---
title: Real Monopole Floer Homology
url: https://www.emergentmind.com/topics/real-monopole-floer-homology
type: topic
---

# Real Monopole Floer Homology

Searching arXiv for recent and foundational papers on real, involutive, and \(\mathrm{Pin}(2)\)-equivariant monopole Floer homology.
Real monopole Floer homology is a conjugation-sensitive refinement of monopole Floer theory in which Seiberg–Witten configurations are endowed with an anti-linear symmetry and the resulting Floer package is built from the corresponding invariant sector. In the literature, the phrase refers to several closely related constructions rather than a single universal formalism. One strand treats self-conjugate \(\mathrm{spin}^c\) structures on closed \(3\)-manifolds and produces involutive or \(\mathrm{Pin}(2)\)-equivariant monopole Floer groups from the conjugation symmetry on the configuration space [1610.08866], [1404.4561]. Another, more recent strand studies links via branched double covers equipped with the deck involution and defines genuinely real groups \(\widehat{\HMR}\), \(\widecheck{\HMR}\), \(\overline{\HMR}\), and a based-link mapping-cone refinement \(\widetilde{HMR}\) [2304.01742], [2406.00152]. Across these variants, the defining feature is that ordinary monopole Floer theory is augmented by data invisible to the purely \(S^1\)-equivariant theory, typically coming from conjugation, quaternionic symmetry, or real \(\mathrm{spin}^c\) structures.

## 1. Definition and geometric input

The geometric source of real monopole Floer theory is the conjugation symmetry of Seiberg–Witten theory. In the involutive formulation, the basic map on configurations is
\[
\jmath:\mathcal C(Y,s)\to \mathcal C(Y,\bar s), \qquad \jmath(B,\Psi)=(\bar B,\Psi\cdot j),
\]
where \(B\) is a connection, \(\bar B\) is the conjugate connection, and \(\Psi\cdot j\) uses the quaternionic structure on the spinor representation [1610.08866]. When \(s=\bar s\), this becomes an involutive symmetry of a single Floer complex up to continuation, and the corresponding theory is a \(\mathbb Z/2\)-equivariant enhancement of ordinary monopole Floer homology [1610.08866].

In the \(\mathrm{Pin}(2)\)-equivariant formulation, one works with a closed oriented \(3\)-manifold \(Y\) equipped with a self-conjugate spin\(^c\) structure \(\mathfrak s\). The extra symmetry arises because the spinor bundle is quaternionic, and the family of Dirac operators over the torus of reducibles becomes equivariant under the involution induced by conjugation [1708.07879]. The relevant symmetry group is
\[
\mathrm{Pin}(2)=S^1\cup j\cdot S^1\subset \mathbb H,
\]
and the bar version \(\overline{HS}_*(Y,\mathfrak s)\) is a module over
\[
\mathcal{R}= \mathbb{F}[V,Q]/Q^3,\qquad \deg(V)=-4,\qquad \deg(Q)=-1
\]
with \(\mathbb F=\mathbb Z/2\mathbb Z\) [1708.07879].

The recent link-theoretic theory is formulated directly for an involutive \(3\)-manifold. If \(Y\) is a closed oriented \(3\)-manifold with an orientation-preserving involution \(\iota:Y\to Y\), a real spin\(^c\) structure on \((Y,\iota)\) is a spin\(^c\) structure \(\mathfrak s=(S,\rho)\) together with an anti-linear involution
\[
\tau:S\to S
\]
covering \(\iota\), compatible with the Hermitian metric and Clifford multiplication [2304.01742]. For a link \(K\subset S^3\), the main example is the branched double cover
\[
Y=\Sigma_2(S^3,K)
\]
with deck involution \(\iota_{\mathrm{deck}}\). In this setting, the real Floer groups are defined on the \(\tau\)-invariant configuration space and its blown-up counterpart [2304.01742]. This construction is then refined for based links \((K,p)\) by the tilde theory \(\widetilde{HMR}_\bullet(K,p)\) [2406.00152].

A common misconception is to identify “real” with real coefficients or characteristic-zero de Rham models. That interpretation is explicitly excluded in the integral computation of \(\overline{HM}_*\), which is not a Real, involutive, quaternionic, or \(\mathrm{Pin}(2)\)-equivariant theory [2108.00984]. The same distinction applies to several boundary or Fueter-type monopole theories, which are monopole-based but not real in the conjugation-equivariant sense [2305.09456], [2005.04333].

## 2. Ordinary monopole Floer theory as the underlying model

Real refinements are built on the blow-up formalism of ordinary monopole Floer homology. For a closed oriented \(3\)-manifold \(Y\) with spin\(^c\) structure \(s\), the configuration space is
\[
\mathcal C(Y)=\Omega^1(Y;i\mathbb R)\oplus \Gamma(S),
\]
with Seiberg–Witten vector field
\[
X(a,\phi)=(*da+\tau(\phi,\phi),D_a\phi),
\]
the \(L^2\)-gradient of the Chern–Simons–Dirac functional [1603.00582]. Because the gauge action is not free at reducibles, one passes to the blown-up configuration space
\[
C^\sigma(Y)=\{(a,s,\phi)\mid s\ge 0,\ \|\phi\|_{L^2}=1\},
\]
whose quotient is a manifold with boundary [1603.00582].

The resulting chain complex has three types of generators: irreducibles \(\mathfrak C^o\), boundary-stable reducibles \(\mathfrak C^s\), and boundary-unstable reducibles \(\mathfrak C^u\). The “to” complex is
\[
\check C(Y,s,\mathfrak q)=C^o\oplus C^s,
\]
with differential
\[
\check{\partial}= \begin{bmatrix} \partial^o_o & -\partial^u_o\bar\partial^s_u\\ \partial^o_s & \bar\partial^s_s-\partial^u_s\bar\partial^s_u \end{bmatrix}
\]
[1603.00582]. The \(U\)-action is defined by counting cut-down trajectories and lowers degree by \(2\) [1603.00582].

This ordinary theory admits a spectrum-level interpretation. For rational homology spheres,
\[
\check{HM}_*(Y,s)\cong \widetilde H^{S^1}_*(SWF(Y,s))
\]
as absolutely graded \(\mathbb Z[U]\)-modules [1603.00582]. This is structurally important for real refinements: it provides a template in which enhanced symmetry should be encoded by a more equivariant Floer spectrum, with the homology theory obtained by applying the corresponding equivariant functor [1603.00582]. This suggests that real monopole Floer homology is best viewed not merely as an involution on \(\check{HM}\), but as an equivariant shadow of a stronger spectrum-level object.

## 3. Involutive and \(\mathrm{Pin}(2)\)-equivariant constructions

The simplest real enhancement is involutive monopole Floer homology. For a self-conjugate spin\(^c\) structure, the conjugation map induces a chain endomorphism \(\tilde\iota\), and the involutive chain complex is defined as the mapping cone
\[
\widetilde{CI}(Y,p,s,q,\hat q,k)=\operatorname{Cone}\!\bigl( Q(1+\tilde\iota): C(Y,p,s,q,\hat q)\to Q\cdot C(Y,p,s,q,\hat q) \bigr),
\]
over \(\mathbb F[Q]/Q^2\), with \(Q\) of degree \(-1\) [1610.08866]. The resulting homology \(\widetilde{HMI}_*(Y,p,s)\) fits into an exact triangle relating it to the ordinary based tilde theory \(\widetilde{HM}_*(Y,p,s)\) [1610.08866]. For non-self-conjugate spin\(^c\) structures, the involution exchanges \(s\) and \(\bar s\), and the involutive theory reduces to
\[
\widetilde{HMI}_*(Y,[s])\cong \widetilde{HM}_*(Y,s)\oplus \widetilde{HM}_*(Y,s)[-1],
\]
with trivial \(Q\)-action [1610.08866].

The fuller equivariant refinement is \(\mathrm{Pin}(2)\)-monopole Floer homology. In the Morse–Bott approach, one generalizes Kronheimer–Mrowka’s construction to gradient flows with Morse–Bott singularities and then restricts to the special case of self-conjugate spin\(^c\) structures, where the conjugation symmetry produces the \(\mathrm{Pin}(2)\)-equivariant package [1404.4561]. The resulting groups
\[
\widetilde{HS}_\bullet(Y,\mathfrak{s}),\qquad \widehat{HS}_\bullet(Y,\mathfrak{s}),\qquad \overline{HS}_\bullet(Y,\mathfrak{s})
\]
are modules over
\[
\mathcal R=\mathbb F[[V]][Q]/(Q^3),\qquad \deg V=-4,\ \deg Q=-1
\]
and fit into a long exact sequence analogous to the usual monopole Floer triangle [1404.4561]. The Morse–Bott formalism is essential because \(\jmath\)-equivariant perturbations do not make all critical points Morse nondegenerate; instead, one obtains Bott families, especially reducible \(S^2\)-families over fixed reducibles [1404.4561].

The bar version \(\overline{HS}_*(Y,\mathfrak s)\) admits a concrete topological classification in terms of ordinary cohomological data plus Rokhlin data. For a closed oriented \(3\)-manifold with self-conjugate spin\(^c\) structure \(\mathfrak s\), \(\overline{HS}_*(Y,\mathfrak s)\) is determined, up to overall grading shift, by the triple cup product \(\cup^3_Y\) and the Rokhlin map \(\mu_{\mathfrak s}\) on the set of spin structures inducing \(\mathfrak s\) [1708.07879]. The fixed points of the involution on the torus of flat connections are precisely the spin connections, and there are exactly \(2^{b_1(Y)}\) of them [1708.07879]. The extra equivariant information is torsion in \(KQ^1(\mathbb T)\), identified by mod \(2\) spectral flow and hence by Rokhlin invariants [1708.07879].

A further structural development is the \(A_\infty\)-package for \(\mathrm{Pin}(2)\)-monopole Floer chains. The chain complex of \(S^3\) carries a partially defined \(A_\infty\)-algebra structure, and for every \(Y\) the chain complex is a partially defined \(A_\infty\)-module over it [1605.03137]. The connected sum formula is then expressed by a quasi-isomorphism
\[
\Phi: C(Y_0,s_0)\boxtimes C(Y_1,s_1)^{\mathrm{opp}} \longrightarrow C(Y_0\#Y_1,s_0\#s_1)\langle -1\rangle,
\]
and the associated Eilenberg–Moore spectral sequence has
\[
E^2_{*,*}\cong \operatorname{Tor}^{R}_{*,*}\big(\widetilde{HS}_\bullet(Y_0,s_0),\widetilde{HS}_\bullet(Y_1,s_1)\big)
\]
[1605.03137].

## 4. Real monopole Floer homology for links and branched double covers

The link-theoretic theory \(\HMR^\circ\) is a real version of monopole Floer homology built from the fixed-point locus of an involution. For a closed oriented \(3\)-manifold \(Y\) with orientation-preserving involution \(\iota\), a real spin\(^c\) structure \((\mathfrak s,\tau)\) determines the real Seiberg–Witten configuration space
\[
\mathcal C(Y,\mathfrak s,\tau)=\mathcal A(Y,\mathfrak s,\tau)\times \Gamma(S)^\tau,
\]
where \(\mathcal A(Y,\mathfrak s,\tau)\) is modeled on the \(\iota^*\)-anti-invariant imaginary-valued \(1\)-forms, and \(\Gamma(S)^\tau\) consists of the \(\tau\)-invariant spinors [2304.01742]. The corresponding blown-up quotient \(\mathcal B^\sigma(Y,\mathfrak s,\tau)\) is a manifold with boundary, and the real chain complexes are defined exactly in the same three-flavor pattern as in ordinary monopole Floer homology [2304.01742].

For a link \(K\subset S^3\), one writes
\[
\HMR^\circ_*(K;\mathfrak s):=\HMR^\circ_*(\Sigma_2(S^3,K),\iota_{\mathrm{deck}};\mathfrak s,\tau),
\]
since every spin\(^c\) structure on the branched double cover supports a unique compatible real structure up to equivalence [2304.01742]. For links with \(n\) components, the completed groups are modules over
\[
\mathcal R_n=\frac{\mathbb F_2[[\upsilon_1,\dots,\upsilon_n]]}{\upsilon_i^2=\upsilon_j^2},
\]
with each \(\upsilon_i\) of degree \(-1\) and \(U=\upsilon_i^2\) [2304.01742].

The based-link refinement is the tilde real monopole Floer homology
\[
\widetilde{HMR}_\bullet(K,p),
\]
defined as the mapping cone of the degree \((-1)\) operator
\[
\upsilon_p:HMR^\circ_\bullet(K)\to HMR^\circ_\bullet(K)
\]
[2406.00152]. On the chain level,
\[
\widetilde C(K)=\widecheck C(K)\oplus \widecheck C(K), \qquad \widetilde\partial= \begin{bmatrix} \check\partial & 0\\ \check m(\upsilon\mid [0,1]\times K) & \check\partial \end{bmatrix},
\]
and its homology is \(\widetilde{HMR}_\bullet(K,p)\) [2406.00152]. This yields a long exact sequence
\[
\cdots \to \widetilde{HMR}_{*}(K) \to \widecheck{HMR}_{*}(K) \xrightarrow{\upsilon} \widecheck{HMR}_{*}(K)\to \cdots
\]
[2406.00152].

The real theory also has an absolute grading in the torsion case. If \(K\subset S^3\), \(Y=\Sigma_2(S^3,K)\), and \(\mathfrak s\) is torsion, the paper defines
\[
\gr^{\mathbb Q}([\mathfrak a])= -\ind_z([\mathfrak a_0],\Sigma,[\mathfrak a]) +\frac18\big(c_1(\mathfrak s)^2-\sigma(W)\big)-\iota(\Sigma)
\]
for a critical point \([\mathfrak a]\), where \(\Sigma\) is a cobordism from the unknot, \(W\) is its branched double cover, and
\[
\iota(\Sigma)=\frac{\chi(W)+\sigma(W)+b_1(Y_+)-b_1(Y_-)}{2}
\]
[2304.01742]. This grading leads to a real Frøyshov invariant \(h_R(K,\mathfrak s)\), defined from the lowest absolute grading in the image of
\[
i_*:\overline{\HMR}_\bullet(K,\mathfrak s)\to \widecheck{\HMR}_\bullet(K,\mathfrak s)
\]
[2304.01742].

## 5. Algebraic structures, exact triangles, and computational consequences

A recurrent structural theme is that the real refinements introduce new coefficient algebras and exact sequences. In ordinary monopole Floer theory the dominant formal variable is \(U\) of degree \(-2\); in involutive theory one adds \(Q\) with \(Q^2=0\) [1610.08866]; in \(\mathrm{Pin}(2)\)-theory the ring becomes
\[
R=\mathbb F[[V]][Q]/(Q^3)
\]
with \(\deg(V)=-4\), \(\deg(Q)=-1\) [1605.03137], [1404.4561]; in the link-theoretic real theory one instead obtains the \(\upsilon_i\)-module structure with \(\deg(\upsilon_i)=-1\) [2304.01742].

The \(\mathrm{Pin}(2)\)-package has a Gysin exact triangle relating \(\widetilde{HS}\) to ordinary \(\widetilde{HM}\), with \(Q\) acting on \(\widetilde{HS}\) and \(V\) acting on \(\widetilde{HM}\) as \(U^2\) [1809.01637]. This triangle is a basic computational tool because many Massey products in \(\widetilde{HS}\) can be identified in terms of the usual \(U\)-action on \(\widetilde{HM}\) [1809.01637]. The paper gives explicit formulas such as
\[
\langle Q,x,V\rangle=\Phi_1(x),\qquad \langle x,Q,Q^2\rangle=\Phi_2(x),\qquad \langle x,Q^2,Q\rangle=\Phi_3(x)
\]
under the stated annihilation hypotheses [1809.01637].

These higher products are not formal artifacts. The \(\mathcal A_\infty\)-algebra underlying \(\mathrm{Pin}(2)\)-monopole Floer homology is non-formal, and a basic witness is the four-fold Massey product
\[
\langle Q,Q^2,Q,Q^2\rangle = V
\]
[1809.01637]. This non-formality is already visible for \(S^3\) and controls higher differentials and extension problems in the connected-sum Eilenberg–Moore spectral sequence [1605.03137], [1809.01637]. A plausible implication is that the “real” information carried by \(\mathrm{Pin}(2)\)-theory is encoded not only in module structure but also in higher coherences.

For the bar version \(\overline{HS}\), the spectral sequence description in terms of the Rokhlin map is particularly concrete. If \(x_1,\dots,x_n\) is a basis of \(H^1(Y;\mathbb Z)\otimes\mathbb F\) and \(s_I\) denotes the spin structure corresponding to a subset \(I\), then there is a spectral sequence converging to \(\overline{HS}_*(Y,\mathfrak s)\) with
\[
E^1=\bigoplus_I \widetilde{\mathcal{R}\langle -2\mu(s_I)+|I|\rangle},
\]
and the differential \(d^1\) is nonzero exactly when \(\mu(s_I)\neq \mu(s_{I'})\) for \(I'\subset I\) and \(|I'|=|I|-1\), in which case it is, up to grading shift, multiplication by \(Q^2\) [1708.07879].

## 6. Computations, relationships, and scope

Several families admit explicit calculations. For involutive monopole Floer homology,
\[
\widetilde{HMI}_*(S^3,p)=\mathbb F[Q]/Q^2
\]
up to the stated grading normalization, and
\[
\widetilde{HMI}_*(S^2\times S^1)\cong \Lambda^*(\mathbb F\langle\gamma\rangle)\otimes \widetilde{HMI}_*(S^3)
\]
with \(\gamma\) in degree \(-1\) [1610.08866]. For \(\mathrm{Pin}(2)\)-theory, if \(b_1=0\), then up to grading shift
\[
\overline{HS}_*(Y,\mathfrak s)=\widetilde{\mathcal R},
\]
while for \(b_1=1\) there are exactly two possibilities depending on whether the two spin structures have the same Rokhlin invariant or not [1708.07879]. For \(Y=T^3\), seven spin structures have Rokhlin invariant \(0\) and one has Rokhlin invariant \(1\), and the corresponding \(\overline{HS}\) can be written explicitly in terms of the cubic Rokhlin map [1708.07879].

Mapping tori provide a different bridge between ordinary and real refinements. For an automorphism \(\varphi\) of a compact Riemann surface \(\Sigma\) with quotient \(\mathbb P^1\), \(\varphi\)-invariant theta characteristics \(L\) correspond naturally to self-conjugate spin\(^c\) structures \(\mathfrak s_L\) on the mapping torus \(M_\varphi\) [2205.04351]. The monopole Floer homology of \((M_\varphi,\mathfrak s_L)\) is explicitly determined by the eigenvalues of the lifted action of \(\varphi\) on \(H^0(L)\), and the paper remarks that the chain-level computation respects the extra symmetry and can be used to compute \(\widetilde{HS}_\bullet(M_\varphi,\mathfrak s_L)\) as well [2205.04351]. This suggests a close relationship between algebraic geometry of theta characteristics and the self-conjugate sectors in real or \(\mathrm{Pin}(2)\)-equivariant Floer theory.

The recent link-theoretic real theory adds two major computational results. First, \(\widetilde{HMR}\) satisfies an unoriented skein exact triangle for skein triples \((K_0,K_1,K_2)\) [2304.01742], [2406.00152]. Second, it is the target of a spectral sequence
\[
E_2 \cong Khr(\overline K;F_2)\Longrightarrow \widetilde{HMR}(K,p),
\]
where \(Khr(\overline K)\) is the reduced Khovanov homology of the mirror [2406.00152]. The \(E_1\)-page is assembled from unlink resolutions, and for the \((n+1)\)-component unlink one has
\[
\widetilde{HMR}_\bullet(\mathcal U_{n+1},p)\cong H_*(T^n;F_2)\cong \Lambda^*[x_1,\dots,x_n]
\]
[2406.00152]. The Euler characteristic of \(\widetilde{HMR}\) is equal to Miyazawa’s invariant:
\[
|\chi(\widetilde{HMR}_\bullet(K,p,\mathfrak s,\tau))|=|\deg(K)|
\]
for nonzero determinant links [2406.00152]. For the pretzel knot \(P(-2,3,7)\), the paper computes
\[
|\chi(\widetilde{HMR}_\bullet(K))|=3,
\]
while
\[
\left|\chi(\widetilde{HM}_\bullet(-\Sigma(2,3,7)))\right|=1,
\]
showing that the real theory is not merely ordinary monopole Floer homology of the branched double cover [2406.00152].

The scope of the subject is therefore twofold. On one side, “real monopole Floer homology” denotes involutive and \(\mathrm{Pin}(2)\)-equivariant refinements of monopole Floer theory for self-conjugate spin\(^c\) structures, governed by conjugation, Rokhlin invariants, and higher algebra [1610.08866], [1708.07879], [1605.03137]. On the other, it denotes a link-theoretic theory built from branched double covers with deck involution and real spin\(^c\) structures, endowed with skein exact triangles, Frøyshov-type invariants, and a Khovanov spectral sequence [2304.01742], [2406.00152]. What unifies these constructions is the principle that anti-linear symmetry in Seiberg–Witten theory yields Floer-theoretic information inaccessible to the ordinary \(S^1\)-equivariant package.

Source: https://www.emergentmind.com/topics/real-monopole-floer-homology