Relating 't Hooft Anomalies of 4d Pure Yang-Mills and 2d $\mathbb{CP}^{N-1}$ Model

It has recently been shown that a center-twisted compactification of the four-dimensional pure $SU(N)$ Yang-Mills theory on a three-torus gives rise to the two-dimensional $\mathbb{CP}^{N-1}$-model on a circle with a flavor-twisted boundary condition. We verify the consistency of this statement non-perturbatively at theta angle $\theta=\pi$, in terms of the mixed 't Hooft anomalies for flavor symmetries and the time-reversal symmetry. This provides further support for the approach to the confinement of four-dimensional Yang-Mills theory from the two-dimensional $\mathbb{CP}^{N-1}$-model.


Introduction
The two-dimensional sigma model with the target space CP N −1 (the CP N −1 -model [1]) has long been studied as a toy model for the four-dimensional Yang-Mills theory; both theories are asymptotically free and has a dynamically-generated mass gap, for example.
Despite the similarities between the two, it has been long unclear if there could be more direct and quantitative relations between the two. Is the CP N −1 -model only a toy model? Or could we use the CP N −1 -model to actually solve the four-dimensional Yang-Mills theory, e.g. for an analytical demonstration of confinement of the latter theory? 1 A positive result in this direction has been recently given in the recent work of the author and K. Yonekura [4]. As we will explain below, it was shown that a center-twisted compactification of four-dimensional pure SU(N) Yang-Mills theory on a three-torus T 3 = S 1 ×T 2 , when the size of the two-torus T 2 is small, gives rise to the two-dimensional CP N −1model 2 on the residual circle S 1 , with a flavor-twisted boundary condition studied in [5]. As emphasized in [4] this provides a well-defined weakly-coupled setup with no notorious IR problems (e.g. the Linde problem [6]). This means that we in principle have a hope of analytically-continuing back to the flat space R 4 -we first start in the weakly-coupled region and then sum up the perturbative as well non-perturbative contributions into a welldefined function (e.g. with the help of the Borel-Écalle resummation of the trans-series [7], as applied to an infinite-dimensional setup), and then adiabatically/analytically continue back to the flat R 4 by smoothly changing the size of the torus. 3 It would be, however, rather non-trivial to carry out this procedure in full generality, and before venturing into the detailed computation one might wish to further check the consistency of the result of [4], preferably in a non-perturbative manner.
The goal of this short note is to verify the consistency of this proposal by matching the 't Hooft anomalies [11] of the two theories (in four and two dimensions). While the standard anomaly is lost upon dimensional reduction due to contamination from high-energy states, we can study the recently-found mixed 't Hooft anomaly between time reversal and center symmetry in four dimensions, present for the special value of the theta angle θ = π [9]. We find that this 't Hooft anomaly, upon a twisted compactification on a three-torus T 3 , precisely reproduces to the 't Hooft anomaly for a twisted compactification of the twodimensional CP N −1 model on a circle, as recently derived in [10].

From 4d to 2d
Let us first recapitulate the some of the crucial statements from [4]. Consider fourdimensional pure SU(N) Yang-Mills theory on the geometry R × T 3 We denote the circumference of the circles by L A , L B , L C , and we choose the periodicity of In [4] the size of the circles are taken to be In this parameter region the Yang-Mills gauge field along the two-torus T 2 AB = S 1 A × S 1 B is given by the flat connection on the two-torus, which is known to be parametrized by a point of the complex projective space CP N −1 [12]. 4 After reduction along the two-torus T 2 AB we find that the resulting two-dimensional theory on R × S 1 A is given by the two-dimensional CP N −1 -model, where the theta angle of the four-dimensional theory is identified with that of the two-dimensional theory. In this paper we denote the homogeneous coordinate of Now the crucial ingredient of [4] is to include a Z N -center symmetry twist in the boundary conditions-we include 't Hooft discrete magnetic flux [13] along the two-torus It was then shown that this reproduces the twisted boundary condition of the two-dimensional theory along the residual circle S 1 C , given by an element of the flavor symmetry of the CP N −1 -model; In four-dimensional language, this Z N (zero-form) symmetry arises from the four-dimensional Z N (one-form) center symmetry by compactification along the circle S 1 B (and hence the notation). 6 Similarly, the four-dimensional Z N center symmetry compactified on the another cycle S 1 A generates another zero-form global symmetry of the two-dimensional symmetry. This is given by [4] (3) 4 The complex projective space CP N −1 arising from the moduli space of flat connections do not have the canonical Fubini-Study metric, and in particular has singularities where new degrees of freedom of Wbosons emerge. The analysis of [4], however, was done at the classical values of the flat connection which are away from these singularities. Such an analysis was enough to demonstrate the existence of fractional instantons and the dynamical recovery of the center symmetry. For this reason we expect that the subtlety coming from the singularities of the CP N −1 does not affect the discrete anomalies discussed in this paper. 5 Beaware that this is different from the other two-torus T 2 The center symmetry is a one-form symmetry [15], and hence upon a circle compactification generates a zero-form symmetry in addition to a one-form symmetry.
As we will see, this symmetry will play a crucial role in what follows.

Twisted Compactification of 't Hooft Anomaly
Let us now come to the 't Hooft anomalies.
As already mentioned above, the four-dimensional pure SU(N) Yang-Mills theory has the Z N center one-form symmetry; we denote the corresponding two-form discrete Z N gauge field as B. In addition the theory has a Z 2 time-reversal symmetry T.
The results of [9] shows that the pure Yang-Mills theory with theta angle θ = π has a mixed 't Hooft anomaly between the center symmetry and the time-reversal symmetry. This means that the partition of the theory Z 4d θ=π [(A, B)] at theta angle θ = π, regarded as a function of the background gauge fields A and B, is not invariant under the time-reversal symmetry: Let us next compactify the theory onto the geometry R × S 1 . We take the limit L A , L B , L C → 0, while still keep the hierarchy of scales as in (1). By decomposing the two-form gauge field B into components, we find the decomposition where B AB are one-forms and zero-forms on the residual R-direction, respectively, and none of them have any non-trivial dependence along the three-torus T 3 .
The one-forms B (1) and the zero-forms B (0) play the role of the 'electric field' and 'magnetic field' for the discrete Z N center symmetry. The electric objects are Wilson lines (holonomies) around the non-trivial cycles of the three-torus. The magnetic gauge field, on the other hand, represents the 't Hooft magnetic flux along a two-torus [13,14]. For example, the zero-form field B (0) BC represents the Aharanov-Bohm-type phase penetrating through the two-torus T BC = S 1 B × S 1 C , making the holonomies along S 1 B and S 1 C noncommutative.
Let us substitute the decomposition (5) into the expression for the mixed anomaly (4). After trivially integrating over the small three-torus directions, we find that the mixed anomaly (4) now is expressed as an integral over the residual R-direction: The expression appearing here is an analog of the Poynting vector of electromagnetism, but now for the discrete Z N center symmetry.
To this point we have not used any information regarding the choice of the boundary conditions. Recall from section 2 we turn on the 't Hooft magnetic flux is turned on along the two-torus T 2 BC = S 1 B ×S 1 C directions, and not in other directions involving S 1 A . Moreover, the value of B (0) BC can be derived from the fact that we have one unit of the 't Hooft discrete magnetic flux [4]: where U B and U C denotes the holonomy along the S 1 B and S 1 C . 7 We thus obtain and we arrive at the 't Hooft anomaly In other words, under the time-reversal symmetry the one-dimensional partition function Z 1d at θ = π, as a function of the background gauge field B (1) A , transforms non-trivially as It turns out that this is exactly the 't Hooft anomaly derived in [10], which discussed the Z N -twisted circle compactification of the two-dimensional anomaly discussed in [16]. Indeed, recall that the one-form gauge field B (1) A (5) arises from the reduction of the twoform field along the circle S 1 A , and hence should be associated with the zero-form global symmetry Z (A) N . As we have seen before, this symmetry acts on the homogeneous coordinates of CP N −1 by (3). This is nothing but the Z N 'shift symmetry' (denoted by (Z N ) S ) in [10], and hence our result (9) coincides with (3.13) of [10]. This completes our discussion of the twisted compactification of the 't Hooft mixed anomalies.

Discussion
The result of this note proves that the vacua of the two theories, namely four-dimensional pure SU(N) Yang-Mills theory on a center-twisted three-torus on the one hand, and twodimensional CP N −1 model on a flavor-twisted circle on the other, are constrained by the same 't Hooft anomaly. In particular we find that neither theory has a trivial vacuum.
It would be interesting to extend the analysis to four-dimensional Yang-Mills theory coupled with matters, say with adjoint or fundamental/anti-fundamental matters.
Our results provides a rather non-trivial non-perturbative consistency check of the proposal of [4], and make it even more plausible the optimistic scenario that the setup of [4] provides a right direction towards an analytic demonstration of the confinement and the mass gap of the asymptotically-free pure Yang-Mills theory, the holy grail of the subject.
The finding of this note also supports the claim in [4] that four-dimensional pure Yang-Mills theory with theta angle has N metastable vacua, 8 as expected from the presence of the N classical vacua in the CP N −1 -model [5,18,4]. In addition to theoretical curiosity, this has an interesting implication to the observability of the tensor modes in the recently-proposed axion-type model of inflation [19].