Monopole deformations of 3d Seiberg-like dualities with adjoint matters

In this section, we first review the Seiberg-like duality of a 3d $U(N_{c})$ gauge theory with one adjoint and fundamental matters Kim:2013cma , which we call the Kim–Park duality.

• •

  Theory A is the 3d $\mathcal{N}=2$ $U(N_{c})$ gauge theory with $N_{f}$ pairs of fundamental $Q^{a}$ and anti-fundamental $\tilde{Q}^{\tilde{a}}$, one adjoint chiral multiplet $X$ and the superpotential

  $\displaystyle W_{A}=\mathrm{Tr}X^{n+1}\,.$

  (1)

• •

  Theory B is the 3d $\mathcal{N}=2$ $U(nN_{f}-N_{c})$ gauge theory with $N_{f}$ pairs of fundamental $q_{\tilde{a}}$ and anti-fundemental $\tilde{q}_{a}$, one adjoint $\hat{X}$, and $nN_{f}{}^{2}+2n$ gauge singlet chiral multiplets ${M_{i}}^{\tilde{a}a}$ and $V_{i}^{\pm}$ for $a,\tilde{a}=1,\dots N_{f}$ and $i=0,\ldots,n-1$. The superpotential is given by

  $\displaystyle W_{B}=\mathrm{Tr}\hat{X}^{n+1}+\sum_{i=0}^{n-1}M_{i}\tilde{q}\hat{X}^{n-1-i}q+\sum_{i=0}^{n-1}\left(V_{i}^{+}\hat{V}_{n-1-i}^{-}+V_{i}^{-}\hat{V}_{n-1-i}^{+}\right)$

  (2)

  where $\hat{V}_{i}^{\pm}$ are the monopole operators of Theory B.

The global symmetry and charges are summarized in Table 1.

One can also generalize this duality to have the different numbers of fundamental and anti-fundamental matters, possibly with a nonzero Chern–Simons term, by giving real mass to some flavors Hwang:2015wna . In this paper, we focus on the same number of the fundamental and anti-fundamental matters for simplicity.

The superpotential (1) imposes the F-term condition $X^{n}=0$ on the adjoint field $X$ such that the traces of powers of $X$ higher than $n$ are truncated in the chiral ring. In 4d, the duality of a theory with such a superpotential, later called the $A_{n}$ theory Intriligator:2003mi , was studied in Kutasov:1995ve ; Kutasov:1995np ; Kutasov:1995ss . It was also discussed that two $A_{n}$ and $A_{n^{\prime}}$ theories with $n>n^{\prime}$ are connected by an RG flow triggered by the extra superpotential Kutasov:2003iy ; Intriligator:2016sgx

$\displaystyle\sum_{i=n^{\prime}}^{n-1}\mathrm{Tr}X^{i+1}\,.$

(3)

This deformation leads to a discrete set of the vacuum expectation values of $X$, breaking the theory as follows:

$\displaystyle A_{n}\quad\longrightarrow\quad A_{n^{\prime}}+\left(n-n^{\prime}\right)A_{1}\,,$

(4)

where the $A_{1}$ theory is an ordinary SQCD without the adjoint because it contains the mass term for $X$. Especially, once we turn on $\Delta W_{X}=\mathrm{Tr}\left[f_{n}(X)\right]$ where $f_{n}(x)$ is a generic polynomial of degree $n$ in $x$, the VEV of $X$ is parameterized by

$\displaystyle\left<X\right>\sim\oplus_{k=1}^{n}x_{k}\mathbf{1}_{r_{k}}\,,\qquad\sum_{k=1}^{n}r_{k}=N_{c}$

(5)

where $x_{k}$’s are the $n$ distinct solutions to the equation

$\displaystyle(n+1)x^{n}+f_{n}^{\prime}(x)=0\,.$

(6)

This breaks the gauge group $U(N_{c})$ into

$\displaystyle\prod_{k=1}^{n}U(r_{k})$

(7)

where each $U(r_{k})$ sector only has $N_{f}$ pairs of fundamental and anti-fundamental fields. The adjoint field becomes massive because $f_{n}^{\prime\prime}(x_{k})\neq 0$. We will call this deformed theory the broken theory because the gauge group is partially broken.

For the original Theory A, the moduli space of vacua is parametrized by the VEV of chiral operators $M_{i}\equiv\tilde{Q}X^{i}Q$, $\mathrm{Tr}X^{i+1}$, and $V_{i}^{\pm}$ for $i=0,\dots,n-1$. While the branch described by $\mathrm{Tr}X^{i+1}$ is lifted after turning on the polynomial superpotential $\Delta W_{X}=\mathrm{Tr}\left[f_{n}(X)\right]$, the dimensions of the other branches remain unchanged. For instance, both the original theory and the broken theory have $2n$-dimensional Coulomb branches, which are respectively described by $V_{i}^{\pm}$ for $i=0,\dots,n-1$ and by $V^{(k)}{}^{\pm}$ for $k=1,\dots,n$ where $V^{(k)}{}^{\pm}$ are a pair of monopole operators of each $U(r_{k})$ sector.

One can consider the same deformation for Theory B. The corresponding deformation on the dual side should be the same polynomial superpotential $\Delta W_{\hat{X}}=\mathrm{Tr}\left[f_{n}(\hat{X})\right]$ of degree $n$ now in $\hat{X}$ because the term $\mathrm{Tr}X^{k}$ is mapped to $\mathrm{Tr}\hat{X}^{k}$. The VEV of $\hat{X}$ is parameterized by

$\displaystyle\left<\hat{X}\right>\sim\oplus_{k=1}^{n}\hat{x}_{k}\mathbf{1}_{\tilde{r}_{k}}\,,\qquad\sum_{k=1}^{n}\tilde{r}_{k}=\tilde{N}_{c}=nN_{f}-N_{c}$

(8)

where $\hat{x}_{k}$’s are the $n$ solutions to the equation

$\displaystyle(n+1)\hat{x}^{n}+f_{n}^{\prime}(\hat{x})=0\,.$

(9)

This VEV breaks the dual gauge group $U(\tilde{N}_{c})$ into

$\displaystyle\prod_{k=1}^{n}U(\tilde{r}_{k})$

(10)

where now one can see that $\tilde{r}_{k}$ should satisfy

$\displaystyle\tilde{r}_{k}=N_{f}-r_{k}$

(11)

so that it is consistent with the duality of the broken theory. Each $U(\tilde{r}_{k})$ sector then has $N_{f}$ pairs of fundamental and anti-fundamental fields $(q^{(k)},\tilde{q}^{(k)})$ and $N_{f}{}^{2}+2$ singlets $M^{(k)},\,V^{(k)}{}^{\pm}$ interacting via the superpotential:

$\displaystyle W_{B}^{(k)}=M^{(k)}\tilde{q}^{(k)}q^{(k)}+V^{(k)}{}^{+}\hat{V}^{(k)}{}^{-}+V^{(k)}{}^{-}\hat{V}^{(k)}{}^{+}\,,$

(12)

which is the Aharony dual theory Aharony:1997gp of the $U(r_{k})$ sector of the broken theory. Hence, once we turn on the polynomial superpotential $\Delta W_{X}=\mathrm{Tr}\left[f_{n}(X)\right]$, the Kim–Park duality lands on the Aharony duality of each broken sector. In the next subsection, we will see that such polynomial superpotential deformation is also useful for investigating new dualities with monopole superpotentials.

In 3d, the Coulomb branch of the vacuum moduli space is described by the VEVs of monopole chiral operators. If such monopole operators are relevant, the theory can also be deformed by them and lead to new fixed points in the IR. For example, it is known that a 3d $U(N)$ SQCD can be deformed by the monopole superpotential $\Delta W=V^{+}V^{-}$ Aharony:2013dha and flow to an IR fixed point distinct from the original one without the monopole superpotential. Interestingly, such a new fixed point can be reached from a 4d $U(N)$ SQCD put on a circle, whose effective theory is 3-dimensional. Similarly, one can also consider the compactification of a 4d $USp(2N)$ SQCD, which flows to the same fixed point as the 3d $USp(2N)$ theory deformed by the monopole superpotential $\Delta W=V$ Aharony:2013dha . Moreover, this theory further flows to the 3d $U(N)$ theory with linear monopole superpotential $\Delta W=V^{+}+V^{-}$ Benini:2017dud once we turn on some real mass breaking $USp(2N)$ into $U(N)$. Note that all those fixed points are distinct from those of the original 3d theories without the monopole superpotentials if the deformation is relevant.

In addition, such monopole deformation also leads to new IR dualities. For instance, Benini:2017dud shows that the 3d $U(N_{c})$ theory with $N_{f}$ flavors deformed by linear monopole superpotential $\Delta W=V^{+}+V^{-}$ enjoys the following Seiberg-like duality.

• •

  Theory A is the 3d $\mathcal{N}=2$ $U(N_{c})$ gauge theory with $N_{f}$ flavors $(Q,\tilde{Q})$ and the superpotential

  $\displaystyle W_{A}=V^{+}+V^{-}$

  (13)

  where $V^{\pm}$ are the monopole operators of Theory A.

• •

  Theory B is the 3d $\mathcal{N}=2$ $U(N_{f}-N_{c}-2)$ gauge theory with $N_{f}$ flavors $(q,\tilde{q})$, $N_{f}{}^{2}$ gauge singlets $M$, and the superpotential

  $\displaystyle W_{B}=M\tilde{q}q+\hat{V}^{+}+\hat{V}^{-}$

  (14)

  where $\hat{V}^{\pm}$ are the monopole operators of Theory B.

There are also studies of the monopole deformation of adjoint SQCDs and their dualities Amariti:2018wht ; Amariti:2019rhc . As reviewed in the previous subsection, the $U(N_{c})$ theory with an adjoint matter and the superpotential $W=\mathrm{Tr}\,X^{n+1}$ has $2n$ monopole operators $V_{i}^{\pm}$ for $i=0,\dots,n-1$. Therefore, one can consider a large variety of monopole superpotentials and corresponding dualities.

In this subsection, we examine the deformation of the single adjoint theory by the following linear monopole superpotential:

$\displaystyle\Delta W_{A}=V_{\alpha}^{+}+V_{\alpha}^{-}\,.$

(15)

For $\alpha=0$, this type of monopole superpotential has been discussed in Amariti:2018wht , which shows that the adjoint theory with such monopole superpotential can be obtained by compactifying a 4d $USp(2N_{c})$ SQCD with an antisymmetric matter and turning on suitable real mass breaking $USp(2N_{c})$ into $U(N_{c})$. Most importantly, this 4d $USp(2N_{c})$ theory enjoys the Intriligator–Leigh–Strassler duality Intriligator:1995ax , which in turn descends to a duality for the 3d $U(N_{c})$ SQCD with an adjoint matter and the superpotential

$\displaystyle W_{A}^{\alpha=0}=\mathrm{Tr}X^{n+1}+V_{0}^{+}+V_{0}^{-}\,.$

(16)

The dual theory is given by the $U(nN_{f}-N_{c}-2n)$ theory with the superpotential

$\displaystyle W_{B}^{\alpha=0}=\mathrm{Tr}\hat{X}^{n+1}+\sum_{i=0}^{n-1}M_{i}\tilde{q}\hat{X}^{n-1-i}q+\hat{V}_{0}^{+}+\hat{V}_{0}^{-}\,,$

(17)

which also includes the linear superpotential of the dual monopole operators $\hat{V}_{0}^{\pm}$. Note that the monopole superpotential lifts the Coulomb branch of the moduli space.

We attempt to generalize this duality for $\alpha>0$. Since $V_{0}^{\pm}$ have the smallest conformal dimension among the $2n$ monopole operators of the single adjoint theory, it is often the case that $V_{0}^{\pm}$ decouple in the IR and its linear deformation breaks supersymmetry. Nevertheless, even in such cases, there could exist interacting $V_{\alpha}^{\pm}$ for higher $\alpha>0$, which would give rise to some relevant deformation of the theory. Hence, it is natural to ask if one can find new dualities by turning on the monopole superpotential (15) for $\alpha>0$.

Our proposal is as follows.

• •

  Theory A is the 3d $\mathcal{N}=2$ $U(N_{c})$ gauge theory with $N_{f}$ pairs of fundamental $Q^{a}$ and anti-fundamental $\tilde{Q}^{\tilde{a}}$, one adjoint chiral multiplet $X$ and the superpotential

  $\displaystyle W_{A}^{mon}=\mathrm{Tr}X^{n+1}+V_{\alpha}^{+}+V_{\alpha}^{-}\,.$

  (18)

  where $V_{\alpha}^{\pm}$ are monopole operators of Theory A.

• •

  Theory B is the 3d $\mathcal{N}=2$ $U(nN_{f}-N_{c}-2n+2\alpha)$ gauge theory with $N_{f}$ pairs of fundamental $q_{\tilde{a}}$ and anti-fundemental $\tilde{q}_{a}$, one adjoint $\hat{X}$, and $nN_{f}{}^{2}$ gauge singlet chiral multiplets ${M_{i}}^{\tilde{a}a}$ for $a,\tilde{a}=1,\dots,N_{f}$ and $i=0,\ldots,n-1$. The superpotential is given by

  $\displaystyle W_{B}^{mon}=\mathrm{Tr}\hat{X}^{n+1}+\sum_{i=0}^{n-1}M_{i}\tilde{q}\hat{X}^{n-1-i}q+\hat{V}_{\alpha}^{+}+\hat{V}_{\alpha}^{-}\,.$

  (19)

  where $\hat{V}_{\alpha}^{\pm}$ are the monopole operators of Theory B.

Due to the monopole superpotential, monopole operators $V_{j}^{\pm}$ of Theory A and $\hat{V}_{j}^{\pm}$ of Theory B become massive for $j\geq\alpha$. The duality map of the remaining monopole operators is given by

$\displaystyle V_{i}^{\pm}\quad\longleftrightarrow\quad\hat{V}_{i}^{\pm}\,,\qquad i=0,\dots,\alpha-1\,.$

(20)

Since the superpotential for $\alpha>0$ is not obtained from the compactification, we should take an alternative approach to obtain the duality deformed by those dressed monopole operators. For this reason, although our main interest is $\alpha>0$, let us first revisit the $\alpha=0$ case using such an altenative approach and then discuss $\alpha>0$ subsequently. We first note that the extra superpotential (15) with $\alpha=0$:

$\displaystyle\Delta W_{A}=V_{0}^{+}+V_{0}^{-}$

(21)

is mapped to the superpotential of the same form on the dual side

$\displaystyle\Delta W_{B}=V_{0}^{+}+V_{0}^{-}$

(22)

where $V_{0}^{\pm}$ are now elementary gauge singlets rather than monopole operators of the theory. Together with the original superpotential of the dual theory (2), the total superpotential of the deformed dual theory is given by

	$\displaystyle W_{B}^{\prime}$	$\displaystyle=W_{B}+\Delta W_{B}$
		$\displaystyle=\mathrm{Tr}\hat{X}^{n+1}+\sum_{i=0}^{n-1}M_{i}\tilde{q}\hat{X}^{n-1-i}q+\sum_{i=0}^{n-1}\left(V_{i}^{+}\hat{V}_{n-1-i}^{-}+V_{i}^{-}\hat{V}_{n-1-i}^{+}\right)+V_{0}^{+}+V_{0}^{-}\,,$		(23)

which gives rise to non-zero VEVs of the dual monopole operators $\hat{V}_{n-1}^{\pm}$.

While the direct analysis of the corresponding Higgs mechanism is complicated, we can take a shortcut using another deformation of the theory: we go back to the original side and turn on the polynomial superpotential of the adjoint field:

$\displaystyle\Delta W_{X}=\mathrm{Tr}\left[f_{n}(X)\right].$

(24)

As seen in the previous subsection, this leads to the nonzero VEV of $X$, which breaks the theory into the $n$ sectors consisting of the $U(r_{k})$ gauge theories without an adjoint matter for $k=1,\dots,n$. Each sector has a pair of monopole operators $V^{(k)}{}^{\pm}$ and the extra monopole superpotential (21) we turn on is expected to descend to the following linear superpotential for each sector:

$\displaystyle W_{A}^{(k)}=V^{(k)}{}^{+}+V^{(k)}{}^{-}\,.$

(25)

Note that this is exactly the linear monopole superpotential (13) discussed in Benini:2017dud . Thus, the dual theory of each sector is given by the $U(N_{f}-r_{k}-2)$ theory with the superpotential:

$\displaystyle W_{B}^{(k)}=M^{(k)}\tilde{q}^{(k)}q^{(k)}+\hat{V}^{(k)}{}^{+}+\hat{V}^{(k)}{}^{-}\,.$

(26)

We find that the collection of those dual sectors is nothing but the effective low energy description of the $U(nN_{f}-N_{c}-2n)$ theory with the superpotential:

$\displaystyle W=W_{B}^{\alpha=0}+\Delta W_{\hat{X}}\,,\qquad\Delta W_{\hat{X}}=\mathrm{Tr}\left[f_{n}(\hat{X})\right]$

(27)

where $W_{B}^{\alpha=0}$ is given in (17). As argued in the previous subsection, $\Delta W_{\hat{X}}$ and the first term of $W_{B}^{\alpha=0}$ result in $n$ distinct vacuum solutions governed by the $U(N_{f}-r_{k}-2)$ theories without the adjoint $\hat{X}$. Then the remaining terms of $W_{B}^{\alpha=0}$ become the superpotential (26) for each $U(N_{f}-r_{k}-2)$ sector. Hence, the $U(nN_{f}-N_{c}-2n)$ theory with the superpotential (27) flows to the collection of the $U(N_{f}-r_{k}-2)$ theories with the superpotential (26).

One should note that $\Delta W_{\hat{X}}$ is dual to $\Delta W_{X}$, the extra superpotential we introduce on the original side. Thus, once we turn off both of them, we have a duality between the $U(N_{c})$ theory with the superpotential (16) and the $U(nN_{f}-N_{c}-2n)$ theory with the superpotential (17), which should be the one obtained from the theory with (2.2) after Higgsing. Indeed, this is exactly the duality proposed in Amariti:2018wht obtained by compactifying the 4d duality. The duality and its deformation by the adjoint polynomial superpotential can be summarized as follows:

$\displaystyle\begin{array}[]{ccc}U(N_{c})&\quad\longleftrightarrow&U(nN_{f}-N_{c}-2n)\\ \big{\downarrow}&&\big{\downarrow}\\ {\displaystyle\prod_{k=1}^{n}U(r_{k})}&\quad\longleftrightarrow&{\displaystyle\prod_{k=1}^{n}U(N_{f}-r_{k}-2)}\end{array}$

(31)

where a horizontal arrow denotes the duality between the two theories, while a vertical arrow indicates the deformation by the polynomial superpotential of the adjoint field.

Notice that the Coulomb branch of each broken sector on the right hand side is completely lifted by the monopole superpotential (25). Therefore, we expect that the monopole superpotential (21) also lifts the entire Coulomb branch on the left hand side. This will be confirmed by the superconformal index shortly in section 2.3.

Now let us move on to the generalization to $\alpha>0$. We first recall that the theory with an adjoint matter has the $2n$-dimensional Coulomb branch described by $V_{i}^{\pm}$ for $i=0,\dots,n-1$, whose dimension is not affected even if we turn on the polynomial superpotential $\Delta W_{X}=\mathrm{Tr}\left[f_{n}(X)\right]$. Once the polynomial superpotential is turned on, the theory is decomposed into $n$ sectors, each of which has a 2-dimensional Coulomb branch described by their own monopole operators $V^{(k)}{}^{\pm}$. Therefore, the total Coulomb branch is still a $2n$-dimensional one, now described by $V^{(k)}{}^{\pm}$ for $k=1,\dots,n$.

We expect that the monopole superpotential (21) completely lifts this $2n$-dimensional Coulomb branch because it descends to the monopole superpotential (25) of each $U(r_{k})$ sector of the broken theory. Now let us imagine we turn on the monopole superpotential only for $m<n$ sectors of the broken theory. In that case, we still have un-lifted Coulomb branch for $n-m$ sectors, which enjoy the Aharony duality Aharony:1997gp rather than the Benini–Benvenuti–Pasquetti duality Benini:2017dud . Namely, the dual relations of the broken sectors are given by

$\displaystyle\left\{\begin{array}[]{cccc}U(r_{k})&\quad\longleftrightarrow&U(N_{f}-r_{k})&\qquad\text{without the monopole superpotential,}\\ U(r_{k})&\quad\longleftrightarrow&U(N_{f}-r_{k}-2)&\qquad\text{with the monopole superpotential,}\end{array}\right.$

(34)

depending on the presence of the monopole superpotential. Now if we turn off the adjoint polynomial superpotential so that the left hand side is restored to the $U(N_{c})$ gauge theory, the dual sectors also combine into a $U(nN_{f}-N_{c}-2m)$ gauge theory for $m<n$.²²2Given the fixed number of the sectors with the monopole superpotential, there is a subtlety in choosing which sector has the monopole superpotential and which sector doesn’t. However, this is not significant for our purpose as the full Weyl symmetry of $U(N_{c})$ will be restored once the adjoint polynomial superpotential is turned off, which is what we are interested in. From this relation, one would expect that there exists a monopole deformation of the original theory lifting the Coulomb branch only partially such that the remaining Coulomb branch is $2(n-m)$-dimensional.

We propose that the deformation (15) plays such a role. Namely, if we turn on the monopole superpotential

$\displaystyle\Delta W_{A}=V_{\alpha}^{+}+V_{\alpha}^{-}\,,$

(35)

we expect that this lifts the components of the Coulomb branch described by $V_{j}^{\pm}$ for $j\geq\alpha$. In the next subsection, we will provide nontrivial evidence of this claim using the superconformal index. Furthermore, now the duality of each broken sector combines to predict a new duality between a $U(N)$ theory and a $U(nN_{f}-N_{c}-2n+2\alpha)$ theory with monopole superpotentials, which is exactly what we proposed at the beginning of this subsection.

We need to check if new monopole terms in the superpotentials (18) and (19) are consistent with the global charges shown in Table 1. First we note that the monopole terms in (18) break the $U(1)_{A}$ and $U(1)_{T}$. In addition, they also demand the $R$-charge of the monopole operators $V_{\alpha}^{\pm}$ to be 2; i.e.,

$\displaystyle(1-\Delta_{Q})N_{f}-\frac{2}{n+1}(N_{c}-1-\alpha)=2\,,$

(36)

which is satisfied only if the $R$-charge $\Delta_{Q}$ of $Q$ and $\tilde{Q}$ is given by

$\displaystyle\Delta_{Q}=\frac{(n+1)N_{f}-2N_{c}-2n+2\alpha}{(n+1)N_{f}}\,.$

(37)

The $R$-charge $\Delta_{q}$ of $q$ and $\tilde{q}$ is then determined by

$\displaystyle\Delta_{q}=\frac{2}{n+1}-\Delta_{Q}=\frac{-(n-1)N_{f}+2N_{c}+2n-2\alpha}{(n+1)N_{f}}\,.$

(38)

This requires the $R$-charge of the dual monopole operators $\hat{V}_{\alpha}^{\pm}$ to be

	$\displaystyle\quad\left(1-\Delta_{q}\right)N_{f}-\frac{2}{n+1}\left(\tilde{N}_{c}-1-\alpha\right)$
	$\displaystyle=\left(1-\frac{2}{n+1}+\Delta_{Q}\right)N_{f}-\frac{2}{n+1}(nN_{f}-N_{c}-2n+\alpha-1)$
	$\displaystyle=2\,,$		(39)

which is consistent with the monopole terms in the dual superpotential (19). The resulting global charges are summarized in Table 2.

In this subsection, we attempt to support our proposal by providing nontrivial evidence using the superconformal index. We will see that the evaluated indices show the perfect agreement under the proposed duality. Furthermore, from the index, we also show that if the monopole superpotential

$\displaystyle\Delta W_{A}=V_{\alpha}^{+}+V_{\alpha}^{-}\,.$

(40)

is turned on, the monopole operators $V_{j}^{\pm}$ for $j\geq\alpha$ become $Q$-exact and vanish in the chiral ring, captureing the algebraic structure of the moduli space, as we claimed in the previous subsection.

The definition of the 3d superconformal index is given by Bhattacharya:2008zy ; Bhattacharya:2008bja :

$\displaystyle I=\mathrm{Tr}(-1)^{F}e^{\beta^{\prime}\{Q,S\}}x^{E+j}\prod_{i}t_{i}^{F_{i}}\,.$

(41)

Here $F$ is the fermion number operator. $Q$ and $S=Q^{\dagger}$ are supercharges chosen such that only the BPS states saturating

$\displaystyle\{Q,S\}=E-j-R\geq 0$

(42)

contribute to the index, where energy $E$, angular momentum $j$, and $R$-charge $R$ are three Cartans of the bosonic subgroup of the 3d $\mathcal{N}=2$ superconformal group $SO(3,2)\times SO(2)$. $F_{i}$ are the Cartan charges of the global symmetry.

This index can be evaluated using the localization technique, either localization on the Coulomb branch Kim:2009wb ; Imamura:2011su or that on the Higgs branch Fujitsuka:2013fga ; Benini:2013yva leading to the factorized form of the index Hwang:2012jh .³³3The factorized index is also closely related to the notion of holomorphic block Beem:2012mb . Indeed, the index of the single adjoint theory without the monopole superpotential was computed using the Coulomb branch formula in Kim:2013cma and using the factorization formula in Hwang:2015wna . For the index without the monopole superpotential, the $R$-charges of the monopole operators $V_{0}^{\pm}$ are free parameters, whose values at the IR fixed point should be independently determined by the $F$-maximization Jafferis:2010un because the $R$-symmetry can vary along the RG-flow as it mixes with other abelian global symmetries. On the other hand, the index with the monopole superpotential can be obtained using the same formulas but with fixed $R$-charges and fugacities because they are constrained by the given monopole superpotential. For examples, since the monopole superpotential (40) breaks $U(1)_{A}$ and $U(1)_{T}$, there is no $U(1)$ global symmetry that can be mixed with $U(1)_{R}$ along the RG-flow unless there is an emergent $U(1)$ symmetry. In such a case, the superconformal $R$-charge at the IR fixed point is the same as the UV value, which is fixed by the monopole superpotential.

In order to compute the index for the single adjoint theory with the monopole superpotential, we use the factorization formula derived in Hwang:2015wna and insert the $R$-charge value $\Delta_{Q}$ determined by (37), which makes the $R$-charge of $V_{\alpha}^{\pm}$ two. A few examples of the results we obtain are shown in Table 3, whereas more results can be found in appendix A.

In all the cases we have examined, the indices of each dual pair show the exact agreement, which is strong evidence of the duality we propose.

Capturing the spectrum of the chiral operators, the superconformal index tells us many important aspects of the theory. For example, the algebraic structure of the vacuum moduli space can be deduced from the index because it is parametrized by the VEV of chiral operators. Especially, we can see how the moduli space is deformed once we turn on the monopole superpotential (40). First recall that, in general, the moduli space of the theory without the monopole superpotential is parameterized by the following chiral operators:

$\displaystyle\begin{aligned} \tilde{Q}X^{i}Q\quad&\longleftrightarrow\quad M_{i}\,,\qquad\qquad i=0,\dots,n-1\,,\\ \mathrm{Tr}X^{i}\quad&\longleftrightarrow\quad\mathrm{Tr}\hat{X}^{i}\,,\qquad\quad i=1,\dots,n-1\,,\\ V_{i}^{\pm}\quad&\longleftrightarrow\quad V_{i}^{\pm}\,,\qquad\quad\,\,\,\,\,i=0,\dots,n-1\,,\end{aligned}$

(43)

where the right hand side shows the corresponding dual operators, which describe the same moduli space. Note that $V_{i}^{\pm}$ on the left hand side are monopole operators, while $V_{i}^{\pm}$ on the right hand side are elementary gauge singlet fields.

If we turn on the linear monopole superpotential (40), as we will show shortly, we find that some of the monopole operators vanish in the chiral ring as follows:

$\displaystyle V_{j}^{\pm}\quad\sim\quad 0\,,\qquad\quad j=\alpha,\dots,n-1$

(44)

because they combine with some fermionic operators and become $Q$-exact. In addition, on the dual side, the singlets $V_{i}^{\pm}$ for all $i$, which used to describe the Coulomb branch of the moduli space, are now all massive. Instead, we have dual monopole operators $\hat{V}_{i}^{\pm}$ for $i=0,\dots,\alpha-1$, which are mapped to the monopole operators of the original theory as follows:

$\displaystyle V_{i}^{\pm}\quad$

$\displaystyle\longleftrightarrow\quad\hat{V}_{i}^{\pm}\,,\qquad\quad\,\,\,\,\,i=0,\dots,\alpha-1\,.$

(45)

Those monopole operators parameterize the unlifted Coulomb branch of the moduli space, which now has dimension $2\alpha$ rather than $2n$.

Therefore, in generic cases, the vacuum moduli space is parameterized by the VEVs of the following gauge invariant operators:

	$\displaystyle\tilde{Q}X^{i}Q\quad$	$\displaystyle\longleftrightarrow\quad M_{i}\,,\qquad\qquad i=0,\dots,n-1\,,$		(46)
	$\displaystyle\mathrm{Tr}X^{i}\quad$	$\displaystyle\longleftrightarrow\quad\mathrm{Tr}\hat{X}^{i}\,,\qquad\quad i=1,\dots,n-1\,,$		(47)
	$\displaystyle V_{i}^{\pm}\quad$	$\displaystyle\longleftrightarrow\quad\hat{V}_{i}^{\pm}\,,\qquad\quad\,\,\,\,\,i=0,\dots,\alpha-1$		(48)

where the right hand side shows the corresponding dual operators describing the same moduli space. In addtion, we will also see that if the gauge ranks are bounded in a certain range, there are additional effects of the monopole superpotential as follows:

$\displaystyle\begin{aligned} \mathrm{Tr}X^{i}\quad&\sim\quad 0\,,\qquad\qquad i=\mathrm{min}(N_{c}+1,nN_{f}-N_{c}-2n+2\alpha+1),\dots,n-1\,,\\ V_{i}^{\pm}\quad&\sim\quad 0\,,\qquad\qquad i=\mathrm{min}(N_{c},nN_{f}-N_{c}-2n+2\alpha),\dots,\alpha-1\end{aligned}$

(49)

if $\mathrm{min}(N_{c},nN_{f}-N_{c}-2n+2\alpha)$ is less than either $n-1$ or $\alpha$.

To see the truncation of the monopole operators (44), let us examine the case with $(n,N_{f},N_{c})=(3,5,5)$ and $\Delta W_{A}=V_{1}^{+}+V_{1}^{-}$ as an example. Before turning on the monopole superpotential $\Delta W_{A}=V_{1}^{+}+V_{1}^{-}$, the index is given by⁴⁴4While we have shown the terms up to $x^{2}$ to avoid clutter, the next terms up to $x^{3}$ can be easily obtained by taking the plethystic exponential of (2.3).

		$\displaystyle I=1+x^{\frac{1}{2}}+\mathbf{5}_{t}\mathbf{5}_{u}\tau^{2}x^{\frac{3}{5}}+2x+2\,\mathbf{5}_{t}\mathbf{5}_{u}\tau^{2}x^{\frac{11}{10}}+\left(\mathbf{15}_{t}\mathbf{15}_{u}+\mathbf{10}_{t}\mathbf{10}_{u}\right)\tau^{4}x^{\frac{6}{5}}$
		$\displaystyle\quad+\left(2+\tau^{-5}\left(w+w^{-1}\right)\right)x^{\frac{3}{2}}+4\,\mathbf{5}_{t}\mathbf{5}_{u}\tau^{2}x^{\frac{8}{5}}$
		$\displaystyle\quad+\left((\mathbf{15}_{t}+\mathbf{10}_{t})(\mathbf{15}_{u}+\mathbf{10}_{u})+\mathbf{15}_{t}\mathbf{15}_{u}+\mathbf{10}_{t}\mathbf{10}_{u}\right)\tau^{4}x^{\frac{17}{10}}$
		$\displaystyle\quad+\left(\mathbf{40}_{t}\mathbf{40}_{u}+\mathbf{35}_{t}\mathbf{35}_{u}+\overline{\mathbf{10}}_{t}\overline{\mathbf{10}}_{u}\right)\tau^{6}x^{\frac{9}{5}}+\left(1+2\tau^{-5}\left(w+w^{-1}\right)-\mathbf{24}_{t}-\mathbf{24}_{u}\right)x^{2}+\dots$		(50)

where $\mathbf{n}_{t}$ and $\mathbf{n}_{u}$ are the characters of the representation $\mathbf{n}$ of $SU(N_{f})_{t}$ and that of $SU(N_{f})_{u}$ respectively, and $\tau$ and $w$ are the $U(1)_{A}$ and $U(1)_{T}$ fugacities. While we have used a trial UV value of $R$-charge $\Delta_{Q}=\frac{(n+1)N_{f}-2N_{c}-2n+2\alpha}{(n+1)N_{f}}$, which is the same as the one determined in (37), the superconformal IR value can be obtained by shifting $\tau\rightarrow\tau x^{\alpha}$ where the mixing coefficient $\alpha$ should be determined by the $F$-maximization. To read the chiral ring relations, it is convenient to take the plethystic logarithm Benvenuti:2006qr , which gets the contribution from the single trace operators, both bosonic and fermionic, as well as their relations. For the current example, the plethystic log of the index is given by

		$\displaystyle\quad(1-x^{2})\,\mathrm{PL}\left[I\right]$
		$\displaystyle=x^{\frac{1}{2}}+\mathbf{5}_{t}\mathbf{5}_{u}\tau^{2}x^{\frac{3}{5}}+x+\mathbf{5}_{t}\mathbf{5}_{u}\tau^{2}x^{\frac{11}{10}}+\tau^{-5}\left(w+w^{-1}\right)x^{\frac{3}{2}}+\mathbf{5}_{t}\mathbf{5}_{u}\tau^{2}x^{\frac{8}{5}}$
		$\displaystyle\quad+\left(\tau^{-5}\left(w+w^{-1}\right)-\mathbf{24}_{t}-\mathbf{24}_{u}-2\right)x^{2}+\left(\tau^{-5}\left(w+w^{-1}\right)-\mathbf{24}_{t}-\mathbf{24}_{u}-2\right)x^{\frac{5}{2}}$
		$\displaystyle\quad+\left(-\mathbf{24}_{t}-\mathbf{24}_{u}-3\right)x^{3}+\dots\,,$		(51)

where both sides are multiplied by $(1-x^{2})$ to remove the contribution of the descendants derived by the derivative operator. From this plethystic logarithm of the index, we can easily find the contribution of the monopole operators $V_{i}^{\pm}$ for $i=0,1,2$:

$\displaystyle\tau^{-5}\left(w+w^{-1}\right)x^{\frac{3+i}{2}}\,,$

(52)

which are all independent operators describing the Coulomb branch of the moduli space. On the other hand, we are also interested in the fermionic operators giving the following contributions:

	$\displaystyle\psi_{Q}^{\dagger}X^{i-1}Q\,$	$\displaystyle:\qquad-\left(\mathbf{24}_{t}+1\right)x^{\frac{3+i}{2}}\,,$		(53)
	$\displaystyle\tilde{Q}X^{i-1}\psi_{\tilde{Q}}^{\dagger}\,$	$\displaystyle:\qquad-\left(\mathbf{24}_{u}+1\right)x^{\frac{3+i}{2}}$		(54)

for $i=1,2$, which play important roles when we turn on the monopole superpotential. Indeed, if we turn on the monopole superpotential $\Delta W_{A}=V_{1}^{+}+V_{1}^{-}$,⁵⁵5Here we assume all the terms in the superpotential are relevant in the IR so that the theory flows to the IR fixed point with the expected symmetry, which is $SU(N_{f})^{2}$ in this case. However, this should be independently checked using, e.g., the $F$-maximization, which we wasn’t able to conduct for this and subsequent examples due to limited computing resources. Nevertheless, regardless of the actual relevance of the superpotential for those particular examples, the explanation here can be regarded as a demonstration of the general structure of the indices and the chiral rings of the theories when their superpotentials are relevant. this breaks $U(1)_{A}$ and $U(1)_{T}$, whose fugacities $\tau$ and $w$ thus have to be 1. Once we set $\tau=w=1$ in (2.3), we see that the monopole contribution (52) for $i=1,\,2$ is exactly canceled by $-2x^{\frac{3+i}{2}}$, the contribution of the trace parts of $\psi_{Q}^{\dagger}X^{i-1}Q$ and $\tilde{Q}X^{i-1}\psi_{\tilde{Q}}^{\dagger}$. In contrast, the monopole contribution (52) for $i=0$ still remains nontrivial. This means that the lowest monopole operators $V_{0}^{\pm}$ remain nontrivial in the chiral ring and still parametrize the Coulomb branch when we turn on the monopole superpotential $\Delta W_{A}=V_{1}^{+}+V_{1}^{-}$, while the other monopole operators $V_{i}^{\pm}$ for $i\geq 1$ combine with the trace parts of the fermionic operators $\psi_{Q}^{\dagger}X^{i-1}Q$ and $\tilde{Q}X^{i-1}\psi_{\tilde{Q}}^{\dagger}$ and become a long multiplet along the RG-flow. Therefore, the components of the Coulomb branch described by $V_{j}^{\pm}$ for $j\geq 1$ are now lifted quantum mechanically.