\}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{def:B_ij}}}{{=}}\sum_{1\leqslant
i<j\leqslant n}\mathcal{B}_{ij}v$ $\displaystyle=\mathcal{B}v.$ (165)

###  B.4 Reaction operators

The reaction operators

$\displaystyle\mathcal{R}^{(k)}$
$\displaystyle=\frac{1}{k!}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{k}\\\
\beta_{1},\dots,\beta_{k}\end{subarray}}\left\langle\bigotimes_{i=1}^{k}u_{\alpha_{i}}^{*},\Lambda^{(k)}\bigotimes_{j=1}^{k}u_{\beta_{j}}\right\rangle\,\,\prod_{i=1}^{k}a^{+}\\{u_{\alpha_{i}}\\}\prod_{j=1}^{k}a^{-}\\{u_{\beta_{j}}^{*}\\},$
(166a) $\displaystyle\mathcal{R}^{(k,l)}$
$\displaystyle=\frac{1}{k!}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{l}\\\
\beta_{1},\dots,\beta_{k}\end{subarray}}\left\langle\bigotimes_{i=1}^{l}u_{\alpha_{i}}^{*},\Lambda^{(k,l)}\bigotimes_{j=1}^{k}u_{\beta_{j}}\right\rangle\prod_{i=1}^{l}a^{+}\\{u_{\alpha_{i}}\\}\prod_{j=1}^{k}a^{-}\\{u_{\beta_{j}}^{*}\\}$
(166b)

can also be written in terms of their symmetric basis:

$\displaystyle\mathcal{R}^{(k)}$
$\displaystyle=\frac{1}{k!}\sum_{\begin{subarray}{c}\alpha_{1}\leqslant\dots\leqslant\alpha_{k}\\\
\beta_{1}\leqslant\dots\leqslant\beta_{k}\end{subarray}}\frac{1}{s_{\beta_{1},\dots,\beta_{k}}}\left\langle\mathcal{S}_{k}\left(\bigotimes_{i=1}^{k}u_{\alpha_{i}}\right)^{*},\Lambda^{(k)}\bigotimes_{j=1}^{k}u_{\beta_{j}}\right\rangle\,\,\prod_{i=1}^{k}a^{+}\\{u_{\alpha_{i}}\\}\prod_{j=1}^{k}a^{-}\\{u_{\beta_{j}}^{*}\\},$
(167a) $\displaystyle\mathcal{R}^{(k,l)}$
$\displaystyle=\frac{1}{k!}\sum_{\begin{subarray}{c}\alpha_{1}\leqslant\dots\leqslant\alpha_{l}\\\
\beta_{1}\leqslant\dots\leqslant\beta_{k}\end{subarray}}\frac{1}{s_{\beta_{1},\dots,\beta_{k}}}\left\langle\mathcal{S}_{k}\left(\bigotimes_{i=1}^{l}u_{\alpha_{i}}\right)^{*},\Lambda^{(k,l)}\bigotimes_{j=1}^{k}u_{\beta_{j}}\right\rangle\prod_{i=1}^{l}a^{+}\\{u_{\alpha_{i}}\\}\prod_{j=1}^{k}a^{-}\\{u_{\beta_{j}}^{*}\\},$
(167b)

where $s_{\beta_{1},\dots,\beta_{n}}$ is defined on eq. 33. The proofs of
these expansions work analogously to the proof in section B.3. The
coefficients $s_{\beta_{1},\dots,\beta_{k}}$ and the coefficients
$s_{\alpha_{1},\dots,\alpha_{k}}$ (or $s_{\alpha_{1},\dots,\alpha_{l}}$) that
appear when commuting symmetrization with duality compensate for the duplicate
terms in the non-symmetrized expansion.

##  Appendix C Local conservation of probabilities and dissipativity of the
CDME operator

The dissipativity of the reaction operator is closely related to the local
conservation of probability, expressed by eq. 75, which we will prove first.
This can be done for each reaction separately, such that we can restrict to
the case of a single reaction with fixed stoichiometric coefficients $k$ and
$l$. Then each “column” $m$ of the reaction operator
$\mathcal{R}=(\mathcal{Q}_{nm})$ has at most two non-zero entries (see eq. 74)
and the column sum reduces to

$\displaystyle\sum_{n=0}^{\infty}\mathcal{J}_{nm}(\rho_{m})$
$\displaystyle=\mathcal{J}_{m-k+l,m}(\rho_{m})+\mathcal{J}_{mm}(\rho_{m})$
$\displaystyle=\int_{\mathbb{X}^{m-k+l}}\,\mathcal{R}^{(k,l)}\rho_{m}dx^{(m-k+l)}-\int_{\mathbb{X}^{m}}\,\mathcal{R}^{(k)}\rho_{m}dx^{(m)}$
(168)

for $m\geqslant k$ and dropping the subscript $r$ on the reaction operators.
(For $m<k$, the reaction does not occur and $\mathcal{J}_{nm}(\rho_{m})=0$,
irrespective of the value of $n$.) For the sum in (C) to be zero it has to
hold

$\int_{\mathbb{X}^{m}}(\mathcal{R}^{(k)}\rho_{m})(x^{(m)})dx^{(m)}=\int_{\mathbb{X}^{m-k+l}}(\mathcal{R}^{(k,l)}\rho_{m})(x^{(m-k+l)})dx^{(m-k+l)}\,,$
(169)

for $\rho_{m}\in\mathcal{S}_{m}H^{\otimes m}$, which means that the considered
reaction induces the probability outflow from the $m$-particle space given on
the l.h.s., which is balanced by the inflow into the space with $m-k+l$
particles on the r.h.s.

We here give the proof of eq. 169 for the case $k=2$ and $l=1$, i.e., the
reaction $\text{A}+\text{A}\to\text{A}$; the general case for arbitrary $k,l$
follows analogously. The reaction operators read [eqs. 69 and 71b]:

$\displaystyle\mathcal{R}^{(2)}$
$\displaystyle=\frac{1}{2}\sum_{\begin{subarray}{c}\alpha_{1},\alpha_{2}\\\
\beta_{1},\beta_{2}\end{subarray}}\langle u_{\alpha_{1}}^{*}\otimes
u_{\alpha_{2}}^{*},\Lambda^{(2)}(u_{\beta_{1}}\otimes
u_{\beta_{2}})\rangle\,\,a^{+}\\{u_{\alpha_{1}}\\}a^{+}\\{u_{\alpha_{2}}\\}a^{-}\\{u_{\beta_{1}}^{*}\\}a^{-}\\{u_{\beta_{2}}^{*}\\},$
(170) $\displaystyle\mathcal{R}^{(2,1)}$
$\displaystyle=\frac{1}{2}\sum_{\alpha}\sum_{\beta_{1},\beta_{2}}\langle
u_{\alpha}^{*},\Lambda^{(2,1)}(u_{\beta_{1}}\otimes u_{\beta_{2}})\rangle
a^{+}\\{u_{\alpha}\\}a^{-}\\{u_{\beta_{1}}^{*}\\}a^{-}\\{u_{\beta_{2}}^{*}\\},$
(171)

with the coefficients given in terms of the same reaction rate function
$\lambda(x^{(1)},x^{(2)})$ as [eqs. 65 and 66]:

$\displaystyle\left(\Lambda^{(2)}(u_{\beta_{1}}\otimes
u_{\beta_{2}})\right)(x^{(2)})$ $\displaystyle=(u_{\beta_{1}}\otimes
u_{\beta_{2}})(x^{(2)})\int_{\mathbb{X}^{1}}\lambda(x^{(1)},x^{(2)})dx^{(1)},$
(172) $\displaystyle\left(\Lambda^{(2,1)}(u_{\beta_{1}}\otimes
u_{\beta_{2}})\right)(x^{(1)})$
$\displaystyle=\int_{\mathbb{X}^{2}}(u_{\beta_{1}}\otimes
u_{\beta_{2}})(x^{(2)})\lambda(x^{(1)},x^{(2)})dx^{(2)}.$ (173)

Integrating the remaining degrees of freedom of each equation and using
Fubini’s theorem, we obtain the relation

$\displaystyle\int_{\mathbb{X}^{2}}\left(\Lambda^{(2)}(u_{\beta_{1}}\otimes
u_{\beta_{2}})\right)(x^{(2)})dx^{(2)}=\int_{\mathbb{X}^{1}}\left(\Lambda^{(2,1)}(u_{\beta_{1}}\otimes
u_{\beta_{2}})\right)(x^{(1)})dx^{(1)}.$ (174)

With this, we conclude eq. 169 as follows:

$\displaystyle\mathrlap{\int_{\mathbb{X}^{m}}\mathcal{R}^{(2)}\rho_{m}\,dx^{(m)}}\hskip
5.0pt$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\eqref{eq:appC_R2}}}}{{=}}\int_{\mathbb{X}^{m}}\frac{1}{2}\sum_{\begin{subarray}{c}\alpha_{1},\alpha_{2}\\\
\beta_{1},\beta_{2}\end{subarray}}\langle u_{\alpha_{1}}^{*}\otimes
u_{\alpha_{2}}^{*},\Lambda^{(2)}(u_{\beta_{1}}\otimes
u_{\beta_{2}})\rangle\,\,a^{+}\\{u_{\alpha_{1}}\\}a^{+}\\{u_{\alpha_{2}}\\}a^{-}\\{u_{\beta_{1}}^{*}\\}a^{-}\\{u_{\beta_{2}}^{*}\\}\rho_{m}dx^{(m)}\,,$
inserting the definition of the creation operator [(51a)],
$\displaystyle=\int_{\mathbb{X}^{m}}\frac{1}{2}\sum_{\begin{subarray}{c}\alpha_{1},\alpha_{2}\\\
\beta_{1},\beta_{2}\end{subarray}}\langle u_{\alpha_{1}}^{*}\otimes
u_{\alpha_{2}}^{*},\Lambda^{(2)}(u_{\beta_{1}}\otimes
u_{\beta_{2}})\rangle\,\,\mathcal{S}_{n}(u_{\alpha_{1}}\otimes
u_{\alpha_{2}}\otimes
a^{-}\\{u_{\beta_{1}}^{*}\\}a^{-}\\{u_{\beta_{2}}^{*}\\}\rho_{m})dx^{(m)}\,,$
omitting the symmetrization under the integral,
$\displaystyle=\int_{\mathbb{X}^{m}}\frac{1}{2}\sum_{\begin{subarray}{c}\alpha_{1},\alpha_{2}\\\
\beta_{1},\beta_{2}\end{subarray}}\langle u_{\alpha_{1}}^{*}\otimes
u_{\alpha_{2}}^{*},\Lambda^{(2)}(u_{\beta_{1}}\otimes
u_{\beta_{2}})\rangle\,\,u_{\alpha_{1}}\otimes u_{\alpha_{2}}\otimes
a^{-}\\{u_{\beta_{1}}^{*}\\}a^{-}\\{u_{\beta_{2}}^{*}\\}\rho_{m}dx^{(m)}\,,$
carrying out the sums over $\alpha_{1},\alpha_{2}$ upon recognizing the
expansion of $\Lambda^{(2)}(u_{\beta_{1}}\otimes u_{\beta_{2}})$ in the
2-particle basis [eq. 28],
$\displaystyle=\int_{\mathbb{X}^{m}}\frac{1}{2}\sum_{\beta_{1},\beta_{2}}\Lambda^{(2)}(u_{\beta_{1}}\otimes
u_{\beta_{2}})\otimes
a^{-}\\{u_{\beta_{1}}^{*}\\}a^{-}\\{u_{\beta_{2}}^{*}\\}\rho_{m}dx^{(m)}\,,$
interchanging the sums and integrals,
$\displaystyle=\frac{1}{2}\sum_{\beta_{1},\beta_{2}}\int_{\mathbb{X}^{2}}\Lambda^{(2)}(u_{\beta_{1}}\otimes
u_{\beta_{2}})dx^{(2)}\int_{\mathbb{X}^{n-2}}a^{-}\\{u_{\beta_{1}}^{*}\\}a^{-}\\{u_{\beta_{2}}^{*}\\}\rho_{m}dx^{(m-2)}\,,$
making use of the identity (174),
$\displaystyle=\frac{1}{2}\sum_{\beta_{1},\beta_{2}}\int_{\mathbb{X}}\Lambda^{(2,1)}(u_{\beta_{1}}\otimes
u_{\beta_{2}})dx^{(1)}\int_{\mathbb{X}^{m}}a^{-}\\{u_{\beta_{1}}^{*}\\}a^{-}\\{u_{\beta_{2}}^{*}\\}\rho_{m}dx^{(m-2)}\,,$
and finally expanding $\Lambda^{(2,1)}(u_{\beta_{1}}\otimes u_{\beta_{2}})$ in
the 1-particle basis [eq. 28],
$\displaystyle=\frac{1}{2}\sum_{\beta_{1},\beta_{2}}\int_{\mathbb{X}}\sum_{\alpha}\langle
u_{\alpha}^{*},\Lambda^{(2,1)}(u_{\beta_{1}}\otimes u_{\beta_{2}})\rangle
a^{+}\\{u_{\alpha}\\}dx^{(1)}\int_{\mathbb{X}^{n-2}}a^{-}\\{u_{\beta_{1}}^{*}\\}a^{-}\\{u_{\beta_{2}}^{*}\\}\rho_{m}dx^{(m-2)}$
$\displaystyle=\int_{\mathbb{X}^{n-1}}\frac{1}{2}\sum_{\begin{subarray}{c}\alpha\\\
\beta_{1},\beta_{2}\end{subarray}}\langle
u_{\alpha}^{*},\Lambda^{(2,1)}(u_{\beta_{1}}\otimes u_{\beta_{2}})\rangle
a^{+}\\{u_{\alpha}\\}a^{-}\\{u_{\beta_{1}}^{*}\\}a^{-}\\{u_{\beta_{2}}^{*}\\}\rho_{m}dx^{(n-1)}$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\eqref{eq:appC_R21}}}}{{=}}\int_{\mathbb{X}^{n-1}}\mathcal{R}^{(2,1)}\rho_{m}dx^{(n-1)}.$
(175)

This implies (169) for the considered reaction. Using equivalent argumentation
for other reactions and combining the results delivers eq. 75.

For the dissipativity of the reaction operator $\mathcal{R}$ one has to show
that

$\|\mu\rho-\mathcal{R}\rho\|\geqslant\mu\|\rho\|\quad\text{for all $\mu>0$ and
$\rho\in\operatorname{dom}(\mathcal{R})$}.$ (176)

Here, $\operatorname{dom}(\mathcal{R})\subset F(H)$ denotes the domain of
$\mathcal{R}$, which is taken such that
$\rho\in\operatorname{dom}(\mathcal{R})$ satisfies
$\|\mathcal{R}\rho\|<\infty$ and
$\sum_{n=0}^{\infty}|\mathcal{J}_{nm}(\rho_{m})|<\infty$ for each
$m\in\mathbb{N}_{0}$. The last condition together with eq. 75 implies the
relation

$\sum_{n=0}^{\infty}\int_{\mathbb{X}^{n}}(\mathcal{R}\rho)_{n}dx^{(n)}=\sum_{n=0}^{\infty}\int_{\mathbb{X}^{n}}\sum_{m=0}^{\infty}(\mathcal{Q}_{nm}\rho_{m})dx^{(n)}=\sum_{m=0}^{\infty}\left(\sum_{n=0}^{\infty}\mathcal{J}_{nm}(\rho_{m})\right)\stackrel{{\scriptstyle\eqref{eq:conservation}}}{{=}}0,$
(177)

noting that only finitely many of the $\mathcal{Q}_{nm}$ are non-zero for
fixed $n$ (see eq. 74), which allows us to interchange the $m$-sum and the
integral over $\mathbb{X}^{n}$ and finally the order of summation. With this
relation, it is straightforward to infer the dissipativity of $\mathcal{R}$:

$\displaystyle\|\mu\rho-\mathcal{R}\rho\|$
$\displaystyle=\sum_{n=0}^{\infty}\|\mu\rho_{n}-(\mathcal{R}\rho)_{n}\|_{n}$
$\displaystyle\geqslant\Big{|}\sum_{n=0}^{\infty}\int_{\mathbb{X}^{n}}[\mu\rho_{n}-(\mathcal{R}\rho)_{n}]\,dx^{(n)}\Big{|}$
$\displaystyle=\Big{|}\sum_{n=0}^{\infty}\int_{\mathbb{X}^{n}}\mu\rho_{n}dx^{(n)}-\sum_{n=0}^{\infty}\int_{\mathbb{X}^{n}}(\mathcal{R}\rho)_{n}\,dx^{(n)}\Big{|}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Rrho-zero}}}{{=}}\mu\|\rho\|,$
(178)

making use of the triangle inequality in the second line and that both sums
are absolutely convergent in the last-but-one step.

Finally, the calculation in (178) can directly be transferred to the
reaction–diffusion operator $\mathcal{D}+\mathcal{R}$, noticing that due to
the reflecting boundary conditions at the boundary of $\mathbb{X}$, it holds

$\int_{\mathbb{X}^{n}}(\mathcal{D}\rho_{n})(x^{(n)})dx^{(n)}=0$ (179)

for any $n\in\mathbb{N}$. In total, we obtain the dissipativity of the CDME
operator $\mathcal{A}=\mathcal{R}+\mathcal{D}$.

##  Appendix D Spatial discretization

The crucial step in performing the spatial discretization is to consistently
define the respective diffusion and reaction operators, which are expressed in
terms of creation and annihilation operators, acting on the basis elements of
the copy number representation. We lay a basis for doing so in the following.

###  D.1 Creation and annihilation operators acting on a discrete space

The action of the creation and annihilation operators on the Galerkin-
projected Fock space $\hat{F}$ is particularly transparent for basis elements
of the copy number representation. Using the short-hand notation (100), it
holds

$\displaystyle a_{k}^{+}\ket{N_{1},\dots,N_{M}}$
$\displaystyle=\hphantom{N_{k}}\ket{N_{1},\dots,N_{k}+1,\dots,N_{M}},$ (180)
$\displaystyle a_{k}^{-}\ket{N_{1},\dots,N_{M}}$
$\displaystyle=N_{k}\ket{N_{1},\dots,N_{k}-1,\dots,N_{M}}.$ (181)

Relation (180) follows directly from the definition of the operator
$a_{k}^{+}$. In order to prove the second relation, eq. 181, we make the
following observation. From the commutator relation
$\left[a^{-}\\{f\\},a^{+}\\{w\\}\right]=\left\langle
f,w\right\rangle\operatorname{\mathcal{I}}$, see eq. 54, we get

$a^{-}_{i}a^{+}_{j}=a^{+}_{j}a^{-}_{i}+\left\langle\xi_{i}^{*},\xi_{j}\right\rangle\operatorname{\mathcal{I}}=a^{+}_{j}a^{-}_{i}+\delta_{ij}\operatorname{\mathcal{I}},$
(182)

where $\delta_{ij}$ is the Kronecker delta. More generally, for $n\geqslant
1$, we have

$\displaystyle a_{j}^{-}(a_{i}^{+})^{n}$
$\displaystyle=a_{i}^{+}a_{j}^{-}(a_{i}^{+})^{n-1}+\delta_{ij}(a_{i}^{+})^{n-1}$
$\displaystyle=(a_{i}^{+})^{2}a_{j}^{-}(a_{i}^{+})^{n-2}+2\delta_{ij}(a_{i}^{+})^{n-1}$
$\displaystyle=\ldots$
$\displaystyle=(a_{i}^{+})^{n}a_{j}^{-}+n\delta_{ij}(a_{i}^{+})^{n-1},$

and thus

$\left[a_{j}^{-},(a_{i}^{+})^{n}\right]=n\delta_{ij}(a_{i}^{+})^{(n-1)}.$
(183)

We can use this formula to calculate the following identities:

$\displaystyle a_{k}^{-}(a_{1}^{+})^{N_{1}}\ldots(a_{M}^{+})^{N_{M}}$
$\displaystyle=(a_{1}^{+})^{N_{1}}\ldots(a_{M}^{+})^{N_{M}}a_{k}^{-}+\sum_{i=1}^{M}N_{i}\delta_{ik}(a_{1}^{+})^{N_{1}}\ldots(a_{i}^{+})^{N_{i}-1}\ldots(a_{M}^{+})^{N_{M}}$
$\displaystyle=(a_{1}^{+})^{N_{1}}\ldots(a_{M}^{+})^{N_{M}}a_{k}^{-}+N_{k}(a_{1}^{+})^{N_{1}}\ldots(a_{k}^{+})^{N_{k}-1}\ldots(a_{M}^{+})^{N_{M}},$
(184)

which, applied to $\rho_{vac}$, yields eq. 181. By repeated application of eq.
181 if follows directly that

$a_{k}^{-}a_{l}^{-}\ket{N_{1},\dots,N_{M}}=N_{k}N_{l}\ket{N_{1},\ldots,N_{k}-1,\ldots,N_{l}-1,\ldots,N_{M}}$
(185)

for $k\neq l$, and

$a_{k}^{-}a_{k}^{-}\ket{N_{1},\dots,N_{M}}=N_{k}(N_{k}-1)\ket{N_{1},\ldots,N_{k}-2,\ldots,N_{M}},$
(186)

which will be used in the next section for deriving the projected reaction
operators of mutual annihilation.

###  D.2 Diffusion and reaction operators

For the calculation of the diffusion and reaction operators we need to apply
the properties of the creation and annihilation operators given in section D.1
and subsequently shift indices. In order to simplify notation in this index
shift, we define $p_{N_{1},\ldots,N_{M}}:=0$ for all
$(N_{1},\dots,N_{M})\notin\mathbb{N}_{0}^{M}$ (especially for negative
indices).

##### Diffusion.

For the projected diffusion operator $\hat{\mathcal{D}}$ given in eq. 96 and
specified in eq. 116 we find, using the copy number representation (101) as
well as the properties (180) and (181),

$\displaystyle\hat{\mathcal{D}}\hat{\rho}=$
$\displaystyle\sum_{N_{1},\dots,N_{M}}p_{N_{1},\ldots,N_{M}}\sum_{i,j}d_{ij}a_{i}^{+}a_{j}^{-}\ket{N_{1},\dots,N_{M}}$
$\displaystyle=$
$\displaystyle\sum_{N_{1},\dots,N_{M}}p_{N_{1},\ldots,N_{M}}\sum_{\begin{subarray}{c}i,j\\\
i\neq
j\end{subarray}}d_{ij}N_{j}\cdot\ket{N_{1},\dots,N_{i}+1,\ldots,N_{j}-1,\ldots,N_{M}}$
$\displaystyle\quad+\sum_{N_{1},\dots,N_{M}}p_{N_{1},\ldots,N_{M}}\sum_{i}d_{ii}N_{i}\cdot\ket{N_{1},\ldots,N_{M}}$
$\displaystyle=$
$\displaystyle\sum_{N_{1},\dots,N_{M}}\sum_{\begin{subarray}{c}i,j\\\ i\neq
j\end{subarray}}p_{N_{1},\ldots,N_{i}-1,\ldots,N_{j}+1,\ldots,N_{M}}d_{ij}(N_{j}+1)\cdot\ket{N_{1},\ldots,N_{M}}$
$\displaystyle\quad+\sum_{N_{1},\dots,N_{M}}\sum_{i}p_{N_{1},\ldots,N_{M}}d_{ii}N_{i}\cdot\ket{N_{1},\ldots,N_{M}}.$
(187)

##### Degradation.

Analogously, we obtain for the conserving one-particle reaction operator
$\hat{\mathcal{R}}_{d}^{(1)}=\sum_{i,j}\langle\xi_{i}^{*},\Lambda^{(1)}_{d}\xi_{j}\rangle
a_{i}^{+}a_{j}^{-}$ of degradation $\text{A}\to\varnothing$ by using
definition (121):

$\displaystyle\hat{\mathcal{R}}_{d}^{(1)}\hat{\rho}$
$\displaystyle=\sum_{N_{1},\dots,N_{M}}p_{N_{1},\ldots,N_{M}}\sum_{i,j}\lambda_{d}^{ij}a_{i}^{+}a_{j}^{-}\ket{N_{1},\ldots,N_{M}}$
$\displaystyle=\sum_{N_{1},\dots,N_{M}}p_{N_{1},\ldots,N_{M}}\sum_{\begin{subarray}{c}i,j\\\
i\neq
j\end{subarray}}\lambda_{d}^{ij}N_{j}\cdot\ket{N_{1},\ldots,N_{i}+1,\ldots,N_{j}-1,\ldots,N_{M}}$
$\displaystyle\qquad+\sum_{N_{1},\dots,N_{M}}p_{N_{1},\ldots,N_{M}}\sum_{i}\lambda_{d}^{ii}N_{i}\cdot\ket{N_{1}\ldots,N_{M}}$
$\displaystyle=\sum_{N_{1},\dots,N_{M}}\sum_{\begin{subarray}{c}i,j\\\ i\neq
j\end{subarray}}p_{N_{1},\ldots,N_{i}-1,\ldots,N_{j}+1,\ldots,N_{M}}\lambda_{d}^{ij}(N_{j}+1)\cdot\ket{N_{1},\ldots,N_{M}}$
$\displaystyle\qquad+\sum_{N_{1},\dots,N_{M}}\sum_{i}p_{N_{1},\ldots,N_{M}}\lambda_{d}^{ii}N_{i}\cdot\ket{N_{1},\ldots,N_{M}},$
(188)

and for the non-conserving one-particle reaction operator
$\hat{\mathcal{R}}_{d}^{(1,0)}=\sum_{i}\langle
1,\Lambda^{(1,0)}_{d}\xi_{i}\rangle a_{i}^{-}$ of degradation, by means of
definition (121):

$\displaystyle\hat{\mathcal{R}}_{d}^{(1,0)}\hat{\rho}$
$\displaystyle=\sum_{N_{1},\dots,N_{M}}p_{N_{1},\ldots,N_{M}}\sum_{i}\lambda_{d}^{i}\,a^{-}_{i}\ket{N_{1},\ldots,N_{M}}$
$\displaystyle=\sum_{N_{1},\dots,N_{M}}p_{N_{1},\ldots,N_{M}}\,\sum_{i}\lambda_{d}^{i}N_{i}\cdot\ket{N_{1},\ldots,N_{i}-1,\ldots,N_{M}}$
$\displaystyle=\sum_{N_{1},\dots,N_{M}}\sum_{i}p_{N_{1},\ldots,N_{i}+1,\ldots,N_{M}}\,\lambda_{d}^{i}(N_{i}+1)\cdot\ket{N_{1},\ldots,N_{M}}.$
(189)

##### Creation.

For the creation reaction $\varnothing\to I$ the conserving part is given by

$\hat{\mathcal{R}}_{c}^{(0)}\hat{\rho}=\sum_{N_{1},\dots,N_{M}}p_{N_{1},\ldots,N_{M}}\,\lambda_{c}\ket{N_{1},\ldots,N_{M}},$
(190)

and the nonconserving one-particle reaction operator acts as:

$\displaystyle\hat{\mathcal{R}}_{c}^{(0,1)}\hat{\rho}$
$\displaystyle=\sum_{N_{1},\dots,N_{M}}p_{N_{1},\ldots,N_{M}}\,\sum_{i}\lambda_{c}^{i}a^{+}_{i}\ket{N_{1},\ldots,N_{M}}$
$\displaystyle=\sum_{N_{1},\dots,N_{M}}p_{N_{1},\ldots,N_{M}}\,\sum_{i}\lambda_{c}^{i}\ket{N_{1},\ldots,N_{i}+1,\ldots,N_{M}}$
$\displaystyle=\sum_{N_{1},\dots,N_{M}}\sum_{i}p_{N_{1},\ldots,N_{i}-1,\ldots,N_{M}}\lambda_{c}^{i}\,\ket{N_{1},\ldots,N_{M}},$
(191)

where we applied definition (127).

##### Mutual annihilation.

For the reaction $\text{A}+\text{A}\to\varnothing$ of mutual annihilation, the
projected conserving operator $\hat{\mathcal{R}}^{(2)}$ is given by

$\hat{\mathcal{R}}^{(2)}=\frac{1}{2}\sum_{\begin{subarray}{c}i,j\\\
k,l\end{subarray}}\langle\xi^{*}_{i}\otimes\xi^{*}_{j},\Lambda^{(2)}(\xi_{k}\otimes\xi_{l})\rangle
a_{i}^{+}a_{j}^{+}a_{k}^{-}a_{l}^{-}$ (192)

see eq. 86a. For the rescaled indicator functions given in eq. 106, the factor

$\langle\xi^{*}_{i}\otimes\xi^{*}_{j},\Lambda^{(2)}(\xi_{k}\otimes\xi_{l})\rangle=\int_{\mathbb{X}^{2}}(\xi^{*}_{i}\otimes\xi^{*}_{j})(x^{(2)})\lambda(x^{(2)})(\xi_{k}\otimes\xi_{l})(x^{(2)})dx^{(2)}$
(193)

is for $k=i$ and $l=j$ equal to $\lambda^{ij}$ defined in eq. 131, and zero
otherwise.

We thus obtain

$\displaystyle\hat{\mathcal{R}}^{(2)}\hat{\rho}$
$\displaystyle=\frac{1}{2}\sum_{N_{1},\dots,N_{M}}\sum_{i,j}p_{N_{1},\ldots,N_{M}}\lambda^{ij}\,a^{+}_{i}a^{+}_{j}a^{-}_{i}a^{-}_{j}\ket{N_{1},\ldots,N_{M}}$
$\displaystyle=\frac{1}{2}\sum_{N_{1},\dots,N_{M}}\Bigg{[}\sum_{i}p_{N_{1},\ldots,N_{M}}\lambda^{ii}N_{i}(N_{i}-1)\,\ket{N_{1},\ldots,N_{M}}$
$\displaystyle\hskip 70.0001pt+\sum_{i\neq
j}p_{N_{1},\ldots,N_{M}}\lambda^{ij}N_{i}N_{j}\,\ket{N_{1},\ldots,N_{M}}\Bigg{]}$
$\displaystyle=\sum_{N_{1},\dots,N_{M}}\Bigg{[}\sum_{i}p_{N_{1},\ldots,N_{M}}\frac{1}{2}\lambda^{ii}N_{i}(N_{i}-1)\,\ket{N_{1},\ldots,N_{M}}$
$\displaystyle\hskip
70.0001pt+\sum_{i<j}p_{N_{1},\ldots,N_{M}}\lambda^{ij}N_{i}N_{j}\,\ket{N_{1},\ldots,N_{M}}\Bigg{]},$
(194)

where the last equality follows from the fact that the reaction rate function
$\lambda(x^{(2)})$ is symmetric, which implies $\lambda^{ij}=\lambda^{ji}$.

For the nonconserving part it holds

$\hat{\mathcal{R}}^{(2,0)}=\frac{1}{2}\sum_{i,j}\langle
1,\Lambda^{(2,0)}(\xi_{i}\otimes\xi_{j})\rangle a_{i}^{-}a_{j}^{-},$ (195)

where

$\displaystyle\langle 1,\Lambda^{(2,0)}(\xi_{i}\otimes\xi_{j})\rangle$
$\displaystyle=\Lambda^{(2,0)}(\xi_{i}\otimes\xi_{j})$
$\displaystyle=\int_{\mathbb{X}^{2}}\lambda(x^{(2)})(\xi_{i}\otimes\xi_{j})(x^{(2)})dx^{(2)}$
$\displaystyle=\lambda^{ij}$ (196)

again for $\lambda^{ij}$ defined in eq. 131, such that

$\displaystyle\hat{\mathcal{R}}^{(2,0)}\hat{\rho}$
$\displaystyle=\frac{1}{2}\sum_{N_{1},\dots,N_{M}}\sum_{i,j}p_{N_{1},\ldots,N_{M}}\lambda^{ij}\,a^{-}_{i}a^{-}_{j}\ket{N_{1},\ldots,N_{M}}$
$\displaystyle=\frac{1}{2}\sum_{N_{1},\dots,N_{M}}\Bigg{[}\sum_{i}p_{N_{1},\ldots,N_{M}}\lambda^{ii}N_{i}(N_{i}-1)\,\ket{N_{1},\ldots,N_{i}-2,\ldots,N_{M}}$
$\displaystyle\hskip 50.00008pt+\sum_{i\neq
j}p_{N_{1},\ldots,N_{M}}\lambda^{ij}N_{i}N_{j}\,\ket{N_{1},\ldots,N_{i}-1,\ldots,N_{j}-1,\ldots,N_{M}}\Bigg{]}$
$\displaystyle=\sum_{N_{1},\dots,N_{M}}\Bigg{[}\sum_{i}p_{N_{1},\ldots,N_{i}+2,\ldots,N_{M}}\frac{1}{2}\lambda^{ii}(N_{i}+2)(N_{i}+1)\,\ket{N_{1},\ldots,N_{M}}$
$\displaystyle\hskip
50.00008pt+\sum_{i<j}p_{N_{1},\ldots,N_{i}+1,\ldots,N_{j}+1,\ldots,N_{M}}\lambda^{ij}(N_{i}+1)(N_{j}+1)\,\ket{N_{1},\ldots,N_{M}}\Bigg{]}.$
(197)

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capbtabboxtable[][]

# A new rank-2 Argyres-Douglas theory

Justin Kaidi1 Mario Martone1,2 1\. Simons Center for Geometry and Physics,
Stony Brook University  2\. C. N. Yang Institute for Theoretical Physics,
Stony Brook University, Stony Brook, NY 11794-3840, USA

###### Abstract

We provide evidence for the existence of a new strongly-coupled four
dimensional ${\mathcal{N}}=2$ superconformal field theory arising as a non-
trivial IR fixed point on the Coulomb branch of the mass-deformed
superconformal Lagrangian theory with gauge group $G_{2}$ and four fundamental
hypermultiplets. Notably, our analysis proceeds by using various geometric
constraints to bootstrap the data of the theory, and makes no explicit
reference to the Seiberg-Witten curve. We conjecture a corresponding VOA and
check that the vacuum character satisfies a linear modular differential
equation of fourth order. We also propose an identification with existing
class ${\mathcal{S}}$ constructions.

##  I Introduction

Among 4d ${\mathcal{N}}=2$ superconformal field theories (SCFTs), of
particular interest are those with some chiral ring generators having
fractional scaling dimension. Such theories are necessarily non-Lagrangian and
as such are not amenable to study by the most naive means. The original
examples of such theories, due to Argyres and Douglas Argyres:1995jj , can be
obtained by tuning to special points in the moduli space of mass-deformed
Lagrangian theories, where mutually non-local dyons become massless. Since the
early works, several other means of obtaining such theories have been
developed, and in particular constructions in class ${\mathcal{S}}$ and
geometric engineering Xie:2012hs ; Xie:2015rpa ; Wang:2018gvb  have allowed
one to circumvent the Lagrangian starting point altogether, giving rise to a
wealth of Argyres-Douglas (AD) type theories which are not connected in any
obvious way to the moduli space of known Lagrangian theories.

In this note, we proceed in the original spirit of Argyres and Douglas.
Namely, we will identify a non-Lagrangian theory obtainable from a mass-
deformation of the Lagrangian $G_{2}$ gauge theory with four fundamental
hypermultiplets. However, unlike in the original works, our analysis will make
no explicit use of the Seiberg-Witten curve. Instead, we will use a variety of
geometric constraints on the structure of the Coulomb and Higgs branches to
bootstrap the data of the theory. This will allow us to construct a consistent
candidate moduli space, as well as a corresponding VOA. We will also make
connection to known class ${\mathcal{S}}$ constructions, identifying the
resulting theory with the $A_{4}$ 6d (2,0) theory compactified on a sphere
with irregular puncture and outer-automorphism twist. Our example serves as a
proof of principle that a bottom-up, geometric approach to bootstrapping
general ${\mathcal{N}}=2$ SCFTs in 4d is feasible.

##  II Geometric Analysis: Higgs and Coulomb branch

Our starting point is a Lagrangian SCFT with $G_{2}$ gauge group and four
hypermultiplets in the fundamental ($\mathbf{7}$) representation. The moduli
space data of this theory can be conveniently summarized in the Hasse diagrams
of Figure 1. Here $\mathbbm{H}^{n}$ represents the theory of $n$ free
hypermultiplets, and the red (resp. blue) lines represent series of singular
strata in the Coulomb (resp. Higgs) branches Martone:2021ixp . Each Higgs
stratum is the minimal nilpotent orbit of a Lie algebra $\mathfrak{g}$, and is
denoted by the same symbol. Following the convention of Giacomelli:2020jel ,
the theory ${\mathcal{T}}^{(1)}_{G,1}$ denotes the rank-1 theory obtained on
the worldvolume of a D$3$-brane probing a $G$-type exceptional 7-brane. In
particular, ${\mathcal{T}}^{(1)}_{D_{4},1}$ is a $\mathfrak{su}(2)$ gauge
theory with $N_{f}=4$ flavors in the fundamental, while
${\mathcal{T}}^{(1)}_{A_{2},1}$ and ${\mathcal{T}}^{(1)}_{A_{1},1}$, to be
introduced momentarily, are the rank-1 AD theories with flavor symmetry
$\mathfrak{su}(3)$ and $\mathfrak{su}(2)$, also known as $H_{2}$ and $H_{1}$.

$\mathfrak{g}_{2}+4F$$[I_{1},\varnothing]{\times}\mathbbm{H}^{4}$
$[I^{*}_{3},\mathfrak{sp}(8)]$$[I_{1},\varnothing]{\times}\mathbbm{H}^{4}$
$\big{[}u^{3}+v=0\big{]}$$\big{[}v=0\big{]}$$\mathbbm{H}^{14}$$\mathfrak{a}_{1}$${\mathcal{T}}^{(1)}_{D_{4},1}$$\mathfrak{a}_{5}$$\mathfrak{su}(3)+6F$$\mathfrak{c}_{4}$$\mathfrak{g}_{2}+4F$
Figure 1: Hasse diagrams for the CB and HB of $\mathfrak{g}_{2}+4F$.

The AD theory which we will describe here is a mass-deformation of the $G_{2}$
gauge theory by

$\displaystyle\delta{\mathcal{L}}=M^{ij}Q_{i}^{\alpha}\Sigma_{{\alpha}{\beta}}Q_{j}^{\beta}~{},$
(1)

where $Q_{i}^{\alpha}$ is a hypermultiplet with flavor index $i=1,\dots,4$ and
fundamental index ${\alpha}=1,\dots,7$. The matrix $\Sigma_{{\alpha}{\beta}}$
is taken to project onto the trivial representation in
$\mathbf{7}\otimes\mathbf{7}$ and $M^{ij}$ is a mass matrix which breaks
$\mathfrak{sp}(8)$ to a subgroup $\mathfrak{sp}(4)$.111One could imagine a
deformation to a rank-3 flavor symmetry group $\mathfrak{sp}(6)$, but we have
not found any theories of this type. Rather than directly studying the effect
of this mass deformation, we will proceed by constructing the result from the
bottom-up using a variety of geometric constraints.

Given two SCFTs ${\mathcal{T}}_{0}$ and ${\mathcal{T}}_{1}$ which are
connected by a mass deformation, it is expected that the moduli space of
${\mathcal{T}}_{1}$ has the same structure of stratification as
${\mathcal{T}}_{0}$, but with the theory on each leaf being replaced with a
mass-deformation thereof Martone:2021drm .222There is a subtle but important
difference between a symplectic leaf and a symplectic stratum, the latter
being the closure of the former. We recall here the following sequences of
mass deformations Eguchi:1996vu ; Argyres:2015ffa ,

$\displaystyle\mathfrak{su}(3)+6F$ $\displaystyle\rightarrow
D_{2}(SU(5))\rightarrow(A_{1},D_{6})\rightarrow\dots$
$\displaystyle{\mathcal{T}}^{(1)}_{D_{4},1}$
$\displaystyle\rightarrow{\mathcal{T}}^{(1)}_{A_{2},1}\rightarrow{\mathcal{T}}^{(1)}_{A_{1},1}\rightarrow\dots$
(2)

where the series of theories dubbed $D_{2}(SU(2N+1))$ were studied in
Cecotti:2012jx ; Cecotti:2013lda ; Xie:2016evu , while the generalized AD
theories $(G_{1},G_{2})$ were introduced in Cecotti:2010fi . This motivates us
to propose the Higgs branch Hasse diagram shown in Figure 3 for our mass-
deformed theory. Note that as per footnote 1 we have gone two steps down the
sequence of mass deformations. Because we have assumed that the mass-
deformation breaks the $\mathfrak{sp}(8)$ flavor symmetry of the starting
theory to $\mathfrak{sp}(4)$, we refer to the theory as $\rm
AD({{\mathfrak{c}}_{2}})$.333We are using the convention in which
$\mathfrak{sp}(2n)$ is rank $n$. Note also that while AD theories have been
historically labeled by the gauge group of the theory on whose Coulomb branch
they arise, we are not using this convention. Here ${{\mathfrak{c}}_{2}}$
refers to the flavor symmetry. The $(A_{1},D_{6})$ theory is taken to be
supported on the minimal nilpotent orbit of this $\mathfrak{sp}(4)$. That this
is the correct thing to do will be confirmed shortly.

[7cm]  $\rm AD({{\mathfrak{c}}_{2}})$ $({\Delta}_{u},{\Delta}_{v})$
($\frac{4}{3},\frac{10}{3}$) $24a$ 48 $12c$ 26 $\mathfrak{f}_{k}$
$\mathfrak{sp}(4)_{\frac{13}{3}}$ $d_{\rm HB}$ 4 $h$ 2 $T({\bf 2}{\bm{h}})$ 1

Figure 2:  Relevant conformal data for $\rm AD({{\mathfrak{c}}_{2}})$.

Let us denote the unbroken symmetry on a generic point of the nilpotent orbit
of $\mathfrak{f}$ by $\mathfrak{f}^{\natural}$, and the symmetry (level)
realized by the theory on the corresponding leaf by
$\mathfrak{f}_{\mathrm{IR}}\,\,(k_{\mathrm{IR}})$. Furthermore, say that upon
spontaneously breaking $\mathfrak{f}$ to $\mathfrak{f}^{\natural}$ one
produces Goldstone bosons in a representation $\mathfrak{R}$. In terms of this
data, the flavor level of the original theory with symmetry $\mathfrak{f}$ is
given by Beem:2019tfp ; Beem:2019snk ; Giacomelli:2020jel

$\displaystyle
k_{\mathfrak{f}}={T_{2}(\mathfrak{R})+k_{\mathrm{IR}}I_{\mathfrak{f}_{\mathrm{IR}}\hookrightarrow\mathfrak{f}}\over
I_{{\mathfrak{f}^{\natural}}\hookrightarrow\mathfrak{f}}}$ (3)

where $T_{2}(\mathfrak{R})$ is the Dynkin index of the representation
$\mathfrak{R}$ and $I_{\mathfrak{g}\hookrightarrow\mathfrak{f}}$ is the
embedding index of $\mathfrak{g}$ into $\mathfrak{f}$.

In the case of interest to us, $\mathfrak{f}=\mathfrak{sp}(4)$ while
$\mathfrak{f}^{\natural}=\mathfrak{f}_{\mathrm{IR}}=\mathfrak{sp}(2)\cong\mathfrak{su}(2)$.
The Goldstone bosons appearing upon spontaneous breaking of $\mathfrak{sp}(4)$
to $\mathfrak{sp}(2)$ are in the $\mathbf{2}$. A standard mathematical
exercise then gives $I_{\mathfrak{sp}(2)\hookrightarrow\mathfrak{sp}(4)}=1$
and we choose a normalization for the Dynkin index such that
$T_{2}(\mathbf{2})=1$. Combined with the known flavor level of
$(A_{1},D_{6})$, i.e. $k_{\mathfrak{sp}(2)}={10\over 3}$, we conclude that the
tentative new theory has

$\displaystyle k_{\mathfrak{sp}(4)}={13\over 3}~{}.$ (4)

The structure of the Higgs branch may also be used to constrain the central
charges of the new theory. In particular, consider a spontaneous breaking of a
flavor symmetry $\mathfrak{f}$ which takes a theory ${\mathcal{T}}_{0}$ to a
theory ${\mathcal{T}}_{\rm Higgs}$ supported on a leaf $\mathfrak{S}$. Using
anomaly matching, the central charges of the two theories can be related by
Giacomelli:2020jel ; CCLMW2021

$\displaystyle 12\,c_{{\mathcal{T}}_{0}}=12\,c_{{\mathcal{T}}_{\rm
Higgs}}+2\left({3\over
2}k_{\mathfrak{f}}-1\right)+\mathrm{dim}_{\mathbbm{H}}\overline{\mathfrak{S}}-1~{}.$
(5)

where the bar indicates the closure of $\mathfrak{S}$
($\overline{\mathfrak{S}}$ is then a symplectic stratum). For our purposes, we
need only note that the closure of the minimal nilpotent orbit of
$\mathfrak{sp}(4)$ has quaternionic dimension
$\mathrm{dim}_{\mathbbm{H}}\overline{\mathfrak{S}}=2$. This together with
$k_{\mathfrak{sp}(4)}$ computed above, as well as with the central charge
$12c_{(A_{1},D_{6})}=14$ for the $(A_{1},D_{6})$ theory, allows us to compute

$\displaystyle 12\,c_{\rm AD({{\mathfrak{c}}_{2}})}=26~{}.$ (6)

[7cm][] $\rm
AD({{\mathfrak{c}}_{2}})$$[I_{1},\varnothing]{\times}\mathbbm{H}^{2}$
$[I^{*}_{1},\mathfrak{sp}(4)]$$\big{[}u^{5}+v^{2}=0\big{]}$$\big{[}v=0\big{]}$$\mathbbm{H}^{4}$$\mathfrak{a}_{1}$${\mathcal{T}}^{(1)}_{A_{1},1}$$\mathfrak{a}_{1}$$(A_{1},D_{6})$$\mathfrak{c}_{2}$$\rm
AD({{\mathfrak{c}}_{2}})$

Figure 3:  Hasse diagrams for the moduli space of $\rm
AD({{\mathfrak{c}}_{2}})$.

Since the dimension of the stratum
$\mathrm{dim}_{\mathbbm{H}}\overline{\mathfrak{S}}$ is the difference of the
quaternionic dimensions of the parent and daughter Higgs branches, i.e.
$\mathrm{dim}_{\mathbbm{H}}\overline{\mathfrak{S}}=d_{\mathrm{HB}}({\mathcal{T}}_{0})-d_{\mathrm{HB}}({\mathcal{T}}_{\rm
Higgs})$, and using the fact that $d_{HB}=2$ for $(A_{1},D_{6})$, we conclude
that the Higgs branch dimension of $\rm AD({{\mathfrak{c}}_{2}})$ must be

$\displaystyle d_{\mathrm{HB}}=4~{}.$ (7)

Finally, for theories which flow on the generic point of the Higgs branch to
$d_{\mathrm{HB}}$ free hypers, the $c$ and $a$ anomalies are constrained to
satisfy

$\displaystyle 24(c-a)=d_{\mathrm{HB}}~{},$ (8)

from which we compute

$\displaystyle 24\,a_{\rm AD({{\mathfrak{c}}_{2}})}=48~{}.$ (9)

The data which we have obtained so far is summarized in Table 2.

Thus far, everything has followed completely from the proposed structure of
the Higgs branch. We next turn towards an analysis of the Coulomb branch. The
Coulomb branch of a rank-2 theory is a complex two-dimensional space with
coordinates $u$ and $v$. These coordinates have scaling dimensions
$\Delta_{u}$ and $\Delta_{v}$ under a certain $\mathbb{C}^{*}$-action
descending from spontaneously broken dilatations and $\mathfrak{u}(1)_{r}$.
The Coulomb branch again has various strata, which are complex codimension-1
varieties specified by the vanishing loci of polynomials in $u,v$ homogeneous
under the $\mathbb{C}^{*}$-action. Such homogenous polynomials come in two
qualitative forms, “unknotted” ($u=0,v=0$) and “knotted” ($u^{p}+\beta
v^{q}=0$ for $\Delta_{v}/\Delta_{u}=p/q$ and $(p,q)=1$) Argyres:2018zay .

Following the notation in Martone:2020nsy , we will denote the degree of
homogeneity of the polynomial corresponding to the $i$-th stratum
${\mathcal{I}}_{i}$ by $\Delta^{\mathrm{sing}}_{i}$. This is $\Delta_{u}$ or
$\Delta_{v}$ for unknotted strata and $p\Delta_{u}$ for knotted strata. The
stratum ${\mathcal{I}}_{i}$ hosts a rank-1 theory, and we will denote the
dimension of this theory’s Coulumb branch coordinate by $\Delta_{i}$.
Likewise, the central charge, level, and extended Coulomb branch dimension of
the theory will be denoted by $c_{i},k_{i},h_{i}$.

Let us denote the set of all Coulomb branch strata by
${\mathcal{I}}=\left\\{{\mathcal{I}}_{i}\right\\}$, and the set of strata
carrying the UV flavor symmetry by
${\mathcal{I}}_{\mathfrak{f}}\subset{\mathcal{I}}$. It is a remarkable fact
that the central charges and level of the theory at the origin of the Coulomb
branch are determined entirely by the data of these strata Martone:2020nsy

$\displaystyle 12\,c$
$\displaystyle\,=\,4+h+\sum_{{\mathcal{I}}_{i}\in{\mathcal{I}}}{\Delta_{i}^{\mathrm{sing}}\over\Delta_{i}}(12\,c_{i}-h_{i}-2)~{},$
$\displaystyle k_{\mathfrak{f}}$
$\displaystyle\,=\,\sum_{{\mathcal{I}}_{i}\in{\mathcal{I}}_{\mathfrak{f}}}{\Delta_{i}^{\mathrm{sing}}\over\Delta_{i}}[k_{i}-T(2h_{i})]+T(2h)~{},$
(10)

together with the relation of Shapere and Tachikawa Shapere:2008zf ,

$\displaystyle 4(2a-c)=2(\Delta_{u}+\Delta_{v})-2~{}.$ (11)

Conversely, given $(c,a,\mathfrak{f},k_{\mathfrak{f}})$ of the theory at the
origin, these equations give constraints on the allowed Coulomb branch
geometries.

In addition, there are three more Coulomb branch constraints which must be
implemented Argyres:2018urp ; Argyres:2020wmq . First, the Coulomb branch of
any rank-2 theory (excluding products of rank-1 theories), must have at least
one knotted stratum. Second, an unknotted stratum $u=0$ (resp. $v=0$) can be
present only if $\Delta_{v}$ (resp. $\Delta_{u}$) is an allowed rank-1 Coulomb
branch dimension,

$\displaystyle\Delta_{\mathrm{rank-}1}\in\left\\{6,\,4,\,3,\,2,\,{3\over
2},\,{4\over 3},\,{6\over 5}\right\\}~{}.$ (12)

Finally, according to the UV-IR flavor condition Martone:2020nsy , the full
flavor symmetry $\mathfrak{f}$ of the theory at the origin must be realized on
at least one stratum, i.e. ${\mathcal{I}}_{\mathfrak{f}}$ is non-empty.

Together, these provide an extremely strict set of constraints, which for
generic $(c,a,\mathfrak{f},k_{\mathfrak{f}})$ do not admit a solution. One can
interpret our ability to construct a solution at all as non-trivial evidence
for our proposal.

We begin our construction of the Coulomb branch with the UV-IR flavor
condition. This requires that at least one of the rank-1 theories describing
the effective low-energy theory on the codimension-1 singular loci has to
carry flavor symmetry $\mathfrak{sp}(4)$. There exist only two theories with
this property Argyres:2015ffa ; Argyres:2015gha ; Argyres:2016xua ;
Argyres:2016xmc ; Argyres:2016yzz , namely
$[I_{2},\mathfrak{su}(4)]_{\mathbb{Z}_{2}}$ and
$[I_{1}^{*},\mathfrak{sp}(4)]$.444The former is a $\mathbb{Z}_{2}$ gauging of
$\mathfrak{u}(1)$ gauge theory with $4$ hypermultiplets, while the latter is
$\mathfrak{su}(2)$ gauge theory with two hypermultiplets in the $\mathbf{3}$.
The latter has a Witten anomaly for the flavor symmetry, while the former does
not. Only the latter can be obtained by deformation of the
$[I_{3}^{*},\mathfrak{sp}(8)]$ stratum in Figure 1. Hence this is the case we
must choose. We make the minimal guess that there is only a single stratum
which carries the $\mathfrak{c}_{2}$ flavor symmetry, and that this stratum is
unknotted, say $v=0$; this is indeed the case for all known rank-2 theories
with simple flavor symmetry Martone:2021ixp . With this, (11) and the second
equation of (II) can be used to determine the dimensions of the Coulomb branch
parameters,

$\displaystyle\Delta_{u}={4\over 3}~{},\hskip 36.135pt\Delta_{v}={10\over
3}~{}.$ (13)

Here we have used the known data of the $[I_{1}^{*},\mathfrak{sp}(4)]$ theory,
i.e. $(12c_{i},k_{i},T(2h_{i}))=\left({39\over 4},3,1\right)$. Note that
$\Delta_{u}$ is indeed in the allowed list (12) of rank-1 scaling dimensions,
as required by our second Coulomb branch constraint.

To satisfy the first Coulomb branch constraint, we are required to have at
least one knotted stratum, and the minimal assumption is to have exactly one
knotted stratum carrying no flavor symmetry. There is in fact a unique such
possibility compatible with (II), namely the theory $[I_{1},\varnothing]$.
This completes the construction of the Coulomb branch stratification.

As a final point, note that the dimension $h$ of the extended Coulomb branch
of ${\rm AD({{\mathfrak{c}}_{2}})}$ is equal to that of the theory supported
on the stratum realizing the SCFT’s flavor symmetry, thus

$\displaystyle h=2~{}.$ (14)

This value matches with that expected from the Higgs branch point of view.
Indeed, the dimension of the extended Coulomb branch is equal to the total
dimension of the symplectic leaves supporting theories of the same rank as our
initial SCFT. In the current case, because $(A_{1},D_{6})$ is of the same rank
as AD($\mathfrak{c}_{2}$), we expect the extended Coulomb branch to be non-
trivial and of dimension equal to the dimension of the leaf where
$(A_{1},D_{6})$ is supported, i.e. 2.

In conclusion, the Hasse diagrams in Figure 3 give a consistent picture of the
moduli space for a tentative new theory with data given in Table 2. These
diagrams pass all non-trivial consistency checks that we know of.

##  III Algebraic analysis: Schur index and VOA

Having used geometric techniques to derive the data of our tentative theory,
we will now make use of a variety of algebraic techniques to check the
consistency of this data.

###  III.1 Schur Index and Higgsing

The Schur index Gadde:2011uv ; Gadde:2011ik  of our proposed theory is given
by

$\displaystyle{\mathcal{I}}_{\rm AD({{\mathfrak{c}}_{2}})}$
$\displaystyle={\rm P.E.}\left[{1\over 1-q}\sum_{i\in
3\mathbbm{N}}\left(\chi^{\mathfrak{c}_{2}}_{\rm
adj}\,q^{i+1}+\chi^{\mathfrak{c}_{2}}_{\bf
4}\,q^{\frac{3}{2}+i}\right.\right.$
$\displaystyle\qquad\quad\qquad\qquad\left.\left.-\chi^{\mathfrak{c}_{2}}_{\bf
4}\,q^{\frac{5}{2}+i}-\chi^{\mathfrak{c}_{2}}_{\rm
adj}\,q^{3+i}\right)\right]$ (15)

where ${\rm P.E.}$ is the plethystic exponential ${\rm
P.E.}[f(x)]:=\exp(\sum_{n}\tfrac{1}{n}f(x^{n}))$ and
$\chi^{\mathfrak{c}_{2}}_{R}$ is the character for the representation $R$ of
$\mathfrak{sp}(4)$.

A first check of this expression is that it reduces to the known Schur index
of $(A_{1},D_{6})$ upon spontaneous breaking of
$\mathfrak{sp}(4)\rightarrow\mathfrak{su}(2)$. To see this, consider giving a
vev to a moment map operator ${\mathcal{O}}_{{\alpha}_{2}}$ associated to the
long root ${\alpha}_{2}=(-2,2)$ of $\mathfrak{sp}(4)$. The contribution of
this operator to the index is $qy^{{\alpha}_{2}}:=qy_{1}^{2}y_{2}^{-2}$. Upon
Higgsing we remove this contribution, as well as those from
$({\mathcal{O}}_{{\alpha}_{2}})^{n}$ for all $n\in\mathbb{N}$. This roughly
amounts to multiplying by
$(\sum_{n}(qy^{{\alpha}_{2}})^{n})^{-1}=(1-qy^{{\alpha}_{2}})$ and setting
$qy^{{\alpha}_{2}}=1$. More precisely, if $y_{2}^{*}$ is the solution to
$qy^{{\alpha}_{2}}=1$, then the index after Higgsing is given by
Gaiotto:2012xa ; Beem:2014rza ; Nishinaka:2018zwq

$\displaystyle{\mathcal{I}}_{\rm Higgs}={\mathcal{I}}_{\rm
vec}\lim_{y_{2}\rightarrow
y_{2}^{*}}\left((1-qy^{{\alpha}_{2}})\,{\mathcal{I}}_{\rm
AD({{\mathfrak{c}}_{2}})}\right)$ (16)

where ${\mathcal{I}}_{\rm vec}=(q;q)_{\infty}^{2}$ is the index of a free
vector multiplet and $(a,q)_{\infty}:=\prod_{i=0}^{\infty}(1-aq^{i})$ is the
_q_ -Pochhammer symbol. Stripping off the contribution from two free
hypermultiplets and setting $y_{1}=1$, this gives an index with $q$-expansion

$\displaystyle{\mathcal{I}}_{(A_{1},D_{6})}$
$\displaystyle=(\sqrt{q},q)_{\infty}^{2}\,{\mathcal{I}}_{\rm Higgs}$
$\displaystyle=1+4q+4q^{3\over 2}+14q^{2}+16q^{5\over 2}+\dots$ (17)

which agrees precisely with the results of Buican:2015ina .

###  III.2 VOA

As is well-known Beem:2013sza , the Schur index gives the vacuum character of
the corresponding VOA via
$\chi_{\mathrm{vac}}(q)=q^{\tfrac{c_{4d}}{2}}{\mathcal{I}}_{\rm
AD({{\mathfrak{c}}_{2}})}$. In this discussion we will only consider the
unrefined version of the index. Setting the flavor fugacities to 1 in (III.1)
gives the following $q$-expansion:

$\chi_{\rm
vac}(q)=q^{\frac{13}{12}}(1+10\,q+4\,q^{\tfrac{3}{2}}+65\,q^{2}+40\,q^{\tfrac{5}{2}}+\dots)~{}.$
(18)

This vacuum character satisfies a degree-4 twisted modular differential
equation ${\mathcal{D}}\chi_{\mathrm{vac}}(q)=0$, where

${\mathcal{D}}:=D^{(4)}+\left(\frac{1}{4}\Theta_{1,1}-\frac{11}{24}\Theta_{0,2}\right)D^{(2)}+\left(\frac{1}{72}\Theta_{0,3}-\frac{7}{144}\Theta_{1,2}\right)D^{(1)}+\left(\frac{13}{2304}\Theta_{0,4}-\frac{13}{576}\Theta_{1,3}+\frac{13}{768}\Theta_{2,2}\right)~{}.$
(19)

Here $D^{(n)}$ are the $n$-th order Serre derivatives and $\Theta_{r,s}$ with
$s\geq r$ are a basis of weight $k=r+s$ modular forms of
${\Gamma}^{0}(2)\subset SL(2,\mathbbm{Z})$. They can be written in terms of
Jacobi theta functions $\theta_{2}$ and $\theta_{3}$ as follows,

$\displaystyle\Theta_{r,s}(\tau)$
$\displaystyle:=\theta_{2}(\tau)^{4r}\theta_{4}(\tau)^{4s}+\theta_{2}(\tau)^{4s}\theta_{3}(\tau)^{4r}~{}.$
(20)

For more details on these definitions, see Appendix A of Beem:2017ooy  and
references therein.

The minimal degree modular differential equation satisfied by
$\chi_{\mathrm{vac}}(q)$ is degree-3 and is non-monic. This means that the
corresponding VOA is a three-character theory with non-zero Wronskian index.
The relevant data for the characters is found to be
$(c_{2d},h_{1},h_{2})=\left(-26,-{4\over 3},-{7\over 6}\right)$. As can be
motivated further from the class ${\mathcal{S}}$ construction below, a
candidate VOA is an extension of affine $\widehat{\mathfrak{sp}(4)}_{-{13\over
6}}$ by four dimension $3\over 2$ generators. As a first order check, one sees
that the usual formula

$\displaystyle c_{2d}={k_{2d}\,{\rm dim}G\over k_{2d}+h^{\vee}}={10k_{2d}\over
k_{2d}+3}=-26$ (21)

indeed holds for $k_{2d}=-{1\over 2}k_{4d}=-{13\over 6}$. Since this
particular level is _admissible_ in the language of Arakawa:2010ni , it
follows that the associated variety is given by the closure of the principal
nilpotent orbit of $\mathfrak{sp}(4)$.

The Higgs branch identified in Figure 3 differs from the principal nilpotent
orbit of $\mathfrak{sp}(4)$ only in the last stratum, which is hosted on
$\mathfrak{a_{1}}=\mathbb{C}^{2}/\mathbb{Z}_{2}$ instead of the expected
$\mathbb{C}^{2}/\mathbb{Z}_{4}$ Arakawa:2010ni . This difference confirms that
the VOA is not simply $\widehat{\mathfrak{sp}(4)}_{-{13\over 6}}$, but rather
an extension thereof. Indeed, the minimal 4d interpretation of the four
operators corresponding to the dimension-$\tfrac{3}{2}$ generators featuring
in the extension is as Higgs branch operators of $\mathfrak{su}(2)_{R}$ charge
$\frac{3}{2}$. These operators are not expected to be nilpotent, and hence are
expected to modify the associated variety of
$\widehat{\mathfrak{sp}(4)}_{-{13\over 6}}$.555We thank Leonardo Rastelli and
Christopher Beem for illuminating discussions on this point.

###  III.3 High-temperature limit

It is by now well-known that the high temperature limit of the
${\mathcal{N}}=1$ superconformal index is determined by $c-a$ DiPietro:2014bca
, from which the high-temperature behavior of the Schur index follows,

$\lim_{i\tau\to 0}{\rm log}\,{\mathcal{I}}_{\rm Schur}(q)\sim\frac{4\pi
i(c-a)}{\tau}+{\rm subleading}~{}.$ (22)

Note that this formula is really only applicable for non-chiral theories with
$c>a$ Ardehali:2015bla ; DiPietro:2016ond . The former is assumed to always be
true in the presence of ${\mathcal{N}}=2$ supersymmetry, while the latter is
obviously true in the case of AD$(\mathfrak{c}_{2})$. As explained in
Beem:2017ooy , the modular differential equation can be used to compute the
high-temperature limit of ${\mathcal{I}}_{\rm Schur}$. Denoting by
${\alpha}_{\rm min}$ the most negative solution of the indicial equation of
the S-transformed modular differential equation, the high-temperature limit is
given by

$\lim_{i\tau\to 0}{\rm log}\,{\mathcal{I}}_{\rm Schur}(q)\sim-\frac{4\pi
i}{\tau}\frac{{\alpha}_{\rm min}}{2}+{\rm subleading}~{}.$ (23)

We refer to the original literature Beem:2017ooy  for details.

The expected high-temperature limit of ${\mathcal{I}}_{{\rm
AD}(\mathfrak{c}_{2})}$ can immediately be calculated using (22) and the
values of $a$ and $c$ in Table 2 to give

$\lim_{i\tau\to 0}{\rm log}\,{\mathcal{I}}_{{\rm
AD}(\mathfrak{c}_{2})}\sim\frac{4\pi i}{\tau}\frac{1}{6}+{\rm subleading}~{},$
(24)

which predicts ${\alpha}_{\rm min}=-\tfrac{1}{3}$. This can indeed be shown to
be the most negative solution of the indicial equation of the S-transform of
the modular differential equation (19), providing yet another check of our
index.

##  IV Class ${\mathcal{S}}$ construction

Finally, we close by noting that our theory is obtainable via the class
${\mathcal{S}}$ constructions of Wang:2018gvb . In particular, consider the
type $A_{4}$ 6d (2,0) theory compactified on a two-punctured sphere with one
irregular puncture and one maximal regular puncture. To obtain a non-simply
laced flavor group, we introduce a $\mathbb{Z}_{2}$ outer automorphism twist,
with the twist line connecting the two punctures. This engineers a class of 4d
${\mathcal{N}}=2$ SCFTs denoted by $C^{\mathrm{anom}}_{2}$, and we choose the
one labelled by $k^{\prime}=-4$ in the notation of Wang:2018gvb . The
superscript “$\mathrm{anom}$” here refers to the fact that the
$\mathfrak{sp}(4)$ flavor symmetry has a Witten anomaly, as follows from
Tachikawa:2018rgw . With this choice the central charges, flavor levels, and
Higgs branch obtained from the formulas in Wang:2018gvb  match precisely with
those for the theory $\rm AD({{\mathfrak{c}}_{2}})$ discussed here.

It should however be noted that there is a slight mismatch between the VOA
predicted by Wang:2018gvb  and the one we find. The analysis of Wang:2018gvb
suggests that the VOA is a non-extended $\widehat{\mathfrak{sp}(4)}_{-{13\over
6}}$, whereas we predict an extension of this algebra by four
dimension-$3\over 2$ generators. In fact, a similar discrepancy arises in the
$C^{\rm anom}_{1}$ class, where the theory engineered by the irregular
singularity with $k^{\prime}=-1$ corresponds to the rank-2 $H_{0}$ theory.
Again, the VOA of this theory is at an admissible level and the analysis of
Wang:2018gvb  would predict $\widehat{\mathfrak{sp}(2)}_{-{17\over 10}}$ as
its VOA. However, the actual VOA of the rank-2 $H_{0}$ theory was worked out
explicitly in Beem:2019snk  and found to instead be an extension of
$\widehat{\mathfrak{sp}(2)}_{-{17\over 10}}$ by two dimension-$\tfrac{5}{2}$
generators. We take this mismatch as a suggestion that the correct VOA
associated to our AD($\mathfrak{c}_{2}$) is indeed the extended
$\widehat{\mathfrak{sp}(4)}_{-{13\over 6}}$.

###### Acknowledgements.

We would like to thank Christopher Beem, Cyril Closset, Simone Giacomelli,
Leonardo Rastelli, Shlomo Razamat, Yifan Wang, Gabi Zafrir, and especially
Yuji Tachikawa for helpful discussions and comments on the draft. MM is
supported by NSF grants PHY-1151392 and PHY-1620610.

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