It has been conjectured that fermions minimally coupled to a Chern-Simons gauge
field define a conformal field theory (CFT) that is level-rank dual to Chern-Simons
gauged Wilson-Fisher Bosons. The CFTs in question admit relevant deformations
parametrized by a real mass. When the mass deformation is positive, the duality of
the two deformed theories has previously been checked in detail in the large

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|c_B|}\end{array}\right.\ . \end{equation} In this range of parameters the off-shell fermionic free energy can also be recast into bosonic language; we find \begin{align} \label{prdfi} v_B(|c_B|,\rho_B)=& \,\frac{N_B}{6\pi} \bigg[ -\frac{|\hat{m}|^3}{|\lambda_B| }+3|\lambda_B| |\hat{m}| \cS^2+2|\lambda_B| ^2 \cS^3 \\ \nonumber & -|c_B|^3 \!+\! 3 \!\int_{-\pi}^{\pi}\!\! \rho_B(\alpha) d\alpha\!\int_{|c_B|}^{\infty} \!\! dy y\left(\log\left(1\!+\!e^{-y-i\alpha -\nu}\right)\!+\log\left(1\!+\!e^{-y+i\alpha + \nu}\right)\right) \! \bigg], \end{align} The equations~\eqref{mdrfooi} and~\eqref{prdfi} may be thought of as the predictions of duality for the gap equation and free energy of the bosonic theory in the condensed phase, i.e.\ when~\eqref{dlcbinta} is not obeyed. In this paper we will reproduce both these results from a direct evaluation of the bosonic path integral in the condensed phase. The exact agreement under duality of the bosonic and fermionic results may be thought of as a detailed new check of the duality conjecture. ]]>

|\nu|$. Next we plug our solution for $\Sigma^{\mu\nu}$ back into our large $N$ expression for the partition function. On-shell, we find that the final result for the free energy $v_B$ perfectly matches the prediction~\eqref{prdfi}. ]]>

|\nu|$. See appendix~\ref{review} for a discussion.) We now turn to simplifying the first term on the r.h.s.\ of~\eqref{sal}. Using the fact that $\Sigma^{33}=\Sigma^{++}=0$ (see~\eqref{sigmacomp}) it follows that that \begin{align} \label{sigalphcont} &\Sigma^{\nu\mu}(q) \alpha_{\mu\nu}(-q) = \\ &\quad\Sigma^{--}(q) \alpha_{--}(-q)+2( \Sigma^{-3}(q) \alpha_{3-}(-q)+\Sigma^{-+}(q) \alpha_{+-}(-q)+\Sigma^{+3}(q) \alpha_{3+}(-q) )\ .\nonumber \end{align} In order to further simplify~\eqref{sigalphcont} we now plug in the explicit expressions for $\Sigma$ and $\alpha$ obtained above (i.e.~\eqref{sigmacomp} and~\eqref{alphaexpr} with the particular values of $F_1 \ldots F_4$ solved for above). The dependence of the resultant expression on $q_3$ is very simple; it is given by a polynomial of degree one in $q_3$ times $\frac{1}{\detr Q}$. (Of course $q_3$ is discretized and holonomy shifted version at finite temperature). The linear term in this Polynomial yields a vanishing contribution when summed over the full range of discrete values of $q_3$ and simultaneously integrated over the holonomy.\footnote{We use here that the eigenvalue distribution function $\rho(\alpha)$ is an even function of $\alpha$.} For this reason we simply ignore the term linear in $q_3$. With this understanding --- omitting the terms discussed above --- we have \begin{equation}\label{sigmaalpha} \begin{aligned} \Sigma^{\nu\mu}(q) \alpha_{\mu\nu}(-q)=& -\frac{\lambda_B }{(2 \pi )^3 \text{detQ}} i m \mathcal{L}(w)\\ \mathcal{L}(w)= &\,\frac{2 g(0)^3 m^2+3 \mathcal{I}(w)}{3 g(w)^2}+\frac{1}{3} \left(-2 g(0)^3 m^2-9 g(0)^2 m^2-3 \mathcal{I}(w)-6 w\right)\\ & -\frac{4}{3} m^2 g(w)^3+m^2 g(w)^2+\frac{1}{3} g(w) \left(6 g(0)^2 m^2+4 m^2+6 w\right) \end{aligned} \end{equation} (the functions $g$ and ${\cal I}$ were defined in~\eqref{defg} and~\eqref{defI} above). The dependence of~\eqref{sigmaalpha} on the discretized and holonomy shifted version of $q_3$ is entirely through the factor of $\frac{1}{\detr Q }$. Performing the sum over the discrete momenta in $q_3$ we find \begin{equation}\label{procfe} \begin{aligned} -\frac{i}{2\lambda_B} \int \frac{\td^3q}{(2 \pi)^3} \Sigma^{\nu\mu}(q) \alpha_{\mu\nu}(-q) &=\int \frac{\td^3q}{(2 \pi)^3} \frac{-m}{2 (2\pi)^3 \text{detQ} } \mathcal{L}(w) \\ &= \frac{-m}{2 (2\pi)^3 } \int \frac{q_s dq_s}{2\pi} \frac{1}{\beta} \sum_{q_3} \frac{1}{-\frac{m}{(2\pi)^3}(q^2+M^2)}\mathcal{L}(w)\\ &= \frac{m}{4 \pi} \frac{-1}{2 (2\pi)^3 } \int dw \chi(w)\mathcal{L}(w) =\frac{m}{4 \pi} \int dw \xi'(w)\mathcal{L}(w)\\ &=\frac{m }{4 \pi}\int dw \ \xi'(w) \sum_n \mathcal{L}_n(w)(-\lambda_B)^n\ . \end{aligned} \end{equation} where we have used~\eqref{intdet} and~\eqref{intchi}. In the last line of~\eqref{procfe} we have simply Taylor expanded $\mathcal{L}$ in all explicit factors of $\lambda_B$ according to the following rule. We see from~\eqref{sigmaalpha} that $\mathcal{L}$ depends on the functions $g$ and $\mathcal{I}$. We use the equation~\eqref{defg} to rewrite $g$ in terms of $\chi$ using \begin{equation} \label{gchi} g(w)=1+ \lambda_B\xi(w)\,,\quad g'(w) = -\frac{\lambda_B}{2(2\pi)^3} \chi(w)\ . \end{equation} In a similar fashion we use~\eqref{defI} to write $\mathcal{I}$ in terms of integrals of $\xi$: \begin{equation} \label{defiii} {\cal I}(w)= \int_0^w dz \left( 1+ \lambda_B\xi(z) \right)\ . \end{equation} We then Taylor expand ${\cal L}$ treating $\xi$ and all its integrals as independent of $\lambda_B$; with this understanding ${\cal L}_n$ are defined by \begin{equation} \label{mcl} \mathcal{L}= \sum_{n=0}^\infty \mathcal{L}_n(w)(-\lambda_B)^n. \end{equation} The various coefficient functions ${\cal L}_n$ are easily worked out. We find\footnote{$\mathcal{L}_0(w)=0$ is just the statement that this contribution is present only when interactions with gauge fields are turned on.} \begin{equation} \label{lnn} \begin{aligned} \mathcal{L}_0(w)&=0, \ \ \ \mathcal{L}_1(w)=2 m^2 \xi[0], \\ \mathcal{L}_2(w)&=-2 \mathcal{I}_\xi(w) \xi (w)-m^2 \xi (0)^2-m^2 \xi (w)^2+3 w \xi (w)^2\\ \mathcal{L}_3(w)&=-3 \mathcal{I}_\xi(w) \xi (w)^2+4 m^2 \xi (w)^3-6 m^2 \xi (0) \xi (w)^2+2 m^2 \xi (0)^2 \xi (w)+4 w \xi (w)^3\\ \mathcal{L}_n(w)&=\frac{1}{3} \xi (w)^{n-3} \Big(-6 n \xi (w)^2 m^2 \xi (0)-2 m^2 (n-2) \xi (0)^3+6 m^2 (n-1) \xi (0)^2 \xi (w)\\ & \quad +(n+1) 2 m^2 \xi(w)^3\Big) + \Big((n+1) w \xi(w)^n - n \xi(w)^{n-1} \mathcal{I}_\xi(w)\Big)\quad\text{for}\quad n\geq 4 \end{aligned} \end{equation} where \begin{equation} \label{ii} \mathcal{I}_\xi(w)=\int_0^w \xi(z) dz\ . \end{equation} The integral~\eqref{procfe} over the last two terms in the expression for ${\cal L}_n$, $n\geq 4$ in~\eqref{lnn} can be simplified using \begin{equation} \begin{aligned} dw \ \xi '(w) \left(( n+1) \xi (w)^n w-n \xi (w)^{n-1} \mathcal{I}_\xi(w)\right)=d( \xi (w)^{n+1} w-\xi (w)^{n} \mathcal{I}_\xi(w)) \end{aligned} \end{equation} It follows that the integral over those terms reduces to surface terms which vanish in the dimensional regularization scheme\footnote{The fact that $\xi(\infty )|_{DR}=0$, $\mathcal{I}_\xi(0)=0$ is used here.} so that \begin{equation} \begin{aligned} \int dw \ \xi '(w) \left(( n+1) \xi (w)^n w-n \xi (w)^{n-1} \mathcal{I}_\xi(w)\right)=0\ . \end{aligned} \end{equation} The integral over all remaining terms in ${\cal L}_n$ for all $n$ are of the form $$\int d\xi\, f(\xi)\,,$$ where the functions $f$ are all simple polynomials of $\xi$. As a consequence all remaining integrals are easily performed and we find \begin{equation} \begin{aligned} \int dw \ \xi '(w) \mathcal{L}_n(w)=0 \quad \text{for $n \geq 3$ }\ . \end{aligned} \end{equation} The only non-zero contributions are \begin{align} &\int dw \ \xi '(w) \mathcal{L}_1(w)=-2 m^2 \xi[0]^2\,,\\ &\int dw \ \xi '(w) \mathcal{L}_2(w)=\frac{4}{3} m^2 \xi[0]^3\ . \end{align} Putting all these together we get, for the $\Sigma\cdot\alpha$ piece, \begin{equation}\label{intp} \begin{aligned} -N_B \mathcal{V}_3 \frac{i}{2\lambda_B} \int \Sigma^{\nu\mu}(q) \alpha_{\mu\nu}(-q)=& N_B\frac{m \mathcal{V}_3}{4 \pi}\(2 m^2 \xi[0]^2 \lambda_B+\frac{4}{3} m^2 \xi[0]^3 \lambda_B^2\)\\ =& N_B\frac{ \mathcal{V}_2 T^2}{6 \pi} \(3 \mathcal{S}^2 |\hat{m}| |\lambda_B|+2 \mathcal{S}^3 \lambda_B^2\)\,, \end{aligned} \end{equation} where we use $\sgn{(m)}=\sgn{(\kappa_B)}=\sgn{(\lambda_B)}$. Combining~\eqref{intp} and~\eqref{detexp} we obtain \begin{align} \label{nnnh} v_B(|c_B|,\rho_B) =&\, \frac{N_B}{6 \pi}\bigg(3 |\lambda_B| |\hat{m}| \mathcal{S}^2 + 2 |\lambda_B|^2 \mathcal{S}^3 \\ \non & -|c_B|^3 + 3 \int_{-\pi}^{\pi} \rho_B(\alpha) d\alpha\int_{|c_B|}^{\infty}dy y \left(\log(1-e^{-y -i\alpha -\nu})+ \log(1-e^{-y+i\alpha + \nu})\!\!\right) \bigg) \end{align} This matches precisely with the prediction for the bosonic free energy from the fermionic result presented in~\eqref{prdfi}. In other words the free energy of the bosonic theory exactly matches the free energy of the fermionic theory under the duality map, as we set out to~show. It is not difficult to promote the expression~\eqref{nnnh} to an offshell free energy. Consider the quantity \begin{equation} \label{osfee} \begin{split} F_B[ \rho_B(\alpha),c_B] =&\,\frac{N_B}{6\pi} {\Bigg[} - \frac{ \left(\lambda_B-{\rm sgn}(\lambda_B) - {\rm sgn} (X_B) \right) }{\lambda_B}|c_B|^3 +\frac{3}{2} {\hat m}_B^{\rm cri} c_B^2 + \alpha \left({\hat m}_B^{\rm cri}\right)^3\\ &+3 \int_{-\pi}^{\pi} \rho(\alpha) d\alpha\int_{|c_B|}^{\infty}dy y\left(\ln\left(1-e^{-y-i\alpha}\right)+\ln\left(1-e^{-y+i\alpha}\right) \right) {\Bigg]},\\ \end{split} \end{equation} where $\alpha$ is an unknown pure number (see below for a discussion). In the case that ${\rm sgn} (X_B) =-{\rm sgn}(\lambda_B)$, $F_B$ reduces to $v_B(\rho)$ reported in~\eqref{offshellfe}. It follows that~\eqref{osfee} is the correct offshell free energy in the unHiggsed phase. Let us now consider the opposite case ${\rm sgn} (X_B) ={\rm sgn}(\lambda_B)$. In this case the expression for $F_B$ in~\eqref{osfee} simplifies to \begin{equation} \label{osfeeh} \begin{split} F_B[ \rho_B(\alpha),c_B] =&\;\frac{N_B}{6\pi} {\Bigg[} - \frac{ \left(\lambda_B-2{\rm sgn}(\lambda_B) \right) }{\lambda_B}|c_B|^3 +\frac{3}{2} {\hat m}_B^{\rm cri} c_B^2+ \alpha\left({\hat m}_B^{\rm cri}\right)^3\\ &+3 \int_{-\pi}^{\pi} \rho(\alpha) d\alpha\int_{|c_B|}^{\infty}dy y\left(\ln\left(1-e^{-y-i\alpha}\right)+\ln\left(1-e^{-y+i\alpha}\right) \right) {\Bigg]},\\ \end{split} \end{equation} It is not difficult to verify that the condition of stationarity of variation of~\eqref{osfeeh} w.r.t.\ $|c_B|$ yields the gap equation~\eqref{fge}. It is also not difficult to verify that when~\eqref{osfeeh} reduces to~\eqref{nnnh} when evaluated onshell (upto the term proportional to $\alpha$: see below) i.e.\ on a solution to~\eqref{osfeeh}. It follows that~\eqref{osfeeh} is an offshell free energy for the critical boson theory in the Higgsed phase, and so also that~\eqref{osfee} is an offshell free energy for the critical boson in either phase --- Higgsed or unHiggsed. Finally, it is not difficult to verify that~\eqref{osfee} maps to the regular fermionic offshell free energy reported in~\eqref{offshellfe} (once we identify $|c_B|$ with $|c_F|$). Let us now return to a discussion of the parameter $\alpha$ in~\eqref{osfeeh}. As this term is independent of $c_B$ it does not affect the variation of the action w.r.t.\ $c_B$ and so does not contribute to the gap equations. This term shifts $\ln Z$ of the theory ($Z$ is the finite temperature partition function) by $- \frac{ V (m_B^{\rm cri})^3 \alpha }{T}$ where $V$ is the volume of space and $T$ is the temperature. This shift can be absorbed into a shift of the ground state energy of the theory by $V \alpha (m_B^{\rm cri})^3$, or equivalently by a shift proportional to $\alpha ( m_B^{\rm cri})^3$ of the cosmological constant counterterm of the original field theory. In other words the parameter $\alpha$ can only be determined once we have made a particular choice of the cosmological constant counterterm. In the absence of such a choice $\alpha$ is ambiguous. We will leave $\alpha$ above as a free parameter in our final result. As we have explained in the introduction, the quantity $v_B$ reported in~\eqref{nnnh} (or equivalently~\eqref{prdfi}) defines the integrand of an integral over unitary matrices $U$. The result of this integral over $U$ is the finite temperature partition function $\mc{Z}$ \begin{equation} \label{pfh} \mc{Z}=\text{Tr } e^{-\beta H}\,, \end{equation} where $H$ is the Hamiltonian. In the Higgsed phase the Hamiltonian $H$ may be obtained by canonically quantizing the action~\eqref{asb} --- the starting point of our path integral evaluation of the free energy. The spectrum of~\eqref{asb} is particularly simple in the limit $\lambda_B=0$ with $|m_W|=\left|\frac{\lambda_B m_B^{\text{cri}}}{2}\right|$ held fixed. In this limit the gauge fields $A_\mu$ are very weakly coupled, and the the partition function~\eqref{pfh} may be evaluated by enumerating the spectrum of effectively free massive $W$ (and $Z$) bosons, subject only to the `Gauss Law' constraint that asserts that all physical states are gauge singlets (see~\cite{Aharony-ml-2003sx} and references therein). It is easy to see that our explicit results~\eqref{mdrfooi} and~\eqref{prdfi} are consistent with this expectation. In this limit~\eqref{mdrfooi} reduces to $|c_B|= |\hat{m}_W|$. In other words the thermal mass of the $W$ bosons agrees with their bare mass at all temperatures, as expected in a free theory. Moreover, after dropping irrelevant constants, the expression~\eqref{prdfi} reduces, in this limit to \begin{equation} \label{prdfd} v_B(|c_B|,\rho_B)=\frac{N_B}{2\pi} \int_{-\pi}^{\pi} \rho_B(\alpha) d\alpha\int_{|\hat{m}_W|}^{\infty}dy y\left(\log\left(1\!+\!e^{-y-i\alpha -\nu}\right)+\log\left(1\!+\!e^{-y+i\alpha + \nu}\right)\right), \end{equation} which is precisely $v_B$ of a free complex bosonic degree of freedom (in this case the $W$ bosons) in the fundamental representation.\footnote{We thank D. Radicevic for a very useful discussion on this point.} ]]>

\pi |\lambda_B| $ and $\rho(\alpha)= \frac{1}{2 \pi |\lambda_B|}$ for $|\alpha|< \pi |\lambda_B|$. } However the distribution will deviate from this universal form away from the large volume limit, giving rise to a rich phase structure with many interesting phase transitions (generalizing the analysis of~\cite{Jain-ml-2013py}). It should be possible to generalize the computations presented in this paper to the study of the partition functions of the regular boson --- critical fermion duality (see e.g.~\cite{Minwalla-ml-2015sca}) and of theories with with both a bosonic and a fermionic field (\cite{Jain-ml-2013gza, xyz}). It would also be interesting to use the techniques of this paper to generalize the S-matrix computations of~\cite{Jain-ml-2014nza,Dandekar-ml-2014era,Inbasekar-ml-2015tsa,Yokoyama-ml-2016sbx,Inbasekar-ml-2017ieo,Inbasekar-ml-2017sqp} to evaluate the bosonic S-matrices in the Higgsed phase, and to match the final results with the fermionic S-matrices as predicted by duality. The techniques of this paper could also permit the computation of the quantum effective action of the scalar theories as a function of the gauge covariant field $\phi^a$ (in a suitable gauge). This computation could prove useful in analysing the vacuum stability of these theories. We hope to turn to several of these issues in the near future. ]]>

|\nu|$. It follows that $\cS$ is also an analytic function of $\nu$ for $|c_B| > |\nu|$. At $|c_B|=\nu$, on the other hand, the arguments of one of the two logarithms in this equation goes to zero at $\alpha=0$. For $|\nu|>|c_B|$, the contour integral passes through the cut of the logarithm. These observations suggest that $\cS(|c_B|, \nu)$ --- viewed as a function of $\nu$ at fixed $|c_B|$ --- might well be non-analytic at $\nu=\pm |c_B|$. Equation~\eqref{dualofq1} --- together with the fact that $\cC$ is analytic at $\nu=\pm |c_B|$ --- tells us that this is indeed the case. Indeed the function $\cS$ must have precisely the singularity needed to cancel that of the function $\frac{\sgn(\lambda_B)}{2} {\rm max}(|c_B|, |\nu|) $ on the r.h.s.\ of the second of~\eqref{dualofq1}. The discussion of the last paragraph motivates us to define the analytic function ${\tilde \cS}$ \begin{equation}\label{deftcs} {\tilde \cS}= \left\{\begin{array}{cc} \cS & {{\rm when} ~~|\nu|<|c_B|}\\ \cS - \frac{1}{2 |\lambda_B|} (|\nu|-|c_B|) & {{\rm when} ~~|\nu|>|c_B|}\end{array}\right.\ . \end{equation} When expressed in terms of ${\tilde \cS}$ the relations between $\cS$ and $\cC$ in~\eqref{dualofq1} become the single relation \begin{equation}\label{ccm} \lambda_F \cC=\lambda_B {\tilde \cS}- \frac{\sgn(\lambda_B)}{2}|c_B|\ . \end{equation} Roughly speaking ${\tilde \cS}$ can be thought of as being defined by the same integral as that for $\cS$ in the second of~\eqref{ss} except that one is instructed to perform the integral over a contour that is deformed to avoid cutting the branch cut of the logarithmic functions. It follows from~\eqref{ccm} that under duality the quantity $X_F = 2\lambda_F \cC + \hat{m}_F^{\text{reg}}$ defined in~\eqref{XFXC} maps to $X_B$ where \begin{equation}\label{dbe} X_B= 2 \lambda_B {\tilde \cS} -\lambda_B \hat{m}_B^{\text{cri}} -{\rm sgn}( \lambda_B) |c_B|\ . \end{equation} Notice that on-shell (i.e.\ on a solution to the bosonic gap equations) \begin{equation}\label{xb} X_B= -{\rm sgn}(\lambda_B)~{\rm max}(|c_B|, |\nu|)\,, ~~~ {\rm so ~~that }~~~-\lambda_B X_B \geq 0\ . \end{equation} In other words all solutions to the bosonic gap equations have $\lambda_B X_B\leq 0$ i.e.\ $\lambda_F X_F \geq 0$. It follows that any solution of the fermionic gap equations that violates this inequality does not have a bosonic dual. We will now see how this works in more detail. Inserting~\eqref{ccm} into the fermionic gap equation~\eqref{mdrf} we obtain \begin{equation}\label{mdrff} |c_B|=\sgn(X_B) \left( 2\lambda_B {\tilde \cS}- ~{\rm sgn} (\lambda_B)|c_B| -\lambda_B \hat{m}_B^{\text{cri}}\right), \end{equation} Equivalently \begin{equation}\label{mdrfnn} |c_B|\left( 1+{\rm sgn} (\lambda_B) \sgn(X_B) \right) =\sgn(X_B) \left( 2\lambda_B {\tilde \cS} -\lambda_B \hat{m}_B^{\text{cri}}\right), \end{equation} Let us first suppose that ${\rm sgn} (\lambda_B) \sgn(X_B)=-1$. In this case~\eqref{mdrfnn} reduces to the equation \begin{equation}\label{mbe} 2{\tilde \cS} = \hat{m}_B^{\text{cri}} \end{equation} This equation matches perfectly with~\eqref{mdcb} when $|c_B|>|\nu|$.\footnote{Eqs.~\eqref{mbe} and~\eqref{mdcb} differ when $|\nu|>|c_B|$, because, in this regime, ${\tilde \cS}$ differs from $\cS$. However the difference between the two equations is quite minor --- as we have explained above ${\tilde \cS}$ and $\cS$ are defined by the same integrals but over slightly different contours. It is possible that the derivation of~\eqref{mdcb} has a subtlety when $|\nu|>|c_B|$ and the correct equation picks out the contour that changes $\cS$ to ${\tilde \cS}$. We leave an exploration of this to future work.} On the other hand when ${\rm sgn} (\lambda_B) \sgn(X_B)= +1$,~\eqref{mdrfnn} becomes \begin{equation}\label{mdrfoo} 2|c_B| =\left( 2|\lambda_B| {\tilde \cS} -|\lambda_B| \hat{m}_B^{\text{cri}}\right)\ . \end{equation} This is a completely new bosonic gap equation that --- at least superficially --- seems different from the bosonic gap equation~\eqref{mdcb}. It has been speculated that this equations governs the dynamics of the critical boson theory in its Higgsed phase. In the rest of this paper we demonstrate that this is indeed the case by directly deriving~\eqref{mdrfoo} from an analysis of the bosonic theory. We can also use the fermionic free energy (the second of~\eqref{offshellfe}) together with the duality map to obtain a prediction for the free energy in Higgsed phase (we focus on the case for boson for $|c_B|>|\nu|$; when $|c_B|<|\nu|$ there is a potential subtlety as in the unHiggsed~phase). \noindent For later use we present our results in terms of a quantity \begin{equation}\label{convn} m = - \frac{\lambda_B m_B^{\text{cri}}}{2} \implies |m|= - \frac{|\lambda_B| m_B^{\text{cri}}}{2}\,, \end{equation} (and correspondingly for the dimensionless hatted quantities.) Note that in the phase under consideration $m_B^{\text{cri}}<0$. The quantity $|m|$ would then correspond to the mass of the $W$ boson in this phase. Using~\eqref{mdrfoo} we find \begin{equation}\label{cFtoB} |c_B| = |\lambda_B| \cS + |\hat{m}|\ . \end{equation} Substituting~\eqref{cFtoB} in~\eqref{offshellfe}, we have (dropping zero temperature contributions) \begin{align*} v_F &=\frac{N_F}{6\pi} {\Bigg[} |c_F|^3 \frac{\left(|\lambda_F| + 1\right)}{|\lambda_F|} -\frac{3}{2|\lambda_F|} |\hat{m}_F^{\text{reg}}| c_F^2 -\\ &\quad-3 \int_{-\pi}^{\pi} \rho_F(\alpha) d\alpha\int_{|c_F|}^{\infty}dy y\left(\log\left(1+e^{-y-i\alpha -\nu}\right)+\log\left(1+e^{-y+i\alpha + \nu}\right) \right) {\Bigg]}, \\ &=\frac{N_B}{6\pi} {\Bigg[} |c_B|^3 \frac{\left(2 - |\lambda_B|\right)}{|\lambda_B|} -\frac{3}{ |\lambda_B|} |\hat{m}| c_B^2 + \\ &\quad +3 \int_{-\pi}^{\pi} \rho_B(\alpha) d\alpha\int_{|c_B|}^{\infty}dy y\left(\log\left(1+e^{-y-i\alpha -\nu}\right)+\log\left(1+e^{-y+i\alpha + \nu}\right) \right) {\Bigg]}\,,\\ &=\frac{N_B}{6\pi} {\Bigg[} \frac{2 |c_B|^3 - 3|\hat{m}| c_B^2}{|\lambda_B|} + \nonumber\\ &\quad - |c_B|^3 + 3 \int_{-\pi}^{\pi} \rho_B(\alpha) d\alpha\int_{|c_B|}^{\infty}dy y\left(\log\left(1+e^{-y-i\alpha -\nu}\right)+\log\left(1+e^{-y+i\alpha + \nu}\right) \right) {\Bigg]}\nonumber\\ &=\frac{N_B}{6\pi} {\Bigg[}-\frac{|\hat{m}|^3}{|\lambda_B| }+3|\lambda_B| |\hat{m}| \cS^2+2|\lambda_B| ^2 \cS^3 \nonumber\\ &\quad + 3 \int_{-\pi}^{\pi} \rho_B(\alpha) d\alpha\int_{|c_B|}^{\infty}dy y\left(\log\left(1+e^{-y-i\alpha -\nu}\right)+\log\left(1+e^{-y+i\alpha + \nu}\right)\right) -|c_B|^3 {\Bigg]}\,, \end{align*} where we have used the following duality maps in the first step: \begin{equation} \begin{split} \!\!\hat{m}_F^{\text{reg}} = 2 \hat{m}\,,\quad \frac{N_F}{|\lambda_F|} = \frac{N_B}{|\lambda_B|}, \quad |\lambda_F| = 1\!-\!|\lambda_B|\,,\quad |\lambda_F| \rho_F(\alpha) = \frac{1}{2\pi} - |\lambda_B| \rho(\pi - \alpha)\,. \end{split} \end{equation} In the second and third steps, we have rearranged terms in the expression in order to put it in a form which will match term by term with the free energy obtained by direct calculation in the Higgsed phase. The first term in the bracket is a zero temperature contribution and can be ignored in this context. ]]>