Topological sectors for heterotic M5-brane charges under Hypothesis H

Assuming Fiorenza-Sati-Schreiber's Hypothesis H, on the charge quantization of M-theory's $C$-field, the topological sectors of the resulting $String^{c_2}(4)$-valued higher gauge theory on a heterotic M5-brane are classified by homotopy classes of maps from the worldvolume $\Sigma_{M5}$ to $BString^{c_2}(4)$. This note calculates the sectors in a number of examples of M5-brane topology, including examples considered in the 3d-3d correspondence, the emergence of skyrmions from higher-dimensional instantons and Witten's analysis of the S-duality of 4d Yang-Mills theory.


Introduction
Non-abelian higher gauge theory is having an increasing impact on string and M-theory [15]. Whereas the interpretation of the B-field as the curving on a gerbe is well-established [14,6], higher gauge theories generalising non-abelian gauge theories like Yang-Mills theory are less developed from the point of view of applications and examples, even if there is plenty of theoretical development (for example, but not limited to, [9,16]). Moreover, the appearance of higher non-abelian gauge theory can be quite unfamiliar, where the 'fields' are much more exotic objects than just Lie algebra-valued forms, or sections of certain vector bundles (even the more familar RR fields are, when fully analysed [8], really valued in differential K-theory, a generalised cohomology theory). This means it may be less obvious what the content of the gauge theory is, and what can be calculated about it.
Consider for the sake of analogy, Yang-Mills theory in 4D. If one is considering just topological information, as in this article, then we can ignore questions of metric signature. The topological sectors for (classical) Yang-Mills with structure group G on spacetime X are classified by continuous maps c : X → BG, where BG is the classifying space of G. More precisely, the sectors are in bijection with homotopy classes of such maps, where two maps c 0 , c 1 are homotopic if there is a continuous family of maps c t : X → BG, 0 ≤ t ≤ 1. The topologically trivial sector is represented by the map defined by c(x) = * for all x ∈ X, where * ∈ BG is a fixed basepoint.
In the case of the BPST SU (2)-instanton on R 4 , there is an additional constraint, namely that as |x| → ∞, c BPST (x) → * . Since this is purely topological, the rate of convergence is not important, unlike the case of thinking of the instanton as a gauge field, and requiring its energy to be finite. As a consequence, c BPST extends to a continuous map on the compactification of R 4 , namely S 4 → BSU (2), sending ∞ → * ∈ BSU (2). However, we know that homotopy classes of maps S 4 → BSU (2) are in bijection with the homotopy group π 4 (BSU (2)) ≃ π 3 (SU (2)) ≃ Z. Thus the topological sectors are labelled by instanton charge, as is well-known.
For higher gauge theory, the Lie group G is replaced by an n-group: a higher categorical object that has grouplike structure. The case of n = 2 can be specified by giving a Lie crossed module (see the review in [3]), with the most famous being the String 2-groups String G , for G a compact simple simply-connected Lie group [2] (for G = Spin(n), we write String(n) instead of String Spin(n) ). We will only be considering 2-groups in this article, and do not need the specifics of such structures; all that is needed is the fact that 2-groups also have a classifying space [4], analogous to BG. For the topological sector analysis we present here, knowing the classifying space is sufficient, as we need to calculate the homotopy classes of maps to B String c2 (4).
The 2-group we are interested in is a modified version of String(4), analogous in one sense to how Spin c (n) is a modified version of Spin(n) [18]. Whereas Spin(n) → SO(n) is onto with kernel {±1}, Spin c (n) → SO(n) is onto with kernel the larger group U (1). In the same way, String(n) → Spin(n) is onto with kernel the 2-group BU (1), but String c2 (n) is onto with kernel the larger group String Sp (1) . See [19, §2.2.2] for an extended discussion of the formal definition of these twisted String 2-groups, and their relevance to M-theory.
The reason that this family of 2-groups, or specifically the 2-group String c2 (4), is of interest, is that from Hypothesis H [20,11] it follows that a heterotic M5-brane [17] automatically carries fields from the higher gauge theory with String c2 (4) as the structure 2-group [12]. Hypothesis H itself is about the true nature of the higher gauge theory of the C-field in M-theory: Hypothesis H (Fiorenza-Sati-Schreiber). The M-Theory C-field is chargequantized in J-twisted cohomotopy theory.
This purely mathematical hypothesis has been shown [5,21,22,10,11,12,23] to imply a host of anomaly cancellation and other consistency conditions previously proposed in the literature on physical grounds.
So while we do not work directly with the content of Hypothesis H here, we will calculate the possible topological sectors of the String c2 (4) higher gauge theory on the simplest examples of M5-branes.

Remark 1.
In what follows, we do not consider the quantum version of such a higher gauge theory, just as the analysis of the topological sectors of YM theory above merely considers the classical field theory.
Just as in the case of ordinary gauge theory, the topological sectors for String c2 (4) higher gauge theory on a manifold M are given by homotopy classes of maps from M to the classifying space B String c2 (4). To do this, we need to understand the homotopy type of the classifying space B String c2 (4), at least in low dimensions. This will be considered in Section 3.1 below, before the calculations for each case of M5-brane topology in Section 3.2.
It is worth outlining exactly how this higher gauge theory arises, since this is reflected in the construction of its classifying space, and in the classication calculations below. We shall only give the barest outline, as the proof is rather involved, with many geometric objects coming together in exceptional ways.
Following [12], we assume the spacetime manifold is of the form R 2,1 × X 8 , for some 8-manifold X 8 . There is an embedding map Σ M5 φ − → R 2,1 × X 8 for the M5-brane. The spacetime manifold carries the C-field, living in J-twisted cohomotopy, which can be represented by a map C : R 2,1 ×X 8 → S 4 / / Spin(5) ≃ B Spin(4). By composition, this gives a map Σ M5 (4). The M5brane carries, among other things, an Sp(1)-valued gauge field, represented by a map Σ M5 → B Sp(1). Together, these satisfy an compatibility condition [18], so that the two maps, Σ M5 → B Spin(4) and Σ M5 → B Sp(1), assemble into a higher gauge field Σ M5 → B String c2 (4). Thus one can think of the String c2 (4) higher gauge theory on the M5-brane as having a component inherited from the ambient spacetime, as well as a component arising from a gauge field living on the brane itself. We shall discuss below for specific examples how to view the classification of the sectors in Section 2.1, in light of this picture.

Results
The results for each M5-brane topology are given in Table 1, with a reference to where it is calculated in Section 3.2. In the next subsection, we give some (mildly speculative) discussion about how these classifications relate to the physics literature, in a sample of cases.

Sector classification
Some of the cases in Table 1 have boundary conditions, and this imposes extra constraints on the solutions, so that sectors can appear or disappear, even though the brane topology is identical. We include the signature of the metric on the worldvolumes listed in the table, to keep things clear, but nothing relying on the metric will enter into the calculations.

Boundary conditions
Sectors classified by Details

Discussion
Some of the above topologies for Σ M5 are rather simple, and do no support nontrivial higher gauge sectors. This is worth checking, just in case there are hidden surprises, or to confirm physical intution that there may not be anything topologially nontrivial, or to avoid fruitless searching for phenomenology when it cannot appear. But some of the cases carry many interesting essentially independent topological sectors, and even torsion sectors, for instance the case of 5d instantons (the second and last rows of Table 1). Even more interesting is when these nontrivial results arise on topologies that are already considered in the literature. Only a few will be pointed out here; the reader can check their own favourite model against the table.
• The case of Σ M5 = R 2,1 × S 3 appears in the 3d-3d correspondence (see [7] for a review). Here we find that what is needed for nontrivial sectors is some spatial boundary condition, so that the gauge field extends to the (spatially) compactified S 2 × S 3 A similar case to consider (after Wick rotation) is Σ M5 = S 3 × S 3 . This is a more complicated calculation without some boundary condition on one of those S 3 -factors, involving the homotopy type of a space of arbitrary maps on the 3-sphere.
• The case of Σ M5 = R 0,1 ×S 3 ×T 2 was considered in [28], who outlined how S-duality of 4d YM-theory should arise as the residual SL(2, Z)-action on T 2 -compactifications of an M5-brane. Here T 2 is really (when the geometry is put back in) an elliptic curve (over C), but topologically is just a torus. Interestingly, Witten expresses a "doubt" in [28] that the 6d "quantum nonabelian gerbe theory" he considers (on a stack of coincident M5-branes) is the quantisation of a classical system; what we are considering here is a single M5-brane, and the (nonabelian) higher gauge theory arises in a completely different way.
While we are not going to discuss each topology in detail, it might be worth considering exactly what these classes of maps are, in an illustrative case. Unlike more familiar cases of classifications of instantons by integers, such as the 4d Yang-Mills instanton outlined above, we find that the higher gauge theory on R 1,1 × S 4 with no boundary conditions has sectors labelled by a pair of integers. These two copies of Z play different, asymetric roles. Recall that up to homotopy, the gauge field on S 4 (as we can ignore the R 1,1 -factor) is given by a map S 4 → B String c2 (4). That is, we are wrapping an S 4 inside the classifying space, which as in Section 3.1 beelow, is constructed from a B Spin(4) ≃ B Sp(1) L × B Sp(1) R and another B Sp(1), which we shall temporary denote by Sp(1) G . The former is linked the geometry of the ambient spacetime and the latter to a gauge field on the M5-brane. Recall first that π 4 (B Sp(1)) = Z, so that a 4-sphere can wrap around a copy of B Sp(1) an integer number of ways, up to homotopy. Because of the compatibility condition that gives us maps to B String c2 (4), we can think of the sectors in this case as being labelled by the pair of integers (n L , n R ), where the S 4 wraps around the B Sp(1) L n L times, around the B Sp(1) R n R times, and the B Sp(1) G n L times.
The appearance of torsion classes in, for example, the case Σ M5 = S 5 , is also less familiar. Here the interpretation of such classes is perhaps a bit more speculative. One option might be an old observation of Witten [27], that links the classes in π 4 (SU (2)) ≃ Z 2 with whether a quantised soliton is a boson or a fermion. Since homotopy classes of maps S 5 → B String c2 (4) are assembled (similar to the previous paragraph) from compatible maps S 5 → B Spin(4) and S 5 → BSp(1) G = BSU (2), the ordinary SU (2) gauge field on S 5 lives in one of two sectors, corresponding to π 5 (BSU (2)) ≃ π 4 (SU (2)). This can be thought of as an analogue of the instanton/skyrmion picture, where the instanton charge becomes the baryon number. There is still, of course, two more factors of Z 2 , but these arise from spacetime geometry, which we are not considering here; their interpretation in this boson/fermion picture remains to be seen.
While the analysis here gives all possible sectors for a String c2 (4) higher gauge theory on Σ M5 , it ignores the complete picture of [12], whereby there will be additional aspects relating to the topology of spacetime itself, in which Σ M5 sits. One may visualise the situation as follows. where the C-field is valued in J-twisted 4-cohomotopy, and the B-field is valued in twisted 3-cohomotopy. In principle, this map could be neither surjective nor injective. If it is not surjective, then there are additional constraints on the higher gauge theory coming from the spacetime topology, and the choice of M5brane embedding. This would tend to lead to fewer sectors, potentially removing some of the zoo of torsion phenomena in Table 1. However, this has to be balanced against a potential increase in sectors coming from a lack of injectivity: this would result from multiple, topologially inequivalent combinations of C-and B-fields (in this twisted cohomotopy picture) giving rise to higher gauge fields on Σ M5 in the same topological sector. This type of analysis is beyond the scope of the current article, though we note the superficially similar analysis of a forgetful map in [5], related to fractional D-brane charges.
However, if one wishes to engage in a little more speculation, taking the higher gauge fields on the M5-brane as a pure exercise in field theory, with no relation to the M-theory spacetime picture in [12], then there may be additional interpretations for some of these torsion sectors.
So far, we have not particularly discussed the group structure on the set of topological sectors. For the case of SU (2)-instantons on S 4 , we can take the sector label in Z to represent total change, and this is additive on combining instantons with disjoint support. For the case of the String c2 (4) higher gauge fields as calculated here, we don't just have the result that there are exactly eight topological sectors, but that they form the finite group (Z 2 ) 3 . The three sectors (1, 0, 0), (0, 1, 0) and (0, 0, 1) are independent, and could be interpreted as being a kind of quasi-particle excitation of the M5-brane on which they live. The addition in the group (Z 2 ) 3 corresponds to combining gauge fields with disjoint support, so that these three quasi-particles are their own antiparticles, as (1, 0, 0) + (1, 0, 0) = (0, 0, 0), and similarly for the other two cases. This should be interpreted in the sense that combining two copies of (1, 0, 0) ceases to be topologically stable and can decay to the trivial solution. It remains to be seen, though, whether such a "higher quasi-particle" interpretation stands up to scrutiny with more a physical mindset. The results of this section use standard results in elementary homotopy theory, but we gather them here for easy reference. A standard text for more background of this section is the first few chapters of [25]. The classifying space of String c2 (4) is defined [19] to be the homotopy pullback B String c2 (4) where 1 2 p 1 and c 2 are the universal fractional first Pontryagin class and the universal second Chern class, respectively 1 . We can present the map c 2 as a fibration without changing the homotopy type of B Sp(1) L , so that the above square can be assumed to be an honest pullback, and π will thus also be a fibration. Moreover, the fibre of c 2 , and hence π, is B String Sp(1)L The main tool we will use is the homotopy long exact sequence of a pointed fibration If we are given a map f : B ′ → B, we have a pullback square of fibrations such that the fibre of E ′ := B ′ × B E → B ′ is canonically isomorphic to F (we shall silently make this identification in what follows). If we are given a basepoint b ′ ∈ B, then this becomes a square of pointed maps, where p ′ := (b ′ , p) ∈ E ′ , and there is then a diagram of homotopy groups where the rows are exact.
which again will give π 3 (B String c2 (4)) = 0. For n = 4 we get (4)) → Z 2 → 0 and so π 4 (B String c2 (4)) ≃ Z 2 . For n = 5 we have the commuting diagram of abelian groups where the rows are exact. Since the leftmost square commutes, we see that the map (a) is the zero map, so that by exactness of the top row, (b) is injective, and we thus have a short exact sequence Exactness in the bottom row implies that the map (c) is the identity map. Our diagram has now been reduced to / / with exact rows. The map (e) is trivial, so an identical analysis to the case n = 5 means that the group π 6 (B String c2 (4)) fits into the same short exact sequence, again with a left splitting. Thus π 6 (B String c2 (4)) ≃ (Z 2 ) 3 , and this completes the proof.
We need two other results about B String c2 (4), which follow from general results about topological groups. In the lemma, and below, we use LX to denote the space of all continuous loops in the space X, with the compact-open topology.
Lemma 2. Let G be a path-connected topological group. Then there is a weak homotopy equivalence BLG → LBG, in the sense that for all k ≥ 0, π k (BLG) ≃ π k (LBG).
This result appears to be folklore, particularly well-known in the case that G is a connected Lie group. The second result deals with a topological group LG, where the group structure is given by pointwise multiplication. Given a topological group G, there is a sequence of (continuous) homomorphisms where ev, a fibration, is evaluation at the basepoint of the circle, and ΩG denotes the subgroup of loops based at the identity of G.
Lemma 3. The evaluation map ev has a continuous splitting G → LG including G as the subgroup of constant loops, so that LG ≃ G ⋉ ΩG (in particular, as topological spaces, is just a product).
We also know that despite String c2 (4) being a 2-group, there is a topological group G such that B String c2 (4) ≃ BG [4] (for example the geometric realisation of a certain simplicial Lie group arising from a crossed module representing String c2 (4)). Abusing notation, we shall also denote this topological group by String c2 (4), as we are only interested in its homotopy type. The homotopy type of the 2-group and the topological group are the same, in any case, and the topological group String c2 (4) will be path-conncted, by virtue of Spin(4), and Sp(1) and any K(Z, 3) space being path-connected. Thus it makes sense to say LB String c2 (4) is weakly homotopy equivalent to BL String c2 (4), and that L String c2 (4) ≃ String c2 (4) ⋉ Ω String c2 (4) as topological groups. Moreover, we can use the standard isomorphisms π k+1 (BG) ≃ π k (G) ≃ π k−1 (ΩG) and the fact that the functors π k , k ≥ 0 preserve products, to calculate homotopy groups of the form π k (LBG).
The last fact we need is that for simply-connected spaces X, there is an isomorphism [S k , X] ≃ π n (X, x) for any x ∈ X [26, §III.1]. We shall use this result repeatedly below, with one variation in 3.2.7.

Calcuating the topological sectors
With these homotopical tools in hand, we can return to the considering the topological sectors for a String c2 (4) higher gauge theory on the worldvolume Σ M5 of a single heterotic M5-brane.

Trivial cases
It is worth noting that for an M5-brane worldvolume of the form R 5,1 , with no boundary conditions, there are no topologically non-trivial sectors, because in this case the sectors are classified by homotopy classes of arbitrary maps R 5,1 → B String c2 (4), and R 5,1 is contractible (we recall that no metric information enters into this, or the other calculations here).

Unwrapped M5-brane, spatial boundary conditions
We will consider the case that Σ = R 0,1 ×R 5 , and will also assume that the gauge fields are stable in time, so that they do not become trivial at past or future infinity. This case is equivalent to Σ = R 0,1 × S 5 , so that also covered by this calculation. Because R 0,1 is contractible, and we are imposing no constraints in the time direction, we can ignore this here and later, as appropriate. We will assume that the gauge fields vanish at spatial infinity, or rather, as |x| → ∞, the function c : R 5 → B String c2 (4) converges to the fixed basepoint * in B String c2 (4). Analogous to the case of the BPST instanton on R 4 , this means c extends to a continuous map S 5 → B String c2 (4), and the topological sectors are given by [S 5 , B String c2 (4)] * = π 5 (B String c2 (4)) ≃ (Z 2 ) 3 .
Here [−, −] * denotes based homotopy classes of based maps: those that take the canonical basepoint ∞ ∈ S 5 to the basepoint * in the classfying space.

Wrapped around one circle, spatial boundary conditions
In this case we are considering (homotopy classes of) maps R 4,1 ×S 1 → B String c2 (4), such that at the limit of spatial infinity, the map approaches the basepoint * in the classifying space. We can again ignore the factor of R 0,1 , as there are no boundary conditions in the time direction. The set of maps we need to consider is then isomorphic to The set of such maps up to homotopy is isomorphic to [S 4 , ΩB String c2 (4)] * = π 4 (ΩB String c2 (4)) ≃ π 4 (ΩB String c2 (4)) = (Z 2 ) 3 as needed.

Wrapped around a 3-sphere, no spatial boundary conditions
Because there are no boundary conditions in the non-compact directions, the calcuation is much simpler: ≃ π 3 (B String c2 (4)) = 0.
where X ∼ denotes the universal covering space of X.

Conclusion
We have in this article calculated the possible topological sectors for String c2 (4) higher gauge theories on various M5-brane topologies. Many nontrivial sectors arise on examples of M5-brane topologies that have been considered in the literature, including those for 5d instantons/4d skyrmions for SU (2) = Sp(1) Yang-Mills on M5-branes wrapped around an S 1 . There are a number of torsion sectors, whose physical interpretation is not yet clear. It should be noted, however, that in the context of [12], the higher gauge fields here are not arbitrary, but determined by other data, including the embedding of the M5-brane into spacetime. The sector classification in this article only considers the topology of the M5-brane, and not the spacetime topology, which will potentially add additional nontrivial constraints. This may lead to fewer sectors, or introduce more sectors due to more than one true sector on the whole M5-brane/spacetime system giving rise to the same topological sector just on the M5-brane. Future work will also have to examine the torsion sectors that have shown up here, to see if they survive to the full sector classification arising from Hypothesis H.