KEK-TH-2557
Effective Brane Field Theory with
Higher-form Symmetry

We discuss how to construct the brane field $\psi[C_{p}]$. We consider a $d$-dimensional spacetime manifold $\Sigma_{d}$ with the metric $g_{\mu\nu}$. We employ the Minkowski metric signature for $d$-spacetime dimensions as $(-,+,+,\cdots)$. $C_{p}$ is a subspace in $\Sigma_{d}$, which can be expressed by an embedding function $S^{p}\to\Sigma_{d}$, i.e., $\{X^{\mu}(\xi)\}_{\mu=0}^{d-1}$, where $S^{p}$ is a $p$-dimensional space. Therefore, as mentioned in Introduction, $\psi[C_{p}]$ can be thought of a functional of $\{X^{\mu}(\xi)\}_{\mu=0}^{d-1}$. Since we are interested in a brane as a $p$-dimensional object at a given time of some specific time choice, we will restrict $C_{p}$ to spacelike objects. $C_{p}$ may have a boundary; however, we mainly focus our discussion on the case where $C_{p}$ has no boundary.

In general, $\psi[C_{p}]$ can take any functionals of $\{X^{\mu}(\xi)\}_{\mu=0}^{d-1}$, but we restrict it by imposing the following conditions as in the ordinary field theory:

Spacetime diffeomorphism

: $\psi[C_{p}]$ is a scalar under the spacetime diffeomorphism $X^{\mu}\rightarrow X^{\prime\mu}=f^{\mu}(X)$:

$\displaystyle\psi^{\prime}[C_{p}^{{}^{\prime}}]=\psi[\{X^{\prime\mu}(\xi)\}]=\psi[\{X^{\mu}(\xi)\}]=\psi[C_{p}]~{}.$

(1)

Reparametrization invariance

: $\psi[C_{p}]$ is invariant under the reparametrization on $C_{p}$: $\xi^{i}\rightarrow\xi^{\prime i}=g^{i}(\xi)$:

$\displaystyle\psi^{\prime}[C_{p}^{{}^{\prime}}]=\psi[\{X^{\mu}(\xi^{\prime})\}]=\psi[\{X^{\mu}(\xi)\}]=\psi[C_{p}]~{}.$

(2)

We note that we imposed a scalar condition (1) on the spacetime diffeomorphism to simplify the argument; more generally, it could take a covariant form. Typical examples that satisfy the above conditions are the functionals of various differential forms:

$\displaystyle\psi[C_{p}]=\psi\left(\left\{\int_{C_{p}}d^{p}\xi\sqrt{h}A^{(a)}(X(\xi))\right\}\right)~{},$

(3)

where $A^{(a)}(X)$ is a spacetime scalar, and $h=\det(h_{ij})$ is the determinant of the induced metric,

$\displaystyle h_{ij}(\xi)=\frac{\partial X^{\mu}(\xi)}{\partial\xi^{i}}\frac{\partial X^{\nu}(\xi)}{\partial\xi^{j}}g_{\mu\nu}(X(\xi))=e^{\mu}_{i}(\xi)e^{\nu}_{j}(\xi)g_{\mu\nu}~{},\quad\det(h_{ij})>0~{}.$

(4)

Besides, the index $a$ represents various types of volume integrals. Note that for a given scalar $A^{(a)}(X)$ we can always rewrite the volume integral in Eq. (3) by a $p$-form integration as

$\displaystyle\int d^{p}\xi\sqrt{h}A^{(a)}(X(\xi))=\int_{C_{p}}A_{p}^{(a)}~{},$

(5)

where $A_{p}^{(a)}=\frac{1}{p!}A^{(a)}_{\mu_{1}\cdots\mu_{p}}(X(\xi))dX^{\mu_{1}}\wedge\cdots\wedge dX^{\mu_{p}}$ is the $p$-form satisfying

$\displaystyle\frac{1}{p!}\eta^{i_{1}\cdots i_{p}}e_{i_{1}}^{\mu_{1}}(\xi)\cdots e_{i_{p}}^{\mu_{p}}(\xi)A^{(a)}_{\mu_{1}\cdots\mu_{p}}(X(\xi))=A^{(a)}(X(\xi))~{}.$

(6)

Here, $\eta^{j_{1}\cdots j_{p}}$ is the totally anti-symmetric tensor defined by Eq. (144) in Appendix A. $A^{(a)}_{\mu_{1}\cdots\mu_{p}}(X(\xi))$ can be explicitly written as

	$\displaystyle A^{(a)}_{\mu_{1}\cdots\mu_{p}}(X(\xi))$	$\displaystyle=g_{\mu_{1}\nu_{1}}\cdots g_{\mu_{p}\nu_{p}}E^{\nu_{1}\cdots\nu_{p}}(\xi)A^{(a)}(X(\xi))$
		$\displaystyle=E_{\mu_{1}\cdots\mu_{p}}(X(\xi))A^{(a)}(X(\xi))~{},$		(7)

where

$\displaystyle E^{\mu_{1}\cdots\mu_{p}}(X(\xi))\coloneqq\eta^{i_{1}\cdots i_{p}}e_{i_{1}}^{\mu_{1}}(\xi)\cdots e_{i_{p}}^{\mu_{p}}(\xi)~{}.$

(8)

By introducing a $p$-form,

$\displaystyle E_{p}\coloneqq\frac{1}{p!}E_{\mu_{1}\cdots\mu_{p}}(X(\xi))dX^{\mu_{1}}\wedge\cdots\wedge dX^{\mu_{p}}~{},$

(9)

we can also check

$\displaystyle E_{\mu_{1}\cdots\mu_{p}}E^{\mu_{1}\cdots\mu_{p}}=p!\quad\leftrightarrow\quad E_{p}\wedge\star E_{p}=\sqrt{-g}dX^{0}\wedge\cdots\wedge dX^{d-1}~{},$

(10)

and

$\displaystyle\mathrm{Vol}[C_{p}]=\int_{C_{p}}E_{p}~{},$

(11)

where $\mathrm{Vol}[C_{p}]$ is the volume of $C_{p}$.

In general, a variation of the brane field $\psi[C_{p}]$ for an arbitrary change of the manifold $\delta C_{p}=\{\delta X^{\mu}(\xi)\}_{\mu=0}^{d-1}$ is given by the functional derivative as

$\displaystyle\delta\psi[C_{p}]=\int d^{p}\xi\sqrt{h}~{}\delta X^{\mu}(\xi)\frac{\delta\psi[C_{p}]}{\delta X^{\mu}(\xi)}~{}.$

(12)

In particular, when $\delta C_{p}$ has a small support around some point $\xi$ and is given by the boundary of a $(p+1)$-dimensional subspace, i.e., ${}^{\exists}\delta D_{p+1},~{}\partial\delta D_{p+1}=\delta C_{p}$ (see Fig. 1 as an example), Eq. (12) can be also written as

$\displaystyle\delta\psi[C_{p}]=\frac{1}{(p+1)!}\sigma^{\mu_{1}\cdots\mu_{p+1}}(\delta C_{p})\frac{\delta\psi[C_{p}]}{\delta\sigma^{\mu_{1}\cdots\mu_{p+1}}(\xi)}~{},$

(13)

where²²2When $p=0$, $\sigma^{\mu_{1}\cdots\mu_{p+1}}(\delta C_{p})$ is just $\delta X^{\mu}$.

	$\displaystyle\sigma^{\mu_{1}\cdots\mu_{p+1}}(\delta C_{p})$	$\displaystyle=\int_{\delta D_{p+1}}dX^{\mu_{1}}\wedge\cdots\wedge dX^{\mu_{p+1}}~{}$
		$\displaystyle=\int_{\delta D_{p+1}}d(\delta X^{[\mu_{1}}\wedge\cdots\wedge dX^{\mu_{p+1}]})~{}$
		$\displaystyle=\int_{\delta C_{p}}\delta X^{[\mu_{1}}dX^{\mu_{2}}\wedge\cdots\wedge dX^{\mu_{p+1}]}~{}.$		(14)

Here, we have used the Stokes theorem in the third line of Eq. (14), and introduced the antisymmetrization,

$\displaystyle A^{[\mu_{1}\mu_{2}\cdots\mu_{p}]}:=\frac{1}{p!}\sum_{\sigma\in S_{p}}\mathrm{sgn}(\sigma)A^{\sigma(\mu_{1})\sigma(\mu_{2})\cdots\sigma(\mu_{p})}~{},$

(15)

where $S_{p}$ is the symmetric group of degree $p$, and $\mathrm{sgn}(\sigma)$ is the sign of permutations. We call $\delta\psi[C_{p}]/\delta\sigma^{\mu_{1}\cdots\mu_{p+1}}(\xi)$ the $p$-th order area derivative, which is the generalization of the area derivative for one-dimensional objects [70, 71, 69, 68] to higher-dimensional ones. The definition of the are derivative (13) is abstract, and we should clarify its relation to the ordinary functional derivative (12). Equation (14) can be more explicitly written as

$\displaystyle\sigma^{\mu_{1}\cdots\mu_{p+1}}(\delta C_{p})$

$\displaystyle=\int_{\delta C_{p}}d^{p}\xi\sqrt{h}\delta X^{[\mu_{1}}(\xi)E^{\mu_{2}\cdots\mu_{p+1}]}(\xi)~{},$

(16)

which leads to the following expression of $\delta\psi[C_{p}]$:

$\displaystyle\delta\psi[C_{p}]=\frac{1}{(p+1)!}\int_{\delta C_{p}}d^{p}\xi\sqrt{h}\delta X^{[\mu_{1}}(\xi)E^{\mu_{2}\cdots\mu_{p+1}]}(\xi)\frac{\delta\psi[C_{p}]}{\delta\sigma^{\mu_{1}\cdots\mu_{p+1}}(\xi)}~{}.$

(17)

This coincides with Eq. (12) by the identification

$\displaystyle\frac{\delta\psi[C_{p}]}{\delta X^{\mu_{1}}(\xi)}=\frac{1}{(p+1)!}E^{\mu_{2}\cdots\mu_{p}+1}(\xi)\frac{\delta\psi[C_{p}]}{\delta\sigma^{\mu_{1}\cdots\mu_{p+1}}(\xi)}~{}.$

(18)

Note that for more general variation $\delta C_{p}$, the above relation does not necessarily hold, and additional terms can appear on the right-hand side³³3For example, for $p=1$, there is also another term called the path derivative which corresponds to the variation $\delta C_{1}$ such that an infinitesimal path $\delta\Gamma$ is added to a point $\{X^{\mu}(\xi_{0})\}$ of the original loop. In this case, the functional derivative becomes [78] $\delta/\delta X^{\mu}(\xi)=\partial_{\mu}|_{X=X(\xi_{0})}\delta(\xi-\xi_{0})$. As a result, the functional derivative is given by the sum of the area derivative and path derivative for $p=1$. .

In particular, the area derivative of the volume integral of a differential $p$ form $A_{p}^{(a)}$ can be calculated in the following way. Under the infinitesimal change $C_{p}+\delta C_{p}$, the variation of Eq. (5) is

	$\displaystyle\int_{\delta C_{p}}A_{p}^{(a)}=\int_{\delta D_{p+1}}dA_{p}^{(a)}$	$\displaystyle=\frac{1}{(p+1)!}\int_{\delta D_{p+1}}dX^{\mu_{p}}\wedge\cdots\wedge dX^{\mu_{p+1}}F_{\mu_{1}\cdots\mu_{p+1}}^{(a)}(X)$
		$\displaystyle=\frac{1}{(p+1)!}\sigma^{\mu_{1}\cdots\mu_{p+1}}(\delta C_{p})F_{\mu_{1}\cdots\mu_{p+1}}^{(a)}(X)~{},$		(19)

where

$\displaystyle F_{p+1}^{(a)}=dA_{p}^{(a)}=\frac{1}{(p+1)!}F_{\mu_{1}\cdots\mu_{p+1}}^{(a)}(X)dX^{\mu_{1}}\wedge\cdots\wedge dX^{\mu_{p+1}}~{}.$

(20)

The above equation implies

$\displaystyle\frac{\delta}{\delta\sigma^{\mu_{1}\cdots\mu_{p+1}}(\xi)}\left(\int_{C_{p}}A_{p}^{(a)}\right)=F_{\mu_{1}\cdots\mu_{p+1}}^{(a)}(X)~{}.$

(21)

Thus, as long as we consider the brane field whose functional form is given by Eq. (3), the area derivative is given by

$\displaystyle\frac{\delta\psi[C_{p}]}{\delta\sigma^{\mu_{1}\cdots\mu_{p+1}}(\xi)}=\sum_{a}F^{(a)}_{\mu_{1}\cdots\mu_{p+1}}(X(\xi))\frac{\partial\psi(\{z^{a}\})}{\partial z^{a}}\bigg{|}_{z^{a}=\int_{C_{p}}A_{p}^{(a)}}~{}.$

(22)

It is also convenient to introduce the $(p+1)$-form of the area derivative:

$\displaystyle D\psi[C_{p}]\coloneqq\frac{1}{(p+1)!}\frac{\delta\psi[C_{p}]}{\delta\sigma^{\mu_{1}\cdots\mu_{p+1}}(\xi)}dX^{\mu_{1}}\wedge\cdots\wedge dX^{\mu_{p+1}}~{}.$

(23)

Let us see a few important examples below.

See Fig. 2 as an example. In this case, we only regard the boundary subspace $C_{p}$ as a physical variable. Since $\mathrm{Vol}[M_{p+1}]$ is given by the $(p+1)$-form integral on $M_{p+1}$, this case does not belong to Eq. (22), but we can calculate its area derivative as follows. A general variation $\delta C_{p}$ can be constructed by adding infinitesimal small loop $\delta C_{p}^{\rm Loop}$ at each point on $C_{p}$. By representing the bulk of $\delta\Gamma_{p}$ as $\delta D_{p+1}$, we have

$\displaystyle\delta\mathrm{Vol}[M_{p+1}]=\int_{\delta D_{p+1}}E_{p+1}~{},$

(28)

which corresponds to the expression (2.2). Thus, one can see that $E_{p+1}$ corresponds to $F_{p+1}$ in this case, and we have

$\displaystyle D\mathrm{Vol}[M_{p+1}]=E_{p+1}~{}.$

(29)

Now, we define the brane field action with global $\mathrm{U}(1)$ $p$-form symmetry. At the leading order of the functional-derivative expansion, the action takes the following form,

$\displaystyle S[\psi]$

$\displaystyle={\cal N}\int[dC_{p}]\left(-\frac{1}{\mathrm{Vol}[C_{p}]}\int d^{p}\xi\sqrt{h}\frac{\delta\psi^{\dagger}[C_{p}]}{\delta X^{\mu}(\xi)}\frac{\delta\psi[C_{p}]}{\delta X_{\mu}(\xi)}-V(\psi^{\dagger}[C_{p}]\psi[C_{p}])\right)~{},$

(30)

where Vol$[C_{p}]$ is the world volume (26) and ${\cal N}$ is a normalization factor determined later. As we mentioned in the previous section, the functional derivative contains various area derivatives of lower degrees in general. In this paper, we focus on the $p$-th order area derivative and consider the simplified version of Eq. (30):

	$\displaystyle S_{0}[\psi]$	$\displaystyle={\cal N}\int[dC_{p}]\left(-\frac{1}{\mathrm{Vol}[C_{p}]}\int_{\Sigma_{d}}\delta(C_{p})D\psi^{\dagger}[C_{p}]\wedge\star D\psi[C_{p}]-V(\psi^{\dagger}[C_{p}]\psi[C_{p}])\right)$
		$\displaystyle={\cal N}\int[dC_{p}]\Biggl{(}-\frac{1}{(p+1)!\mathrm{Vol}[C_{p}]}\int_{C_{p}}d^{p}\xi\sqrt{h}\frac{\delta\psi^{\dagger}[C_{p}]}{\delta\sigma^{\mu_{1}\cdots\mu_{p+1}}(\xi)}\frac{\delta\psi[C_{p}]}{\delta\sigma_{\mu_{1}\cdots\mu_{p+1}}(\xi)}$
		$\displaystyle\qquad-V(\psi^{\dagger}[C_{p}]\psi[C_{p}])\Biggr{)}~{},$		(31)

where $\delta(C_{p})$ is defined as

$\displaystyle\delta(C_{p})\coloneqq\int_{C_{p}}d^{p}\xi\sqrt{-\frac{h}{g}}\prod_{\mu=0}^{d-1}\delta\left(X^{\mu}-X^{\mu}(\xi)\right)~{}.$

(32)

Besides, the (path-)integral measure is defined by

$\displaystyle[dC_{p}]={\cal D}Xe^{-T_{p}\mathrm{Vol}[C_{p}]}~{},$

(33)

where $T_{p}$ is the $p$-brane tension, and ${\cal D}X$ is the induced measure by the diffeomorphism invariant norm [79],

$\displaystyle||\delta X||^{2}\coloneqq\int d^{p}\xi\sqrt{h}g_{\mu\nu}(X(\xi))\delta X^{\mu}(\xi)\delta X^{\nu}(\xi)~{}.$

(34)

The weight in the (path-)integral measure (33) that we choose is nothing but the $p$-brane action [80], which suppresses large branes. Besides, when $\{X^{\mu}(\xi)\}$ represents an embedding of a closed subspace $C_{p}$, its translation $\{X^{\mu}(\xi)+X_{0}^{\mu}\}$ also represents another closed subspace, which means that there are always zero mode integrations in Eq. (33). Equation (2.3) is a straightforward generalization of the action for mean string-field theory [68] to $p$-dimensional brane.

The brane action (2.3) is invariant under the global $\mathrm{U}(1)$ $p$-form transformation,

$\displaystyle\psi[C_{p}]~{}\rightarrow~{}\exp\left(i\int_{C_{p}}\Lambda_{p}\right)\psi[C_{p}]~{},\quad d\Lambda_{p}=0~{}.$

(35)

Note that if $C_{p}$ is a boundary, i.e., $C_{p}=\partial C_{p+1}$, the contribution of $\Lambda_{p}$ to the phase vanishes from the Stokes theorem. Even when the topology of space-time is trivial, this transformation is useful as a symmetry, e.g., it leads to Ward-Takahashi identities as in ordinary $p=0$ symmetry.

We can also promote the global symmetry to the gauge symmetry by introducing a $(p+1)$-form gauge field,

$\displaystyle B_{p+1}=\frac{1}{(p+1)!}B_{p+1,\mu_{1}\cdots\mu_{p+1}}(X)dX^{\mu_{1}}\wedge\cdots\wedge dX^{\mu_{p+1}}~{}$

(36)

and replacing the derivative with the covariant derivative,

$\displaystyle\frac{D_{G}\psi[C_{p}]}{\delta\sigma^{\mu_{1}\cdots\mu_{p+1}}(\xi)}\coloneqq\left(\frac{\delta}{\delta\sigma^{\mu_{1}\cdots\mu_{p+1}}(\xi)}-iB_{p+1,\mu_{1}\cdots\mu_{p+1}}(X(\xi))\right)\psi[C_{p}]~{}.$

(37)

The gauge transformation is given by

$\displaystyle\psi[C_{p}]~{}\rightarrow~{}e^{i\int_{C_{p}}\Lambda_{p}}\psi[C_{p}]~{},\quad B_{p+1}~{}\rightarrow~{}B_{p+1}+d\Lambda_{p}~{},$

(38)

where we have used Eq. (25). Note that the action (2.3) is invariant under the spacetime diffeomorphism and the reparametrization on $C_{p}$ by the construction of $\psi[C_{p}]$.

We should comment on the interactions of the brane field. The potential $V(\psi^{\dagger}[C_{p}]\psi[C_{p}])$ in Eq. (2.3) corresponds to contact interactions in ordinary field theory, but we can also consider more general interactions such as [68]

$\displaystyle\int[dC_{p}^{1}]\int[dC_{p}^{2}]\int[dC_{p}^{3}]\delta(C_{p}^{1}-C_{p}^{2}-C_{p}^{3})\psi^{\dagger}[C_{p}^{1}]\psi[C_{p}^{2}]\psi[C_{p}^{3}]+{\rm h.c.}~{},$

(39)

which represents the splitting or merging of branes⁴⁴4This interaction still preserves the $\mathrm{U}(1)$ $p$-form symmetry due to the delta function. It may seem strange to have $\mathrm{U}(1)$ symmetry while the number of branes is changing; however, $\mathrm{U}(1)$ $p$-form symmetry does not represent the conservation of the number of branes itself, but rather the conservation of the winding number on a space with nontrivial topology. . Such interactions seem to alter the mean-field dynamics of the brane field significantly as the behaviors of phase transition in the ordinary Landau theory change by adding odd potential terms. In this paper, we simply neglect these interactions and focus on the model (2.3).

As with ordinary symmetry, when $p$-form global symmetry is continuous, we have a current $(p+1)$-form $J_{p+1}$, which is conserved as

$\displaystyle d\star J_{p+1}=0~{}.$

(40)

The corresponding conserved charge is given as

$\displaystyle Q_{p}[\Sigma_{d-p-1}]=\int_{\Sigma_{d-p-1}}\star J_{p+1}~{},$

(41)

where $\Sigma_{d-p-1}$ is a $(d-p-1)$-dimensional (closed) subspace.

We can calculate $J_{p+1}$ in the brane field theory as follows. Instead of the global $p$-form transformation, we consider an infinitesimal local $p$-form transformation

$\displaystyle\psi[C_{p}]\quad\rightarrow\quad e^{i\int_{C_{p}}\Lambda_{p}}\psi[C_{p}]~{},\quad d\Lambda_{p}\neq 0~{}.$

(42)

Then, the variation of the action in the linear order of $\Lambda_{p}$ should have the form

$$\delta S=-\int_{\Sigma_{d}}d\Lambda_{p}\wedge\star J_{p+1}~{},$$

(43)

since the action needs to vanish if $d\Lambda_{p}=0$. $J_{p+1}$ is nothing but the Noether current. From the integrating by part, we obtain

$$\delta S=(-)^{p}\int_{\Sigma_{d}}\Lambda_{p}\wedge d\star J_{p+1}~{}.$$

(44)

If $\psi$ satisfies the equation of motion, the action is stationary, so the divergence of the current vanishes $d\star J_{p+1}=0$.

Let us derive the explicit form of $J_{p+1}$ for the action (2.3). The variation of the action is calculated as

	$\displaystyle\delta S_{0}[\psi]$	$\displaystyle={\cal N}\int[dC_{p}]\frac{i}{\mathrm{Vol}[C_{p}]}\int_{\Sigma_{d}}\delta(C_{p})\left(d\Lambda_{p}\wedge\star\psi^{\dagger}D\psi-\psi D\psi^{\dagger}\wedge\star d\Lambda_{p}\right)$
		$\displaystyle={\cal N}\int[dC_{p}]\frac{i}{\mathrm{Vol}[C_{p}]}\int_{\Sigma_{d}}\delta(C_{p})d\Lambda_{p}\wedge\star(\psi^{\dagger}D\psi-\psi D\psi^{\dagger})~{},$		(45)

where we have used $\eta\wedge\star\omega=\omega\wedge\star\eta$. Comparing Eqs. (43) and (45), we obtain

$\displaystyle J_{p+1}(X)=-{\cal N}\int[dC_{p}]\frac{\delta(C_{p})}{\mathrm{Vol}[C_{p}]}i(\psi^{\dagger}D\psi-\psi D\psi^{\dagger})~{}.$

(46)

This expression shows that the $(p+1)$-form current is given by the integral over all the brane configurations. Note that $X$ dependence of the current $J_{p+1}(X)$ comes from $\delta(C_{p})$ in Eq. (32). We can also check that $Q_{p}$ generates the $p$-form transformation (35) in the following way. We formally define the quantum theory of the present brane field by the following path integral:

$\displaystyle\langle{\cal O}(X(\xi))\rangle\coloneqq\frac{1}{Z}\int{\cal D}\psi{\cal D}\psi^{\dagger}{\cal O}(X(\xi))e^{iS[\psi]}~{}.$

(47)

Consider the expectation value of $\psi[C_{p}]$,

$$\langle\psi[C_{p}]\dots\rangle~{},$$

(48)

where “$\dots$” represents other arbitrary operators. We choose $\Lambda_{p}$ in the field transformation of Eq. (42) as

$$\Lambda_{p}=\epsilon(-1)^{p(d-p-1)}\delta_{p}(D_{d-p})$$

(49)

with $\partial D_{d-p}=\Sigma_{d-p-1}$, and an infinitesimal parameter $\epsilon$. Here, $\delta_{p}(D_{d-p})$ is the Poincaré-dual form of $D_{d-p}$ such that

$$\int_{D_{d-p}}f_{d-p}=\int_{\Sigma_{d}}f_{d-p}\wedge\delta_{p}(D_{d-p})~{},$$

(50)

for an arbitrary $(d-p)$-form, $f_{d-p}$. $\delta_{p}(D_{d-p})$ can be thought of as a generalization of the delta function.

In this parametrization, the variance of the action (43) is

$$\delta S=\epsilon\int_{\Sigma_{d-p-1}}\star J=\epsilon Q_{p}[\Sigma_{d-p-1}]~{},$$

(51)

while $\psi[C_{p}]$ transforms as

$$\psi[C_{p}]\rightarrow e^{i\int_{C_{p}}\Lambda_{p}}\psi[C_{p}]=e^{i\epsilon(-1)^{p}\mathrm{Link}[\Sigma_{d-p-1},C_{p}]}\psi[C_{p}]~{}.$$

(52)

Here, we have defined the linking number as

$$\mathrm{Link}[\Sigma_{d-p-1},C_{p}]\coloneqq\int_{\Sigma_{d}}\delta_{d-p}(C_{p})\wedge\delta_{p}(D_{d-p})~{}.$$

(53)

See the left figure in Fig. 3 for an example of the configuration of links, where we take $d=3$ and $p=1$. We assume other operators in “$\dots$” do not have support on $D_{d-p}$, so that “$\dots$” does not transform under the field transformation with Eq. (49). In the path-integral, the field transformation (42) is merely a redefinition of the integral variables. Therefore, assuming the path-integral measure is invariant under the field transformation (42), it leads to the identity,

$$\langle\psi[C_{p}]\dots\rangle=\langle e^{i\epsilon Q_{p}[\Sigma_{d-p-1}]+i\epsilon(-1)^{p}\mathrm{Link}[\Sigma_{d-p-1},C_{p}]}\psi[C_{p}]\dots\rangle~{}.$$

(54)

This implies

$$Q_{p}[\Sigma_{d-p-1}]\psi[C_{p}]=-(-1)^{p}\mathrm{Link}[\Sigma_{d-p-1},C_{p}]\psi[C_{p}]~{}.$$

(55)

This is the relation in the path integral. To derive the relation in the operator formalism, consider the Cauchy surface labeled by time $t$. Let $C_{d-p-1}(t_{0})$ be $(d-p-1)$-dimensional subspace on the Cauchy surface at $t=t_{0}$. We choose that $\Sigma_{d-p-1}=C_{d-p-1}(t_{0}+\eta)\cup\overline{C_{d-p-1}}(t_{0}-\eta)$ with an infinitesimal parameter $\eta$. Here, $\overline{C_{d-p-1}}$ is the $(d-p-1)$-dimensional subspace with the opposite orientation of $C_{d-p-1}$. We also choose that $C_{p}$ is a $p$-dimensional subspace on the Cauchy surface at $t=t_{0}$. See the right figure in Fig. 3 for the configuration. In this configuration, the left-hand side in Eq. (55) becomes

$$Q_{p}[\Sigma_{d-p-1}]\psi[C_{p}]=Q_{p}[C_{d-p-1}(t_{0}+\eta)]\psi[C_{p}]-Q_{p}[C_{d-p-1}(t_{0}-\eta)]\psi[C_{p}]~{}.$$

(56)

In operator formalism, the ordering of the operator product corresponds to the time ordering, so $Q_{p}[C_{d-p-1}(t_{0}+\eta)]\psi[C_{p}]-Q_{p}[C_{d-p-1}(t_{0}-\eta)]\psi[C_{p}]\to[Q_{p}[C_{d-p-1}],\psi[C_{p}]]$; thus, we find that the Noether charge $Q_{p}$ generates the symmetry transformation,

$$[iQ_{p}[C_{d-p-1}],\psi[C_{p}]]=-i\mathrm{I}[C_{d-p-1},C_{p}]\psi[C_{p}]~{}.$$

(57)

Here, $\mathrm{I}[C_{d-p-1},C_{p}]$ represents the intersection number between $C_{d-p-1}$ and $C_{p}$ on the Cauchy surface, which can be obtained by evaluating the right-hand side in Eq. (55) with $\Sigma_{d-p-1}=C_{d-p-1}(t_{0}+\eta)\cup\overline{C_{d-p-1}}(t_{0}-\eta)$.