Motivic Coaction of Multiple Zeta Values

• The topic defines motivic coaction as the mechanism through which the motivic Galois group acts on multiple zeta values via a graded Hopf algebra structure.
• It highlights a non-commutative f‐alphabet representation that linearizes intricate MZV relations through deconcatenation and canonical mapping.
• The framework underpins amplitude expansions in quantum field and string theory by integrating depth filtrations, derivations, and period polynomials.

Motivic coaction of multiple zeta values (MZVs) refers to the structure arising from the action of the motivic Galois group on the algebra generated by motivic versions of MZVs, captured via a Hopf algebra coaction. This structure refines the known algebraic relations among MZVs, connects to deep phenomena in quantum field and string theory, and encodes Galois symmetries fundamental to mathematics and mathematical physics.

1. Motivic MZVs and the Hopf Algebra Framework

The lift from classical MZVs $\zeta(n_1,\ldots,n_r)$ to motivic MZVs $\zeta^m(n_1,\ldots,n_r)$ is effected by passing from periods to objects in a graded commutative Hopf algebra $\mathcal{H}$. These elements retain additional motivic information and admit a canonical coaction—dually, a Hopf algebra coproduct—that reflects the Galois symmetries of mixed Tate motives over $\mathbb{Z}$ (Schlotterer et al., 2012, Brown, 2014).

The basic motivic iterated integral, representing a motivic MZV, is $I^m(0; a_1,\ldots,a_n;1)$ for $a_i \in \{0,1\}$ (or in cyclotomic settings, roots of unity). The coproduct on such integrals is given by

$$\Delta I^m(0; a_1,\ldots,a_n;1) = \sum_{0 = i_0 < \cdots < i_k = n} I^m(0; a_{i_1},\ldots, a_{i_k}; 1) \otimes \prod_{p=0}^{k-1} I^m(a_{i_p}; a_{i_p+1}, ..., a_{i_{p+1}-1}; a_{i_{p+1}})$$

which decomposes the depth and weight structure in a way retaining all algebraic relations (Schlotterer et al., 2012, Brown, 2014).

2. Structure, Coaction, and the Non-Commutative Hopf Algebra

Motivic MZVs form a graded commutative Hopf algebra, but a central technical advance is the existence of a canonical isomorphism to a non-commutative polynomial algebra in generators $f_{2n+1}$ (for odd $n\geq1$) and $f_2$. The isomorphism $\phi$ maps each motivic MZV to non-commutative words in these generators (f-alphabet), linearizing the web of identities among MZVs independent of basis choice,

$$\phi(F^m) = \sum_{k=0}^\infty f_2^k P_{2k} \cdot \sum_{p\geq0} \sum_{i_1,\ldots,i_p} f_{i_1} \ldots f_{i_p} M_{i_1} \ldots M_{i_p}$$

where $M_{i_j}$ and $P_{2k}$ encode kinematic (matrix) data in amplitude expansions (Schlotterer et al., 2012).

The coproduct (coaction) in the motivic setting is compatible with this structure. It generalizes the symbol of a transcendental function, capturing not only iterated integral data but also relations among pure and composite MZVs. The coaction in the f-alphabet is simply deconcatenation:

$$\Delta(f_{i_1} f_{i_2} \cdots f_{i_k}) = \sum_{j=0}^k f_{i_1} f_{i_2} \cdots f_{i_j} \otimes f_{i_{j+1}} \cdots f_{i_k}$$

every relation among motivic (hence numerical) MZVs is thus realized as a Hopf algebra identity.

3. Depth Filtration, Derivation Operators, and Period Polynomials

MZVs possess a depth filtration: for $\zeta(n_1,\ldots,n_r)$, the depth is $r$ and weight $w = \sum n_i$. Motivic coaction acts compatibly with depth, and its infinitesimal version yields derivations $d_{2n+1}$ and $c_{2k}$ that extract the primitive components corresponding to odd and even weight MZVs,

$d_{2n+1} \sim M_{2n+1}, \quad c_{2k} \sim P_{2k}$

matching precisely the operators in the open superstring $\alpha'$-expansion (Schlotterer et al., 2012).

Depth-graded considerations reveal that, while the depth-1 part is canonically generated by odd zeta elements, higher-depth spaces (notably from depth 4) may require further generators. These are constructed from period polynomials of modular forms—homogeneous polynomials $P(X,Y)$ satisfying:

$$P(X,Y) + P(Y,X) = 0, \quad P(X,Y) + P(X-Y, X) + P(-Y, X-Y) = 0, \quad P(X,0) = 0,$$

corresponding to the "exceptional" generators in the motivic Lie algebra (Brown, 2013).

4. Motivic Coaction and Amplitude Expansions

The motivic coaction structure linearizes and encodes all algebraic identities among the MZVs which appear as coefficients in the $\alpha'$-expansion of open superstring amplitudes:

$$F(\alpha') = 1 + \zeta(2) P_2 + \zeta(3) M_3 + \zeta(2)^2 P_4 + \zeta(5) M_5 + \cdots$$

By lifting all coefficients to their motivic versions, one obtains a motivic expansion whose structure maps isomorphically to words in the non-commutative alphabet and whose amplitude relations are automatically dictated by the coaction (Schlotterer et al., 2012). All rational coefficients multiplying zeta values in the expansion are incorporated by the Hopf algebra structure, independent of any preferred MZV basis.

In higher-point amplitudes and at high transcendental weight, motivic coaction encodes the intricate shuffle, stuffle, and other relations—including those coming from modular forms—not visible in the classical period data.

5. Relation to the Depth-Graded Motivic Lie Algebra

The depth-graded motivic Lie algebra, conjecturally free on odd-weight generators and "exceptional" period polynomial classes, is controlled by the coaction structure. The linearized double shuffle relations, when considered motivically, are given in terms of functional equations in the depth-graded polynomial representations. The homology of this Lie algebra is conjectured to be as simple as

$$H_1(\mathfrak{ls};\mathbb{Q}) \simeq \bigoplus_{n\geq1} (\mathbb{Q}\cdot \sigma_{2n+1} \oplus e(S_{2n})), \quad H_i(\mathfrak{ls};\mathbb{Q}) = 0 \ \forall i\geq3$$

(Brown, 2013), establishing a minimal set of generators and supporting the Broadhurst-Kreimer and Zagier dimension conjectures for MZVs.

The motivic coaction thus encodes, via its compatibility with the depth filtration and period polynomial inclusions, the deep connections between the algebra of MZVs, modular forms, and Galois symmetries.

6. Generalizations, Computations, and Applications

The motivic coaction can be explicitly computed for all (motivic) MZVs and generalizes to cyclotomic MZVs, $p$-adic analogues, and period variants arising from generalized iterated integrals (Jarossay, 2015, Li, 2020). Explicit formulas—in various "f-alphabet" settings or as derivations on iterated integral words—enable the direct, combinatorial computation of the motivic content of amplitudes, periods, and their relations.

In practical applications, the motivic coaction underpins the automorphic structure of scattering amplitudes, guides the search for canonical bases for period computations, and predicts rigidity in all functionally allowed relations among high-weight MZVs in physics computations (Schlotterer et al., 2012).

7. Significance and Broader Impact

The motivic coaction of multiple zeta values serves as a unifying algebraic and Galois-theoretic framework that:

• Encodes all polynomial relations among MZVs, including those from modular forms and amplitude computations;
• Provides the means to linearize and "basis-independent" manipulate the transcendental structure in quantum field theory and string theory;
• Allows reconstruction of all amplitude coefficients and their relations from motivic data alone, integrating modular and period polynomial information;
• Bridges number theory, arithmetic geometry, and mathematical physics through the explicit and computable action of the motivic Galois group.

In summary, the motivic coaction, through its Hopf-algebraic, non-commutative, and depth-filtered structures, organizes the complexity of MZV relations, exposes deep connections to modular forms, and manifests in the structural simplicity of physical amplitude expansions (Schlotterer et al., 2012, Brown, 2013, Brown, 2014).