Two-Dimensional Gauge Theory and Matrix Model

We study a matrix model obtained by dimensionally reducing Chern-Simon theory on S^3. We find that the matrix integration is decomposed into sectors classified by the representation of SU(2). We show that the N-block sectors reproduce SU(N) Yang-Mills theory on S^2 as the matrix size goes to infinity.


Introduction
Matrix models have been proposed as non-perturbative formulation of superstring or Mtheory [1][2][3]. Since low energy physics predicted by string theory depends on topological aspects of compactification, it is relevant to investigate how they are incorporated in matrix models. The topological field theories have been developed to efficiently describe the topological aspects of field theories. It is, therefore, worthwhile to study realization of the topological field theories in matrix models.
Hinted by the work [4], the authors of [5] found the following classical relationships among Chern-Simons (CS) theory on S 3 , two-dimensional Yang-Mills (2d YM) on S 2 and a matrix model. The latter two theories are obtained by dimensionally reducing the first theory. The theory around each multiple monopole background of 2d YM is obtained by expanding the matrix model around a certain multiple fuzzy sphere background in the continuum limit (see also [6]). CS theory is obtained by applying an extension of compactification in matrix models developed in [4,6,7] to the theory around a multiple monopole background of 2d YM. Eventually, CS theory is obtained by expanding the matrix model around a certain multiple fuzzy sphere background and imposing the orbifolding condition. 2d YM is also viewed as BF theory with a mass term on S 2 . The matrix model takes the form of the superpotential for N = 1 * theory. The classical relationships between CS on S 3 and 2d YM on S 2 are generalized to those between CS theory on a U(1) bundle over a Riemann surface Σ g of genus g and 2d YM on Σ g .
In this Letter, we show that 2d YM on S 2 is obtained from the matrix model also at quantum level. We find that the matrix integration is decomposed into sectors classified by the representation of SU (2). We show that the N-block sectors reproduce the partition function of 2d SU(N) YM on S 2 .
It has been already shown in [8,9] that different types of matrix models give 2d YM on S 2 . Moreover, the authors of [9] have shown that the localization works also for the matrix model in the same way as it works for the continuum 2d YM. We hope to elucidate the relation of our work with [8,9] in the future.
This Letter is organized as follows. In section 2, we briefly review part of the results in [5], which are associated with the present work. In section 3, we reduce the pathintegral in the matrix model to the integral over the eigenvalues of a single matrix, which is decomposed into the sectors classified by the representation of SU (2). In section 4, we show that part of the above sectors reproduce 2d YM on S 2 . Section 5 is devoted to conclusion and outlook. In appendix, we summarize some useful properties of S 3 and S 2 .
2 Classical relationships among CS theory, 2d YM and a matrix model In this section, we briefly review only part of the results in [5] which are concerned with the present Letter. We start with CS theory on S 3 with the gauge group U(M): We expand the gauge field in terms of the right-invariant 1-form defined in (A.3) as Then, we rewrite (2.1) as where L i is the Killing vector dual to E i and defined in (A.7).
By dropping the derivative of the fiber direction y, we obtain a gauge theory on S 2 : where g 2 BF = 1/k 1 and L (0) i are the angular momentum operators on S 2 given in (A. 13) with q = 0. In order to see that (2.4) is BF theory with a mass term, we define L Then, it is easy to see that L (0)µ i and N i satisfy the following relations: where g µν and ǫ µν can be read off from (A.12). We expand X i as [6, 10] a µ and χ turn out to be the gauge field and a scalar filed on S 2 , respectively. By using the relations (2.6), we can show that (2.4) is equivalent to is the field strength. Indeed, the first term is the BF term and the second term is a mass term.
By integrating χ out in (2.8), we obtain 2d YM on S 2 : . By dropping all the derivatives in (2.4) and rescale X i as X i → µX i , we obtain N = 1 * matrix model: where 1/g 2 = 4π/g 2 BF . In the sense of the Dijkgraaf-Vafa theory [11], this matrix model is regarded as a mass deformed superpotential of N = 4 supersymmetric Yang-Mills theory (super-YM), which gives the so-called N = 1 * theory. We call the matrix model (2.10) the N = 1 * matrix model in this Letter.
Inversely, we can obtain the BF theory with the mass term from the matrix model as follows. The matrix model (2.10) possesses the following classical solution, where L [js] i (s = 1, · · · , N) are the spin j s representation of the SU(2) generators obeying k , and the relation N s=1 (2j s + 1) = M is satisfied. We label the blocks by s. We put 2j s + 1 = N 0 + n s with N 0 and n s integers and take the limit in which where A = 4π/µ 2 is the area of S 2 . Then, we can show classically [5] that the theory around (2.11) is equivalent to the theory around the following classical solution of (2.4), where q s = n s /2, and L (qs) i are the angular momentum operators in the presence of a monopole with the monopole charge q s , which are given in (A.13). This theory can also be viewed as the theory around the following classical solution of (2.8), where the upper sign is taken in the region 0 ≤ θ < π and the lower sign in the region 0 < θ ≤ π, andâ θ andâ ϕ represent the monopole configuration.

Exact integration of the partition function
In this section, we evaluate the partition function of (2.10). We reduce the path-integral in the matrix model to the integral over the eigenvalues of a single matrix. In (2.10), we redefine the matrices as and (3.1), we define the partition function of N = 1 * matrix model (2.10) by where we introduce the '−iǫ' term in the action to make the integral converge. Integral over Z and Z † leads to a one matrix model with respect to Φ [11][12][13] where [Φ, ·] represents an adjoint action. Furthermore, if we diagonalize Φ as Φ = diag(φ 1 , φ 2 , · · · , φ M ), the matrix integral reduces integrals over the eigenvalues φ i in the numerator of the integrand comes from the Vandermonde determinant owing to the diagonalization of Φ.
As a simple example, we consider the M = 2 case. In this case, (3.3) is explicitly written as In what follows, we frequently use the identity where P.V. stands for Cauchy's principal value of an integral. Applying (3.5) to (3.4) leads to We generalize the above calculation to the case of arbitrary M. We apply (3.5) to the factor in the integrand of (3.3), and obtain the sum of the terms, each of which includes some delta functions. It is easily seen that any term giving non-vanishing contribution must be proportional to where we have reordered and relabeled the eigenvalues of Φ, φ i (i = 1, · · · , M)), as 1 , · · · , φ with N s=1 (2j s + 1) = M, such that the form of (3.8) is obtained. φ degree of freedom in each block. We label the irreducible representation by r and denote the U(1) part in the s-th block by a s , putting a s ≡ φ (s) 2js+1 + j s . Then, we find that the contribution of (3.8) to (3.2) is where N r = 1 (♯ of blocks with the same length)! . (3.11) and the other factor in (3.10) is obtained from the following calculation: We further do some algebra for the exponent in (3.10): (3.14) Gathering all the above results, we eventually find that (3.3) results in where L [js] i is the spin j s representation of the SU(2) generators seen in (2.11). Thus the partition function of the N = 1 * matrix model is decomposed into the sectors classified by the irreducible representation of SU (2). Indeed, it is ensured by P.V. that the whole integral region of a s are decomposed into these sectors without overlap, which means that the full matrix integral over X 1 , X 2 , X 3 is decomposed into these sectors without overlap.

Relation to Continuum Field Theory
In this section, we reproduce 2d YM on S 2 from the N = 1 * matrix model in the large matrix size limit. As we will see, the number of the matrix blocks in the irreducible representation of SU (2) where we put d s = 2j s + 1 and λ is a Lagrange multiplier for the constraint N s=1 d s = M. This potential is minimized at a s = 0 and d s = M/N for ∀ s, that is, a configuration of almost equal size blocks is dominated.
Thus we now consider the fluctuation around the dominated configuration where M = NN 0 and N s=1 n s = 0. In the large matrix size limit, we take the limit (2.12) with fixed N, which reduces the N-block sectors to where a ′ s = a s − 1 N a, a = s a s and the integral over a has been performed. Irrelevant constants and divergences are absorbed into a renormalized constant C. In this limit, the poles in the integral measure have disappeared, then we have taken integral domains as whole space of integral variables a ′ s . By rescaling a ′ s by y s ≡ 2a ′ s and making an analytical continuation g 2 Y M → −ig 2 Y M , we finally obtain

Conclusion and discussion
In this Letter, we study the N = 1 * matrix model which is obtained by dimensionally reducing CS theory on S 3 . We decompose the matrix integral into the sectors classified 2 Note that (3.7) in [16] represents the partition function of U (N ) YM on S 2 . By applying the procedure in [16] to the partition function of SU (N ) YM in [18], it is easy to see that the corresponding expression of the partition function of SU (N ) YM takes the form (4.2).
by the representation of SU (2). We show that the N-blocks sectors reproduce 2d SU(N) YM on S 2 in the large matrix size limit.
We reproduced the partition function of 2d YM on S 2 from the N = 1 * matrix model.
It is relevant to investigate whether the correlation functions of the physical observables in 2d YM on S 2 can be reproduced from the matrix model. For instance, the vev of where the trace is taken over a representation R of the matrix Φ, is easily calculated in the matrix model . This kind of the observables should be interpreted as a Wilson loop-like operator in 2d YM.
Our result suggests that the localization also works for the N = 1 * matrix model as for 2d YM theory. It has been discussed in [9] by using a different matrix model. We need further investigation on the localization mechanism of N = 1 * matrix model and relationship of our work to [9].
We expect that CS theories on S 3 and the lens space S 3 /Z q are obtained from the N = 1 * matrix model also at quantum level as N = 4 super-YM on R × S 3 is obtained from the plane wave matrix model [19][20][21]. In this case, the operator Tr R e Φ in the matrix model should correspond to the Wilson loop operator in CS theory [5], and hopefully the knot invariant is derived from the matrix model.
The N = 1 * matrix model is also interesting from the point of view of 4d super-YM theory, since the large N limit of the matrix model describes the effective superpotential of N = 1 * theory which is mass deformed theory from N = 4 theory [11,13]. The different sectors of the SU(2) representations that we have investigated should be related to the different Higgs branches of the N = 1 * theory. The effective superpotential in the different Higgs branches can be investigated by using the direct integration of the matrix model partition function.
While we have extracted the N-block sectors 'by hand' in the present Letter, we may expect that the large N limit with the large N 0 limit realizes 2d large N YM on S 2 naturally, as the planar limit of N = 4 super-YM is realized in [19]. The large N limit seems relevant for the following reason. It has been already pointed out that the 1/N expansion of 2d YM describes the genus expansion of (non-critical) string theory [22]. One can deduce a world-sheet description of string theory from the partition function of 2d YM.
On the other hand, in this Letter, we have derived the partition function of 2d YM from the N = 1 * matrix model in the large matrix size limit. Matrix models are often regarded as regularization of (non-critical) string theory, giving world-sheet description in the large matrix size limit. Our investigation strongly suggests the relationship between the matrix models in the large matrix size limit, gauge theory in the large N limit and string theory and the N = 1 * matrix model is a good example to understand the relationship (see also [23,24]). Further investigation of the N = 1 * matrix model may shed lights on nonperturbative definition of string theory.
A S 3 and S 2 In this appendix, we summarize some useful facts about S 3 and S 2 (See also [6,19]). S 3 is viewed as the SU(2) group manifold. We parameterize an element of SU(2) in terms of the Euler angles as where 0 ≤ θ ≤ π, 0 ≤ ϕ < 2π, 0 ≤ ψ < 4π. The periodicity with respect to these angle variables is expressed as The isometry of S 3 is SO(4) = SU(2) × SU(2), and these two SU(2)'s act on g from left and right, respectively. We construct the right-invariant 1-forms, where the radius of S 3 is given by 2/µ. They are explicitly given by and satisfy the Maure-Cartan equation The metric is constructed from E i as The Killing vector dual to E i is given by where M = θ, ϕ, ψ and E M i are inverse of E i M . The explicit form of the Killing vector is Because of the Maure-Cartan equation (A.5), the Killing vector satisfies the SU(2)algebra One can also regard S 3 as a U(1) bundle over S 2 = SU(2)/U(1). S 2 is parametrized by θ and ϕ and covered with two local patches: the patch I defined by 0 ≤ θ < π and the patch II defined by 0 < θ ≤ π. In the following expressions, the upper sign is taken in the patch I while the lower sign in the patch II. where q is quantized as q = 0, ± 1 2 , ±1, ± 3 2 , · · · , because y is a periodic variable with the period 4π. These operators act on the local sections on S 2 and satisfy the SU(2) algebra k , Note that when q = 0, these operators are reduced to the ordinary angular momentum operators (A.13) on S 2 (or R 3 ), which generate the isometry group of S 2 , SU(2). The SU(2) acting on g from left survives as the isometry of S 2 .