A class of matrix models that arises as a partition function in U() Chern–Simons matter theories on the three-sphere is investigated. Employing the standard technique of expansion we solve the system beyond the planar limit. In particular, we study a case where the matrix model potential has correction and give a general solution thereof up to the order of . We confirm that the general solution correctly reproduces the past exact result of the free energy up to the order in the case of pure Chern–Simons theory. We also apply to the matrix model of Chern–Simons theory with arbitrary numbers of fundamental chiral multiplets and anti-fundamental ones, which does not admit Fermi gas analysis in general.
B04B35B831. Introduction
Recent progress in supersymmetric Chern–Simons matter theories has been made on the basis of the exact result by means of supersymmetric localization. This technique allows one to compute the partition function of supersymmetric theories exactly by the steepest descent method, which reduces the path integral calculation to that of a certain matrix model [1]. This calculation was done generically on [2–4] (see also Refs. [5,6]) and on —called the superconformal index [7,8] (see also Refs. [9–11]). These exact results were in precise agreement with the prediction from AdS/CFT duality [12,13] (see also Refs. [14,15]). See Ref. [16] for a review and further references.
On the other hand, progress in non-supersymmetric Chern–Simons matter theories has also been made not relying on the localization technique but on the expansion technique by restricting the class of matter fields to vector fields. It was conjectured that such a system is exactly soluble in the ’t Hooft large- limit [17,18]. The thermal partition function in Chern–Simons vector models on was determined exactly in the leading order of the expansion near the critical high temperature [19] (see also Refs. [20,21]), which was used to show the three-dimensional bosonization duality [22–24].
In contrast, the exact large- analysis of the three-sphere partition function for any non-supersymmetric Chern–Simons matter theory has not been performed due to its technical difficulty. So far, analysis of the three-sphere partition function has been done perturbatively near the weak coupling limit of the Chern–Simons coupling constant [25] to confirm that the system obeys the F-theorem [3]. Perturbative analysis is, however, not enough to provide evidence for duality, and exact large- analysis is keenly awaited.
In this situation we change gears to study a class of matrix models that is to be obtained as the three-sphere partition function of Chern–Simons matter theories in order to capture a generic feature of such a class of matrix models toward the bigger goal of showing bosonization duality on the three-sphere. We are also interested in analyzing such a class of matrix models beyond the planar limit because the bosonization duality is expected to hold at the subleading order in the expansion [21]. (See Refs. [26–28] for recent arguments.)
To illustrate the kind of matrix models that are to be studied, let us consider the partition function of generic U Chern–Simons matter theories on the three-sphere with unit radius:
where denotes all the matter fields collectively and is the action for the matter fields. This may be computed perturbatively as follows (see Ref. [29] for details). We first expand the gauge field by the vector spherical harmonics
We take the Lorenz gauge , which kills the modes with 0$]]>. The residual gauge can kill the mode except for its Cartan part, which we denote by . We expand the matter fields in a similar manner. Taking into account the Faddeev–Popov determinant, we integrate out all the massive modes such as and all the modes coming from the matter fields, which are massive on the three-sphere. Then the partition function reduces to the finite-dimensional integration of the effective action over :
where the ellipsis is some function of generated by integrating out the massive modes. When the matter fields vanish, the effective action can be exactly computed by, for example, supersymmetric localization or cohomological localization [30] by adding the auxiliary fields to complete the gauge field in the vector multiplet. The result is [31]
Then, by denoting the correction of the matter fields to the effective action by , the partition function is such that
In this note we analyze this class of matrix models by restricting the form of the potential to consist of single trace operators: .1 The goal of this paper is to solve this class of matrix models incorporating the standard technique of expansion developed in the study of ordinary Hermitian matrix models [32] beyond the spherical limit [33,34]. See Refs. [35,36] for reviews and further references.
The rest of this paper is written as follows. In Sect. 2 we perform some preliminary analysis of the matrix models. In Sect. 3 we derive the loop equation for this class of matrix models. In Sect. 4 we solve the loop equation by using the expansion. We give a general solution for the planar limit (Sect. 4.1) and for the genus one (Sect. 4.3). Then we apply this solution to a few examples in Sect. 5. We first apply it to pure Chern–Simons theory and compare with the known exact result to test the validity of the presented framework (Sect. 5.1). We then apply it to Chern–Simons theory with arbitrary numbers of fundamental and anti-fundamental chiral multiplets (Sect. 5.2). Sect. 6 is devoted to discussion and future direction. In the appendix we give a brief review of the partition function of pure Chern–Simons theory on the three-sphere in order for this paper to be self-contained.
2. Matrix model of Chern–Simons matter theories
Throughout this paper we investigate a class of matrix models such that
where is a normalized constant and is a non-singular function of , which can be determined at least perturbatively by integrating out the massive modes for the original Chern–Simons matter theory. When the original theory has supersymmetry, the matrix model potential can be determined exactly by using the localization method [2]. For example, for U Chern–Simons theory with fundamental chiral multiplets with the canonical R-charge, the partition function is of the form [2,3]
where , and is the FI parameter, which is equivalent to the real mass parameter. Adding the same number of anti-fundamental chiral multiplets to this system, the partition function becomes
See Refs. [37,38] for detailed analysis of this type of matrix model.
For explicit computation we write the form of the potential in Eq. (6) as
where are parameters. In this parametrization the partition function of pure Chern–Simons theory on , which we briefly review in the appendix, is given by
See Eq. (A4) for the normalization. The first example in Eq. (7) is formally given by
where .2 The second example in Eq. (8) is
The matrix model of Eq. (6) can be recast in the same form as Hermitian matrix models with positive eigenvalues. By changing the integration variables so that , the partition function becomes
where represents the positive real axis and
where is analytic on the positive real axis. Note that the matrix model potential has the logarithmic cut on the negative real axis.
As a result, the partition function can be written by using a positive definite Hermitian matrix as
with . This suggests that this class of matrix model can be analyzed by using the standard technique employed in ordinary Hermitian matrix models. In what follows we show that the free energy and correlators of some sector can be determined in order in the expansion.
3. Loop equation
There is a well-known method to determine the free energy in matrix models by using the so-called resolvent, which is in the current situation defined by the vacuum expectation value of the generating function of regular single trace operators:
We remark that the resolvent is well defined around infinity and formally behaves as in the vicinity of infinity, though this asymptotic behavior is not guaranteed due to the fact that the potential in Eq. (14) has the logarithmic cut, which ends at infinity. This suggests that the behavior of the resolvent around infinity generally gets a logarithmic correction. Still, we can expect that there exists a limit approaching infinity such that the resolvent behaves as on a certain patch. This will be important in determining the resolvent later.
Once the resolvent is determined, the coupling dependence on of the free energy given by is determined by
where . This can be seen from the fact that .
In order to determine the resolvent systematically, we first derive the Schwinger–Dyson equation for a generic operator . The vacuum expectation value of is defined by
Consider a one-to-one transformation on denoted by . Then we obtain the identity such that
Suppose the (infinitesimal) transformation . Expanding the right-hand side in terms of , we find that the zeroth-order term cancels the left-hand side and the equation at the linear order gives the Schwinger–Dyson equation
t}{{\cal O}[\phi] \over \phi_s - \phi_t}(\delta\phi_s - \delta\phi_t) - \sum_{s=1}^N {N {\cal O}[\phi] {\partial W(\phi_s) \over \partial \phi_s} \delta\phi_s}}\right\rangle = 0.
\label{SD}
\end{align}
]]>
To derive the loop equation from the Schwinger–Dyson equation let us choose , . Then the transformation becomes one-to-one on , because , and 0$]]> for 0$]]>. Then the left-hand side in the Schwinger–Dyson equation in Eq. (19) is computed as
t}{1 \over (z - \phi_s)(z - \phi_t) } - \sum_{s=1}^N N {\partial W(\phi_s) \over \partial \phi_s}{1 \over z - \phi_s}}\right\rangle + \left\langle{ \sum_{s=1}^N N{\partial W(\phi_s) \over \partial \phi_s}}\right\rangle\!. \notag
\end{align}
]]>
The first two terms are computed as
t}{1 \over (z - \phi_s)( z - \phi_t )} }\right\rangle
=\sum_{s,t=1}^N\left\langle{ {1 \over (z - \phi_s)( z - \phi_t )} }\right\rangle
= {d\over dT_z} \omega(z) + N^2 \omega(z)^2. \notag
\end{align}
]]>
The third term is
where we define the density function
Hereafter we assume that is outside the support of the density function in order to exclude the case where Eq. (20) is divergent. The density function satisfies , and can be computed by evaluating the discontinuity of the resolvent across the real axis. Indeed, by using the formula
where is a real number, is an infinitely small positive number, and denotes the principal value, the discontinuity of the resolvent between is computed as
By using this relation, Eq. (20) can be rewritten as
where denotes a circle encircling counterclockwise. The fourth term vanishes because
Collecting these, we obtain the loop equation
Note that this is of the same form as that of the ordinary matrix models except for the integration region.
Once the resolvent is determined by the loop equation, so are the density function by Eq. (23) and the coupling dependence on of the free energy by Eq. (17), and similarly for the other coupling dependence. For example, when , the other coupling dependence on of the free energy is determined by
where and is the vacuum expectation value of the generating function of singlet operators of the form :
which is computed by using the resolvent as
Similarly, the dependence of the free energy is determined as
Acting and on the free energy gives correlators of singlet operators such that
with 1$]]>, where the subscript conn means the connected part of the correlator. It is also possible to compute correlators including some number of the operator by differentiating the free energy several times with respect to .
We remark that it is not guaranteed and has to be confirmed that correlators computed in this way agree with those computed from the original theory by using the path integral.3 In other words, the potential of the matrix model generally depends on operators inserted in the path integral. This can easily be seen by considering a partition function of some supersymmetric theory computed by using the localization method. Since correlators of non-supersymmetric operators cannot be computed by the exact method, the potential of the matrix model cannot be reused to compute correlators of non-supersymmetric operators. In this context the matrix model potential is available when operators inserted in the path integral or parameters deforming the original theory maintain supersymmetry.
4. Solution of the loop equation
In this section we solve the loop equation in Eq. (26) in the expansion. The analysis will depend on the large- behavior of the potential. Generically it can be written as
where do not depend on . For explicit calculation maintaining a certain extent of the generality we study the case where the potential is given by Eq. (14) with of order one:
In the examples of supersymmetric Chern–Simons theory in Sect. 2, this case corresponds to the ’t Hooft limit with the number of flavors of order . Accordingly, the consistent expansion of the resolvent will be such that
Plugging this into the loop equation and expanding with respect to , the loop equation is decomposed as follows. For ,
which we call the genus-zero and genus-half loop equations respectively for convenience, and for ,
where we define
From these equations, can be determined in order from . Once the resolvent is determined at the order in the expansion, so is the density function from Eq. (23) as
where . The coupling dependence of the free energy on is determined from Eq. (17) as
where and
4.1. Planar solution
Let us solve the planar loop equation of Eq. (35). First we show that the planar loop equation contains the saddle point equation of the starting matrix model in the large- limit. For this purpose we compute the discontinuity of both sides in Eq. (35) between and . The discontinuity of the left-hand side is
where we used Eq. (39). That of the right-hand side is
Therefore we obtain
with in the support of the density function in the leading order of the expansion. This is the same as the saddle point equation derived from the starting matrix model in Eq. (13) in the large- limit.
Suppose that the support of the density function consists of distinct connected intervals, , where . Taking account of the fact that the loop planar equation in Eq. (35) is quadratic, we make each interval correspond to a square root cut of the solution. Under this ansatz we solve Eq. (41). Let us consider a trial function that sees the deviation of the resolvent from the -cut square root function :
As mentioned in the previous section, we solve the loop equation so that the resolvent behaves as in the limit approaching infinity. This suggests that the trial function behaves as up to the signature, and thus is analytic around infinity. Therefore, using the Cauchy theorem we find that4
where is a complex number outside and is an infinitely large circle. Assuming further that the trial function is analytic except for the support of the leading density function, we can compute the left-hand side by deforming the contour to the non-analytic region:
where we used Eq. (41) in advance and denotes a circle encircling the intervals counterclockwise. Therefore the trial function is determined as
so is the planar resolvent:
Then the planar density function is computed from Eq. (39) as
with and for .
The endpoints of the cuts are determined in the following way. Assume that the solution obtained above behaves asymptotically as approaching infinity. This is satisfied if and only if
where the signature is chosen suitably. These give constraints for endpoints of the cuts, which is not sufficient unless . For the case, the residual conditions are provided by stability against the tunneling of eigenvalues between different cuts [39]. We demonstrate the residual condition following Ref. [40]. First we write the total matrix model potential in terms of the density function in the large- limit:
where is a Lagrange multiplier and is the dynamical planar density function determined by the saddle point equation
Differentiating this with respect to leads to Eq. (41), which we solved as Eq. (46). This suggests that integrating Eq. (41) with respect to does not get back to Eq. (50), because the density function is not analytic on the edges of the cuts so integration of Eq. (41) takes different values on each interval in general. Requiring those values to be the same (as ) gives a non-trivial condition.5 To write down the condition we define a function on by
where is defined by the analytic continuation of from an interval to the whole positive real axis. Then the function takes pure imaginary values outside . Differentiating with respect to gives , which suggests that the derivative of vanishes on each cut and thus is constant on each cut, as expected. The condition for all of these constants to be equal can be written as . Since
where is a circle encircling the interval counterclockwise, we obtain6
. These yield the residual constraint equations to fix the endpoints of the cuts.
There is a comment on the solution in Eq. (46). By using the condition in Eq. (48), the solution can be rewritten in a different form such as
where . On the other hand, this form of solution can be obtained directly by starting with a different equation from Eq. (43) as mentioned in footnote 4. Then the cuts can be determined not only by the asymptotic condition with , but also the fact that is non-singular around . These two conditions give the condition in Eq. (48).
The free energy in the leading order of the expansion is easily determined as
with Eq. (50). From this expression, we can reproduce the derivatives of the free energy with respect to the coupling constants obtained in the previous section. For example, acting on Eq. (54) yields
where we used Eq. (50), , and Eq. (33). This is nothing but the equation in Eq. (40).
4.2. Hole correction
The hole correction of the resolvent is determined by the genus-half loop equation in Eq. (36). Let us derive the saddle point equation at this order from Eq. (36). For this purpose let us rewrite Eq. (36) as
As in the planar case, we compute the discontinuity of the left-hand side between and . The discontinuity of the first term is computed as
That of the second term is
Therefore we obtain
with . Combining this with the planar saddle point equation in Eq. (35) we obtain
with , where we set . This is the same form as the planar saddle point equation in Eq. (35), replacing with , and the previous argument to solve this equation holds without any modification. Therefore a solution of the usual saddle point equation is correct up to the order of hole correction! This is a nice simplification, while the caveat is the region that Eq. (58) holds. That is, the region where we need to solve Eq. (58) is on . However, when we solve Eq. (58) as done in Sect. 4 the cut appears as the support of the density function including the hole correction, . This small discrepancy may imply that the loop equation can be solved by assuming that the support of the planar density function matches the one including the hole correction. This assumption may be important to separate out the genus-half one from . We expect that the discussion above will hold in more general matrix models such as two-matrix models. We leave the proof of this conjecture to future work.
4.3. Genus-one correction
Let us determine the genus-one correction of the resolvent. For simplicity we first study the case where , so . In this case the genus-one loop equation, Eq. (37) with , reduces to
This can be solved in the same manner as in the ordinary Hermitian matrix model [33,34]. Let us first compute the right-hand side:
Then the second term is computed as
The third term is
where we set
Therefore we obtain
In order to compute , we act on the constraint equations of the edges of the cuts, Eqs. (48) and (52):
where we used . These are computed as
where . The solution can be written as
where are determined by plugging this back in and setting the coefficients of polynomials with respect to to zero. The determining equations of are:
Substituting Eq. (66) into Eq. (61), we find
This can be rewritten as the image of the linear operator in such a way that
where satisfies for . is constructed inductively as follows. Start with the identity
. Acting on both sides and computing the right-hand side results in
Equivalently,
The case with implies that . Assuming that is constructed so that is valid for , we can rewrite the right-hand side as . Therefore we find that
Hence, if we define by
then is valid for . By using this function we finally solve the genus-one loop equation as
up to terms in the kernel of the operator such as with . Note that behaves at most as , and thus the leading asymptotic behavior of the total resolvent is unchanged; so is the cut. From Eq. (77), the coupling dependence of the genus-one free energy can be determined. For example, the dependence on is determined by Eq. (40), and that on is by Eq. (30).7
Next we consider the case where the matrix model potential contains the correction: . In this case, as derived in Eq. (37), the genus-one loop equation is corrected by the genus-half resolvent so that This equation is more involved and there may be some simplification in the way that the planar resolvent and the genus-half one can be determined at the same time as shown in Sect. 4.2. To see this, consider the deviation of the resolvent from the solution and substitute this into the original loop equation of Eq. (26). We obtain
Since is of order one, is of order : . Therefore, at the leading order of the expansion this reduces to
where we define by
This equation is of the same form as the one without the hole correction by replacing with , respectively. Since the above argument to solve this equation holds as it is by performing the replacement, a solution of Eq. (79) is given in the same form as Eq. (77), where and are replaced with the ones including the hole correction. We emphasize that this simplification happens only at the genus-one order, and at higher order one may need to solve Eq. (37) in general.
5. Applications
In this section we apply the presented formulation developed in the previous section to a few examples. First we apply it to the three-sphere partition function in U pure Chern–Simons theory in order to test the presented framework by comparing with the exact result known for pure Chern–Simons theory as reviewed in the appendix. Secondly, we apply it to U Chern–Simons theory with fundamental chiral multiplets and anti-fundamental ones. This system does not admit the Fermi gas analysis in general, and there may be no systematic way to study the system beyond the spherical limit except for our formulation at present.
5.1. Pure Chern–Simons theory
The matrix model potential for pure Chern–Simons theory is given by Eq. (13) with Eq. (10). For simplicity we first study the case where there is no hole correction. Then .
Let us first determine the planar resolvent. For 0$]]>, the potential has only one stable minimum so we have only to consider a solution with one cut: with . In order to simplify the integration we start with a solution of the form of Eq. (53) with :
where . Inflating the contour we can compute the right-hand side as
where we computed the first term as
The edges of the cut, , are determined by the asymptotic behavior around infinity and the regularity around the origin. can approach when , which is achieved if and only if is regular at the origin if and only if These can be solved as Note that . Then the planar resolvent can be simplified as
The planar density function is computed as
This solution matches the one given in Ref. [36], where the solution is expressed in the original coordinates. We plot the density function as well as the potential in Fig. 1(a).
(a) The blue and yellow curves depict the matrix model potential and the planar density function, respectively, when . The eigenvalues tend to clump around the potential minimum. (b) The differentiation of the planar free energy with respect to is plotted. The blue curve depicts the result obtained by the resolvent method and the yellow one is from the past exact result. They almost coincide.
The planar free energy is given by Eq. (54) and its derivative is Eq. (30):
We could not perform the integration on the right-hand side analytically, so instead we evaluated it numerically. The numerical result is in good agreement with the past exact result of Eq. (A19), as can be seen in Fig. 1(b).
Next we study the genus-one correction. Now we consider the one-cut solution, so the genus-one correction of the resolvent is given by
where are defined by Eq. (76). are computed by Eq. (60). As in the computation of the resolvent, the integration can be simplified by using Eq. (48):
By inflating the contour to infinity, the first term is computed as
and the second term vanishes. Therefore In the same way, are computed as
The genus-one correction of the free energy is computed by using Eq. (30):
This time we could perform the integral analytically:
This result is in precise agreement with the past exact result of Eq. (A20) with .
Next we consider the case where the matrix model potential has a hole correction by the “FI parameter”:8 with , which is still integrable as shown in the appendix. As discussed in Sect. 4.2, we solve the usual saddle point equation , which is correct up to the hole order. To emphasize the difference from the previous computation we denote the cut with a prime, so that . Then a solution of this saddle point equation, , is given by
where . This can be computed in the same way as previously and we obtain9
The edges of the cut are also determined in the same way. The result is , where and are the same as previously. By using this, the resolvent up to the hole order is simplified as , where . As argued in Sect. 4.3, the genus-one resolvent is given by
where is given by Eq. (76) with replaced by . By using , the genus-one resolvent including the FI term can be written as . Finally, we compute the differentiation of the total free energy with respect to :
where the ellipsis represents the terms of order . Then the second term is computed as
The third term is
As a result, we obtain
This is in perfect agreement with the past exact result of Eq. (A20).
5.2. <inline-formula><tex-math notation="LaTeX" id="ImEquation274"><![CDATA[${\cal N}=2$]]></tex-math></inline-formula> Chern–Simons theory with arbitrary numbers of fundamental and anti-fundamental chiral multiplets
As another example we consider the matrix model of Chern–Simons theory with fundamental chiral multiplets and anti-fundamental ones with the canonical R-charge. Let us set
Without losing generality, we can assume that . The matrix model potential of this system is given by combining Eqs. (7) and (8):
Its derivative is
where we used . Thus it is clear that the matrix model potential takes complex values for a general number of chiral multiplets.
In order to determine the number of cuts in the resolvent by identifying that of the potential minimum, we regard the matrix model potential as an analytic function with all the parameters. When is pure imaginary and , , and are real, the potential becomes real. We fix the number of cuts of the resolvent in this situation.
We consider the large- limit, holding its ratio and the other parameters fixed. In this limit the potential has only one stable minimum, as in the case of pure Chern–Simons theory, so we have only to consider a solution with one cut: with .
We compute the resolvent up to the hole correction by Eq. (53) with :
where . Inflating the contour we can compute the right-hand side by picking up the pole as
where the first term is the contribution of the logarithmic branch cut , the second one is that of the pole at , and the third one is at . We compute the integrations such that
where
Then the resolvent becomes
The edges of the cut are determined by the asymptotic behavior around infinity and the regularity around the origin. can approach when , which is achieved if and only if
is regular at the origin if and only if
From these equations the edges of the cut are determined order by order in .
As argued in Sect. 4.2, the planar resolvent should be determined so as to have the same cut as that of . Since the leading part of the potential in the large- limit is unchanged, the form of the planar resolvent is unchanged except for the edges of the cut: Eq. (82) with . The genus-half resolvent is determined before the edges of the cut are expanded in the power series and given by
Then, as argued in Sect. 4.3, the genus-one resolvent is given by
where the are given by Eq. (76) with replaced by . The derivative of the free energy up to the genus-one order is given by
It is known that this system has the dual description known as Seiberg-like duality [41]. The dual theory is U Chern–Simons theory with fundamental and anti-fundamental chiral multiplets, with mesonic operators as well as some monopole operators with a suitable superpotential, where depends generally on , , , and , which is still in the class investigated in this paper. It would be interesting to test the duality from our general solution. We hope to come back to this problem in a future publication.
6. Discussion
In this paper we have performed a general analysis of a class of matrix models describing Chern–Simons matter theories on the three-sphere incorporating the standard technique of expansion developed in the study of ordinary Hermitian matrix models. We have derived the loop equation for all orders in the expansion and presented its explicit solution up to the genus-one order when the potential has a correction. We have applied the formulation to pure Chern–Simons theory and confirmed that the presented solution reproduces the exact result known in the past. We have also applied the framework to Chern–Simons theory with arbitrary numbers of fundamental and anti-fundamental chiral multiplets, and obtained a formal expression for the solution up to the genus-one order in the expansion.
This paper mainly focused on the construction of the framework to solve a class of matrix models. We are very much interested in applying the formula obtained in this paper to a duality pair of Chern–Simons matter systems and testing that the bosonization duality holds at the next leading order in the expansion. In particular, it would be interesting to develop the presented large- technique in a class of unitary matrix models which arises as a partition function of Chern–Simons matter theories on . For a class of Chern–Simons vector models, the effective matrix model potential was determined exactly in the leading order of the large- limit [19], and the three-dimensional bosonization was confirmed at that order. We hope that the formulation developed in this paper is useful for future study in this direction.
In this paper, in order to study beyond the planar limit we adopted the iterative procedure given in Refs. [33,34]. Another iterative approach has been proposed, using the Feynman graph of the trivalent vertexes [42,43]. It would be interesting to reformulate the formula presented in this note in terms of the different approach.
Another interesting question is whether this class of matrix models has the equivalent description of some two-dimensional CFTs as ordinary Hermitian matrix models [44–46] (see also Ref. [47]). Naively, the answer seems to be no due to the fact that the degrees of freedom in a three-dimensional system are much bigger than those of a two-dimensional one in a generic situation. However, we have a suspicion that the answer could be yes for a certain matrix model of this kind, intuitively because vector models coupling to Chern–Simons theory appear as an effective field theory of an anyonic system [48,49], and the wave function describing a quantum Hall state known as the Laughlin wave function [50] is given by a correlator of certain two-dimensional (rational) CFTs [51]. In fact, it was shown that this answer becomes yes for a similar class of matrix models to the one studied in this paper [52], where the corresponding CFT is identified with a -deformed one. Exploring this question is left for future work.
There is a straightforward generalization of the presented formulation to a different gauge group [53] or two matrices. This generalization to two matrices is important for the application to higher supersymmetric Chern–Simons matter theories such as the ABJM theory [54]. The correction of the free energy in the ABJM theory was computed in Refs. [55–57]. In this development, a new technique called the Fermi gas approach was invented [56]. This approach is powerful for studying non-perturbative aspects of the ABJM theory from the partition function [58,59]. It is an important problem to test whether the traditional techniques of matrix models can reproduce the results obtained by new ones in recent developments beyond the spherical limit.
We hope to come back to these issues in the near future.
Acknowledgements
The author would like to thank S. Sugimoto and T. Takayanagi for valuable discussions and comments on the draft. The author would also like to thank Y. Imamura for a helpful comment on the first version of this paper.
Funding
Open Access funding: SCOAP.
Appendix. Partition function of pure Chern–Simons theory on <inline-formula><tex-math notation="LaTeX" id="ImEquation329"><![CDATA[$\mathbf S^3$]]></tex-math></inline-formula>
In this appendix we give a brief overview of the three-sphere partition function in U pure Chern–Simons theory, and a derivation of its large- expansion as used in the main text.
The partition function is defined formally by a path integral over the gauge field on such that10
The classic paper Ref. [60] demonstrated explicitly in the case of SU(2) that this can be exactly determined as a function of the Chern–Simons level without performing the path integral by clarifying its relation to a modular transformation matrix of the characters in the corresponding affine Lie algebra. Generalization to an arbitrary gauge group is straightforward. Since modular transformation matrices had already been determined in general affine Lie algebras [61], the exact result of the partition function for U pure Chern–Simons theory was given by
After this exact result was studied in terms of the and expansions [62,63], it was insightfully observed that the Chern–Simons partition function of Eq. (A2) exactly matches that of the topological string theory on a Calabi–Yau three-fold background by identifying the string coupling constant with the pure imaginary Chern–Simons level [64]. This led to the conjecture of gauge/geometry duality [65] between Chern–Simons and topological string theories [66,67].
It was subsequently pointed out that the partition function in Eq. (A1) reduces to a matrix model such that [31]
This matrix model was extensively studied in relation to topological string theory [68,69]. With the help of the Weyl denominator formula, the matrix integral was explicitly performed by Gaussian integration in perfect agreement with Eq. (A2) [2]. This matrix model was also evaluated exactly by the orthogonal (or characteristic) polynomial method in accordance with Eq. (A2) [70]. The orthogonal polynomials associated with this matrix model were found to be Stiltjes–Wigert polynomials.
Let us compute the matrix model in a Fermi-gas-like approach [56] including the “FI term”:
For this purpose we rewrite the partition function as a determinant by using the Weyl denominator formula
t}2\sinh\left({\sigma_s - \sigma_t \over 2}\right) =\underset{s,t}\det\left[ \exp{\left\{\sigma_s\left(t-{N+1 \over 2}\right)\right\}} \right]\!.
\label{Weylden}
\end{equation}
]]>
From this formula we can show that
which enables us to rewrite the partition function as
The inside of the determinant is computed by Gaussian integration as follows:
Plugging this back in gives
t}2\sin\left({\pi (s-t) \over k}\right) \!,
\notag
\end{align}
]]>
where in the second equation we used and the Weyl denominator formula in Eq. (A5). By using the formula t}2\sin\left({\pi (s-t) \over k}\right) = \prod_{I=1}^N (2\sin{\pi I \over k})^{N-I}$]]>, we obtain
Then the free energy is computed as
The expansion was done as follows [64,66]. The expansion coefficients are determined as functions of :
By using
the second term in Eq. (A10) can be expanded as
where is the zeta function defined by . The second term in Eq. (A13) can be expressed by using Barn’s function as , whose large- expansion is known:
Here, is the th Bernoulli number defined by11
The summation in the last term in Eq. (A13) can be done by using a formula such that12
Plugging these back in, we obtain the free energy as
As a result, the coefficients in the expansion of the form in Eq. (A11) are determined as
A few comments are in order. The leading term in the expansion, , can be obtained directly from Eq. (A10) by taking the large- limit [62]:
where in the last equation we used the definition of the Riemann integral. This can be further computed by using Eq. (A12) as
where is the polylogarithm defined by . The next-to-leading term can be simplified by using Eq. (A12) as follows:
Footnotes
1 This restriction corresponds to the representation of the matter fields excluding higher-dimensional representations such as the adjoint one.
2 This expansion may not be useful for practical computation, though.
3 The author would like to thank S. Sugimoto for discussion on this point.
4 One may more generally conclude that, for example, where are polynomials of . The resolvent obtained from this form in the same way as described below at first looks different from Eq. (46), but reduces to the same form by using the boundary condition, which is the same as Eq. (48). We give a comment on this point below.
5 It is possible to consider a case where the Lagrange multiplier in Eq. (49) takes different values on each interval. In this case their values become parameters of the theory and play the role of a kind of chemical potential.
6 The chemical potentials mentioned in footnote 5 can be added such that for .
7 The differential equation in Eq. (40) will be solved as in the original Hermitian matrix model for the one-cut case [33] and the two-cut case [34], though such explicit solutions of the genus-one free energy do not contain the information about the dependence on other coupling constants such as .
8 The usage of this terminology can be justified by adding some auxiliary fields into the pure Chern–Simons theory so that the theory has supersymmetry.
9 The planar resolvent and the hole one are determined from this by
which should be done before the edges of the cuts are determined.
10 Here, is the renormalized Chern–Simons coupling constant so that , where is the level of the corresponding WZW model.
11 Our definition of the Bernoulli number is different from the one adopted in several past works such as Refs. [63,64,66]. The difference is .
12 This can be proved by using
where is the integer part of .
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