Matrix Model and Refined Wall-Crossing Formula
aa r X i v : . [ h e p - t h ] N ov Preprint typeset in JHEP style - HYPER VERSION
Matrix Model and Refined Wall-Crossing Formula
Haitao Liu
Department of Mathematics and Statistics,University of New Brunswick, Fredericton, Canada, E3B 5A3andTheoretical Physics Group, The Blackett Laboratory,Imperial College London, London, UKandDepartment of Applied Mathematics, Hebei University of Technology, Tianjin, ChinaEmail: [email protected]
Jie Yang
School of Mathematical Sciences, Capital Normal University, Beijing, ChinaandINFN, Sezione di Trieste, via Bonomea 265, Trieste, ItalyEmail: [email protected]
Jian Zhao
SISSA and INFN, Sezione di Trieste, via Bonomea 265, Trieste, ItalyEmail: [email protected]
Abstract:
In this paper, we show how to get matrix models corresponding to the refinedBPS states partition functions of C , resolved conifold and C / Z by inserting the identityoperator at a proper position in the fermionic expression of the refined BPS states partitionfunctions. Keywords:
Matrix model, Refined BPS states partition function, different chambers,Vertex operators in 2d free fermions. ontents
1. Introduction 12. Matrix model and wall-crossing formula 2 C C / Z
3. refined Matrix model and refined wall-crossing formula 7 C C / Z
4. Conclusion and discussion 12A. Refined matrix model for different chambers of the resolved conifold 13
A.1 The chamber (
R > , < n < B < n + 1) 13A.2 The chamber ( R > , n − < B < n ≤
0) 14
1. Introduction
Lately there has been progress in understanding the space of BPS states, H BP S , in typeIIA string compactifications on Calabi-Yau threefolds. In general, such compactificationsgive rise to the effective N = 2 theories in four dimensions. H BP S is a special subspaceof the full Hilbert space which is the one-particle representation of the d = 4 , N = 2supersymmetry algebra. It contains lots of information about the Calabi-Yau threefold X and can be viewed as a bridge connecting the black hole physics and topological strings[1]. Due to the existence of the universal hypermultiplets, H BP S ( γ ) has the followingdecomposition H BP S ( γ ) = ( , ; ) ⊗ H ′ BP S ( γ ) , (1.1)where γ is given by the generalized Mukai vector of the stable coherent sheaves correspond-ing to the D6/D4/D2/D0 branes γ = ch ( E ) q ˆ A ( X ) = p + P + Q + q ∈ H ⊕ H ⊕ H ⊕ H D D D D H ′ BP S depends on the asymptotic boundary conditionsin the four-dimensional spacetime, where the boundary conditions in IIA compactificationare the complexified K¨ahler moduli u = iJ + B of the Calabi-Yau threefold X [2]. Roughlyspeaking, H ′ BP S ( γ, u ) ∼ H ∗ ( M ( γ, u )), where M ( γ, u ) is the moduli space of stable coherentsheaves with the generalized Mukai vector γ under certain u -dependent stability condition[3]. The Spin(3) action on H BP S gives rise to the following refined index of H ′ BP S [4], afterfactorizing the contribution of the universal hypermultiplets,Ω ref ( γ, u, y ) := Tr H ′ BPS ( γ,u ) ( − y ) J ′ , (1.3)where J ′ is the reduced angular momentum [2]. Ω( γ, u, y ) is conjectured to be related tothe Poincar´e polynomial of the BPS states moduli space [4]. Like the unrefined case wemay define the refined BPS states partition function [4] by Z refBP S ( q, Q, y, u ) := X β ∈ H ( X ; Z ) n ∈ Z ( − q ) n Q β Ω ref ( γ β,n , u, y ) . (1.4)In [5], we have shown how to use the vertex operators in 2d free fermions and the crystalcorresponding to the Calabi-Yau threefold X to reproduce the wall-crossing formula ofthe refined BPS states partition function. In [5], we also conjecture that for the toric CYwithout any compact four-cycles we have the following formulas Z refBP S ( q , q , Q ) | chamber = Z reftop ( q , q , Q ) Z reftop ( q , q , Q − ) | chamber . (1.5)In this paper, we present a connection between the matrix model with the Z refBP S byemploying the method in [6] to insert the identity operator at a proper position to get a one-matrix model corresponding to the refined BPS states partition function. In section 2 wereview the work of [6]. In section 3 we show how to get the matrix model corresponding tothe refined BPS states partition function. In section 4 we give the summary and discussionon future research directions.
2. Matrix model and wall-crossing formula
In this section we will review the matrix model for three dimensional toric Calabi-Yaugeometry without any compact four-cycles arising from a triangulation of a strip [7, 6]. Letus denote the Euler characteristic of the Calabi-Yau as χ . Then the number of base P ofa toric CY 3-fold will be χ − A − ( x ):= χ Y i =1 Γ s i − x i − Y j =0 q j , (2.1)and A + ( x ):= χ Y i =1 Γ s i + xq i − Y j =0 q − j , (2.2)– 2 – igure 1: Toric diagram for Calabi-Yau threefold without compact four-cycles arises from a tri-angulation of a strip copied from [6]. where s i = 1 or −
1, and q is defined in terms of the eigenvalues q i of all color operators as q := χ − Y i =0 q i . (2.3)The convention of Γ matrices we will use isΓ s i =+1 ± ( x ) = Γ ± ( x ) , Γ s i = − ± ( x ) = Γ ′± ( x ) , (2.4)where the vertex operators Γ are derived from two dimensional free fermion theory [7, 8]and they satisfy the following commutation relation:Γ s + ( x )Γ s − ( y ) = (1 − s s xy ) − s s Γ s − ( y )Γ s + ( x ) . (2.5)In terms of free fermions, the BPS partition functions can be expressed as correlationfunctions of the vertex operators in 2d free fermions [7]. The ket and bra states of theNCDT chamber are generated by the creation and annihilation operators as follows: | Ω − i := ∞ Y r =0 A − ( q r ) | i , h Ω + | := h | ∞ Y l =0 A + ( q l ) . (2.6)Therefore the partition function for the NCDT chamber is Z = h Ω + | Ω − i . (2.7)The corresponding matrix model partition function is obtained by inserting the identityoperator I of Hermitian matrix models, namely Z matrix = h Ω + | I | Ω − i , (2.8)– 3 –here I = Z dU ∞ Y i =1 Γ ′− ( u i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y j =1 Γ ′ + ( u − j ) . (2.9)Here dU is the unitary measure for U ( ∞ ) and u i = e iφ i are the eigenvalues of U : dU = Y k dφ i Y i Toric diagram for C . According to [7], we may define A − ( x ) := Γ s − ( x ) and A + ( x ) := Γ s + ( xq ). We can splitthe integrand of the matrix model partition function into * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y l =0 A + (cid:16) q l (cid:17) ∞ Y i =1 Γ ′− ( u i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = ∞ Y i =1 ∞ Y l =0 (cid:16) su i q l +1 (cid:17) s , (2.11)and * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y j =1 Γ ′ + ( u − j ) ∞ Y r =0 A − ( q r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = ∞ Y j =1 ∞ Y r =0 (cid:16) su − j q r (cid:17) s . (2.12)Therefore the integrand of the matrix model is ∞ Y j =1 ∞ Y r =0 (cid:16) su − j q r (cid:17) s ∞ Y i =1 ∞ Y l =0 (cid:16) su i q l +1 (cid:17) s = Det ∞ Y r =0 (cid:0) U − q r (cid:1) (cid:0) U q r +1 (cid:1)! , s = 1Det − ∞ Y r =0 (cid:0) − U − q r (cid:1) (cid:0) − U q r +1 (cid:1)! , s = − . (2.13)where U ∈ U ( ∞ ) whose eigenvalues are u i .– 4 – igure 3: Toric diagram for the resolved conifold O ( − ⊕ O ( − −→ P . Figure 3 is the toric diagram for the resolved conifold O ( − ⊕ O ( − −→ P . Accordingto the discussion of the vertex on a strip [9], two C are connected by a ( − , − 1) curve,thus s = − s . Therefore we can choose s = s and s = − s . In the NCDT chamberwe denote q = q q , and Q = − q . Thus we can produce the BPS partition function bycounting the pyramid model [10, 7, 11].According to [7], in the NCDT chamber, the creation operator and the annihilationoperator are defined as follows: A − ( x ):= χ Y i =1 Γ s i − x i − Y j =0 q j = Γ s − ( xq ) Γ − s − ( xq ) , (2.14)and A + ( x ):= χ Y i =1 Γ s i + xq i − Y j =0 q − j = Γ s + ( xq ) Γ − s + ( x ) . (2.15)After inserting the matrix identity I we get two matrix elements * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y j =1 Γ ′ + (cid:16) u − j (cid:17) ∞ Y r =0 A − ( q r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = ∞ Y j =1 ∞ Y r =0 (cid:16) su − j q r q (cid:17) s (cid:16) − su − j q r +1 (cid:17) − s , (2.16)and * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y l =0 A + (cid:16) q l (cid:17) ∞ Y i =1 Γ ′− ( u i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = ∞ Y i =1 ∞ Y l =0 (cid:16) − su i q l (cid:17) − s (cid:16) su i q l q (cid:17) s . (2.17)Then the partition function of the matrix model in the NCDT chamber is Z dU Det s ∞ Y k =1 (cid:0) − sU − q k +1 Q − (cid:1) (cid:0) − sU q k Q (cid:1) (1 − sU − q k +1 ) (1 − sU q k ) ! . (2.18) Here s can be chosen to be either +1 or − – 5 –or chamber R > 0, 0 < n < B < n + 1 the wall crossing operator is defined in [7] as W p =1 ( x ) = Γ s − ( x ) ˆ Q Γ − s + ( x ) ˆ Q . (2.19)Therefore the partition function of the matrix model in the chamber ( R > , < n < B Toric diagram for the resolved C / Z . Figure 4 is the toric diagram of O P ( − , 0) which is the resolved C / Z . The vertexstrip in this case is different from the resolved conifold by s = s rather than s = − s .In the NCDT chamber, we define the creation operator A − ( x ):= χ Y i =1 Γ s i − x i − Y j =0 q j = Γ s − ( xq ) Γ s − ( xq ) , (2.22)– 6 –nd the annihilation operator A + ( x ):= χ Y i =1 Γ s i + xq i − Y j =0 q − j = Γ s + ( xq ) Γ s + ( x ) . (2.23)The insertion of the matrix identity I results in two matrix elements * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y j =1 Γ ′ + (cid:16) u − j (cid:17) ∞ Y r =0 A − ( q r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = ∞ Y j =1 ∞ Y r =0 (cid:16) su − j q r q (cid:17) s (cid:16) su − j q r +1 (cid:17) s , (2.24)and * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y l =0 A + (cid:16) q l (cid:17) ∞ Y i =1 Γ ′− ( u i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = ∞ Y i =1 ∞ Y l =0 (cid:16) su i q l (cid:17) s (cid:16) su i q l q (cid:17) s . (2.25)Then the partition function of the matrix model is Z dU Det s ∞ Y k =1 (cid:16) − sU − q k +1 Q − (cid:17) (cid:16) − sU q k Q (cid:17) (cid:16) sU − q k +1 (cid:17) (cid:16) sU q k (cid:17)! . (2.26)The wall crossing operator is defined in [7] as W ( x ) = Γ s + ( x ) ˆ Q Γ s − ( x ) ˆ Q . (2.27)Therefore the partition function of the matrix model for chamber R > 0, 0 < n < B < n +1is Z BP S | chamber n = Z dU * Ω + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N Y i =1 Γ ′− ( u i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N Y j =1 Γ ′ + (cid:16) u − j (cid:17) (cid:0) W (1) (cid:1) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ω − + . (2.28)Similarly we can get the matrix models in all the chambers as previous section. 3. refined Matrix model and refined wall-crossing formula In this section, we will use techniques introduced in previous section and the refined BPSstates partition functions proposed in [5] to obtain refined matrix model for several typicaltoric Calabi-Yau 3-folds. C In [5], we define the creation and annihilation operators as A − ( x ):= ˆ Q − δ , − Γ − ( x ) ˆ Q + δ , − = Γ − (cid:18) xq − δ (cid:19) ˆ Q , − , (3.1) A + ( x ):= ˆ Q − δ , + Γ + ( x ) ˆ Q + δ , + = ˆ Q , + Γ + (cid:18) xq + δ (cid:19) , (3.2)– 7 –nd states h Ω + | := h | A + (1) · · · A + (1) = h | ∞ Y i =1 Γ + ( q i − + δ ) , (3.3) | Ω − i := A − (1) · · · A − (1) | i = ∞ Y j =1 Γ − ( q j − − δ ) | i . (3.4)In the general convention, we can rewrite h Ω s + | and | Ω s − i as follows h Ω s + | := h | ∞ Y i =1 Γ s + ( q i − + δ ) = h | Q ∞ i =1 Γ + ( q i − + δ ) if s = 1 , h | Q ∞ i =1 Γ ′ + ( q i − + δ ) if s = − , (3.5) | Ω s − i := ∞ Y j =1 Γ s − ( q j − − δ ) | i = Q ∞ j =1 Γ − ( q j − − δ ) | i if s = 1 , Q ∞ j =1 Γ ′− ( q j − − δ ) | i if s = − . (3.6)Then the refined BPS states partition function is Z refBP S = h Ω s + | Ω s − i = M δ ( q , q ) (3.7)where the refined MacMahon function M δ ( q , q ) is defined by M δ ( q , q ) = ∞ Y i,j =1 (1 − q i − + δ q j − − δ ) − . (3.8)In order to get a matrix model we insert the identity operator I into the formula (3.7) Z refBP S = * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y i =1 Γ s + ( q i − i + δ ) I ∞ Y j =1 Γ s − ( q j − − δ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = M δ ( q , q ) . (3.9)Due to the formula (2.9) of I , the matrix elements are * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y j =1 Γ ′ + (cid:16) u − j (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ω s − + = * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y j =1 Γ ′ + (cid:16) u − j (cid:17) ∞ Y k =1 Γ s − (cid:18) q k − − δ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = ∞ Y k =1 ∞ Y j =1 (cid:18) su − j q k − − δ (cid:19) s , (3.10)and * Ω s + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y i =1 Γ ′− ( u i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y l =1 Γ s + (cid:18) q l − + δ (cid:19) ∞ Y i =1 Γ ′− ( u i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = ∞ Y l =1 ∞ Y i =1 (cid:18) su i q l − + δ (cid:19) s . (3.11)Therefore the matrix model integrand isDet s " ∞ Y k =1 (cid:18) sU − q k − − δ (cid:19) (cid:18) sU q k − + δ (cid:19) . (3.12)– 8 –inally we have Z refBP S = Z dU Det s " ∞ Y k =1 (cid:18) sU − q k − − δ (cid:19) (cid:18) sU q k − + δ (cid:19) , (3.13)where dU denotes the unitary measure for U ( ∞ ). It is given by dU = Y k dφ k Y i 0) is Z refBP S (cid:12)(cid:12)(cid:12) ( R> ,n − , ≤ n < B < n + 1) is Z refBP S (cid:12)(cid:12)(cid:12) ( R> , ≤ nn +1 (1 − q i q j − Q − ) − (3.27)Now we insert the identity operator I as follows: h | ∞ Y k =1 Γ s + h q k + δ q δ ( − Q ) i Γ s + h q k + δ q δ ( − Q ) − i · I ∞ Y k =1 Γ s − h q k − − δ q − δ ( − Q ) − i Γ s − h q k − − δ q − δ ( − Q ) i | i . (3.28)Then the matrix elements are * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y k =1 Γ s + h q k + δ q δ ( − Q ) i Γ s + h q k + δ q δ ( − Q ) − i ∞ Y i =1 Γ ′− ( u i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = ∞ Y i =1 ∞ Y k =1 (cid:16) su i ( − Q ) − q k + δ q δ (cid:17) s (cid:16) su i ( − Q ) q k + δ q δ (cid:17) s , (3.29)and * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Y j =1 Γ ′ + (cid:16) u − j (cid:17) ∞ Y k =1 Γ s − h q k − − δ q − δ ( − Q ) − i Γ s − h q k − − δ q − δ ( − Q ) i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = ∞ Y j =1 ∞ Y k =1 (cid:16) su − j ( − Q ) − q k − − δ q − δ (cid:17) s (cid:16) su − j ( − Q ) q k − − δ q − δ (cid:17) s . (3.30)– 11 –herefore the partition function of the matrix model for the ( R > , ≤ n < B < n + 1)chamber is Z refBP S (cid:12)(cid:12)(cid:12) ( R> , ≤ n
4. Conclusion and discussion In this paper, we use the free fermion version of refined BPS states partition functions toobtain their corresponding matrix models. But there still are some subtle problems hiddenin calculations.The first one is the choice of s , the “type” of the first vertex in a strip. As argued in [6],the final results should have an analytic continuation. Here we want to give some simplearguments on this analytic continuation. The key formula in this paper is the equation(2.5), from which we may see if we change one of the s ’s into its opposite one, then variableswill go from numerator to denominator or from denominator to numerator. This is similaras the analytic continuation on P , which has two patches and we can construct the analyticcontinuation from one patch to the other.Another subtle problem is the choice of δ in the refined MacMahon function M δ ( q , q ).In fact δ is an arbitrary constant set up by hand if we just want to get the generatingfunction of 3d partition function. While it is not clear to us whether the choice of δ in thepaper [12] is unique or not, how to get the refined MacMahon function appearing in [13]which gives the mathematical rigid refined BPS partition functions for the D0 branes, andwhether those different refined MacMahon functions in [13] and [5] are physically identical.The third subtle problem comes from the position of insertion of the identity opera-tor. Apparently, a different inserting position will give rise to a different action of matrixmodel. But just as in QFT, an identity operator means summation over complete set ofintermediate states, and inserting an identity operator at different positions just means weobserve different stages of interactions. Actually, if we want to get a multi-matrix modelrather than a one-matrix model we may insert more identity operators in the correspondingcorrelation function of refined BPS partition function.In [6] besides getting the matrix models corresponding to the BPS partition function,the authors also find the following interesting property of the BPS partition function:the matrix model for the BPS counting on the CY X is related to the topological stringpartition function for another CY Y , whose K¨ahler moduli space M ( Y ) contains two copiesof M ( X ), e.g. the partition function of matrix model corresponding to the BPS partitionfunction on the conifold will be related to the topological string partition function on theSPP geometry. It would be interesting to see if the matrix model proposed in this paper– 12 –s related to the refined topological string partition function on another CY. This work isunder consideration.Along the line of techniques discussed in the paper, we can also obtain refined matrixmodels for any strip like toric CY quickly, and what’s more, if we insert an identity in theequation (150) of [12] we can get a matrix model for the refined topological vertex. Sincethe refined topological vertex is the element to generate 5d instanton partition function, wecan obtain a matrix model for U ( N ) N = 2 instanton partition function. But the matrixmodels obtained by using this method have too many matrices and are very difficult todeal with.As in [6], in addition to inserting the identity operator, there is also another wayto obtain matrix model from BPS partition functions, namely the non-intersecting pathmethod introduced in [14, 15, 16, 6]. It would be interesting to see how to reproducethe matrix models presented in this paper by using non-intersecting path method andhow to use this method to get matrix models of refined topological vertex. We hope thatthe investigation on the relationship between the non-intersecting paths and the refinedtopological vertex will deepen our understanding on the refined topological vertex andrefined BPS partition function. This work is in progress. Acknowledgments H. Liu thanks the hospitality of the theoretical physics group of Imperial College whilepreparing this paper and also appreciates Professor Jack Gegenberg for his support. Theresearch of H. Liu is partially supported by the “Pam and John Little Overseas Scholar-ship” in the University of New Brunswick, Canada. J. Yang is supported by the ScientificResearch Foundation for The Excellent Youth Scholars of Capital Normal University, Bei-jing. J. Zhao thanks G. Bonelli and A. Tanzini for many useful discussions in related topicsand giving him much support in research. A. Refined matrix model for different chambers of the resolved conifold A.1 The chamber ( R > , < n < B < n + 1)The refined partition function for chamber ( R > , < n < B < n + 1) of the resolvedconifold is Z refBP S (cid:12)(cid:12)(cid:12) ( R> , 0) of the resolvedconifold is Z refBP S (cid:12)(cid:12)(cid:12) ( R> ,n − ,n − ,n − ,n −
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