Free energy and defect $C$-theorem in free fermion

In this section, we summarise coordinate systems for a sphere $\mathbb{S}^{d}$, a hemisphere $\mathbb{HS}^{d}$, a hyperbolic space $\mathbb{H}^{d}$ and $\mathbb{H}^{p+1}\times\mathbb{S}^{q-1}$ and conformal maps among a flat space $\mathbb{R}^{d}$ and them.

Let us consider DCFT_d on $\mathbb{R}^{d}$,

$\displaystyle\mathrm{d}s^{2}=\mathrm{d}x_{a}^{2}+\mathrm{d}y_{i}^{2}\,,\qquad(a=1,\cdots,p,i=p+1,\cdots,d)$

(9)

where a $p$-dimensional defect sits at $y_{i}=0$. For later convenience, we introduce a codimension of the defect,

$\displaystyle q=d-p\,.$

(10)

By using the polar coordinate for the $y_{i}$-coordinate,

$\displaystyle\mathrm{d}y_{i}^{2}=\mathrm{d}z^{2}+z^{2}\mathrm{d}s_{\mathbb{S}^{q-1}}^{2}\,,$

(11)

the metric of the flat space becomes

$\displaystyle\begin{aligned} \mathrm{d}s^{2}&=\mathrm{d}x_{a}^{2}+\mathrm{d}z^{2}+z^{2}\mathrm{d}s_{\mathbb{S}^{q-1}}^{2}\\ &=z^{2}\left(\frac{\mathrm{d}x_{a}^{2}+\mathrm{d}z^{2}}{z^{2}}+\mathrm{d}s_{\mathbb{S}^{q-1}}^{2}\right)\,.\end{aligned}$

(12)

After a Weyl rescaling, the metric (12) reduces to a geometry $\mathbb{H}^{p+1}\times\mathbb{S}^{q-1}$ with radius $R$,

$\displaystyle\mathrm{d}s^{2}=R^{2}\left(\frac{\mathrm{d}x_{a}^{2}+\mathrm{d}z^{2}}{z^{2}}+\mathrm{d}s_{\mathbb{S}^{q-1}}^{2}\right)\,.$

(13)

Now the defect sits at the boundary of $\mathbb{H}^{p+1}$. We can also use the global coordinate for $\mathbb{H}^{p+1}$,

$\displaystyle\mathrm{d}s^{2}=R^{2}\left(\mathrm{d}\rho^{2}+\sinh^{2}\rho\,\mathrm{d}s_{\mathbb{S}^{p}}^{2}+\,\mathrm{d}s_{\mathbb{S}^{q-1}}^{2}\right)\,,$

(14)

where the defect sits at $\rho=\infty$. Introducing a new variable $\varphi$, $\tan\varphi=\sinh\rho$, the metric (14) becomes

$\displaystyle\mathrm{d}s^{2}=\frac{R^{2}}{\cos^{2}\varphi}\left(\mathrm{d}\varphi^{2}+\sin^{2}\varphi\,\mathrm{d}s_{\mathbb{S}^{p}}^{2}+\cos^{2}\varphi\,\mathrm{d}s_{\mathbb{S}^{q-1}}^{2}\right)\,.$

(15)

After a Weyl rescaling, the metric (15) can be mapped to the sphere metric with radius $R$

$\displaystyle\mathrm{d}s^{2}=R^{2}\left(\mathrm{d}\varphi^{2}+\sin^{2}\varphi\,\mathrm{d}s_{\mathbb{S}^{p}}^{2}+\cos^{2}\varphi\,\mathrm{d}s_{\mathbb{S}^{q-1}}^{2}\right)\,,$

(16)

where $0\leq\varphi<\pi$ and the defect sits at $\varphi=\pi/2$. The hemisphere $\mathbb{HS}^{d}$ has the same metric of the sphere (16). A different point is that the range of $\varphi$ is $0\leq\varphi\leq\pi/2$ and the boundary sits at $\varphi=\pi/2$.

In this section, we classify allowed boundary conditions of a massless fermion on $\mathbb{H}^{p+1}\times\mathbb{S}^{q-1}$. A free fermion on a curved background which is conformally equivalent to $\mathbb{R}^{d}$ is studied in e.g. Camporesi:1992tm ; Camporesi:1995fb ; Lewkowycz:2012qr ; Herzog:2019bom ; Klebanov:2011uf . See Chester:2015wao for the notation of a fermion on $\mathbb{H}^{2}\times\mathbb{S}^{2}$. The action of a massive Dirac fermion is given by³³3The massive fermion is not conformal. However, we introduce a mass term for later convenience.

$\displaystyle I=\int\!\mathrm{d}^{d}x\sqrt{g}\,\left(\textrm{i}\,\psi^{\dagger}\,\Gamma^{a}\nabla_{a}\psi+M\psi^{\dagger}\psi\right)\,,$

(17)

where we assume $M\geq 0$. The rank of the gamma matrix for $d\geq 2$ is

$\displaystyle r_{d}=2^{\lfloor\frac{d}{2}\rfloor}\,,$

(18)

and the gamma matrix satisfies the anti-commutation relation,

$\displaystyle\{\Gamma^{a},\Gamma^{b}\}=2\delta^{ab}\mathds{1}\,.$

(19)

The covariant derivative $\nabla_{a}$ and the spin connection $\omega_{\mu bc}$ are defined as

$\displaystyle\begin{aligned} \nabla_{a}&=e_{a}^{\mu}\nabla_{\mu}\,,\qquad\nabla_{\mu}=\partial_{\mu}+\frac{1}{2}\sigma^{bc}\omega_{\mu bc}\,,\\ \sigma^{ab}&=\frac{1}{4}[\Gamma^{a},\Gamma^{b}]\,,\qquad\omega_{\mu bc}=e_{b}^{\nu}(\partial_{\mu}e_{c\nu}-\Gamma_{\nu\mu}^{\alpha}e_{c\alpha})\end{aligned}$

(20)

using a frame field $e_{a}^{\mu}$, which satisfies

$\displaystyle e_{a}^{\mu}e_{b}^{\nu}g_{\mu\nu}=\delta_{ab}\,.$

(21)

The covariant derivative satisfies a relation,

$\displaystyle(\Gamma^{a}\nabla_{a})^{2}=\nabla^{2}-\frac{1}{4}\mathcal{R}\,,$

(22)

where $\mathcal{R}$ is a Ricci scalar.

For $\mathbb{H}^{d}$, a solution of the equation of motion for the massive fermion behaves

$\displaystyle\psi\sim z^{\Delta_{\pm}}$

(23)

near the boundary at $z=0$, where $\Delta_{\pm}$ is given by

$\displaystyle\Delta_{\pm}-\frac{d-1}{2}=\pm MR\,.$

(24)

Then, $\Delta_{\pm}$ become degenerate in the massless limit, and the allowed asymptotic behaviour is unique for a massless fermion.

For the product space $\mathbb{H}^{p+1}\times\mathbb{S}^{q-1}$, we consider a massless fermion from the beginning. We first decompose the fermionic field as

$\displaystyle\psi(z,x,\theta)=\sum_{\ell}\psi_{\mathbb{H}^{p+1}}(z,x)\otimes\psi_{\ell,\mathbb{S}^{q-1}}(\theta)\,,$

(25)

and the gamma matrices are also decomposed similarly. For even $p$ and even $q\geq 4$, the decomposition of the fermion (25) has an additional $U(1)$ as is clear from that the rank of the spinor representation are different in both sides of (25).⁴⁴4We are indebted to D. Rodriguez-Gomez and J. G. Russo for the decomposition of the fermion. Mathematically, this is equivalent to a decomposition of a representation of $SO(\frac{p+q}{2})$ to that of $SO(\frac{p}{2})\times SO(\frac{q-2}{2})\times U(1)$. The spherical part $\psi_{\ell,\mathbb{S}^{q-1}}(\theta)$ satisfies the equation,

$\displaystyle(\Gamma\cdot\nabla)_{\mathbb{S}^{q-1}}\psi_{\ell,\mathbb{S}^{q-1}}(\theta)=\pm\,\textrm{i}\,\frac{\ell+\frac{q-1}{2}}{R}\psi_{\ell,\mathbb{S}^{q-1}}(\theta)\,,$

(26)

where we decomposed the covariant derivative on $\mathbb{H}^{p+1}\times\mathbb{S}^{q-1}$ into the covariant derivatives on $\mathbb{H}^{p+1}$ and $\mathbb{S}^{q-1}$ appropriately. Then, the equation of motion reduces to

$\displaystyle\left(\textrm{i}\,(\Gamma\cdot\nabla)_{\mathbb{H}^{p+1}}\pm\frac{\ell+\frac{q-1}{2}}{R}\right)\psi_{\mathbb{H}^{p+1}}(z,x)=0\,.$

(27)

The solution of this equation of motion behaves as

$\displaystyle\psi_{\mathbb{H}^{p+1}}(z,x)\sim z^{\Delta_{\pm}^{\ell}}$

(28)

near the boundary, $z=0$, and $\Delta_{\pm}^{\ell}$ is given by

$\displaystyle\Delta_{\pm}^{\ell}=\frac{p}{2}\pm\left(\ell+\frac{q-1}{2}\right)\,.$

(29)

The parameter $\Delta_{\pm}^{\ell}$ can be understood as the conformal dimensions of operators localising on a $p$-dimensional conformal defect at the boundary of $\mathbb{H}^{p+1}$. Then not all the operators with conformal dimensions (29) are allowed to exist due to the unitarity bound in $p$ dimensions Minwalla:1997ka :

$\displaystyle\Delta\geq\frac{p-1}{2}\,.$

(30)

$\Delta_{+}^{\ell}$ is always above the unitarity bound, while $\Delta_{-}^{\ell}$ is not necessary to satisfy the unitarity bound unless

$\displaystyle\ell\leq 1-\frac{q}{2}\ .$

(31)

Hence the mode with $\ell=0$ for $q=2$ is allowed to have the boundary conditions corresponding to $\Delta_{-}^{\ell}$.

The allowed boundary conditions for the massless fermion are classified as follows and are listed in table 1.

For a massless fermion on $\mathbb{S}^{d}$, the free energy is given by⁵⁵5As noted in Nishioka:2021uef ; Monin:2016bwf , there is an ambiguity to decompose the logarithmic function into two parts. We require two conditions to remove the ambiguity: the free energy in the Schwinger representation should be convergent and the resulting zeta function does not depend on the cutoff scale $\tilde{\Lambda}$. In (35), the two decomposed logarithmic functions are the same and the free energy in the Schwinger representation (40) is convergent in the $\ell\to\infty$ limit.

$\displaystyle\begin{aligned} F[\mathbb{S}^{d}]&=-\frac{1}{2}\text{tr}\log\left[-\tilde{\Lambda}^{-2}(\Gamma^{a}\nabla_{a})^{2}\right]\\ &=-\sum_{\ell=0}^{\infty}g^{(d)}(\ell)\,\log\left(\frac{\nu_{\ell}^{(d)}}{\tilde{\Lambda}R}\right)\,,\end{aligned}$

(35)

where we use the equation,

$\displaystyle\Gamma^{a}\nabla_{a}\psi_{\ell}=\pm\,\textrm{i}\,\frac{\nu_{\ell}^{(d)}}{R}\psi_{\ell}$

(36)

with the eigenvalues,

$\displaystyle\nu_{\ell}^{(d)}=\ell+\frac{d}{2}\,,\qquad\ell=0,1,2,\cdots\,.$

(37)

The degeneracy for each sign is given by

$\displaystyle g_{\pm}^{(d)}(\ell)=\frac{r_{d}\Gamma(\ell+d)}{\Gamma(d)\Gamma(\ell+1)}\,,$

(38)

where $r_{d}$ (18) is the rank of the gamma matrix as before. For later convenience, we introduce a notation

$\displaystyle g^{(d)}(\ell)=2g_{\pm}^{(d)}(\ell)=\frac{2r_{d}\Gamma(\ell+d)}{\Gamma(d)\Gamma(\ell+1)}\,.$

(39)

We write the free energy (35) in the Schwinger representation,

$\displaystyle F[\mathbb{S}^{d}]=\int_{0}^{\infty}\,\frac{\mathrm{d}t}{t}\,\sum_{\ell=0}^{\infty}g^{(d)}(\ell)\,\mathrm{e}^{-t\nu_{\ell}^{(d)}/(\tilde{\Lambda}R)}\,,$

(40)

and we introduce the regularised free energy Vassilevich:2003xt to remove the divergence in the integral,

$\displaystyle\begin{aligned} F_{s}[\mathbb{S}^{d}]&=\int_{0}^{\infty}\,\frac{\mathrm{d}t}{t^{1-s}}\,\sum_{\ell=0}^{\infty}g^{(d)}(\ell)\,\mathrm{e}^{-t\nu_{\ell}^{(d)}/(\tilde{\Lambda}R)}\\ &=\frac{1}{2}(\tilde{\Lambda}R)^{s}\,\Gamma(s)\,\zeta_{\mathbb{S}^{d}}(s)\,,\end{aligned}$

(41)

where the zeta function $\zeta_{\mathbb{S}^{d}}(s)$ is defined by

$\displaystyle\zeta_{\mathbb{S}^{d}}(s)\equiv 2\sum_{\ell=0}^{\infty}g^{(d)}(\ell)\,\left(\nu_{\ell}^{(d)}\right)^{-s}\ .$

(42)

Then the (unregularised) free energy is obtained in the $s\to 0$ limit:

$\displaystyle F_{s}[\mathbb{S}^{d}]=\frac{1}{2}\left(\frac{1}{s}-\gamma_{\text{E}}+\log(\tilde{\Lambda}R)\right)\,\zeta_{\mathbb{S}^{d}}(0)+\frac{1}{2}\partial_{s}\zeta_{\mathbb{S}^{d}}(0)+\mathcal{O}(s)\ ,$

(43)

which is divergent due to the pole at $s=0$. Here $\gamma_{\text{E}}$ is the Euler constant. By removing the pole, the remaining part becomes the renormalized free energy

$\displaystyle F_{\text{ren}}[\mathbb{S}^{d}]\equiv\frac{1}{2}\partial_{s}\zeta_{\mathbb{S}^{d}}(0)+\frac{1}{2}\log(\Lambda R)\,\zeta_{\mathbb{S}^{d}}(0)\ ,$

(44)

where $\Lambda=\mathrm{e}^{-\gamma_{\text{E}}}\,\tilde{\Lambda}$.

To compute the zeta function, we expand the gamma functions in the degeneracy (39),

$\displaystyle\frac{\Gamma\left(\nu_{\ell}^{(d)}+\frac{d}{2}\right)}{\Gamma\left(\nu_{\ell}^{(d)}-\frac{d}{2}+1\right)}=\begin{dcases}\sum_{n=0}^{\frac{d}{2}}(-1)^{\frac{d}{2}+n}\,\alpha_{n,d+1}\,\left(\nu_{\ell}^{(d)}\right)^{2n-1}&d:\text{even}\,,\\ \sum_{n=0}^{\frac{d-1}{2}}(-1)^{\frac{d-1}{2}+n}\,\beta_{n,d+1}\,\left(\nu_{\ell}^{(d)}\right)^{2n}&d:\text{odd}\,,\end{dcases}$

(45)

where we used the awkward suffix for $\alpha_{n,d+1}$ and $\beta_{n,d+1}$ to use the same notation in the scalar case Nishioka:2021uef . Since $\alpha_{0,d+1}=0$, we can omit the $n=0$ term in the summation for even $d$ whenever we want. Using the asymptotic expansion ((5.11.14) in NIST:DLMF )

$\displaystyle\frac{\Gamma(x+a)}{\Gamma(x+b)}=\sum_{k=0}^{\infty}\,\left(x+\frac{a+b-1}{2}\right)^{a-b-2k}\,\binom{a-b}{2k}\,B_{2k}^{(a-b+1)}\left(\frac{a-b+1}{2}\right)\ ,$

(46)

where $B_{k}^{(m)}(x)$ is the generalized Bernoulli polynomial, and comparing both sides, we find

	$\displaystyle\text{Even }d:\qquad\alpha_{n,d+1}$	$\displaystyle=(-1)^{\frac{d}{2}+n}\,\binom{d-1}{d-2n}\,B^{(d)}_{d-2n}\left(\frac{d}{2}\right)\,,$		(47)
	$\displaystyle\text{Odd }d:\qquad\beta_{n,d+1}$	$\displaystyle=(-1)^{\frac{d-1}{2}+n}\,\binom{d-1}{d-1-2n}\,B^{(d)}_{d-1-2n}\left(\frac{d}{2}\right)\,.$		(48)

Next, let us consider the free energy on the hemisphere. At the boundary of $\mathbb{HS}^{d}$, a mixed boundary condition is imposed Dowker:1995sw ,

$\displaystyle P_{+}\psi=0\,.$

(61)

Here $P_{+}$ is a projection operator,

$\displaystyle P_{+}=\frac{1}{2}\left(1-\textrm{i}\,\Gamma^{\ast}\Gamma^{a}e_{a}^{\mu}n_{\mu}\right)$

(62)

with a chirality matrix $\Gamma^{\ast}$ and an incoming normal vector $n_{\mu}$. See appendix A in Dowker:1995sw for the detail of a construction of the chirality matrix $\Gamma^{\ast}$. The mixed boundary condition preserves a conformal symmetry.

A degeneracy with the mixed boundary condition is given by (38) which is just a half of the degeneracy of $\mathbb{S}^{d}$ (39). Then, the free energy on $\mathbb{HS}^{d}$ is nothing but half of that on $\mathbb{S}^{d}$,

$\displaystyle F_{\text{ren}}[\mathbb{HS}^{d}]=\frac{1}{2}F_{\text{ren}}[\mathbb{S}^{d}]\,.$

(63)

This formula correctly reproduces the known results Rodriguez-Gomez:2017aca .⁶⁶6In Rodriguez-Gomez:2017aca , anomaly coefficients of a ball $\mathbb{B}^{d}$ which is conformally equivalent to $\mathbb{H}^{d}$, is obtained.

In this section, we extend a computation of the zeta function for a massive fermion on $\mathbb{H}^{d}$ for $d=3,4$ in Bytsenko:1994bc to general dimensions.

The free energy of the massive Dirac fermion with mass $M=m/R$ is given by

$\displaystyle\begin{aligned} F[\mathbb{H}^{d}](m)&=-\frac{1}{2}\text{tr}\log\left[-\tilde{\Lambda}^{-2}\left((\Gamma^{a}\nabla_{a})^{2}-M^{2}\right)\right]\\ &=-\frac{1}{2}\int_{0}^{\infty}\!\mathrm{d}\omega\,\mu^{(d)}(\omega)\left[\log\left(\frac{\omega+\textrm{i}\,m}{\tilde{\Lambda}R}\right)+\log\left(\frac{\omega-\textrm{i}\,m}{\tilde{\Lambda}R}\right)\right]\,,\end{aligned}$

(64)

where $\tilde{\Lambda}$ is the UV cutoff scale introduced to make the integral dimensionless. The parameter $\omega$ is an eigenvalue of the equation,

$\displaystyle\Gamma^{a}\nabla_{a}\psi_{\omega}=\textrm{i}\,\frac{\omega}{R}\psi_{\omega}\,,\qquad\omega\geq 0\,,$

(65)

and the Plancherel measure of a fermion on $\mathbb{H}^{d}$ of unit radius takes the form Camporesi:1995fb

$\displaystyle\begin{aligned} \mu^{(d)}(\omega)&=c_{d}\,r_{d}\,\left|\frac{\Gamma\left(\frac{d}{2}+\textrm{i}\,\omega\right)}{\Gamma\left(\frac{1}{2}+\textrm{i}\,\omega\right)}\right|^{2}\\ &=c_{d}\,r_{d}\,\begin{dcases}\prod_{j=\frac{1}{2}}^{\frac{d-2}{2}}(\omega^{2}+j^{2})&d:\text{odd}\,,\\ \omega\,\coth(\pi\omega)\,\prod_{j=1}^{\frac{d-2}{2}}(\omega^{2}+j^{2})&d:\text{even}\,,\end{dcases}\end{aligned}$

(66)

with the coefficient

$\displaystyle c_{d}=\frac{\text{Vol}(\mathbb{H}^{d})}{2^{d-1}\pi^{d/2}\Gamma(d/2)}$

(67)

and the rank of the gamma matrix $r_{d}$ (18). The regularised volume of the hyperbolic space is given by

$\displaystyle\text{Vol}(\mathbb{H}^{d})=\pi^{\frac{d-1}{2}}\Gamma\left(\frac{1-d}{2}\right)=-\frac{\pi^{\frac{d+1}{2}}}{\sin\left(\pi\frac{d-1}{2}\right)\Gamma\left(\frac{d+1}{2}\right)}\,.$

(68)

The hyperbolic volume is finite for even $d$ but divergent for odd $d$ due to the pole of the sine function. By introducing a small cutoff parameter $\epsilon$, the sine function can be replaced by the logarithmic divergence,

$\displaystyle-\frac{1}{\sin\left(\pi\,\frac{d-1}{2}\right)}=\begin{dcases}(-1)^{\frac{d-1}{2}}\frac{2}{\pi}\,\log\left(\frac{R}{\epsilon}\right)&\qquad d:\text{odd}\,,\\ (-1)^{\frac{d}{2}}&\qquad d:\text{even}\,.\end{dcases}$

(69)

Then, the coefficient becomes

$\displaystyle c_{d}=-\frac{1}{\sin\left(\pi\frac{d-1}{2}\right)\Gamma(d)}=\frac{1}{\Gamma(d)}\begin{dcases}(-1)^{\frac{d-1}{2}}\,\frac{2}{\pi}\,\log\left(\frac{R}{\epsilon}\right)&\qquad d:\text{odd}\,,\\ (-1)^{\frac{d}{2}}&\qquad d:\text{even}\,.\end{dcases}$

(70)

Since the Plancherel measure (66) needs to satisfy the square integrability condition, the free energy is defined for $m\geq 0$. That is, equation (64) represents a free energy with a boundary condition, $\Delta=(d-1)/2+m$. We add a mass term in the free energy (64) for the regularisation of the zero mode, and we take a massless limit to obtain free energies of the conformal fermion.

By using the Schwinger representation, the free energy can be written as

$\displaystyle F[\mathbb{H}^{d}](m)=\frac{1}{2}\int_{0}^{\infty}\!\frac{\mathrm{d}t}{t}\int_{0}^{\infty}\!\mathrm{d}\omega\,\mu^{(d)}(\omega)\,\left(\mathrm{e}^{-t(\omega+\textrm{i}\,m)/\tilde{\Lambda}R}+\mathrm{e}^{-t(\omega-\textrm{i}\,m)/\tilde{\Lambda}R}\right)\,.$

(71)

To remove the divergence in the integral, we introduce the regularised free energy

$\displaystyle\begin{aligned} F_{s}[\mathbb{H}^{d}](m)&=\frac{1}{2}\int_{0}^{\infty}\!\frac{\mathrm{d}t}{t^{1-s}}\int_{0}^{\infty}\!\mathrm{d}\omega\,\mu^{(d)}(\omega)\,\left(\mathrm{e}^{-t(\omega+\textrm{i}\,m)/\tilde{\Lambda}R}+\mathrm{e}^{-t(\omega-\textrm{i}\,m)/\tilde{\Lambda}R}\right)\\ &=\frac{1}{2}(\tilde{\Lambda}R)^{s}\Gamma(s)\zeta_{\mathbb{H}^{d}}(s,m)\,,\end{aligned}$

(72)

where the zeta function is defined as

$\displaystyle\zeta_{\mathbb{H}^{d}}(s,m)=\int_{0}^{\infty}\!\mathrm{d}\omega\,\mu^{(d)}(\omega)\,\left((\omega+\textrm{i}\,m)^{-s}+(\omega-\textrm{i}\,m)^{-s}\right)\,.$

(73)

Then, the (unregularised) free energy is obtained in the $s\to 0$ limit,

$\displaystyle F_{s}[\mathbb{H}^{d}](m)=\frac{1}{2}\left(\frac{1}{s}-\gamma_{\text{E}}+\log(\tilde{\Lambda}R)\right)\zeta_{\mathbb{H}^{d}}(s,m)+\frac{1}{2}\partial_{s}\zeta_{\mathbb{H}^{d}}(s,m)+\mathcal{O}(s)\,.$

(74)

By removing the pole at $s=0$, we obtain the renormalized free energy

$\displaystyle F_{\text{ren}}[\mathbb{H}^{d}](m)=\frac{1}{2}\partial_{s}\zeta_{\mathbb{H}^{d}}(0,m)+\frac{1}{2}\log(\Lambda R)\zeta_{\mathbb{H}^{d}}(0,m)\,,$

(75)

where $\Lambda=\mathrm{e}^{-\gamma_{\text{E}}}\tilde{\Lambda}$.