Circle compactification and ’t Hooft anomaly
Abstract
Anomaly matching constrains lowenergy physics of stronglycoupled field theories, but it is not useful at finite temperature due to contamination from highenergy states. The known exception is an ’t Hooft anomaly involving oneform symmetries as in pure YangMills theory at . Recent development about large volume independence, however, gives us a circumstantial evidence that ’t Hooft anomalies can also remain under circle compactifications in some theories without oneform symmetries. We develop a systematic procedure for deriving an ’t Hooft anomaly of the circlecompactified theory starting from the anomaly of the original uncompactified theory without oneform symmetries, where the twisted boundary condition for the compactified direction plays a pivotal role. As an application, we consider twisted sigma model and massless QCD, and compute their anomalies explicitly.
1 Introduction
Quantum field theory (QFT) provides us a universal description about collective quantum phenomena that appear in huge varieties of physical systems from particle and nuclear physics to condensed matter physics. However, we often encounter the situation where things we observe at low energies look completely different from microscopic degrees of freedom describing QFT. That happens when QFTs of our interest is strongly coupled, and nonperturbative aspects of QFTs are still in big mystery.
One of the possible approaches to tackle this situation is to find a rigorous nature of QFTs, especially related to symmetries and topologies. Symmetry has always played a key role in the development of QFTs; for instance, the idea of spontaneous symmetry breaking classifies traditional phases of matter following Landau’s characterization landau1937theory (); ginzburg1950theory (). In order to refine the data of QFTs related to symmetry, one can try to promote global symmetry to local gauge symmetry, but sometimes topology related to the symmetry gives an obstruction. Such obstruction is called an ’t Hooft anomaly, which is of great importance because of its preservation under the renormalization group flow tHooft:1979rat (); Frishman:1980dq (); Coleman:1982yg (): An ’t Hooft anomaly computed by the lowenergy effective theory must be equal to that of microscopic degrees of freedom. It provides us an important consistency check to determine the structure of vacuum and its lowenergy excitations and we can use it irrespective of QFTs of our interest being strongly coupled or not. Originally, anomaly matching was proposed for studying chiral symmetries of gauge theories with massless fermions tHooft:1979rat (); Frishman:1980dq (); Coleman:1982yg (). Recent development on topological phases of matter pushes that notion further Vishwanath:2012tq (); Wang:2014pma (); Kapustin:2014lwa (); Kapustin:2014zva (); Cho:2014jfa () and it is now applicable also for systems with discrete symmetries, higherform symmetries, and so forth, to derive nontrivial consequences on vacuum structures Witten:2015aba (); Seiberg:2016rsg (); Witten:2016cio (); Tachikawa:2016nmo (); Gaiotto:2017yup (); Wang:2017txt (); Tanizaki:2017bam (); Komargodski:2017dmc (); Komargodski:2017smk (); Cho:2017fgz (); Shimizu:2017asf (); Wang:2017loc (); Metlitski:2017fmd (); Kikuchi:2017pcp (); Gaiotto:2017tne ().
Although anomaly matching is a powerful technique to study nonperturbative physics, it cannot uncover details of dynamical aspects of QFTs and just provides us a consistency condition. For example, Yang–Mills theory is believed to exhibit confinement in fourdimensional spacetime, and an ’t Hooft anomaly can tell us about some additional information on vacuum assuming confinement, but it does not show how confinement can happen. Analytic computation of confinement in four dimensions is currently impossible because of its strong coupling nature. Still, it is found that the confinement of YangMills theory is realizable on by adding several massive adjoint fermions or deformations of the action itself, and this confinement is calculable with reliable semiclassical computations Unsal:2007vu (); Kovtun:2007py (); Unsal:2007jx (); Unsal:2008ch (); Shifman:2008ja (); Shifman:2009tp (). What is more interesting is that this semiclassical confinement is argued to be adiabatically connected to the confinement in the stronglycoupled regime by decompactifying the circle especially in the large limit. This finding motivated many studies of various asymptoticallyfree field theories by compactifying one direction to a circle with an appropriate boundary condition, and it is expected to map the stronglycoupled dynamics into the semiclassical regime without losing its essential information Cossu:2009sq (); Cossu:2013ora (); Argyres:2012ka (); Argyres:2012vv (); Dunne:2012ae (); Dunne:2012zk (); Poppitz:2012sw (); Anber:2013doa (); Basar:2013sza (); Cherman:2013yfa (); Cherman:2014ofa (); Misumi:2014raa (); Misumi:2014jua (); Misumi:2014bsa (); Dunne:2016nmc (); Misumi:2016fno (); Cherman:2016hcd (); Fujimori:2016ljw (); Sulejmanpasic:2016llc (); Yamazaki:2017ulc (); Buividovich:2017jea (); Aitken:2017ayq ().
In this paper, we would like to make a connection between these two recent developments of nonperturbative QFTs; adiabatic circle compactification and ’t Hooft anomaly matching. If the vacuum structures of the original and circlecompactified theories are really adiabatically connected, it is natural to think that both vacuum structures reproduce the same ’t Hooft anomaly matching condition. However, there is the following difficulty in this idea: Anomaly is renormalization group invariant, and it is matched by the vacuum or its lowenergy excitations. Since other highenergy states do not produce the anomaly, the effect of anomaly would disappear once those highenergy states give dominant contributions at finite temperature. How can this observation be consistent with the story about adiabatic continuity? In order to understand the situation better, we consider two quick examples.
Let us consider a threedimensional free Dirac fermion, which has a symmetry and timereversal symmetry . Consider the partition function under the gauge field , then
(1) 
Here, is the eta invariant Witten:2015aba (), which is roughly the level Chern–Simons action but is gaugeinvariant modulo , and itself has no ’t Hooft anomaly. With the background gauge field, the timereversal symmetry is broken because
(2) 
That is, and has a mixed ’t Hooft anomaly and it is characterized by the ChernSimons action. Now, we consider the threedimensional manifold of the form , and let be small enough. We want to check the fate of above anomaly in the twodimensional effective theory, whose symmetry is and . To gauge the symmetry of this twodimensional theory, we set with independent , which is a connection on . The anomaly vanishes with this gauge field , since . As we can see in this example, the anomaly is characterized by a topological invariant of the background gauge field , and if we make be independent of one compactified direction the topological invariant vanishes identically. Physical interpretation is that thermal fluctuation appears after circle compactification and information of topology is lost because of it. This observation seems to be generic and shows the fundamental difficulty in making a connection between vacuum structures of the original theory and the circlecompactified theory from the viewpoint of ’t Hooft anomaly matching.
What is recently found is that if the ’t Hooft anomaly involves a oneform symmetry then it does survive even at finite temperatures Gaiotto:2017yup (). The known example is the Yang–Mills theory at . Yang–Mills theory has the oneform symmetry that acts on Wilon lines. At the theory also has the timereversal symmetry . If we consider the partition function with the background twoform gauge field for the center symmetry, the timereversal symmetry is broken:
(3) 
Therefore, there is a mixed ’t Hooft anomaly between the oneform symmetry and timereversal symmetry. At , either of them must be broken spontaneously if we assume the mass gap at .
Now, we compactify one direction and set the fourdimensional manifold as , where the size of is sufficiently small, i.e., the temperature is sufficiently high. In this case, in addition to oneform symmetry, there exists zeroform symmetry that acts on Polyakov loop . In order to gauge these symmetries, we introduce the twoform gauge field and also the oneform gauge field . In the fourdimensional language, these gauge fields for threedimensional effective theory can be regarded as
(4) 
Substituting this form into the anomaly relation, we obtain
(5) 
This suggests that there is a mixed ’t Hooft anomaly among zeroform, oneform, and timereversal symmetries. Even at finite temperatures, one of these three symmetries must be spontaneously broken. The intuitive difference between anomalies involving only ordinary symmetry and containing oneform symmetries is the following: If the anomaly involves oneform symmetry, the line operator wrapping around is affected by the compactified direction even if that direction is small, and information of topology survives in the circlecompactified theory. This provides positive support to the idea of adiabatic continuity for YangMills theory with adjoint matters Unsal:2007vu (); Kovtun:2007py (); Unsal:2007jx (); Unsal:2008ch (); Shifman:2008ja (); Shifman:2009tp () because we can claim that vacuum structures of the original and circlecompactified theories are controlled by the same ’t Hooft anomaly.
What happens if the gauge theory contains some matter fields not in the adjoint representation like quantum chromodynamics (QCD)? Typically, such theories do not have oneform symmetries, but still its vacuum property is sometimes constrained by ’t Hooft anomalies. The same situation occurs in some twodimensional nonlinear sigma models, such as the model, which has no oneform symmetries and an ’t Hooft anomaly exists. There is a circumstantial evidence that ’t Hooft anomalies should survive under circle compactifications even for these cases from the viewpoint of adiabatic continuity: Adiabatic continuity of twodimensional sigma models seems to be valid under specific boundary conditions on Dunne:2012ae (); Dunne:2012zk (); Cherman:2013yfa (); Misumi:2014jua (); Misumi:2014bsa (), and it is rigorously proven for or sigma models in large limit Sulejmanpasic:2016llc (). It is elucidated in the large limit that the twisted boundary condition eliminates most of contributions to the partition function from highenergy states Basar:2013sza (); Sulejmanpasic:2016llc () as seen in Witten index of supersymmetric theories Witten:1982im (), and the property of the vacuum is correctly captured at any size of circle compactification.
In this paper, we positively answer the question whether there is any example in which an ’t Hooft anomaly only of ordinary symmetries survives after circle compactification. We develop a systematic procedure generating the anomaly in the circle compactification starting from the anomaly of the original uncompactified theory, where the twisted boundary condition plays a pivotal role in our construction of anomaly. Although no oneform symmetry exists, the above calculation for the case with oneform symmetry Gaiotto:2017yup () gives us a strong motivation of our construction. Theories covered by our procedure contain sigma model with twisted boundary condition, massless QCD with with twisted boundary condition (QCD) Kouno:2012zz (); Sakai:2012ika (); Kouno:2013zr (); Kouno:2013mma (); Poppitz:2013zqa (); Iritani:2015ara (), and so on. Our systematic procedure is valid even away from large limit so long as an ’t Hooft anomaly exists.
The paper is organized as follows. In Section 2, we establish a systematic procedure to compute the ’t Hooft anomaly of the circle compactified theory when the original one has no oneform symmetry. We there give a concrete construction of the anomaly, and the importance of appropriately twisted boundary condition is clarified. In Section 3, we demonstrate our method in twodimensional model. Starting from twodimensional ’t Hooft anomaly of model at , we derive the anomaly of twisted model on . In Section 4, we discuss an anomaly of massless QCD and derive it starting from fourdimensional massless QCD. We compare the ’t Hooft anomaly with results of previous studies, and discuss the application of anomaly matching to the phase diagram. Section 5 is devoted to conclusion and discussion.
2 Formalism
In this section, we develop a systematic procedure for deriving an ’t Hooft anomaly of circlecompactified theories starting from the ’t Hooft anomaly of an original theory. Only when there exists oneform symmetry, it has been already wellunderstood that ’t Hooft anomaly survives even at finite temperatures Gaiotto:2017yup (). Our procedure given below derives the anomaly of circlecompactified theories when the original theory has no oneform symmetries. It turns out that the boundary condition twisted by global symmetry plays an important role for nonvanishing anomaly.
2.1 Systematic procedure for anomaly with circle compactification
We consider a dimensional quantum field theory (QFT), and assume that the QFT has two symmetries and acting on its physical Hilbert space faithfully. We call as a flavor symmetry, and consider the case when with and . It is straightforward to extend the discussion for the case and (center of ). However, it makes notations for the following discussion complicated, and we try to make our explanation as simple as possible.
We further assume that and have a mixed ’t Hooft anomaly but and have no anomaly. In order to see the anomaly, we introduce the background gauge field for the flavor symmetry . gauge field consists of two ingredients Kapustin:2014gua (); Gaiotto:2014kfa (); Aharony:2013hda ():

gauge field that is locally a oneform in dimensions.

gauge field that is locally a twoform in dimensions.
The physical interpretation is that the electric oneform symmetry appears after gauging and its charged object is the Wilson line
(6) 
Gauging this oneform symmetry by , the Wilson line is no longer a genuine line operator and we obtain gauge theory instead of gauge theory. We denote the partition function of this QFT under the background gauge field as . The above assumption on anomaly implies that
(7) 
where , is the transformation of gauge field , and is a dimensional topological gauge theory
In order to consider the circle compactification, we set dimensional manifolds as , and the size of its dimensional part is much larger than the circle of size . At this stage, all the fields of QFT obey the periodic boundary condition along . To describe the theory at first, we turn off and dimensional components of , i.e., . In this process, we can still fix the Polyakovloop matrix of along to a nontrivial one, and denote it as
(8) 
For each that is uniform on , we obtain a dimensional QFT on , and denote its partition function as . This is equivalent to imposing a twisted boundary condition on fields of QFT along by performing a boundaryconditionchanging gauge transformation, but we keep the periodic boundary condition with nontrivial holonomy during our explanation of the general strategy.
Before introducing the twoform gauge field , we originally have dimensional oneform symmetry , and it induces dimensional zeroform and oneform symmetries after circle compactification when is dynamical.
The zeroform symmetry acts as with some , and we thus identify
In order to have a nontrivial anomaly on , we need to have a symmetry involving the above zeroform transformation, . We specify the holonomy such that there exists satisfying
(9) 
Since is not an element of the center of by definition, . Recall that acts faithfully on the physical Hilbert space, and it means that generates a faithful symmetry of QFT, which we call a “shift symmetry”. When the shift symmetry acts on fields, the holonomy matrix is changed to . The requirement (9) states that the symmetry generated by is intertwined with the zeroform symmetry , , in order to maintain the holonomy , and the symmetry of is obtained: We denote this zeroform symmetry generated by as in order to distinguish it from the original one, . Because of (9), cannot be proportional to the identity matrix: Typical example of and satisfying (9) is ()
(10) 
Since the flavor symmetry of dimensional theory must commute with , flavor symmetry might be explicitly broken to a maximal Abelian subgroup as . Symmetry with the faithful representation is again given by the quotient . Let us assume that is not explicitly broken by fixing , then the dimensional effective theory has three symmetries; shift symmetry , flavor symmetry , and .
Let us try to introduce the background gauge fields for and . We denote the gauge field as that is locally a oneform on . The gauge field consists of two ingredients:

gauge field that is locally a oneform on .

gauge field that is locally a twoform on .
When is gauged, the dimensional oneform symmetry emerges, so we can introduce which is a twoform on . Using these gauge fields, we define the gauge field on as
(11) 
The field in the expression (11) may require some explanations: When it appears in , it is regarded as a gauge field for the subgroup of the flavor symmetry, while it should be regarded as a gauge field for the zeroform symmetry induced from the oneform symmetry when it appears inside . We use this slightly abused notation in order to emphasize that these two transformations are intertwined as a symmetry of . Let be the partition function with the background gauge fields for and , then the given construction of the dimensional theory shows that
(12) 
where and are given in (11).
We can now derive the mixed dimensional anomaly among , , and as follows: Using the dimensional expression of the dimensional theory, (11) and (12), we obtain
(13)  
Here, we use the anomaly relation (7) in dimensions. Since and have no dependence on , the compactified direction of can be integrated out in a trivial manner. As a result, becomes the dimensional topological gauge theory of and , and this defines the anomaly of the dimensional QFT . That is, the dimensional QFT has a mixed ’t Hooft anomaly among , , and , and the trivial gapped state is forbidden.
2.2 Comments on choice of the background holonomy
One of the most important part in our construction of dimensional anomaly via circle compactification is the choice of nontrivial holonomy background . In order to clarify the importance of the condition (9), let us take the trivial one as a “bad” example.
Whether or not (9) is satisfied, the zeroform transformation acts on holonomy as . However, any elements commutes with , and then this zeroform transformation on the Polyakov loop has no connection with the symmetry of dimensional system. Since it is not a faithful symmetry of the system, we must set in the trivial choice . Instead, the flavor symmetry is unbroken, and the corresponding gauge field on is denoted as . As a result, the anomaly relation (13) becomes
(14) 
Since is a gaugefield on while is a dimensional topological theory, we obtain . This means that the trivial boundary condition eliminates the anomaly of dimensions. Twoform gauge fields on are not enough for anomaly.
In our construction, the zeroform transformation on Polyakovloop is translated into the faithful symmetry generated by on fields of QFT via (9). The appearance of the faithful zeroform symmetry allows us to introduce the gauge field . Since and is intertwined in dimensions, it is built into the dimensional twoform gauge field as a form . Therefore, the ’t Hooft anomaly can survive even if the original theory has no oneform symmetry.
3 twisted sigma model
As a demonstration of the systematic procedure in Sec. 2, we calculate the anomaly of twisted model starting from the twodimensional ’t Hooft anomaly. Twodimensional sigma model can be realized as a gauged linear sigma model,
(15) 
where is an component complex vectorvalued fields, and is the gauge field
3.1 ’t Hooft anomaly and global inconsistency in two dimensions
The symmetry of this theory (15) consists of the following Komargodski:2017dmc ():

Flavor symmetry, , which is given by with .

Timereversal symmetry at .
Since the symmetry is gauged, the center elements of cannot act faithfully on gaugeinvariant operators. The flavor symmetry with the faithful representation is thus given by . In the notation used in Sec. 2, we have the following correspondence: , , , and .
We introduce the background gauge field for flavor symmetry. Such a background gauge field consists of two ingredients: gauge field and twoform gauge field . To explain it, let us first gauge the symmetry, then the action obtained by the minimal coupling becomes
(16) 
where . Flavor symmetry is gauged and no longer a global symmetry, but the theory acquires the oneform symmetry: Considering the and Wilson lines,
(17) 
then the theory has a symmetry under the simultaneous rotation,
(18) 
The Wilson lines charged under this oneform symmetry must be dropped from the spectrum of genuine line operators if we appropriately gauge the flavor symmetry Kapustin:2014gua (); Gaiotto:2014kfa (). For this purpose, we introduce the twoform gauge field , and then we obtain
(19) 
At , we consider the timereversal transformation under the background flavor gauge field , and we obtain Komargodski:2017dmc ()
(20) 
We should check whether this anomaly is genuine or fake, so we consider whether it can be canceled by local counter terms of . The topological twoform gauge theory is given by with some integer modulo , and it is a candidate for the counterterm. The transformation after adding this counterterm behaves as
(21) 
Thus, anomaly is fake if and only if modulo . For even , this is impossible and we find the ’t Hooft anomaly between the flavor and timereversal symmetries.
For odd , we can eliminate the anomaly by choosing modulo , and no ’t Hooft anomaly exists. If we do the same computation at , the timereversal symmetry is respected by choosing , and thus there is no common counterterm that respects both at . This is the global inconsistency condition, and we can derive a nontrivial consequence although it is slightly weaker than ’t Hooft anomaly matching (see Ref. Kikuchi:2017pcp () for global inconsistency condition).
In two dimensions, Coleman–Mermin–Wagner theorem Coleman:1973ci (); mermin1966absence () tells us that flavor symmetry with a continuous parameter cannot be broken. This naturally gives us the nonperturbative data that timereversal symmetry at is spontaneously broken for twodimensional model.
3.2 twisted model and its anomaly
We consider the circle compactification from to , where the circumference of the circle , i.e. , is regarded to be small. We impose the twisted boundary condition,
(22) 
where ()
(23) 
We call this as twisted sigma model, and we denote its partition function at as .
For our purpose, it is better to regard this twisting matrix as a holonomy of gauge field along compactified direction. Indeed, the boundaryconditionchanging gauge transformation makes the boundary condition of periodic (up to gauge symmetry) and the price to be paid is the background holonomy . When is gauged, there is a oneform symmetry and it induces the zeroform symmetry, . What is special for this choice of the nontrivial holonomy is that the above transformation induces the symmetry given by
(24) 
We call this a shift symmetry, and it is a faithful transformation on the physical spectrum. As an example, a gaugeinvariant operator is mapped to another gaugeinvariant operator . When the transformation is performed as , the boundary condition for the transformed field becomes
(25) 
That is, the zeroform symmetry on the Polyakov loop is intertwined with the shift symmetry generated by in order to maintain the boundary condition (22), and we call it . The symmetry acts on local operators on as
(26) 
We give a short summary of the situation: The compactified theory obtained here has a zeroform symmetry, and it is induced by the oneform symmetry in two dimensions when is gauged. Continuous part of the flavor symmetry is explicitly broken to , but it is not relevant for the following discussion and we do not introduce gauge fields for it.
As a result, the ’t Hooft anomaly (or global inconsistency) in two dimensions has the same meaning in the twisted model on . The twisted model has the shift symmetry and the timereversal symmetry at . We introduce the oneform gauge field for gauging , which is independent of . Since is intertwined with the zeroform symmetry acting on the and Polyakov loops, we can embed it into the twoform gauge field in the twodimensional language by setting . The twodimensional anomaly (20) tells us that
(27)  
We find that and at has an ’t Hooft anomaly (or global inconsistency depending on even or odd ), and either of them must be spontaneously broken.
For usual periodic boundary condition, it means that we put and thus there is no room to introduce . Therefore, we cannot obtain ’t Hooft anomaly in such cases. The emergence of symmetry by the twisted boundary condition is essential for a deep connection with twodimensional anomaly.
3.3 Comparison with previous studies and Discussion
The angle dependence of model in two dimensions is studied in large limit Witten:1978bc (); Affleck:1979gy (), and the ground state energy should behave as
(28) 
This behavior matches the ’t Hooft anomaly (20), because the timereversal symmetry is spontaneously broken at . Our derivation of the anomaly (27) for twisted model claims that the same multibranch structure would naturally appear under adiabatic circle compactification.
Indeed, dependence of twisted model is studied in Refs. Dunne:2012ae (); Dunne:2012zk (). Under the twisted boundary condition, there are types of fractional instantons which has the topological charge . As a result, the quasiground states are composed of states and the th groundstate energy behaves as
(29) 
The ground state energy is thus given by minimum of these,
(30) 
We can see that the timereversal symmetry is spontaneously broken at , which satisfies matching of ’t Hooft anomaly or global inconsistency (27). What we have shown in this paper is that these two behaviors (28) and (29) are both consistent with anomalies, and those anomalies have essentially the same origin.
We argue that this observation gives a positive support for the adiabatic continuity. For twisted model, it is rigorously shown that expectation values of any invariant operators does not depend on in the large limit Sulejmanpasic:2016llc (), and our consideration on anomaly gives a complementary and consistent analysis.
4 Massless Qcd
As another demonstration, we consider a fourdimensional example: YangMills theory with massless Dirac fermions in the fundamental representation, i.e. massless QCD with . The action of this theory is given by
(31) 
where is the color gauge field, is the covariant derivative, is the gauge field strength, and are matrixvalued Dirac fermions. The color group acts on from left, and the flavor group acts on from right, i.e., for ; the quark field is in the bifundamental representation of the color and flavor groups.
This theory possesses various symmetries, but we pay attention only to the vectorlike flavor symmetry, , and the anomalyfree discrete subgroup of the axial symmetry, , in our demonstration. The complete analysis involving other symmetries will be discussed at future opportunity. We first compute the ’t Hooft anomaly of the above symmetries in four dimensions. Using the fourdimensional computation, we derive the anomaly of the circlecompactified theory with the twisted boundary condition, QCD.
Our discussion can be generalized to the case when color and flavor have different numbers so long as they have a nontrivial common divisor, . For notational simplicity, we only consider the case in this paper.
4.1 Fourdimensional ’t Hooft anomaly of massless flavor QCD
Here, we start with explanation on the vectorlike flavor symmetry and . In the notation used in Sec. 2, we have the following correspondence: , , , and .
Quark field is in the bifundamental representation, for , and thus the subgroup generated by does not act on faithfully (). Therefore, the group acting faithfully on is . Since the color group symmetry is gauged, the flavor symmetry with the faithful representation on the physical Hilbert space is given by .
Since the quark field is massless, there is a symmetry , , at the Lagrangian level, but the fermion integration measure generates the additional term due to the index theorem. Therefore, it is a symmetry only when is quantized to , and is explicitly broken to by quantum anomaly.
We shall derive the mixed ’t Hooft anomaly between and , and we introduce the background gauge field of for that purpose.
We first introduce the flavor gauge field , then the minimalcoupling procedure
(32) 
where the covariant derivative is replaced by
(33) 
The theory (32) has oneform symmetry that does not exist in the original massless QCD. It acts on the color and flavor Wilson lines, and , as the simultaneous rotation of phase,
(34) 
This symmetry does not arise in the gauge theories, and we have to introduce twoform gauge field Kapustin:2014gua (); Gaiotto:2014kfa (), which is a twoform gauge field satisfying
(35) 
with a certain oneform gauge field . This constraint respects the oneform gauge invariance under and , and respecting this gauge invariance prevents us from adding extra degrees of freedom to the theory. We introduce the gauge fields made of gauge fields , , and a gauge field , as
(36) 
and define their gauge field strengths as
(37) 
These field strengths transform under the oneform gauge transformation as and , and we obtain the gauge invariant combinations, and . This tells us that the introduction of twoform gauge field changes the action (32) as
(38) 
We can shortly summarize the set of above procedures as follows: Introducing the flavor gauge field , the theory becomes bifundamental QCD, which has oneform symmetry, so we again introduce the twoform gauge field Tanizaki:2017bam (); Shimizu:2017asf ().
Let us perform the axial rotation, then the Lagrangian is invariant again, but the action acquires the additional topological term due to the fermion measure:
(39) 
The last equality holds modulo
(40) 
For , this additional phase is nontrivial, and the ’t Hooft anomaly exists between the flavor symmetry and the discrete axial symmetry . Fourdimensional QCD is believed to break the chiral symmetry spontaneously, which also breaks axial symmetry to spontaneously ( is the fermion number operator), and the ’t Hooft anomaly is matched.
4.2 Massless QCD and its anomaly
We compactify onedirection, and derive the associated threedimensional effective theory. We fix the holonomy as
(41) 
Equivalently, we introduce the boundary condition on the quark field as
(42) 
The extended gauge transformation eliminates the holonomy, but the quark field obeys the twisted boundary condition. This is called QCD, and we denote its partition function as .
Circle compactification induces zeroform transformation, , from the oneform symmetry (34), but it changes the boundary condition and maps a theory to another theory . We should intertwine it with the flavor rotation , where is defined in (10), in order to maintain the boundary condition. This generates the zeroform symmetry of , and we call this as the shift symmetry, , which acts on local operators on as
(43) 
To obtain the threedimensional anomaly, we gauge the shift symmetry and denote the corresponding gauge field as . Because of the holonomy , the explicit breaking of the flavor symmetry occurs , and the faithful flavor symmetry is . We introduce the background gauge field and threedimensional twoform gauge field . The twisted partition function under these backgrounds are given by
(44) 
We thus obtain under the transformation as