Anomaly inflow of Rarita–Schwinger field in 3 dimensions

Abstract

We study the anomaly inflow of the Rarita–Schwinger field with gauge symmetry in 3 dimensions. We find that global anomalies of the Rarita–Schwinger field are obtained by the spectral flow, which is similar to Witten’s SU(2) global anomaly for a Weyl fermion. The Rarita–Schwinger operator is shown to be a self-adjoint Fredholm operator, and its spectral flow is determined by a path on the set of self-adjoint Fredholm operators with the gap topology. From the spectral equivalence of the spectral flow, we find that the spectral flow of the Rarita–Schwinger operator is equivalent to that of the spin-3/2 Dirac operator. From this fact, we confirm that the anomaly of the three-dimensional Rarita–Schwinger field is captured by the anomaly inflow. Finally, we find that there are no global anomalies of gauge–diffeomorphism transformations on spin manifolds with any gauge group. We also confirm the anomalous phase of the partition function which corresponds to the generator of $\Omega _4^{\textrm{Pin}^+}(pt)=\mathbb {Z}_{16}$ is $\exp (3i\pi /8)$ for the Rarita–Schwinger theory on unorientable $\textrm{Pin}^+$ manifolds without gauge symmetry.

Appendices

A 1.5 order formalism

In this section, we will briefly summarize the 1.5 order formalism. Let us consider a three-dimensional spin manifold. We denote $x^\mu$ the local coordinate, the metric as $g_{\mu \nu }$. We denote $e^a(x)=e^a{}_\mu (x)\partial ^\mu$ the frame field, where $e^a{}_\mu$ is the vielbein. The curvature is defined by using the spin connection $\omega _{\mu ab}$ as

$$\begin{aligned} R_{\mu \nu ab}:=[D_\mu ,D_\nu ]_{ab}=\partial _\mu \omega _{\nu ab}-\partial _\nu \omega _{\mu ab}+\omega _{\mu ac}\omega _\nu {}^c{}_b-\omega _{\nu ac}\omega _\mu {}^c{}_b. \end{aligned}$$

(A.1)

Here, $D_\mu$ is the covariant derivative on the spinor bundle. The torsion is defined by $T^a:=De^a:=de^a+\omega ^a{}_b\wedge e^b=:T_{\mu \nu }{}^adx^\mu \wedge dx^\nu$, where

$$\begin{aligned} T_{\mu \nu }{}^a=\partial _\mu e^a{}_\nu -\partial _\nu e^a{}_\mu +\omega _\mu {}^a{}_b e^b{}_\nu -\omega _\nu {}^a{}_b e^b{}_\mu . \end{aligned}$$

(A.2)

The spin connection is determined by the torsion and vielbein [33]:

$$\begin{aligned} \omega _{\mu ab} =&-\frac{1}{2}e^\nu {}_ae^\rho {}_b(\partial _\mu g_{\nu \rho }+\partial _\nu g_{\mu \rho }-\partial _\rho g_{\mu \nu }) +e^\nu {}_a\partial _\mu e_{b\nu } \nonumber \\&-\frac{1}{2}\left( T_{\mu \nu }{}^ce_{c\rho }-T_{\nu \rho }{}^ce_{c\mu }+T_{\rho \mu }{}^ce_{c\nu }\right) e^\nu {}_ae^\rho {}_b. \end{aligned}$$

(A.3)

Below, we will choose the torsion such that $\delta S^E_\textrm{SUGRA}/\delta \omega _{\mu ab}=0$, where $S^E_\textrm{SUGRA}$ is the Rarita–Schwinger action defined in Eq. (3.2). This formalism is called Palatini formalism or 1.5 order formalism. The condition $\delta S^E_\textrm{SUGRA}/\delta \omega _{\mu ab}=0$ is equivalent to

$$\begin{aligned} -T_{a\rho }{}^\rho e_b^\nu +T_{b\rho }{}^\rho e_a^{\nu } +T_{ab}{}^\nu = \kappa ^2\epsilon ^{cde}\epsilon _{abf} e^\mu _ce^\nu _de^\rho _e\psi ^\dag _\mu \sigma ^f\psi _\rho . \end{aligned}$$

(A.4)

The following choice of the torsion satisfies Eq. (A.4) (see [33]):

$$\begin{aligned} T_{ab}{}^\nu =\kappa ^2( \psi _a^\dag \gamma ^\nu \psi _b -\psi ^\dag _b\gamma ^\nu \psi _a). \end{aligned}$$

(A.5)

Substitute Eq. (A.5) into Eq. (A.3), the corresponding spin connection is obtained as

$$\begin{aligned} \omega _{\mu ab} =&e^\nu {}_a\partial _\mu e_{b\nu } +\frac{\kappa ^2}{2} e_\mu ^c\epsilon _{abc}\epsilon ^{def} (\psi _d^\dag \gamma _e\psi _f). \end{aligned}$$

(A.6)

In this paper, we use this torsion and spin connection. With the spin connection Eq. (A.6), the action Eq. (3.2) is invariant under Eq. (3.4). However, when we choose this spin connection Eq. (A.6), we cannot construct any covariant derivatives on a spinor bundle, because the corresponding covariant derivative cannot be linear.

The covariant derivative $D_\mu$ of the metric $g_{\mu \nu }$ is determined by:

$$\begin{aligned} (D_\mu g)_{\nu \rho }:=\partial _\mu g_{\nu \rho }-\Gamma ^\sigma {}_{\mu \nu }g_{\sigma \rho }-\Gamma ^\sigma {}_{\mu \rho }g_{\nu \sigma }. \end{aligned}$$

(A.7)

Here, $\Gamma ^\sigma {}_{\mu \nu }$ is a connection on the tangent bundle. We choose the Affine connection:

$$\begin{aligned} \Gamma ^\rho {}_{\mu \nu }=\frac{1}{2}g^{\rho \sigma }(\partial _\mu g_{\sigma \nu }+\partial _\nu g_{\mu \sigma }-\partial _\sigma g_{\mu \nu }) +\frac{1}{2}\left( T_{\mu \nu }{}^ae_{a}{}^\rho -T_\nu {}^\rho {}^ae_{a\mu }+T^\rho {}_{\mu }{}^ae_{a\nu }\right) . \end{aligned}$$

(A.8)

Then, we find

$$\begin{aligned} (D_\mu g)_{\nu \rho }=(D_\mu e)^a{}_\nu =0. \end{aligned}$$

(A.9)

B Self-adjoint Fredholm operators and spectral flow

We will review the spectral flow of self-adjoint Fredholm operators. We will first explain the Dirac operator and the Rarita–Schwinger operator are elliptic Fredholm operators, and then define a distance on the space of self-adjoint Fredholm operators and introduce the distance phase on this space. Then, we review an important feature of the spectral flow, which we use in Sect. 3.

Let us consider a vector space V with a complex inner product $\left<\quad ,\quad \right>:V\times V\rightarrow \mathbb {C}$. We say a norm space V is a Hilbert space if there exists an element $v\in V$ such that $v_n\rightarrow v$ for each Cauchy sequence $\{v_n\}_n$. For any vector space V with a complex inner product, there exists a vector space ${\mathcal H}(V)$ with a complex inner product and a norm-preserving map $\tau :V\rightarrow {\mathcal H}(V)$ which satisfies the following conditions:

• (1) ${\mathcal H}(V)$ is Hilbert space.

• (2) $\tau (V)$ is dense in ${\mathcal H}(V)$.

${\mathcal H}(V)$ is unique up to isometry, which is the completion of V. In particular, we denote $L^2(E)$ the completion of $C^\infty (E)$, where $C^\infty (E)$ is the vector space of the smooth sections of a smooth vector bundle $E\rightarrow X$ on X whose inner product is given by the Lebesgue inner product Eq. (3.1). By definition, $L^2(E)$ is a Hilbert space.

An operator T on a Hilbert space ${\mathcal H}$ is Fredholm if $\textrm{Ker}(T)$ and $\textrm{codim}(T)$ are both finite dimensions, where the codimension of an operator T on a Hilbert space ${\mathcal H}$ is defined as

$$\begin{aligned} \textrm{codim}(T):=\textrm{dim}({\mathcal H}/T({\mathcal H})). \end{aligned}$$

(B.1)

For any self-adjoint Fredholm operator ${\mathcal F}$ on a Hilbert space ${\mathcal H}$, the reduced $\eta$-invariant $\eta ({\mathcal F})$ is defined as follows [11]:

$$\begin{aligned} \eta ({\mathcal F}):=&\frac{1}{2}\textrm{lim}_{s\rightarrow 0}\sum _{j,\;\lambda _j\ne 0} \textrm{sign}(\lambda _j)|\lambda _j|^{-2s}+\frac{1}{2}\textrm{dim}\,\textrm{ker}({\mathcal F}). \end{aligned}$$

(B.2)

Here, $\lambda _j\in \mathbb {R}$ are eigenvalues of ${\mathcal F}$, $\textrm{sign}(\lambda )=+1$ ($\lambda >0$) and $\textrm{sign}(\lambda )=-1$ ($\lambda <0$). By the theorem 1.3.2 of [49], all elliptic operators on a compact manifold are Fredholm. To show that the Dirac operator and the Rarita–Schwinger operator are elliptic, we will write down the definition of the d-th order pseudo-partial differential operator and the elliptic operator (see the definition 1.3.3, definition 1.3.5, definition 1.3.9, and definition 1.3.22 of [49]).

• d -th order pseudo-partial differential operator: Let U be a bounded open subset of $\mathbb {R}^m$. Let $p(x,\xi )$ be smooth on $U\times \mathbb {R}^m$ and compact map on U, where $(x,\xi )$ is the coordinate on $U\times \mathbb {R}^m$. Let $S^d(U)$ be the set of smooth maps on $U\times \mathbb {R}^m$ which are compact maps on U that satisfies the following condition for a constant $C(\alpha ,\beta ,p)$:

  $$\begin{aligned} |D^\alpha _xD^\beta _\xi p|\le C(\alpha ,\beta ,p)(1+|\xi |)^{d-|\beta |},\,\forall \alpha :=(\alpha ,\cdots ,\alpha _m),\,\beta :=(\beta _1,\cdots ,\beta _m). \end{aligned}$$

  (B.3)

  Here, $\alpha _j,\beta _j\in \mathbb {N}$, and we use the following notation:

  $$\begin{aligned} |\beta |:=&\,\beta _1+\cdots +\beta _m, \nonumber \\ d_x^\alpha :=&\,\left( \frac{\partial }{\partial x_1}\right) ^{\alpha _1}\cdots \left( \frac{\partial }{\partial x_m}\right) ^{\alpha _m}, \nonumber \\ D_x^\alpha :=&\,(-\sqrt{-1})^{|\alpha |}d_x^\alpha . \end{aligned}$$

  (B.4)

  We find that $S^{d_1}(U)\subset S^{d_2}(U)$ if $d_1<d_2$. For a $p\in S^d(U)$ and $f\in C^\infty _0(U)$, we define the associated pseudo-differential operator P of order d by

  $$\begin{aligned} P:\,&C_0^\infty (U)\rightarrow C^\infty _0(U);\qquad f\mapsto Pf; \nonumber \\ Pf(x):=&\int e^{\sqrt{-1}(x-y)\cdot \xi }p(x,\xi )f(y)dyd\xi . \end{aligned}$$

  (B.5)

  Here, $x,y\in U$ and $\xi \in \mathbb {R}^m$. We denote $C^\infty _0(U)$ as the space of smooth function on U with compact support in U. We denote $\sigma (P):=p$ and call $\sigma (P)$ the symbol of the pseudo-differential operator Pf(x).

• Elliptic Operator: Let $E\rightarrow X$ be a vector bundle on a manifold X and we denote P a pseudo-differential of order d on $C^\infty (V)$ with leading symbol p. We say that P is elliptic if there exists $\epsilon >0$ such that

  $$\begin{aligned}&\sqrt{\left| \left<p(x,\xi )\cdot f,\,p(x,\xi )\cdot f\right>\right| }\nonumber \\ {}&\quad \ge \epsilon \cdot \sqrt{|\left<f,f\right>|}\cdot |\xi |^d\,\textrm{for}\,|\xi |\ge \epsilon ^{-1}\,\textrm{and}\,\forall f\in C^\infty (E). \end{aligned}$$

  (B.6)

  Here, $\left<\quad ,\quad \right>$ is the Lebesgue inner product.⁷

To see that the Rarita–Schwinger operator Eq. (3.8) is elliptic, we use the following theorem:

• Theorem (B1): For an elliptic operator P and a pseudo-differential operator of order $d-1$ on $C^\infty (V)$ q, $P+Q$ is elliptic (see the explanation in the definition 1.3.22 in [49]).

Now let us show the following fact by using theorem (B1):

• Lemma (B2): The Rarita–Schwinger operator defined in Eq. (3.8) and spin-3/2 Dirac operator are both elliptic.

Proof

We will first confirm that the spin-3/2 Dirac operator is elliptic. is a pseudo-differential operator of order-1 whose symbol is

(B.7)

Here, $\Gamma _\mu$ is the Affine connection and $\omega _\mu$ the spin connection. We find that Eq. (B.7) satisfies Eq. (B.6) with $d=1$:

$$\begin{aligned} \left<p_D(x,\xi )\cdot \psi ,p_D(x,\xi )\cdot \psi \right> =&\int d^3x |\xi +A(x)+\omega (x)|^2\psi ^{\mu \dag }\psi _\mu +\Gamma _\nu {}^\mu {}_\rho \Gamma ^\nu {}_{\mu \kappa }\psi ^{\dag \rho }\psi ^\kappa \nonumber \\ \ge&\int d^3x\left\{ |\xi |^2-|A(x)+\omega (x)|^2 -|\Gamma _\nu {}^\mu {}_\rho \Gamma ^\nu {}_{\mu \rho }|\right\} \psi ^{\dag \sigma }\psi _\sigma \nonumber \\ \ge&\left\{ |\xi |^2-|M|^2\right\} \left<\psi ,\psi \right>. \end{aligned}$$

(B.8)

Here, $\left<\quad ,\quad \right>$ is the Lebesgue inner product Eq. (3.1), and $|M|^2$ is the maximal value of $|A(x)+\omega (x)|^2 +|\Gamma _\nu {}^\mu {}_\rho \Gamma ^\nu {}_{\mu \rho }|$ as a function of x. $\psi$ is a section of the tensor product of the tangent bundle and a spinor bundle on a spin manifold X. Therefore, the symbol Eq. (B.7) satisfies Eq. (B.6), and we find is elliptic. In the same way, the spin-1/2 Dirac operator is elliptic. Next, we will consider the Rarita–Schwinger operator. The Rarita–Schwinger operator ${\mathcal R}$ defined in Eq. (3.8) can be written as , where

$$\begin{aligned} (Q\psi )^\mu :=-\gamma ^\mu \gamma ^\nu \psi ^\rho (i\partial _\nu \gamma _\rho ). \end{aligned}$$

(B.9)

This operator Q is a pseudo-differential operator and the corresponding symbol is $\sigma (Q):=Q$ of order zero. Therefore, by using theorem (B1), we find that the Rarita–Schwinger operator Eq. (3.8) is elliptic. $\square$

Let us introduce a phase on the set of the self-adjoint Fredholm operators. Consider a manifold X and a vector bundle $E\rightarrow X$ on X. We define

$$\begin{aligned} {\mathcal CF}^{sa}(E):=\{T:\textrm{closed}\,\textrm{densely}\,\textrm{defined}\,\mathrm{self-}\,\textrm{adjoint}\,\textrm{Fredholm}\,\textrm{operator}\,\textrm{on}\,L^2(E)\}. \end{aligned}$$

(B.10)

We said an operator $T:L^2(E)\rightarrow L^2(E)$ is densely defined if and only if the domain of the operator D(T) is a dense subset of $L^2(E)$, and we said an operator $T:L^2(E)\rightarrow L^2(E)$ is closed if and only if for any arbitrary converges low $\{x_n\}_{n\in \mathbb {N}}\subset D(T)$ that satisfies $x_n\rightarrow x\in L^2(E)$ ($n\rightarrow \infty$) and $Tx_n\rightarrow y\in L^2(E)$, $x\in D(T)$ and $Tx=y$. We define the gap metric on ${\mathcal CF}^{sa}(E)$ (see section 1.1 in [32]), and the gap metric induces the following distance function on ${\mathcal CF}^{sa}(E)$ (see Theorem 1.1 in [32]):

$$\begin{aligned} d(T_1,T_2):=||\kappa (T_1)-\kappa (T_2)||, \end{aligned}$$

(B.11)

where the Cayley transform $\kappa :{\mathcal CF}^{sa}(E)\rightarrow \{U:\textrm{unitary}|U-I\,\textrm{is}\,\textrm{injective}\}$ is defined as

$$\begin{aligned} \kappa (T):=(T-i)(T+i)^{-1}=I-2i(T+i)^{-1}. \end{aligned}$$

(B.12)

The image $\kappa (T)$ is a bounded operator [50], which means that

$$\begin{aligned} ||\kappa (T)u||\le c||u||,\qquad u\in L^2(E). \end{aligned}$$

(B.13)

for a constant $c>0$. Then, we define the resolvent norm [50]:

$$\begin{aligned} ||\kappa (T)||:=\textrm{sup}_{u\ne 0}\frac{||\kappa (T)u||}{||u||}. \end{aligned}$$

(B.14)

Then, we can define a distance phase ${\mathcal O}$ on ${\mathcal CF}^{sa}(E)$ by this distance function Eq. (B.11):

$$\begin{aligned} {\mathcal O}(E)&:=\{U_\epsilon (T)\subset {\mathcal CF}^{sa}(E)|\,T\in {\mathcal CF}^{sa}(E),\,U_\epsilon (T)\nonumber \\ {}&:=\{T'\in {\mathcal CF}^{sa}(E)|\;d(T,T')\le \epsilon \}\}. \end{aligned}$$

(B.15)

With this gap phase, the set ${\mathcal CF}^{sa}(E)$ is path connected (see Theorem 1.10 of [32]). The spectral flow is as follows (see definition 2.12 and Lemma 2.9 of [32]):

• Let $f:[0,1]\rightarrow {\mathcal CF}^{sa}(E)$ be a continuous path. By the proposition 2.10 in [32], we can choose a partition, $\{0=t_0<t_1<\cdots <t_n=1\}$ of the interval [0, 1] and a real number $\epsilon _j>0$, $j=1,\cdots , n$ such that for each $j=1,\cdots ,n$ the following map is continuous and finite rank on the interval $[t_{j-1},t_j]$:

  $$\begin{aligned} E_j:\,[t_{j-1},t_j]\rightarrow \{\textrm{Bounded}\,\textrm{linear}\,\textrm{operators}\};\qquad t\mapsto \frac{1}{2\pi i}\int _{\Gamma _j}(\lambda -f(t))^{-1}d\lambda . \end{aligned}$$

  (B.16)

  Here, we denote $\Gamma _j$ the circle of radius $(\epsilon _j-\epsilon _{j-1})$ and center $(\epsilon _j+\epsilon _{j-1})/2$. Then, we define the spectral flow of $f:[0,1]\rightarrow {\mathcal CF}^{sa}(E)$ to be

  $$\begin{aligned} \textrm{SF}(f):=\sum _{j=1}^n\left\{ \textrm{dim}(E_j^\ge (t_j))-\textrm{dim}(E_j^{\ge }(t_{j-1}))\right\} . \end{aligned}$$

  (B.17)

  Here, we denote $E_j^\ge (t)$ the space of positive eigenmodes of the operator $E_j(t)$.

By proposition 2.13 in [32], this definition of the spectral flow does not depend on the partition $\{0=t_0<t_1<\cdots <t_n=1\}$ of the interval [0, 1] and a real number $\epsilon _j>0$, $j=1,\cdots , n$. The spectral flow only depends on the map $f:[0,1]\rightarrow {\mathcal CF}^{sa}(E)$.

• Lemma (B3): The spectral flow is homotopy equivalence (see prop 2.16 in [32]).

We will finally explain the change of the reduced $\eta$-invariant of a Fredholm operator by a gauge–diffeomorphism transformation captured by the spectral flow. By the theorem 1.10 of [32], the space ${\mathcal CF}^{sa}(E)$ is a path connected with the gap phase. Therefore, there exists at least one path that connects two different points on ${\mathcal CF}^{sa}(E)$. We consider a continuous path $f:[0,1]\rightarrow {\mathcal CF}^{sa}(E)$ that connect two different elliptic self-adjoint operators ${\mathcal F}_1$ and ${\mathcal F}_2$ that have the same spectral. If we can choose the path f on the set of elliptic self-adjoint operators, we obtain the difference of the reduced $\eta$-invariant of these operators as follows (see proposition 2.12, eq.(7.1), p.79, and p.94 of [11], where the reduced $\eta$-invariant is dented as $\xi$):

$$\begin{aligned} \eta ({\mathcal F}_2)-\eta ({\mathcal F}_1)=\textrm{SF}(f). \end{aligned}$$

(B.18)

Here, $\textrm{SF}(f)$ is the spectral flow of f defined in Eq. (B.17).