π_*(TMF)-Valued Invariants in Topology & Physics

• π_*(TMF)-valued invariants are algebraic-topological quantities derived from topological modular forms, encoding information on modular forms, stable homotopy, and periodic phenomena.
• Descent theory and spectral sequences, including the Atiyah-Hirzebruch spectral sequence, provide the methodology to compute these invariants and classify exotic invertible modules.
• Field-theoretic constructions using supersymmetric models and Dirac–Ramond index theorems yield invariants that extend to low-dimensional topology, linking manifold invariants with TMF-modules.

${\pi}_*({\rm TMF})$-valued invariants are algebraic-topological quantities associated to topological modular forms (TMF), typically arising from the interplay of stable homotopy theory, derived algebraic geometry, low-dimensional topology, and mathematical physics. These invariants encode rich structural information accessible via descent-theoretic methods, spectral sequences, field-theoretic constructions, and physical models, with connections to the modular forms ring, the geometry of elliptic curves, and the classification of invertible modules and field theory phases.

1. Foundations: Topological Modular Forms and Homotopy Groups

Topological modular forms (TMF) is an $\mathbf{E}_\infty$-ring spectrum associated with the moduli stack of elliptic curves and embodies a powerful invariant in stable homotopy theory. Its homotopy groups, denoted ${\pi}_*({\rm TMF})$, are closely linked to the ring of integral modular forms and organize various periodic phenomena in the stable homotopy groups of spheres. The spectrum TMF plays a central role in multiplicative theory, classified via its Picard group and periodicity properties.

The homotopy groups:

• ${\pi}_*({\rm TMF})$: graded abelian groups with modular form structure;
• ${\pi}_*({\rm tmf})$: connective version encoding filtered modular data.

Invertible TMF-modules are detected via Picard groups: $\mathrm{Pic}(\mathrm{TMF}) \cong \mathbb{Z}/576$, generated by suspension; for the non-periodic variant, $$\mathrm{Pic}(\mathrm{Tmf}) \cong \mathbb{Z} \oplus \mathbb{Z}/24$$, admitting exotic modules not realized by suspension alone (Mathew et al., 2014).

2. Descent Theory and Picard Group Computations

The paper of TMF-valued invariants employs descent-theoretic techniques. An $\mathbf{E}_\infty$-ring $R$ is resolved by simpler $\mathbf{E}_\infty$-rings via cosimplicial diagrams from étale or Galois covers of a moduli stack, allowing algebraic determination of the Picard spectrum: $$\mathrm{pic}(R) \simeq \mathrm{Tot}\{\mathrm{pic}(R_i)\}$$ The descent spectral sequence is a principal computational device: $$E_2^{s,t} = H^s(X, \mathcal{V}^t) \implies \pi_{t-s} \mathrm{pic}(R)$$ where $X$ is, e.g., the moduli stack of elliptic curves. The coefficient sheaves $\mathcal{V}^t$ align with modular form line bundles, with $t=0$ giving $H^s(X, \mathbb{Z}/2)$ and odd $t \geq 3$ supplying powers of the Hodge bundle $\omega$.

A stable range identification equates truncated connective covers: $\tau_{[n, 2n-1]} R \simeq \tau_{[n, 2n-1]} gl_1(R)$ extending similarly for $\mathrm{pic}(R)$. A notable formula for descent differentials is: $\dot{d}_{t+1} (\bar{x}) = (d_{t+1}(x) + x^2)^-$ which introduces quadratic corrections to the unstable differential, a key ingredient in extension problems and the identification of exotic invertible modules.

For $\mathrm{TMF}$, the periodicity forces all invertible modules to be suspensions, yielding the cyclic Picard group. By contrast, $\mathrm{Tmf}$ exhibits extra torsion, reflected in the $24$-torsion summand, and exotic invertible modules corresponding to non-algebraic clutched line bundles.

3. Spectral Sequence Structures and Mahowald Invariants

Periodic families in stable homotopy theory, such as the $2$-primary homotopy $\beta$-family, are encoded via Mahowald invariants, which lift elements in low stems to higher ones. TMF-based approximations are constructed by computing Mahowald invariants in the tmf-spectrum, exploiting the splitting: $$\mathrm{tmf}^{tC_2} \simeq \prod_{i \in \mathbb{Z}} \Sigma^{8i} \mathrm{bo}$$ Calculations leverage the Atiyah-Hirzebruch spectral sequence (AHSS) for $\mathrm{tmf}^{tC_2}$, with differentials determined by cellular and Adams filtrations and Toda bracket relationships. The filtered Mahowald invariant machinery permits a stepwise refinement:

• The first nontrivial algebraic tmf-based invariant agrees with the first nonzero HF$_2$-filtered invariant.
• Higher filtrations are systematically excluded via lifting arguments.

Explicit formulas for tmf-based Mahowald invariants exhibit pronounced periodicity and are integral to organizing the $\beta$-family and Greek letter periodic elements: $$\begin{align} M_{\mathrm{tmf}}(2^0) &= 1[0], \ M_{\mathrm{tmf}}(2^1) &= \nu[-2], \ M_{\mathrm{tmf}}(2^3) &= (c_4 + \epsilon)[-4], \ M_{\mathrm{tmf}}(\beta) &= \bar{\kappa}[-12] \end{align}$$ with grading shifts tracking AHSS filtration levels (Quigley, 2019).

4. $\pi_*({\rm TMF})$-Valued Invariants from Field Theoretic Constructions

Supersymmetric field theories, particularly $2|1$-dimensional Euclidean field theories, facilitate the construction of TMF-valued invariants by exploiting geometric data of bordisms and superconnections. Field theory functors on categories of supermanifolds yield deformation invariants valued in (height $\leq 1$) approximations of TMF, represented by cocycles built from superconnection eigenbundles and analytic index data: $Z = T_n(\phi) = \sum_k q^k \cdot \mathrm{Ch}(A_k)$ where $A_k$ denotes the $k$th eigenspace. Restriction to nearly constant field theories produces invariants sensitive to torsion. In particular, the family index of the Dirac-Ramond operator recovers torsion classes in degrees $4k-1$, e.g., for 3d string manifolds, the invariant $\sigma(X)$ detects the generator of $\pi_3(\mathrm{TMF}) \cong \mathbb{Z}/24$ (Berwick-Evans, 2023).

Further, the elliptic Euler class is realized field-theoretically as

$$\mathrm{Euler}(V) = \varphi(q)^{-\dim V} \cdot [V \otimes \bigotimes_k \Lambda_{q^k} V]$$

relating the infinite free fermion construction to Riemann–Roch corrections.

Constructions of supersymmetric sigma models with string manifold targets use the Bismut–Ramond superconnection formalism, generating analytic index invariants that refine the Witten genus and align with TMF-index theorems.

5. TMF-Modules Associated to Low-dimensional Topology and Physics

Recent advances define families of $\pi_*({\rm TMF})$-valued invariants for closed 3-manifolds and 4-manifolds via TMF-module assignments. For a closed 3-manifold $Y$ bounded by a simply-connected 4-manifold $X$, the intersection pairing on $H_2(X)$ presents $H_1(Y)$ and encodes the torsion subgroup $\tau H_1(Y)$ via non-degenerate linking pairings: $$\lambda: \tau H_1(Y) \times \tau H_1(Y) \to \mathbb{Q}/\mathbb{Z}$$ This linking pairing, in concert with the free rank, determines the stable (±-equivalence) class of the corresponding bilinear form up to additions of hyperbolic summands. Invariants of unimodular bilinear forms take values in $\pi_*({\rm TMF})$, conjecturally generalizing the classical theta function and enabling gluing properties under 4-dimensional cobordisms.

For 4-manifold cobordisms, the difference of the linking pairings across boundaries is computed by intersection numbers in the interior, controlled by the annihilator of the torsion class: $$\lambda(a_0, b_0) - \lambda(a_1, b_1) \equiv -\frac{[\tilde{X}] \cdot [\tilde{Y}]}{\Delta^2} \mod 1$$ This framework supports applications in quantum field theory, such as distinguishing phases via TMF-module data (Gukov et al., 15 Sep 2025).

6. Implications and Applications

${\pi}_*({\rm TMF})$-valued invariants fundamentally intertwine the homotopical, geometric, and physical aspects of topology:

• They classify invertible modules, periodic families, and torsion phenomena in stable homotopy.
• Spectral and descent-theoretic methods enable explicit calculations and the detection of exotic modules.
• Field-theoretic and superconnection-based constructions realize expected invariants such as the elliptic Euler class and torsion index theorems.
• Topological quantum field theories leverage TMF-modules for manifold invariants and phase classification.

A plausible implication is that further development of TMF-valued TQFTs and their connection to 6-dimensional superconformal field theories may uncover new invariants for 4-manifolds, enhancing the interplay between algebraic geometry, topology, and physics.

7. Summary Table: Key Structural Features

Structure	Example Value	Defining Property
$\mathrm{Pic}(\mathrm{TMF})$	$\mathbb{Z}/576$	Cyclic, generated by suspension
$\mathrm{Pic}(\mathrm{Tmf})$	$\mathbb{Z} \oplus \mathbb{Z}/24$	Exotic invertible modules
Mahowald Invariant (tmf-based)	$M_{\mathrm{tmf}}(\beta) = \bar{\kappa}[-12]$	Chromatic height two approximation
Torsion Detecting Invariant	$\sigma(X) \in \mathbb{Z}/24$	Dirac–Ramond index for string manifold
Linking Pairing on $3$-manifolds	$$\lambda: \tau H_1(Y) \times \tau H_1(Y) \to \mathbb{Q}/\mathbb{Z}$$	Stable (±)-equivalence

These invariants serve as foundational tools for classification and detection problems in topology and mathematical physics. Their calculations are intimately tied to the interplay of descent theory, spectral sequences, field-theoretic index theorems, and algebraic geometry, forming a basis for advanced advancements in the paper of TMF and its associated homotopical invariants.