Differential $KO$-theory via gradations and mass terms

Here, we explain the physical motivations mentioned above. Our models of differential extensions of the $KO$-theory and the $K$-theory, especially $\widehat{KO}_{-}$ and $\widehat{K}_{-}$ in terms of skew-adjoint operators, can be regarded as classifying “fermionic mass terms”. It is known that fermionic mass terms on the $n$-dimensional Minkovski spacetime are classified topologically by $KO^{n-3}\simeq KO_{-}^{1,n}$ (see for example [Freed19, Section 10.2]), and our differential model $\widehat{KO}_{-}^{1,n}$ refines this classification on the differential level.

First we explain the mathematical theory of fermionic mass terms, following [Freed19, Section 10.2] and [Freed:2016rqq, Section 9.2]²²2 Remark that the sign convention on the Clifford algebras used there is differerent from ours (see (LABEL:eq_sign_C(V))). In our convention, $Cl_{p,q}$ has the negative generators $\alpha_{1},\cdots,\alpha_{p}$ and the positive generators $\beta_{1},\cdots,\beta_{q}$. . Let $n$ be the dimension of the spacetime. We start from an ungraded $Cl_{1,n-1}^{0}$-module $S$, without any specified inner product. Let $\mathrm{Spin}_{1,n-1}\subset Cl_{1,n-1}^{0}$ be the Lorentz spin group. Then there exists a $\mathrm{Spin}_{1,n-1}$-invariant symmetric nonnegative bilinear pairing

(1.1)

$\displaystyle\Gamma\colon S\otimes S\to\mathbb{R}^{1,n-1},$

uniquely up to a contractible choice. Here the nonnegativity means that $\Gamma(s,s)$ is timelike for all $s\in S$. Such $\Gamma$ induces a unique compatible $\mathbb{Z}_{2}$-graded $Cl_{1,n-1}$-module structure on $S\oplus S^{*}$, where the grading operator is given by $\gamma_{S\oplus S^{*}}:=\mathrm{id}_{S}\oplus(-\mathrm{id}_{S^{*}})$.

In [Freed19], mass forms $m_{\mathrm{form}}$ on $S$ are defined as nondegenerate skew-symmetric $\mathrm{Spin}_{1,n-1}$-invariant bilinear forms

(1.2)

$\displaystyle m_{\mathrm{form}}\colon S\otimes S\to\mathbb{R}.$

Here we remark that such $m_{\mathrm{form}}$ is called “mass terms” in [Freed19]. We use the above terminology and notation in order to distinguish it from our definition of mass terms in terms of skew-adjoint operators. Then it is shown in [Freed:2016rqq, Lemma 9.55] that the existence of such $m_{\mathrm{form}}$ is equivalent to the existence of a $\mathbb{Z}_{2}$-graded $Cl_{2,n-1}$-module structure on $S\oplus S^{*}$ which extends the $Cl_{1,n-1}$-module structure above.

We now explain how this formulation fits into our picture. Since the differential $KO$-groups should remember the differential, not just topological, information on mass terms, we do not want the ambiguity such as “contractible choice” above. Our model $\widehat{KO}_{-}$ is given in terms of Clifford modules with inner products (which are compatible with the Clifford action, see Footnote 1), and skew-adjoint invariant operators on them. Suppose we have $S$ and a pairing $\Gamma$ as in (1.1). Since the action on $S\oplus S^{*}$ by the negative Clifford generator $\alpha_{1}\in Cl_{1,n-1}$ anticommutes with $\gamma_{S\oplus S^{*}}$, the restriction defines a linear isomorphism

$\displaystyle\alpha_{1}|_{S}\colon S\to S^{*},$

and defines a symmetric bilinear form $(\cdot,\cdot)_{S}\colon S\otimes S\to\mathbb{R}$ by

(1.3)

$\displaystyle(s_{1},s_{2})_{S}:=\langle\alpha_{1}|_{S}(s_{1}),s_{2}\rangle,$

where the right hand side is the duality pairing $\langle\cdot,\cdot\rangle\colon S^{*}\otimes S\to\mathbb{R}$. Using the positivity of $\Gamma$, we see that the form (1.3), extended to $S\oplus S^{*}$ in the canonical way, defines an inner product on $S\oplus S^{*}$ which is compatible with the $\mathbb{Z}_{2}$-graded $Cl_{1,n-1}$-module structure induced by $\Gamma$. Now, we state a lemma essentially contained in the proof of [Freed:2016rqq, Lemma 9.55] in the form we need, where we let $Cl_{1,(n-1)+1}$ act on $S\oplus S^{*}$ by the action of $Cl_{1,n-1}$ and $\gamma_{S\oplus S^{*}}$ noting that a $\mathbb{Z}_{2}$-graded $Cl_{1,n-1}$-module structure is equivalent to an ungraded $Cl_{1,(n-1)+1}$-module structure.

The bijection is simply given as follows. Assume we are given an element $m\in\mathrm{Skew}_{1,n}^{*}(S\oplus S^{*})$. Since $m$ anticommutes with $\gamma_{S\oplus S^{*}}$, the restriction of $m$ to $S$ is a linear isomorphism

$\displaystyle m|_{S}\colon S\to S^{*}.$

We define the associated mass form $m_{\mathrm{form}}\colon S\times S\to\mathbb{R}$ by

(1.5)

$\displaystyle m_{\mathrm{form}}(s_{1},s_{2}):=\langle m|_{S}(s_{1}),s_{2}\rangle,$

where $\langle\cdot,\cdot\rangle$ is the duality pairing. Then we can check that this assignment $m\mapsto m_{\mathrm{form}}$ gives the desired bijection. For details see the proof of [Freed:2016rqq, Lemma 9.55].

Elements of our differential model $\widehat{KO}^{1,n}_{-}(X)$ is represented by a quadruple $(S,m_{0},m_{1},\eta)$, where $S$ is a $Cl_{1,n}$-module with inner product, $m_{0},m_{1}\in C^{\infty}(X,\mathrm{Skew}_{1,n}^{*}(S))$, and $\eta\in\Omega^{4\mathbb{Z}+n}(X)/\mathrm{Im}d$ an additional data of a differential form. Suppose that we are given a quadruple

$\displaystyle(S,\Gamma,m_{\mathrm{form},0},m_{\mathrm{form},1}),$

where $S$ and $\Gamma$ are as above and $m_{\mathrm{form},i}=\{m_{\mathrm{form},i}(x)\}_{x\in X}$, $i=0,1$, are smooth families of mass forms on $S$ parametrized by $X$. Then, by Lemma 1.4, we get an element

(1.6)

$\displaystyle[S\oplus S^{*},m_{0},m_{1},0]\in\widehat{KO}_{-}^{1,n}(X).$

In this way, our groups $\widehat{KO}^{1,n}_{-}(X)$ can be regarded as classifying pairs of smooth families of (nondegenerate) fermionic mass terms on $n$-dimensional Minkovski spacetime.³³3Typically in the physics literature, we often have a fixed constant mass term $m_{*}$. In such a case, we set $m_{0}:=m_{*}$ and regard $m_{1}$ as a single variable. Our model $\widehat{KO}^{1,n}$ is a differential extension of the topological $KO$-theory $KO_{-}^{1,n}\simeq KO^{n-3}$. On the topological level, the element (1.6) corresponds to the element $[S\oplus S^{*},m_{0},m_{1}]\in KO_{-}^{1,n}(X)\simeq KO^{n-3}(X)$, which recovers the well-known topological classifications of mass terms by the $KO$-theory.

Now we explain further perspectives. We expect that the further development of our differential $KO$ and $K$-theories, in particular the theory of differential pushforwards, would lead to an understanding of the long-range effective theories of massive fermions in terms of the differential refinement of the Anderson dual to the Atiyah-Bott-Shapiro maps.

A belief in the community of physicists is that deformation classes of invertible field theories should be classified by generalized cohomology theories. This idea is proposed in a lecture of Kitaev as reviewed in [GaiottoFreyd2019], and is further developed in [freed2014shortrange] and [Freed:2016rqq] from a mathematical viewpoint. Moreover, it has also been noticed that differential cohomology theories give refined classifications of such theories.

In the case of the theory on massive fermions, assume we have data of $S$, $\Gamma$ as in the last subsubsection, and fix a mass term $m_{*}$ (footnote 3). By the process of the Wick rotation, we produce the corresponding theory on Euclidean Spin manifolds as follows. We consider the complexification $Cl_{1,n-1}\otimes_{\mathbb{R}}\mathbb{C}=\mathbb{C}l_{n}$, which has the Riemannian Clifford algebra $Cl_{0,n}$ as a subalgebra. Then the Riemannian Spin group $\mathrm{Spin}_{n}$ acts on the complexification $S^{\mathbb{C}}$ of $S$, and the $\mathbb{Z}_{2}$-graded $Cl_{1,n-1}$-module structure on $S\oplus S^{*}$ explained in Subsubsection 1.1.1 induces the $\mathbb{Z}_{2}$-graded $Cl_{0,n}$-module structure on $S_{\mathbb{C}}\oplus S_{\mathbb{C}}^{*}$. If we have a closed $n$-dimensional $\mathrm{Spin}_{n}$-manifold $X$ with a Spin-connection $\nabla$ (regarded as a “Wick-rotated spacetime”), we form the associated bundle $\not{S}_{\mathbb{C}}$ to $S$, and the Dirac operator is given by

(1.7)

$\displaystyle\not{D}=c\circ\nabla\colon C^{\infty}(X;\not{S}_{\mathbb{C}}\oplus\not{S}_{\mathbb{C}}^{*})\to C^{\infty}(X;\not{S}_{\mathbb{C}}\oplus\not{S}_{\mathbb{C}}^{*}),$

where $c$ is the Clifford multiplication. Since $\not{D}$ anticommutes with $\gamma_{\not{S}_{\mathbb{C}}\oplus\not{S}_{\mathbb{C}}^{*}}$, it restricts to an operator from $\not{S}_{\mathbb{C}}$ to $\not{S}_{\mathbb{C}}^{*}$. Given a mass term $m=\{m(x)\}_{x\in X}$ parametrized by $X$, we get the massive Dirac operator,

(1.8)

$\displaystyle\not{D}+m\colon C^{\infty}(X;\not{S}_{\mathbb{C}})\to C^{\infty}(X;\not{S}_{\mathbb{C}}^{*}),$

which gives a formally skew-symmetric operator. Then the associated Lagrangian density on $X$ is

(1.9)

$\displaystyle L(\psi,m)=\frac{1}{2}\left\langle(\not{D}+m)\psi,\psi\right\rangle|dvol_{X}|$

for $\psi\in C^{\infty}(X;\not{S}_{\mathbb{C}})$. Physicists believe that the following expression makes sense and call the partition function for the massive fermions,

(1.10)

$\displaystyle\mathcal{Z}(m)=\frac{\int\mathcal{D}\psi\exp\left(-\int_{X}L(\psi,m)\right)}{\int\mathcal{D}\psi\exp\left(-\int_{X}L(\psi,m_{*})\right)},$

which is formally equal to the quotient of the Pfaffians of the massive Dirac operators,

(1.11)

$\displaystyle\frac{\mathrm{Pf}(\not{D}+m)}{\mathrm{Pf}(\not{D}+m_{*})}.$

The nondegeneracy of the mass term $m$ implies that this theory is gapped, and the long-range limit is an invertible theory. The corresponding element $[S\oplus S^{*},m_{*},m,0]\in\widehat{KO}_{-}^{1,n}(X)$ in (1.6) in our model is regarded as classifying this invertible theory.

Moreover, we expect that the theory of differential pushforwards in our model $\widehat{KO}_{-}$ should give a mathematical interpretation of the complex phase of the partition function (1.10) (1.11) in the long-range limit. For the differential $K$-theory, in the vector-bundle model by Freed-Lott [FL2010] and the geometric-cycle model by Bunke-Schick [BunkeSchicksmoothK], the differential pushforward along the map $p_{X}\colon X\to\mathrm{pt}$ for $(2k-1)$-dimensional closed Spin${}^{c}$ manifolds $X$ with Spin connections $\nabla$,

$\displaystyle(p_{X},\nabla)_{*}\colon\widehat{K}^{0}(X)\to\widehat{K}^{-(2k-1)}(\mathrm{pt})\simeq\mathbb{R}/\mathbb{Z},$

is given by the reduced eta invariants of twisted Dirac operators. Indeed, in a simplest case where $m$ is constant, in the limit $m\to\infty$ the quantity (1.11) is known to be given in terms of the eta invariants of the (massless) Dirac operator [Witten:2019bou]. Abstractly, we have a canonical way to define pushforwards in multiplicative differential refinements of $KO$-theories for Spin-oriented proper submersions with connection [Yamashita2021, Appendix]. The problem is how to describe the pushforward explicitly; for example such a description is obtained in the model by Grady-Sati [GradySatiDiffKO]. We expect that the differential pushfowards in $\widehat{KO}_{-}$ for $n$-dimensional Spin manifolds $X$,

(1.12)

$\displaystyle(p_{X},\nabla)_{*}\colon\widehat{KO}_{-}^{1,n}(X)\simeq\widehat{KO}^{n-3}(X)\to\widehat{K}^{-3}(\mathrm{pt})\simeq\mathbb{R}/\mathbb{Z},$

would be described, and the complex phase of (1.11) can be understood in terms of the image of the element $[S\oplus S^{*},m_{*},m,0]\in\widehat{KO}_{-}^{1,n}(X)$. For example such a picture is compatible with the principle that, in invertible theories, the variation of partition functions under smooth variation of the geometric structures on manifolds is given by an integration of some locally-constructed differential forms. Indeed, it is a general feature, called the Bordism formula [Bunke2013, Problem 4.235] of pushforwards in differential cohomology theory, that the image under the differential pushforwards varies by integrations of appropriate characteristic forms.

Finally we comment on the relation with the Anderson duality. Freed and Hopkins [Freed:2016rqq, Conjecture 8.37] conjectured that the deformation classes of Wick-rotated, fully extended reflection positive $n$-dimensional invertible theories on $G$-manifolds are classified by $(I\Omega^{G})^{n+1}(\mathrm{pt})$, where $I\Omega^{G}$ is a generalized cohomology theory called the Anderson dual to $G$-bordism theory. Motivated by this conjecture, in [YamashitaYonekura2021] Yonekura and one of the authors of the present paper gave a model $\widehat{I\Omega^{G}_{\mathrm{dR}}}$ for a differential extension of $I\Omega^{G}$ with a physical interpretation. An element in $(\widehat{I\Omega^{G}_{\mathrm{dR}}})^{n+1}(X)$ is given by a pair $(\omega,h)$, where $h$ is a map which assigns an $\mathbb{R}/\mathbb{Z}$-value to an $n$-dimensional differential $G$-manifold with a map to $X$ and plays the role of partition functions, and $\omega$ is an element in $\Omega^{*}_{\mathrm{clo}}(X)\otimes_{\mathbb{R}}H^{*}(MTG;\mathbb{R})$ of total degree $(n+1)$ which describes the variations of $h$. Moreover, it is shown in [Yamashita2021] that, in terms of differential pushforwards, we can construct a transformation of differential cohomology theories which refines the Anderson dual of multiplicative genera.

In the case of the theory on massive fermions, the conjecture by Freed-Hopkins [Freed:2016rqq, Conjecture 9.70] states that, on the topological level, the map which assigns the class of the data $(S,\Gamma,m_{*},m)$ the deformation class of the corresponding invertible theory as above should coincide with the composition

(1.13)

$\displaystyle KO^{n-3}(\mathrm{pt})\xrightarrow{\gamma_{KO}}IKO^{n+1}(\mathrm{pt})\xrightarrow{I\mathrm{ABS}}(I\Omega^{\mathrm{Spin}})^{n+1}(\mathrm{pt}),$

where $\gamma_{KO}\colon KO^{*}\simeq IKO^{*+4}$ is the Anderson duality of $KO$ which shifts the degree by $4$, and $I\mathrm{ABS}$ is the Anderson dual to the Atiyah-Bott-Shapiro map $\mathrm{ABS}\colon MT\mathrm{Spin}\to KO$.

Actually, the general theory in [Yamashita2021] applied to this case provides the differential refinement of (1.13). The construction in [Yamashita2021] gives the transformation

(1.14)

$\displaystyle\Phi_{\mathrm{ABS}}(-\otimes\gamma_{KO})\colon\widehat{KO}^{n-3}(X)\to\widehat{I\Omega^{\mathrm{Spin}}}^{n+1}(X),$

which refines (1.13). For an element in $\widehat{KO}^{n-3}(X)$, the map (1.13) assigns the element whose partition function is given in terms of the pushforwards of the pullbacks of this element in $\widehat{KO}$ to Spin manifolds with connections. This is indeed the expectation that we explained above for differential pushforwards (1.12) in $\widehat{KO}_{-}$. In this way, the authors believe that the understanding of the differential pushfowards in $\widehat{KO}_{-}$, combined with the general results in [YamashitaYonekura2021] and [Yamashita2021], would give a verification and a differential refinement of the statement in [Freed:2016rqq, Conjecture 9.70]. This deserves a promising future work.