PREPRINT
國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University
www.math.ntu.edu.tw/ ~ mathlib/preprint/2012- 01.pdf
BPS States and the P = W Conjecture
W.-Y. Chuang, D.-E. Diaconescu, and G. PAN
February 9, 2012
arXiv:1202.2039v1 [hep-th] 9 Feb 2012
W.-Y. CHUANG1, D.-E. DIACONESCU2, G. PAN2
Abstract. A string theoretic framework is presented for the work of Hausel and Rodriguez-Vilegas as well as de Cataldo, Hausel and Migliorini on the cohomology of character varieties. The central element of this construction is an identification of the cohomology of the Hitchin moduli space with BPS states in a local Calabi-Yau threefold. This is a summary of several talks given during the Moduli Space Program 2011 at Isaac Newton Institute.
1. Introduction
Consider an M-theory compactification on a smooth projective Calabi-Yau three- fold Y . M2-branes wrapping holomorphic curves in Y yield supersymmetric BPS states in the five dimensional effective action. These particles are electrically charged under the low energy U (1) gauge fields. The lattice of electric charges is naturally identified with second homology lattice H2(Y, Z). Quantum states of massive particles in five dimensions also form multiplets of the little group SU (2)L× SU (2)R ⊂ Spin(4, 1), which is the stabilizer of the time direction in R5. The unitary irreducible representations of SU (2)L× SU (2)R may be labelled by pairs of half-integers (jL, jR) ∈ 12Z2
. The half-integers (jL/2, jR/2) are the left, respectively right moving spin quantum numbers. In conclusion, the space of five dimensional BPS states admits a direct sum decomposition
HBP S(Y ) ≃ M
β∈H2(Y,Z)
M
jL,jR∈12Z
HBP S(Y, β, jL, jR).
The refined Gopakumar-Vafa invariants are the BPS degeneracies N (Y, β, jL, jR) = dim HBP S(Y, β, jL, jR).
The unrefined invariants are BPS indices, N (Y, β, jL) = X
jR∈12Z
(−1)2jR+1(2jR+ 1)N (Y, β, jL, jR).
String theory arguments [12] imply that BPS states should be identified with cohomology classes of moduli spaces of stable pure dimension sheaves on Y . More specifically, let M(Y, β, n) be the moduli space of slope (semi)stable pure dimension one sheaves F on Y with numerical invariants
ch2(F ) = β, χ(F ) = n.
Suppose furthermore that (β, n) are primitive, such that there are no strictly semistable points. If M(Y, β, n) is smooth, the BPS states are in one-to-one cor- respondence with cohomology classes of the moduli space. This identification still holds [15] when M(Y, β, n) is a singular variety provided that singular cohomology
1
is replaced by intersection cohomology. In such cases, there is a geometric con- struction of the expected SL(2)L× SL(2)R action on the BPS Hilbert space if in addition there is a natural Hitchin map h : M(Y, β, n) → B to a smooth projective variety B. Then [15] the action follows from decomposition theorem [2], [6] and as well as the relative hard Lefschetz theorem [7]. In particular the action of the positive roots JL+, JR+ should is given by cup product with a relative ample class ωh, respectively the pull back of an ample class ωB on the base. One then obtains a decomposition of intersection cohomology of the form
IH∗(M(Y, β, n)) ≃ M
(jL,jR)∈Z2
R(jL, jR)⊕d(jL,jR)
where R(jL, jR) is the irreducible representation of SL(2)L× SL(2)Rwith highest weight (jL, jR). A priori the multiplicities d(jL, jR) should depend on n for a fixed curve class β. Since no such dependence is observed in the low energy theory, one is lead to further conjecture that the d(jL, jR) are in fact independent of n, as long as the numerical invariants (β, n) are primitive. Granting this additional conjecture, the refined BPS invariants are given by N (Y, β, jL, jR) = d(jL, jR).
In more general situations no rigorous mathematical construction of a BPS co- homology theory is known. There is however a rigorous construction of unrefined GV invariants via stable pairs [30, 31] which will be briefly reviewed shortly. It is worth noting that the BPS cohomology theory would have to detect the scheme structure and the obstruction theory of the moduli space as is the case in [30, 31].
Concrete examples where the moduli space M(Y, β, n) is smooth are usually encountered in local models, in which case Y is a noncompact threefold.
Example 1.1. LetS be a smooth Fano surface and Y the total space of the canon- ical bundle KS. Then any semistable pure dimension one sheafF must be scheme theoretically supported on the zero section. Therefore there is a natural Hitchin map to a linear system onS. For primitive numerical invariants (β, n), the moduli space is smooth and the Hitchin map is projective.
Example 1.2. LetX be a smooth projective curve and D an effective divisor on X, possibly trivial. LetY be the total space of the rank two bundle OX(−D) ⊕ KX(D).
Note that H2(Y ) ≃ Z is generated by the class σ of the 0 section. Let M(Y, d, n) be the moduli space of stable pure dimension sheaves F on Y with compact support and numerical invariants
ch2(F ) = dσ, χ(F ) = n.
Let Xre(X) be the moduli space of rank r ≥ 1, degree e ∈ Z stable Hitchin pairs on X. Then it is easy to prove the following statements.
a) If D = 0 and (d, n) = 1, there is an isomorphism M(Y, d, n) ≃ Hdn+d(g−1)(X) × C.
b) If D 6= 0 and (d, n) = 1, there is an isomorphism M(Y, d, n) ≃ Hn+d(g−1)d (X).
As mentioned above unrefined GV numbers can be defined via stable pair or Donaldson-Thomas invariants. From a string theoretic point of view, this has been explained in [8] using IIA/M-theory duality. Let ZDT(Y, q, Q) be the (unrefined) reduced Donaldson-Thomas theory of Y defined in [24], or, equivalently, the stable
pair theory of Y defined in [30]. In a IIA compactification on Y , ZDT(Y, q, Q) is the generating function for the degeneracies of BPS states corresponding to bound states of one D6-brane and arbitrary D2-D0 brane configurations on Y . According to [12, 8], M-theory/IIA duality yields an alternative expression for this generating function in terms of the five dimensional BPS indices N (Y, β, jL). Then
(1.1) ZDT(Y, q, Q) = exp (FGV(Y, q, Q)) , where
(1.2)
FGV(Y, q, Q) =X
k≥1
X
β∈H2(Y ),β6=0
X
jL∈12Z
Qkβ
k (−1)2jLN (Y, β, jL)q−2kjL+ · · · + q2kjL (qk/2− q−k/2)2 . Relation (1.1) can be either inferred from [12] relying on the GW/DT correspon- dence conjectured in [24], or directly derived on physical grounds from Type IIA/M- theory duality [8].
According to [18], a similar relation is expected to hold between refined stable pair invariants and the GV numbers N (Y, β, jL, jR). As explained in [9] refined sta- ble pair invariants are obtained as a specialization of the virtual motivic invariants of Kontsevich and Soibelman [21]. Then one expects [18] a relation of the form (1.3) ZDT,Y(q, Q, y) = exp (FGV,Y(q, Q, y)) ,
where
(1.4)
FGV,Y(q, Q, y) =X
k≥1
X
β∈H2(Y ),β6=0
X
jL,jR∈12Z
Qkβ
k (−1)2jL+2jRN (Y, β, jL, jR)
q−k(q−2kjL + · · · + q2kjL)(y−2kjR+ · · · + y2kjR) (1 − (qy)−k)(1 − (qy−1)−k) . The expression (1.4) was written in [18] in different variables, (q−1y, q−1y−1).
The main goal of this note is to point out that the refined GV expansion (1.3) for a local curve geometry is related via a simple change of variables to the Hausel- Rodriguez-Villegas formula for character varieties. There a few conjectural steps involved in this identification. First, it relies on a explicit conjectural formula for the refined stable pair theory of a local curve derived in section (3) from geometric engineering and instanton sums. In fact, it is expected that a rigorous construc- tion of motivic stable pair theory of local curves should be possible following the program of Kontsevich and Soibelman [21]. A conjectural motivic formula general- izing equation (3.4) has been recently written down by Mozgovoy [27]. Second, as explained in detail in section (4), the refined GV invariants of the local curve are in fact perverse Betti numbers of the Hitchin moduli space. Therefore, the conver- sion of the HRV formula into a refined GV expansion relies on the identification between the weight filtration on the cohomology of character varieties and the per- verse filtration on the cohomology of the Hitchin system conjectured by de Cataldo, Hausel and Migliorini [5]. This will be referred to as the P = W conjecture. From a physicist’s perspective, the connection found here provides a natural explanation as well as strong evidence for this conjecture. Finally, note that further evidence for all the claims of the present paper comes from the recent rigorous results of [27, 25, 26]. In [27] it is rigorously proven that the refined theory of the local curve implies the HRV conjecture for the Poincar´e polynomial of the Hitchin system via
motivic wallcrossing while [25, 26] prove expansion formulas analogous to (1.3) for families of irreducible reduced plane curves.
Acknowledgements. We are very grateful to Tamas Hausel and Fernando Rodriguez- Villegas for illuminating discussions on their work. We would also like to thank Jim Bryan, Ugo Bruzzo, Ron Donagi, Oscar Garcia-Prada, Lothar G¨ottsche, Jochen Heinloth, Dominic Joyce, Ludmil Katzarkov, Bumsig Kim, Melissa Liu, Davesh Maulik, Greg Moore, Sergey Mozgovoy, Kentaro Nagao, Alexei Oblomkov, Rahul Pandharipande, Tony Pantev, Vivek Shende, Artan Sheshmani, Alexander Schmitt, Jacopo Stoppa, Balazs Szendroi, Andras Szenes, Michael Thaddeus, Richard Thomas, and Zhiwei Yun for very helpful conversations. D.-E.D. would like to thank the or- ganizers of the Moduli Space Program 2011 at Isaac Newton Institute for the partial support during completion of this work, as well as a very stimulating mathemat- ical environment. The work of D.-E.D. was also supported in part by NSF grant PHY-0854757-2009.
2. Hausel-Rodriguez-Villegas formula and P = W
Let X be a smooth projective curve over C of genus g ≥ 1, and p ∈ X an arbitrary closed point. Let γp ∈ π1(X \ {p}) be the natural generator associated to p. For any coprime integers r ∈ Z≥1, e ∈ Z, the character variety Cre(X) is the moduli space of representations
φ : π1(X \ {p}) → GL(r, C), φ(γp) = e2iπe/rIr
modulo conjugation. Cre(X) is a smooth quasi-projective variety, and its rational cohomology H∗(Cre(X)) carries a mixed Hodge structure
(2.1) W0k ⊂ · · · Wik ⊂ · · · ⊂ W2kk = Hk(Crn(X)).
According to [13], W2ik = W2i+1k for all i = 0, . . . , 2k, hence one can define the virtual Poincar´e polynomial
(2.2) W (Cre(X), z, t) =X
i,k
dim(Wik/Wi−1k )tkzi/2.
Moreove the virtual Poincar´e polynomial is independent on e for fixed r, therefore it will be denoted below by Wr(z, t). Obviously Wr(1, t) is the usual Poincar´e poly- nomial. The opposite specialization, Pr(X, z, 1) is identified with the E-polynomial with compact support via Poincar´e duality
Hck(Crn(X)) × H2d−k(Crn(X)) → C.
Using number theoretic considerations Hausel-Rodriguez-Villegas [13] derive a con- jectural formula for the Poincar´e polynomials Wr(z, t) as follows.
2.1. Hausel-Rodriguez-Villegas formula. The conjecture formulated in [13] ex- presses the generating function
FHRV(z, t, T ) = X
r,k≥1
Br(zk, tk)Wr(zk, tk)Tkr k ,
Br(z, t) = (zt2)(1−g)r(r−1)
(1 − z)(1 − zt2), as
(2.3) FHRV(z, t, T ) = ln ZHRV(z, t, T )
where ZHRV(z, t, T ) is a sum of rational functions associated to Young diagrams.
Given a Young diagram µ as shown below
• ✲
❄ a l
let µi be the length of the i-th row, |µ| the total number of boxes of µ, and µtthe transpose of µ. For any box ✷ = (i, j) ∈ µ let
a(✷) = µi− j, l(✷) = µtj− i, h(✷) = a(✷) + l(✷) + 1, be the arm, leg, respectively hook length. Then
ZHRV(z, t, T ) =X
µ
Hµg(z, t)T|µ|
where
Hµg(z, t) = Y
✷∈µ
(zt2)l(✷)(2−2g)(1 − zh(✷)t2l(✷)+1)2g (1 − zh(✷)t2l(✷)+2)(1 − zh(✷)t2l(✷)).
The main observation in this note is that equation (2.3) can be identified with the expansion of the refined Donaldson-Thomas series of a certain Calabi-Yau threefold in terms of numbers of BPS states.
2.2. Hitchin system and P = W . Let Her(X) be the moduli space of stable Higgs bundles (E, Φ) on X, where Φ is a Higgs field with coefficients in KX. For coprime (r, e) this is a smooth quasi-projective variety equipped with a projective Hitchin map
h : Her(X) → B to the affine variety
B = ⊕ri=1H0(KX⊗i).
The decomposition of the direct image Rh∗Q into perverse sheaves yields [6, 5] a perverse filtration
0 = P0k⊂ P1k⊂ · · · ⊂ Pkk = Hk(Her(X))
on cohomology. It is well known that Cre(X) and Her(X) are identical as smooth real manifolds. More precisely the complex structures are related by a hyper-K¨ahler rotation. Therefore there is a natural identification H∗(Cre(X)) = H∗(Her(X)).
Then it is conjectured in [5] that the two filtrations Wjk, Pjk coincide, W2jk = Pjk
for all k, j. This is proven in [5] for Hitchin systems of rank r = 2.
For future reference note that an h-relatively ample class ω yields an hard Lef- schetz isomorphism [7]
ωl: GrPd−lHk ∼−→Grd+lP Hk+2l.
This is known under the name of relative hard Lefschetz theorem. In particular, one then obtains a splitting of the perverse filtration.
Note that granting the P = W conjecture, equation (2.3) yields explicit formulas for the perverse Poincar´e polynomial of the Hitchin moduli space. In particular, by specialization to z = 1 it determines the Poincar´e polynomial of the Hitchin moduli space of any rank r ≥ 1.
3. Refined stable pair invariants of local curves
Let Y be the total space of the rank two bundle OX(−D) ⊕ KX(D) where D is an effective divisor of degree p ≥ 0 on X as in Example (1.2). Note that H2(Y ) ≃ Z is generated by the class σ of the zero section. Following [30], stable pairs on Y are two term complexes P = (OY s
−→F ) where F is a pure dimension one sheaf and s a generically surjective section. Since Y is noncompact, in the present case, it will be also required that F have compact support, which must be necessarily a finite cover of X. The numerical invariants of F will be
ch2(F ) = dσ, χ(F ) = n.
Then according to [30], there is a quasi-projective fine moduli space P(Y, d, n) of pairs of type (d, n) equipped with a symmetric perfect obstruction theory. The moduli space also carries a torus action induced by the C×action on Y which scales OX(−D), KX(D) with weights −1, 1. Virtual numbers of pairs can be defined by equivariant virtual integration. According to [1], the resulting invariants coincide with the Behrend Euler numbers of the moduli spaces,
P (d, n) = χB(P(Y, d, n)).
Let
ZP T(Y, q, Q) = 1 +X
d≥1
X
n∈Z
P (d, n)Qdqn.
Applying the motivic Donaldson-Thomas formalism of Kontsevich and Soibelman, one obtains a refinement Pref(d, n, y) of stable pair invariants modulo foundational issues. The Pref(d, n, y) are Laurent polynomials of the formal variable y with integral coefficients. In a string theory compactification on Y these coefficients are numbers of D6-D2-D0 bound states with given four dimensional spin quantum number. The resulting generating series will be denoted by ZP Tref(Y, q, Q, y).
3.1. TQFT formalism. A TQFT formalism for unrefined Donaldson-Thomas the- ory of a local curve has been developed in [29], in parallel with a similar construction [3] in Gromov-Witten theory. Very briefly, the final result is that the generating series of local invariants is obtained by gluing vertices corresponding to a pair of pants decomposition of the Riemann surface X. Each such vertex is a rational func- tion Pµi(q) labelled by three partitions µi, i = 1, 2, 3 corresponding to the three boundary components. In the equivariant Calabi-Yau case a nontrivial result is obtained only for identical partitions, µi= µ, i = 1, 2, 3, in which case
Pµ(q) = Y
✷∈µ
(qh(✷)/2− q−h(✷)/2).
Then the generating function is given by (3.1) ZDT(Y, q, Q) =X
µ
(−1)p|µ|q−(g−1−p)κ(µ)(Pµ(q))2g−2Q|µ|
where
κ(µ) =X
✷∈µ
(i(✷) − j(✷)).
3.2. Refined invariants from instanton sums. Although the refined stable pair invariants are not rigorously constructed for higher genus local curves, string duality leads to an explicit conjectural formula for the series ZP Tref(Y, q, Q, y). This follows using geometric engineering [19] of supersymmetric five dimensional gauge theories.
In the present case, the threefold Y yields a U (1) gauge theory with g adjoint hypermultiplets. By analogy with previous toric models [22, 10, 28, 16, 17, 11, 14, 20, 23, 18], it is expected that the refined stable pair invariants of invariants of Y will be completely determined by the instanton partition function of this theory [28]. The instanton partition function of U (1) five dimensional gauge theory with g adjoint hypermultiplets is constructed in terms of quivariant K-theoretic invariants of the Hilbert scheme of points in C2as follows [4].
Let Hilbd(C2) denote the Hilbert scheme of length d ≥ 1 zero dimensional sub- schemes of C2. It is smooth, quasi-projective and carries a natural tautological vector bundle Vd whose fiber at a point [Z] is the space of global sections H0(OZ).
For each d ≥ 1 let
Ed = T∗Hilbd(C2)⊕d⊗ det(Vd)−1.
Note that Ed carries a natural equivariant structure with respect to the C×× C× induced by the scaling action on C2. Let q1, q2 be the characters of the basic one dimensional representations of C××C×. Then the equivariant K-theoretic partition function is defined by
(3.2) Zinst(q1, q2, eQ, ey) =X
k≥0
χey(T∗Hilbk(C2)⊕g⊗ det(Vd)1−g−p) eQk
where
χye(Ed) =X
i,j
(−ey )j(−1)ich Hi(∧jEd)
is the equivariant χy˜-genus of Ed. A fixed point theorem gives an explicit formula for Zinst(q1, q2, eQ, ey) as a sum over partitions.
Zinst(q1, q2, eQ, ey ) =X
µ
Y
✷∈µ
(q1−l(✷)q2−a(✷))g−1+p (1 − eyq−l(✷)1 qa(✷)+12 )g(1 − eyq1l(✷)+1q−a(✷)2 )g
(1 − q−l(✷)1 qa(✷)+12 )(1 − q1l(✷)+1q−a(✷)2 ) Qe|µ|
The resulting conjectural expression for the refined stable pair partition function is then [4]
(3.3) ZP Tref(Y, q, Q, y) = Zinst(q−1y, qy, (−1)g−1y2−gQ, y−1).
A straightforward computation shows that (3.4) ZP Tref(Y, q, Q, y) =X
µ
Ωµg,p(q, y)Q|µ|
where
Ωµg(q, y) = (−1)p|µ| Y
✷∈µ
ql(✷)−a(✷)y−(l(✷)+a(✷))p
(qy−1)(2l(✷)+1)(g−1)
(1 − q−h(✷)yl(✷)−a(✷))2g
(1 − q−h(✷)yl(✷)−a(✷)−1)(1 − q−h(✷)yl(✷)−a(✷)+1)
.
The change of variables in (3.3) does not have a conceptual derivation. This con- jecture is supported by extensive numerical computations involving wallcrossing for refined invariants in [4]. Further supporting evidence for the formula (3.3) is obtained by comparison with the unrefined TQFT formula (3.1) for local curves.
Specializing the right hand side of (3.3) at y = 1, one obtains ZP Tref(Y, q, Q, 1) =X
µ
Q|µ|Y
✷∈µ
(−1)p|µ|q(g−1+p)(l(✷)−a(✷))(qh(✷)/2− q−h(✷)/2)2g−2. Agreement with (3.1) follows from the identity
X
✷∈µ
(l(✷) − a(✷)) =X
✷∈µ
(j(✷) − i(✷)) = −κ(µ).
Finally, note that the expression (3.3) with p = 0 is related to the left hand side of the HRV formula by
(3.5) ZHRV(z, t, T ) = ZP Tref(Y, (zt)−1, (zt2)g−1T, t).
4. HRV formula as a refined GV expansion
This section spells out in detail the construction of refined GV invariants of a local curve geometry as in Example (1.2) with p = deg(D) = 0 in terms of the perverse filtration on the cohomology of the Hitchin moduli space. Using the conjectural formula (3.3), it will be shown that equation (1.3) yields the HRV formula by a monomial change of variables. As observed in Example (1.2), the moduli space of slope stable pure dimension one sheaves F on Y with compact support and numerical invariants
ch2(F ) = rσ, χ(F ) = n
is isomorphic to C × Hn+r(g−1)r (X) provided that (r, n) = 1. Therefore, following the general arguments in the introduction, one should be able to define refined GV invariants using the decomposition theorem for the Hitchin map h : Hre(X) → B, e = n + r(g − 1). However, since the base of the Hitchin fibration is a linear space, there will not exist an SL(2)L× SL(2)R action on cohomology as required by M- theory. In this situation one can only define an SL(2)L× C×R-action where C×R can be thought of as a Cartan subgroup of SL(2)R. This action can be explicitly described in terms of the perverse sheaf filtration constructed in [5, Sect. 1.3].
Note that a relative ample class ωh for the Hitchin map yields an hard Lefschetz isomorphism [7]
ωl: GrPd−lHk(Her(X))−→Gr∼ d+lP Hk+2l(Her(X)).
This yields a splitting
Hk(HX(r, e)) ≃M
p
Hpk
of the perverse filtration, where Hpk≃ M
i+2j=p
Qi,j;k, Qi,j;k= ωhjQi,0;k−2j.
Let Qi,j =L
k≥0Qi,j;k. By construction, for fixed 0 ≤ i ≤ d, there is an isomor- phism
Md−i j=0
Qi,j≃ R⊕dim(Q(d−i)/2i,0)
where RjL is the irreducible representation of SL(2)L with spin jL ∈ 12Z. The generator JL+ is represented by cup-product with ωh, and the Qi,jis an eigenspace of the Cartan generator JL3 with eigenvalue j − (d − i)/2. Note that cup-product with ωhpreserves the grading k −d−2j therefore one can define an extra C×-action on H∗(Hre(X)) which scales Qi,j;k with weight d + 2j − k. This torus action will be denoted by C×R× H∗(Her(X)) → H∗(Her(X)). Note also that
d + 2j − k ≥ −d since j ≥ 0 and k ≤ −2d.
In conclusion, in the present local curve geometry the SL(2)L×SL(2)Raction on the cohomology of the moduli space of D2-D0 branes is replaced by an SL(2)L×C×R action. This is is certainly puzzling from a physical perspective since the BPS states are expected to form five-dimensional spin multiplets. The absence of a manifest SL(2)R symmetry of the local BPS spectrum is due to noncompactness of the moduli space. This is simply a symptom of the fact that there is no well defined physical decoupling limit associated to a local higher genus curve as considered here in M-theory. In principle, in order to obtain a physically sensible theory, one would have to construct a Calabi-Yau threefold Y containing a curve X with infinitesimal neighborhood isomorphic to Y so that the moduli space MY(r[X], n) is compact and there is an embedding H∗(MY(r, n)) ⊂ H∗(MY(r[X], n)). The cohomology classes in the complement would then provide the missing components of the five- dimensional spin multiplets. Such a construction seems to be very difficult, and it is not in fact needed for the purpose of the present paper.
Given the SL(2)L× C×R action described in the previous paragraph, one can define the following local version of the refined Gopakumar-Vafa expansion (1.4).
(4.1)
FGV,Y(q, Q, y) =X
k≥1
X
r≥1
Xd/2 jL=0
X
l≥−d
Qkr
k (−1)2jL+lNr((jL, l)) q−k(q−2kjL+ · · · + q2kjL)ykl
(1 − (qy)−k)(1 − (qy−1)−k). where
Nr(jL, l) = dim(Qd−2jL,0;d+l).
Making the same change of variables as in equation (3.5) yields
(4.2) FGV,Y((zt)−1, (zt2)g−1T, t) =X
k≥1
X
r≥1
Tkr
k Br(zk, tk)Pr(zk, tk)
where Br(z, t) is defined above equation (2.3), and Pr(z, t) =
Xd j=0
X
l≥0
(−1)j+lNr((j − d)/2, l − d) tl(1 + · · · + (zt)2j).
Now it is clear that the change of variables
(q, Q, y) = ((zt)−1, (zt2)g−1T, t)
identifies the HRV formula (2.3) with the refined GV expansion (1.3) for a local curve provided that
(4.3) Pr(z, t) = Wr(z, t).
However, given the cohomological definition of the refined GV invariants Nr(jL, l), relation (4.2) follows from the P = W conjecture of [5]. This provides a string theoretic explanation as well as strong evidence for this conjecture.
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1 Department of Mathematics, National Taiwan University, Taipei, Taiwan
2 NHETC, Rutgers University, Piscataway, NJ 08854-0849 USA
