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Large-N Duality, Lens Spaces and the Chern-Simons Matrix Model

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Large-N Duality, Lens Spaces and the Chern-Simons Matrix Model Published by Institute of Physics Publishing for SISSA/ISAS Received: January 10, 2004 Accepted: April 6, 2004 Large-N duality, Lens spaces and the Chern-Simons matrix model JHEP04(2004)014 Nick Halmagyi,ab Takuya Okudac and Vadim Yasnova aDepartment of Physics and Astronomy, University of Southern California Los Angeles, CA 90089, U.S.A. bKavli Institute for Theoretical Physics, University of California Santa Barbara, CA 93106, U.S.A. cCalifornia Institute of Technology Pasadena, CA, 91125, U.S.A. E-mail: [email protected], [email protected], [email protected] Abstract: We demonsrate that the spectral curve of the matrix model for Chern-Simons 3 theory on the Lens space S =Zp is precisely the Riemann surface which appears in the mirror to the blownup, orbifolded conifold. This provides the ¯rst check of the A-model ¤ 3 large-N duality for T (S =Zp), p > 2. Keywords: 1/N Expansion, Chern-Simons Theories, Matrix Models. °c SISSA/ISAS 2004 http://jhep.sissa.it/archive/papers/jhep042004014 /jhep042004014.pdf Contents 1. Introduction 1 2. The matrix model spectral curve 2 3. The orbifold of the resolved conifold 4 A. Toric variety described by a fan 7 1. Introduction JHEP04(2004)014 The conifold transition is an example of an open/closed string duality. In the topological A-model, this is a duality between the open A-model on T ¤(S3) (which is equivalent to 3 1 Chern-Simons (CS) theory on S [1]), and Kahler gravity on O¡1 + O¡1 ! P . This was originally studied by taking the partition function of large-N Chern-Simons (CS) theory on S3 expanded in a 't Hooft limit and presenting it ¯rst in the form of an open string theory [2] (i.e. an expansion in genus and holes) and then summing over the holes to get a closed string theory [3]. A worldsheet proof of this duality has since been provided [4]. ¤ 3 It is of some interest to extend this to a duality between T (S =Zp) and a Zp orbifold of the resolved conifold. Whilst an explicit form of the partition function for CS theory 3 on S =Zp is known [5], this form includes the summation over all vacua in the theory, yet for the purposes of string theory we want only the contribution from a single vacuum. For ¤ 3 perhaps these reasons it has not been possible to exhibit the large-N duality of T (S =Zp) ¤ 3 at the level of partition function however the worldsheet proof does generalize to T (S =Zp) and other geometries [6]. 3 In [7, 8] it was shown that CS theory on S =Zp has a matrix model description. In [8] it was also shown that Holomorphic Chern-Simons (HCS) theory reduced to P1's inside ¤ 3 the mirror (call it X) to T (S =Zp) has a matrix model description. Further, these matrix models are identical. For the case p = 2 the partition function of this matrix model was calculated perturbative ely and was shown to agree with the Kodaira-Spencer theory [9] predictions from the large-N dual geometry, providing solid evidence for the proposed duality. For CS theory on S3 the matrix model was solved to all genus using orthogonal polynomials in [10] and the orientifold of the conifold was studied in [11]. The manifold X is given by the blowup of u v e xy = Fp(e ; e ) ; (1.1) where u v v v+pu Fp(e ; e ) = (e ¡ 1)(e ¡ 1) (1.2) and by the general arguments of Dijkgraaf-Vafa theory [12, 13], the spectral curve of the corresponding matrix model should be a complex structure deformation of Fp = 0. In [14] { 1 { two of the current authors found an expression for the spectral curve of the matrix model 3 for CS theory on S =Zp. This involved ¯rst showing that although this matrix model looks similar to a p-matrix model, it does in fact have square root branch cuts, there- fore its spectral curve has only two sheets. This led to an explicit expression for the resolvent, depending on p ¡ 1 parameters di which in principle could be found perturba- tively by performing the A-cycle integrals. The spectral curve can be read o® from the resolvent and the di correspond to complex structure moduli. This will be reviewed in section 2. In section 3 we use toric geometry to construct a resolution of the Zp orbifold of the 1 resolved conifold. This is a particular Ap ¯bration over P . Then using the Hori-Vafa mirror map we can write down the mirror geometry and ¯nd that after a suitable co- ordinate rede¯nition, the non-trivial Riemann surface inside this threefold is precisely the ¤ 3 JHEP04(2004)014 spectral curve found in [14]. This explicitly identi¯es the large-N dual of T (S =Zp) for all p > 1. The matching of the geometries proves the equivalence of the leading order (in gs) free energy between the matrix model and the closed A-model on this particular ¯bration. To our knowledge this is the ¯rst check of this large-N duality for p > 2. 2. The matrix model spectral curve 3 The partition function of CS theory on the Lens space S =Zp [8] can be written as a matrix integral over p sets of eigenvalues, which we label by an index I 2 f0; ::; p ¡ 1g. The Ith set contains NI eigenvalues. The measure is a product of two factors, a self interaction term (¢1) and a term containing the interaction between di®erent sets of eigenvalues (¢2), 2 uI ¡ uI ¢ (u) = 2 sinh i j (2.1) 1 2 6 à à !! YI Yi=j I J IJ 2 ui ¡ uj + d ¢2(u) = 2 sinh ; (2.2) à à 2 !! IY<J Yi;j where dIJ = 2¼i(I ¡ J)=p. The potential has an overall factor of p compared to the S 3 case, 2 (uI ) V (u) = p i : (2.3) 2 XI;i In the above notations the CS partition function becomes p¡1 NI I 1 Z » dui ¢1(u)¢2(u)exp ¡ V (u) : (2.4) gs Z IY=0 Yi=1 µ ¶ We de¯ne individual resolvents for each set of the eigenvalues by z ¡ uI ! (z) = g coth i (2.5) I s 2 Xi µ ¶ { 2 { and the total resolvent, which we are most interested in is 2¼iI !(z) = ! z ¡ : (2.6) I p XI µ ¶ Anticipating taking the large-N limit we also introduce 't Hooft parameters SI = gsNI and S = I SI . The equation of motion for each eigenvalue is P I I I J IJ I ui ¡ uj ui ¡ uj + d pui = gs coth + gs coth : (2.7) à 2 ! à 2 ! Xj=6 i JX=6 I Xj From the large-N limit of this equation we can derive 1 2¼iI 2¼iI !2(z) ¡ p z ¡ ! z ¡ = f(z) ; (2.8) 2 p I p JHEP04(2004)014 XI µ ¶ µ ¶ where f(z) is a regular function 2¼iI z ¡ uI ¡ 2¼iI=p S2 f(z) = p g uI + ¡ z coth i + : (2.9) s i p 2 2 XI Xi µ ¶ µ ¶ Given the large-N limit of the equations of motion it is possible to ¯nd the total resolvent !(z). This method has been developed and checked in [14] which we will now review. We assume that the eigenvalues spread only along the real line. For general multi matrix models this is not true [15, 16, 17]. However, this assumption leads to the correct result for our case. Note that we do not make any assumption on the type of the cuts. In the total resolvent !(z), the individual resolvents come with relative shifts of the argument by 2¼iI=p. Therefore the cuts in the total resolvent are now separated by 2¼iI=p. For example, if !0(z) jumps at the point z, all other individual resolvents !I(z ¡ 2¼iI=p) with I 6= 0 are regular at this point. This means that on Ith cut the total resolvent jumps only due to the resolvent !I(z). From this it follows that 1 2¼iI 2¼iI ! z + + ! z + = pz; (I0th cut) (2.10) 2 + p ¡ p µ µ ¶ µ ¶¶ and so every cut is a square root. We label the contour around the Ith cut as AI . From (2.6) it is clear that lim !(z) = S ; (2.11) z!1 lim !~(z) = ¡S ; (2.12) z!¡1 where !~(z) is the value of the resolvent on the second sheet. From (2.5) we also have that 1 !(z)dz = 2¼iS : (2.13) 2 I IAI Since the integral over the A = I AI cycle is ¯xed by (2.12), there are only p ¡ 1 independent periods. Now we construct a regular function, P g(Z) = e!=2 + Zpe¡!=2 ; (2.14) { 3 { which has the limiting behavior, lim g(Z) = e¡S=2Zp (2.15) Z!1 lim g(Z) = e¡S=2 (2.16) Z!0 and is thus of the form, ¡S=2 p p¡1 g(Z) = e (Z + dp¡1Z + ¢ ¢ ¢ + d1Z + 1) : (2.17) The function g(Z) depends on p ¡ 1 moduli dn, which could be found by evaluating the period integrals (2.13). Since we have already ¯xed the integral over the cycle A = I AI by (2.16), there are only p ¡ 1 independent A-periods. P We can solve (2.14) for !(Z) to get JHEP04(2004)014 !(Z) 1 = log g(Z) ¡ g2(Z) ¡ 4Zp ; (2.18) 2 2 µ ¶ ³ p ´ the function under the square root is a polynomial of the degree 2p, it has 2p distinct roots that depend on only p ¡ 1 parameters.
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