Bubbling Calabi-Yau geometry from matrix models
aa r X i v : . [ h e p - t h ] F e b EFI-07-32NSF-KITP-07-192
Bubbling Calabi-Yau geometryfrom matrix models
Nick Halmagyi
Enrico Fermi Institute, University of ChicagoChicago, IL 60637, USA andTakuya Okuda
Kavli Institute for Theoretical Physics, University of CaliforniaSanta Barbara, CA 93106, USA
Abstract
We study bubbling geometry in topological string theory. Specifically, we analyse Chern-Simons theory on both the 3-sphere and lens spaces in the presence of a Wilson loop ofan arbitrary representation. For each three manifold, we formulate a multi-matrix modelwhose partition function is the Wilson loop vev and compute the spectral curve. Thisspectral curve is closely related to the Calabi-Yau threefold which is the gravitationaldual of the Wilson loop. Namely, it is the reduction to two dimensions of the mirror tothe Calabi-Yau. For lens spaces the dual geometries are new. We comment on a similarmatrix model relevant for Wilson loops in AdS/CFT. ontents S from a matrix model 4 S . . . . . . . . . . . . . . . . . . . . 42.2 Physical derivation of the matrix model . . . . . . . . . . . . . . . . . . . 52.3 Algebraic derivation of the matrix model . . . . . . . . . . . . . . . . . . 82.4 Spectral curve as the bubbling geometry dual to a Wilson loop in S . . 92.5 Eigenvalue distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 A Summary of Young tableau data 21B Area of the annulus diagrams 22C Alternative matrix models for a Wilson loop in S C.1 Physical derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23C.2 Solving (C.65) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25C.3 Solving (C.66) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
D An improved matrix model for N = 4 Yang-Mills 27
A useful aspect of duality between a gauge theory and a gravitational system is theemergence of spacetime through dynamics of gauge theory. Deeper understanding ofemergent geometry should help us find new formulations of string theory and quantumgravity that may be used to address fundamental questions in physics.In gauge/gravity duality, the vacuum state corresponds to a certain background space-time, and inserted operators to excitations. The fields of gauge theory backreact sig-nificantly to the insertion of some operators. The corresponding gravitational dual is a1ew geometry that shares the asymptotics with the original background. A bubble ofnew cycles supported by flux appears, and the new spacetime is thus called the bubblinggeometry . The bubbling phenomenon was originally found for local operators [1], andwas generalized to Wilson loops [2, 3, 4] in AdS/CFT. It is useful to introduce a ma-trix model which captures the dynamics of all the relevant fields that respond to theoperator insertion [7, 8]. One is able to visualize the backreaction in terms of eigenvaluedistributions, which in turn encode the bubbling geometry on the gravity side.The current work studies the topological string version of bubbling phenomena [5],which naturally extend the Gopakumar-Vafa gauge/gravity duality [6]. More specificallywe consider U ( N ) Chern-Simons theory on S or lens space L ( p,
1) = S / Z p with Wilsonloop insertions. The Wilson loop operator is defined as W R ≡ Tr R e H A (1.1)where A is the gauge field and is integrated along the unknot. For S / Z p we take theunknot that generates the fundamental group. The trace is evaluated in an arbitraryrepresentation R of U ( N ). Throughout the paper the symbol R also denotes the corre-sponding Young tableau, and we parametrize it as in Figure 1. Each edge length be it n I or k I , will correspond to the size of a new cycle in the bubbling geometry. R PSfrag replacements n k n k m − n m k m n m +1 Figure 1: The Young tableau R , shown rotated and inverted, is specified by the lengths n I and k I of the edges. Equivalently, n I and k I denote the lengths of the black andwhite regions that are obtained by vertically projecting down the edges in R onto thehorizontal line. n m +1 is defined by P m +1 I =1 n I = N .Building on the earlier work [9, 10], we formulate a matrix model whose partition func-tion is the vev of the Wilson loop in S or S / Z p . We then study the eigenvalue dynamicsin the large N limit and derive the spectral curve. For S the spectral curve is preciselythe mirror of the bubbling toric Calabi-Yau geometry identified as the gravitational dualof the Wilson loop in [5]. The topology of this threefold depends on the data encoded inthe Young tableau R : its toric web diagram is shown in Figure 2(a)2or the lens spaces S / Z p , the backreaction of the fields to the Wilson loop leads toadditional classical vacua, and the path-integral splits into sectors corresponding to thedifferent vacua. Because the matrix model we formulate computes the Wilson loop vevin each sector, we propose that for given N, p , and R , a single Wilson loop insertion isdual to a sum over bubbling geometries. Each term in the sum is the toric Calabi-Yauthat is mirror to the spectral curve which we derive. The summed geometries have thesame toric data shown in Figure 2(b) except different values of K¨ahler moduli. As inthe S case, the topology of the geometry depends on the Young tableau data.(a) (b)Figure 2: (a) The toric web diagram for the bubbling Calabi-Yau dual to the Wilson loop W R in S . It has 2 m + 1 copies of P . (b) The web diagram for the bubbling Calabi-Yaudual to W R in lens space S / Z p with p = 3. The diagram is a chain of m + 1 basic units.The paper is organized as follows. Section 2 focuses on the S case. In subsection 2.1we present the matrix model for a Wilson loop in S . In subsection 2.2 we derive thematrix model from physical arguments. Specifically we present it as an open string fieldtheory of a D-brane configuration that realizes the Wilson loop. Then we algebraicallyderive the matrix model in subsection 2.3. In subsection 2.4, we solve the matrix modelin the large N limit and derive the spectral curve, which is the mirror of the bubblingCalabi-Yau found in [5].Section 3 deals with lens space S / Z p , and is structured in parallel with section 2.For each vacuum of the gauge theory with Wilson loop insertion, we derive the spectralcurve. We propose that the mirror toric Calabi-Yau is the bubbling geometry dual tothe Wilson loop.Appendix A summarizes the notation regarding the Young tableau data. In appendixC we study alternative matrix models that compute the Wilson loop vev. The models arethe direct analog of the matrix models for N = 4 Yang-Mills considered in [11]. AppendixD is targeted at readers interested in AdS/CFT. We use the algebraic techniques insubsection 2.3 to formulate a matrix model, whose partition function is the vev of thesupersymmetric circular Wilson loop in N = 4 Yang-Mills. In this formulation it is veryeasy to derive the eigenvalue distributions for the Wilson loop found in [12, 13, 14, 11]. To take the limit p → SL (2 , Z ) transformation. Bubbling Calabi-Yau for S from a matrix model S The realization that the open topological A-model can be reduced to a matrix modelfirst appeared in Marino’s work [9], and a B-model version of this idea was subsequentlyderived by Dijkgraaf and Vafa [15]. Both derivations are of course mirror to each otheras was demonstrated for certain examples in the nice work [10]. We are interested herein the A-model, which is of course equivalent to Chern-Simons theory [16], possibly withinstanton corrections [16, 17, 18].Marino’s observation for Chern-Simons theory on S with the gauge group G was thatthe partition function is Z = Z d H ue − gs Tr u = Z N ! N Y i =1 du i Y i PSfrag replacements n m +1 n n m n PSfrag replacements n m +1 n n m n (a) (b) (c)Figure 3: (a) The web diagram for the deformed conifold. α and β degenerate along thehorizontal and vertical lines respectively. The dashed line represents S that N D-braneswrap. The other dashed line ending on the vertical solid line represents a non-compactcycle L = R × S that P non-compact D-branes wrap. (b) P non-compact branes aredistributed along the horizontal line where α degenerates.First, the above brane configuration is equivalent to another system that has a new setof non-compact D-branes, distributed along the locus where α degenerates [25]. The newsystem has only N − P (= n m +1 ) D-branes wrapping the S . As we review in AppendixB, a stack of n I non-compact branes sits at distance a I = g s ( L I − L m +1 ) away from the S for I = 1 , . . . , m . See Figure 3(b).Second, by considering the new ambient geometry of Figure 3(c) with more complex Recall that P is the number of rows in R . z z = w, z z = ( µ − w ) m Y I =1 (1 − w/µ I ) , (2.8)the non-compact branes can be compactified without changing the physics. This isa legitimate maneuver since it reduces to the deformed conifold (2.7) by making thecomplex structure moduli µ I infinite and A-model depends only on K¨ahler moduli. Theresult is the D-brane system from which we can derive the matrix model (2.5).We now have a daisy chain of Chern-Simons theories all of them on an S and there arethen annulus instantons which connect them [18]. The representation R of the Wilsonloop determines all the necessary data, in particular the I -th Chern-Simons theory hasgauge group U ( n I ), I = 1 , . . . , m + 1. We get annulus instantons by integrating out themassive bifundamental open strings [23]. Since the mass of the string between the I -thand the J -th spheres is a I − a J , the interactions generated from such annulus instantonsare summarized as h W R i ∼ Z m +1 Y I =1 [ DA I ] e iS CS ( A I ) Y I 10f ( X , X , . . . , X m +1 ) invariant under the process. The symmetric polynomial is regularon all of the cuts, and the only singularities are at Z = ∞ . Let us now recall the definitionof the j -th elementary symmetric polynomials E j : E j ( x , . . . , x n ) = X i < ···
I < J. (2.30)Because the last term in (2.15) becomes constant we have u ( I ) i = g s X j = i − e u ( I ) j − u ( I ) i + g s m X J =1 k J − I X J =1 n J + m +1 X J = I +1 n J ! . (2.31)We expect that when g s n I is large, the eigenvalues of u ( I ) spread over a large region,allowing us to approximate the function 1 / (1 − e x ) in (2.31) by a step function. If weorder the eigenvalues so that u ( I ) i < u ( I ) j for any i < j , it follows that u ( I ) i = 2 g s i + g s m X J =1 k J − I X J =1 n J + m +1 X J = I +1 n J ! , i = 1 , . . . , n I . (2.32)Along the I -th cut that has width 2 g s n I , the eigenvalues of u ( I ) are distributed uniformly.The I -th and I + 1 cuts are distance g s k I apart from each other. We can thus justifythe approximations above when g s n I and g s k I are all large. See Figure 5. As discussedabove, this sheet is connected to other m + 1 sheets through the m + 1 cuts as shown inFigure 4(a). Since u is the holonomy along α , its eigenvalue distribution is different from the distribution (B.63)of holonomy H β A . In particular the eigenvalues are quantized in unit of 2 g s . It should be possible tophysically explain (2.32) using the fact that the matrix model captures the Wilson loop in a non-canonicalframing [10]. m + 1 cuts on the cylinder parametrizedby z . A simple generalization of the topological A-model on T ∗ S is the orbifold X p ≡ T ∗ ( S / Z p )[10]. The particular orbifold action is such that S / Z p is the lens space L ( p, | z | + | z | = 1 (3.33)for complex variables z and z , together with identification( z , z ) ∼ ( e πi/p z , e − πi/p z ) . (3.34)We study the Wilson loop W R = Tr R P e H A (3.35)along a circle that is the generator of the fundamental group. We assume that the circleis the unknot.The U ( N ) Chern-Simons theory on L ( p, 1) has many vacua. Since the equation ofmotion is solved by a flat connection, the vacua are in one-to-one correspondence withthe N -dimensional representations of π ( S / Z p ) = Z p . The group Z p is abelian, so anysuch representation is a sum of one-dimensional ones. A one-dimensional representationis specified by an integer a = 1 , . . . , p . Thus a vacuum is specified by a partition of N : N = N + N + . . . + N p . (3.36)Here N a is the number of times the a -th irrep appears. The contribution of this vacuumto the partition function is given by Z p = Z N Y i =1 du i Y i We now derive the matrix model from a D-brane configuration that realizes the Wilsonloop in a lens space. 15et us recall that X p is a Z p orbifold of the deformed conifold given by (2.7). Theorbifold action is generated by( z , z , z , z ) → ( e − πi/p z , e πi/p z , e πi/p z , e − πi/p z ) , (3.42)and the Z p action on the S given by z = z ∗ , z = − z ∗ (so | z | + | z | = µ ) defines thelens space L ( p, 1) = S / Z p . Since the Z p only acts on the phases, X p is still a fibrationof T × R over R . Let us redefine α to be the 1-cycle corresponding to the generatorof the fundamental group, and β the 1-cycle given by the 2 π phase rotation of z . Weuse the axes of the two cylinders (given by z z = const ., z z = const . ) and the Re( w )direction as the base R . The cycle β degenerates at w = µ and so does β ′ ≡ − pα + β at w = 0. The cycle α never degenerates.PSfrag replacements β − pα + β PSfrag replacements n n n m n m +1 PSfrag replacements n n n m n m +1 (a) (b) (c)Figure 6: (a) The cycle β degenerates along the vertical line while − pα + β degeneratesalong the other line. If a linear combination qα + rβ degenerates, it does so along a linein the ( q, r ) direction. (b) P non-compact branes are distributed along the line where − pα + β degenerates. (c) There are m + 1 copies of S / Z p .We engineer U ( N ) Chern-Simons theory by wrapping N D-branes on the S / Z p . Toinsert a Wilson loop along the knot α , we consider P D-branes that wrap the non-compactcycle L = R × S in which β is contractible. See Figure 6(a). The boundary condition h R | on the P branes picks out the Wilson loop insertion in representation R , as explainedin [5]. The boundary condition induces holonomy I β = pα + β ′ A = diag (cid:18) g s (cid:18) R i − i + 12 ( P + N + 1) (cid:19)(cid:19) Pi =1 (3.43)along the contractible cycle β = pα + β ′ . By fibering the T over a semi-infinite lineending on the locus where β ′ degenerates, we obtain a 3-manifold in which β is non-contractible. We can consider a configuration of D-branes wrapping this 3-manifold. Is16he configuration equivalent to the one we started with, as in the S case? We assumeit is, and we will see evidence below. The basic nontrivial cycle in the new 3-manifold is α , and the holonomy along it is given by Z α A = 1 p diag (cid:18) g s (cid:18) R i − i + 12 ( P + N + 1) (cid:19)(cid:19) Pi =1 (3.44)because β ′ is contractible. See Figure 6(b).As in the S case, it is natural to split the P non-compact branes into m stacks with the I -th stack containing n I branes. We can now replace X p by the Z p orbifold of the large N dual geometry given by the equations (2.8). This is possible because (2.8) are invariantunder the orbifold action. The non-compact branes are now replaced by compact oneswrapping copies of lens space S / Z p . Thus we reach the desired system of D-branes,whose world-volume theory is m + 1 copies of Chern-Simons theory on lens space S / Z p ,interacting via Ooguri-Vafa operators. The system is shown in Figure 6(c).To write down the matrix model, we need to choose the vacuum of the theory. Wehave a U ( n I ) Chern-Simons theory on the I -th lens space. As reviewed in the previoussubsection, the theory has many vacua corresponding to the choice of a flat connection.Let us choose the vacuum specified by the partition n I = P a N Ia . Then according to theprescriptions in [10], the contribution to the Wilson loop vev from this vacuum is givenby h W R i ( N Ia ) S / Z p ∼ Z Y I,a,i d H u ( Ia ) Y I,a
2+ 12 g s X J = I,b,i coth u ( Ia ) i − u ( Jb ) j , (3.50)and so we first define several resolvents v ( Ia ) ( z ) = g s N Ia X i =1 e u ( Ia ) i e u ( Ia ) i − e z , v ( I ) ( z ) = p X a =1 v ( Ia ) ( z ) , v ( z ) = m +1 X I =1 v ( I ) ( z ) . (3.51)In terms of these we can write (3.50) as an equation on the ( Ia )-cut: pz + v ± ( z ) = − v ( I ) ∓ ( z ) + g s m X J = I k J + m +1 X J = I n J ! + 2 πia. (3.52)Following the same procedure as in section 2.4 we define some new variables X = Z p e v ,X I = A I e − v ( I ) , I = 1 . . . , m + 1 , (3.53)where Z = e z , A I = exp g s (cid:0)P mJ = I k J + P m +1 J = I n J (cid:1) . Then the spectral curve is againgiven by (recalling once more that ( Y, Z ) are C ∗ valued variables) f ( Y, Z ) = 0 (3.54)where f ( Y, X , . . . , X m +1 ) = m +2 Y j =0 ( Y − X j )= m +2 X j =0 ( − ) j Y m +2 − j E j ( X , . . . , X m +1 ) . (3.55) Despite identical nomenclature these variables are of course unrelated to those in section 2.4. E ( X , . . . , X m +1 ) = 1 ,E j ( X , . . . , X m +1 ) = p X i =0 a j,i Z i for j = 1 , . . . , m + 1 ,E m +2 ( X , . . . , X m +1 ) = A . . . A m +1 Z p . (3.56)Some coefficients are easily determined: a j, = X ≤ I < ···
N.H. and T.O. acknowledge the hospitality of the Simons Workshop at Stony Brook aswell as the summer program at the Aspen Center for Physics, where part of this work wasdone. T.O. also thanks the Michigan Center for Theoretical Physics and the Enrico FermiInstitute for hospitality. The research of T.O. is supported in part by the NSF grantsPHY-05-51164 and PHY-04-56556 and that of N.H. is supported by a Fermi-McCormickFellowship and NSF Grants PHY-0094328 and PHY-0401814. Appendix A Summary of Young tableau data The Young tableau R has n I rows of length K I such that K > K > . . . > K m >K m +1 ≡ 0. It also has k I columns of length N m − I +1 such that N > N > . . . > N m . Wealso define n m +1 ≡ N − P mI =1 n I , N ≡ N , and K m +1 ≡ 0. The integers n I , k I , N I , and K I satisfy the relations N I = m − I +1 X J =1 n J for I = 0 , , . . . , m, (A.57)and K I = m X J = I k J for I = 1 , , . . . , m, K m +1 = 0 . (A.58)See also Figure 1. We also denote by P the number of rows in R , so P = N .Other useful sets of quantities are L I = m X J = I k J − I − X J =1 n J + 12 m +1 X J = I +1 n J , (A.59) a I = g s (cid:18) K I − (cid:18) n + . . . + n I − + 12 n I (cid:19) + 12 ( P + N ) (cid:19) = g s ( L I − L m +1 ) , (A.60)21nd A I = exp g s m X J = I k J + m +1 X J = I n J ! , (A.61) B I = exp g s m X J = I k J + m +1 X J = I +1 n J ! . (A.62) B Area of the annulus diagrams Here we explain the identification a I = g s ( L I − L m +1 ) in subsection 2.2.The P non-compact D-branes with the boundary condition h R | has background holon-omy [25] (gauge equivalent to the position relative to the N − P compact branes on S ) I β A = diag (cid:18) g s ( R i − i + 12 ( P + N + 1)) (cid:19) Pi =1 (B.63)along the β cycle. When we split the P non-compact branes into m stacks, the averagevalue of the holonomy in the I -th stack is a I = g s (cid:18) K I − (cid:18) n + . . . + n I − + 12 n I (cid:19) + 12 ( P + N ) (cid:19) , I = 1 , . . . , m. (B.64)Since this is the distance from the S , it is natural to define a m +1 ≡ 0. The parameters a I ( I = 1 , . . . , m + 1) are then the positions of m + 1 copies of S in the new geometrygiven by (2.8). a I − a m +1 is the area of the annulus between the S and the I -th stackof non-compact branes. See Figure 3(b). Note that (B.64) can be written as a I = g s ( L I − L m +1 ) = g s L I +( I -independent). C Alternative matrix models for a Wilson loop in S S . The first is h W R i = Z d H u dU (1) dU (2) . . . dU ( m ) e − gs Tr( u ) (det U (1) ) k m (det U (2) ) k m − . . . (det U ( m ) ) k × − e u ⊗ U (1) − ) 1det(1 − U (1) ⊗ U (2) − ) . . . − U ( m − ⊗ U ( m ) − ) . (C.65)22ere U ( I ) is an N I × N I unitary matrix. The second is h W R i = Z d H udU (1) dU (2) . . . dU ( m ) e − gs Tr( u ) (det U (1) ) n (det U (2) ) n . . . (det U ( m ) ) n m × det(1 + e u ⊗ U (1) − ) 1det(1 − U (1) ⊗ U (2) − ) . . . − U ( m − ⊗ U ( m ) − ) , (C.66)for which U ( I ) is a K I × K I unitary matrix.These models are obtained from (2.4) by the same algebraic manipulations that led tosimilar multi-matrix models for N = 4 Yang-Mills in [11]. C.1 Physical derivation Here we give a physical derivation of the matrix model (C.65) from a D-brane configu-ration.We begin with the configuration of N compact and P = N non-compact D-branes(Figure 3(a)) that we discussed in subsection (2.2). On the non-compact branes weimpose the boundary condition h R | to picks out the Wilson loop W R from the annulusdiagrams between the branes.We now consider a new configuration that realizes the Wilson loop insertion. We mod-ify the geometry and introduce another locus on which β degenerates. By fibering the T over a line interval that connects the two loci where β degenerates, we get a cycle of topol-ogy S × S . We wrap N D-branes around this cycle while placing external fundamentalstrings in an appropriate configuration. This configuration of the fundamental stringsis that they insert the Wilson loop in the one-dimensional representation A ⊗ k m N [25]. Additionally we place N non-compact D-branes that end on the second locus where β shrinks. We choose the boundary condition to be h Q (2) | , where the Young tableau Q (2) isobtained from R by removing the first k m columns (Figure 9). The external strings andannulus diagrams from the non-compact branes insert to the S × S branes the Wilsonloop Tr A ⊗ kmN e H A Tr Q (2) P e H A = Tr R P e H A . (C.67)Since S × S is obtained by gluing two copies of solid torus by identifying their bound-aries, the path-integral there reduces to the inner product. Thus from the annulus di-agrams between the S and S × S , the path-integral picks out the combination thatinserts the Wilson loop W R into S . See Figure 8(a). A N is the rank N totally anti-symmetric representation of U ( N ) and is one-dimensional. | Q (2) i S × S N N N PSfrag replacements N N N N m (a) (b)Figure 8: (a) The P = N non-compact D-branes in Figure 3 (a) are compactified bymodifying the Calabi-Yau geometry without changing the topological string amplitudes.The state | R i specifying the boundary condition is implemented by placing externalstring world-sheets that insert the Wilson loop Tr R exp H A . (b) The geometry andthe configuration of D-branes and non-compact string world-sheets that give rise to themulti-matrix model (C.65). Each horizontal dashed line represents D-branes wrappinga Lagrangian submanifold of topology S × S . The cylinder ending on the I -th dashedhorizontal line represents fundamental strings in a configuration that inserts a Wilsonloop in the representation A ⊗ k m − I +1 N I for I = 1 , . . . , m .We can repeat this process (Figure 9) and show that the following configuration isequivalent to the Wilson loop insertion. The total geometry is given by the same equation(2.8) as in subsection 2.2, with one locus where α shrinks, and m + 1 parallel loci where β shrinks. N D-branes wrap the original S . We also wrap N I D-branes on the S × S between the I -th and ( I + 1)-th loci where β shrinks. Finally we place fundamentalstrings, along the I -th locus, that insert the Wilson loop in the representation A ⊗ k m − I N I +1 into the I -th S × S . See Figure 8(b).PSfrag replacements R ≡ Q (1) Q (2) Q ( m − Q ( m ) N N N m − N m Figure 9: A shrinking sequence of Young tableaux R ≡ Q (1) ⊃ Q (2) ⊃ . . . ⊃ Q ( m ) .Using the prescriptions in [10], we obtain the matrix model (C.65) from this D-braneconfiguration. There is no Gaussian factor for the Chern-Simons on S × S since the24ath-integral is simply the inner product. The external fundamental strings insert thedeterminant factors.It is also easy to extend the derivation to (C.66), this time using anti-branes insteadof D-branes. This explains the appearance of one determinant, rather than the inverseof it, in (C.66). C.2 Solving (C.65) Now that we know the physical origin of the matrix model (C.65), let us here solve it inthe large N limit. In terms of the eigenvalues, the matrix model can be written as h W R i ∝ Z N Y i =1 du i m Y I =1 N I Y i =1 du ( I ) i exp " − g s N X i =1 u i + m X I =1 k m − I +1 N I X i =1 u ( I ) i + X i 0. Theseequations state that the following quantities are permuted as one goes through a cut: X ≡ e v + z , X I ≡ A I e − v ( m − I +1) + v ( m − I +2) for I = 1 , . . . , m + 1 , (C.72) The quantities u ( I ) i and v ( I ) in this subsection are not to be confused with the quantities denotedby the same symbols in other parts of the paper. A I is the familiar quantity defined in (A.61). The asymptotic behavior of X I as z → ±∞ is the same as that of X I in subsection 2.4. The rest of the analysis then goesexactly in the same way, leading to the spectral curve (2.24). In particular X I here canbe identified with the quantity denoted by the same symbol there. It was found therethat X has m + 1 branch cuts, while X I with I = 1 , . . . , m + 1 shares with X just the I -th cut. One can now show using (C.72) that v ( m − I +1) ( z ) shares with v ( z ) the first I ofthese cuts, and thus the I -th cut consists of the eigenvalues of u , u (1) ,. . . , and u ( m − I +1) .How do we interpret the different kinds of eigenvalues that lie along the same cut? Webelieve that these eigenvalues form bound states due to attractive forces, as explained in[11] for a matrix model that describes a Wilson loop in N = 4 super Yang-Mills. The I -th cut has n I ( u - u (1) -. . . - u ( m − I +1) ) bound states. C.3 Solving (C.66) Let us also solve (C.66), which in terms of eigenvalues reads h W R i ∝ Z N Y i =1 du i m Y I =1 K I Y i =1 du ( I ) i exp " − g s N X i =1 u i + m X I =1 n I K I X i =1 u ( I ) i + X i 26n the u (1) -cuts, and (cid:0) v ( I − ( z ) − v ( I ) ( z ) (cid:1) ± = (cid:0) v ( I ) ( z ) − v ( I +1) ( z ) − g s ( n I + k I ) (cid:1) ∓ (C.77)on the u ( I ) -cuts for I = 2 , . . . , m , where we defined v ( m +1) ≡ 0. From these equations wesee that the following quantities are permuted as one goes through a cut: X ′ ≡ e v + z , X ′ ≡ A e − v − v (1) , X ′ I ≡ A I e v ( I − − v ( I ) for I = 2 , . . . , m + 1 , (C.78)where A I are defined in (A.61). The asymptotic behavior of X ′ I as z → + ∞ is that of X I ,but as z → −∞ , X ′ behaves like X m +1 , and X ′ I like X I − for I = 2 , . . . , m + 1. X ′ and X share the same asymptotics, hence so do E j ( X ′ , . . . , X ′ m +1 ) and E j ( X , . . . , X m +1 ).One concludes that the spectral curve of this model is the one found in subsection 2.4.What is the explanation of the difference between X ′ I and X I ? The functions X I areall holomorphic on the zero-th sheet except on the m + 1 cuts along the real axis. While( X ′ , X ′ , . . . , X ′ m +1 ) = ( X , X , . . . , X m +1 ) (C.79)for Re( z ) that is positively large enough, for negatively large Re( z ) we have( X ′ , X ′ , X ′ , . . . , X ′ m +1 ) = ( X , X m +1 , X , . . . , X m ) . (C.80)Thus X ′ I are not continuous, and we believe that the discontinuities arise due to the v ( I ) -cuts ( I = 1 , . . . , m ) that lie in the imaginary direction as in [11]. D An improved matrix model for N = 4 Yang-Mills This appendix is targeted at readers who are interested in Wilson loops in the AdS/CFTcontext.It is believed [8, 32] that the correlation functions of circular loops in N = 4 Yang-Mills are captured by the Gaussian matrix model. The precise correspondence states inparticular that (cid:28) Tr R P exp I ( A + θ i X i ds ) (cid:29) U ( N ) = 1 Z Z dM exp (cid:18) − Nλ Tr M (cid:19) Tr R e M . (D.81)The left-hand side is the normalized expectation value of the circular supersymmetricWilson loop in the Yang-Mills with gauge group U ( N ). 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