A Concrete Construction of a Topological Operator in Factorization Algebras

Abstract

1. Introduction

Quantum field theory is a central topic in theoretical physics. It is valid for describing various kinds of phenomena and also useful in the generation of some mathematical conjectures. In spite of such effectiveness, we still do not understand the complete mathematical formulation of quantum field theories.

In the last 10 years, Kevin Costello and Owen Gwilliam have developed a new formulation based on factorization algebras [1,2]. This formulation works well in perturbative field theories, conformal field theories, and topological quantum field theories. These days, some people are working on formulations of nonperturbative aspects of quantum field theories [3,4].

In such a context, nonperturbative Noether theorem is also proposed in Ref. [5], motivated by recent developments of generalized symmetries [6,7,8]. The dogma of generalized symmetries is

A topological operator with |$\mathcal {U}(U)$| some support U is invariant under homotopic transformations of U.

A question arises: what does this invariance mean? We will show that |$\mathcal {U}(U_1)$| and |$\mathcal {U}(U_2)$| are in the same equivalence class of Batalin–Vilkovisky cohomology when |$U_1$| and |$U_2$| are homotopic. We will propose that this is the definition of the topological operators.

In this paper, we focus only on the case of 1D free massless scalar theory (i.e. quantum mechanics without potential terms). This theory has a shift symmetry |$\Phi \mapsto \Phi +\alpha$| where |$\Phi$| is a field and |$\alpha$| is a real number. We give a concrete construction of the topological operator representing this shift |$\mathbb {R}$|-symmetry. One virtue of the topological operator is that we can perform gauging of the subsymmetry |$\mathbb {Z}\subset \mathbb {R}$|⁠. By this |$\mathbb {Z}$|-gauging, we can obtain a compact scalar theory. This is an interesting theory because it has instanton effects. Work has been done on compact |$U(1)$|  p-form gauge theory [4] motivated by S-duality. The case of |$p=0$| is essentially the same as this paper in terms of using |$\mathbb {Z}$|-gauging. However, our construction is more concrete, and we will also discuss the relation to generalized symmetry notions and the |$\theta$|-vacuum.

This paper is organized as follows. In Section 2, we will review the formulation by Costello and Gwilliam. In Section 3, we will propose a concrete construction of the topological operator. In Section 4, we will see the |$\theta$|-degree and the formulation. Section 5 is devoted to the conclusion and discussion.

2. Review of factorization algebra of free real scalar theory

2.1. Observable space

In this section, we will define an observable |$\mathcal {O}$|⁠. Let M be a d-dimensional manifold and a real scalar field |$\Phi$| be in |$C^\infty (M)$|⁠. |$\mathcal {O}$| must have two properties:

• |$\mathcal {O}$| is a functional; i.e. |$\mathcal {O}$| is a map from a field |$\Phi$| to a number |$\mathcal {O}(\Phi )$|⁠.

• |$\mathcal {O}$| has a concept of locality. In other words, |$\mathcal {O}$| has support |$U\subset M$|⁠.

Here U is an open subset of M. If the shape of U is like a d-dimensional open ball |$B^d$|⁠, |$\mathcal {O}$| is a point operator with a UV cutoff. In the case of U being |$B^{d-1}\times S^1$|⁠, it is a loop operator.

Motivated by the above two properties, we will find an example of observables: a linear observable,

where f is a |$\mathbb {C}$|-function with compact support in U. The integral |$\int _M f\Phi$| refers to only the |$\Phi$| on U; hence in this sense |$\mathcal {O}_{\rm linear}$| has locality. We identify |$\mathcal {O}_{\rm linear}$| with f, then define linear observable space on U as

A more general observable can be regarded as a polynomial of functions with compact supports: |$\mathcal {O}=c+f+f_1*f_2+\cdots$|⁠, where |$*$| is a formal symmetric product, |$c\in \mathbb {C}$|⁠, and |$f,f_1,f_2,\cdots \in C_{\rm c}^\infty (U,\mathbb {C})$|⁠. |$\mathcal {O}$| acts |$\Phi$| as

2.2. Classical derived observable space

Roughly speaking, derived means that we will add a concept of degree to the observable space |${\rm Obs}(U)$|⁠. This formulation was originally given by Batalin and Vilkovisky [9,10]:

The first one is a degree-0 linear observable, and the second one is a degree-|$(-1)$| linear observable, which corresponds to an antifield in the physics literature. For simplicity, we denote |$(f,0)\in C_{\rm c}^\infty (U)^{0}$| and |$(f,-1)\in C_{\rm c}^\infty (U)^{-1}$| as f and |$f^\star$|⁠. The symmetric product |$*$| is defined for them as

By Eq. (8), we can rewrite classical derived observable space |${\rm Obs}^{\rm cl}(U)$|⁠:¹

This is a decomposition about the degree. The first line is degree |$(-2)$|⁠, second one is degree |$(-1)$|⁠, and third one is degree 0. Actually |$\Delta _{\rm BV}^{\rm cl}$| is defined as a map from degree |$(-n)$| to degree |$(-n+1)$|⁠:

The concrete form of |$\Delta _{\rm BV}^{\rm cl}$| depends on what theory we want. In this paper, we are interested in free theory; thus we will define it as follows.

By some calculations, we have

• Leibniz rule: |$\Delta _{\rm BV}^{\rm cl}(A*B) = \Delta _{\rm BV}^{\rm cl}(A)*B + (-1)^{|A|}A*\Delta _{\rm BV}^{\rm cl}(B)$|⁠,

• Nilpotency: |$\left(\Delta _{\rm BV}^{\rm cl}\right)^2=0$|⁠.

Nilpotency means that

is a chain complex. This is called the classical Batalin–Vilkovisky complex. The cohomology of the classical Batalin–Vilkovisky complex is called classical Batalin–Vilkovisky cohomology:

The Leibniz rule means that |$H^{*}\left({\rm Obs}^{\rm cl}(U)\right)$| has a product |$*$|⁠, because the product of |$A+\Delta _{\rm BV}^{\rm cl}(\cdots )$| and |$B+\Delta _{\rm BV}^{\rm cl}(\cdots )$| can be rewritten as |$A*B+\Delta _{\rm BV}^{\rm cl}(\cdots )$|⁠.²

Let us consider the physical meanings of the cohomology |$H^0\left({\rm Obs}^{\rm cl}(U)\right)$|⁠. We take 0-degree observables |$\mathcal {O}_1,\mathcal {O}_2$| and assume that these are the same in the cohomology, i.e. |$\exists X\ {\rm s.t.}$|⁠:

Let |$\Phi _{\rm cl}$| be a solution of the equation of motion |$(-\Delta +m^2)\Phi =0$|⁠.³ We have

Hence we can see |$H^0\left({\rm Obs}^{\rm cl}(U)\right)$| as the on-shell evaluation of observables. However, |$[\mathcal {O}_1](=[\mathcal {O}_2])$| is not a number. Then we define a map called a state  |$\langle -\rangle$|⁠.

2.3. Concrete calculations of classical Batalin–Vilkovisky cohomology

In some situations, we can calculate |$H^{*}\left({\rm Obs}^{\rm cl}(M)\right)$| explicitly.

Theorem 1 tells us that the state |$\langle -\rangle$| can be given as an isomorphism; i.e. the state |$\langle -\rangle$| is canonically determined. On the other hand, for the cases of Theorems 2 and 3, there is room to choose the state |$\langle -\rangle$|⁠. Why is there such a difference? In the massive case, we have a reasonable interpretation for this difference.

In the case of the noncompact manifold |$M_{\rm non}$|⁠, the boundary condition is necessary to compute the path integral. If not, the scalar field |$\Phi$| might diverge on |$\partial \overline{M_{\rm non}}$|⁠. Then the path integral calculation fails. In order to compute the value of the path integral, we must choose a boundary condition. Reflecting this, we must choose a map |$\langle -\rangle :H^0\left({\rm Obs}^{\rm cl}(M_{\rm non})\right)\rightarrow \mathbb {C}$|⁠.

On the other hand, in the case of the compact manifold |$M_{\rm com}$|⁠, the scalar field |$\Phi$| never diverges on |$\partial M_{\rm com}$|⁠. Thus we can decide the value of the path integral. Similarly, we find the state |$\langle -\rangle$| canonically.

The above discussion is valid for the massive case. However, in the massless case, even for a compact manifold, |$H^0\left({\rm Obs}^{\rm cl}(M_{\rm com})\right)$| is not isomorphic to |$\mathbb {C}$|⁠. In other words, we cannot decide the expectation value naturally, though there is no |$\Phi$| divergence on the boundary |$\partial M_{\rm com}$|⁠. In order to understand this, we have to see another factor to interrupt the path integral computation: IR divergence. We will propose a treatment of IR divergence in a forthcoming paper (M. Kawahira, manuscript in preparation).

2.4. Definition of position observable Q and momentum observable P

We define a position observable Q as a function in |$C_{\rm c}^\infty (I)$| satisying

By the second condition, we obtain |$\int _{I} Q(x)\phi _p(x)\ {d}x=0$|⁠. We define a momentum observable as

Now we see why we call Q and P the position and momentum. If |$Q,P$| act on field |$\Phi \in C^\infty (\mathbb {R})$|⁠, we get

Especially in the massless case, |$Q(\cdot )$| is a “smeared |$\delta$|-function” because we have

by using |$\phi _q=1$|⁠. Hence, if we narrow the width of the interval, Q gets close to the |$\delta$|-function, and |$Q(\Phi )$| and |$P(\Phi )$| approach |$\Phi (t)$| and |$\Phi ^{\prime }(t)$|⁠.

In the massive case, Q is also a kind of smeared |$\delta$|-function. To see this, we remind ourselves that

Then we have

Therefore, we also expect Q to get close to the |$\delta$|-function in the massive case.

Q and P have the following important property.

2.5. Quantum derived observable space

Note that |$C_{\rm c}^\infty (U)*\mathbb {C}=C_{\rm c}^\infty (U)$|⁠; we can rewrite this as a formal series of |$\hbar$|⁠:

|$\delta$| is a quantum correction and will be defined as a map from |$-n$| degree to |$-n+1$| degree:

The definition of |$\delta$| is slightly complicated. We thus take three steps.

|$\delta$| is nilpotent⁴ and anticommutes with |$\Delta _{\rm BV}^{\rm cl}$|⁠:⁵  |$\delta ^2=0,\ \Delta _{\rm BV}^{\rm cl}\delta =-\delta \Delta _{\rm BV}^{\rm cl}$|⁠. Then we have

This means that the following one is a chain complex. This is called the quantum Batalin–Vilkovisky complex:

|${\rm Obs}^{\rm q}(U)$| does not have a Leibniz rule. Instead, we have the following theorem.

Here |$\lbrace -,-\rbrace$| is the antibracket, which is defined below.⁶ By this theorem, |$H^{*}\left({\rm Obs}^{\rm q}(U)\right)$| does NOT have a product |$*$| in contrast to |$H^{*}\left({\rm Obs}^{\rm cl}(U)\right)$|⁠.

However, in special situations we can define a product |$*$| for quantum cohomology! Let A and B have compact support in |$U_1$| and |$U_2$| respectively and assume that |$U_1$| and |$U_2$| are disjoint open subsets of U. Then

The subscript U denotes that this is computed in U. This is because the computation of |$\lbrace A,B\rbrace _U$| reduces to the integrals

Equation (56) means that we have a Leibniz rule for such |$A,B$|⁠; thus we can define a product |$*$| for them.

The cohomology is a formal series of |$\hbar$|⁠; hence the definition of a state changes slightly.

2.6. Weyl algebra in a 1D system

In the case of |$M=\mathbb {R}$|⁠, the quantum cohomology forms a Weyl algebra (the canonical commutation relation).

The aim of this section is to explain what the generators of Weyl algebra are. Roughly speaking, the generators of Weyl algebra are |$\hbar ,[Q],[P]$|⁠.⁷ However, this seems strange, because |$Q,P$| have the same support |$I=(-1/2,1/2)$|⁠; thus we cannot define the products of them.

In order to take the products, we will define the modified position observable  |$\mathcal {Q}_t$| and modified momentum observable  |$\mathcal {P}_t$|⁠.

Modified observables have the following special property:

The support of |$\mathcal {Q}_t$| and |$\mathcal {P}_t$| is |$I_t=(-1/2+t,1/2+t)$|⁠. Therefore we can define the products of them like

where |$I_t$| and |$I_s$| denote being in |$H^{*}({\rm Obs}^{\rm q}(I_t))$| and |$H^{*}({\rm Obs}^{\rm q}(I_s))$|⁠. Note that this product is independent for the choice t and s because of Theorem 8.

3. Construction of the topological operator

3.1. Shift symmetry

In the case of |$m=0$|⁠, we have a shift symmetry

for the equation of motion |$\Delta \Phi =0$|⁠. Reflecting this, |$P_t$| must be conserved. This is true because |$\mathcal {P}_t$| is conserved and in the massless case

In other words, |$P_t$| is the Noether charge.

3.2. Construction of the topological operator

First of all, we will review the usual construction of the topological operators. Let |$\hat{q}$| and |$\hat{p}$| be the generators of Weyl algebra; i.e. they satisfy |$[\hat{q},\hat{p}]=\hbar$|⁠. In the physics literature, the topological operator of the shift symmetry is⁹

Obviously we have

Then the naive construction of the topological operator in the Batalin–Vilkovisky formalism is

The supports of |$I_{s_0+t},I_{s_0+t},\cdots$| are disjoint. Hence each term of |$[\mathcal {V}_{\alpha ,t}]$| works well in terms of “topologicalness” and “action for |$Q_0$|”.

• Topologicalness

  By using Theorem 8 and |$\mathcal {P}_t=P_t$|⁠,

  $$\begin{eqnarray} \frac{{d}}{{d}t} [P_{s_0+t}*P_{s_1+t}*\cdots *P_{s_n+t}]=0. \end{eqnarray}$$

  (88)

• Action for |$Q_0$|

  Similar to Eq. (86), we have

  $$\begin{eqnarray} \left[ Q_0* P_{s_0+t_+}*P_{s_1+t_+}*\cdots *P_{s_n+t_+} \right] &=& \left[ P_{s_0+t_-}*P_{s_1+t_-}*\cdots *P_{s_n+t_-} *Q_0 \right] \\ &&+\,\, n\hbar \left[ P_{s_0+t_-}*P_{s_1+t_-}*\cdots *P_{s_{n-1}+t_-} \right] \end{eqnarray}$$

  (89)

  where |$t_+$| is sufficiently large and |$t_-$| is sufficiently small.

However, there is a problem. The support of |$\mathcal {V}_\alpha$| is infinitely wide. This is not good for the topological operator because we cannot define the action.

Therefore we define the topological operator in another way.¹⁰

Instead of the finite support, we lost the clarity of the following properties:

• Topologicalness

  $$\begin{eqnarray} \left[\frac{{d}}{{d}t}\mathcal {U}_{\alpha ,t}\right]_{I_t}=0 \end{eqnarray}$$

  (91)
• Action for |$Q_0$|  

  $$\begin{eqnarray} [Q_0*\mathcal {U}_{\alpha ,t}] = [\mathcal {U}_{\alpha ,s}*(Q_0+\alpha \hbar )],\ \ (t\gt 1,-1\gt s). \end{eqnarray}$$

  (92)

In the following sections, we will prove these properties.

3.3. Proof of topologicalness

3.4. Proof of action for operators

4. Gauging and the compact scalar

In this section, we will give an application of the topological operator. Some discussion is not established rigorously, but is physically natural.

4.1. |$\mathbb {Z}$|-gauging

Topological operators have the advantage over Noether charge; thus we can define discrete gauging.¹² This is because topological operators describe finite group transformations, while Noether charges only describe infinitesimal transformations. Hence now we have a group action like

The subgroup |$\mathbb {Z}\subset \mathbb {R}$| also acts as

Hence we can define¹³

The generators are

where |${\rm Exp}(\bullet ):=\sum _n\frac{1}{n!}\bullet ^{*n}$|⁠. They satisfy

This is the same as |$[\hat{p},e^{\pm i\hat{q}}]=\hbar e^{\pm i\hat{q}}$| in the usual notation.

We can say

similarly to Theorem 7.

4.2. Compact scalar and |$\theta$|-vacuum

After |$\mathbb {Z}$|-gauging, massless scalar theory becomes compact scalar theory, as explained in Refs. [11,12]. In compact scalar theory, we have |$\theta$|-degree. We can express |$\theta$| as a |$\theta$|-term or |$\theta$|-vacuum. However, implementation of the |$\theta$|-term is difficult in this formalism, because it is based on an equation of motion and the equation of motion does not have a |$\theta$|-term. On the other hand, implementing a |$\theta$|-vacuum is relatively easy since a vacuum is just a map from the cohomology to |$\mathbb {C}[\hbar ]$|⁠:

One way to give a state is to think of a representation of |$H^{*}({\rm Obs}^{\rm q}(\mathbb {R}))/\mathbb {Z}$|⁠. For simplicity, we use |$e^{\pm i\hat{q}}$| and |$\hat{p}$| rather than |$[{\rm Exp}(\pm iQ)]$| and |$[P]$|⁠. Take the eigenvector |$v_p$|⁠:

Then we find that p must be quantized. To see this, consider a vector |$e^{i\hat{q}}v_p$|⁠. This satisfies

Therefore |$e^{\pm i\hat{q}}$| are ladder operators. The spectrum of |$\hat{p}$| is

which depends on r. Obviously the |$r+1$|-spectrum is equivalent to the r-spectrum. Physicists denote r as |$\theta /2\pi$|⁠. At |$\theta =0$| and |$\theta =\pi$|⁠, we can see the |$\mathbb {Z}_2$|-symmetry of the spectrum: |$p\mapsto -p$|⁠. Hence the map (130) is given by a trace:

We can see the |$\theta$|-dependence in the state.

5. Conclusion and discussion

We have seen the construction of a topological operator |$\mathcal {U}_{\alpha ,t}$|⁠. The point of the construction is the order: first taking products |$*$|⁠, then taking a cohomology |$[\cdot ]$| (we call this order A). If we take reversed order (we call this order B), we have

and it has infinitely wide support. Order A versus order B is similar to path integral formalism versus operator formalism. In path integral formalism, we can formally think of multiplications of operators like |$\mathcal {O}_1(\Phi )\mathcal {O}_2(\Phi )\cdots \mathcal {O}_n(\Phi )$| because these are just c-numbers. On the other hand, in operator formalism, in order to give multiplications of operators, we need to get rid of UV divergences like |$:\mathcal {O}_1\mathcal {O}_2\cdots \mathcal {O}_n:$| since they are q-numbers. In order A, observables |$\mathcal {O}$| are just c-numbers; thus we can freely take the products |$\mathcal {O}_1*\mathcal {O}_2*\cdots *\mathcal {O}_n$|⁠. This is the same as path integral formalism. However, in order B, products of |$[\mathcal {O}]$| cannot be taken freely; therefore this corresponds to operator formalism. In order to take products, the support of |$[\mathcal {O}_1]*[\mathcal {O}_2]*\cdots *[\mathcal {O}_n]$| need to be wider than the original support. This is a kind of UV divergence.

In addition, we have seen the |$\theta$|-dependence of the state |$\langle -\rangle$| by representation theory. It is interesting to construct the map in another way. For example, the state of the massive scalar theory on |$M=\mathbb {R}$| can be given by the embedding

where |$\mathcal {S}(\mathbb {R})$| is a set of Schwartz functions on |$\mathbb {R}$|⁠. By considering |${\rm Obs}_{\mathcal {S}}:={\rm Sym}(\mathcal {S}(\mathbb {R})^{-1}\rightarrow \mathcal {S}(\mathbb {R})^{0})$|⁠, we obtain |$H^0({\rm Obs}^{\rm q}_{\mathcal {S}}(\mathbb {R})))\cong \mathbb {C}[\hbar ]$|⁠. Then f induces the state

It is attractive to give a state in a similar way in the case of the compact scalar. However, in order to achieve this, we need a technique to remove the IR divergence because the compact scalar is massless. Such a technique is discussed in our forthcoming paper (M. Kawahira, manuscript in preparation).

There is another way to consider the compact scalar theory. It is to focus on factorization homology. The factorization algebra of the compact scalar is essentially a functor that assigns a periodic Weyl algebra to each open subset. Hence, if we consider this system in |$S^1$|⁠, the factorization homology is the same as the Hochschild homology of the periodic Weyl algebra. To extract the |$\theta$|-information from the Hochschild homology is fascinating work.

Acknowledgment

We would like to thank Yuji Ando, Hayato Kanno, Keisuke Konosu, Hiroaki Matsunaga, and Jojiro Totsuka-Yoshinaka for the discussions at the early stages of this work. We would also like to thank Jun Maeda and Yuto Moriwaki for helpful discussions. The work of M.K. is supported by a Grant-in-Aid for JSPS Fellows No. 22KJ1989.

Funding

Open Access funding: SCOAP³.

Footnotes

References