aa r X i v : . [ m a t h . A T ] J un Topological Mathieu Moonshine
Theo Johnson-Freyd
Perimeter Institute for Theoretical PhysicsWaterloo, ON, CANADA
E-mail: [email protected]
Abstract:
We explore the Atiyah–Hirzebruch spectral sequence for the tmf • [ ]-cohomologyof the classifying space B M of the largest Mathieu group M , twisted by a class ω ∈ H ( B M ; Z [ ]) ∼ = Z . Our exploration includes detailed computations of the F -cohomologyof M and of the first few differentials in the AHSS. We are specifically interested in thevalue of tmf • ω ( B M )[ ] in cohomological degree −
27. Our main computational result is thattmf − ω ( B M )[ ] = 0 when ω = 0. For comparison, the restriction map tmf − ω ( B M )[ ] → tmf − (pt)[ ] ∼ = Z is nonzero for one of the two nonzero values of ω .Our motivation comes from Mathieu Moonshine. Assuming a well-studied conjecturalrelationship between TMF and supersymmetric quantum field theory, there is a canonically-defined Co -twisted-equivariant lifting [ V f♮ ] of the class { } ∈ TMF − (pt), for a specificvalue ω of the twisting, where Co denotes Conway’s largest sporadic group. We conjecturethat the product [ V f♮ ] ν , where ν ∈ TMF − (pt) is the image of the generator of tmf − (pt) ∼ = Z , does not vanish Co -equivariantly, but that its restriction to M -twisted-equivariantTMF does vanish. We explain why this conjecture answers some of the questions in MathieuMoonshine: it implies the existence of a minimally supersymmetric quantum field theory withM symmetry, whose twisted-and-twined partition functions have the same mock modularityas in Mathieu Moonshine. Our AHSS calculation establishes this conjecture “perturbatively”at odd primes.An appendix included mostly for entertainment purposes discusses “ ℓ -complexes,” inwhich the differential D satisfies D ℓ = 0 rather than D = 0, and their relation to SU(2)Verlinde rings. The case ℓ = 3 is used in our AHSS calculations. Keywords: supersymmetry, topological modular forms, mock modular forms, sporadicgroups, moonshine, group cohomology, Mathieu group, Steenrod powers, higher complexes. ontents N = (0 , SQFTs 6 • as an Ω-spectrum 102.3 Equivariant SQFT • and ’t Hooft anomalies 162.4 Twisted and twined shadows 222.5 TMF • and tmf • M • (M ; F ) 303.2 Cohomology of P + ǫr acting on H • (M ; F ) 34 tmf • ω ( B M ) • (pt)[ ] 414.3 Differentials for p ≥ p = 3 454.5 Running the spectral sequence 50 A Higher complexes 54
A.1 ℓ -complexes in characteristic not dividing ℓ ℓ -complexes in characteristic ℓ By writing the elliptic genus of an N = (4 ,
4) K3 sigma model in terms of characters of thechiral N = 4 superalgebra, Eguchi, Ooguri, and Tachikawa [EOT11] discovered a specificweight- mock modular form (for Γ = SL ( Z )) with shadow 24 η ( τ ) : H ( τ ) = 2 q − / (cid:0) − q + 231 q + 770 q + 2277 q + . . . (cid:1) Physics readily explains the mock modularity and integrality of H . It does not, however,explain why the coefficients of H are dimensions of representations of Mathieu’s largest groupM , and more generally raises the following mysteries:– 1 – uestion 1.1. Whenever a finite group G acts on a K3 sigma model preserving N = (4 , supersymmetry, the elliptic genus can be twisted and twined by a commuting pair of elements g, h ∈ G . This produces twisted-twined versions H g,h ( τ ) of H ( τ ) with interesting (mock)modularity properties, with multiplier that depends on the ’t Hooft anomaly of G . The group G = M does not act on any K3 sigma model [GHV12], but nevertheless the functions H g,h ( τ ) exist for all commuting pairs g, h ∈ M [GPRV13, Gan16]. Why? Question 1.2.
A priori, the supertrace in the elliptic genus allows for a large cancelation ofbosonic and fermionic modes. In particular, the coefficients of g H g ( τ ) = H e,g ( τ ) are au-tomatically virtual characters of G , but have no reason to be honest characters. Nevertheless,except for the constant term − , these coefficients are honest characters [Gan16]. Why? Question 1.3.
The functions H g,h ( τ ) enjoy a mock-modular analogue of the “genus-zeroproperty” from monstrous moonshine [CD12]. Why? The goal of this note is to suggest a solution to Question 1.1. We will not provide acomplete solution—some calculations are too hard—but our suggestion will at least answerwhat type of quantum field theory it is that can produce the functions H g,h ( τ ). We willhave nothing to say about Question 1.2. We will briefly comment in Conjecture 2.8 aboutQuestion 1.3.The first step is to recast the problem as a question in stable homotopy theory. Asexplained in Section 2, compact minimally supersymmetric (1+1)-dimensional quantum fieldtheories are the cocycles for an extraordinary cohomology theory SQFT • . This statementis not mathematically rigorous: even the set of “(1+1)-dimensional quantum field theories”is not mathematically defined (although [ST11] comes close), and topologizing this set willsurely be subtle, but the construction is physically straightforward. This cohomology theoryconnects directly with mock modularity [GJF19a]: if S is an SQFT of cohomological degree1 − k representing the trivial class in SQFT − k (pt), then any nullhomotopy of S determinesa (generalized) mock modular form with shadow determined by S . We will call the theory S = ∂ F the boundary of its nullhomotopy F . (Note that this is not a “boundary condition,”where the boundary is on the worldsheet. Rather, it should be thought of as a boundary in“field space” or “target space,” because if F is a sigma model with target M , then ∂ F is asigma model with target ∂M .)If the boundary SQFT S furthermore admits an action by a finite group G of flavoursymmetries, and if the nullhomotopy is G -equivariant, then the same construction producesmock modular forms depending on commuting pairs ( g, h ). The level structure depends onthe orders of g and h , and the multiplier system depends on the ’t Hooft anomaly ω ∈ H (M ; U(1)) ∼ = H (M ; Z ) of the G -action. (For the purposes of this introduction, we willignore the fact that ’t Hooft anomalies for fermionic QFTs live in “supercohomology” andnot in ordinary cohomology.) In algebrotopological language, the fact that it makes senseto talk about deformations of SQFTs with G -flavour symmetry and anomaly ω means thatthe cohomology theory SQFT • has a twisted equivariant enhancement, allowing us to define– 2 –wisted equivariant cohomology groups SQFT • ω ( B G ) for any finite group G and anomaly ω ∈ H ( G ; Z ). Here and throughout, we will write B G for the classifying stack of G ; a morestandard name for SQFT • ω ( B G ) is SQFT • G,ω (pt).For example, the direct sum Fer(3) ⊕ of 24 copies of the antiholomorphic supercon-formal field theory Fer(3) (three antichiral Majorana–Weyl fermions, with supersymmetryencoding the structure constants of su (2)) is nullhomotopic [GJFW19], and the correspond-ing mock modular form is H ( τ ). We can let M act on Fer(3) ⊕ by permuting the sum-mands. Writing for the standard degree-24 permutation representation of M , we willcall the corresponding M -equivariant SCFT ⊗ Fer(3). Because the M -symmetry spon-taneously breaks to M , and because H (M ; Z ) = 0, we can think of the M action on ⊗ Fer(3)as having any ’t Hooft anomaly that we want (see § ⊗ Fer(3)] = [ ] ⊗ [Fer(3)] ∈ SQFT − ω ( B M ) for every ω . If one of them were nullho-motopic, then the nullhomotopy, with its corresponding mock modular forms, might explainMathieu Moonshine.Unfortunately, we will show in Proposition 2.5 that ⊗ Fer(3) is not M -equivariantlynullhomotopic (for any value of the ’t Hooft anomaly). Rather, the boundary SQFT that wewill use is S = V f♮ ⊗ Fer(3), where V f♮ is the holomorphic SCFT constructed in [Dun07].The automorphism group of V f♮ is Conway’s largest sporadic group Co , which contains M as a subgroup; the computations in [JFT18] show that the anomaly ω of the correspondingM -action on S agrees with the anomaly for Mathieu Moonshine computed in [GPRV13].Cohomological degrees in SQFT • are determined by the central charges of the representingQFTs, and this S represents a class in cohomological degree −
27. Our suggested answer toQuestion 1.1 is:
Conjecture 1.4.
The antiholomorphic SCFT S = V f♮ ⊗ Fer(3) represents the trivial class [ S ] = 0 in SQFT − ω ( B M ) . Without further information about SQFT • , it seems impossible to test this conjecture.But in fact there is a rather clear idea of the structure of SQFT • , with evidence continuingto amass [Seg88, HK04, ST04, Che06, Seg07, ST11, BE15, BE16, BET18, GJF18, BET19,GJFW19, GJF19a] in favour of the following conjecture: Conjecture 1.5.
The spectrum
SQFT • represents the universal elliptic cohomology theory TMF • of “topological modular forms” described in [Lur09, DFHH14]. Under this equivalence, the class [Fer(3)] ∈ SQFT − (pt) corresponds to the class usuallydenoted ν ∈ TMF − (pt) = π TMF, the image under the Hurewicz map of the 3-sphere S = SU(2) with its Lie group framing [GJFW19], and the class [ V f♮ ] ∈ SQFT − (pt) is { } ∈ TMF − (pt) [GJF18], where ∆ = ( c − c ) / { } are there because ∆ itself is not a class in TMF − (pt).)Recently a complete definition of equivariant TMF has become available [Lur19, GM20].Assuming Conjecture 1.5, Conjecture 1.4 becomes:– 3 – onjecture 1.6. There is a distinguished refinement of { } ∈ TMF − (pt) to a classin TMF − ω ( B Co ) , and, after multiplying by ν and restricting along M ⊂ Co , the class { } ν vanishes in TMF − ω ( B M ) . Note that the M -action on V f♮ , and hence on S = V f♮ ⊗ Fer(3), extends to a Co -action. However, we do not believe that [ S ] = { } ν vanishes Co -equivariantly. It is worthemphasizing that, in order to define { } ν ∈ TMF − ω ( B M ), one would need to show thatthe nonequivariant class { } ∈ TMF − (pt) admits an equivariant refinement to a classin TMF − ω ( B Co ). The existence of such a refinement is implied by Conjecture 1.5, but ithas not been shown mathematically rigorously. Furthermore, in § Conjecture 1.7. { } refines to a class in Tcf − ω ( B Co ) , the space of (twisted) Co -equivariant topological cusp forms , and the restriction of { } ν vanishes in Tcf − ω ( B M ) . Unfortunately, this author is not aware of techniques for computing twisted equivariantTMF • (let alone Tcf • ) groups. Instead, as evidence in favour of Conjecture 1.6, we willattempt to compute the related group tmf − ω ( B M ). There are two changes involved. First,we have replaced the genuinely equivariant problem with the Borel-equivariant one. Anygroup G has a classifying space BG , and for any cohomology theory E • , Borel-equivariant E • -cohomology studies cohomology of BG in place of B G . As with Atiyah–Segal completion forK-theory [AS04, AS06], one expects in general that E • ( BG ) is an approximation of E • ( B G ),but the latter may include more information than the former. (In fact, the “completion”story for TMF is subtle, and typically fails for Lie groups [GM20], but seems to hold forfinite groups.) Second, we have replaced the spectrum TMF • by the related spectrum tmf • .Speaking very roughly (see § • corresponds to the modularforms which are bounded at the cusp τ = i ∞ and TMF • corresponds to the modular formswhich are meromorphic at the cusp; on homotopy groups, TMF • (pt) = tmf • [∆ − ], and ifa class in tmf • vanishes, then its image in TMF • also vanishes. There is no known physicaldescription of tmf • , and there is not expected to be one.Actually, computing tmf • ω ( B M ) is still too hard, because the 2-local structure of tmf • is complicated and the 2-local cohomology of M is not known. So we will attempt onlytmf • ω ( B M )[ ]. After further inverting 3, the spectrum tmf • [ ] becomes the spectrum called“ Eℓℓ ” in [Tho94], where it is shown that tmf • ω ( B M )[ ] (which is independent of ω ) issupported only in even degrees. As such, our computation is interesting only at the prime 3.After a detailed study of H • (M ; F ) in Section 3, in Section 4 we investigate the Atiyah–Hirzebruch spectral sequence for tmf • ω ( B M )[ ]. Note that we are particularly interested inthe groups tmf − ω ( B M ), which houses the image under the completion map tmf( B M ) → tmf( B M ) of the equivariant enhancement of { } ν , and tmf − ω ( B M ), which houses theimage under completion of ν . Our main mathematical result is:– 4 – heorem 1.8. If ω ∈ H (M ; Z [ ]) ∼ = Z is nonzero, then tmf − ω ( B M ) = 0 . For com-parison, for one of the two nonzero values of ω , and not the other one, the restriction map tmf − ω ( B M )[ ] → tmf − (pt)[ ] is nonzero. Spectral sequences in general, and the Atiyah–Hirzebruch spectral sequence in particular,are the homotopy algebraist’s version of perturbation theory. Indeed, a physicist should thinkof the difference between TMF • ω ( B G ) and TMF • ω ( BG ) as the difference between nonperturba-tive and perturbative field theory. One can pull back along the map BG → B G to produce amap TMF • ω ( B G ) → TMF • ω ( BG ). The domain, hypothetically, encodes deformation classes ofSQFTs with G -flavour symmetry, and in particular their behaviours on worldsheets equippedwith arbitrary G -bundle, whereas the codomain remembers only the physics “near the trivial G -bundle.” (Since G is finite, the stack of G -bundles has no perturbative structure “over C ,”but it does have perturbative structure p -locally for any prime p dividing the order of G .)To end the paper, Appendix A describes some of the theory of “chain complexes” in whichthe “differential” does not satisfy D = 0 but rather D ℓ = 0 for ℓ >
2. Some of this theory,for ℓ = 3, is important in our calculations. The larger story connects in intriguing ways tothe Verlinde ring for SU(2) at level k = ℓ −
2, and some readers may find it entertaining.
We will write Z n for the cyclic group of order n . This name is reasonably standard inthe physics literature; mathematicians may prefer C n or Z /n Z . For other finite groups, wegenerally follow ATLAS naming conventions [CCN + p , we will write Z ( p ) forthe ring of p -adic integers. In the mathematics literature this name is sometimes used insteadfor the subring of Q consisting of fractions with denominator coprime to p . If G is a finitegroup, then H • ( G ; Z ( p ) ) is independent of which meaning of “ Z ( p ) ” is used, and so we will notworry about the difference. The finite field with q = p n elements is F q ; a generic field is K .We will always use cohomological degree conventions, with degrees always written assuperscripts. For example, the homotopy groups of a spectrum E • are E • (pt) = π E • = π −• E .If E • is connective (e.g. tmf • ), then these groups are supported in nonpositive cohomologicaldegree. Without care, this paper would devolve into alphabet soup. So, for example, Bock-stein maps will be denoted (cid:3) rather than β . We will sometimes write the group cohomologyof a finite group G , with coefficients in an abelian group A , as H • ( G ; A ), and sometimesas H • ( BG ; A ), with “ BG ” denoting the classifying space of G . For an extraordinary coho-mology theory E • , we will always use the latter name: E • ( BG ) is the E • -cohomology of thespace BG . If E • also admits an equivariant refinement, then we can evaluate E • on the clas-sifying stack B G of G ; by definition E • ( B G ) = E • G (pt) is the G -equivariant cohomology of apoint. When E • = H • ( − ; A ) is ordinary cohomology, the groups H • ( BG ; A ) and H • ( B G ; A )agree, justifying our use of simply H • ( G ; A ). I thank D. Berwick-Evans for detailed and helpful comments on a draft of this paper. Sec-tions 2.1 and 2.2 are based on ideas developed by the author jointly with D. Gaiotto, whom– 5 – thank for ongoing discussions about this circle of ideas. Research at Perimeter Instituteis supported in part by the Government of Canada through the Department of Innovation,Science and Economic Development Canada and by the Province of Ontario through the Min-istry of Colleges and Universities. The Perimeter Institute is located within the HaldimandTract, land promised to the Six Nations (Haudenosaunee) peoples. N = (0 , SQFTs
The starting point of our analysis is the following question: What type(s) of quantum fieldtheories produce mock modular forms? The (an?) answer has been well-investigated for morethan a decade [EST07, MM10, Tro10, ES11, Sug12, Mur14, ADT14, GM17, GJF19a, DJR19,Sug20, DPW20]: A (1+1)-dimensional quantum field theory can produce mock modularinstead of modular forms if it is noncompact.We will not in this paper attempt to define “quantum field theory.” We will always assumeour QFTs to be unitary, so that we have access to Wick rotation to Euclidean signature(imaginary time). The physics literature does not seem to include a complete definition of“compactness” for a QFT, but the consensus is that it should be a “spectral condition,”since in the case of sigma-models what distinguishes compact from noncompact target is thatthe former lead to Hamiltonians with discrete spectrum, whereas the latter have continuousspectrum. We propose the following: a ( d +1)-dimensional QFT is compact if its Wick-rotated partition function “converges absolutely” on all closed spacetimes: in Lagrangianformalism, we imagine an “absolutely convergent path integral” (in spite of the fact that notall QFTs have path integral descriptions, and most spaces of fields do not support measuresof integration in the mathematical sense); in Hamiltonian formalism, we are asking that theWick-rotated evolution operator tr(exp( − τ ˆ H )) should be trace-class. This latter conditionoccurs when the spectrum of the Hamiltonian ˆ H is bounded below, discrete, and does notgrow too slowly. Compactness is a nontopological version of asking whether a functorialtopological field theory is defined on all cobordisms, or if it is only partially defined.Badly noncompact QFTs might even fail to assign Hilbert spaces of states to all closed d -dimensional spaces. The most mild type of noncompactness is when the Hilbert spaces are allwell-defined, but the Wick-rotated partition function converges only conditionally. The valueof a conditionally-convergent sum or integral can depend on the method used to evaluate it,and so the partition function of a mildly noncompact QFT is not quite well-defined. This is theorigin of phenomena like mock modularity in noncompact QFTs: modular transformationsmay not be compatible with the chosen evaluation method.Focusing on the case we care most about, let F be a (1+1)-dimensional QFT, and write Z ( F ) for its partition function on flat oriented 2-dimensional tori (these being the only flatclosed oriented 2-manifolds). If F has fermions, then Z ( F ) depends on a choice of spinstructure on the worldsheet. We will care most about the case of nonbounding spin structure,which is to say the Ramond spin structure along both the A - and B -cycles; we will thus call– 6 –his the “Ramond-Ramond” partition function Z RR ( F ). The space of flat tori (with RR spinstructure) is 3-real dimensional: the local coordinates are the complex structure ( τ, ¯ τ ) and thearea a . If F is compact, then Z ( F ) is a well-defined function of these three real variables, andwe assume that it is real-analytic. (As with essentially all analytic questions about QFT, thisis an assumption, and we must fold it into some aspect of the definition of “compact QFT.”)Because different values of ( τ, ¯ τ , a ) describe the same torus, a Z RR ( τ, ¯ τ , a ) is a real-analyticfamily of real-analytic SL ( Z )-modular functions. In the conformal case, of course, there isno a -dependence.Now suppose that F is not just a compact QFT, but also is equipped with an N = (0 , Z RR ( F ) depends only on τ . Thisargument is so familiar that we will not review it, except to make a few comments:1. The statement only holds in the Ramond-Ramond spin structure.2. Let ¯ Q denote the supercurrent for the N = (0 ,
1) supersymmetry. (It is usually called“ ¯ G ,” but we will soon want the letter G to stand for a finite group.) This supercurrent isa worldsheet spinor, and so has two components, which explain the two nondependencies(on ¯ τ and on a ). Given coordinates z, ¯ z on the worldsheet, we can write the twocomponents of ¯ Q as ¯ Q z and ¯ Q ¯ z . The former is the “trace” of ¯ Q , and vanishes if F issuperconformal.3. The growth rate of Z RR ( F )( τ ) as τ → i ∞ is not worse than exp( τ c/ c isthe central charge of F , and τ = ( τ − ¯ τ ) / i is the imaginary part of τ . As such, Z RR ( F )( τ ) is a weakly holomorphic modular function, meaning a modular functionwhich is holomorphic for finite τ , and meromorphic at the cusp τ = i ∞ .4. F has a gravitational anomaly if its left and right central charges c L , c R do not match.The difference w = c R − c L is always a half-integer. When w = 0, Z RR ( F ) suffers amultiplier under T -transformations: T [ Z RR ( F )] = e − w πi Z RR ( F ) . This multiplier can be absorbed by adjusting Z RR ( F ) Z ′ RR ( F ) = Z RR ( F ) η ( τ ) w . This adjusted partition function Z ′ RR ( F ) is then a weakly holomorphic modular form ofweight w and trivial multiplier. It matches better with the mathematics conventions forWitten genera. In [GJF19a], this adjustment (up to a factor of i w that we will ignore)is interpreted in terms of “spectator” Majorana–Weyl fermions that are added to F tocure the anomaly. In § n = − w as the cohomologicaldegree of F .5. The q -expansion of Z RR ( F ) is the index of the S -equivariant supersymmetric quantummechanics model H ( F ) formed by compactifying F on an A -cycle with Ramond spin– 7 –tructure, thus explaining why Z RR ( F ) ∈ Z (( q )). The q -expansion of Z ′ RR ( F ) is builtby adjusting the SQM model by some spectator fermions.When F is not compact, the standard arguments can break down, as we have alreadyindicated. Following [GJF19a], we will focus on a particularly mild noncompactness, whichis when F has “cylindrical ends.”In order to give the definition, we will need the following construction. Let Φ be aself-adjoint operator in the SQFT F , thought of as a “function” Φ : F → R . There is astraightforward way to construct the “fibre” of Φ over x ∈ R , which we will denote F (Φ = x ), or F ( x ) when Φ is implicit. Namely, add to F a chiral Majorana–Weyl fermion λ ,which will serve as a Lagrange multiplier, to produce the QFT F ⊗
Fer(1). Now deform thesupersymmetry on
F ⊗
Fer(1) by adding a superpotential equal to W = λ (Φ − x ) . This results in an adjustment of the Lagrangian like λ ¯ Q [Φ − x ] + (Φ − x ) . In the IR, oneexpects this F (Φ = x ) to flow to an SQFT in which Φ takes the constant value x .Conversely, one expects to be able to recover F from the R -family of SQFTs x
7→ F (Φ = x ) by dynamicalizing the parameter x . This dynamicalization procedure involves replacingthe parameter x by a scalar field φ and also introducing its (right-moving) superpartner ψ ,so that together ( φ, ψ ) is a scalar supermultiplet for the N = (0 ,
1) algebra. We will writethe result of this dynamicalization as F ( x ) Z φ,ψ F ( φ ) . The SQFT F is then said to have cylindrical ends if it can be equipped with a Φ suchthat the SQFTs F ( x ) are all compact, and if supersymmetry is spontaneously broken when x ≪
0, and if the theories F ( x ) stabilize to some fixed SQFT ∂ F when x ≫
0. We will call ∂ F the boundary of F . Note that this is a boundary in field space, and not a “boundarycondition” that can be assigned to boundaries of the worldsheet.For example, if F consists of a massless scalar φ (i.e. a noncompact nonchiral boson),together with its superpartner ψ (an antichiral Majorana–Weyl fermion), and if Φ = φ , then F ( x ) picks up a quartic potential ( φ − x ) and a Yukawa coupling λφψ . If x < F ( x )has spontaneous supersymmetry breaking, whereas if x > F ( x ) has two massive vacua,with fermion masses of opposite signs, and so the two vacua differ by a relative Arf invariant(c.f. § R , has cylindrical boundary equal to (trivial TQFT) ⊕ (Arf TQFT).Suppose that F has cylindrical ends, parameterized by an operator Φ. Then the partitionfunction of F has no reason to converge absolutely. But if the partition function of theboundary vanishes, Z RR ( ∂ F ) = 0 , – 8 –hen one expects that the path integral description of Z RR ( F ) will converge conditionally,because the end R ≫ × ∂ F contributes a term like 0 × vol( R ≫ ), which we take to be 0. In thisway we can define Z RR ( F ). (The value of Z RR ( F ) might depend on the parameterization Φ.)Because we never chose coordinates on the worldsheet, Z RR ( F )( τ, ¯ τ , a ) is manifestlySL ( Z )-modular, and we will assume that it is real-analytic. However, it is not automat-ically holomorphic, because the standard argument for holomorphicity requires compact-ness. Rather, Z RR ( F )( τ, ¯ τ , a ) can suffer a holomorphic anomaly. One of the main resultsof [GJF19a] is a precise formula for this holomorphic anomaly. The justification given in thatpaper is a combination of heuristic arguments about path integrals and applying Stokes’ for-mula in field space, together with carefully-worked examples to fix the proportionality factors.A more detailed proof in the case of sigma models is given in [DPW20]. But in the generalcase the arguments from [GJF19a] would need further improvements in order to provide a“theorem” even at physicists’ level of rigour, and so we will call it here a “conjecture”: Conjecture 2.1 ([GJF19a]) . Suppose F is an N = (0 , SQFT with cylindrical ends andboundary ∂ F , and that Z RR ( ∂ F ) = 0 . Then deformations of F that are “compactly sup-ported,” i.e. that don’t deform the end F ≫ , do not effect the value of Z RR ( F ) . Moreover, the ¯ τ - and a -dependence of the conditionally convergent partition function Z RR ( F )( τ, ¯ τ , a ) are de-termined entirely by the boundary ∂ F . In particular, if ∂ F is superconformal, then Z RR ( ∂ F ) has no a -dependence, and its ¯ τ -dependence is governed by the holomorphic anomaly equation (up to convention-dependent factors of √− ): √− τ η ( τ ) ∂∂ ¯ τ Z RR ( F ) = ( torus one-point function of ¯ Q ¯ z in ∂ F )Thus, if ∂ F is superconformal, the adjusted partition function Z ′ RR ( F ) = Z RR ( F ) η ( τ ) w is a real-analytic, but not holomorphic, modular form of weight w , where w = c R − c L is thegravitational anomaly of F . Any real-analytic modular form ˆ f ( τ, ¯ τ ) has a holomorphic part f ( τ ), defined by analytically continuing and then taking a limit f ( τ ) = lim ¯ τ →− i ∞ ˆ f ( τ, ¯ τ ) , assuming the limit exists. This is an example of a generalized mock modular form , with shadow the complex conjugate of √− τ ∂ ˆ f∂ ¯ τ . It is honestly mock modular if the shadow is (weakly)holomorphic. In particular, suppose that ∂ F is a purely antiholomorphic SCFT (i.e. all of itsfields are antichiral). Then the torus one-point function of ¯ Q = ¯ Q ¯ z is antiholomorphic, andso f ( τ ) = η ( τ ) lim ¯ τ →− i ∞ Z RR ( F )( τ, ¯ τ )will be a weight-1 / Q in ∂ F .The analysis in [GJF19a] furthermore suggests:– 9 – onjecture 2.2 ([GJF19a]) . Suppose that F as in Conjecture 2.1, with ∂ F superconformal.Then the holomorphic part of Z RR ( F ) exists (the limit converges). Its q -expansion is the indexof the S -equivariant SQM model H ( F ) formed by compactifying F on an appropriate Ramondcircle called the Tate curve . (The compactification explicitly breaks SL ( Z ) -modularity.) Thisindex lives in Z (( q )) up to a correction given by an Atiyah–Patodi–Singer invariant of ∂ F .Because of the extra time-reversal symmetry of H ( F ) (c.f. § ∂ F . The following example was the primary motivation for [GJFW19, GJF19a]. Take a K3surface with 24 punctures, and arrange a B-field on M so that its flux near each puncturesatisfies R S H/ π = 1. Now form an N = (0 ,
1) sigma model with target this noncompact 4-manifold. (The (0 ,
1) worldsheet supersymmetry enhances to (0 ,
4) by using the hyperka¨ahlerstructure. The B-field is needed to cancel an anomaly that would otherwise be present becauseof the mismatched fermions [MN85].) The result is a noncompact SCFT F with cylindricalends. The boundary theory ∂ F = Fer(3) ⊕ is a direct sum of 24 copies of the same theory, onefor each puncture. The contribution from each puncture is a purely antiholomorphic SCFTFer(3) consisting of three antichiral Majorana–Weyl fermions ¯ ψ , ¯ ψ , ¯ ψ and supersymmetry¯ G = : ¯ ψ ¯ ψ ¯ ψ :, up to convention-dependent factors of √−
1. The torus one-point functionof ¯ G in each summand is η (¯ τ ) . Thus the K3 surface produces a mock modular form withshadow 24 η ( τ ) , namely the function H ( τ ) that we started with in Section 1.Fer(3) ⊕ is not the only possible boundary theory for producing H ( τ ), and is not theone we will end up using. There is a famous holomorphic SCFT called V f♮ constructedin [Dun07], with automorphism group Aut( V f♮ ) = Co and central charge c L = 12 (and c R = 0). We will work instead with its reflection to an antiholomorphic SCFT V f♮ . Thesupersymmetry together with the antiholomorphicity imply that the Ramond-sector partitionfunction Z RR ( V f♮ ) simply counts the Ramond-sector ground states, of which there are 24.Thus the torus one-point function of ¯ Q in V f♮ ⊗ Fer(3) is h ¯ Q i V f♮ ⊗ Fer(3) = h i V f♮ h ¯ Q i Fer(3) + h ¯ Q i V f♮ h i Fer(3) = 24 η (¯ τ ) + 0 . As we will explain in § V f♮ ⊗ Fer(3) is a boundary of an N = (0 ,
1) SQFT F with cylindrical ends, thus providing another source of mock modularforms with shadow 24 η ( τ ) . SQFT • as an Ω -spectrum The story in the previous section applies in the presence of a finite group G of flavour symme-tries. Namely, suppose that F is a noncompact SQFT with cylindrical ends ∂ F and G -flavoursymmetry. After averaging, we may assume that Φ is G -invariant. Then G acts on ∂ F , andso the right-hand side of the holomorphic anomaly equation 2.1 may be twisted and twined byelements of G , and we predict that it will be the holomorphic anomaly for the correspondingtwisted-twined partition function of F . After taking holomorphic parts, we would produce a(generalized) mock modular form valued in characters of G .– 10 –hus we can answer Question 1.1 if we can produce an SQFT S which is not just theboundary of an SQFT F with cylindrical ends, but is such a boundary compatibly with anM -flavour symmetry. By exchanging F with the R -family x
7→ F ( x ), we are equivalentlyasking whether S can be deformed continuously, in an M -equivariant way, to an SQFT withspontaneous supersymmetry breaking: whether S is in the “spontaneous-supersymmetry-breaking phase” of SQFTs with M flavour symmetry, or whether its supersymmetry is“protected” by the M -symmetry.This question—whether some object can be continuously deformed into some otherobject—is the fundamental question of homotopy theory, and we will try to answer it byadopting homotopical techniques. Specifically, we will see that the space of SQFTs is notmerely a topological space, but rather has extra structure making it into a “spectrum.” Thealgebraic topology of spectra is more rigid than the algebraic topology of spaces, and thereare more tools available. The construction of a spectrum SQFT • described in this section isclosely related to a construction in [BE16] (see also [ST04, ST11, DH11]).In order to describe this spectrum structure, we will need to discuss in a bit more detailthe gravitational anomalies that (1+1)-dimensional (S)QFTs can enjoy. Specifically, we willdistinguish two versions of the word “quantum field theory,” which we call “absolute” versus“anomalous.” For related recent discussion, see e.g. [Fre19, JF20].For a QFT to be absolute , it must come with extra data which is of debatable physicalcontent. An absolute QFT has an absolutely-defined partition function, with no ambiguityabout, say, the “zero” in the energy scale, or about the normalization of the path integralmeasure. An absolute QFT should have well-defined (super) Hilbert spaces, with no projec-tivity in the action by isometries: vectors in this Hilbert space have well-defined phases. Sincethe partition function is part of the data of an absolute QFT, symmetries of absolute QFTsnever have ’t Hooft anomalies. The group of symmetries of an absolute quantum mechanicsmodel is a subgroup of the unitary group rather than the projective unitary group. The usualfunctorial definition of topological QFTs, building on the original definition of [Ati88], is anattempt to model absolute TQFTs.For comparison, an anomalous QFT is one that tolerates many phase ambiguities in itsvalues. To tolerate an ambiguity in the meaning of “zero” energy, anomalous QFTs have“partition functions” that are not functions, but rather sections of possibly-nontrivial linebundles. To tolerate an ambiguity in the phase of a pure state, anomalous QFTs assignprojective Hilbert spaces rather than honest Hilbert spaces. Symmetries of anomalous QFTstypically have nontrivial ’t Hooft anomalies. QFTs defined in terms of their algebras ofoperators are typically anomalous, and more data would to be needed in order to promotethem to absolute QFTs. The simplest example of this is the Stone–von Neumann theorem,which says that an algebra of observables determines the Hilbert space functorialy only upto a phase ambiguity, i.e. only as a projective Hilbert space.In the case of (1+1)-dimensional QFTs, the obstruction to promoting from anomalous toabsolute is the gravitational anomaly w = c R − c L mentioned in § § Z , Z , Z , , Z .This spectrum is called “fGP ×≤ ” in [GJF19b], and the convention in that paper is that thehomotopy groups live in degrees(fGP ×≤ ) (pt) = Z , (fGP ×≤ ) (pt) = Z , (fGP ×≤ ) (pt) = Z , (fGP ×≤ ) (pt) = 0 , (fGP ×≤ ) − (pt) = Z , where by definition (fGP ×≤ ) • (pt) = π −• fGP ×≤ , and, as mentioned already in § c R − c L is a half-integer) in the Z in cohomological degree3 = (1+1) + 1, and the Z in cohomological degree 2 corresponds to the two choices forpromoting a QFT with vanishing gravitational anomaly to an absolute QFT.The simplest example of a (1+1)-dimensional QFT whose gravitational anomaly does notvanish is the theory Fer(1) of a single chiral Majorana–Weyl fermion. This is a holomorphicconformal field theory, with central charges c L = and c R = 0. It can be made into an N = (0 ,
1) superconformal field theory by declaring that the supersymmetry operator actstrivially. Since c R = c L , this SCFT cannot be promoted to an absolute QFT. For example,its “partition function” is not a function, but rather a section of a nontrivial line bundleon the moduli space of spin Riemann surfaces called the Pfaffian line [Fre87, Bor92]. (Itis in fact best thought of as a bundle of superlines, with fibres isomorphic either to C | or C | depending on whether the spin Riemann surface is or is not the boundary of a spin3-manifold.) The tensor product (aka stacking) of n copies of Fer(1) produces an N = (0 , n ) = Fer(1) ⊗ n with ( c L , c R ) = ( n , Definition 2.3.
SQFT n is the space of compact unitary N = (0 , SQFTs whose anomalyis identified with the anomaly for
Fer( n ) . For example, SQFT is the space of absolute SQFTs. The gravitational anomaly of F ∈
SQFT n is w = c R − c L = − n , but to give a point in SQFT n requires more datathan just an anomalous SQFT with this gravitational anomaly: one must give some “parity”information about the “Ramond-sector Hilbert space” of F , which is not a Hilbert space butrather an object of a possibly non-trivial category determined by Fer( n ) (namely, the categoryof Ramond-sector vertex modules for the chiral algebra of Fer( n )).The symmetric group acts naturally on Fer( n ) by permuting the constituent free fermions,and hence on SQFT n . Indeed, as an anomalous SQFT, Fer( n ) carries an action by theorthogonal group O( n ). (The group acting on Fer( n ) with trivialized ’t Hooft anomaly is– 12 –pin( n ).) More generally, one can functorially define a holomorphic CFT Fer( V ) for any realvector space V with positive-definite inner product, and so we could have defined a spaceSQFT V for any V , which is noncanonically isomorphic to SQFT dim V .There is a canonical isomorphism [DH11]Fer( V ) ⊗ Fer( W ) ∼ = Fer( V ⊕ W ) . This implies that tensor product (stacking) of SQFTs provides a commutative and associativeoperation ⊗ : SQFT V × SQFT W → SQFT V ⊕ W which is compatible with the actions by O( V ) × O( W ) ⊂ O( V ⊕ W ). We warn that the“commutativity” is subtle. Indeed, given F ∈
SQFT V and F ′ ∈ SQFT W , to compare F ⊗ F ′ with F ′ ⊗ F , one must use the isomorphism SQFT V ⊕ W ∼ = SQFT W ⊕ V coming from theisomorphism Fer( V ⊕ W ) ∼ = Fer( W ⊗ V ) that permutes the fermions. Even if V = W , thisisomorphism is nontrivial, and may have a nontrivial anomaly.Thus one can think of SQFT • all together as a sort of “graded commutative monoid.”With a bit of work, one can define a direct sum operation on each SQFT n , so that SQFT • is a graded commutative ring-without-negation. Rather than trying to define direct sumsdirectly, we will see that each SQFT n is in fact a commutative group up to homotopy: wewill give SQFT • the structure of a (commutative orthogonal) Ω-spectrum, with one smallmodification.By definition, an Ω -spectrum E • is a sequence of spaces E , E , E , . . . , each equippedwith a basepoint 0 ∈ E n , together with homotopy equivalences E n ∼ → Ω E n +1 , where Ω E n +1 means the space of loops in E n +1 that start and end at the basepoint 0. In particular, each E n is an infinite loop space (i.e. a homotopically coherent abelian group). The spectrum E • is orthogonal when the grading is not just by integers but by real vector spaces V as above,and the homotopy equivalence E V ∼ → Ω E V ⊕ R is compatible with the O( V ) action. Let X bea space. The E • -cohomology of X is by definition E • ( X ) = π maps( X, E • ) . This is an abelian group because of the homotopy equivalence E n ∼ → Ω E n +1 , which provides,for any s ≥
0, an isomorphism E • ( X ) ∼ = π s maps( X, E • + s ) . For our spectrum SQFT • , we want to choose the basepoint 0 ∈ SQFT n to be the “zeroQFT.” This is the TQFT that assigns “0” to every nonempty input: its partition function iszero, its Hilbert space is zero-dimensional, etc. This can be thought of as having any anomalythat one so chooses. When a physicist says “supersymmetry is spontaneously broken in F ,” amathematician should hear “ F flows to 0 under RG flow,” where “RG flow” is a canonically-defined action by the monoid R ≥ on SQFT n (and, debateably, by the group R ), and “ F flows to F IR ” means that F IR is the limit of the RG-flow starting at F .– 13 –ut some physicists will rightly quibble with the idea of the “zero QFT,” and will insteadtake the phrase “supersymmetry is spontaneously broken” as a primitive notion. Moreover,some mathematical attempts to define the notion of “quantum field theory,” including thefunctorial ones suggested in [ST04], include this zero QFT as a point on SQFT n , but othersof a more operator-algebraic nature (e.g. [DH11]) do not. (Indeed, if one follows the ideasof [DH11], then the definition of SQFT n should be the space of operator-algebraically-definedSQFTs equipped with a Morita equivalence to Fer( n ). There is a “zero” operator algebra,but it is not Morita equivalent to a nonzero algebra.) For this reason, we will modify ournotion of spectrum to tolerate a subspace of basepoints, rather than a single basepoint. Theloop space Ω E n +1 then should consist of loops that begin and end inside this subspace, andthe homotopy groups defining E • -cohomology should be relative homotopy groups. Otherwisethere is no real difference. And if the reader’s model of “quantum field theory” includes thezero QFT, then the reader may use the usual notion of Ω-spectrum in what follows.Let us parameterize paths by the real line R : a point in Ω SQFT n is an R -family x
7→ F ( x )in SQFT n such that supersymmetry is spontaneously broken for all x ≪ x ≫ x → ±∞ .One can promote the former type of loop to the latter by turning on an RG flow whose strengthincreases as x → ±∞ .)Then the map SQFT n → Ω SQFT n +1 couldn’t be simpler. As above, let Fer(1) denotethe holomorphic CFT of a single chiral Majorana–Weyl fermion λ . Above we promoted thisCFT to an N = (0 ,
1) SCFT by declaring that the supersymmetry operator was 0. But, atthe cost of conformal invariance, we may give it other N = (0 ,
1) structures. Specifically,the supercurrent ¯ Q = xλ defines an N = (0 ,
1) supersymmetry on Fer(1), which is notsuperconformal. Let Fer(1)( x ) denote the SQFT (Fer(1) , ¯ Q = xλ ). Comparing with § x ) is exactly the “fibre” over − x of the operator Φ = 0 in the vacuum theory 1 ∈ SQFT (with one-dimensional Hilbert space and partition function identically equal to 1).Then the map SQFT n → Ω SQFT n +1 is: F 7→ F ⊗
Fer(1)( x ) . If x = 0, supersymmetry spontaneously breaks in Fer(1)( x ) and hence in F ⊗
Fer(1)( x ). Sothis family x
7→ F ⊗
Fer(1)( x ) is indeed a point in Ω SQFT n +1 . (In fact, it is a point evenfor the stricter version where “0” is a meaningful QFT: the action of RG flow on Fer(1)( x )simply rescales x e s x , where s → ∞ is the IR limit, and so the x → ±∞ limits of Fer(1)( x )are both the zero QFT.)We must now prove that this map F 7→ F ⊗
Fer(1)( x ) is a homotopy equivalence. Con-sider the “dynamicalization” map Ω SQFT n +1 → SQFT n that takes a family x
7→ F ( x )in Ω SQFT n +1 and promotes the parameter x to a dynamical scalar multiplet, producingthe SQFT that, as in § R φ,ψ F ( φ ). We claim without proof that R φ,ψ F ( φ )is compact for ( x
7→ F ( x )) ∈ Ω SQFT n +1 : the justification is that, since supersymmetryis spontaneously broken, this is essentially a compactly-supported family; but more work– 14 –ould need to be done to justify this claim, and one may have to first modify the family byRG-flowing F ( x ) by some finite amount that grows as x → ±∞ .To prove that F 7→ { x
7→ F ⊗
Fer(1)( x ) } is a homotopy equivalence, it suffices toprove that its two compositions with R φ,ψ are both homotopic to the identity. We do notneed to confirm any higher homotopy coherence: in particular, we do not need to show thatthe homotopies to the identity are compatible in any way. (We would need to prove suchcompatibilities if we wanted to claim that R φ,ψ was the homotopy-coherent inverse to tensoringwith Fer(1)( − ).)First, consider the composition F 7→ F ⊗
Fer(1)( x ) Z φ,ψ F ⊗
Fer(1)( φ ) . The copy of F comes out of the integral, and so it suffices to show that R φ,ψ Fer(1)( φ ) iscontinuously deformable to the vacuum theory 1 ∈ SQFT . This is a special case of thephilosophy mentioned in § R φ,ψ Fer(1)( φ ) contains the followingfields. First, there is the chiral fermion λ ∈ Fer(1). Next, there is a full scalar boson φ ,which is the bosonic component of the superfield that dynamicalizes x . Finally, there isthe superpartner ψ of φ , which is an antichiral fermion. The supersymmetry operator incomponents is ¯ Q = λ ¯ ∂φ + ψ∂φ. The first summand is from the supersymmetry λx in Fer(1)( x ), and the second summand saysthat ψ is the superpartner of φ . The Lagrangian contains the standard massless terms λ ¯ ∂λ , φ ∆ φ , and ψ∂ψ . It also contains a correction coming from ¯ Q , which ends up being λψ + φ .(The Lagrangian for Fer(1)( x ) had a correction like λ ¯ Q [ x ]+ x , and when we replace x with φ ,¯ Q [ x ] becomes ψ .) All together, we can recognize R φ,ψ Fer(1)( φ ) as the free theory consistingof a massive Majorana fermion and a massive scalar boson. This free theory is well-knownto flow to the trivial vacuum theory in the IR, which is to say that RG flow implements ahomotopy R φ,ψ Fer(1)( φ ) ≃ F ( x ) Z φ,ψ F ( φ ) Z φ,ψ F ( φ ) ⊗ Fer(1)( x ) . We have not tried to be precise about the meaning of “family of SQFTs.” For the purposesof this article, let us suppose that the field content (and any other kinematical information)of F ( x ) is independent of x , and only the Lagrangian and the supersymmetry (and any otherdynamical information) varies with x . This is not unreasonable: if there is a field that existsonly for certain values of x , one can extend it to a field that exists for all x but is very massiveexcept at the values of x for which it was earlier defined. Assuming we have topologized thespace of SQFTs in a way that cares primarily about the effective low-energy field theory,turning on very massive fields should be a very small deformation, and so should not change– 15 –he homotopy type of the family F ( − ). Then the field content on the right-hand side consistsof: the fields in F ( − ), the scalar φ , the antichiral fermion ψ , and the chiral fermion λ . Writing L LHS ( x ) and ¯ Q LHS ( x ) for the Lagrangian and supersymmetry operators in F ( x ), and writing¯ Q ′ LHS ( x ) = ∂∂x ¯ Q LHS , the Lagrangian and supersymmetry operators on the right-hand sideare: L RHS = L LHS ( φ ) + ¯ Q ′ LHS ( φ ) ∂ψ + φ ∆ φ + ψ∂ψ + λ ¯ ∂λ, ¯ Q RHS = ¯ Q LHS ( φ ) + ψ∂φ + xλ. Now consider deforming this SQFT by the superpotential W = f ( φ ) λ for some polynomial f ∈ R [ x ]. This deformation is allowed: it does not destroy compactness, nor does it destroythe spontaneous supersymmetry breaking. The deformation changes the Lagrangian to: L deformed = L RHS + ¯ Q RHS [ W ] + (cid:18) ∂W∂λ (cid:19) . Since the original ¯ Q LHS ( x ) was a function in x , neither W nor ¯ Q LHS ( φ ) have any ∂φ -dependence, and so commute. Thus we have: L deformed = L RHS + ψf ′ ( φ ) λ + xf ( φ ) + f ( φ ) . Taking f ( φ ) = − φ gives L deformed = (cid:2) L LHS ( φ ) + ¯ Q ′ LHS ( φ ) ∂ψ (cid:3) + (cid:2) φ ∆ φ + ψ∂ψ + λ ¯ ∂λ + ψλ + − xφ + φ (cid:3) . Focusing on the second bracketed expression, we see that φ now has a mass with vacuumexpectation value x and the full fermion ( λ, ψ ) is also massive. So, performing the pathintegral in those variables first, the IR behaviour of the deformed theory is described simplyby setting φ = x and λ = ψ = 0, and we recover the original theory F ( x ).In summary, we have outlined a proof of the following result. We call a “conjecture”because we did not attempt to mathematically define or topologize the spaces SQFT • , andbecause even at a physicists’ level of rigour we left some questions about the details of thesespaces. Conjecture 2.4.
The spaces
SQFT V of compact unitary N = (0 , SQFTs with anomalyidentified with the anomaly of
Fer( V ) compile into a commutative ring orthogonal Ω -spectrum SQFT • . SQFT • and ’t Hooft anomalies Let G be a finite group (or a Lie group or . . . , but we will need only the finite group case). Thediscussion in the previous section applies equally well if one considers SQFTs, and familiesthereof, which are equipped with a nonanomalous G -flavour symmetry. The correspondingspectrum SQFT • G is a G -equivariant enhancement of the spectrum SQFT • : using it, one canassign cohomology groups to spaces equipped with G -action. The fundamental reason thatSQFT • admits an equivariant enhancement is that SQFTs admit automorphisms, and so the– 16 –ollection SQFT n of SQFTs with a given gravitational anomaly is not merely a space, butrather a groupoid or stack. If X is any stack, then we can consider the space of maps ofstacks from X to SQFT n , and evaluate its homotopy groups. Taking X = B G the classifyingstack of the group G gives: SQFT • G = maps( B G, SQFT • ) , SQFT • ( B G ) = SQFT • G (pt) = π maps( B G, SQFT • ) . We may also consider SQFTs with G -flavour symmetry and prescribed ’t Hooft anomaly ω .These compile into an orthogonal Ω-spectrum SQFT • G,ω . Because ’t Hooft anomalies add un-der stacking (i.e. under tensor product of SQFTs), SQFT • G,ω is not a ring spectrum, but itis a module spectrum for SQFT • G . The algebrotopologists’ name for introducing an ’t Hooftanomaly ω is twisting : the homotopy groupsSQFT • G,ω (pt) = SQFT • ω ( B G ) = π SQFT • G,ω are the “ ω -twisted G -equivariant SQFT • -cohomology of a point.”Where do ’t Hooft anomalies live? For (1+1)-dimensional fermionic QFTs, they live inthe (extended) supercohomology SH ( B G ) of the group G [GW14, WG17], which is the 3-layerspectrum described in § (pt) = Z , SH (pt) = Z , SH (pt) = 0 , SH − (pt) = Z , and Postnikov k -invariants Sq : Z → Z and (cid:3) Z ◦ Sq : Z → Z , where Sq denotes thesecond Steenrod squaring operation and (cid:3) Z denotes the integral Bockstein (for the shortexact sequence Z → Z → Z ).By definition there is a map H • +1 ( G ; Z ) → SH • ( B G ). A long exact sequence shows thatthis map is an injection in degree • ≤ • ). It is asurjection if H •− ( G ; Z ) and H •− ( G ; Z ) both vanish. In particular, if G is the Schur coverof simple group, then SH ( B G ) = H ( G ; Z ).Actually, as emphasized in [GJF19b], it is best to think of SH ( G ) instead as the reduced group cohomology of G with coefficients in the 4-level spectrum fGP ×≤ mentioned in § → B G gives a canonical isomorphism(fGP ×≤ ) ( B G ) = SH ( B G ) ⊕ SH (pt) , and SH (pt) ∼ = Z indexes the gravitational anomaly n = 2( c L − c R ). This is consistentwith the general story of twistings of cohomology theories: fGP ×≤ controls all the anoma-lies, both ’t Hooft and gravitational, for the spectrum SQFT • , and so algebrotopologistssometimes write the twisted cohomology groups as SQFT • + ωG ( − ), with • + ω a total class in(fGP ×≤ ) ( B G ).To build SQFT • G,ω completely correctly, one should rigidify the ’t Hooft anomaly by choos-ing some representative SQFT V ω with anomaly ω and then asking that points in SQFT • G,ω – 17 –ave ’t Hooft anomaly identified with the anomaly of V ω . If one just asks that the anomalyof an SQFT F be equal to ω in cohomology, then there is an ambiguity in this identification(parameterized by the reduced cohomology group f SH ( B G )), analogous to the ambiguity inpromoting an anomalous SQFT with c L = c R to an absolute SQFT. Given that we mod-elled SQFT • as an orthogonal spectrum, identifying gravitational anomalies with anomaliesof Fer( V ) for real vector spaces V , one might try now to set V ω ? = Fer( V ) for some realrepresentation V : G → O( n ). The anomaly of G acting on Fer( V ) is a characteristic class p ( V ) ∈ SH ( B G ) called the fractional Pontryagin class . (It is in the image of H ( G ; Z )whenever the representation V is Spin.)Thus one can ask: How many of the classes in SH ( B G ) arises as fractional Pontryaginclasses of real representations? This is a supercohomological version of the classical questionof understanding which classes in H ev ( G ; Z ) arise as Chern classes of complex representations.The answer is: it depends on the group G . To illustrate this, consider the case when G isa Schur cover of a sporadic group. The calculations in [JFT18, JFT19, JF19] show thatSH ( B G ) = H ( G ; Z ) vanishes if G is one of { M , , J , Ly } and does not vanish but consists entirely of fractional Pontryagin classes for G in { M , , , M , , Co , Co , , , , He } . On the other hand, for the groups G in { , Mon } , it is shown in those papers that SH ( B G ) = H ( G ; Z ) is not generated by fractional Pontrya-gin classes. The calculations for the other sporadic groups have not been completed.This might lead the reader to worry that perhaps there is no good representative V ω in general. Fortunately, the main result of [EG18] is that there is, for any finite group G and anomaly ω ∈ H ( G ; Z ), a bosonic holomorphic conformal field theory V ω with G -flavoursymmetry and ’t Hooft anomaly ω . The CFT V ω is not canonical, and is of very high centralcharge. Although [EG18] focuses on bosonic CFTs, the construction extends to the fermioniccase for any ω ∈ SH ( B G ). Any holomorphic conformal field theory can be thought of as an N = (0 ,
1) superconformal field theory by simply declaring the supersymmetry operator tobe trivial. Thus we can construct representatives V ω as required.The two examples from the end of § ⊕ and V f♮ ⊗ Fer(3), each carry naturalM -actions. The action on the former permutes the 24 summands, and so (as in the introduc-tion) we will call it ⊗ Fer(3); we will write “ ” for both the standard degree-24 permutationrepresentation of M as well as its enhancement to a TQFT with 24 massive vacua permutedby M . The action on the latter is the restriction of the action by Co = Aut( V f♮ ). Thuswe find classes [ ⊗ Fer(3)] ∈ SQFT − ω ( B M ) and [ V f♮ ⊗ Fer(3)] ∈ SQFT − ω ′ ( B M ), where ω, ω ′ are the ’t Hooft anomalies of the various actions. In both cases the action of M on– 18 –er(3) is trivial, and so these classes are the products of classes [ ] ∈ SQFT ω ( B M ) and[ V f♮ ] ∈ SQFT − ω ′ ( B M ) by the class [Fer(3)] ∈ SQFT − (pt). (The ring SQFT • (pt) acts onall twisted equivariant cohomologies SQFT • ω ( X ).) One of the main results of [GJFW19] isthat there is a well-defined class in SQFT − n (pt) for each cobordism class of n -dimensionalmanifolds with String structure (and in particular for each class of n -dimensional framedmanifolds), and the 3-sphere S = SU(2) with its Lie group framing determines the class[Fer(3)]. In algebraic topology, generalized cohomology classes determined by S -with-its-Lie-group-framing are conventionally named “ ν .” Following this convention, we can writeour SCFTs as [ ] ν ∈ SQFT − ω ( B M ) , [ V f♮ ] ν ∈ SQFT − ω ′ ( B M ) . What are the ’t Hooft anomalies ω, ω ′ ? The answer in the former case is complicated, andso we will do it second. For [ V f♮ ] ν , the anomaly of M is restricted from the anomaly of theCo -action on V f♮ . The main result of [JFT18] implies that SH (Co ) ∼ = Z is cyclic, gen-erated by the fractional Pontryagin class p of the 24-dimensional projective representationof Co . (The paper [JFT18] calculates instead the ordinary cohomology of the Schur coverCo = 2 . Co , but it is not hard to show that the canonical maps SH (Co ) → SH (Co ) andH (Co ; Z ) → SH (Co ) are both isomorphisms.) Up to a sign convention in the definition of“’t Hooft anomaly,” p is precisely the anomaly of Co acting on V f♮ [JF19]. Furthermore,Theorem 8.1 of [JFT18] asserts that, upon restriction to M ⊂ Co , this anomaly restrictsto − α , where α is the anomaly of Mathieu Moonshine computed in [GPRV13]. Actually,since different authors might reasonably disagree on the sign of “the anomaly,” it is usefulthat [CHVZ18] has compared the multipliers for some elements acting in Mathieu Moonshineversus in V f♮ (see Table 3 therein). Multipliers depend linearly on the anomaly, and in allcases checked the anomaly for V f♮ restricts to minus the anomaly from [GPRV13]. Togetherwith the computer calculation H (M ; Z ) = Z from [DSE09] (confirmed using elemen-tary methods in Theorem 5.1 of [JFT18]), and further calculations from [GPRV13], thesecomparisons are enough to establish that(anomaly of V f♮ ) | M = − (anomaly of Mathieu Moonshine) . But the anomalies of V f♮ and V f♮ have opposite signs. Writing α for the Mathieu Moonshineanomaly from [GPRV13], we find:[ V f♮ ] ν ∈ SQFT − α ( B M ) . Turning to [ ] ν ∈ SQFT − ω ( B M ), we must answer the question: What is the ’t Hooftanomaly of the M -action on the TQFT ? The na¨ıve answer, “zero,” misses an importantsubtlety, which is that the question is badly posed: ’t Hooft says that there is an anomaly whena partition function or other datum, which was expected to be G -invariant, in fact changesby a phase; but in our case those data are often zero because of the vacuum degeneracy.More precisely, the M -symmetry on spontaneously breaks to a trivial M -symmetry.– 19 –his trivial M -symmetry is definitely nonanomalous. But we may consider the total M -symmetry to have any anomaly that we choose in the kernel of SH (M ) = H (M ; Z ) → SH (M ) = H (M ; Z ). Remarkably, H (M ; Z ) = 0 [Mil00] (that paper in fact showsthat H • (M ; Z ) = 0 for • ≤
5, and provides further information about H • (M ; Z ); thelow-cohomology results are confirmed computationally in [DSE09], and the H calculation isconfirmed with elementary methods in [JFT18]). Thus we may consider the M -action on[ ] ν as having any anomaly that we want:[ ] ν ∈ SQFT − ω ( B M ) for any desired ω ∈ SH (M ) = H (M ; Z ) = Z . The same argument can be rephrased algebrotopologically in terms of pushforwards . If f : H → G is a homomorphism of finite groups, then an SQFT with G -symmetry and’t Hooft anomaly ω ∈ SH ( B G ) determines, by forgetting some information, an SQFT with H -symmetry and ’t Hooft anomaly f ∗ ω ∈ SH ( B H ). This provides a pullback map f ∗ : SQFT • G,ω → SQFT • H,f ∗ ω . This map has an “adjoint” f ∗ : SQFT • H,f ∗ ω → SQFT • G,ω . To construct it, note that any map f : H → G factors canonically as a surjection followed by an injection: H ։ im( f ) ֒ → G. Thus it suffices to describe f ∗ when f : H → G is either surjective or injective.Suppose first that f : H → G is a surjection with kernel K = ker( f ). Then f ∗ ω ∈ SH ( B H ) restricts trivially to K , and so if F is an SQFT with H -symmetry, the K -action isnonanomalous and may be gauged. Furthermore, because the anomaly f ∗ ω of the H -actionis pulled back from G , there is no “mixed anomaly.” It follows that the gauged theory F (cid:12) K carries a G -action with anomaly ω . The pushforward map f ∗ is f ∗ : F 7→ F (cid:12) K. Note the repeated use of the fact that the anomaly is pulled back from G . If all we knewwas that H acted on F with some anomaly ω ′ ∈ SH ( H ), and that ω ′ | K = 0, then therewould be an ambiguity in the meaning of the gauged theory: there would be f SH ( K )-manytheories that deserve the name “ F (cid:12) K ,” parameterized by the f SH ( K )-many trivializationsof ω ′ | K . (Here and throughout, f SH • denotes reduced supercohomology.) In our case, we canchoose a canonical gauging because, since f ∗ ω is restricted from G , it trivializes canonicallyon K . Gauging uses up the K -symmetry, but produces a new “magnetic dual” action by f SH ( K ) ∼ = hom( K, U(1)), and in general the remaining G -action could be extended by thissymmetry (c.f. [BT17] or § f ∗ ω is pulled back from G . If the extension were trivializable but not canonically so, then thedifferent trivializations might lead to different G -actions with different anomalies. But, againbecause f ∗ ω is pulled back from G , the extension is canonically trivializable, and the resulting G -action has anomaly ω . – 20 –uppose now that f : H → G is an injection, and let X = G/H denote the space of cosets.If F is an SQFT with H -symmetry and anomaly f ∗ ω , then the direct sum (aka disjoint union)of X -many copies of F can be given a G -action that permutes the copies, and that acts as H on each copy. Physically, this is an SQFT where the G -symmetry spontaneously breaks toan H -symmetry. This is the pushforward map.In both cases, the pushforward f ∗ can be described as a type of finite path integral.Indeed, gauging a K -symmetry is the same as integrating over K -gauge fields, which aremaps from the worldsheet to B K , which is the fibre of B H → B G in the case when H → G is an injection. When H → G is a surjection, then the fibre of B H → B G is the set X , andagain we are taking an integral over maps to this fibre. This explains the general structure: f ∗ implements a finite path integral over the space of maps from the worldsheet to the fibre X of the map B H → B G .As an example, suppose that f : H ֒ → G is an inclusion, and ω ∈ SH ( G ) is an anomalysuch that f ∗ ω = 0 ∈ SH ( H ). Then we have a pushforward map f ∗ : SQFT • ( B H ) → SQFT • ω ( B G ) . The domain is a commutative ring (the codomain is not, if ω = 0), with unit 1 ∈ SQFT • ( B H )represented by the trivial “vacuum” SQFT with trivial H -symmetry. The pushforward f ∗ (1)is simply the (1+1)-dimensional TQFT with X = G/H many ground states, permuted bythe G -symmetry, and no other structure. (In terms of functorial field theories valued in the2-category of algebras and bimodules, f ∗ (1) corresponds to the algebra L X C .) For G = M and H = M , this is the TQFT that we called “ ” above. If instead we had chosensome F ∈
SQFT • (pt), equipped with the trivial H -action (equivalently, pulled back along B H → pt), then f ∗ ( F ) = f ∗ (1) ⊗ F = ⊗ F .For the purposes of explaining Mathieu Moonshine, this looks pretty good. When re-stricted along pt ⊂ M , the SQFT ⊗ Fer(3) restricts to Fer(3) ⊕ , which we alreadysaw is nullhomotopic and produces the mock modular form H ( τ ). If ⊗ Fer(3) were M -equivariantly nullhomotopic, then we would produce mock modular forms H g,h ( τ ) as desired.Unfortunately, it is not: Proposition 2.5. 24 ⊗ Fer(3) is not M -equivariantly nullhomotopic, for any ’t Hooftanomaly ω .Proof. If [ ⊗ Fer(3)] = [ ] ⊗ ν were trivial in SQFT − ω ( B M ), then its restriction to M would also be trivial. Since SH (M ) = 0, this restriction has trivial gauge anomaly, and sowe may gauge the M -action. If ⊗ Fer(3) were equivariantly nullhomotopic, then so wouldbe this gauged theory (by gauging the M -action on the nullhomotopy). In algebrotopologicallanguage, writing f : M ֒ → M and p : M → pt, we wish to compute p ∗ f ∗ f ∗ ⊗ ν .This is purely a TQFT computation. Our goal is to compute the (1+1)-dimensionalTQFT which counts maps from the worldsheet to the quotient stack (cid:12) M . – 21 –ut, restricted to M , splits as ⊔ , and so (cid:12) M = B M ⊔ (cid:12) M = B M ⊔ B M . In other words, the TQFT p ∗ f ∗ f ∗ p ∗ , and pure gauge theory for M .For any finite group G , pure G -gauge theory in (1+1)-dimensions is described by thegroup algebra C [ G ] of G , which is Morita equivalent to the direct sum of G/G )-manycopies of C , where G/G ) means the number of conjugacy classes in G . The group M has17 conjugacy classes, and the group M has 12 conjugacy classes. Thus p ∗ f ∗ f ∗ p ∗ , and so p ∗ f ∗ f ∗ p ∗ ⊗ ν = 29 ν , represented by the SQFT Fer(3) ⊕ . But 29 is not divisible by24, and so Fer(3) ⊕ is not nullhomotopic by [GJF19a].One could wonder if perhaps the day would be saved by somehow squeezing in somediscrete torsion, i.e. nontrivial Dijkgraaf–Witten action, into the M gauge theory, since f SH (M ) = H (M ; Z ) = Z is nontrivial. This effects a change from the group algebra C [M ] to a twisted group algebra. The twisted group algebras of M are Morita equivalentto a sum of 10 or 11 copies of C , depending on the twisting, and neither 17 + 10 nor 17 + 11is divisible by 24. Proposition 2.5 means that we will not be able to answer Question 1.1 by working justwith the permutation representation of M . There is another reason to reject it as ananswer. Suppose, contradicting Proposition 2.5, that ⊗ Fer(3) were M -equivariantlynullcobordant, and choose a nullcobordism F . For each commuting pair g, h ∈ M , we maytwist and twine F , thereby producing a partition function Z RR ( F ) g,h ( τ, ¯ τ ). Following thelogic of Conjecture 2.1, the holomorphic part of Z RR ( F ) g,h ( τ, ¯ τ ), normalized with a factor of η ( τ ), will be a mock modular form (for some subgroup of SL ( Z ) depending on g, h ) whoseshadow is (the complex conjugate of) the torus one-point function of ¯ Q in ⊗ Fer(3), twistedand twined by g and h .Since g and h do not act on Fer(3), this shadow factors asputative shadow = Z RR ( ) g,h η ( τ ) . The computation of Z RR ( ) g,h is very easy, because itself is very easy, being simply theTQFT of maps from the worldsheet to the standard permutation representation of M .To have a map to this set from a torus twisted and twined by g and h , the value of the mapmust be fixed by both g and h , and we discover: Z ( ) g,h ∝ number of common g, h fixed points in . – 22 –f we were treating as a nonanomalous M -equivariant TQFT, then the two sides wouldbe equal. We have written only that they are proportional because of the possibility of anontrivial anomaly ω . Indeed, the presence of ω means that the twisted-twined partition“function” is not really a function at all, but rather a section of a flat line bundle on thespace of spin tori with G -bundles. Under modifying a 3-cocycle representative of ω by d ξ , forsome 2-cochain ξ on G , the “function” Z ( ) g,h changes by a factor of ξ ( g,h ) ξ ( h,g ) .When g = e is the identity, Z RR ( ) e,h is simply the trace of the h -action on , whichagrees with the shadows in Mathieu Moonshine (compare [CDH14a]). More generally, if thesubgroup of M generated by g and h is cyclic (for example, if g and h have coprime order),then Z ( ) g,h = Z ( ) ,x = tr ( x ), where x is any generator of the cyclic group. This isagain consistent.However, if the subgroup generated by g and h is not cyclic, then this putative shadowis not the shadow of the mock modular form H g,h ( τ ) from (generalized) Mathieu Moonshine.Indeed, [GPRV13] finds that H g,h ( τ ) has trivial shadow (i.e. it is holomorphic modular) assoon as g and h do not generate a cyclic group, and for most such pairs H g,h simply vanishes.But there are many rank-2 subgroups of M which do have fixed points. A list of all conjugacyclasses of rank-2 abelian subgroups of M is available in Table 1 of [GPRV13]. The first entryon that list, for example, is a Klein-4 subgroup Z which acts with 8 fixed points in .Instead, as in Conjecture 1.4, we conjecture that V f♮ ⊗ Fer(3) is nullhomotopic. Ifit is, then the twisted and twined partition functions of its nullhomotopy would have, astheir shadows, the functions Z RR ( V f♮ ) g,h η ( τ ) . The antiholomorphicity of V f♮ means that Z RR ( V f♮ ) g,h is just an integer: the signed trace of h acting on the ground states of the g -twisted Ramond sector of V f♮ . When g = 1, these ground states form the Leech latticerepresentation Leech ⊗ R of Co = 2 . Co . (The double cover comes from the “Gu–Wenlayer” of the anomaly of the Co -action on V f♮ .) This representation restricts over M tothe permutation representation, and so Z RR ( V f♮ ) ,h = tr ( h ) = Z RR ( ) ,h , which is thedesired value. More generally, if g, h generate a cyclic group, with cyclic generator x , then Z RR ( V f♮ ) g,h = Z RR ( V f♮ ) ,x = Z RR ( ) ,x = Z RR ( ) g,h , simply because these two integersare related by a modular transformation. On the other hand, when g, h generate a rank-2group, then Z RR ( V f♮ ) g,h and Z RR ( ) g,h may not agree. In fact: Theorem 2.6. If g, h ∈ M generate a rank-2 abelian group, then Z RR ( V f♮ ) g,h = 0 . ThusConjecture 1.4 is consistent with the shadows found by [GPRV13]. The calculations of [CdLW19] suggest that there may be an elegant proof of this theorem,but the author did not find one. Rather, we will prove the theorem by computing all cases.
Proof.
The integer Z RR ( V f♮ ) g,h = Z RR ( V f♮ ) g,h depends only on the conjugacy class of therank-2 abelian group h g, h i . It transforms with nontrivial multiplier under some congruencesubgroup, and hence must be zero, as soon as the ’t Hooft anomaly restricts nontrivially to h h i , or to any other generator of h g, h i . This leaves only the groups where h g, h i consists ofelements with M -conjugacy classes 1 , do not participate in rank-2 abelian groups.) In Table 1 of [GPRV13], these are theentries numbered 1 , , , , , , , , , g -twisted Ramond-sectors of V f♮ for g of M conjugacy class 2A, 3A, 4B, and 6A. In fact, the only rank-2 subgroups inM that include a 6A element are generated by a 6A element and a 2A element, and so, byswitching g and h , we do not need to consider the last case.In order to study these twisted sectors, let us recall a bit about the holomorphic SVOA V f♮ . It is a lattice SVOA for the D +12 lattice : D +12 = n λ = ( λ , . . . , λ ) ∈ Z ⊔ (cid:0) Z + (cid:1) such that X λ i ∈ Z o . It has a canonical translation coset inside R :( D +12 ) R = n λ = ( λ , . . . , λ ) ∈ Z ⊔ (cid:0) Z + (cid:1) such that X λ i ∈ Z + 1 o . The Ramond sector V f♮R is built from ( D +12 ) R in the same way that the Neveu–Schwarz sectoris built from D +12 . Namely, V f♮R is generated over the Heisenberg algebra Bos(12) by a state Γ λ for each λ ∈ ( D +12 ) R . There is no canonical way to assign a fermion parity operator “( − f ”to the R-sector of a holomorphic SVOA, but the relative parity is well-defined, and we willarbitrarily declare the absolute parity by saying that Γ λ is bosonic (resp. fermionic) if λ ∈ Z (resp. ( Z + ) ).The N =0 automorphism group (i.e. the automorphism group as an SVOA, ignoringthe supersymmetry) of V f♮ is the Lie group SO + (12), defined as the image of Spin(12) inthe positive half-spin representation. Because of the ’t Hooft anomaly, this group acts onlyprojectively on the Ramond sector: the group that acts linearly on V f♮R is Spin(12) itself.Since SO + (12) is connected, any g ∈ SO + (12) can be conjugated into the maximal torus T ∼ = R /D +12 ⊂ SO + (12). We will abusively also call this torus element “ g ,” but we willwrite the group law in T additively. Any g ∈ T determines a translated lattice D +12 + g ⊂ R ,from which the g -twisted sector V f♮g is built. A special case is when g = R is the centralelement in SO + (12), in which D +12 + R = ( D +12 ) R is the canonical translated lattice, andthe notation “ V f♮R ” is consistent. As an element of R /D +12 , R can be represented by thevector R = (1 , , . . . ,
0) (mod D +12 ). More generally, the g -twisted R-sector deserves the name“ V f♮R + g .” For the symmetries g ∈ M ⊂ SO + (12) that we are interested in, the vectors are:M name cycle structure g ∈ T = R /D +12
2A 1 ( , , , , , , , , , , , ( , , , , , , , , , , , ( , , , , , , , , , , , L acts on the state Γ λ with eigenvalue | λ | /
2. If g preserves a supersymmetry operator Q , then in the g -twisted R-sector the Hamiltonian L − c/
24 is a square (of the zero mode of Q , up to a normalization convention) and so takes– 24 –nly nonnegative values. We are interested in the space of ground states, which are thus inbijection with those λ ∈ V f♮R + g with | λ | = 1 (since c = 12 for V f♮ ). Such a state contributesa bosonic or fermionic mode according to whether λ − g is integral or half-integral. Each ofthe vectors g ∈ T listed above contains at least five 0s. Thus a vector λ ∈ g + ( Z + ) will have at least five entries with absolute value ≥ , and so | λ | ≥ >
1. It followsthat the g -twisted R-sector has only bosonic ground states. The number of ground states isthen, by modularity, equal to the trace of g acting on the ground states in V f♮R , which is justthe 24-dimensional representation of M . This trace tr ( g ) is easily read from the abovetable: it is the exponent of 1 in the cycle structure. (A priori, there could be both bosonicand fermionic ground states in the g -twisted R-sector, and only their signed count is equalto tr ( g ).)For any g acting on any holomorphic conformal field theory V , the g -twisted sectors V g and V R + g carry projective actions of the centralizer C ( g ) inside the automorphism group of V . As we have remarked already, the anomaly for the M -action on V f♮ is (up to a sign)the same as the anomaly computed in [GPRV13]. This anomaly determines the projectivityof the action of C ( g ) on the g -twisted sectors. (See [GPRV13] for a nice explanation.) Thecentralizers C ( g ) ⊂ M of the elements g listed above, and their projective character tables,are listed in the appendix of [GPRV13].When g = 2A, we have C (2A) = 2 . (2 :L (2)) in the ATLAS notation. Its action on V f♮R +2A is genuinely projective. The eight ground states must compile into an 8-dimensionalmodule. This is the smallest dimension of any projective representation on C (2A). It remainsto identify the correct representation: they are listed under the names χ , χ , χ , χ in theappendix of [GPRV13]. But look at the class called 4A therein. It has a nontrivial anomaly,and so its trace vanishes. Thus we find that the ground states of V f♮R +2A correspond to thecharacter χ . The only nonzero entries in the χ correspond to elements h ∈ C (2A) such that h and g = 2A together generate a cyclic group. This establishes the Theorem for the groupsnumbered 1 , , , , , , ,
29 in Table 1 of [GPRV13].When g = 3A, we have C (3A) = 3 A , the exceptional Schur cover of the alternatinggroup A . The ground states of V f♮R +3A form a six-dimensional linear representation M ;because H ( C (3A); U(1)) = 0, there is a canonical trivialization of the projectivity, and so nophase ambiguities in the actions of elements of C (3A) on M . The centralizer of g inside the fullautomorphism group Aut N =1 ( V f♮ ) = Co is a group of shape 3 . U (3) . +
85, Wil83].This acts through a double cover (3 × . U (3) . V f♮R +3A , and 6 is the smallest dimension ofany simple representation thereof. The central g ∈ (3 × . U (3) . ± ± √−
3. It follows that M breaks up over 3 A as a sum of thecharacters labeled χ , χ , χ , χ , χ , χ in [GPRV13]. For all of these modules, the elementlabeled 3A acts with trace 0. This establishes the Theorem for the group numbered 33 inTable 1 of [GPRV13].Finally, we have the groups h g, h i numbered 25 and 26, for which g and h are both ofclass 4B. According to [GPRV13], C (4B) is a group of shape ((4 × V f♮R +4B . For both groups (numbers 25 and 26), the– 25 –entralizer of h g, h i has order 16. It follows that h is one of the conjugacy classes named“4B ,” “4B ,” “4B ,” and “4B ” in the appendix of [GPRV13]. But for h = 4B or 4B , thegroup h g, h i contains an element of class 4A, which has an anomaly (they correspond to thegroups numbered 23 and 24 in Table 1 of [GPRV13]). The classes h = 4B and 4B act withtrivial trace on all genuinely projective representations of C (4B).The mock modular forms H g,h from [GPRV13] are integral in the sense that their co-efficients, as functions of h , they are all virtual characters of projective representations ofthe centralizer of g in M . (Indeed, they are mostly zero.) Thus the equivariant versionof the invariant from [GJF19a] vanishes for V f♮ ⊗ Fer(3): that invariant does not obstructConjecture 1.4.It was observed early in the development of Mathieu Moonshine [GHV12, Gan16] that thecharacters that appear (i.e. the coefficients of the q -expansion of H ,h ( τ )) are all restrictionsof virtual characters of projective represenations of Co . However, it is unlikely that thefunctions H ,h , let alone H g,h , have (mock) modular integral extensions to Co . Said anotherway, it is likely that the invariant from [GJF19a] is strong enough to prove: Conjecture 2.7. V f♮ ⊗ Fer(3) is not Co -equivariantly nullhomotopic. TMF • and tmf • The main conjecture of [ST04, ST11] (our Conjecture 1.5) is that the spectrum SQFT • isequivalent to the “universal elliptic cohomology” spectrum TMF • called Topological ModularForms . There is quite a lot of evidence in favour of this conjecture, and versions of it werepredicted as early as [Wit87, Wit88, Seg88]. Notably, Witten explained in the first of thosepapers that the then-recently-discovered “elliptic genus” of Landweber, Stong, and Ochaninearises as the Z -twisted partition function of the N = (1 ,
1) sigma model (with Z -actionthat breaks the the left-moving supersymmetry), and also introduced what is now known asthe Witten genus by using instead an N = (0 ,
1) sigma model.Another piece of evidence supporting Conjecture 1.5 is that the corresponding statementin (0+1) dimensions is understood [Che06, HST10, Mar10, Ulr19]. Indeed, the construc-tion from § N =1 supersymmetric quantum mechanics models in place of (1+1)-dimensional QFTs produces a spectrum SQM • . If the SQM models are not required tosupport a time-reversal symmetry, then the resulting spectrum SQM • is known to modelthe complex K-theory spectrum KU • . If the SQM models are equipped with a time-reversalsymmetry then SQM • models orthogonal K-theory KO • . (The time-reversal symmetry mustsatisfy T = +1, corresponding to Pin − under Wick rotation. The Pin + case T = ( − f isnot compatible with the dynamicalization procedure leading to a spectrum structure.) KO • isthe universal cohomology theory of chromatic height 1, whereas TMF • is the universal coho-mology theory of chromatic height 2.Quantum mechanics has a complete mathematical axiomatization in terms of Hilbertspaces and von Neumann algebras, and the statement “SQM • = K • ” is a mathematical– 26 –heorem. Presuming that a complete mathematical axiomatization of unitary, compact (1+1)-dimensional quantum field theory can be found, Conjecture 1.5 offers an analytic model ofTMF • , for which so far the only known models are homotopy-algebraic. (Progress towardsproving Conjecture 1.5 is available in [Che06, BET18, BET19], which establish versions “overthe Tate curve” and, equivariantly, over C .)There are in fact three closely-related spectra that go under the name “topological mod-ular forms,” distinguished by their capitalizations. The first, TMF • , is the space of “weaklyholomorphic topological modular forms”: it is a homotopical refinement of the ring MF • ofintegral modular forms that are holomorphic for finite values values of τ , but possibly mero-morphic at the cusp τ = i ∞ . (By “integral,” we mean that the q -expansion lives in Z (( q )).Modular forms of weight w are assigned cohomological degree n = − w .) The algebrotopo-logical definition of TMF • is a “derived” version of MF • . Specifically, there is a “derivedstack” M derell which refines the stack M ell of smooth elliptic curves. It carries a “derivedstructure sheaf” O der whose fibre at an elliptic curve E ∈ M derell is the spectrum presenting E -elliptic cohomology. The homotopy sheaf π w O der = L ⊗ w is the line bundle whose sec-tions are weight- w modular forms. (Constructing these derived algebrogeometric objects ishard [GH04, Lur09, Goe10, HM14, Lur18a, Lur18b, Lur19].) Then TMF • is the spectrum ofderived global sections of O der . This is the spectrum that appears in Conjecture 1.5.The second spectrum, Tmf • , is the space of “holomorphic topological modular forms,”analogous to the ring mf • of modular forms that are bounded at τ = i ∞ . Its definitionparallels TMF • and mf • : compactify M derell to a derived stack M derell that allows elliptic curveswith nodal singularities, extend the derived structure sheaf, and take derived global sections.Because of the derived nature of these constructions, the homotopy groups of both TMF • andTmf • include information about the cohomology of the line bundles L ⊗ w . In particular, eventhough there are no holomorphic modular forms of negative weight (positive cohomologicaldegree), the line bundles L ⊗ w for negative even w do have cohomology over M ell , leading tonontrivial classes in π −• Tmf = Tmf • (pt) for • > • is not yet clear: there probably is an analogue ofConjecture 1.5 for Tmf • , but a satisfactory one has not yet been proposed. One approach issuggested in [Mar10], but this author doubts that that method can be made physically sensiblein (1+1) dimensions. A more direct approach would involve a spectral constraint on theoperator L in the Ramond sector which is strong enough to assure that the adjusted partitionfunction Z ′ RR ( τ ) = Z RR ( τ ) η c R − c L ) ( τ ) converges as τ → i ∞ . For example, it should rule outthe holomorphic SCFT V f♮ , since that SCFT represents the class { − } ∈ TMF (pt),which is not in the image of Tmf • → TMF • . On the other hand, Tmf (pt) ∼ = Z is generatedby a class that deserves the name { − ν } , which is is in the kernel of Tmf • → TMF • . Theauthor believes that the generator { − ν } should be represented by the N =(1 ,
1) SCFT V f♮ ⊗ Fer(3), but it is not clear how to tune the constraint so as to allow this.Third, the spectrum tmf • is defined to be the connective cover of Tmf • : as an Ω-spectrum,tmf n = Tmf n = Ω − n Tmf for n ≤
0, but tmf n = B n Tmf for n >
0. Said another way, tmf • is built by keeping only the 0-space Tmf of the Tmf • -spectrum, which is automatically an– 27 –nfinite loop space, and then interpreting infinite loop spaces as a special class of spectra. Inhomotopy, we have tmf • (pt) = ( Tmf • (pt) , • ≤ , , • > . More generally, if X is a space, then tmf • ( X ) = Tmf • ( X ) if • ≤
0, but not for • >
0. We areinterested in a particular M -equivariant TMF-class [ V f♮ ] ν of cohomological degree • = − V f♮ ] = { } ∈ TMF − (pt) is in the image of Tmf − (pt), and webelieve that this holds M -equivariantly as well. Nonequivariantly { } ν = 0 ∈ tmf − (pt),and hence in TMF − (pt). Together with Conjecture 1.5, this implies that [ V f♮ ] ν = 0 ∈ SQFT − (pt).It is not expected that tmf • itself will admit a natural physical description. The reasonis that any physical description in terms of spaces of SQFTs will naturally lead to a gen-uinely equivariant enhancement (by working with SQFTs with a given flavour symmetry),but tmf • is not expected to admit a genuinely equivariant enhancement. A better calcula-tion than we will attempt in Section 4 would be to work out the equivariant cohomologyTmf − α ( B M ), and perhaps show, as suggested by Theorem 1.8, that it vanishes away fromthe prime p = 2. (Perhaps it even vanishes at p = 2.) But there is no Atiyah–Hirzebruchspectral sequence for computing equivariant cohomology groups like Tmf − α ( B M ), and sowe will not attempt such a calculation. Rather, in Section 4 we will attempt Tmf − α ( B M ),approximating the classifying stack B G by its classifying space BG , and the above remarksidentify Tmf − α ( B M ) ∼ = tmf − α ( B M ), since B M is just a space, not a stack.Physically, the difference between B G and BG is the following. As explained in § • ω ( B G ) is represented by a compact SQFT with G flavour symmetry (andanomaly ω ). A class in SQFT • ω ( BG ) is instead represented by a family of SQFTs over thespace BG . There is a map SQFT • ω ( B G ) → SQFT • ω ( BG ) which uses the G -action on anelement F ∈
SQFT • ω ( B G ) to prescribe the monodromies of a family over BG which is locallyconstant with value F . This map is definitely not an isomorphism of spaces, because its imageconsists of families which are locally constant, whereas a typical family over BG may vary alot. But we don’t need it to be, since we care only about homotopy classes. With some work,a family of SQFTs over BG can be “integrated” to a G -equivariant SQFT, and so one mightexpect that SQFT • ω ( B G ) → SQFT • ω ( BG ) is a homotopy equivalence. The problem is that BG is infinite-dimensional, and so this “integral” will usually fail to produce a compact SQFT. (Atopological space is “finite-dimensional” if it is homotopy equivalent to a finite cell complex.Except for spaces homotopy equivalent to finite sets, no space is both finite-dimensionaland finite in homotopy.) Indeed, there are sequences of G -equivariant SQFTs which divergein SQFT • ω ( B G ) because their limits are noncompact, but which converge in SQFT • ω ( BG )because this noncompactness can be concentrated “near infinity” in BG : as you go out alonga cell decomposition of BG , the family stays compact but becomes larger and larger. As such,one expects SQFT • ω ( BG ) to be a “completion” of SQFT • ω ( B G ), analogous to the famous resultfrom [AS69] describing KU • ( BG ) as a completion of KU • ( B G ). A completion statement for– 28 –MF is known for finite abelian groups G [Lur19]. But we warn that direct computationsin [GM20] show that for G = U(1), the map TMF • ( B U(1)) → TMF • ( B U(1)) is far from acompletion. (There is a more sophisticated “completion” statement that holds for U(1) atthe level of sheaves over M ell . The failure of TMF • ( B U(1)) → TMF • ( B U(1)) can then betraced to the non-affineness of the stack of elliptic curves with U(1)-bundle.)In addition to the rings MF • and mf • of weakly holomorphic and holomorphic modularforms, number theorists care also about the ideal cf • ⊂ mf • of cusp forms , which are theholomorphic modular forms which vanish at τ = i ∞ . Like modular forms, cusp forms admita topological enhancement to a spectrum Tcf • of topological cusp forms . Summarizing a fairamount of hard work, the idea is to promote the restriction map mf • → O (cusp) = Z to amap Tmf • → KO • , which was done in [HL16]. Then Tcf • is defined as the homotopy fibre ofTmf • → KO • . The topology literature seems to contain very little investigation of Tcf • , and,just like for Tmf • , the physical significance it not yet clear. We will mention one interestingfact about Tcf • , which makes its behaviour different from the classical case of cf • . Namely, cf • is the principal ideal inside mf • generated by ∆, and as such it represents (up to suspension,aka degree-shift) the trivial class in the Picard group of mf • . But ∆ does not lift to an elementin Tmf • (pt), and Tcf • is not isomorphic to a suspension of Tmf • . Rather, in unpublishedwork L. Meier has identified the class of Tcf • with the exotic 24-torsion element in the Picardgroup of Tmf • called Γ( J ) in [MS16]. (That paper shows that Γ( J ) and suspension togethergenerate Pic(Tmf • ) ∼ = Z × Z . Other exotic elements are studied in [HM17, MO20].)The genus-zero property in Monstrous Moonshine is reformulated in [CDH14b] in termsof an optimal growth condition which provides the “moonshine” part of Umbral (and inparticular Mathieu) Moonshine. The condition (for M ) is simply that the mock modularforms H g,h ( τ ) grow no worse than q − / as τ → i ∞ , and are bounded near other cusps.Recall from § H g,h ( τ ) is the holomorphic part of η ( τ ) Z RR ( F ) g,h ( τ, ¯ τ ),for some SQFT F with with cylindrical ends ∂ F = V f♮ ⊗ Fer(3) of cohomological degree n = −
28, whereas the homotopy theory convention for Witten genera is to work with theadjusted partition function Z ′ RR ( τ, ¯ τ ) = Z RR ( τ, ¯ τ ) η ( τ ) − n . The optimal growth condition thenbecomes the statement that the adjusted function H g,h ( τ ) η ( τ ) = O ( q ), i.e. it is a “mockcusp form.” This leads us to propose the following answer to Question 1.3: Conjecture 2.8.
The M -equivariant class [ V f♮ ] ν , refining the cusp form { } ν = 0 ∈ Tmf − (pt) , is M -equivariantly nullhomotopic in the spectrum Tcf • of topological cuspforms. M In order to run the Atiyah–Hirzebruch spectral sequence for tmf • ω ( B M )[ ], we will needgood control over the ordinary group cohomology rings H • (M ; Z [ ]) and H • (M ; F ). Wefind it necessary to invert the prime 2 simply because the 2-local computations are too hard.These rings were computed in [Tho94, Gre96]. We will review and extend that analysis: ourgoal is to have explicit control over the action of the first Steenrod cube P on H • (M ; F ).– 29 –s with any finite group, the computation of H • (M ; Z [ ]) factors prime-by-prime, andat the prime p , the computation is controlled by the Sylow p -subgroup. The primes p ≥ p -subgroup is cyclic, from which it already follows thatH • (M ; Z [ ]) is supported in even degrees. Since tmf • (pt)[ ] is supported in even degreesand has no torsion, we learn immediately that tmf • ω ( B M )[ ], which is independent of thetwisting ω since ω is 12-torsion, is also supported only in even degrees. A stronger statementis proved in [Tho94]: tmf • ω ( B M )[ ] is generated by elliptic Chern classes. (The cohomologytheory tmf • [ ] is called Eℓℓ • in [Tho94].) H • (M ; F )Thus the interesting computation is at the prime 3, where the Sylow subgroup is nonabelian,being isomorphic to the extraspecial group S = 3 of order 27 and exponent 3. The ringsH • (M ; Z (3) ) and H • (M ; F ) are computed in [Gre96]. We will report the main result, butchange some letters: Theorem 3.1 ([Gre96]) . The graded commutative ring H • (M ; Z (3) ) has a presentation withfour generators, of cohomological degree and additive order as follows:Name Degree Additive order r s
12 9 t
16 3 u
11 3
The only relations are u = 0 (which follows from the Koszul sign rules) and rt = 0 .We will use the same names for classes in H • (M ; Z (3) ) as for their mod-3 reductions in-side H • (M ; F ) . The Ext term in the universal coefficient theorem implies that each genera-tor “ x ” of H • (M ; Z (3) ) of cohomological degree n also determines a generator of H • (M ; F ) of degree ( n − , which will be denoted “ X .” They are related by the mod-3 Bockstein (cid:3) forthe extension Z → Z → Z : (cid:3) R = r, (cid:3) S = 3 s = 0 , (cid:3) T = t, (cid:3) U = u. Note that (cid:3) S = 0 in H • (M ; F ) because s ∈ H • (M ; Z (3) ) has additive order and not .The following is a complete list of relations for H • (M ; F ) as an F -algebra: Ru = rU, T S = T u = tU, tS = tu,u = R = T = U = rt = rT = uU = tR = RU = RT = U T = 0 ,rS = uS = RS = U S = S = 0 . A standard lemma (see e.g. Section XII.8 of [CE56]) implies that the restriction mapsH • (M ; Z (3) ) → H • ( S ; Z (3) ) and H • (M ; F ) → H • ( S ; F ) are injections onto direct sum-mands, where S = 3 ⊂ M is the Sylow 3-subgroup. The rings H • ( S ; Z (3) ) and H • ( S ; F )– 30 –re computed in [Lew68, Lea91, Lea92]. In particular, the third of those articles confirmsthe following result due to [MT95] (the order of final publication did not match the order inwhich the preprints were originally circulated): Proposition 3.2 ([MT95]) . Let A , A , A , A denote the four maximal abelian subgroupsof S = 3 , each isomorphic to Z × Z . The total restriction map H • ( S ; F ) → Y i H • ( A i ; F ) is an injection. Since H • (M ; F ) ⊂ H • ( S ; F ), it follows in particular that we can compute insideH • (M ; F ) by computing restrictions to subgroups isomorphic to Z × Z . Let us de-scribe these subgroups. First, M has two conjugacy classes of elements of order 3. Class 3Aconsists of those elements in M that act in the degree-24 permutation representation withcycle structure 1 ; class 3B acts with cycle structure 3 . The central Z ⊂ S consists of 3A-elements. There are also two conjugacy classes of subgroups Z × Z ⊂ M . Both subgroupsare maximal-abelian. One of them, which we will call simply Z × Z , consists entirely of3A-elements. The other is Z × Z . In addition to the identity element, it contains two3A-elements (forming a Z -subgroup), and the other six elements are of class 3B. The “Weylgroups” W ( A ) = N ( A ) /A of these maximal abelian subgroups of M are as large as possiblegiven the conjugacy classes of elements: W ( Z × Z ) = N ( Z × Z ) Z × Z ∼ = GL ( F ) , W ( Z × Z ) = N ( Z × Z ) Z × Z ∼ = D . By D we mean the dihedral group of order 12, isomorphic to the upper Borel (cid:0) ∗ ∗ ∗ (cid:1) ⊂ GL ( F ). These and other claims about M are easily checked in the computer algebraprogram GAP [GAP].For any finite group and any abelian subgroup A ⊂ G and for any ring R , the restrictionmap H • ( G ; R ) → H • ( A ; R ) lands within the Weyl-invariant subring H ( W ( A ); H • ( A ; R )). Wetherefore conclude:H • (M ; F ) ⊂ H (GL ( F ); H • ( Z × Z ; F )) × H ( D ; H • ( Z × Z ; F )) . The next step is to understand the right-hand side. Let us choose “coordinates” on Z × Z and Z × Z , writing Y , resp. Z , for the homomorphisms onto the “fibre” Z , resp. andonto the “base” Z or Z . After identifying Z = F , these coordinates give classes inH ( Z × Z ; F ). The full algebra H • ( Z × Z ; F ) is then a graded polynomial algebraH • ( Z × Z ; F ) = F [ Y, Z, y, z ]where y = (cid:3) Y and z = (cid:3) Z are in degree 2, and the only relations are the ones imposed bythe Koszul sign rules: Y = Z = Y Z + ZY = 0. Note also that mod-3 reduction identifies– 31 – • ( Z × Z ; Z ) with ker (cid:3) ⊂ H • ( Z × Z ; F ) (except in degree 0), and that this ring isgenerated by y and z and the degree-3 element w = yZ − Y z = (cid:3) ( Y Z ). In particular, thesubring of H • ( Z × Z ; F ) consisting of even-degree elements with integral lifts is F [ y, z ].The D -action on F [ Y, Z, y, z ] is generated by the following three automorphisms:(
Y, Z, y, z ) ( − Y, z, − y, z ) , ( Y, Z, y, z ) ( Y, − z, y, − z ) , ( Y, Z, y, z ) ( Y + Z, Z, y + z, z ) . To generate the full GL ( F )-action, it suffices to include also the automorphism( Y, Z, y, z ) ( Z, Y, z, y ) . Lemma 3.3.
The subring of F [ y, z ] invariant under ( y, z ) ( y + z, z ) is F [ z, c ] where c = y ( y + z )( y − z ) = y − yz is of cohomological degree . The subring of F [ y, z, w ] is F [ z, w, c ] . Note that F [ y, z, w ] = H • ( Z × Z ; Z ) except in degree 0. Proof.
Certainly z , c , and w are invariant. Suppose p ( y, z ) = p z n + p yz n − + · · · + p n y n is an invariant homogeneous polynomial of polynomial degree n (cohomological degree 2 n ).Then the largest i with p i = 0 must be divisible by 3. Indeed, otherwise under y y + z ,the coefficient on y i − z n − i +1 will change by ip i = 0. So the space of invariant degree- n polynomials is at most (1 + ⌊ n ⌋ )-dimensional. But this is the dimension of the space ofdegree- n polynomials in F [ z, c ]. The second claim follows from the first together with thefact that F [ y, z, w ] = F [ y, z ] ⊕ w F [ y, z ], since w = 0 by the Koszul sign rules.We note also that z and c are not invariant under y ↔ z , but that z + c = y + y z + y z + z and z c = y z + y z + y z are.We can now work out the restrictions to Z × Z and Z × Z of the integralgenerators r, s, t, u . First, r has cohomological degree 4. There are no GL ( F )-invariantdegree-4 classes, and so r | Z × Z = 0. Then Proposition 3.2 implies that r | Z × Z = 0,and so is proportional to z , as that is the only degree-4 class invariant under y y + z .Changing the sign of r if necessary, we learn: r | Z × Z = 0 , r | Z × Z = z . The extension class R is also immediate, since it is an invariant degree-3 class satisfying (cid:3) R = r . R | Z × Z = 0 , R | Z × Z = Zz.
Note that w is a degree-3 class invariant under z y + z and in the kernel of (cid:3) , but it picksup a sign under some of the D transformations, and so cannot appear here.The next classes worth considering are the generators T, t , since rT = rt = 0. Thusboth of these classes restrict trivially to Z × Z , since z is not a zero-divisor. Thus– 32 – and t restrict nontrivially to Z × Z by Proposition 3.2. Since t is an integral classof cohomological degree 16, its restriction to Z × Z must a polynomial in y and z ofpolynomial degree 8 invariant under all of GL ( F ). By using Lemma 3.3, it is easy to seethat the only such polynomial is c z = y z + y z + y z . Changing the sign of t as necessary,we have: t | Z × Z = y z + y z + y z , t | Z × Z = 0 . As for T | Z × Z , we need a GL ( F )-invariant degree-15 class satisfying (cid:3) T = t . One couldworry that there could be multiple choices. By Lemma 3.3, the only degree-15 class in thekernel of (cid:3) which is GL ( F )-invariant up to a sign is w ( c + z ), but this class changes signunder some of the involutions in GL ( F ). So T is uniquely determined by invariance and (cid:3) T = t . We have: T | Z × Z = Y ( yz − y z ) + Z ( y z − y z ) , T | Z × Z = 0 . We next consider the degree-12 generator s . It must have nontrivial restrictions to both Z × Z and Z × Z , since neither rs nor st vanishes. The restriction to Z × Z is a polynomial in y and z of degree 6, invariant under all of GL ( F ), and so must be c + z = y + y z + y z + z , after possibly changing the sign of s . The restriction to Z × Z is some linear combination of c and z . But note that, upon further restrictionto the fibre Z , the two restrictions must agree. On the other hand, we have the freedomto change s s ± r without changing the presentation of H • (M ; Z ) in Theorem 3.1. Wewill choose the modification so that s | Z = 0. Thus we may assume: s | Z × Z = y + y z + y z + z , s | Z × Z = y + y z + y z . We will consider the two degree-11 generators S and u at the same time. It will beconvenient to replace u with v = u − S . Note that (cid:3) S = (cid:3) v = 0, and so the restrictions ofboth must land in F [ y, z, w ], where as above w = (cid:3) ( Y Z ) = yZ − Y z . Theorem 3.1 provides rS = 0 and tv = 0. Therefore S | Z × Z = 0 and v | Z × Z = 0. By Lemma 3.3, the otherrestrictions are of the form wp ( z, c ), where the polynomial p ( z, c ) is of homogeneous degree4 in y and z . After tracking the behaviour under z
7→ − z and y
7→ − y , the only option is p ( z, c ) = zc = y z − yz , and so: S | Z × Z = Y ( yz − y z ) + Z ( y z − y z ) , S | Z × Z = 0 . The signs of S and u are not independent if we want to preserve the relation tS = tu fromTheorem 3.1, and the author was not able to identify the correct sign for the restriction of v = u − S . We do have: v | Z × Z = 0 , v | Z × Z = ± (cid:0) Y ( yz − y z ) + Z ( y z − y z ) (cid:1) . Last, we have the degree-10 generator U , which must satisfy (cid:3) U = u = v + S . Since U is of even degree and not in the kernel of (cid:3) , its restriction must be of the form p ( y, z ) Y Z , for– 33 –ome polynomial p of polynomial degree 4. Invariance then gives the answer: U | Z × Z = Y Z ( y z − yz ) , U | Z × Z = ± Y Z ( y z − yz ) . The sign is the same as above, set by (cid:3) U = v + S .In summary, we have shown: Proposition 3.4.
With notation as in Theorem 3.1, and writing v = u − S , the generatorsof H • (M ; F ) have the following restrictions to the maximal abelian subgroups Z × Z and Z × Z :Generator Z × Z Z × Z R Zzr z U Y Z ( y z − yz ) ± Y Z ( y z − yz ) v ± (cid:0) Y ( yz − y z ) + Z ( y z − y z ) (cid:1) S Y ( yz − y z ) + Z ( y z − y z ) 0 s y + y z + y z + z y + y z + y z T Y ( yz − y z ) + Z ( y z − y z ) 0 t y z + y z + y z The sign ± is the same throughout. P + ǫr acting on H • (M ; F )Our next goal is to understand in detail the action of the first Steenrod cube P on H • (M ; F ).This is a degree-4 universal cohomology operation defined on all F -cohomology rings. Oftenthe notation P k is used to denote the k th Steenrod cube. We will avoid that notation becauseit does not mean the k th power of P , writing instead P ( k ) for the k th Steenrod cube, and P ◦ k for the k -fold composition.Among the defining properties of P are that it is a derivation, that it vanishes in cohomo-logical degree 1, and that it is the cube in cohomological degree 2. For example, on Z × Z ,it satisfies P ( Y ) = P ( Z ) = 0 , P ( y ) = y , P ( z ) = z . Propositions 3.2 and 3.4 then provide enough information to work out the action of P on thegenerators of H • (M ; F ): g P ( g ) | Z × Z P ( g ) | Z × Z R Zz r − z U v ± (cid:0) Y ( − y z + yz ) + Z ( y z − z z ) (cid:1) S Y ( − y z + yz ) + Z ( y z − z z ) 0 s y z + y z + y z y z + y z + y z T t Proposition 3.5.
With notation as in Proposition 3.4, the first Steenrod cube P acts as: P ( R ) = Rr, P ( U ) = 0 , P ( S ) = T, P ( T ) = 0 , P ( r ) = − r , P ( v ) = vr ± Rs, P ( s ) = sr + t, P ( t ) = 0 . Recalling that u = v + S , we note that P ( u ) = vr ± Rs + T = ( u − S ) r ± Rs + T = ur + T ± Rs, since Sr = 0. Proof.
The most interesting case is P ( v ); the other cases are left to the reader. Recall that v , and hence P ( v ), vanishes when restricted to Z × Z . Their other restrictions are v | Z × Z = ± (cid:0) Y ( yz − y z ) + Z ( y z − y z ) (cid:1) , P ( v ) | Z × Z = ± (cid:0) Y ( yz − y z ) + Z ( y z − y z ) (cid:1) , where for example we use P ( yz − y z ) = y z +4 yz − y z − y z = yz − y z . Comparing P ( v ) | Z × Z with vr | Z × Z = vz , we find a discrepency:( P ( v ) − vr ) | Z × Z = ± Z (cid:0) ( y z − y z ) − z ( y z − y z ) (cid:1) = ± Z ( y z + z z + y z ) . Factoring out Zz = R | Z × Z gives ± s | Z × Z .Fix some ǫ ∈ F . The specific understanding that we seek is the following: we willcalculate the “cohomology” of the D = P + ǫr . This operator is not a differential in the usualsense: D ◦ D 6 = 0. Rather, it is a 3 -differential in the sense that its cube is zero:
Lemma 3.6.
Let X be a space, and choose x ∈ H ( X ; F ) . Then the operator D = P + x on H • ( X ; F ) satisfies D ◦ = 0 .Proof. Expanding ( P + x ) ◦ , we have: D ◦ = P ◦ + P ◦ ◦ x + P ◦ x ◦ P + x ◦ P ◦ + P ◦ x + x ◦ P ◦ x + x ◦ P + x . The Adem relations provide P ◦ = 0 and P ◦ = −P (2) , where P (2) denotes the secondSteenrod power. The statement “ P is a derivation” may be summarized as: P ◦ x = x P + P ( x ) . Here and in the sequel, by “ x P ” we mean of course x ◦ P , i.e. apply P and then multiplyby x , whereas “ P ( x )” means multiplication by P ( x ). Thus we find: D ◦ = 0 + ( P ◦ ( x ) + 2 P ( x ) P + x P ◦ ) + ( P ( x ) P + x P ◦ ) + x P ◦ + (2 x P ( x ) + x P ) + ( x P ( x ) + x P ) + x P + x . – 35 –ince we are in characteristic 3, everything cancels to D ◦ = P ◦ ( x ) + x . But P ◦ ( x ) = −P (2) ( x ) = − x since x has degree 4.Appendix A contains an extended discussion of 3- and higher differentials. For ourpurposes, it suffices to record the following. A usual differential on a K -vector space makes itinto a module for the algebra K [ D ] / ( D ); the usual cohomology is the result of decomposingthe module as a direct sum of indecomposables, and discarding the free summands. We haveinstead an action of F [ D ] / ( D ) on a vector space V , and its cohomology H ∗ ( V, D ) is the resultof decomposing V as a sum of indecomposable F [ D ] / ( D )-modules and discarding the freesummands. Whereas in the usual case the non-free indecomposable module is unique, over F [ D ] / ( D ) there are two isomorphism classes of non-free indecomposable modules, of F -dimensions 1 and 2. So H ∗ ( V, D ) is not just a vector space, but rather picks up a Z -grading inaddition to any cohomological grading on V . One of the punchlines of Appendix A is that this Z -grading is really a fermionic grading: H ∗ converts the two-dimensional indecomposable F [ D ] / ( D )-module into an odd line F | . The other punchline is that H ∗ is functorial andsymmetric monoidal.To illustrate this, and as a warm-up to the M -case that we care about, let us studythe cohomology of P + ǫa on H • ( Z ; F ) = F [ A, a ], where A has degree 1 and a = (cid:3) A hasdegree 2. Since P ( A ) = 0, multiplication by A is an isomorphism of 3-complexes betweenthe even-degree cohomology F [ a ] and the odd-degree cohomology A F [ a ]. So it suffices tounderstand the cohomology of D = a ∂∂a + ǫa on F [ a ], where ǫ ∈ F . On monomials, wehave D ( a i ) = ( i + ǫ ) a i +2 . For ǫ = 0, this complex looks like: a a a a a a a a a a a a · · · · · · +1 − − − − The non-free summands—the cohomology—are { a } and { a → a } . For ǫ = 1, we haveinstead: a a a a a a a a a a a a · · · · · · +1 − − − − The cohomology is just { a → a } . Finally, for ǫ = −
1, we see: a a a a a a a a a a a a · · · · · · − − − − – 36 –he cohomology is { a } . In all cases, the “tails” are exact: the only cohomology is near the“head” a .We now turn to the case we care about, which is the cohomology of D = P + ǫr onH • (M ; F ). Note that D preserves the cohomological degree mod 4. It also preserves an auxiliary degree defined by assigning auxiliary degree 0 to R and r and auxiliary degree +1to U, v, S, s, T, and t .The following monomials are a basis of the submodule of H • (M ; F ) in cohomologicaldegree 0 (mod 4) and auxiliary degree j : s j , s j r i , < i, s j − i t i , < i ≤ j. The action of D = P + ǫr is: s j s j b s j b s j b s j b · · · s j − t s j − t s j − t s j − t · · · t jj + ǫ j + ǫ − j + ǫ − j + ǫ − j + ǫ − j j − j − j − j − As in the warm-up example, the tails are exact. The cohomology near the head depends onthe values of both j and ǫ mod 3. Going through all nine cases, we find D -cohomology in thefollowing cohomological degrees: j (mod 3)0 1 − ǫ { j } { j → j + 4 } ⊕ { j + 4 } { j + 4 → j + 8 } { j → j + 4 } { j + 4 } ∅− ∅ { j → j + 4 } { j + 4 } For example, when ǫ = 0 and j = 1 (mod 3), cutting off the manifestly exact tails returns s j s j bs j − t The submodule { s j → s j b + s j − t } splits off as a direct summand, and the other summand isone-dimensional, generated by the cohomology class [ s j b ], which is cohomologous to − [ s j − t ].Multiplication by U is an isomorphism between the subspace of H • (M ; F ) of coho-mological degree 0 (mod 4) and the subspace of cohomological degree 2 (mod 4). Since P ( U ) = 0, this isomorphism is in fact an isomorphism of F [ D ] / ( D )-modules. Thus we– 37 –mmediately learn that the D -cohomology in cohomological degree 2 (mod 4) consists of: j (mod 3)0 1 − ǫ { j + 10 } { j + 10 → j + 14 } ⊕ { j + 14 } { j + 14 → j + 18 } { j + 10 → j + 14 } { j + 14 } ∅− ∅ { j + 10 → j + 14 } { j + 14 } Theorem 3.1 implies that H • (M ; F ) vanishes in cohomological degree 1 (mod 4). Theonly remaining case is cohomological degree 3 (mod 4). Monomials of cohomological degree3 (mod 4) are divisible by exactly one of R, v, S, T . The first two vanish when restricted to Z × Z , and the second two vanish when restricted to Z × Z . This allows us to splitthe • = 1 (mod 4) subcomplex of H • (M ; F ) into two summands: the kernel of restrictionto Z × Z and the kernel of restriction to Z × Z . (These kernels are disjoint byProposition 3.2.)The first summand consists of those terms divisible by R or v . In auxiliary degree j + 1 ≥
1, it looks like (the totalization of): s j v s j vr s j vr s j vr · · · s j +1 R s j +1 Rr s j +1 Rr s j +1 Rr · · · ± ± ± ± j + ǫ +1 j + ǫ j + ǫ − j + ǫ − j + ǫ +2 j + ǫ +1 j + ǫ j + ǫ − The sign is the same in all vertical arrows. The reader is invited to check that the cohomologyof this complex: • Vanishes when j + ǫ = − • Has one-dimensional cohomology in degree { j + 15 } when j + ǫ = 1 (mod 3). • Has odd-one-dimensional cohomology in degree { j + 11 → j + 15 } when j + ǫ = 0(mod 3).In auxiliary degree j + 1 = 0, we just see R ǫ +1 −→ Rr ǫ −→ Rr ǫ − −→ · · · which instead has cohomology in degrees: • { → } if ǫ = 0. Since j + 1 = 0, this replaces the j + ǫ = − • ∅ if ǫ = 1. Since j + 1 = 0, this replaces the j + ǫ = 0 entry, which would have been thenonsensical cohomological degrees {− → } in any case. • { } if ǫ = −
1. Since j + 1 = 0, this agrees with the j + ǫ = 1 entry.– 38 –inally, we have the summand consisting of those monomials divisible by S or T . Notethat P ( S ) = T and P ( T ) = 0, and Sr = T r = 0. So, in auxiliary degree j + 1, we may factorthe total complex as a tensor product: { s j j −→ s j − t j − −→ · · · −→ st j − −→ t j } ⊗ { S +1 −→ T } As explained in Appendix A, the functor H ∗ that takes cohomology of 3-complexes is symmet-ric monoidal. H ∗ ( { S +1 −→ T } ) = { S → T } is supported in cohomological degrees { → } .The first tensorand { s j → · · · → t j } is exact except near the head, where its cohomologydepends on j : • If j = 2 (mod 3), then { s j → · · · → t j } is exact, and so { s j → · · · → t j } ⊗ { S → T } isexact. • If j = 1 (mod 3), then H ∗ ( { s j → · · · → t j } ) = { s j → s j − t } is supported in degrees { j → j + 4 } . So H ∗ ( { s j → · · · → t j } ⊗ { S → T } ) = { j + 15 } in cohomologicaldegree. • If j = 0 (mod 3), then H ∗ ( { s j → · · · → t j } ) = { j } and H ∗ ( { s j → · · · → t j } ⊗ { S → T } ) = { j + 11 → j + 15 } .These cohomologies are independent of ǫ .Summarizing the above computations, we have: Proposition 3.7.
Pick ǫ ∈ F , and consider the -differential D = P + ǫr acting on V =H • (M ; F ) . Its cohomology H ∗ ( V, D ) is supported in the following cohomological degreesmodulo : • ǫ = 0 : { } , { → } , { } , { → } , { → } , { → } , { } , { → } , { } , { } , { } , { → } , { → } . • ǫ = 1 : { → } , { → } , { → } , { } , { } , { } , { } , { → } . • ǫ = − { } , { } , { → } , { → } , { → } , { → } , { } , { } , { → } .The repeated terms in the ǫ = 0 line indicate that the cohomology contains two summands inthose degrees. When ǫ = 0 , there is also one nonperiodic cohomology class in degree { → } . Except for that one non-periodic class, all other cohomology is periodic with periodicityelement s .On the subalgebra F [ R, r ] ⊂ H • (M ; F ) , D = P + ǫr has the following cohomology: • ǫ = 0 : { r } in degree and { R → Rr } in degree { → } . • ǫ = 1 : { r → r } in degree { → } . • ǫ = − { R } in degree { } . – 39 – The Atiyah–Hirzebruch spectral sequence for tmf • ω ( B M ) With the ordinary 3-local cohomology of B M understood, we are now ready to analyze the3-local structure of tmf • ω ( B M ). We will do so by analyzing its Atiyah–Hirzebruch spectralsequence. Any space X and spectrum E • determine an Atiyah–Hirzebruch spectral sequence . (The namerefers to [AH61] but the proof there consists essentially of a reference to [CE56], and [Ada74]attributes the construction to unpublished work of Whitehead. Further important early devel-opments are in [Mau63].) Spectral sequences are an algebrotopolical version of perturbationtheory. As with any perturbative calculation, the goal is to approximate some nonperturba-tive object. In the AHSS case, that nonperturbative object is E • ( X ), the E • -cohomology ofthe space X .The rough idea of the AHSS is the following. Imagine that X is a CW complex andthat you have fixed some cochain model for E • . Then a cochain for E n ( X ) assigns, to each m -cell in X , an element of E n − m . The cohomology E n ( X ) is the cohomology for some totaldifferential on this set of cochains. The AHSS perturbatively approximates that total differ-ential. The 0th approximation entirely ignores the topology of X : for a cell x ∈ X and acochain x e ( x ), the 0th approximation is ( d e )( x ) = d E ( e ( x )), where d E is the differentialin (the cochain model for) E • . The 1st approximation includes some attaching informationfor the cells in X . By definition, the E k -page of a spectral sequence is the cohomology of the( k − m of a cell in X and the E -degree n of a cochain. In the AHSS case, the E page is E m,n = H m ( X ; E n (pt)) . As one “turns the page,” one includes higher-order corrections, which take into account howthe homotopy groups E • (pt) are connected. The result is an infinite sequence of finer andfiner approximations to E • ( X ).Any time one works perturbatively, one must worry about two related problems: • Does the perturbative expansion converge at all? • Does the perturbative expansion converge to the object one wishes to compute?In particular, perhaps there are “nonperturbative effects” not seen in the perturbative ex-pansion, so that it does not in fact calculate the desired result.In the case of AHSSs, this second problem is present as soon as one tries to extendfrom spaces to stacks. Indeed, suppose X is a stack with classifying space X . Then thereis an AHSS which tries to approximate E • ( X ), with E page H • ( X ; E • (pt)). But ordinarycohomology does not distinguish stacks from spaces: H • ( X ; E • (pt)) = H • ( X ; E • (pt)). The– 40 –igher differentials also do not distinguish X from X . As a result, this AHSS will at bestconverge to E • ( X ), which in general is not isomorphic to E • ( X ) (compare § E • (pt) is bounded below, or when H • ( X ; Z ) is bounded above. More general convergenceresults are available in [Boa99]. In particular, Theorem 12.4 of that paper says that theAHSS for E • ( X ) does indeed converges “conditionally” to E • ( X ), for any spectrum E andspace X . (The convergence is in the “colimit” topology. We will not here discuss the differenttopologies in which the convergence might hold.) Theorem 7.1 of that paper gives conditionsunder which this “conditional” convergence is in fact “strong,” which is what one wants forapplications. In particular, as explained in the remark following Theorem 7.1 of that paper,for a conditionally convergent spectral sequence to converge strongly, it suffices if each entry E m,n supports only finitely many nonzero differentials, i.e. if there are only finitely many k for which d k : E m,nk → E m + k,n − k +1 k is nonzero. In particular, we find: Lemma 4.1.
Suppose E is a spectrum all of whose homotopy groups E • (pt) are finitelygenerated as abelian groups, and suppose that X = BG is the classifying space of a finitegroup. Then the AHSS H • ( BG ; E • (pt)) ⇒ E • ( BG ) converges strongly.Proof. Since G is finite and E n (pt) is finitely generated, E m,n = H m ( BG ; E n (pt)) is finite if m >
0. It therefore can support only finitely many differentials.And, other than d and d , the m = 0 column supports no differentials at all! Indeed, thed and higher differentials in an AHSS are always stable cohomology operations, and stablecohomology operations always vanish in degree m = 0.The proof of Lemma 4.1 does not automatically apply for cohomology with twisted coef-ficients, because the differentials in the twisted case can involve multiplication and higherMassey products with the twisting parameter. (The m > α ∈ H ( X ; Z ), the d k differentials in the AHSSH • ( X ; KU(pt)) ⇒ KU • α (pt) vanish, and the d k +1 differential is a stable operation plus a k -fold Massey product with α . But Massey products (other than the ordinary product) van-ish in degree m = 0, and so again the m = 0 column supports only finitely many (namely,one) differential, and the AHSS converges strongly.The overall message of [Boa99] is that one should run spectral sequences without worryingtoo much about convergence, and then check convergence at the end. This is because, fora conditionally convergent spectral sequence, strong convergence is simply a property of thesequence itself. Following this advice, we will compute the first few differentials in the AHSSfor tmf • ω ( B M ). This will be enough for us to confirm in Corollary 4.8 that the AHSSconverges. tmf • (pt)[ ]The first step towards constructing the AHSS H • (M ; tmf • (pt)[ ]) ⇒ tmf • ω ( B M )[ ] is tounderstand the coefficient ring tmf • (pt)[ ]. An excellent reference for looking up information– 41 –bout this ring is the chapter [Hen14] of [DFHH14], and [Mat12] provides a nice survey ofhow the computations are done.Write c w for the weight- w Eisenstein series, normalized so that c w ( q ) = 1 + O ( q ). Recallthat the ring mf of integral SL(2 , Z )-modular forms, which are “holomorphic” in the sense ofbeing bounded at the cusp τ = i ∞ , ismf = Z [ c , c , ∆] / ( c − c − . In particular, after inverting 6, we have mf[ ] = Z [ ][ c , c ]. As in § • , with the modular forms of weight w in cohomological degree − w . (Our insistence of working with cohomological gradings means that mf • is supportedin nonpositive degrees.) Justifying the names, there is a ring map tmf • (pt) → mf • . It is anisomorphism away from 6: tmf • (pt)[ ] ∼ → mf • [ ] = Z [ ][ c , c ] . It follows that the map tmf • (pt) → mf • has kernel exactly the torsion in tmf • (pt). It istraditional to name non-torsion classes in tmf • (pt) by their images in mf • .(So far as the author knows, there is no interesting spectrum E • with homotopy groups E • (pt) = mf • : the only such spectrum is a product of Eilenberg–Mac Lane spaces, andrepresents H • ( − ; mf • ). The map on coefficients tmf • (pt) → mf • does not lift to a spectrummap tmf • → H • ( − ; mf • ).)We will keep 2 inverted, and describe tmf • (pt)[ ] in terms of its map to mf • [ ]. Thismap is almost a surjection. In particular, its image contains the Eisenstein series c , c , andhence powers of 27∆ = ( c − c ). (At the prime 2, c is in the image of tmf • (pt), but c isnot: only 2 c is.) In fact, m ∆ k is in the image of tmf • (pt)[ ] if and only if mk = 0 (mod 3):tmf • (pt)[ ] contains nontorsion classes { } , { } , and ∆ . The curly brackets remindthat { } is not divisible by 3 in tmf • (pt)[ ].The kernel of the map tmf • (pt)[ ] → mf • [ ], i.e. the torsion in tmf • (pt)[ ], is periodic ofcohomological degree 72, with periodicity element ∆ . All torsion in tmf • (pt)[ ] has additiveorder 3. A framed compact manifold of dimension n determines a class [ M ] ∈ tmf − n (pt).In the case of a group manifold for a connected simply connected compact group G , thecorresponding class is represented, assuming Conjecture 1.5, by the antiholomorphic SCFTconsisting of n = dim G antichiral free Majorana–Weyl fermions, with supersymmetry en-coding the bracket on the Lie algebra of G [GJFW19]. In the introduction, we mentionedalready the class ν represented by the 3-sphere SU(2). Note that [Hen14] calls this class “ α ”in the section describing the 3-local structure of tmf • (pt) (and “ ν ” in the section describingthe 2-local structure). Another important class is represented by the 10-dimensional groupmanifold Spin(5). For want of a better name, we will call this class “ µ ”; it is called “ β ”in [Hen14]. These classes satisfy ν = µ ν = µ = 0 in tmf • (pt)[ ]. Furthermore, thereis a nontrivial Massey product h ν, ν, ν i = − µ = µ, which the reader is invited to thinkthrough geometrically by decomposing Spin(5) into two pieces. (Hint: use the inclusion– 42 –U(2) = Spin(4) ⊂ Spin(5).) There are two further torsion classes tmf • (pt)[ ] not in thesubring generated by ν and µ . The first is in degree 27, and is called “ { ν ∆ } ,” because itis represented by the product ν ∆ in the elliptic spectral sequence (see § { ν ∆ } µ . These are related to ν and µ by: h ν, ν, µ i = { ν ∆ } , ν { ν ∆ } = µ . Except for the powers of ∆ , the torsion and non-torsion classes in tmf • [ ] do not mix: forexample, { } µ = 0.In summary: Proposition 4.2.
The torsion in tmf • (pt)[ ] is -periodic, with periodicity given by multi-plication by ∆ . In the range ≥ • ≥ − , it looks as follows. The boxed class is nontorsion,and the remaining classes are torsion with additive order . The southwest-to-northeast edgesindicate multiplication by ν , and the northwest-to-southeast edges indicate (up to sign) anontrivial Massey product h ν, ν, −i . (The y -axis is otherwise insignificant.) Degree mod − − − − − − − − µ µ µ µ ν νµ { ν ∆ } { ν ∆ } µ p ≥ • [ ]. We will startfirst with the untwisted case and then add the twistings. This section studies the story afterlocalizing at a prime p ≥
5; the p = 3 story is in the next section.To warm up, let us review the analogous story for connective complex K-theory ku • ,due to [AH61, AS04, AS06]. After localizing at a prime p ≥
3, the coefficient ring isku • ( p ) (pt) = Z ( p ) [ u ], where u has cohomological degree −
2. As remarked in § p -locally, the first stable cohomology operation is the compositionH • ( − ; Z ( p ) ) (mod p ) −→ H • ( − ; F p ) P −→ H • +2 p − ( − ; F p ) (cid:3) Z −→ H • +2 p − ( − ; Z ( p ) ) . Here by “ P ” we mean the first Steenrod p ’th power operation, and by “ (cid:3) Z ” we mean theintegral Bockstein (for the extension Z → Z → Z p ). Write P Z for this total composition.Then the first nontrivial differential in the AHSS H • ( X ; ku • (pt)) ⇒ ku • ( X ) isd p − = P Z ⊗ u p − . – 43 –n the formula, we have identified the E page as H • ( X ; ku • (pt) ( p ) ) ∼ = H • ( X ; Z ( p ) ) ⊗ Z ( p ) [ u ],and “ u p − ” means multiplication thereby. In fact, the same formula works also at the prime p = 2, with P replaced by Sq , so that P Z is the integral lift of Sq . This gives the d differential identified in [AH61]. The higher differentials are similar, with P replaced byhigher Steenrod p th powers.To see that d p − is in fact a differential, note that (mod p ) ◦ (cid:3) Z = (cid:3) p is the mod- p Bockstein (for the extension Z p → Z p → Z p ), and so: P Z ◦ P Z = (cid:3) Z ◦ P ◦ (cid:3) p ◦ P ◦ (mod p ) . But an Adem relation says
P ◦ (cid:3) p ◦ P = P (2) ◦ (cid:3) p + (cid:3) p ◦ P (2) , where P (2) denotes the second Steenrod power, and (cid:3) p ◦ (mod p ) and (cid:3) Z ◦ (cid:3) p both vanish.The occurrence of P (2) here is related to the occurrence of P (2) in the next differential d p − .The twisted story is only slightly more complicated. As explained in [AS04, AS06], theK-theory of a space X can be twisted by any class ω ∈ H ( X ; Z ) (and more generally, at theprime 2, by classes in the supercohomology SH • of [GW14, WG17]). The twisting modifiesthe d differential to e d = d − ω ⊗ u. Higher differentials are also modified, now by Massey products with ω . For example, e d = d − h ω, ω, −i ⊗ u . Note that the Massey product h ω, ω, −i is not well-defined on the E -page of the AHSS, butit is well-defined on the E page.With the K-theory case understood, we can describe the tmf story. This is simplest ifwe work locally at a prime p ≥
5. There is a map of spectra H : tmf • → ku • J q K , which onhomotopy groups takes a nontorsion class in tmf • (pt) to its q -expansion (with the power of u just recording the weight of the corresponding modular form). The name “ H ” is becauseof its physical interpretation as the map SQFT • → KU • (( q )) that sends an SQFT to itsHilbert space, with the parameter q encoding the S -action that rotates the spatial circle.The existence of H forces the values of some differentials in the AHSS for tmf • , since theconstruction of AHSSs depends functorially on the spectrum. Indeed, suppose we are working p -locally, so that the E page is H • ( X ; Z ( p ) )[ c , c ]. As earlier, just because differentials arestable and because tmf • (pt) ( p ) has no torsion, the first possible nonzero differential is d p − .Suppose that ξ ∈ H • ( X ; Z ( p ) )[ c , c ] is some class. Ifd p − ( H ξ ) = ( P Z ⊗ u p − )( H ξ )is not zero in ku • ( X ) ( p ) J q K , then certainly d p − ( ξ ) must also be nonzero. Indeed, we findthat, in the AHSS for tmf • : d p − = P Z ⊗ A – 44 –or some modular form A of weight p − u p − via H . Actually, that’s notquite the requirement: if H ( A ) = u p − , then the q -expansion of A is 1 ∈ Z ( p ) J q K , whichdoes not happen for a modular form of nonzero weight. The trick is that P Z factors throughmod- p reduction, and so its image consists just of p -torsion classes. Thus we do not need H ( A ) = u p − on the nose, but only that H ( A ) ≡ u p − (mod p ). Said another way, the q -expansion of A should be 1 ∈ F p J q K .Working over F p , there is only one weight-( p −
1) modular form with trivial q -expansion,namely the Hasse invariant . When p ≥
5, it is liftable to an integral modular form. Thestandard lift is the weight-( p −
1) Eisenstein series c p − , and so we could set:d p − = P Z ⊗ c p − . But we don’t in fact need to choose a lift: different lifts differ by multiples of p , whereas theimage of P Z is p -torsion, so different lifts give the same d p − differential. Indeed, all we needis a criterion for checking liftability. Sufficient conditions are: Lemma 4.3.
A mod- p modular form of weight w admits an integral lift if H ( M ell ; L ⊗ w ) hasno p -torsion, where M ell is the compactified moduli stack of elliptic curves, and L ⊗ w is theline bundle whose sections are weight- w modular forms.Proof. This is automatic from the cohomology long exact sequence . . . p −→ H ( M ell ; L ⊗ k ) mod p −→ H ( M ell ; L ⊗ k /p ) (cid:3) Z −→ H ( M ell ; L ⊗ k ) p −→ . . . . But the only torsion in H • ( M ell ; L ⊗ k ) is at the primes 2 and 3. Thus, for p ≥
5, in factall mod- p modular forms admit integral lifts. p = 3If we try to repeat the p ≥ p = 3, we run into the following issue.Suppose f ∈ tmf • (pt) is nontorsion, and that x ∈ H • ( X ; Z (3) ). Then the E page of theAHSS contains the class x ⊗ f . The map H sends this class to the class x ⊗ H ( f ) on the E page of the AHSS for ku • ( X ) (3) J q K , which supports a d differential sending it to x ⊗ H ( f ) (cid:3) Z P ( x ) ⊗ u H ( f ) , where P now denotes the first Steenrod cube, and we will leave out from the notation themod-3 reduction. As above, this suggests that x ⊗ f should support a d differential of theform d : x ⊗ f ? (cid:3) Z P ( x ) ⊗ (integral lift of Af ) , where A denotes the mod-3 Hasse invariant.By Lemma 4.3, the obstruction to lifting Af is measured by the class (cid:3) Z ( Af ) = αf ∈ H ( M ell ; L ⊗ k ) (3) , where α = (cid:3) Z ( A ). This cohomology group turns out to vanish except when k = 2 + 12 j , in which case it is a Z generated by α ∆ j . We therefore find that the above d differential is well-defined if f is a multiple of 3, c , c , or their translates by powers of ∆.– 45 –odulo this ideal, the nontorision subring is just F [∆ ], and our d differential is notdefined on classes of the form x ⊗ ∆ j . Conveniently, it doesn’t need to be. The presenceof ν ∈ tmf − (pt) means that the tmf • -AHSS may contain a d -differential equal (up to anirrelevant sign) to d = P ⊗ ν. Indeed, the fact that the map from the sphere spectrum to tmf • is an equivalence in lowdegrees forces the existence of such a differential. (The sphere spectrum is initial among E ∞ ring spectra. This implies that, for the sphere spectrum, any differential which is allowed tobe nonzero is in fact nonzero.) The d differential needs only to be defined on the cohomologyof d , and if (cid:3) Z P ( x ) = 0 so that “d ( x ⊗ ∆ j ) = (cid:3) Z P ( x ) ⊗ (lift of A ∆ j )” is undefined, thend ( x ⊗ ∆ j ) = ( − deg x P ( x ) ⊗ ν ∆ j = 0 . The sign comes from the Koszul sign rules, since ν has odd degree.These d and d differentials are closely related. Indeed, the class α = (cid:3) Z ( A ) representsthe class ν in the following sense. There is a elliptic spectral sequence converging to tmf • (pt) (3) whose E page is H • ( M ell ; L ⊗• ) (3) . In this spectral sequence, α is a permanent cocycle, andits image on the E ∞ page is the associated graded element to ν .Returning to the d differential, we must work out d ( x ⊗ f ) whenever f ∈ tmf • (pt)satisfies νf = 0. The discussion above about lifts of multiplication by the Hasse invariantimplies: d ( x ⊗ c ) = (cid:3) Z P ( x ) ⊗ c , d ( x ⊗ c ) = (cid:3) Z P ( x ) ⊗ c , d ( x ⊗ { } ) = 0 . These almost completely determine the behaviour of d , since it must be linear for the actionby tmf • (pt), and so, if f , f ∈ tmf • (pt) with νf = 0, thend ( x ⊗ f f ) = d ( x ⊗ f ) f . Note that this is consistent with the above rules because (cid:3) Z P ( x ) is 3-torsion. Indeed, for f = c c , we would a priori face a discrepancy liked ( x ⊗ c ) c − d ( x ⊗ c ) c = (cid:3) Z P ( x ) ⊗ ( c − c ) = (cid:3) Z P ( x ) ⊗ { } , but this vanishes since 576 is divisible by 3.These rules do not quite determine the behaviour of d on the whole E page: there isthe possibility of a differential of the formd ( x ⊗ µ ) = (cid:3) Z P ( x ) ⊗ f – 46 –or some f ∈ tmf − (pt) = Span( { } , c ). Because (cid:3) Z P ( x ) is always 3-torsion, this maponly depends on the class of f modulo 3. Furthermore, f must be in the kernel of H :tmf − (pt) / → ku − J q K / F J q K . So the only possibility is, up to sign,d : x ⊗ µ (cid:3) Z P ( x ) ⊗ { } . One hint that there is in fact such a differential comes from the elliptic spectral sequence.The class µ ∈ tmf • (pt) is represented on the E page by a class β ∈ H ( M ell ; L ⊗ ) (3) .Although νµ = 0 in tmf • (pt), αβ = 0 in H ( M ell ; L ⊗ ) (3) ; it is instead the image of ad -differential emitted by ∆. Rather than exploring the elliptic spectral sequence in moredetail, we will give an alternate proof: Proposition 4.4.
The AHSS for tmf • includes a nontrivial d differential supported byclasses of the form x ⊗ µ .Proof. Our strategy is to compare our AHSS with the computations from [Hil07], whichcomputes the 3-local tmf • -homology of the symmetric group S . More specifically, thatpaper computes the 3-local homology tmf • (Σ BS ) (3) , where BS is the classifying space of S , and Σ BS is its suspension. We will focus on the specific valuetmf (Σ BS ) = 0 . Since [Hil07] computes homology, not cohomology, in this proof only we will work withhomological, rather than cohomological, gradings. The homology and cohomology of a pointare related by tmf • (pt) = tmf −• (pt) . But note that the same formula does not hold with pt replaced by other spaces.The AHSS for homology reads:H m (Σ BS ; tmf n (pt)) ⇒ tmf m + n (Σ BS ) . Homology AHSSs converges strongly by Theorem 12.2 of [Boa99]. The differentials in homol-ogy AHSSs are essentially the same as the differentials in cohomology AHSSs, because in bothcases they come form the Postnikov tower of the coefficient spectrum. The only difference isto understand the cohomology operations instead as homology operations. This is easy: thegroups H • ( X ; F ) and H • ( X ; F ) are dual, and so one uses the dual map.We have the following Z (3) - and F -homology of Σ BS :H • (Σ BS ; Z (3) ) = Z , • = 0 , F t k , • = 4 k, k > , , else , H • (Σ BS ; F ) = F t k , • = 4 k, F T k , • = 4 k + 1 , k > , , else , – 47 –hese are easily seen by noting that S = C ⋊ C . We have named basis vectors t k , T k , with t k denoting both the integral class and its mod-3 reduction. The Bockstein is (cid:3) ( T k ) = t k ,and the first Steenrod power is P ( t k ) = ( k − t k − , P ( T k ) = kT k − . In total degree 29, the only nonzero entries on the E page are: t ⊗ νµ ∈ H (Σ BS ; tmf (pt)) , T ⊗ µ ∈ H (Σ BS ; tmf (pt)) . The former is the image of a d differential:d ( t ⊗ µ ) = P ( t ) ⊗ νµ = t ⊗ νµ. The latter class must also be killed by some differential in order to confirm the compu-tation tmf (Σ BS ) = 0 from [Hil07]. It is not the image of a differential. Indeed, the onlyclasses of total degree 30 on the E page that could emit differentials to T ⊗ µ are t ⊗ µ ,which we already saw does not survive d , and T ⊗ νµ = d ( T ⊗ µ ) , and so also does not survive d . Thus T ⊗ µ must emit a differential. But the only degreepossible is d , and so we conclude d ( T ⊗ µ ) = 0 . And so general AHSSs for tmf include a d differential supported by classes of the form x ⊗ µ .After d , d , the next differential allowed by general considerations about degrees of stableoperations is d . To derive its formula, compare with the analysis in [AS06]: d arises as the“reason” that d ◦ d = 0. What is this reason? The vanishing ofd ◦ = P ◦ ⊗ ν has nothing to do with the “ X ” part of the differential, because P ◦ = 0. Rather, it vanishesbecause ν = 0 in cohomology. This means that the d differential will include a Masseyproduct. Indeed, suppose that x ⊗ f ∈ ker(d ) simply because νf = 0, while perhaps P ( x ) = 0. Then, if we imagine working at cochain level, we instead haved ( x ⊗ f ) = ( − deg x P ( x ) ⊗ d ( F ) , where d is the differential computing tmf • (pt) and F is some cochain for which d ( F ) = νf .Imagine a “total differential” d + d . One can find cochain formulas so that d and d commute, but (d + d ) will not vanish. Rather, (d + d ) = d , which at cochain level isd ( x ⊗ f ) = P ◦ ( x ) ⊗ ν d ( F ) = −P ◦ ( x ) ⊗ d ( νF ) . – 48 –o correct this, we include a d differential whose commutator with d is x ⊗ f
7→ −P ◦ ( x ) ⊗ d ( νF ). I.e. we should have d ( x ⊗ f ) ? = −P ◦ ( x ) ⊗ ( νF + N f ) , where N is some cochain such that d ( N ) = ν . This is an okay thing to write becaused ( νF + N f ) = − ν d ( F ) + d ( N ) f = − ν f + ν f = 0 at cochain level.The combination νF + N f is, by definition, the
Massey product h ν, ν, f i . We find:d = −P ◦ ⊗ h ν, ν, −i . We emphasize that this is only well-defined on the d -cohomology. Indeed, h ν, ν, f i doesn’texist unless νf = 0. Furthermore, the class F is defined only modulo cocycles, but if youchange F by a cocycle, then you change h ν, ν, f i by a multiple of ν , and so do not change d onthe d -cohomology. There is no essential ambiguity in the choice of N because tmf − (pt) = 0.In the AHSS for some generic 3-local E ∞ ring spectrum E , there is room for one further termin the d differential, equal to −P ◦ ⊗ λ for some λ ∈ E − (pt). But again we use thattmf − (pt) = 0 to rule out that possibility here.Although we will not need it, we mention that an analysis as in the p ≥ -differential of the form P (2) ⊗ (integral lift of A ). Note that, although the Hasse invariant A itself does not have an integral lift, A lifts to c . Finally, if we twist by an ’t Hooftanomaly ω ∈ H ( X ; Z (3) ), the only difference is that the Steenrod operator P is replaced bythe operator D = P − ω . All together, we find: Proposition 4.5.
The AHSS H • ( X ; tmf • (pt)) ⇒ tmf • ω ( X ) (3) includes the following differ-entials, with D = P − ω : • There is a d differential of the form d = D ⊗ ν. • The multiples of classes c , c , and µ support a d differential of the form d = (cid:3) Z D ⊗ f c f c ,f c f c ,f µ f { } . • There is a d differential of the form d = −D ◦ ⊗ h ν, ν, −i . • There is a d differential of the form d = − (cid:3) Z D ◦ ⊗ c . There are also higher differentials which we will not work out.– 49 – .5 Running the spectral sequence
We are now ready to understand the AHSS H • ( B M ; tmf • (pt)[ ]) ⇒ tmf • ω ( B M )[ ], where ω ∈ H (M ; Z [ ]) = F r , with r as in Section 3. Given Conjecture 1.4, we are interested inthe value of tmf • ω ( B M )[ ] for • ≡ • = − p ≥ E page is H • ( B M ; Z ( p ) )[ c , c ], wherethe ring H • ( B M ; Z ( p ) ) vanishes if p
6∈ { , , , } , and for p ∈ { , , , } it is a polyno-mial ring in a generator x p of cohomological degree 2( p −
1) and additive order p . Thus alldifferentials vanish for degree reasons, and the spectral sequence stabilizes on the E page.The differentials do play a role, however: they lead to extensions. Indeed, let ˜ x p = x p c p − ,which is of total degree 0 on the E ∞ page. Then, by the q -expansion map H to K-theory, wesee that the translates of ˜ x p F p [˜ x p ] on the E ∞ -page compile to copies of the p -adic integers Z ( p ) in tmf • ( B M ) ( p ) . For the purposes of this paper, all that we care about is that, for p ≥
5, tmf • ( B M ) ( p ) is supported in degrees • ≡ p = 2 for being too complicated, leaving only the prime p = 3. Thefirst few differentials for the AHSS H • ( B M ; tmf • (pt)) ⇒ tmf • ω ( B M ) (3) are summarized inProposition 4.5. A typical term on the E page has shape x ⊗ f , where f ∈ tmf • (pt) and x ∈ H • (M ; Z (3) ) if f is nontorsion and x ∈ H • (M ; F ) if f is torsion. With some caveats,the differentials sort into two sets. If f is nontorsion (and not a power of ∆ ), then x ⊗ f only supports d and d differentials. If f is torsion (and not a translate of µ ) then x ⊗ f only supports d and d differentials.By Theorem 3.1, H • (M ; Z (3) ) is supported only in degrees • ≡ , E -page entries x ⊗ f with f nontorsion are also only in degrees • ≡ , (cid:3) Z P ( u ) = (cid:3) Z ( ur + T ± Rs ) = t ± rs = 0 . But, since we care mostly about the case • ≡ x ⊗ f with f ∈ { , ν, µ, νµ, µ , { ν ∆ } , µ , { ν ∆ } µ, µ } or the translates thereof by powers of ∆ . Except for µ , these classes only support d and d differentials. Since d ( x ⊗ µ ) is an integral class times { } , it has degree 3 or 4 (mod 4),and so does not interact with the • ≡ • ω ( B M )[ ] for • ≡ E page of theform H • (M ; F ) ⊗ F { , ν, µ, νµ, µ , { ν ∆ } , µ , { ν ∆ } µ, µ } , and just with the differentialsd = D ⊗ ν, d = −D ◦ ⊗ h ν, ν, −i , where D = P − ω . – 50 –ecall from Proposition 4.2 the action of the maps ν and h ν, ν, −i . Writing H =H • (M ; F ), we therefore find ourselves interested in the following total complex: H D −→ Hν D −→ Hµ D −→ Hνµ D −→ Hµ D −→ H { ν ∆ } D −→ Hµ D −→ H { ν ∆ } µ D −→ Hµ This is an ordinary cochain complex because D is a 3-differential by Lemma 3.6. Its cohomol-ogy is closely related to the cohomology of D itself, which is listed in Proposition 3.7. Indeed,both ker( D ) / im( D ) and ker( D ) / im( D ) vanish whenever D is exact. More generally, bothker( D ) / im( D ) and ker( D ) / im( D ) are of the same total dimension as the cohomology of D (with both {∗} and cohomology {∗ → ∗} thought of as 1-dimensional; this is the totaldimension of the super-vector-space-valued cohomology from Appendix A). The precise co-homological degrees of cohomology classes depends on whether we use ker( D ) / im( D ) orker( D ) / im( D ), but their degrees mod 4 do not depend, since D preserves the cohomologicaldegree mod 4.The cohomology of the above total complex at the entries H Hµ , and Hµ is morecomplicated, but for our purposes irrelevant. Indeed, 1, µ , and µ have cohomologicaldegrees • ≡ H vanishes in degrees • ≡ • ≡ Theorem 4.6.
Let ω = − ǫr with ǫ ∈ F and notation for H • (M ; Z (3) ) as in Theorem 3.1.On the E page of the AHSS H • ( B M ; tmf • (pt)[ ]) ⇒ tmf • ω ( B M )[ ] , the cohomology intotal degree • ≡ is the following. Write Υ = U ⊗ µ , and note that Υ = 0 . Thenwe have: • ǫ = 0 : A free F [ s , ∆ , Υ] / (Υ ) -module generated in cohomological degrees (repeatedentries indicate multiplicity in the generating set): , , , , − , − , − , − , plus a free F [ s , ∆ ] -module generated in degrees , , , , , − , − , − , plus a free F [∆ ] -module generated in degrees − , − . • ǫ = 1 : A free F [ s , ∆ , Υ] / (Υ ) -module generated in cohomological degrees , − , − , − , plus a free F [ s , ∆ ] -module generated in degrees , , , , , , − , − . – 51 – ǫ = − A free F [ s , ∆ , Υ] / (Υ ) -module generated in cohomological degrees , , , − , plus a free F [ s , ∆ ] -module generated in degrees , , , , , , − , − . Proof.
Each entry in Proposition 3.7 produces two free F [ s , ∆ ]-modules the E page.For example, the ǫ = 0 entry in Proposition 3.7 listed as “ { → } ” is represented by[ s b s b ]. (It is also represented by [ st t ].) The E page includes the degree 1 (mod 4)classes s b ⊗ ν , s b ⊗ ν , s b ⊗ { ν ∆ } , and s b ⊗ { ν ∆ } . We haved : s b ⊗ s b ⊗ ν and so s b ⊗ ν does not contribute to cohomology on the E page, but the presence of D -cohomology means that s b ⊗ ν is not in the image of d nor in the kernel of d , and so doescontribute cohomology, as do its translates by s and ∆ . If we look instead at s b ⊗ { ν ∆ } and s b ⊗ { ν ∆ } , we see that the former supports a d -differential but the latter survivesto E .Consider now the ǫ = 0 entry in Proposition 3.7 listed as “ { → } .” It is represented by[ U s b P U s b ], which can be moved to have total degree • ≡ νµ or { ν ∆ } µ . In this way we see E -page classes represented by U s b ⊗ νµ and U s b ⊗ { ν ∆ } µ .Indeed, while proving Proposition 3.7, we noted that the D -cohomology in degree n ≡ D -cohomology in degree n + 10, with the isomorphism givenby multiplication by U , of degree • = 10, whereas the odd-degree torsion classes listed inProposition 4.2 are related by multiplication by µ , of degree • = −
10. Thus we find thateach entry in Proposition 3.7 of degree 0 (mod 4) contributes not just a pair of copies of F [ s , ∆ ], but a pair of copies of F [ s , ∆ , Υ]. The general rule is that an entry like “ { n } ”with n ≡ n − n −
27, whereas an entry like“ { n → n + 4 } ” produces generators in degrees n − n + 4 − • (M ; F ) vanishes in degree 1 (mod 4), the only other entries in Proposition 3.7are of degree 3 (mod 4). These can be combined with µ or µ and a similar analysis can beperformed, but now multiplication by U (and hence by Υ) is zero. Thus we find merely a copyof F [ s , ∆ ]. The entry “ { n } ” with n ≡ n −
10 and n −
30, whereas an entry “ { n → n + 4 } ” produces generators in degrees n + 4 −
10 and n − Corollary 4.7.
In cohomological degree • = − , both twisted cohomology groups tmf − ± r ( B M ) (3) vanish. – 52 – et Φ = s ⊗ ∆ . The E -page approximation to the untwisted cohomology tmf − ( B M ) (3) is tmf − ( B M ) (3) ≈ F [Φ] { ⊗ { ν ∆ } , U ⊗ { ν ∆ } µ } ⊕ F { r ⊗ µ } . By this we mean the abelian group isomorphic to F [Φ] ⊕ F , where the F [Φ] summand isgenerated over F [Φ] by the elements on the E page represented by ⊗ { ν ∆ } and U ⊗ { ν ∆ } µ ,and where the F summand is generated by r ⊗ µ .For comparison, in cohomological degree • = − , the E -page approximations to tmf − ǫr ( B M ) (3) are: tmf − ( B M ) (3) ≈ F [Φ] { ⊗ ν, U ⊗ νµ, s R ⊗ µ , sT ⊗ µ } ⊕ F { rR ⊗ µ } , tmf − − r ( B M ) (3) ≈ F [Φ] { ⊗ ν, U ⊗ νµ, sT ⊗ µ } , tmf − r ( B M ) (3) ≈ F [Φ] { sT ⊗ µ } . Here we have listed just E -page representatives of the E -page generators, and those are ofcourse ambiguous. For example, sT and St are D -cohomologous.In cohomological degree • = 1 , tmf − − ǫr ( B M ) (3) vanishes for ǫ = 0 , , and tmf r ( B M ) (3) ≈ F [Φ] { s b ⊗ { ν ∆ }} . Again note the ambiguity that s b and st are D -cohomologous in this case. To complete the proof of Theorem 1.8, we have:
Corollary 4.8.
For all values ω = ǫr ∈ H (M ; Z (3) ) , the AHSS for tmf • ω ( B M ) (3) con-verges. When ω = 0 or − r , the class ⊗ ν on the E page is a permanent cycle, and representsa class in tmf − ω ( B M ) (3) with nontrivial restriction to tmf − ( B M ) . When ω = + r , therestriction map tmf − ω ( B M ) (3) → tmf − ω ( B M ) (3) vanishes.Proof. The first two statements follow from the claim that, other than d , all differentialsin the AHSS H m ( B M ; tmf n (pt) (3) ) ⇒ tmf m + nω ( B M ) (3) vanish when m = 0. The laststatement is already clear from Corollary 4.7 together with the fact that the restriction mapalong pt → B M is restriction to the m = 0 column, and annihilates r, s, t, u and hence sT ⊗ µ .That the claim implies convergence is explained in § m = 0 that survives d is permanent, and so must representa class in tmf • ω ( B M ) (3) . But when ω = 0 , − r , the class 1 ⊗ ν itself survives d , and hasnontrivial restriction along pt → B M . As explained already in § m = 0 column for the untwisted cohomology ω = 0.The only way a d k -differential can be nontrivial on the m = 0 column is if it contains aterm of that simply multiplies by an element on the E k -page of total cohomological degree 1.For ω = ǫr with ǫ = 0 ,
1, Corollary 4.7 implies that there are no such elements. For ǫ = − E page of cohomological degree 1, and so we need to know thatthey cannot appear. – 53 –owever, the universality of the AHSS means that the multiplying element must be of theform x ⊗ f where f is arbitrary but where x is produced from ω by a cohomology operation.The algebra of 3-local cohomology operations is generated by the Steenrod powers and theBockstein. The Bockstein vanishes on r and the first Steenrod power acts as P ( r ) = − r .The second Steenrod power is simply r r for degree reasons, and the higher powersannihilate r . These, together with the Cartan relation (which says that each Steenrod operatoris a derivation modulo lower Steenrod operators), imply that the only cohomology classes thatcan be produced by r are in the polynomial ring F [ r ].The degree-1 classes Φ k s b are not cohomologous to anything in this ring. Indeed, asshown in Proposition 3.7, for ω = − ǫr with ǫ = − D = P + ǫr is exact on F [ r ]. Thus, by repeating the proof of Theorem 4.6 just on this subring, we see that nothingof total degree 1 in F [ r ] ⊗ tmf • (pt) survives to the E -page, and so there are no elementsthat could appear as multipliers in higher differentials, and so all higher differentials vanishon the m = 0 column. A Higher complexes
Our analysis of the AHSS for tmf • (3) in § § D = P + ǫr is a 3 -differential in the sense that D ◦ = 0. The goal of this Appendixis to tell the general story of higher differentials, and to point out an intriguing connectionto Verlinde rings that the author has not seen stated directly in the literature.Let K be a field, perhaps of positive characteristic, and choose a positive integer ℓ . An ℓ -differential on a K -vector space V is a linear endomorphism D such that D ℓ = 0. For example,a 1-differential is the zero map, and a 2-differential is an ordinary differential. In the ordinarycase, the cohomology of a 2-differential is H ∗ ( V, D ) = ker( D ) / im( D ). There is also a theory ofcohomology of ℓ -differentials for higher ℓ , which dates as early as [May42]; see the introductionof [IKM17] for some history and a number of relevant references. Various authors have triedto define the cohomology of an ℓ -differential as, for example, H ∗ ( V, D ) = ker( D ) / im( D ℓ − ) orker( D ℓ − ) / im( D ), and these definitions are fine for basic purposes. But there is a somewhatricher story, that may be especially entertaining for quantum field theorists.The idea is the following. A vector space with an ℓ -differential is equivalently a modulefor the algebra K [ D ] / ( D ℓ ). This algebra has ℓ indecomposable modules, indexed by their K -dimensions 1 , . . . , ℓ : the one-dimensional module is simple, and the ℓ -dimensional moduleis free. Every K [ D ] / ( D ℓ )-module splits as a direct sum of indecomposable modules. We willsay that D is exact when the module is free, i.e. all of its indecomposable summands are ℓ -dimensional. The cohomology of an ℓ -differential should measure its failure to be exact.Both ker( D ) / im( D ℓ − ) or ker( D ℓ − ) / im( D ) measure this failure coarsely: those two vectorspaces are (noncanonically) isomorphic, and their dimension counts the number of non-freeindecomposable summands. But we can measure things more finely, by recording which non-free summands appear. We will define the cohomology H ∗ ( V, D ) of an ℓ -differential D on V to be the result of: – 54 – Decomposing (
V, D ) as a direct sum of indecomposable K [ D ] / ( D ℓ )-modules. • Discarding the free summands. • Converting the other summands into simple objects of a semisimple category.For example, in the ordinary case, there is one non-free indecomposable K [ D ] / ( D )-module,namely the one-dimensional one. Thus the cohomology in our sense is an object of a semisim-ple category with one simple object, i.e. the category of vector spaces.This procedure is functorial, although it doesn’t look so from our description. To makeit cleaner, we will use the technology of semisimplification developed in [BW99, EO18]. Al-though we care most about the case K = F and ℓ = 3, we will first tell the story when ℓ isnot divisible by the characteristic of K . A.1 ℓ -complexes in characteristic not dividing ℓ The category of K [ D ] / ( D ℓ )-modules is not naturally monoidal (if ℓ is not a power of thecharacteristic of K ). This is clear already when ℓ = 2: the tensor product of ordinarycomplexes is D ( v ⊗ w ) = D ( v ) ⊗ w + ( − deg v v ⊗ D ( w ), which requires at least a Z -grading.To correct this, let us say that a (periodic) ℓ -complex is a Z ℓ -graded vector space with an ℓ -differential that increases the grading by 1. By field-extension if necessary, suppose that K contains a primitive ℓ ’th root of unity q . Then we may define the tensor product of two ℓ -complexes to be their usual tensor product as Z ℓ -graded vector spaces, equipped with thedifferential D ( v ⊗ w ) = D ( v ) ⊗ w + q deg v v ⊗ D ( w ) . We will write C q for this monoidal category. The choice of q identifies it with the category ofmodules for the semidirect product Hopf algebra H q = K [ D ] / ( D ℓ ) ⋊ Z ℓ = K h D, K i / ( D ℓ , K ℓ − , KD − qDK ) , ∆( K ) = K ⊗ K, ∆( D ) = D ⊗ K ⊗ D, which is nothing but the upper Borel inside Lusztig’s small quantum group for SL(2).The monoidal category C q is not braided, but it is spherical, and so has a semisimplifica-tion C q . The defining property of C q is that it is the universal semisimple monoidal categoryreceiving a monoidal (but neither left- nor right-exact) functor C q → C q . To construct it,one follows [BW99] and defines a monoidal ideal N ⊂ C q of “negligible morphisms,” whichare those morphisms in the kernel of the trace pairing hom( X, Y ) ⊗ hom( Y, X ) → K (whichexists in any spherical monoidal category). Then C q is defined to be the quotient category C q / N . Although not obvious, this quotient category is semisimple, and the simple objectsare indexed by the indecomposable objects in C q of nonzero quantum dimension.It is not hard to show that an indecomposable H q -module of K -dimension n has quantumdimension [ n ] q = ( q n − / ( q − C q → C q areprecisely the modules which are free over K [ D ] / ( D ℓ ). We are therefore justified in using– 55 –he name “H ∗ ” for the functor C q → C q , and calling H ∗ ( V, D ) the cohomology of the ℓ -complex ( V, D ) ∈ C q .When K is of characteristic 0, Theorem 5.2 of [EO18] identifies the fusion rules for C q :the fusion ring is isomorphic to the fusion ring of Vec [ Z ℓ ] ⊠ Ver q ⊠ Rep (PGL(2)) . Here
Ver q is the Verlinde category of SL(2) at level k = ℓ −
2. Corollary 5.3 of [EO18] showsthat C q does indeed contain Ver q as a subcategory. The fusion ring for Rep (PGL(2)) is thesame as that of
Rep (OSp(1 | C q . Finally, Vec [ Z ℓ ] ⊂C q is spanned by the images of 1-dimensional H q -modules. It is therefore conjectured in thatpaper that there is an equivalence of spherical fusion categories C q ? ∼ = Vec [ Z ℓ ] ⊠ Ver q ⊠ Rep (OSp(1 | . To check this conjecture requires checking that there are no interesting associators betweenobjects coming from the various tensorands on the right-hand side.There are two extremal cases of this story. When ℓ = 1, the Hopf algebra H q is trivial,and C q = C q = Vec . More interesting is the case ℓ = + ∞ . Then by “primitive ℓ ’th rootof unity” we will mean that q does not solve an algebraic equation with nonnegative integercoefficients. The Hopf algebra H q is then simply K h D, K ± i / ( KD − qDK ). To impose that D act nilpotently, we will say that an ∞ -complex is a direct sum of finite-dimensional H q -modules, and write C q for the category thereof. Again assuming that K is of characteristic 0, C q then contains no objects of zero quantum dimension (since the quantum dimension of anyobject is a polynomial in q with nonnegative integer coefficients), and the semisimplification C q of C q is identified in Proposition 5.1 of [EO18]: C q ∼ = Rep (GL q (2)) , where GL q (2) is the Drinfeld–Jimbo quantum group. Intriguingly, the right-hand side isbraided, even though the left-hand side has no reason to be. Note that when q = 1, werecover the symmetric monoidal category of GL(2)-modules. Indeed, C was the category offinite-dimensional modules for the Borel subgroup B ⊂ GL(2), and the passage B GL(2)is an example of the reductive envelope of [AK02].
A.2 ℓ -complexes in characteristic ℓ Finally, we discuss the case of most importance in this paper, which is when ℓ is prime and K has characteristic ℓ . (We care specifically about the case ℓ = 3.) In this case, we donot need any gradings or q ’s. More precisely, in characteristic ℓ , the unique primitive ℓ ’throot of unity is q = 1, because the definition of “primitive ℓ ’th root” (when ℓ is prime)is “solution to ( q ℓ − / ( q −
1) = q ℓ − + · · · + q + 1,” which in characteristic ℓ factors as q ℓ − + · · · + q + 1 = ( q − ℓ − . As such, the category C ℓ of K [ D ] / ( D ℓ )-modules is symmetricmonoidal. – 56 –s above, the free K [ D ] / ( D ℓ )-module is the only indecomposable of (quantum) dimen-sion 0. Thus we are justified in defining the cohomology of an object ( V, D ) ∈ C ℓ to be itsimage under the semisimplification functor H ∗ : C ℓ → C ℓ . The codomain C ℓ is studied in detailin [Ost15]. It is called therein the universal Verlinde category Ver ℓ , because the fusion ringof C ℓ is precisely the fusion ring of the Verlinde category for SL(2) at level k = ℓ −
2. Notethat
Ver ℓ is symmetric monoidal.(Actually, [Ost15] uses a different symmetric monoidal structure on the category C ℓ of K [ D ] / ( D ℓ )-modules. The one we are using corresponds to the Hopf structure on K [ D ] / ( D ℓ )in which D is primitive, i.e. ∆ D = D ⊗ ⊗ D . But ( D + 1) ℓ = D ℓ + 1 ℓ = 0 + 1 incharacteristic ℓ , and so as a category C ℓ ∼ = Rep ( Z ℓ ); this corresponds to the Hopf structurein which D + 1 is grouplike, i.e. ∆ D = D ⊗ D + D ⊗ ⊗ D . Although these symmetricmonoidal structures on C ℓ are different, they determine the same fusion rules on the semisim-plification C ℓ . Theorem 1.5 of [Ost15] then implies that the two versions produce equivalentsymmetric monoidal structures on C ℓ .)The main examples are the following. When ℓ = 2, Ver = Vec , and we recover theusual cohomology of an ordinary complex. When ℓ = 3, Ver = SVec is the category ofsupervector spaces: the fermionic line K | ∈ Ver is the cohomology of the two-dimensionalindecomposable K [ D ] / ( D ℓ )-module. When ℓ = 5, Ver factors as a tensor product SVec ⊠ Fib , where
Fib is the
Yang–Lee or Fibonacci category, a (symmetric, in characteristic 5)fusion category with simple objects { , X } and fusion rules X ⊗ X = 1 ⊕ X . In general, when ℓ is an odd prime, Ver ℓ factors as SVec ⊠ Ver + ℓ , where Ver + ℓ is the “bosonic part” of Ver ℓ ,and is spanned by the images under H ∗ of the indecomposable K [ D ] / ( D ℓ )-modules of odd K -dimension.Let us end by observing the following. Still working in characteristic ℓ , with ℓ an oddprime, let us say that a (nonperiodic) ℓ -complex is a Z -graded vector space equipped withan ℓ -differential that raises degree by 1. We will not use any Koszul signs when multiply-ing ℓ -complexes: the underlying vector spaces are entirely bosonic. Let us decide that anindecomposable ℓ -complex supported in degrees m, . . . , m + n has spin the average degree m + n . This is either integral or half-integral depending on whether the K -dimension n + 1of the complex is odd or even. This spin is additive under tensor product, and so provides a Z grading to the semisimplification of the category of ℓ -complexes, in which the “bosonic”objects are precisely the ones of integral spin, and the “fermionic” objects are the ones ofhalf-integral spin. The factorization Ver ℓ ∼ = SVec ⊠ Ver + ℓ means that these “fermionic” ob-jects really are fermionic in the sense of Koszul signs. In this way, the category of ℓ -complexessecretly knows about the Koszul sign rules: fermions have “emerged” during the passage fromthe category ℓ -complexes (a “UV” object) to its semisimplification (the “IR”). References [Ada74] J. F. Adams.
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