Circle compactification and ’t Hooft anomaly
Anomaly matching constrains low-energy physics of strongly-coupled field theories, but it is not useful at finite temperature due to contamination from high-energy states. The known exception is an ’t Hooft anomaly involving one-form symmetries as in pure Yang-Mills 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 one-form symmetries. We develop a systematic procedure for deriving an ’t Hooft anomaly of the circle-compactified theory starting from the anomaly of the original uncompactified theory without one-form 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.
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 low-energy 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 low-energy 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, higher-form 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 four-dimensional 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 Yang-Mills 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 strongly-coupled regime by decompactifying the circle especially in the large- limit. This finding motivated many studies of various asymptotically-free field theories by compactifying one direction to a circle with an appropriate boundary condition, and it is expected to map the strongly-coupled 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 circle-compactified 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 low-energy excitations. Since other high-energy states do not produce the anomaly, the effect of anomaly would disappear once those high-energy 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 three-dimensional free Dirac fermion, which has a symmetry and time-reversal symmetry . Consider the partition function under the gauge field , then
Here, is the eta invariant Witten:2015aba (), which is roughly the level- Chern–Simons action but is gauge-invariant modulo , and itself has no ’t Hooft anomaly. With the background gauge field, the time-reversal symmetry is broken because
That is, and has a mixed ’t Hooft anomaly and it is characterized by the Chern-Simons action. Now, we consider the three-dimensional manifold of the form , and let be small enough. We want to check the fate of above anomaly in the two-dimensional effective theory, whose symmetry is and . To gauge the symmetry of this two-dimensional 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 circle-compactified theory from the viewpoint of ’t Hooft anomaly matching.
What is recently found is that if the ’t Hooft anomaly involves a one-form 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 one-form symmetry that acts on Wilon lines. At the theory also has the time-reversal symmetry . If we consider the partition function with the background two-form gauge field for the center symmetry, the time-reversal symmetry is broken:
Therefore, there is a mixed ’t Hooft anomaly between the one-form symmetry and time-reversal symmetry. At , either of them must be broken spontaneously if we assume the mass gap at .
Now, we compactify one direction and set the four-dimensional manifold as , where the size of is sufficiently small, i.e., the temperature is sufficiently high. In this case, in addition to one-form symmetry, there exists zero-form symmetry that acts on Polyakov loop . In order to gauge these symmetries, we introduce the two-form gauge field and also the one-form gauge field . In the four-dimensional language, these gauge fields for three-dimensional effective theory can be regarded as
Substituting this form into the anomaly relation, we obtain
This suggests that there is a mixed ’t Hooft anomaly among zero-form, one-form, and time-reversal 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 one-form symmetries is the following: If the anomaly involves one-form 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 circle-compactified theory. This provides positive support to the idea of adiabatic continuity for Yang-Mills 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 circle-compactified 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 one-form symmetries, but still its vacuum property is sometimes constrained by ’t Hooft anomalies. The same situation occurs in some two-dimensional nonlinear sigma models, such as the model, which has no one-form 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 two-dimensional 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 high-energy 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 one-form symmetry exists, the above calculation for the case with one-form 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 one-form 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 two-dimensional model. Starting from two-dimensional ’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 four-dimensional 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.
In this section, we develop a systematic procedure for deriving an ’t Hooft anomaly of circle-compactified theories starting from the ’t Hooft anomaly of an original theory. Only when there exists one-form symmetry, it has been already well-understood that ’t Hooft anomaly survives even at finite temperatures Gaiotto:2017yup (). Our procedure given below derives the anomaly of circle-compactified theories when the original theory has no one-form 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 one-form in dimensions.
-gauge field that is locally a two-form in dimensions.
The physical interpretation is that the electric one-form symmetry appears after gauging and its charged object is the Wilson line
Gauging this one-form 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
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 Polyakov-loop matrix of along to a nontrivial one, and denote it as
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 boundary-condition-changing -gauge transformation, but we keep the periodic boundary condition with nontrivial holonomy during our explanation of the general strategy.
Before introducing the two-form gauge field , we originally have -dimensional one-form symmetry , and it induces -dimensional zero-form and one-form symmetries after circle compactification when is dynamical.
The zero-form 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 zero-form transformation, . We specify the holonomy such that there exists satisfying
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 zero-form symmetry , , in order to maintain the holonomy , and the symmetry of is obtained: We denote this zero-form 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 ()
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 one-form on . The -gauge field consists of two ingredients:
-gauge field that is locally a one-form on .
-gauge field that is locally a two-form on .
When is gauged, the -dimensional one-form symmetry emerges, so we can introduce which is a two-form on . Using these gauge fields, we define the -gauge field on as
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 zero-form symmetry induced from the one-form 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
where and are given in (11).
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 zero-form transformation acts on holonomy as . However, any elements commutes with , and then this zero-form 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
Since is a gauge-field on while is a -dimensional topological theory, we obtain . This means that the trivial boundary condition eliminates the anomaly of dimensions. Two-form gauge fields on are not enough for anomaly.
In our construction, the zero-form transformation on Polyakov-loop is translated into the faithful symmetry generated by on fields of QFT via (9). The appearance of the faithful zero-form symmetry allows us to introduce the -gauge field . Since and is intertwined in dimensions, it is built into the -dimensional two-form gauge field as a form . Therefore, the ’t Hooft anomaly can survive even if the original theory has no one-form 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 two-dimensional ’t Hooft anomaly. Two-dimensional sigma model can be realized as a gauged linear sigma model,
where is an -component complex vector-valued fields, and is the -gauge field
3.1 ’t Hooft anomaly and global inconsistency in two dimensions
Flavor symmetry, , which is given by with .
Time-reversal symmetry at .
Since the symmetry is gauged, the center elements of cannot act faithfully on gauge-invariant 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 two-form gauge field . To explain it, let us first gauge the symmetry, then the action obtained by the minimal coupling becomes
where . Flavor symmetry is gauged and no longer a global symmetry, but the theory acquires the one-form symmetry: Considering the and Wilson lines,
then the theory has a symmetry under the simultaneous rotation,
The Wilson lines charged under this one-form 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 two-form gauge field , and then we obtain
At , we consider the time-reversal transformation under the background flavor gauge field , and we obtain Komargodski:2017dmc ()
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 two-form 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
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 time-reversal 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 time-reversal 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).
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,
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 boundary-condition-changing 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 one-form symmetry and it induces the zero-form symmetry, . What is special for this choice of the nontrivial holonomy is that the above transformation induces the symmetry given by
We call this a shift symmetry, and it is a faithful transformation on the physical spectrum. As an example, a gauge-invariant operator is mapped to another gauge-invariant operator . When the transformation is performed as , the boundary condition for the transformed field becomes
That is, the zero-form 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
We give a short summary of the situation: The compactified theory obtained here has a zero-form symmetry, and it is induced by the one-form 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 time-reversal symmetry at . We introduce the one-form gauge field for gauging , which is independent of . Since is intertwined with the zero-form symmetry acting on the and Polyakov loops, we can embed it into the two-form gauge field in the two-dimensional language by setting . The two-dimensional anomaly (20) tells us that
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 two-dimensional anomaly.
3.3 Comparison with previous studies and Discussion
This behavior matches the ’t Hooft anomaly (20), because the time-reversal symmetry is spontaneously broken at . Our derivation of the anomaly (27) for -twisted model claims that the same multi-branch 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 quasi-ground states are composed of states and the -th ground-state energy behaves as
The ground state energy is thus given by minimum of these,
We can see that the time-reversal 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 four-dimensional example: Yang-Mills theory with massless Dirac fermions in the fundamental representation, i.e. massless QCD with . The action of this theory is given by
where is the color gauge field, is the covariant derivative, is the -gauge field strength, and are matrix-valued 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 vector-like flavor symmetry, , and the anomaly-free 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 four-dimensional computation, we derive the anomaly of the circle-compactified 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 Four-dimensional ’t Hooft anomaly of massless -flavor QCD
Here, we start with explanation on the vector-like 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 minimal-coupling procedure
where the covariant derivative is replaced by
The theory (32) has one-form 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,
with a certain one-form gauge field . This constraint respects the one-form 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
and define their gauge field strengths as
These field strengths transform under the one-form gauge transformation as and , and we obtain the gauge invariant combinations, and . This tells us that the introduction of two-form gauge field changes the action (32) as
We can shortly summarize the set of above procedures as follows: Introducing the flavor gauge field , the theory becomes bifundamental QCD, which has one-form symmetry, so we again introduce the two-form 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:
The last equality holds modulo
For , this additional phase is nontrivial, and the ’t Hooft anomaly exists between the flavor symmetry and the discrete axial symmetry . Four-dimensional 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 one-direction, and derive the associated three-dimensional effective theory. We fix the holonomy as
Equivalently, we introduce the boundary condition on the quark field as
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 zero-form transformation, , from the one-form 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 zero-form symmetry of , and we call this as the shift symmetry, , which acts on local operators on as
To obtain the three-dimensional 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 three-dimensional two-form gauge field . The -twisted partition function under these backgrounds are given by
We thus obtain under the transformation as