Phase diagram of q-deformed Yang-Mills theory on S2 at non-zero θ-angle

We study the phase diagram of q-deformed Yang-Mills theory on S2 at non-zero θ-angle using the exact partition function at finite N . By evaluating the exact partition function numerically, we find evidence for the existence of a series of phase transitions at non-zero θ-angle as conjectured in [hep-th/0509004].


Introduction
The topological θ-angle plays an important role in the dynamics of gauge theories and σ-models in various dimensions (see e.g. [1][2][3][4][5] and references therein). More than a decade ago, the phase diagram of q-deformed Yang-Mills theory (qYM) on S 2 at non-zero θ-angle was conjectured in [6]. In this paper, we will study the phase diagram of qYM on S 2 using the exact partition function at finite N . In general, exact result is a very powerful tool to analyze the non-perturbative aspects of gauge theories, which are usually beyond reach by other means. This approach was successfully applied to many examples, including the ABJM theory on S 3 [7][8][9][10] and the Gross-Witten-Wadia (GWW) unitary matrix model [11][12][13].
In [6,14,15], it was found that qYM on S 2 in the large N 't Hooft limit has a third order phase transition at some critical value t = t c of the 't Hooft coupling when θ = 0. The physical mechanism of this phase transition is essentially the same as that of ordinary (undeformed) Yang-Mills theory on S 2 found by Douglas and Kazakov [16]. As shown in [6,14,15], at the critical point the instanton contribution becomes comparable with the zero-instanton sector and the phase transition is triggered by the instanton condensation, in much the same way as the undeformed Yang-Mills theory [17].
In [6], it was further conjectured that the phase diagram of qYM at non-zero θ has an intricate structure (see figure 3 in [6]): there is a series of phase transition curves on the t-θ plane which accumulate at the point (t, θ) = (0, π). Each transition curve corresponds to the exchange of dominance of different instanton sectors. In this paper we will examine this JHEP04(2018)059 conjecture using the exact partition function of qYM at finite N . By evaluating the exact partition function numerically, we find evidence for the conjectured phase diagram in [6]. We find that the prefactor of instanton contribution is important for the understanding of the phase structure at non-zero θ. However, it turns out that it is difficult to find the analytic form of this prefactor in the q-deformed case. Instead, in section 5 we consider the undeformed case where the prefactor of 1-instanton contribution is already known [17], and we propose an analytic form of the first two transition curves at non-zero θ and check this proposal numerically.
This paper is organized as follows. In section 2, we write down the exact partition function of qYM on S 2 as a determinant of N ×N matrix. Using this exact result, we study the behavior of free energy at θ = 0 and confirm the phase transition found in [6,14,15]. In section 3, we study the instanton contribution using our exact result at finite N . In section 4, we numerically compute the free energy at non-zero θ and find evidence that indeed there is a series of phase transitions coming from the exchange of dominance of different instanton sectors, as conjectured in [6]. In section 5, we consider the phase diagram of undeformed theory at θ = 0 and propose an analytic form of the first two transition curves. Finally, we conclude in section 6 with some discussion of the future directions.
2 Exact partition function at finite N In this section, we consider the exact partition function of qYM on S 2 at finite N . This section is mostly a review of the known results.
The q-deformed U(N ) Yang-Mills theory on S 2 naturally appears as a worldvolume theory on N D4-branes on a non-compact Calabi-Yau X p where the D4-branes in question are wrapping around the base P 1 and one of the fiber O(−p) of X p [18]. It is argued in [18,19] that the path integral of D4-brane worldvolume theory localizes to the q-deformed Yang-Mills theory on the base P 1 = S 2 . This partition function of D4-brane theory is identified as the partition function of 4d black holes made of the bound states of D4, D2, D0-branes, which in turn is related to the topological string partition function on X p via the OSV conjecture [20]. It is known that qYM on any Riemann surface can be solved exactly [18] in a similar manner as the ordinary undeformed 2d Yang-Mills theory [21]. The partition function of U(N ) qYM on S 2 is given by where [n] denotes the q-integer 3)

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and is given by = 0, (odd N ), 1, (even N ). (2.4) In other words, the summation of n i in (2.2) runs over integers for odd N and half-integers for even N . 1 The overall normalization of the partition function is ambiguous; in (2.2) we followed the convention of [6]. As we will show below, we can rewrite this partition function as a determinant of N × N matrix. To see this, we first notice that the factor i<j [n i − n j ] in (2.2) is basically the Vandermonde determinant and it is rewritten as This relation can also be understood as the Weyl denominator formula of U(N ) gauge group. Squaring the above expression (2.6) we get a sum over two permutations, but one of them can be trivialized by using the invariance of i n 2 i and i n i under the permutation of n i . In this way (2.2) is rewritten as where the Jacobi theta function is defined by Finally, the sum over S N in (2.7) reduces to the determinant This is the main result of this section. 2 This determinant form of Z N is reminiscent of the exact partition function of GWW model [22,23]. However, there is an important difference: the exact partition function of qYM in (2.9) is given by a Hankel determinant det(a i+j ), while the exact partition function of GWW model is given by a Toeplitz determinant det(b i−j ). We are interested in the behavior of this partition function in the large N 't Hooft limit g s → 0, N → ∞, t = g s N : fixed, (2.10) 1 Our ni is related to the rows {λ1, · · · , λN } of Young diagram by It follows that ni ∈ Z + 2 since λi is integer. 2 As far as we know, this expression has not appeared in the literature before.

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and we would like to study the genus expansion of free energy . (2.11) In [6,14,15], it is found that when θ = 0 there is a third order phase transition at the critical value t = t c of the 't Hooft coupling, where t c is given by This phase transition occurs only for p > 2 [6,14,15], and we will assume p > 2 throughout this paper. In the rest of this section, we will consider the behavior of free energy above (t > t c ) and below (t < t c ) the phase transition using the exact partition function Z N at finite N .

Strong coupling phase
Let us first consider the strong coupling phase (t > t c ). In the large N limit, the eigenvalue distribution in this phase is described by a two-cut solution of a certain matrix model and the explicit form of the resolvent was constructed in [6,14,15]. However, it is not so straightforward to compute the genus-zero free energy F 0 (t) from this solution.
To study this phase, it is convenient to regard the partition function (2.2) as a sum over configurations of N non-relativistic fermions (2.13) where n = (n 1 , · · · , n N ) specifies the momentum of N fermions and the total energy E and the total momentum P of fermions are given by (2.14) These N fermions are interacting through the factor i<j [n i − n j ] 2 . In the strong coupling phase, we can compute (2.13) by summing over the fermion configurations from the small total energy E. A similar computation has been performed for the undeformed Yang-Mills case in [16]. One can easily see that the ground state (lowest energy configuration) is given by The energy E 0 and the momentum P 0 of the ground state are given by 16) and the contribution of this ground state is .
(2.17) −n F n F + 1 n F Figure 1. Maya diagram for the ground state. The black nodes (−n F ≤ n ≤ n F ) are occupied by fermions while the gray nodes (|n| > n F ) are empty. One can visualize the configuration of fermions by the so-called Maya diagram as shown in figure 1: the black nodes are occupied by the N fermions and the gray nodes are empty. In the ground state the nodes between n = −n F and n = n F are occupied; the nodes at n = ±n F can be thought of as the Fermi levels.
We can also draw the Maya diagram for excited states as in figure 2. The excitation energy ∆E = E − E 0 and the total momentum of the states (a) and (b) in figure 2 can be easily computed as and their contributions to the partition function are given by There are two more states with the same excitation energy ∆E = N/2 which are obtained by flipping the sign of momenta n i → −n i in figure 2. In this way, we can compute the

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large t expansion of partition function systematically as As a consistency check of our result (2.20), we can take the limit p → ∞, g = g s p, A = gN : fixed, (2.21) in which the partition function of qYM reduces to the partition function of undeformed Yang-Mills theory [6,14,15]. After taking this limit, the free energy becomes Here we have set θ = 0 for simplicity. This can be further expanded in the coupling g as

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where the first three terms read As expected, the above F 0 (A) agrees with the genus-zero free energy of undeformed Yang-Mills theory computed in [16]. 3 Chiral partition function. One can naturally distinguish the excitations as "chiral", "non-chiral", and "anti-chiral", as follows. In the chiral excitation, changes from the ground state are allowed only near the positive Fermi level n = +n F . In other words, a chiral excitation is a configuration where the modes near the negative Fermi level are the same as the ground state n = (−n F , −n F + 1, · · · , n N −1 , n N ). . This type of decomposition was first considered in the undeformed Yang-Mills theory in [24][25][26]. Now we can define the chiral partition function Z + N by summing over the chiral excitations only. Z + N is easily found from the full partition function (2.20) by dropping the non-chiral and anti-chiral terms It turns out that the chiral free energy log Z + N can be organized into a double expansion in terms of Q and Q in the 't Hooft limit, where Q and Q are defined by (2.27) 3 Our F0(A) and that in [16] differ by a factor of A 2 which comes from the different definition of the genus expansion. In [16] the genus expansion is defined as the 1/N expansion, log Z = h≥0 N 2−2h F h (A), while we expand the free energy in terms of g = A/N .

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The terms involving Q correspond to open string amplitudes due to the additional D-brane insertions on X p [18]. The pure closed string amplitude is obtained by discarding the Q dependent terms from (2.26) (2.28) and the closed string free energy F closed = log(Z +,closed N /Z gnd ) is given by One can show that the genus expansion of F closed in (2.29) reproduces the result of topological string on X p [27][28][29].

Weak coupling phase
Next consider the weak coupling phase (t < t c ). To study the weak coupling phase, we should perform the modular S-transformation of the Jacobi theta function in (2.9). Using the formula the exact partition function in (2.9) becomes (2.31) Then, plugging the series expansion of Jacobi theta function into the determinant of (2.31), Z N is written as a sum over integer vectors m = (m 1 , · · · , m N ) with some coefficient Ω( m). This S-dual expression has a natural interpretation as the instanton expansion. As in the case of undeformed Yang-Mills theory, the instanton in question is a classical solution of gauge field where the Dirac monopole configuration is embedded in the Cartan part of gauge field [30,31]. In the weak coupling phase, the most dominant contribution comes from the zero-instanton sector since the instanton contribution m = 0 in (2.33) is suppressed by the factor O(e −1/gs ) which is non-perturbative in g s .

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By setting ϑ 3+ = 1 in (2.31) (or taking the m = 0 term in (2.32)), the perturbative part of Z N in the weak coupling phase is found to be This is exactly the same as the partition function of pure Chern-Simons theory on S 3 up to a rescaling of the coupling g s → g s /p. The genus expansion of (2.34) is easily found from the known result of pure Chern-Simons theory (see e.g. [32]) (2.35)

Phase transition at θ = 0
As shown in [6,14,15], the qYM on S 2 at θ = 0 has a third order phase transition. The mechanism of the phase transition is essentially the same as the undeformed case found by Douglas and Kazakov [16] where the phase transition occurs when the eigenvalue density saturates the bound ρ(h) ≤ 1.
As we have seen in the previous subsection, the zero-instanton sector of the weak coupling phase is described by the pure Chern-Simons theory. From the known eigenvalue density of Chern-Simons matrix model [32] ρ(h) = p π arccos e which leads to the result (2.12) found in [6,14,15]. Now we can numerically study the behavior of free energy at θ = 0 using the exact partition function (2.9) at finite N . The determinant in (2.9) can be evaluated numerically with high precision and we can plot the free energy as a function of t = g s N by varying the coupling g s with fixed N . In figure 3 we show the plot of free energy for p = 3, N = 80 at θ = 0. As we can see from this figure, the behavior of the free energy changes at t = t c from Z weak (2.34) in the weak coupling phase to Z gnd (2.17) in the strong coupling phase, as expected from the result in [6,14,15].

Instanton in the weak coupling phase
It is argued in [6,14,15] that the phase transition of qYM is induced by instantons. As we have seen in the previous section 2.2, the instanton expansion in the weak coupling phase naturally arises after performing the modular S-transformation of Jacobi theta function (2.33). It turns out that when 0 < θ < π the dominant contribution comes from the m = 0 and m = 1 terms in the expansion of Jacobi theta function (2.32). If we keep only those terms in the expression of Z N in (2.31), the partition function is approximated as where ξ is the weight factor of 1-instanton

2)
N is the overall factor and A and B in (3.1) are the following N × N matrices: (i, j = 1, · · · , N ).

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The validity of this approximation (3.1) will be discussed in detail in the next section 4. Note that the sign (−1) N −1 = (−1) of B i,j in (3.4) comes from the sign (−1) m in the expansion of Jacobi theta function (2.32) with m = 1. In this notation, Z weak in (2.34) is written as Now we can define the instanton part of partition function by dividing Z N by Z weak . In the approximation (3.1) the instanton partition function becomes where the matrix M is given by From the explicit form of A and B in (3.4), we find that the matrix element of M has a simple expression where (a; q) k denotes the q-Pochhammer symbol We have checked this relation (3.9) for N ≤ 10, and we believe that this is true for all N .
In what follows we will assume that (3.9) holds for all N . It would be interesting to find a general proof of (3.9). One can expand Z inst in (3.7) as a power series in ξ where Z 0 = 1 and Z k ∝ ξ k . For instance, the 1-instanton term is given by Higher instanton corrections Z k≥2 will be studied in section 4. In the 't Hooft limit, we expect that the 1-instanton correction Z 1 at θ = 0 is characterized by the instanton action S inst (t) computed in [6,14,15] ξ 0 Tr M ∼ e  In [6], it was found that the instanton action is given by the integral of eigenvalue density ρ(h) in (2.36) along the imaginary axis Here the upper bound of integral, h 0 , is determined by the condition ρ(ih 0 ) = 1: (3. 16) This suggests that the large N limit of instanton in the weak coupling phase can be thought of as a complex instanton. A similar phenomenon was observed in the GWW model as well [11,33]. In figure 4 we show the plot of ξ 0 Tr M for p = 3, N = 400 using the exact form of M in (3.9). One can clearly see that the exact result of M nicely reproduces the analytic form of instanton action in (3.15). The instanton action vanishes at t = t c as shown in [6,14,15], which is also reproduced numerically by our exact result of M . This leads to a physical picture of the phase transition that it is triggered by the condensation of instantons, as in the case of undeformed theory [17]. We also observed numerically that the 1-instanton correction ξ 0 Tr M is always positive for both even N and odd N in the weak coupling phase; we emphasize that the sign (−1) N −1 of B in (3.4) is crucial for this positivity of 1-instanton correction.

Z inst in the p → ∞ limit
We expect that Z inst reduces to the known instanton correction of undeformed theory in the limit p → ∞ with g = g s p fixed (2.21). Indeed, we find that the 1-instanton term (3.12) JHEP04(2018)059 where L α n (x) = 1 n! x −α e x ∂ n x (x n+α e −x ) denotes the Laguerre polynomial. As expected, this agrees with the result of 1-instanton correction of undeformed theory [17]. This implies that the 1-instanton term Tr M in qYM can be thought of as a certain q-deformation of the Laguerre polynomial. As shown in [17], the large N limit of L 1 N −1 (4π 2 /g) has a sign (−1) N −1 which is precisely canceled by the overall sign in (3.17) coming from the modular S-transformation of Jacobi theta function. The resulting 1-instanton factor (3.17) in the undeformed theory is always positive for both even N and odd N , which is consistent with the above observation of the positivity of Tr M in the q-deformed theory.
More generally, we find lim p→∞ det(1 + ξM ) = det(1 + ξ M ), (3.18) where the N × N matrix M is given by The right hand side of (3.18) is exactly the generating function of the expectation value of 't Hooft loops in the anti-symmetric representations in 4d N = 4 U(N ) super Yang-Mills theory, 4 up to a change of sign of the coupling g = −g 2 4d [34]. This is expected since the computation of 't Hooft loops in 4d N = 4 super Yang-Mills theory localizes to 2d Yang-Mills theory on S 2 in the instanton sector [34].
We have checked that (3.17) and (3.18) hold for small N , but we do not have a general proof. It would be nice to find a proof of (3.17) and (3.18) for general N .
4 Phase diagram of q-deformed Yang-Mills theory at θ = 0 The phase diagram of qYM at non-zero θ was conjectured in [6]. In this section we will examine this conjecture using our exact result of partition function at finite N .
We first notice that using the symmetry of the exact partition function we can restrict θ to the region 0 ≤ θ ≤ π without loss of generality. As discussed in [36], for general value of θ we should minimize the energy of Z weak in (2.34) under all 2π-shifts of θ

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The minimum is given by = 0 if θ is in the range 0 ≤ θ ≤ π, and hence we can safely use the S-dual expression of Z N in (2.31) and (3.1) in this region of θ. Thus the 1-instanton term for 0 ≤ θ ≤ π is given by where we used (3.13). Then the critical value t = t * (θ) where Z 1 becomes of order one is determined by the condition that the exponent in (4.3) vanishes It is conjectured in [6] that the critical line t = t * (θ) on the t-θ plane is just the first one of such critical lines; there are many critical lines on the t-θ plane which accumulate at (t, θ) = (0, π) (see figure 3 in [6]). The conjectured phase diagram of [6] is based on the following two assumptions: (i) Only the instantons with charges m = (1, 1, · · · , 1, 0 · · · , 0) in the expansion (2.33) are relevant for the phase transition at θ = 0.
(ii) There is a series of phase transitions at θ = 0 where the instantons of the type in (i) exchange dominance.
The assumption (i) amounts to using the approximation in (3.1). We can test this assumption by computing the following ratio numerically where Z weak and Z inst are given by (2.34) and (3.7), respectively. In figure 5, we show the plot of this ratio for p = 3, N = 80 at θ = π 3 . From this figure, one can see that this ratio is very close to 1 r ≈ 1, (4.6) which confirms the assumption (i). Next consider the assumption (ii). In figure 6, we show the plot of the instanton part of free energy F inst = log Z inst and its derivatives for p = 3, N = 80 at θ = π 3 . 5 One can see that the third derivative ∂ 3 t F inst in figure 6d has several jumps at different values of t, and the first jump (or discontinuity of ∂ 3 t F inst ) occurs at t ≈ t * (θ). This is consistent with the assumption (ii). In figure 6 we have used the approximate instanton partition function Z inst in (3.7), but the plot of the exact partition function Z N /Z weak does not change much from figure 6 due to the property r ≈ 1 (4.6). In figure 6d, the third derivative ∂ 3 t F inst has sharp discontinuities only at the first few zeros of ∂ 3 t F inst , but we expect that this is a finite N effect and in the strict large N limit they become sharp phase transitions.
We can collect more evidence for the assumption (ii) by evaluating the k-instanton contribution Z k in (3.11) separately. In figure 7    contribution (k = 1, · · · , 4) for p = 3, N = 80 at θ = π 3 . One can see that the dominant instanton changes from k = 1 to k = 4 as t increases, and the exchange of dominance occurs at different values of t for different instanton number k. In other words, there is a series of phase transitions at t = t k (k = 1, 2, · · · ) where Z k−1 and Z k exchange dominance at t = t k . Our numerical result in figure 7 gives strong evidence for the assumption (ii). Also, we observed numerically that the difference of t k and t k+1 decreases as N becomes large, and the difference scales approximately as 1/N t k+1 − t k ∼ 1/N.   This suggests that the 1/N correction of instanton factor, in particular the prefactor f k of k-instanton, is important for the understanding of the scaling behavior (4.7) (4.8) However, we were unable to find the analytic form of the prefactor f k and hence we could not determine the analytic form of the critical value t = t k . In the next section, we will consider the phase diagram of undeformed Yang-Mills theory, where the instanton prefactor is more tractable analytically. Finally, we can draw the phase diagram of qYM by using our exact result of Z inst in (3.7). To do this, we first observe from figure 6 that the local maximum of the second derivative ∂ 2 t F inst corresponds to the (approximate) discontinuous point of the third derivative ∂ 3 t F inst . Based on this observation, in figure 8 we plot the local maxima of ∂ 2 t F inst for p = 3, N = 80 for several values of θ. Our result in figure 8 agrees with the conjectured phase diagram in [6], at least qualitatively. In particular we can see from figure 8 that the transition curves seem to accumulate at the point (t, θ) = (0, π).
5 Phase diagram of undeformed Yang-Mills theory at θ = 0 In this section, we consider the phase transition curves in the undeformed theory, where the instanton prefactor can be studied analytically. In fact, the analytic form of the prefactor of 1-instanton term has been already obtained in [17].
Let us consider the instanton contributions in the undeformed Yang-Mills theory (3.18) where M is given by (3.19). The coupling in the undeformed theory is defined in (2.21). It is convenient to rescale the coupling g and A in (2.21) aŝ g = g π 2 , a =ĝN = . We plotted the local maxima of ∂ 2 t F inst at fixed θ in the range t * (θ) ≤ t ≤ 1.4t * (θ) and varied θ with step ∆θ/π = 0.05. The orange curve is the line t = t * (θ) given by (4.4).
In this normalization, when θ = 0 the phase transition occurs at a = 1, corresponding to the critical value A = π 2 found in [16]. In terms of these rescaled couplings, M and ξ in ( By expanding Z inst in (5.1) as a power series in ξ we can define the k-instanton term Z k as in the case of qYM in (3.11). As we have seen in (3.17), the 1-instanton term can be written in a closed form The large N limit of Z 1 has been studied in [17] and the result reads where the instanton action is given by 6 S inst (a) = 2 √ 1 − a − a cosh −1 (1/ √ a) . The black dots are the first two local maxima of ∂ 2 t F inst at fixed θ and we varied θ with step ∆θ/π = 0.05. The blue and the orange curves are the lines a = a 1 (θ) and a = a 2 (θ), respectively. The gray dashed curve represents the curve a = a * (θ) without taking into account the effect of instanton prefactors.
where S inst (a) = ∂ a S inst (a) is given by . Now we can draw the phase diagram of undeformed theory in a similar manner as the q-deformed case in figure 8. In figure 9, we plot the first two maxima of ∂ 2 t F inst computed numerically from Z inst in (5.1). One can see that numerical data points fit well on the curves a = a 1 (θ) and a = a 2 (θ) obtained in (5.14). It would also be interesting to study the transition curves a = a k (θ) for higher instanton corrections Z k≥3 , which we will leave for a future problem.

Conclusion and open problems
In this paper we have examined the phase diagram of q-deformed Yang-Mills theory on S 2 conjectured in [6] and found numerical evidence for this conjecture using the exact partition function at finite N (2.9). We found that the 1/N correction to the instanton contribution, in particular the prefactor of instanton, is important for the understanding of the splitting of phase transition curves at non-zero θ. Our analysis heavily relied on numerics and it is desirable to find a more analytic method to study the phase diagram.
There are various open problems. We have seen that the instanton correction in qYM has an interesting connection to a q-deformation of Laguerre polynomials, in the sense