PPrepared for submission to JHEP
On the BPS spectrum of 5d SU (2) super-Yang-Mills Pietro Longhi a a Institute for Theoretical Physics, ETH Zurich, 8093, Zurich, Switzerland
E-mail: [email protected]
Abstract:
We provide a closed-form expression for the motivic Kontsevich-Soibelman in-variant for M-theory in the background of the toric Calabi-Yau threefold K F . This encodesthe refined BPS spectrum of SU (2) 5d N = 1 Yang-Mills theory on S × R , correspondingto rank-zero Donaldson-Thomas invariants for K F , anywhere on the Coulomb branch.January 11, 2021 a r X i v : . [ h e p - t h ] J a n ontents Structures of BPS spectra reflect deep aspects of a gauge theory’s dynamics, a fact thatbecame prominent in theories with eight supercharges with the work of Seiberg and Wit-ten on 4d N = 2 Yang-Mills theory [1]. This paper provides an exact description of theBPS spectrum of the circle uplift of that theory, 5d N = 1 Yang Mills with gauge group SU (2). BPS states of five-dimensional theories with eight supercharges are also interest-ing from a mathematical viewpoint, because their BPS indices correspond to rank-zeroDonaldson-Thomas invariants of certain Calabi-Yau threefolds. The theory consideredhere corresponds to the canonical bundle of the Hirzebruch surface F [2–4]. Enumerativeinvariants of geometries with compact four-cycles are generally very difficult to compute.There are currently no examples where an explicit and exhaustive description of rank-zeroDonaldson-Thomas invariants is available, with the exception of vey recent developmentsfor local P based on scattering-diagram techniques [5].A given geometry typically has not one, but many different BPS spectra, correspondingto different regions of moduli space and being related to each other by wall-crossing [6–9]. A neat way to describe all possible spectra of a given geometry (or gauge theory) atonce, is to compute the wall-crossing invariant of Kontsevich and Soibelman also knownas (motivic) spectrum generator or BPS monodromy in physics. In this paper we will takethis approach, and derive an exact expression for the motivic wall-crossing invariant U for the canonical bundle K F . The standard definition of U involves knowing in advancethe full BPS spectrum at some point in K¨ahler moduli space. When this information isunavailable (as in our case) a different strategy is to find a choice of moduli for which centralcharges are maximally aligned [10]. This is the
Roman locus of [11], where the Kontsevich-Soibelman invariant may be obtained directly via spectral (or exponential [12]) networks[13]. Unfortunately, the conditions for existence of Roman loci are poorly understood evenfor 4d N = 2 theories, and much less is known about the 5d case. In this paper weintroduce a new approach to computing U , based on leveraging information about BPSstates at different points in moduli space studied in [14, 15]. Details of this will be explainedbelow. – 1 –ixing a choice of K¨ahler moduli determines a unique BPS spectrum, with BPS statesof charge γ ∈ Γ characterized by a central charge Z γ ∈ Hom(Γ , C ). Here Γ (cid:39) Z is a latticeof charges endowed with a skew-symmetric bilinear form (cid:104) · , · (cid:105) which corresponds to theDirac-Schwinger-Zwanziger pairing in the gauge theory. CPT symmetry implies that if thespectrum features a state with charge γ , there is a corresponding state with charge − γ .We henceforth focus on studying the half of the spectrum with − π/ ≤ Arg Z γ < π/ γ of all such states can be described as positive-integer linear combinationsof four basic charges γ , . . . , γ , whose definition is fixed by our choice of half-plane. Forthe central charge configurations considered in this note, the γ i correspond to the followingexceptional sheaves on K F , or fractional D-branes in IIA string theory γ γ γ γ O (0 , O (1 , O (1 , O (2 , D D f - D D D b - D f - D D b - D D b (resp. D f ) denotes a D2-brane wrapping the base (resp. fibre) P in F andthe overline denotes the anti-brane. The pairing is (cid:104) γ i , γ i +1 (cid:105) = − i ∈ Z / Z , see [15].The BPS spectrum is encoded by the BPS index Ω( γ ) ∈ Z , a supersymmetry-protectedquantity whose absolute value roughly coincides with the dimension of the BPS Hilbertspace. The Protected Spin Character Ω( γ ; y ) is a Laurent polynomial with integer coeffi-cients, that refines the BPS index by encoding information about the spin of BPS states.The relation between the two is Ω( γ ) = Ω( γ ; y = − U = (cid:37) (cid:89) k ≥ Φ( ˆ Y γ + k ( γ + γ ) )Φ( ˆ Y γ + k ( γ + γ ) ) × (cid:89) n ≥ Φ(( − y ) − ˆ Y n ( γ + γ + γ + γ ) ) − Φ(( − y ) ˆ Y n ( γ + γ + γ + γ ) ) − Φ(( − y ) ˆ Y n ( γ + γ + γ + γ ) ) − × (cid:89) k ≥ Φ(( − y ) − ˆ Y γ + γ + k ( γ + γ + γ + γ ) ) − Φ(( − y ) ˆ Y γ + γ + k ( γ + γ + γ + γ ) ) − × (cid:89) k ≥ Φ(( − y ) − ˆ Y γ + γ + k ( γ + γ + γ + γ ) ) − Φ(( − y ) ˆ Y γ + γ + k ( γ + γ + γ + γ ) ) − × (cid:38) (cid:89) k ≥ Φ( ˆ Y γ + k ( γ + γ ) )Φ( ˆ Y γ + k ( γ + γ ) ) (1.2)where Φ( ξ ) = (cid:81) s ≥ (1 + y s +1 ξ ) − is a variant of the quantum dilogarithm function, ˆ Y γ are quantum-torus variables obeying ˆ Y γ ˆ Y γ (cid:48) = y (cid:104) γ,γ (cid:48) (cid:105) ˆ Y γ + γ (cid:48) , and (cid:37) (respectively (cid:38) ) denotesincreasing (decreasing) values of k to the right. More precisely, | Ω | coincides with the dimension of the Hilbert space for the center-of-mass degrees offreedom, and only under the no-exotics assumption [16–18]. – 2 –n important clarification is now in order. Factorizations of U into products of quan-tum dilogarithms like (1.2) are typically associated with a certain BPS spectrum, whereeach factor Φ(( − y ) m ˆ Y γ ) corresponds to a BPS state of charge γ and spin m , and the or-dering of factors reflects the ordering of phases of Z γ ’s. However this is not the case here,at least not necessarily. In fact, we did not find a point in the moduli space of K F wherecentral charges have this configuration and where the spectrum is so simple. As a conse-quence, the derivation is not straightforward, and will be discussed below. More precisely,factors of lines 3 & 4 are derived directly for k = 0 ,
1. We provide an algorithm to computehigher k , and give strong evidence for the all-order expression, plus extensive checks.An exact expression for U allows to study the BPS spectrum anywhere in moduli space.For this purpose, one must factorize U into a product of terms Φ(( − y ) m ˆ Y γ ). Since variablesˆ Y γ do not commute, the factorization depends on the ordering, which corresponds to thatof Arg Z γ (decreasing from left to right), and is in turn fixed by a generic choice of moduli.Exponents of the factorization correspond to Laurent coefficients of the Protected SpinCharacters. We illustrate this in Section 3, by factorizing U at a point in moduli spacewith fiber-base symmetry, and obtaining the corresponding refined BPS spectrum. We now explain how (1.2) is derived. Suppose we fix a generic choice of moduli, awayfrom walls of marginal stability. Given any angular sector (cid:93) in the complex plane, wemay define U ( (cid:93) ) as the phase-ordered product of Φ(( − y ) m ˆ Y γ ) a m ( γ ) for any γ whose centralcharge has phase within that sector. Here a m ( γ ) ∈ Z are coefficients of the PSC, namelyΩ( γ ; y ) = (cid:80) m ∈ Z ( − y ) m a m ( γ ). We will split the half-plane − π/ ≤ Arg
Z < π/ R + , the sector with positive real an imaginary parts (cid:93) + corresponding to 0 < Arg
Z < π/ (cid:93) − corresponding to − π/ ≤ Arg
Z < U = U ( (cid:93) + ) · U ( R + ) · U ( (cid:93) − ) (2.1)The wall-crossing formula of Kontsevich and Soibelman asserts that U ( (cid:93) ) is invariantunder changes of stability conditions (i.e. central charges [19, 20]), as long as no BPSrays enter or exit the sector (cid:93) . By a BPS ray we mean any locus Z γ R + ⊂ C such thatΩ( γ ; y ) (cid:54) = 0. We will take advantage of this invariance property to compute the three factorsin (2.1) at different points in the moduli space of stability conditions: in particular we willcompute U ( (cid:93) ± ) for a certain configuration of central charges, and U ( R + ) for a differentone. All we need to ensure is that these configurations of central charges are connectedby a variation of central charges that never causes a BPS ray to cross the boundaries ofsectors (cid:93) ± . A path in the moduli space of stability conditions
We consider the mirror geometry of K F , described by a conic bundle over the algebraiccurve 1 − Q b ( x + x − ) + Q f ( y + y − ) = 0 in C ∗ x × C ∗ y . The four basic charges are identifiedwith homology cycles on this curve, see [15] for a detailed description. Central charges are– 3 –etermined by periods of the 1-form λ = (2 π ) − log y dxx , and can be evaluated numerically.For Q b = − , Q f = 2 one finds that π/ > Arg Z γ > Arg Z γ > Z γ = Arg Z γ = − π/
2, and in particular Z γ + γ , Z γ + γ ∈ R + . (2.2)The ray R + contains central charges of all states in the span of γ + γ and γ + γ . Notethat these are mutually local, i.e. (cid:104) γ + γ , γ + γ (cid:105) = 0 (the same holds for charges in theirspan), this ensures we are not on a wall of marginal stability. This is the starting point t = 0 of the path in the moduli space of central charge configurations˜ Z γ ( t ) = (1 − t ) Z γ + tZ γ ˜ Z γ ( t ) = Z γ ˜ Z γ ( t ) = (1 − t ) Z γ + tZ γ ˜ Z γ ( t ) = Z γ (2.3)The path ends at t = 1, where Z (cid:48) γ = Z (cid:48) γ ∈ (cid:93) + Z (cid:48) γ = Z (cid:48) γ ∈ (cid:93) − . (2.4)The path (2.3) has the crucial property that˜ Z γ ( t ) + ˜ Z γ ( t ) = (1 − t )( Z γ + Z γ ) + t ( Z γ + Z γ ) ∈ R + (2.5)is real and positive for all t ∈ [0 , t = 0 is realized at a point in the moduli space of the (mirror) geometry, and werefer to it as a physical stability condition. On the other hand, the configurations ˜ Z ( t ) for t > virtual stabilityconditions. Configurations analogous to t = 0 and t = 1 were considered in [15] and [14]. Computing U ( (cid:93) ± ) at t = 1The virtual stability condition for t = 1 is especially nice because the spectrum in (cid:93) ± hasa simple structure. This was analyzed in [14], whose approach we now review. For U ( R + )we will adopt a different method later, by working at t = 0. The BPS spectrum admits adescription in terms of the representation theory of a BPS quiver with four nodes [21], ourconventions are adapted to [15]. For the choice of half-plane considered in this note, thestructure of the quiver is cyclic, with two arrows from the i -th node to the i + 1-th node.Moreover each node is labeled by a charge: quite simply, node i is labeled by γ i . Tiltingthe choice of half-plane clockwise induces the following mutation sequence µ + = µ ◦ µ ◦ µ ◦ µ (2.6)respectively on nodes 1 , , ,
4. The central charges with largest phase are Z (cid:48) γ , Z (cid:48) γ , for thisreason µ ◦ µ come first. To see the rest of the sequence, recall that a mutation on thenode with label γ changes the labeling as follows [22, 23] γ → − γ γ (cid:48) → γ (cid:48) + [ (cid:104) γ (cid:48) , γ (cid:105) ] + γ (2.7) Numerical evaluation yields Z γ ≈ . . i , Z γ ≈ − . i , Z γ ≈ . . i , Z γ ≈ − . i . The reality condition (2.2) can also be verified by plotting the exponential network at ϑ = 0, where both saddles of γ + γ and of γ + γ appear, see Figure 1a. The structure of the quiver at each step may be recovered by recalling that (cid:104) γ, γ (cid:48) (cid:105) = m > m arrows oriented from the node with charge γ (cid:48) to the node with charge γ . – 4 –here [ x ] + = max(0 , x ). The sequence (2.6) produces the following labels at each stepquiver \ node 1 2 3 4 Q γ γ γ γ µ ◦ Q − γ γ + 2 γ γ γ µ ◦ µ ◦ Q − γ γ + 2 γ − γ γ + 2 γ µ ◦ µ ◦ µ ◦ Q γ + 3 γ − γ − γ − γ γ + 2 γ µ ◦ µ ◦ µ ◦ µ ◦ Q γ + 3 γ − γ − γ γ + 3 γ − γ − γ (2.8)where each time the mutation is on the node with highest Arg Z (cid:48) γ (clockwise tilting).In particular, the quiver µ ◦ µ ◦ µ ◦ µ ◦ Q has the same exact structure as Q , and thestability condition is also analogous. Thus tilting the half-plane further clockwise simplyiterates µ + , infinitely many times. The BPS rays encountered in this way, within sector (cid:93) + , correspond to charges γ + k ( γ + γ ) and γ + k ( γ + γ ) for k ≥
0. Since they appearas charges of nodes, they all have PSC Ω( γ ; y ) = 1 [22, 23]. Similar considerations applyto counter-clockwise rotations of the choice of half-plane. In this case one finds BPS stateswith charges γ + k ( γ + γ ) and γ + k ( γ + γ ), again with Ω( γ ; y ) = 1. Note that thecentral charges of these towers of BPS states both asymptote to R + , but from oppositesides. This leads to the conclusion that there is a single ‘accumulation’ ray along R + .Since the clockwise (resp. counter-clockwise) tilting of the half-plane covers the wholeangular sector (cid:93) + (resp. (cid:93) − ), we can write down U ( (cid:93) ± ) U ( (cid:93) + ) = (cid:37) (cid:89) k ≥ Φ( ˆ Y γ + k ( γ + γ ) )Φ( ˆ Y γ + k ( γ + γ ) ) U ( (cid:93) − ) = (cid:38) (cid:89) k ≥ Φ( ˆ Y γ + k ( γ + γ ) )Φ( ˆ Y γ + k ( γ + γ ) ) (2.9)One should worry that Arg Z (cid:48) γ + k ( γ + γ ) = Arg Z (cid:48) γ + k ( γ + γ ) may cause ordering ambiguitiesin the above formulae. But since (cid:104) γ + k ( γ + γ ) , γ + k (cid:48) ( γ + γ ) (cid:105) = 0 and (cid:104) γ + k ( γ + γ ) , γ + k (cid:48) ( γ + γ ) (cid:105) = 0 when k = k (cid:48) , the corresponding factors commute.As we move from t = 1 to the physical stability condition t = 0, the BPS rays in (2.9)begin to move within (cid:93) ± , however they never exit these sectors. This is clear because,e.g. Arg Z γ + k ( γ + γ ) ( t ) > Arg Z γ + γ ( t ) for all k ≥ Z γ ( t ) > Arg Z γ ( t )and as long as both have positive real part. Since Z γ + γ ( t ) ∈ R + which separates the twosectors (cid:93) ± , it is clear that the BPS rays are confined within each sector separately. As BPSrays move around within (cid:93) ± they may cross each other, and generate new BPS rays bywall-crossing. Any new rays generated in this way must lie in the cone of the two BPS raysthat generated them, ensuring that even these descendants (and their own descendants)must be confined within one of (cid:93) ± as well. Furthermore any BPS rays within R + at t = 1are confined there also for 0 ≤ t <
1, never crossing into (cid:93) ± , we provide a direct derivationin Appendix for | γ | ≤ .
9) obtained for the virtual stability condition at t = 1 must coincidewith the parts of the spectrum generator (2 .
1) for the physical stability conditions at t = 0.– 5 – omputing U ( R + ) at t = 0What is left out by the above analysis is to determine the part of U corresponding to theaccumulation ray. This can be actually obtained quite easily, by plotting the exponentialnetwork at Q b = − , Q f = 2 (corresponding to t = 0) for ϑ = 0, see Figure 1a.To determine the BPS spectrum encoded by the saddle one may use the machinery of[12, 15, 24, 25]. But in this case one can take a shortcut. Note that saddles are dividedinto two disconnected parts. Each set has the same topology as the exponential BPSgraph of O (0) ⊕ O ( − → P , shown in Figure 1b. Recall that (exponential) BPS graphsencode the whole BPS spectrum of a theory [10]. In the case of the half-geometry, thespectrum is known to consist of Ω( n D
0) = − n ≥
1, Ω( D k D
0) = − k ≥
0, andΩ( D k D
0) = − k ≥ U ( R + ) we simply have to identify D D f (cf. [15, Section 5]). Noting that γ + γ = D f and γ + γ = D D f , andtaking into account that Figure 1a contains two disconnected copies of the saddle in Figure1b, we arrive at the following BPS indicesΩ( n ( γ + γ + γ + γ )) = − n ≥ γ + γ + k ( γ + γ + γ + γ )) = − k ≥ γ + γ + k ( γ + γ + γ + γ )) = − k ≥ k > n > U includes also threshold states. To promote BPS indices to PSCs, we note that states with Ω = − γ ; y ) = y + y − . For the states with Ω = − γ + γ + γ + γ is the charge of a pure D n D K F wasrecently argued to be Ω( γ, y ) = y − (1 + y ) in [27]. Taking this into account completesthe description of the BPS states with real central charge, leading to U ( R + ) = (cid:89) n ≥ Φ(( − y ) − ˆ Y n ( γ + γ + γ + γ ) ) − Φ(( − y ) ˆ Y n ( γ + γ + γ + γ ) ) − Φ(( − y ) ˆ Y n ( γ + γ + γ + γ ) ) − × (cid:89) k ≥ Φ(( − y ) − ˆ Y γ + γ + k ( γ + γ + γ + γ ) ) − Φ(( − y ) ˆ Y γ + γ + k ( γ + γ + γ + γ ) ) − × (cid:89) k ≥ Φ(( − y ) − ˆ Y γ + γ + k ( γ + γ + γ + γ ) ) − Φ(( − y ) ˆ Y γ + γ + k ( γ + γ + γ + γ ) ) − (2.11)As discussed earlier, all charges appearing in this expression are mutually local, ensuringno ordering ambiguities. More precisely, this is the exponential network at ϑ = 0 for 1 + y + xy + Qy = 0 with Q = 6. It isequivalent (up to framing, which has no effect on BPS states) to the curve studied in [25, Equation (4.3)]. This interpretation was suggested to me by Michele del Zotto. See [26] for a derivation of the PSC based on the topology of the saddle. – 6 – a) (b)
Figure 1 : (a) is the exponential network of local F at ϑ = 0 for Q b = − , Q f = 2. (b) isthe exponential BPS graph for O (0) ⊕ O ( − → P in quadratic choice of framing. The mirror geometry of K F has a symmetry under exchange of fiber and base moduliwhenevever Q b = ± Q f . The point Q b = − , Q f = 1 was studied extensively in [15], wherecentral charges were evaluated to be Z γ ≈ . Z γ ≈ . i , Z γ ≈ . Z γ ≈− . i . It is tedious but straightforward to compute the corresponding factorization of U , see [10, Appendix E] for an algorithm. Eventually we obtain the following invariants γ Ω( γ ; y )( n, n, n, n ) y + 2 y + y − ( n > , , ,
1) 1(0 , , ,
0) 1(1 , , ,
0) 1(0 , , ,
0) 1(0 , , , y + y − (0 , , , y + y − (0 , , ,
2) 1(0 , , ,
1) 1(0 , , ,
0) 1(0 , , ,
0) 1(0 , , , y + 2 + y − (0 , , , y + y − (0 , , , y + y − (0 , , , y + y + y − + y − (0 , , ,
3) 1(0 , , ,
2) 1(0 , , , y + 2 y + 4 + 2 y − + y − (0 , , , y + 2 y + 4 + 2 y − + y − (0 , , ,
0) 1(0 , , ,
0) 1(1 , , , y + 2 + y − (0 , , ,
2) 1 γ Ω( γ ; y )(0 , , , y + y + y − + y − (0 , , , y + 2 y + 4 y + 4 y − + 2 y − + y − (0 , , , y + 2 y + 4 y + 4 y − + 2 y − + y − (0 , , , y + y + y − + y − (1 , , , y + y − (1 , , , y + y − (0 , , , y + 2 y + 4 y + 4 y − + 2 y − + y − (0 , , ,
4) 1(0 , , ,
3) 1(0 , , ,
4) 1(0 , , , y + 2 y + 4 y + 6 + 4 y − + 2 y − + y − (0 , , , y + y + 2 + y − + y − (0 , , , y + 2 y + 4 + 2 y − + y − (0 , , , y + 2 y + 4 y + 4 y + 5 + 4 y − + 4 y − + 2 y − + y − (0 , , , y + y + 2 + y − + y − (0 , , , y + 2 y + 4 + 2 y − + y − (0 , , , y + 2 y + 4 y + 6 + 4 y − + 2 y − + y − (0 , , ,
0) 1(0 , , ,
1) 1(0 , , ,
0) 1(1 , , , y + 2 + y − (1 , , , y + 3 y + 6 + 3 y − + y − (1 , , , y + 2 + y − (3.1)– 7 –here ( n , n , n , n ) is the shorthand for (cid:80) i =1 n i γ i . The full spectrum is infinite, this listincludes all states up to | γ | ≤
7. Upon specialization y → −
1, the spectrum (3.1) recoversthe unrefined spectrum obtained in [15] up to degree | γ | ≤
6, and predicts several newstates for | γ | > In this note we provided an exact expression for the motivic wall-crossing invariant ofKontsevich and Soibelman for the BPS spectrum of M-theory on K F . This operatorencodes the spectrum of BPS states for any generic choice of K¨ahler moduli. In the languageof gauge theory, this is the motivic spectrum generator for BPS monopole strings andinstanton particles of 5d N = 1 SU (2) Yang-Mills theory on S × R . From the viewpoint ofgeometry, it encodes the spectrum of rank-zero (generalized) Donaldson-Thomas invariantsfor K F .The derivation is based on data on BPS states at two different points in the modulispace of stability conditions, closely analogous to those studied in [15, Sec 5.3] and [14, Sec7.2]. Let us comment on how the expression U obtained here compares with these works.The main difference with [14, Sec 7.2] lies in the factor U ( R + ), where we find additionalinfinite towers of states Ω( γ + γ + kγ D ; y ) = Ω( γ + γ + kγ D ; y ) = y + y − for all k ≥
0, aswell as additional states Ω( nγ D ; y ) = y − (1+ y ) for all n ≥
1. Here γ D = γ + γ + γ + γ is the charge of a D γ + γ is the charge of D f and γ + γ the charge of D D f . A resolution of this apparent discrepancy should be that the authors of [14] onlystudied stable quiver representations, while our results imply that U should also includecontributions from threshold states such as D D Physically, these additional statescan be expected, since they correspond to Kaluza-Klein modes of an M P [28, 29]. The pure D nγ D ) = −
4, weadopted the motivic refinement Ω( nγ D ; y ) = y − (1 + y ) from the recent work [27]. Comparing with [15] we find direct agreement. We factorized U to compute the refinedBPS spectrum at a point with fiber-base symmetry, recovering and extending results from[15] in the limit of the spin fugacity y → −
1. At generic y , our results agree with predictionsfrom the Coulomb branch/Attractor Flow formulae of [31–35], computed with [36]. Byextension our results should also agree with computations of Vafa-Witten invariants basedon these formulae, recently carried out in [37], and confirming earlier predictions of [38] forthe spectrum of stable sheaves of arbitrary rank on Hirzebruch surfaces.Having an exact expression for the wall-crossing invariant, it would be very interestingto use it to study its relation to the 5d superconformal index [39]. Another interestingdirection would be to compare with computations of Vafa-Witten generating functionsbased on different techniques, such as [40–43]. The counting of D physical counting of [30]. The relation between the two has been discussed in [27]. – 8 – cknowledgements I would like to thank Sibasish Banerjee and Mauricio Romo for collaboration on relatedprojects, and Fabrizio del Monte, Michele del Zotto and Boris Pioline for correspondence.This work is supported by NCCR SwissMAP, funded by the Swiss National Science Foun-dation.
Appendix: Real BPS rays from Z symmetry Here we fill a gap left behind in the derivation above. A crucial property of the pathconnecting the physical stability condition t = 0 to the virtual one t = 1, is that no BPSrays must cross the boundaries of (cid:93) ± as one goes from t = 1 to t = 0. We already discussedthe behavior of BPS rays within (cid:93) ± , what remains to be addressed is the behavior of therays on R + .How may a positive-real BPS ray exit R + along the path? Since at t = 1 we have Z γ = Z γ = Z γ = Z γ , the BPS rays in R + are those with charge γ = ( n , n , n , n )such that n + n = n + n . There are ( m + 1) charges with | γ | = 2( n + n ) = 2 m , namely γ = ( m , m , m − m , m − m ) for 0 ≤ m , m ≤ m . We denote Γ the set of these charges.At t = 0 we only found states with n = n and n = n in R + , but principle theremay be BPS rays which violate this condition. Notice that this may include charges whichare not mutually local with those claimed in U ( R + ), implying the potential presence a wallof marginal stability (MS). This would not be a problem, since U is well-defined (in fact itis unchanged) even on MS walls.To keep this general possibility into account, we must find a way to study U ( R + )that would work even on a MS wall. A similar problem was studied in [10], where it wasnoticed that U must exhibit certain discrete symmetries. In the case at hand we expect a Z symmetry, generated by the following relabeling of charges in U expressed as a formalseries in ˆ Y γ σ c : γ → γ → γ → γ → γ . Since U is independent of the stability condition, and only depends on a choice of half-plane,it has the same Z symmetry of the BPS quiver. Another way to argue this symmetry, isto consider the opposite stability condition with Z = Z , Z = Z but Arg Z < Arg Z ,which would produce a spectrum with an identical structure, but precisely the relabeling σ c . Hence the spectrum generator would have the same form, with charges relabeledaccordingly. We also introduce σ d = σ c · σ c which obviously exchanges γ ↔ γ , γ ↔ γ .Let U = U ( (cid:93) + ) U U ( (cid:93) − ), where U is to be determined (and may differ in principlefrom U ( R + ) determined at t = 0). W.l.o.g. we introduce the formal series expansion U = (cid:80) γ ∈ Γ c γ ˆ Y γ where c γ are functions of y to be determined.Notice that σ d is a manifest symmetry of U ( (cid:93) ± ). Since the whole U is invariant, itmeans that U must also be invariant, thus implying c γ = c σ d ( γ ) . Taking this into account, In [10] this was the symmetry group of a BPS graph, which is dual to a quiver [11], making contactwith the discussion here. – 9 –e want to study the equations Eq γ : [ U − σ c ( U )] γ = 0for all γ = ( n , . . . , n ) with n i ≥
0, and where [ F ] γ denotes the coefficient of ˆ Y γ . Equationsfor | γ | = 2 m and | γ | = 2 m + 1 will only include coefficients c γ with | γ | , the system is highlyoverconstrained. For example, to fix all coefficients in U with | γ | = 2 one only needs toconsider the following subset of equations Eq γ of levels | γ | = 2 , γ + γ : c (0 , , , + 1 + y − y = c (0 , , , , Eq γ + γ : y − (cid:18) y c (0 , , , − y − y − c (0 , , , (cid:19) = 0leading to c (0 , , , = − y − y , c (0 , , , = 0. It is straightforward although tedious to proceedto higher orders. Taking into account equations Eq γ to | γ | = 9 we find U = 1 − y − y (cid:16) ˆ Y (1 , , , + ˆ Y (0 , , , (cid:17) + (cid:0) y + y (cid:1) y (1 − y ) (1 + y ) (cid:16) ˆ Y (0 , , , + ˆ Y (2 , , , (cid:17) + c (1 , , , ˆ Y (1 , , , − y (cid:0) y + y (cid:1) (1 − y ) (1 + y + y ) (cid:16) ˆ Y (3 , , , + ˆ Y (0 , , , (cid:17) − (cid:0) − y (cid:1) (cid:0) − y (cid:1) c (1 , , , − (cid:0) y − y (cid:1) y (1 − y ) (cid:16) ˆ Y (1 , , , + ˆ Y (2 , , , (cid:17) + y (cid:0) y + 5 y + 3 y + 2 y + y + y (cid:1) ( y − ( y + 1) (1 + y + 2 y + y + y ) (cid:16) ˆ Y (0 , , , + ˆ Y (4 , , , (cid:17) + y (cid:0) − y − y + y + y (cid:1) c (1 , , , + (cid:0) − y − y + y (cid:1) (cid:0) y (cid:1) (1 − y ) (1 + y ) (1 + y + y ) (cid:16) ˆ Y (1 , , , + ˆ Y (3 , , , (cid:17) + c (2 , , , ˆ Y (2 , , , + O ( ˆ Y | γ | =10 ) . This expression shows that U agrees with U ( R + ), fixing terms in the 3rd and 4th line of(1.2) up to order k = 1, excluding the presence of any other BPS rays on R + up to | γ | ≤ k states includes their appearance in Figure 1a,and the agreement of the spectrum (3.1) with the one of [15]. We have carried out furthertests of this agreement in the unrefined limit y → − | γ | = 12 finding agreementwith their predictions, in addition to new states (predictions of [15] are only exhaustiveup to | γ | = 6). The fact that terms c ( n,n,n,n ) cannot be fixed, is related to the fact thatfactors Φ(( − y ) k ˆ Y ( n,n,n,n ) ) commute with all other factors in U and don’t participate inwall-crossing, since ( n, n, n, n ) is in the kernel of the pairing matrix. To pin down thepure D nD
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