Absence of exotics I. This section will be concluded with an application of the above results to the absence of exotics conjecture formulated in Section 2.5

Therefore (7.20) reduces to

−J(γ; z, w) + DT (γ, 1; z, w, +∞) + DT (γ2, 1; z, w, +∞)J(γ1; z, w) = 0.

According to Appendix E, DT (γ, 1; z, w, +∞) = 0 and DT (γ2, 1; z, w, +∞) = 1.

Moreover J (γ; z, w) = J (γ₁; z, w) = 1 at strong coupling, hence the y→ (−1) limit of formula (7.20) holds in this case.

As an alternative to the recursion formula, note that the spectrum of framed BPS states at δ >> 0 can be also related to the spectrum at δ << 0 applying directly the wall crossing formula of [106]. Again, using the SU (3) strong coupling chamber found in Section 6 as an example, recall that there are 6 unframed BPS states with dimension vectors

γ₁= (1, 0, 0, 0), γ₂= (0, 1, 0, 0), γ₃= (0, 0, 1, 0), γ4= (0, 0, 0, 1), γ5= (1, 1, 0, 0), γ6= (0, 0, 1, 1).

In addition in the chamber δ << 0 there is only one framed BPS states with dimension vector γ = 0 corresponding to the simple module supported at the framing node. Consider the Lie algebra overQ generated by {eγ, f_γ} satisfying

[eγ, eγ^′] = (−1)^χ(γ,γ′)χ(γ, γ^′)eγ+γ^′, [eγ, fγ^′] = (−1)^{dj(γ)+χ(γ,γ′)}(dj(γ)+χ(γ, γ^′))fγ+γ^′. Define Ui, i = 1, . . . , 6 by

U_i= exp(∑ emγ_i

m² ),

Then the framed spectrum in the chamber δ >> 0 is determined by (U4U2U6U5U3U1)⁻¹exp(f0)U4U2U6U5U3U1 .

Note that U2 and U6 commute and U3 and U5 commute. After the algebraic manipulation we obtain the 7 invariants listed in Appendix E, equation (E.7).

7.5. Absence of exotics I. This section will be concluded with an application

wallcrossing formula yields a polynomial wallcrossing formula for Hodge type Donaldson-Thomas invariants:

(7.21)

DT (γ, 1; z, w, δ₋; x, y)(γ)− DT (γ, 1; z, w, δ+; x, y) =

∑

l≥2

1 (l− 1)!

∑

γ1+···+γl=γ γ_s̸=(0,0), 1≤s≤l−1 µ(γ_s)=δ_c, 1≤s≤l−1

C_(−(xy)1/2)((γ_s))DT (γ_l, 1; z, w, δ₊; x, y)

l−1

∏

s=1

J (γ_s; z, w; x, y),

where the J (γ_s; z, w; x, y) are the images of the motivic invariants J^mot(γ_s; z, w) via the virtual Hodge polynomial map. Analogous considerations hold of course for the recursion formula (7.20) which is an iteration of the wallcrossing formula. Using these formulas, absence of exotics for framed and unframed invariants reduces to absence of exotics for the framed asymptotic ones. The latter will then be proven shortly using the results of [116]. Therefore, in short, rationality of both framed and unframed invariants is established, granting the motivic wallcrossing formula of [106] for SU (N ) quivers.

The proof of absence of exotics for asymptotic framed invariants will be based on the main result of [116], where they are expressed in terms of Chow motives of certain affine varieties. In order to apply the results of [116] one first has to check that the potential

W =

N∑−2 i=1

[ri(aici− ci+1bi) + si(aidi− di+1bi)]

of the SU (N ) quiver has a linear factor according to [116, Def. 2.1]. First note that any potential W^′ which differs from W by cyclic permutations in each term is equivalent to W since they define the same relations in the path algebra. Therefore W is equivalent to

W^′=

N∑−2 i=1

(aiciri− birici+1+ aidisi− bisidi+1)

Then note that W^′ has a factorization of the form W^′= LR in the path algebra of the quiver without relations, where

L =

N∑−2 i=1

(ai+ bi), R =

N∑−2 i=1

(ciri− ci+1bi+ aidi− di+1bi).

Since the product is defined by concatenation of paths it is straightforward to check that all terms in the expansion of LR not belonging to W^′are trivial. For example

aici+1bi= 0

since the tail of ai does not coincide with the head of ci+1. Moreover, any two distinct nodes of the quiver are connected by at most one of the arrows ai, bi, 1≤ i ≤ N2 in L. These are precisely the conditions required by [116, Def 2.1].

Then [116, Thm. 7.1] provides an explicit expression for the motivic Donaldson-Thomas invariants DT^mot(γ, 1; z, w, +∞) in terms of Chow motives of general linear groups GL(n,C), n ≥ 1 and Chow motives of “reduced quiver varieties”, which are

constructed as follows. For a fixed dimension vector γ = (di, ei)1≤i≤N−1 let

V(γ) = ⊕

a∈{aj,b_j,c_j,d_j,r_j,s_j}

Hom(C^dt(a),C^dh(a))

be the linear space of all quiver representations. The potential W determines a gauge invariant polynomial function W_γ on V(γ). The “reduced quiver variety”

R(γ) is defined as the zero locus of the F-term equations a_i= b_i = 0, ∂_aiW_γ = 0, ∂_biW_γ = 0 inV(γ). In the present case, one obtains the quadratic equations (7.22) ciri+ disi= 0, rici+1+ sidi+1 = 0.

According to equation (B.1) in Appendix B, [GL(n,C)] =

∏n k=1

(L^k− 1),

hence its virtual Hodge polynomial is∏n

k=1((xy)^k− 1). Therefore, thanks to the above result of [116], in order to prove absence of exotics in the asymptotic chamber it suffices to prove that the virtual Hodge polynomials with compact support of the reduced varietiesR(γ) are polynomial functions on (xy). This will be done below using the compatibility of the virtual Hodge polynomial with motivic decomposi-tions.

Note thatR(γ) admits a natural projection π : R(γ) → B(γ) to the linear space B(γ) =⊕

a∈{cj,d_j}Hom(C^dt(a),C^dh(a)) given by π(ci, di, ri, si)7→ (ci, di).

For each i = 1, . . . , N − 1 the pair of linear maps (ci, d_i) determine a Kronecker module κ_i of dimension vector (e_i, d_i). ThereforeB(γ) is a direct product

B(γ) = ×^Ni=1⁻¹V(ei, d_i)

where V(ei, d_i) denotes the linear space of all Kronecker modules of dimension vector (d_i, e_i). Using equation (C.2), the space of solution (r_i, s_i) to equations (7.22) is isomorphic to the dual extension group of Kronecker modules Ext¹(κi, κi+1)^∨. Therefore the fiber of π over a point (κi)∈ B(γ) is isomorphic to the linear space

N⊕−2 i=1

Ext¹_K(κi, κi+1)^∨,

whereK denotes the abelian category of Kronecker modules.

If the dimension of the fiber π⁻¹(κ1, . . . , κN−1) were constant,R(γ) would be isomorphic to a product of linear spaces, which is obviously rational. This is in fact not the case; the dimensions of the fiber jumps as the point (κ1, . . . , κN1) moves in B(γ). However, suppose there is a finite stratification of B with locally closed strataSα such that the fiber of π has constant dimension p_α over the stratumSα. Then the following relation holds in the ring of motives

[R(γ)] =∑

α

L^pα[Sα].

This yields a similar relation,

P_(x,y)(R(γ)) =∑

α

(xy)^pαP_(x,y)(Sα)

for virtual Hodge polynomials with compact support. Therefore in order to prove the claim it suffices to construct a stratification Sα such that each polynomial P_(x,y)(Sα) is only a function of (xy). It will be shown next that the natural strati-fication ofB(γ) by gauge group orbits satisfies this condition.

Let V(e, d) ≃ Hom(C^e,C^d)^⊕2 be the linear space of all Kronecker modules of dimension vector (e, d). SupposeS is an orbit of the natural GL(e, C) × GL(d, C) action onV(e, d) and let G_S ⊂ GL(e, C) × GL(d, C) be its stabilizer. Given any Kronecker module κ ∈ S corresponding to a point in S, the stabilizer G_S is iso-morphic to the group of invertible elements in the endomorphism algebra End_K(κ).

Recall that K denotes the abelian category of Kronecker modules. According to [94] (see below Def. 2.1), this implies that G_S is special, which means that any principal G_S-bundle over a complex variety is locally trivial in the Zariski topology.

In particular this holds for the natural principal G_S-bundle G_S // GL(e, C) × GL(d, C)

S,

which yields a relation of the form

[GL(e,C)][GL(d, C)] = [G_S][S]

in the ring of motives. Taking virtual Hodge polynomials with compact support one further obtains

P_(x,y)(S)P(x,y)(G_S) =

∏d k=1

((xy)^k− 1)

∏e l=1

((xy)^l− 1).

Note that the right hand side of this identity is a product of irreducible factors (xy− ζ), with ζ a root of unity. Since the polynomial ring is a unique factorization domain, it follows that the same must hold for both factors in the left hand side.

Therefore P_(x,y)(S) is a polynomial function of (xy) as claimed above.

In order to conclude this section, one can ask the question whether absence of ex-otics may hold for the BPS spectrum of any toric Calabi-Yau threefold. We expect this to be the case for general BPS states on toric Calabi-Yau threefolds, based on similar arguments. Using dimer technology [92, 21] any toric Calabi-Yau threefold X has an exceptional collection of line bundles which identifies the derived cate-gory D^b(X) with the derived category of a quiver with potential (Q, , W ). There is moreover a region in the K¨ahler moduli space where one can construct Bridgeland stability conditions where the heart of the underlying t-structure is the abelian category of (Q, W )-modules. In this region, BPS states will be mathematically modeled by supersymmetric quantum states of moduli spaces of stable quiver rep-resentations. Moreover, explicit formulas for motivic Donaldson-Thomas invariants of moduli spaces of framed cyclic representations have been obtained in [121], and they depend only onL^1/2. Therefore one can employ a similar strategy, defining

δ-stability conditions for framed representations, and studying motivic wallcross-ing. This provides a framework for a mathematical study of absence of exotics for dimer models. The details will be left for future work.

A much harder problem is absence of exotics in geometric regions of the K¨ahler moduli space [15] where there are no stability conditions with algebraic t-structures.

In those cases, one has to employ perverse t-structures in the construction of stabil-ity conditions, and the role of framed quiver representations is played by large radius stable pair invariants. Explicit motivic formulas for such invariants are known only in cases where X has no compact divisors [117, 118]. If X has compact divisors, no explicit large radius motivic computations have been carried up to date. However absence of exotics is expected based on the refined vertex formalism [91]. More-over, the cohomology of smooth moduli spaces of semi-stable sheaves on rational surfaces is known to be of Hodge type (p, p) [16, 73], which suggests that also for toric Calabi-Yau’s with compact divisors no exotic BPS states arise. For complete-ness note that absence of exotics is known to fail [4, 44] on non-toric Calabi-Yau threefolds.

8. BPS states and cohomological Hall algebras

This section explains the relation between BPS states and the mathematical formalism of cohomological Hall algebras [108]. Although cohomological and mo-tivic Donaldson-Thomas invariants are known to be equivalent [108], the cohomo-logical construction provides more insight into the geometric construction of the SL(2,C)spin-action on the space of BPS states [105]. Moreover, it also offers a new perspective on the absence of exotics, which is now related to a conjectural Atiyah-Bott fixed point theorem for the cohomology groups defined in [108].

8.1. Cohomological Hall algebras. The algebra of BPS states was first con-structed in [81] in terms of scattering amplitudes for D-brane bound states. In the semiclassical approximation, the algebraic structure is encoded in the overlap of three quantum BPS wave functions on an appropriate correspondence variety.

This formulation can be made very explicit for quiver quantum mechanics. More recently, a rigorous mathematical formalism for BPS states has been proposed in [108] for quivers with potential. A detailed comparison between the physical def-inition of [81] and the formalism of [108] has not been carried out so far in the literature. Leaving this for future work, the construction of [108] will be briefly summarized below.

The most general definition of the corresponding algebraic structure is given in the framework of Cohomological Hall algebra (COHA for short) in [108]. In the loc. cit. the authors defined the category of Exponential Mixed Hodge Structures (EMHS for short) as a tensor category which encodes “exponential periods”, i.e.

integrals of the type ∫

Cexp(W )α, where C is a cycle in an algebraic variety X, W : X → C is a regular function (or even a formal series) and α is a top degree form on C. There are different “cohomology theories” which give “realizations” of EMHS. Every such theory is a tensor functor H from the category EM HS to the category of graded vector spaces. Similarly to the conventional theory of motives there are several standard realizations:

a) Betti realization which is given by the cohomology of pairs H^•(X, W⁻¹(t)), where t is a negative number with a very large absolute value (it is also called

“rapid decay cohomology”);

b) De Rham realization which is given by the cohomology H^•(X, d + dW∧ •) of the twisted de Rham complex (or, better, the hypercohomology of the correspond-ing complex in the Zariski topology on X);

c) critical realization which is given by the cohomology of X with the coefficients in the sheaf of vanishing cycles of W .

It is convenient to combine all those versions of COHA into the following defini-tion proposed in [108].

Definition 8.1. Cohomological Hall algebra of (Q, W ) (in realization H) is an