We establish a linear relation between the
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0 \,. \end{equation} The two-dimensional version was established in~\cite{Zamolodchikov-ml-1986gt}. The four-dimensional $a$-theorem was conjectured in~\cite{Cardy-ml-1988cwa} and further analyzed in~\cite{Osborn-ml-1989td,Jack-ml-1990eb}. A proof was presented in~\cite{Komargodski-ml-2011vj,Komargodski-ml-2011xv}. It utilizes the fact that the anomaly matching conditions for the $a$-anomaly discussed in~\cite{Schwimmer-ml-2010za} lead to a special Wess-Zumino-like interaction term in the effective action for a (dynamical or background) dilaton field, which is interpreted as a Nambu-Goldstone (NG) boson for the spontaneous breaking of conformal symmetry. Another property of the $a$-anomaly in two and four dimensions that is closely related to, but independent of~\eqref{atheorem} is that it is non-negative, \begin{equation}\label{apositivity} a \geq 0 \,, \end{equation} and vanishes if and only if the theory has no local degrees of freedom~\cite{Zamolodchikov-ml-1986gt,Hofman-ml-2008ar}. Prior to~\cite{Komargodski-ml-2011vj,Komargodski-ml-2011xv}, some of the strongest evidence for the four-dimensional $a$-theorem came from supersymmetric RG flows, see for instance~\cite{Anselmi-ml-1997am,Intriligator-ml-2003jj,Kutasov-ml-2003iy,Intriligator-ml-2003mi,Erkal-ml-2010sh}. For $\CN=1$ superconformal field theories (SCFTs) in $d=4$ dimensions, it was shown in~\cite{Anselmi-ml-1997am} that the $a$-anomaly is linearly related to the 't~Hooft anomalies for the $\U(1)_R$ symmetry, \begin{equation}\label{a4d} a = \frac{3}{32} (3 k_{RRR}-k_R) \,. \end{equation} Here $k_{RRR}$ and $k_R$ are the $\Tr \U(1)_R^3$ and $\Tr \U(1)_R$ 't~Hooft anomalies, which appear in the anomaly polynomial,\footnote{See section~\ref{sec:poly} for a brief review of anomaly polynomials.} \begin{equation} {\cal I}_6 = {1 \over 3!} \big(k_{RRR}\,c_1(R)^3 +k_R\,c_1(R)p_1(T)\big) \,. \end{equation} The formula~\eqref{a4d} makes it possible to determine the $a$-anomaly in a large number of strongly interacting examples, by relying on the calculability of 't~Hooft anomalies. Given the two- and four-dimensional results summarized above, it is natural to anticipate similar statements in six dimensions, but the proof of a six-dimensional $a$-theorem remains an open problem. (See~\cite{Myers-ml-2010tj} and references therein for evidence from holography.) Following~\cite{Komargodski-ml-2011vj,Komargodski-ml-2011xv}, the constraints of conformal symmetry on dilaton self-interactions in six dimensions, and their relation to the $a$-anomaly, were analyzed in~\cite{Maxfield-ml-2012aw, Elvang-ml-2012st}, where it was pointed out that $\Delta a$ can in general receive contributions of either sign, even in a unitary theory (see also~\cite{Grinstein-ml-2014xba, Grinstein-ml-2015ina}). The ability to test the six-dimensional $a$-theorem is limited by the fact that the $a$-anomaly has only been computed for a handful of interacting CFTs, and that examples of controlled RG flows between such theories are scarce. All known examples of interacting CFTs in six dimensions are supersymmetric and arise from decoupling limits of string constructions. Nevertheless, they are believed to be quantum field theories~\cite{Seiberg-ml-1996vs,Seiberg-ml-1996qx}. The most well-studied such theories are the maximally supersymmetric $(2,0)$ SCFTs constructed in~\cite{Witten-ml-1995zh, Strominger-ml-1995ac, Witten-ml-1995em}, which are labeled by an ADE Lie algebra $\mathfrak{g}$. The $A$-type theories arise on the worldvolume of parallel M5-branes in M-theory~\cite{Strominger-ml-1995ac}. Their 't~Hooft anomalies have been computed in~\cite{Duff-ml-1995wd,Witten-ml-1996hc,Freed-ml-1998tg,Harvey-ml-1998bx,Intriligator-ml-2000eq,Yi-ml-2001bz,Ohmori-ml-2014kda}. Some of the $(2,0)$ theories come in infinite families that admit a large-$N$ limit with weakly coupled holographic duals, and these have been used to show that the $a$-anomaly scales like $N^3$ at leading order in the $1 \over N$ expansion~\cite{Henningson-ml-1998gx}. Some subleading corrections were discussed in~\cite{Tseytlin-ml-2000sf,Beccaria-ml-2014qea}. The $(2,0)$ theories have a moduli space of vacua on which conformal symmetry is spontaneously broken, and (partially) moving onto this moduli space induces a non-trivial RG flow.\footnote{One might distinguish RG flows onto the moduli space, where conformal invariance is broken spontaneously, from those associated with explicit breaking, which are triggered by adding a relevant operator. In either case, there is a flow along which one integrates out massive degrees of freedom.} The constraints of maximal supersymmetry on such flows were systematically analyzed in~\cite{Cordova-ml-2015vwa}, following~\cite{Maxfield-ml-2012aw}, which lead to an exact calculation of the $a$-anomaly for all $(2,0)$ theories and a proof of the $a$-theorem for all RG flows that preserve $(2,0)$ supersymmetry. A large class of interacting six-dimensional SCFTs with $(1,0)$ supersymmetry have been constructed in string theory, starting with the work of~\cite{Seiberg-ml-1996vs, Seiberg-ml-1996qx}. Further examples were studied using brane constructions~\cite{Ganor-ml-1996mu, Blum-ml-1997mm,Brunner-ml-1997gk,Brunner-ml-1997gf,Hanany-ml-1997gh}. Recently, vast landscapes of $(1,0)$ theories have been systematically constructed in F-theory~\cite{Heckman-ml-2013pva, DelZotto-ml-2014hpa, Heckman-ml-2015bfa}, and a detailed analysis of holographic theories with $(1,0)$ supersymmetry was carried out in~\cite{Apruzzi-ml-2013yva, Gaiotto-ml-2014lca}. Large classes of RG flows in these examples were studied in~\cite{Gaiotto-ml-2014lca,Heckman-ml-2015ola}. All of these flows are induced by moving onto a moduli space of vacua. This is a general feature of all $(1,0)$ SCFTs: using superconformal representation theory, it can be shown that such theories do not contain relevant or marginal operators that can be used to deform the theory while preserving supersymmetry~\cite{Cordova-ml-2016xhm} (see also~\cite{Louis-ml-2015mka}), and hence all supersymmetric RG flows are necessarily moduli-space flows. Since all known six-dimensional interacting CFTs are supersymmetric, it has been a longstanding expectation (see for instance~\cite{Harvey-ml-1998bx}) that supersymmetry should make it possible to compute the $a$-anomaly in these theories by relating it to their 't~Hooft anomalies, in analogy with the known relations~\eqref{a4d} in four dimensions. Such a relation is expected to follow from an anomalous stress-tensor supermultiplet, which embeds the anomalous trace of the stress tensor in~\eqref{Tanomaly} into rigid background supergravity. These anomaly multiplets are currently under investigation~\cite{ctkii}. Another ambitious line of attack would be to directly supersymmetrize the four-point functions of stress-tensors and $R$-currents, in analogy with the results of~\cite{Osborn-ml-1998qu} on stress-tensor three-point functions in four dimensions. In this paper, we assume the existence of a universal linear relation, valid for all $(1,0)$ SCFTs, that relates the $a$-anomaly to the 't~Hooft anomalies. Given this assumption, we then derive the precise formula by combining the constraints of supersymmetry with those from anomaly matching for the Weyl and 't~Hooft anomalies. We find that \begin{equation}\label{ais} a = \frac{16}{7} (\alpha -\beta +\gamma) +\frac{6}{7} \delta \,. \end{equation} Here, and throughout the paper, we use a normalization of the $a$-anomaly, in which a free $(2,0)$ tensor multiplet has $a = 1$. The constants $\alpha$, $\beta$, $\gamma$, $\delta$ are 't~Hooft anomaly coefficients for the $\SU(2)_R$ symmetry and gravitational anomalies of the theory, which enter the anomaly polynomial as follows,\footnote{We follow the conventions of~\cite{Ohmori-ml-2014kda,Intriligator-ml-2014eaa} for anomaly polynomials and characteristic classes.} \begin{equation}\label{anomalyI} \CI_8 = {1 \over 4!} \big(\alpha c_2^2(R) +\beta c_2(R) p_1(T) +\gamma p_1^2(T) +\delta p_2(T)\big) \,. \end{equation} In table~\ref{known}, we summarize the values of the $a$-anomaly and the 't~Hooft anomaly coefficients in~\eqref{anomalyI} for all $(1,0)$ free fields, which are known from~\cite{Fradkin-ml-1983tg, AlvarezGaume-ml-1983ig, Bastianelli-ml-2000hi}, as well as for all $(2,0)$ theories. The negative value of $a$ for a free vector multiplet, which is not a CFT, is obtained by naively applying~\eqref{ais}. Its meaning will be discussed in detail below. \begin{table}[t] \centering \renewcommand{\arraystretch}{1.5} {\small\begin{tabular}{|c|c|c|c|c|c|} \hline \emph{Theory} & $\alpha$ & $\beta$ & $\gamma$ & $\delta$ & $a$ \\ \hline Hypermultiplet & $0$ & $0$ & $\frac{7}{240}$ & $-\frac{1}{60}$ & $\frac{11}{210}$ \\ \hline Tensor multiplet & $1$ & $\half$ & $\frac{23}{240}$ & $-\frac{29}{60}$ & $\frac{199}{210}$ \\ \hline Vector multiplet & $-1$ & $-\half$ & $-\frac{7}{240}$ & $\frac{1}{60}$ & ``$-\frac{251}{210}$'' \\ \hline $(2,0)$ theory with algebra $\mathfrak{g}$ & $h^\vee _{\mathfrak{g}}d_{\mathfrak{g}}+r_{\mathfrak{g}}$ & $\half r_{\mathfrak{g}}$ & $\frac{1}{8}r_{\mathfrak{g}}$ & $-\half r_{\mathfrak{g}}$ & $\frac{16}{7} h_{\mathfrak{g}}^\vee d_{\mathfrak{g}}+r_{\mathfrak{g}}$ \\ \hline \end{tabular}} ]]>
0$ for some Higgs branch flows in these theories, which does not automatically follow from our general arguments. In section~\ref{sec:vectors}, we discuss subtleties in the statement and proof of an $a$-theorem for RG flows that terminate on tensor branches with vector multiplets. Unlike in four dimensions, where free gauge fields constitute an ordinary CFT, in six dimensions the free vector field is scale invariant, but not conformally invariant, i.e.\ it is an SFT\@. (See for instance~\cite{ElShowk-ml-2011gz}.) Such theories possess a well-defined stress tensor $T_{\mu\nu}$, but its trace $T^\mu_\mu \sim \Tr (f^2)$ does not vanish. (Here $f$ is the field strength). String constructions of six-dimensional field theories suggest that RG flows from CFTs to free SFTs abound. To formulate an $a$-theorem for such flows, we must extend the definition of the $a$-anomaly to these theories, and we do so by insisting that~\eqref{ais} holds. With this definition, an $a$-theorem for tensor branch flows continues to hold, but $a$ may no longer be positive (see for instance the vector multiplet in table~\ref{known}). Nevertheless, the $a$-anomaly for the UV conformal field theory turns out to be positive in the examples we consider. In appendix~\ref{app}, we review the GS mechanism for chiral scalars in two dimensions, to supplement the six-dimensional discussion in section~\ref{sec:moduli}. ]]>
0$ unless the dilaton is a free field, in which case $b$ vanishes~\cite{Adams-ml-2006sv} (see also~\cite{Komargodski-ml-2011vj,Komargodski-ml-2011xv,Elvang-ml-2012st}). Similarly, $\Delta a$ appears at $\CO(p^6)$ in dilaton scattering amplitudes. As long as the theory in the deep IR is a conventional CFT, we can treat $\varphi$ as a decoupled field, which only interacts with itself, up to and including six-derivative order. (See~\cite{Komargodski-ml-2011xv,Luty-ml-2012ww} for a discussion of the corresponding statement in four dimensions.) At higher orders in the derivative expansion, we must also take into account possible couplings of $\varphi$ to other massless degrees of freedom in the IR, which can give rise to non-analytic terms in dilaton scattering amplitudes. As was stated in the introduction, many $(1,0)$ SCFTs admit RG flows that do not terminate in conventional CFTs. These are discussed in section~\ref{sec:vectors}. ]]>
0$ unless the flow is trivial, in which case $\Delta a = 0$, thus proving $a$-theorem for RG flows of $(1,0)$ SCFTs onto their tensor branch. Throughout our discussion here, we assume that the massless degrees of freedom that remain in the deep IR for the theory with non-zero expectation value on the tensor branch constitute a genuine SCFT $\CT_{\text{IR}}$. Since the dilaton $\varphi$ is the NG boson of spontaneous conformal symmetry breaking, it follows from Goldstone's theorem that $\CT_{\text{IR}}$ consists of a (possibly interacting) SCFT $\CT_0$ and a free decoupled tensor multiplet $\CT_\varphi$ containing the dilaton $\varphi$, \begin{equation} \CT_{\text{IR}} = \CT_0 +\CT_{\varphi} \,. \end{equation} The assumption that $\CT_0$ is an SCFT excludes tensor branches with massless gauge fields. If the IR theory is not a CFT, both the statement and the proof of an $a$-theorem require additional clarification. For this reason we defer a discussion of tensor branches with gauge fields until section~\ref{sec:vectors}. Prototypical examples of SCFTs without gauge fields on their tensor branch are the $(2,0)$ theories, as well as the $(1,0)$ theories $\CE_N$ describing $N$ small $E_8$ instantons, which will be discussed in section~\ref{sec:E8}. We now turn to the implications of $(1,0)$ supersymmetry for the dilaton effective action~\eqref{ldilaton} on the tensor branch. Here we closely follow the recent discussion of tensor-branch effective actions with $(2,0)$ supersymmetry in~\cite{Cordova-ml-2015vwa}. As was explained in section~\ref{sec:anomaly}, the dilaton $\varphi$ resides in a $(1,0)$ tensor multiplet, together with a symplectic Weyl Fermion $\psi_\alpha^i$ and a self-dual three-form field strength $H$, which can be written as a symmetric bispinor $H_{\alpha\beta} = H_{(\alpha\beta)}$. (Here $\alpha, \beta = 1, \ldots, 4$ are chiral spinor indices and $i = 1,2$ is an $\SU(2)_R$ doublet index.) At the two-derivative level, they all satisfy free equations of motion, \begin{equation}\label{freeeom} \square \varphi = 0 \,, \qquad \d^{\alpha\beta} \psi_\beta^i = 0 \,, \qquad \d^{\alpha\beta} H_{\beta\gamma} = 0 \,. \end{equation} Here $\d^{\alpha\beta} = \d^{[\alpha\beta]}$ is a spacetime derivative in bispinor notation. The fact that $H$ is self dual implies that the standard quadratic Lagrangian $H \wedge * H$ vanishes, so that the free theory needs to be defined with some care (see for instance~\cite{Witten-ml-2007ct,Witten-ml-2009at,moorefklect} and references therein), but this subtlety will not affect our discussion. Since the tensor-branch effective action is supersymmetric, the higher-derivative terms in the pure dilaton Lagrangian~\eqref{ldilaton} for $\varphi$ must be completed by terms involving its superpartners $\psi_\alpha^i$ and $H_{\alpha\beta}$.\footnote{We follow the standard rules for counting derivatives in supersymmetric moduli-space effective actions: spacetime derivatives and the three-form field strength $H$ have weight $1$, supercharges and Fermions have weight $\half$, and the scalar $\varphi$ has weight $0$.} In order to determine whether supersymmetry leads to additional constraints on these terms, we follow the general approach to moduli-space effective actions advocated in~\cite{Cordova-ml-2015vwa,Cordova-ml-2016xhm}. We first expand the dilaton in fluctuations $\delta \varphi$ around a fixed vev, \begin{equation}\label{vevfluc} \varphi = \langle \varphi \rangle +\delta \varphi \,, \end{equation} and view the resulting Lagrangian as a deformation of a free tensor multiplet by higher-derivative local operators constructed out the fields in this tensor multiplet and their derivatives. If some term in this expansion leads to local operators that cannot be embedded in an independent supersymmetric deformation, then that term is constrained by supersymmetry, i.e.\ it satisfies a non-renormalization theorem. To implement this procedure, we now determine the independent supersymmetric deformations of a single free $(1,0)$ tensor multiplet. Since this multiplet constitutes a (free) SCFT, its supersymmetric deformations can be classified using superconformal representation theory, see~\cite{Cordova-ml-2016xhm} and references therein for further details. (Below we will mention another approach, based on scattering superamplitudes.) Unlike the $(2,0)$ case discussed in~\cite{Cordova-ml-2015vwa}, which admits both $F$- and $D$-term deformations, a free $(1,0)$ tensor multiplet can only be deformed by full $D$-terms, i.e.\ descendants formed with all supercharges \begin{equation}\label{dterm} \SL_{D} = Q^{8} (\mathcal{O}) \,, \end{equation} where $\CO$ is constructed out of fields in the tensor multiplet and their derivatives. This is similar to the situation in four-dimensional $\CN=2$ theories, where all higher-derivative operators on the Coulomb branch are full $D$-terms~\cite{Dine-ml-1997nq}. In order for the deformation~\eqref{dterm} to be non-trivial, $\CO$ must be the superconformal primary (i.e.\ the bottom component) of a long multiplet, which does not satisfy any shortening conditions. Both $\CO$ and $\SL_D$ must be Lorentz scalars, and they transform in the same representation of the $\SU(2)_R$ symmetry; the eight supercharges in~\eqref{dterm} are contracted to an $\SU(2)_R$ singlet. The $\SU(2)_R$ symmetry is unbroken on the tensor branch, so the operator $\CO$ must be an $\SU(2)_R$ singlet. The leading interaction in the dilaton effective Lagrangian~\eqref{ldilaton} is the four derivative term, proportional to $b$. Expanding it around a fixed dilaton vev as in~\eqref{vevfluc} gives rise to an infinite series of four-derivative terms involving $n+4$ dilatons, for all $n \in \Z_{\geq 0}$, \begin{equation}\label{fourderivative} b\, \frac{(\partial \varphi)^{4}}{\varphi^{3}} \quad \longrightarrow \quad b \bigg(\frac{1}{\langle \varphi\rangle^{3}} (\partial \delta \varphi)^{4} -3 \,\frac{\delta \varphi}{\langle \varphi \rangle^{4}} (\partial \delta \varphi)^{4} +\cdots +\CO \big((\delta \varphi)^n (\d \delta \varphi)^4\big) +\cdots\bigg) \,. \quad \end{equation} All terms in this expansion can be interpreted as arising from $D$-terms~\eqref{dterm} of the form $Q^8\big((\delta\varphi)^{n+4}\big)$. Therefore, supersymmetry does not constrain the coefficient $b$. We now note that the six-derivative couplings in~\eqref{ldilaton}, which are proportional to $\Delta a$, cannot arise from a $D$-term~\eqref{dterm}. In order to see this, it suffices to list all candidate Lorentz and $\SU(2)_R$ singlet primaries $\CO$, which should contain two derivatives for~\eqref{dterm} to be a six-derivative term. However, no such $\CO$ exists, as can be verified by enumerating all local Lorentz scalar operators containing two derivatives: \begin{itemize} \item $(\delta\varphi)^{n}(\partial \delta\varphi)^{2} $ is not a conformal primary (i.e.\ it is a total derivative), since $\square \varphi = 0$. \item $\psi_{\alpha}^{i}\psi_{\beta}^{j}\partial^{\alpha\beta}(\delta\varphi^{n})$ transforms in the ${\bf 3}$ of $\SU(2)_R$, since $\d^{\alpha\beta} = \d^{[\alpha\beta]}$ is antisymmetric. \item $(\delta \varphi)^{n}\varepsilon^{\alpha \beta \gamma \delta} \psi_{\alpha}^{i}\psi_{\beta}^{j}\psi_{\gamma}^{k}\psi_{\delta}^{\ell}$ transforms in the ${\bf 5}$ of $\SU(2)_R$, since the totally antisymmetric $\ep$-symbol is needed to contract the spinor indices to a Lorentz singlet. \item $H_{\alpha \beta}\partial^{\alpha \beta}\big((\delta\varphi)^{n}\big) = (\delta \varphi)^{n} \ep^{\alpha\beta\gamma\delta} \psi_{\alpha}^{i}\psi_{\beta}^{j}H_{\gamma\delta} = (\delta \varphi)^{n}\ep^{\alpha\beta\gamma\delta} H_{\alpha \beta}H_{\gamma\delta} = 0$ since $H_{\alpha\beta} = H_{(\alpha\beta)}$ is symmetric and $\d^{\alpha\beta} = \d^{[\alpha\beta]}$ is antisymmetric. \end{itemize} The absence of independent six-derivative couplings implies that the six-derivative terms in~\eqref{ldilaton}, and their superpartners, can only be induced by supersymmetrically completing the four-derivative terms~\eqref{fourderivative} proportional to $b$. Thus all six-derivative terms in the dilaton effective Lagrangian must be proportional to $b^2$. See~\cite{Cordova-ml-2015vwa} for various ways of understanding this quadratic relation. Therefore the coefficient $\Delta a$ of the six-derivative term in~\eqref{ldilaton} must be proportional to $b^2$, with a model-independent proportionality constant that is completely fixed by supersymmetry. This coefficient can be determined directly, or via examining any suitable example. For instance, it was shown in the~\cite{Cordova-ml-2015vwa} that \begin{equation}\label{absq} \Delta a = \frac{98304 \pi^3}{7}\, b^2 , \end{equation} for all $(2,0)$ SCFTs. Since these are also particular examples of $(1,0)$ SCFTs, the relation~\eqref{absq} continues to hold on the tensor branch of all $(1,0)$ theories, due to the non-renormalization theorem derived above.\footnote{It is interesting to contemplate the extent to which a relation like~\eqref{absq}, which was derived using spontaneously broken conformal symmetry, as well as supersymmetry, continues to apply if we relax some of these assumptions. See for instance the discussion around equation~(2.7) in~\cite{Maldacena-ml-1997re}. We thank J.~Maldacena for suggesting this possibility, and for related discussions.} As in~\cite{Cordova-ml-2015vwa}, the fact that $\Delta a$ is proportional to $b^2$ can also be understood by examining tree-level scattering amplitudes of the fields in the dilaton multiplet. The implications of $(1,0)$ supersymmetry on local tensor multiplet supervertices were analyzed in~\cite{Chen-ml-2015hpa}, where it was shown that these constraints are incompatible with the existence of an independent six-point, six-derivative supervertex. Any six-point, six-derivative treel-level amplitude of fields in the dilaton multiplet must therefore factorize through a product of four-point, four-derivative amplitudes (see figure~\ref{Factor}), which again leads to the quadratic relation $\Delta a \sim b^2$. Similar techinques have recently been used to argue for supersymmetry relations between various higher-derivative couplings in diverse dimensions~\cite{Wang-ml-2015jna,Lin-ml-2015ixa,Wang-ml-2015aua}, including those previously obtained using the methods of~\cite{Paban-ml-1998qy,Paban-ml-1998ea}. \begin{figure}[t] \[ \xymatrix @R=1pc { &*{\varphi}\ar@{-}[ddr]&&&& *{~\varphi}\ar@{-}[ddl]&\\ \\ {\varphi}\ar@{-}[rr]&&*+[o][F]{ b } \ar@{-}[rr]^{\mathlarger{\varphi}}&& *+[o][F]{ b }\ar@{-}[l]&&{\varphi}\ar@{-}[ll]\\ \\ &{\varphi}\ar@{-}[uur]&&&& {\varphi}\ar@{-}[uul]&} \] ]]>
0$ unless the dilaton is a free field, in which case $b = 0$. Therefore $\Delta a>0$ for all non-trivial RG flows of $(1,0)$ SCFTs onto their tensor branch. This argument for the $a$-theorem is identical to that given in~\cite{Cordova-ml-2015vwa} for tensor-branch flows of $(2,0)$ theories (see also~\cite{Maxfield-ml-2012aw}), since the crucial relation~\eqref{absq} is the same in both cases. ]]>
4$ spacetime dimensions are not genuine CFTs, since their stress tensors have a non-zero trace $T^\mu_\mu \sim \Tr(f^2)$ (see for instance~\cite{ElShowk-ml-2011gz}). (By contrast, free gauge fields are conformally invariant in four dimensions.) They are, in a sense, scale invariant (they possess a conserved dilatation charge), although there is no gauge-invariant scale current. In accord with standard terminology, we will refer to such scale-invariant but non-conformal theories as SFTs. (See for instance~\cite{Luty-ml-2012ww,Fortin-ml-2012hn,Dymarsky-ml-2013pqa,Dymarsky-ml-2014zja,Dymarsky-ml-2015jia} and references therein for a recent discussion of such theories.) Even though it is believed that the $(1,0)$ SCFTs we are considering are conformally invariant, the preceding discussion implies that RG flows onto tensor branches with massless gauge fields terminate in an IR theory that is an SFT, but not a genuine CFT\@. Similar phenomena occur in the deep IR of many supersymmetric RG flows in five dimensions. By contrast, in four dimensions we are not aware of non-trivial RG flows from an interacting CFT in the UV that terminate in an SFT in the IR\@. In three dimensions, the gauge coupling is relevant, and there are many examples of non-trivial RG flows from an SFT in the UV to a CFT in the IR. In light of the preceding discussion, it is natural to search for an extension of the $a$-theorem to six-dimensional RG flows between a CFT in the UV and an SFT in the IR\@. In the remainder of this section, we will formulate and investigate such a generalization, focusing on RG flows of $(1,0)$ SCFTs onto tensor branches with gauge fields. An immediate challenge is that a suitable analogue of the $a$-anomaly need not obviously exist for all SFTs. In CFTs, there are many equivalent definitions: we can define the $a$-anomaly through the anomalous trace of the stress tensor $T_{\mu\nu}$ in a gravitational background, as in~\eqref{Tanomaly}, or in terms of the four-point function of $T_{\mu\nu}$ in flat space. For $(1,0)$ SCFTs, we have argued that the $a$-anomaly may also be expressed in terms of 't~Hooft anomalies as in~\eqref{ais}, which we repeat here, \begin{equation}\label{aisrephere} a = \frac{16}{7} (\alpha-\beta+\gamma) +{6 \over 7}\, \delta \,. \end{equation} By contrast, in an SFT these various candidate definitions may not agree, or even make sense. For instance, in an SFT the stress tensor need not be traceless, while the definition of $a$ via~\eqref{Tanomaly} assumes that $T^\mu_\mu$ is a redundant operator, whose flat-space correlation functions are pure contact terms. For the purposes of our discussion of RG flows in $(1,0)$ theories, we choose to define the value of $a$ in supersymmetric SFTs in terms of their 't~Hooft anomalies, via~\eqref{aisrephere}. We can use the 't~Hooft anomalies in table~\ref{known} to compute the value of $a$ for a free theory of $n_h$ hypermultiplets, $n_t$ tensor multiplets, and $n_v$ vector multiplets, \begin{equation}\label{afrees} a = \frac{1}{210} (11n_{h}+199n_{t}-251n_{v}) \,. \end{equation} The $a$-anomalies for hypermultiplets and tensor multiplets are well defined and were computed in~\cite{Fradkin-ml-1983tg,Bastianelli-ml-2000hi}. According to our definition, the value of $a$ for a free vector multiplet (which follows from~\eqref{aisrephere} and the 't~Hooft anomalies summarized in~table~\ref{known}) is negative. Although surprising, this feature has a precedent in four dimensions. A simple four-dimensional SFT with $\CN=1$ supersymmetry is the theory of a linear multiplet, which is dual to a free chiral multiplet with a shift symmetry. The shift symmetry forces us to assign vanishing $\U(1)_R$ charge to the scalar in the chiral multiplet, so that its fermionic superpartner has $R$-charge $-1$. If we attempt to define the value of the $a$-anomaluy in this theory by extending the relation~\eqref{a4d} between the $a$-anomaly and the $\U(1)_R$ symmetry anomalies that holds in SCFTs, we find that $a = -\frac{3}{16}<0$. Returning to six dimensions, it is tempting to search for an interpretation of the negativity of the $a$-anomaly, e.g.\ by embedding the unitary SFT of a free vector multiplet into a non-unitary CFT as in~\cite{ElShowk-ml-2011gz}.\footnote{A concrete interpretation along similar lines was subsequently proposed in~\cite{Beccaria-ml-2015uta}.} Another puzzling aspect of~\eqref{afrees} is that it assigns a negative value $\Delta a < 0$ to Higgsing, where a vector multiplet pairs up with a hypermultiplet to become massive, so that $\Delta n_h = \Delta n_v = 1$. Given the above, we would like to emphasize that the $a$-anomaly of a genuine unitary CFT in six dimensions is expected to be positive, as has been shown in two and four dimensions~\cite{Zamolodchikov-ml-1986gt,Hofman-ml-2008ar}. In six dimensions, the positivity of $a$ for all unitary CFTs has not yet been established.\footnote{The generalization of the arguments in~\cite{Hofman-ml-2008ar} to six dimensions only constrains the $c$-type Weyl anomalies, but not the $a$-anomaly~\cite{deBoer-ml-2009pn} (see also~\cite{Osborn-ml-2015rna}).} (See~\cite{Myers-ml-2010tj} for a general discussion of positivity constraints on $a$ and $\Delta a$ from holography.) Below, we will consider explicit examples of unitary RG flows from UV CFTs to IR SFTs such that $a_{\text{IR}} < 0$. However, in those examples the anomaly deficit $\Delta a$ between the UV and the IR theories is always sufficiently positive to ensure that $a_{\text{UV}} >0$. We note in passing that the $a$-theorem only requires that $a>0$ for theories that can be deformed to a gapped phase. It follows from the results of~\cite{Cordova-ml-2016xhm} (see also~\cite{Louis-ml-2015mka}) that this is not possible for $(1,0)$ theories while maintaining supersymmetry. (See~\cite{Nakayama-ml-2015bwa} for a related discussion in four dimensions.) \boldmath ]]>
0$ continues to hold for all RG flows of unitary $(1,0)$ SCFT onto their tensor branch, even in the presence of vector multiplets. This follows straightforwardly from the definition~\eqref{aisrephere} of $a$ in terms of 't~Hooft anomalies, and the fact that the GS anomaly-matching mechanism for these anomalies implies that $\Delta \alpha -\Delta \beta +\Delta \gamma$ is a sum of squares, and hence positive (see also section~\ref{sec:highrk}). This argument is not affected by the fact that the IR theory is an SFT. More physically, the change $\Delta a$ as defined by the 't~Hooft anomalies continues to determine the six-point, six-derivative scattering amplitudes of the dilaton, even if there are vector fields on the tensor branch and the dilaton couples to them via $\varphi \Tr(f^2)$, as in~\eqref{phiFF}. This is easy to see by generalizing the discussion at the end of section~\ref{sec:athm}. Recall that the results of~\cite{Chen-ml-2015hpa} imply that there is no independent six-point, six-derivative supervertex for the dilaton, so that this amplitude factorizes through lower-point amplitudes. However, the vertex $\varphi \Tr(f^2)$ cannot contribute to this factorization. Hence, the presence of vector multiplets does not modify the arguments in sections~\ref{sec:athm} and~\ref{sec:relation} that lead to the relation between $\Delta a$ and the changes of 't~Hooft anomalies in~\eqref{deltafinal}. ]]>
0$ for unitary CFTs. In the context of RG flows of $(1,0)$ theories onto their tensor branch, this amounts to \begin{equation}\label{matterrestrict} a_{\text{UV}} = a_{\text{IR}}+\Delta a = \frac{16}{7} (\Delta \alpha -\Delta \beta +\Delta \gamma) +a_{\text{IR}} > 0 \,. \end{equation} Even though we have argued that $\Delta a > 0$, the value of $a_{\text{IR}}$ may be negative if the IR theory contains sufficiently many vector multiplets. In this case~\eqref{matterrestrict} constitutes a non-trivial constraint on the matter content and the GS couplings of the tensor-branch effective theory. An example will be discussed below. ]]>
0$ in all cases, consistent with the interpretation of this system as a unitary SCFT. \begin{table}[t] \centering \renewcommand{\arraystretch}{1.3} {\small\begin{tabular}{|c|c|c|c|c|c|} \hline $\Gamma$ & $\mathbb{Z}_{k}$ & $D_{k}$ & $\mathbb{T}$ & $\mathbb{O}$ & $\mathbb{D}$ \\ \hline $G$ & $\SU(k)$ & $\SO(2k+2)$ & $E_{6}$ & $E_{7}$ & $E_{8}$ \\ \hline $|\Gamma|$ & $k$ & $4k$ & $24$ & $48$ & $120$ \\ \hline $r_{\Gamma}$ & $k-1$ & $k+1$ & $6$ & $7$ & $8$ \\ \hline $d_{\Gamma}$ & $k^{2}-1$ & $2k^{2}+3k+1$ & $78$ & $133$ & $248$ \\ \hline \end{tabular}} ]]>
1$, this expression can be negative, due to the negative contribution of the vector multiplets. However, the change $\Delta a$, which can be computed using~\eqref{aTNGamma}, is positive, as required by our general arguments. The change in the 't~Hooft anomalies between the UV theory at the origin and the IR theory on the tensor branch is \begin{equation} \Delta \alpha = Nk^{2}(N^{2}-1) \,, \qquad \Delta \beta = 0 \,, \qquad \Delta \gamma = 0 \,. \end{equation} The constraint of section~\ref{sec:aposcons} then reads \begin{equation} a_\text{UV} = \frac{16}{7}\Delta \alpha +a_\text{Tensor} > 0 \,, \end{equation} which is indeed satisfied, even though $a_\text{Tensor}$ can be negative. We can also flow onto a mixed branch. This is achieved by a motion of the of the NS5-branes transverse to the singularity as illustrated in figure~\ref{figORB}(c). At low-energies, the resulting theory is described by $N$ free $(1,0)$ tensor multiplets together with $N$ hypermultiplets probing the singularity $\mathbb{C}^{2}/\mathbb{Z}_{k}$. We therefore have \begin{equation} a_\text{Mixed} = N \,, \end{equation} which is positive, since the low-energy theory does not contain any vector multiplets. Note that the naive extension of the $a$-theorem to flows from the tensor to the mixed branche is false because $a_\text{Tensor}< a_\text{Mixed}$. However, as expected, the full RG flow from the UV CFT $\mathcal{T}_{N,\Gamma}$ onto the mixed branch in the IR does satisfy the $a$-theorem $a(\mathcal{T}_{N,\Gamma})>a_\text{Mixed}$. It would be interesting to consider RG flows of other theories onto Higgs and mixed branches (see e.g.~\cite{Gaiotto-ml-2014lca,Heckman-ml-2015ola}), and further investigate the monotonicity properties of $a$. ]]>
0$ is analogous to the matrix $\Omega_{IJ}$ that appears in~\eqref{SBwedgeX} and~\eqref{dixsq}. (Canonically normalized kinetic terms are obtained by setting $\Omega = 1$.) The sign factor $\sigma$ will turn out to determine the chirality of the scalar. This can be seen by varying~\eqref{philag} to obtain the following equation of motion, \begin{equation}\label{eom} (\d_x +i \sigma \d_y) \d_x \phi = 0 \,. \end{equation} We can therefore introduce a one-form $J$ that satisfies \begin{equation}\label{jdef} J = \d_x \phi (dx +i \sigma dy) \,, \qquad * J = -i \sigma J \,, \qquad d J = 0 \,. \end{equation} The (anti-) self-dual one-form $J$ is simply the conserved current corresponding to the $\U(1)$ flavor symmetry carried by the chiral scalar. It is a standard fact that the chiral $\U(1)$ current in~\eqref{jdef} has a non-zero 't~Hooft anomaly, which can be exhibited by coupling $J$ to a background gauge field $A$, \begin{equation}\label{sourceS} \SL_{A} = \SL +\Omega (J_x A_x +J_y A_y) = \frac{\Omega}{2}\, \d_x\phi (\d_x \phi +i \sigma \d_y \phi) +\Omega\, \d_x \phi (A_x +i \sigma A_y) \,. \end{equation} Here we have included a factor of $\Omega$ in the $J$-$A$ coupling to mirror the conventions used in~\eqref{SBwedgeX}. The equations of motion~\eqref{eom} are deformed to \begin{equation}\label{eomii} \d_x \big((\d_x \phi +A_x) +i \sigma (\d_y \phi +A_y)\big) = 0 \,. \end{equation} They are manifestly invariant under background gauge transformations \begin{equation} \delta A = d \lambda \,, \qquad \delta \phi = -\lambda \,, \qquad \lambda = \lambda(x,y) \,. \end{equation} By contrast, the Lagrangian~\eqref{sourceS} is not invariant, \begin{equation}\label{Svar} \delta \SL_A = -\Omega\, \d_x \lambda (A_x +i \sigma A_y) +(\text{total derivative}) \,. \end{equation} As befits an anomaly, this non-invariance cannot be removed using the available local counterterms $A_x^2$, $A_y^2$, and $A_x A_y$. However, they can be tuned to covariantize the variation~\eqref{Svar}, \begin{equation}\label{finaldeltas} \delta \bigg(\SL_A +\frac{\Omega}{2} A_x^2 +\frac{i \sigma \Omega}{2} A_x A_y\bigg) = \frac{i \sigma}{2}\, \Omega\, \lambda\, F +(\text{total derivative}) \,, \qquad F = dA \,. \end{equation} This expression manifests the factor of $\half$ discussed at the beginning of this appendix. Note that the sign of the anomaly depends on the chirality $\sigma$ of the scalar, consistent with the fact that the shift symmetry of a non-chiral scalar does not have an anomaly. The variation~\eqref{finaldeltas} can be accounted for by a contribution to the anomaly four-form polynomial. In the conventions of section~\ref{sec:poly}, \begin{equation} \Delta \CI_4 = \frac{\sigma}{2} \cdot \frac{\Omega}{2 \pi}\, F \wedge F \,. \end{equation} This equation is the two-dimensional analogue of the six-dimensional formula~\eqref{dixsq}. There the overall sign was also fixed by the chirality of the two-form gauge fields in tensor multiplets, which is determined by supersymmetry. ]]>