[2507.12802[hep-th]] Fractional instantons in 2d ℂPN-1 model and 4d Yang-Mills theory with 't Hooft twists

with Yui Hayashi, Tatsuhiro Misumi, Muneto Nitta, Keisuke Ohashi

Yui gave a talk on our work on the monopole-vortex continuity (2405.12402[hep-th]) at the JPS autum meeting in the last year, and Nitta-san proposed to consider its application of this continuity idea for the monopole-vortex composite system in the gauge-Higgs systems, which initiated this project. The effective theory on the vortex is given by the  2d ℂPN-1 sigma model, and we started to think about its fractional instanton solutions under twisted boundary conditions. 

Ohashi-san immediately noticed that the BPS equation can be solved with the theta functions, and he gave two alternative realizations of the analytic formula to identify the global structure of the moduli space. We then find that the moduli space for the topological charge k+p/N is given by the ℂPNk+p-1-fiber bundle over the 2-torus for the translational moduli. 

Yui then pointed out that this global sturcture of the moduli space should be the same also for the fractional instantons for the 4d Yang-Mills theory on the 4-torus with 't Hooft twists. We can perform the dimensional reudction to obtain the 2d non-Fubini-Study ℂPN-1 model with the clock-shift twisted boundary condition, and Yui gave a nice interpretation of 't Hooft's toron solution on the 4-torus with the specific aspect ratio from this 2d viewpoint. Combined with our computation for the moduli space of 2d ℂPN-1 model, we can explicitly confirm that the moduli space of these two theories are actually the same for special choices of the 't Hooft twists. 

[arXiv:2505.16546[hep-th]] Twisted Partition Functions as Order Parameters

with Jun Maeda

This is a paper in collaboration with Jun Maeda, a PhD student at Kyoto University. Last year, we had an intensive lecture on the tensor renormalization group (TRG) by Shinichiro Akiyama, and we learned that it provides a way to directly compute the partition function, which has a sharp contrast with the Monte Carlo simulations. This brings us the idea to study the behaviors of the partition function with the background gauge field in details as an order parameter to diagnose quantum phases of matter. 

We explicitly compute the symmetry-twisted partition function for discrete symmetries for several concrete examples, and we find that it discriminates the spontaneous symmetry breaking (SSB), symmetry-protected topological (SPT) states, and also symmetry-enriched topological (SET) states. In particular, I find it very intersting that the symmetry fractionalization class for the SET states can be nicely extracted by this method. 

By applying this method to study the 4d gapped phases with the Z_N 1-form symmetry, we reproduce the Wilson-'t Hooft classification by the area-versus-perimeter law of the order and disorder operators. The examples of U(1) symmetry are also studied, and we find the importance of the mixed anomaly between the broken symmetry and the emergent solitonic symmetry. 

[2412.17493[cond-mat.str-el]] Infrared properties of two-dimensional SU(N)/H nonlinear σ models at nonzero θ angles

with Philippe Lecheminant and Keisuke Totsuka

In the last summer (July 2023), Philippe visited Yukawa Institute to work with Keisuke, and I had a chance to discuss with them about the 2D flag-manifold sigma model related to one of my previous works. About the flag manifold sigma model, the seminal work by Lajko, Wamer, Mila, and Affleck (1706.06598) predicts that the SU(N)/U(1)N-1 sigma model at the specific θ angles flows to the SU(N) level-1 WZNW CFT, and my previous work (1805.11423) with Tin Sulejmanpasic shows that it satisfies the anomaly matching and that there is actually a continuous local deformation connecting them. Drinking coffee in Keisuke's office, Philippe, Keisuke, and I were wondering if there is a stronger evidence for supporting the massless RG flows by investigating the dynamics in more detail. 

Philippe and Keisuke developed an idea of extending the work by Shanker and Read that considers the lattice strong coupling, and I started considering that idea. But, I encountered the difficulty due to the fact that the low-energy property of the SU(N) antiferromagnetic spin chain itself is unknown unlike the SU(2) case, and I started to consider if there is more field-theoretic approach to study the RG flow. In the previous work with Tin, I built an explicit connection between the flag-manifold sigma model and the WZNW CFT by extending the Affleck-Haldane argument from SU(2) to SU(N). I then came up with the idea of applying the short-distance OPE to the classical potential appearing in Affleck-Haldane's argument to derive the leading perturbation to the CFT fixed point. I quickly confirmed it actually works well for the ℂP1 sigma model at theta=pi by deriving the current-current perturbation of the WZNW CFT including its sign. This determination of the sign is crucial as the current-current perturbation is marginal at the CFT point, i.e. it becomes either marginally relevant or irrelevant depending on its sign. 

I told this field-theoretic idea to Philippe and Keisuke. They found it interesting, but they were also skeptical at the beginning because the result crucially depends on the sign for the current-current operator and then there was a concern that it is easily affected by the subtle Klein factor out of the Abelian bosonization procedures. Philippe and Keisuke then worked on the detailed computations of OPE for various setups. Some models are known to enjoy massless RG flows thanks to the technique of integrability, and we found that this idea always gives the consistent prediction for such special cases. 

[2410.21392[hep-th]] Monopole-Vortex Continuity of N=1 Super Yang-Mills Theory on R2xS1xS1 with 't Hooft Twist

with Yui Hayashi, Tatsuhiro Misumi

After I gave a talk on the 2d center-vortex semiclassics in the QFT workshop at Komaba this March, Tatsu sent me a strong message that we should establish the adiabatic continuity between the 4d confining vacuum and the 2d semiclassical regime. While there is no doubt that it is an important task, it has to be a very tough problem as this is almost equivalent to solving the Yang-Mills mass gap problem. Therefore, as a first step toward this big goal, I suggested working on the "weak-weak" continuity for SYM on R3xS1 and R2xT2 since this was already nontrivial but expected to be easier than the "strong-weak" continuity. 

This setup resembles the work (2405.12402[hep-th]) with Yui on deformed YM theory, but the presence of massless adjoint fermion causes an essential difference: BPS and KK monopoles have two fermionic zero modes, so they contribute to the chiral condensate but does not produce the string tension. The confining-string formation is due to bions, a composite of monopoles that carries no topological charge (such as BPS-KK*). However, BPS and KK are no longer discriminated after the T2 compactification with the 't Hooft twist as they both become the center vortex. It leads to the deconfinement of the 2d Wilson loop on the T2 setup, which is consistent with the anomaly matching. The 2d mixed center-chiral anomaly forces that the Wilson loop should flow to the chiral-symmetry generator for gapped systems (Sec. 3 of 2201.06166[hep-th]), but this poses a puzzle about its continuity to the 3d confining vacuum. We need to get a microscopic understanding of these aspects that were absent in the deformed YM case.

Yui proposed a nice solution to these problems: the double-string picture and the wall-like confining string. With the 't Hooft twist, the confining string out of the Wilson loop forms the wall-like configuration separating the inside and outside that belong to different chiral broken vacua. Therefore, the local 3d (or 4d) physics keeps showing confinement under the T2 compactification, but the asymptotically large 2d Wilson loops show the perimeter law, which explains how two seemingly contradictory behaviors become consistent. Moreover, the quasi-moduli integration of the bion configuration shows that the magnetic bion saddle merges with other saddles and becomes unstable one under the T2 compactification. These results provide the microscopic description for the weak-weak continuity of SYM vacua.

[2405.12402[hep-th]] Unifying Monopole and Center Vortex as the Semiclassical Confinement Mechanism

with Yui Hayashi

One day in April, after Yui gave her group seminar on our recent work of the eta' periodicity, we chatted about the possibility of the weak-weak continuity between two semiclassical descriptions of color confinement: 3d monopole semiclassics on R3xS1 and 2d center-vortex semiclassics on R2xT2. While drawing several figures and staring blankly at the 3d dual-photon Lagrangian, we suddenly noticed that the fundamental monopoles can be smoothly transmuted to the center vortex. 

Over the weekend, Yui checked how this idea works concretely at the level of the partition function, and she identified the classical vacua of the dual photons in the 't Hooft twisted boundary condition serves as the discrete label for the N confinement states in the 2d center-vortex theory. We also solved the classical equation of motion for the monopole in the dual-photon description and found that the magnetic flux gets squeezed into a vortex due to the perturbative mass gap caused by the twisted boundary condition. 

I was wondering if there exists a nice connection between these semiclassical descriptions over two years since I developed the center-vortex semiclassics in 2201.06166[hep-th], and I was surprised and quite fascinated that we could solve it in such a neat way. I find our discovery quite exciting as it opens new directions for the semiclassical theories of quark confinement. 

[2404.07595[hep-lat]] ℤ3 lattice gauge theory as a toy model for dense QCD

with  Yoshimasa Hidaka, Arata Yamamoto

Properties of finite-density QCD are still veiled due to the lack of reliable computational tools. Standard Euclidean lattice formalism suffers from the sign problem, and some people expect that the quantum computation with the Hamiltonian formalism might solve this difficulty. Although I am not sure if this is true, it would be interesting to develop a new technology to tacke difficult problems. 

When considering the Hamiltonian formalism of field theories, we need to decide how to manage its exponentially large Hilbert space. Especially in quantum gauge theories, the gauge fields have the infinite-dimensional Hilbert space even at finite volumes and the truncation is unavoidable. In the autumn of 2023, Arata, Yoshimasa, and I have a chance to discuss this problem in person at the Yukawa institute, and Arata proposed us that we should develop a nice toy model with the finite gauge theory. In our study, we found that certain finite gauge theories exhibit striking similarities with the anticipated behaviors of finite-density QCD, as evidenced by our sketch of the phase diagram of ℤ3 gauge theories.

[2307.16655[hep-lat]] Calculating composite-particle spectra in Hamiltonian formalism and demonstration in 2-flavor QED1+1d

with Etuko Itou and Akira Matsumoto

When we started this project, if I remember correctly, Akira was just graduating the PhD from KEK (which means that this took us more than a year). He is interested in the new approach potentially free from the sign problem, and Etsuko, Akira and I started the collaboration on the DMRG approach. This is an extension of the previous study about the 1-flavor Schwinger model (2210.04237[hep-lat]), but the mass spectrum of 2-flavor theory is highly rich compared with the 1-flavor case. Numerical simulations are mostly done by Akira, and I helped the idea for analyses and gave  theoretical backgrounds. 

We worked on three methods for computing the mass spectrum of gauge theories in the Hamiltonian formalism: (1) correlation-function scheme, (2) one-point function scheme, and (3) dispersion-relation scheme. In all these methods, we find stable composite particles, pions, sigma and eta mesons. Notably, the mass of the sigma meson is lighter than twice the pion mass, making it stable against the decay process σ → ππ, unlike 4d QCD. The numerical findings closely match the analytic prediction based on the WKB approximation, with Mσ/Mπ = √3 pointed out by Coleman long time ago, and it's very curious to me why the semiclassical method work unreasonably so well as parameters seem to be outside the range of its validity.

[2307.00939[hep-th]] Solitonic symmetry as non-invertible symmetry:  cohomology theories with TQFT coefficients

with Shi Chen

This is a followup work of our previous study (2210.13780[hep-th]) that has uncovered the noninvertible structure in the symmetry of topological solitons for the 4d CP^1 sigma model. One day, Shi found that there is a beutiful mathematical story behind the general structure of the solitonic symmetry, which was very striking and unexpected for me. 

We develop a new approach called solitonic cohomology or cohomology with TQFT coefficients to study noninvertible solitonic symmetries, and this perspective allows for the identification of invertible solitonic subsymmetries and sheds light on the topological origin of non-invertibility in solitonic symmetry. To treat it systematically, we need resort to higher category theories,  with which I was completely unfamiliar before doing this study and Shi helped me a lot with detailed explanations. 

[2304.14815[hep-lat]] Magnetic operators in 2D compact scalar field theories on the lattice

with Motokazu Abe, Okuto Morikawa, Soma Onoda, and Hiroshi Suzuki

When I visited Kyushu University in the last December, we also started considering about the properties of magnetic monopoles on the lattice. In the previous study by Motokazu, Okuto and Hiroshi, they defined the topological theta angle for the 4D Maxwell theory. To this end, the admissibility condition plays the important role, but it rules out the magnetic objects such as the 't Hooft loop, and we should develop some new techniques to observe the interesting phenomena, such as Witten effect, on the lattice field theories with well-defined topological properties.

I pointed out that these problems should exist in a simpler model, i.e. 2D compact scalar theories, and we should try to solve them in this setup. By proposing the excision method, we find that many observations in the continuum descriptions are nicely derived at the finite lattice spacing. Moreover, we consider a model with two compact scalars, which accepts the continuous theta angle, and we show that the Witten effect can be reproduced on the lattice in the ultra-local way in this 2D lattice model. 

[2210.13780[hep-th]] Solitonic symmetry beyond homotopy: invertibility from bordism and non-invertibility from topological quantum field theory 

with Shi Chen 

Solitons are topologically stable objects in QFTs, especially in non-linear sigma models, and their classification is believed to obey the homotopy group. We uncover that this is not always the case, and demonstrate that a more sophisticated algebraic structure appears as the solitonic symmetry of 4d CP^1 sigma model. 

4d CP^1 model has the particle-like soliton, called hopfion or Hopf soliton, which name comes from the Hopf fibration. There are also string-like solitons, and their charged are characterized by the U(1) 1-form symmetry. However, we cannot find the local conserved current for the hopfion number. We have shown that the only good symmetry is just Z_2 instead of U(1). 

What is more surprising is that the full structure of the hopfion number symmetry is characterized by the non-invertible symmetry. Because of this, we can prove the stronger selection rule, and the hopfion number is conserved as integers as long as the vortex-line operator is not inserted. Once the vortex operator is inserted, the hopfion number is conserved only in mod 2, and this subtlety is nicely explained by the non-invertibility of the correct hopfion number symmetry.  

Originally, this project started from the study on baryons of SO(N_c) QCD, but we unexpectedly achieved the interesting applications of non-invertible symmetries. 

[2207.03008[hep-th]] Winding θ and destructive interference of instantons 

with Mendel Nguyen and Mithat Ünsal

During my visit to NCSU, Mendel, Mithat, and I discussed what happens if we take the chirally twisted boundary condition to obtain the weak-coupling setup for the asymptotically free QFTs. We noticed that it is related to the case where the theta parameter has the nonzero winding number as a result of the ABJ anomaly, and we decided to work on it first. Moreover, I had been personally interested in this setup from another perspective. By shifting the theta parameter by 2pi, the vacuum is replaced by the different SPT states, and thus the nonzero winding number of theta requires the interface between different SPT states. Since we do not have an anti-interface, it means that the total Hilbert space has 't Hooft anomalous symmetry. 

Our main goal of this paper is to understand the semiclassical realization of the above observation. We pick up the 2d CP^{N-1} sigma model as an example, and then the interface should be in the projective representation of PSU(N) symmetry, and there have to be N-fold degenerate ground states. Although this is the immediate consequence of the anomaly matching condition, it is quite nontrivial from the concrete analysis of semiclassical analysis. To make the dilute gas approximation reliable, we take the flavor twisted boundary condition, and then there are N classical vacua related by (fractional) instantons. Usually, instantons lift the classical degeneracy, but the anomaly tells us that instantons should not do the job at all. 

We have such a situation when fermion exists due to the spurious axial symmetry, but how we can eliminate the instanton effects at all in purely bosonic systems? This is the question we answered in this paper, and the moduli parameter of instanton plays an essential role here. 

[2204.07440[hep-th]] Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly 

with Yui Hayashi

Recently, people have just started to understand non-invertible self-duality symmetry in 4d gauge theories (arXiv:2109.05992, arXiv:2111.01139, and arXiv:2111.01141). Since the electromagnetic duality exchanges the electric and magnetic charges, we cannot naively realize it if there is an imbalance between the electric and magnetic 1-form symmetries. These recent developments overcome this difficulty, by noticing that the duality defects can be constructed by using the half-space gauging procedure. 

This progress reminded me that I could not fully understand the electromagnetic duality of 4d U(1) gauge theory when studying the Cardy-Rabinovici model with Masazumi (arXiv:2009.10183). We could construct the self-duality for the partially gauged CR model, which constrains the low-energy dynamics thanks to the mixed-gravitational anomaly. However, it was not clear if there exists such constraints on the original model. So I thought that this would be a good opportunity to revisit this problem. Yui Hayashi, a Ph.D. student at Chiba Univ., just visited me for discussions so I talked to her about this problem and started this collaboration. 

We immediately noticed that we can have the ST^p duality by gauging the Z_N 1-form symmetry with the level-p discrete topological term. At the self-dual point, this gives the non-invertible symmetry as the fusion rule becomes non-group-like. Moreover, we find that there exists a mixed gravitational anomaly that rules out the trivially gapped phase. 

[2201.06166[hep-th]] Center vortex and confinement in Yang-Mills theory and QCD with anomaly-preserving compactifications 

with Mithat  Ünsal

In this paper, we constructed a novel semiclassical theory of color confinement for 4d gauge theories by considering the T2 compactification with 't Hooft flux. This is the project that took us more than 2 years, as I remember that Mithat and I chatted about this idea when I was doing a postdoc in North Carolina. Interestingly, we already reached the possibility of adding the baryon-number monopole flux to apply the T2  compactification to QCD with fundamental quarks at that time, but we could not be so confident if such a naive idea works to study the strong-coupling nature of QCD. Then, time flew, and this project was stopped for a while. 

 The situation has changed since I got an invitation to talk at the conference "Paths to QFT", and I started rethinking this problem by taking this opportunity. I become confident about the vortex-induced confinement by concretely solving 2d Abelian Higgs models in a semiclassical way and the result is consistent with anomaly matching. We understood the 2d theory well, so the next problem became to apply this idea to 4d gauge theories with compactification. When discussing 4d Yang-Mills theory, what annoyed us was the absence of analytic solutions for the center vortex under compactified setup with 't Hooft flux. Still, there was numerical evidence for its existence and important properties were also numerically justified in previous studies, so we decided to use those center vortices in our semiclassical description.  The results are remarkably successful, and the weak-coupling analysis with center vortices describes many important features of 4d gauge theories including QCD, which turn out to be consistent with 4d strong dynamics.

 Understanding color confinement is one of my biggest goals since I became a graduate student, so I am very pleased that I could write this paper achieving a small step toward it.

[2109.10051[cond-mat.str-el]] Sigma-model analysis of SU(3) antiferromagnetic spins on the triangular lattice

 with Itsuki Takahashi

This is a paper with my student, Itsuki Takahashi, studying the Neel-VBS transition for the (2+1)d SU(3) spin systems on the triangular lattice. I was interested in extending the sigma model analysis for the SU(3) antiferromagnetic spin chain to the 2d quantum antiferromagnets since I did the related previous work (1805.11423) with Tin Sulejmanpasic. I suggested Itsuki study this system using the nonlinear sigma model, and we derived the 3d SU(3)/U(1)^2 nonlinear sigma model with the emergent relativistic invariance. 

To apply this to the Neel-VBS phase transition, it is crucial to study the monopole effective theory when all the electric particles are massive. The possible monopole perturbations are restricted by the destructive interference determined by the Berry phase. We have derived the graphical rule to compute the Berry phase for various tunneling processes, and we show that there is a nontrivial magnetic symmetry even with dynamical monopoles. This leads to the 't Hooft anomaly matching condition, which supports a direct phase transition between Neel and VBS orders. 

[2103.05035[hep-th]] Topological terms of (2+1)d flag-manifold sigma models

 with Ryohei Kobayashi, Yasunori Lee, Ken Shiozaki

Recently, it has been suggested that possible topological terms in QFTs should be regarded as the partition function of SPT phases. According to this proposal, topological terms can be classified by the bordism group with an appropriate structure for the spacetime manifolds. As I have worked on the 2d flag-manifold sigma model, I got interested in its extension to 3d version and wondered what would be the possible topological terms for Lagrangians of such models. 

The last December, Yasunori visited the CREST workshop at YITP organized by Ken, and, after some conversations, Ken and I decided to invite Yasunori and Ryohei for more discussions this January. During discussions with them, I learned a lot about spectral sequences, and we finally decided to work on the above problem using those techniques. With the help of these abstract topology techniques, we determined the explicit form of possible effective Lagrangians and discussed the physical consequences of those topological terms with concrete computations. 

Above all, I enjoyed working on this project a lot as we could discuss in person getting together at YITP. 

[2009.10183[hep-th]] Topological aspects of 4D lattice Abelian gauge theories with the θ parameter

 with Masazumi Honda

Since I started working on recent anomaly matching, I was interested in understanding the oblique confinement phase proposed by 't Hooft, which is a very exotic but interesting scenario for the SU(2) Yang-Mills theory at θ=π. In this paper, with Masazumi, we revisit the model proposed by Cardy and Rabinovici from the recent topological aspects and anomaly matching. 

The Cardy-Rabinovici model is expected to show the rich phase diagram thanks to the presence of both electric and magnetic particles. We show that this model has the anomaly and/or global inconsistency involving the one-form and CP symmetries, and it is the same anomaly with the SU(N) YM theory. As we can understand the microscopic details of this model, we got very excited that this simple model provides various concrete and nontrivial ways to match the 't Hooft anomaly and the global inconsistency conditions. This study not only justifies the conjectured phase diagram but also uncovers the topological aspect of each phase, including the oblique confinement phase. 

It has been thought that the Cardy-Rabinovici model enjoys the SL(2, ℤ) self-duality, which is almost true for the local dynamics. However, we noticed that it cannot be correct for the global nature of the theory if we pay attention that whether the low-energy theories are topological or not. To rescue the original idea, we constructed the SL(2, ℤ) self-dual model by gauging a part of the one-form symmetry of the Cardy-Rabinovici model. We discuss various anomalies, including the mixed anomaly of SL(2, ℤ) and gravity, to constrain the phase diagram of gauged Cardy-Rabinovici model. 

[2005.13822[hep-th]] Finite-density massless two-color QCD at isospin Roberge-Weiss point and 't Hooft anomaly

 with Takuya Furusawa, Etsuko Itou

We studied how the phase diagram of finite-density two-color QCD can be constrained by 't Hooft anomaly. This is along the line of some of my previous studies (1710.08923 and 1711.10487), and it was pointed out by Itou-san that two-color QCD has a chance to show very different behaviors compared with the usual QCD. Moreover, two-color QCD can be tested by numerical lattice simulation even at finite densities, so it is worthwhile to think about the anomaly constraint on the two-color QCD phase diagram in detail. 

Takuya immediately computed the discrete anomaly with full generality for this purpose, and we discussed its physical meanings. It gives a meaningful constraint on the phase diagram with imaginary isospin chemical potentials, but I am surprised as the way to match the anomaly turns to be very different from that of 3-flavor 3-color QCD. What is more interesting to me personally is that the anomaly suggests the similarity between two-color QCD and (2+1)d anti-ferromagnetic systems. It's very curious if this similarity on symmetry and anomaly extends to that of dynamics, too, or not. 

[1912.01033[hep-th]] Modified instanton sum in QCD and higher-group structures

 with Mithat Ünsal

One day in November, I asked Mithat if the instanton sectors are really necessary for resolving the U(1)_A problem. We have indeed worked on a related issue in the 2d case, higher-charge 2d U(1) gauge theory (1905.05781), so motivated by those studies, Mithat and I were wondering if a similar thing occurs in 4d gauge theory. In order to study it, we restrict the possible instanton sectors to multiples of a certain integer p for SU(N) Yang-Mills theory and quantum chromodynamics (QCD). 

To make such restrictions consistent with locality, we have to introduce the topological degrees of freedom to the theory. As a consequence, the theory acquires the 3-form symmetry, although the local dynamics are completely identical with the usual theory. Quite often, this 3-form symmetry has a mixed anomaly, and it leads to the new selection rule prohibiting the domain-wall excitation or the false-vacuum decay. We are enjoying a lot with this stronger superselection rule. 

[1908.09858[hep-th]] Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD

 with Aleksey Cherman, Theodore Jacobson, Mithat Ünsal

After finishing the paper about the Schwinger model, I got more interested in other 2d gauge theories. Aleksey, Theo, and Mithat were already working on 2d adjoint QCD at that time, and I started to join the collaboration. This theory has a history of almost 25 years, and many claims about it sound being mysterious (at least to us). 

We decided to look at what symmetry tells us. This is a system of 1+1d interacting Majorana fermions, which can belong to an interesting nontrivial class of SPT phases if we turn on a negative mass. I adapt the mod 2 index theorem in a suitable form for our purpose, and it enables us to find many mixed 't Hooft anomalies. These anomalies, fortunately, tell us many new things about the confinement and chiral symmetry breaking of this model. Moreover, semiclassical analysis on small cylinders gives a consistent result to those findings. 

Unfortunately, however, these new things are inconsistent with many previous studies, so we had to check carefully how these inconsistencies appear. After those surveys, the new technique, mod 2 index theorem, turns out to play an important role to understand the dynamics of 2d adjoint QCD. In our understandings, the previous studies without it failed to capture the fully non-Abelian dynamics. 

Addendum: About a year later, an interesting paper appears, by Komargodski, Ohmori, Roumpedakis, Seifnashri (2008.07567). There, it has been reported that, at the superrenormalizable points, the 2d adjoint QCD enjoys many non-invertible topological lines. As a result, there are exponentially many vacua for this theory. Therefore, when we include generic four-fermion couplings which do not break any 0-form symmetries, our suggestion is correct. But, when those four-fermion couplings satisfy certain constraints, all the Wilson loops are deconfined as expected in the '90s but with completely different reasonings, and there are much more deconfined line operators as a result of non-invertible symmetry. 

[1904.08570[hep-th]] Effective gauge theories of superfluidity with topological order

 with Yuji Hirono

This is the follow-up paper of our previous paper (1811.10608[hep-th]). In the previous paper, we found it useful to consider the generalized BF theory with non-square charge matrix K when discussing the possibility of topological order under the existence of superfluidity. 

In this paper, Yuji and I analyzed the generalized BF theory in details, and discussed its structure in view of higher-form symmetry. We generally identify the generators of discrete and continuous higher-form symmetries, and give the criteria when the symmetry is spontaneously broken. We revisit the analysis of CFL phase in high-density QCD in this framework and no topological order appears there. We also discuss an example of superfluidity with topological order within our framework. 

[1812.02259[hep-th]] Phase structure of the twisted SU(3)/U(1)^2 flag sigma model on R*S^1

 with Masaru Hongo, Tatsuhiro Misumi

Previously, in 1805.11423, I worked on the two-dimensional flag sigma model to generalize the Haldane conjecture to SU(N) spin chains with Tin Sulejmanpasic. I wanted to check the conjectured phase diagram obtained by anomaly and global inconsistency. Then, Masaru, Tatsuhiro and I worked together on this project by using reliable semiclassical analysis with twisted boundary condition.

Interestingly, the knowledge of fractional instanton on CP^N sigma model becomes quite useful to construct the BPS solutions of the SU(3)/U(1)^2 flag sigma model. By employing the symmetry-twisted boundary condition, the dilute instanton gas approximation (DIGA) gives the reliable answer without suffering from the infrared divergences. We found that the interesting phase structure conjectured by anomaly and global inconsistency can be concretely understood thanks to DIGA. 

We also gave some speculations about what happens beyond DIGA, by estimating the imaginary ambiguity about the complex bion amplitude and assuming the structure of resurgence theory. The size of imaginary ambiguity shows the difference from the one expected from two-dimensional renormalon diagrams. This difference appears for 4d adjoint QCD but does not occur for 2d CP^N model. In this respect, our model might be more similar to adjoint QCD compared with CP^N model. 

[1807.07666[hep-th]] Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking

When I gave a seminar at Toronto University about anomaly matching, Erich pointed out the possibility of application of anomaly to exotic scenarios of chiral symmetry breaking in QCD. This paper is motivated by the discussion at that time, and we exclude the Stern phase based on symmetry and the discrete 't Hooft anomaly of massless QCD. 

Stern phase is the chiral-symmetry broken phase with the ordinary pattern of continuous chiral symmetry breaking, but the discrete axial symmetry remains unbroken there. We prove that the baryon current is anomalously broken for massless QCD if we introduce the background gauge fields for discrete axial symmetry and the projective vector-like flavor symmetry. In the ordinary chiral broken phase, the 3-form Wess-Zumino current (Skyrmion current) reproduces this anomalous breaking, and this result is very interesting to me that the Skyrmions play a pivotal role for anomaly matching. In the Stern phase, however, the anomalous violation of the Skyrmion current takes a different form, and the anomaly matching is not satisfied. We, therefore, rule out the Stern phase from possible QCD vacua. 

The Stern phase was previously excluded from the zero-density QCD phases using the QCD inequality, and our no-go theorem is consistent with that previous result. Moreover, our result extends its applicability to the finite-density zero-temperature QCD. 

[1803.02430[hep-th]] Quantum Distillation of Hilbert Spaces, Semi-classics and Anomaly Matching 

 with Gerald Dunne and Mithat Ünsal 

When I found the construction of persistent anomaly under S^1 compactification with the symmetry-twisted boundary condition (1710.08923), I was wondering its meaning from the viewpoint of physical states of QFT since the construction was quite formal. With Gerald and Mithat, we discuss the role of the symmetry-twisted boundary condition at the KITP workshop to revisit the volume independence via new perspectives. 

In this paper, we emphasize that the choice of the symmetry-twisted boundary condition does not change the theory. Our interpretation is that it is a procedure that makes the ground-state contribution dominant to the partition function, which cannot happen for the usual thermal partition function, and we name this the Quantum Distillation. Since the 't Hooft anomaly constrains the property of ground states by anomaly matching, it is quite reasonable for the persistence of anomaly under S^1 compactification when the symmetry-twisted boundary condition distills the ground state. 

[1801.10283[cond-mat.mes-hall]] Giant photocurrent in asymmetric Weyl semimetals from the helical magnetic effect 

 with Dmitri E. Kharzeev, Yuta Kikuchi, Rene Meyer 

One day, Dima suggested to us that there may be a photocurrent induced similarly with the Chiral Magnetic Effect (CME). Using the chiral kinetic theory, we can immediately find that such current vanishes for usual Weyl semimetals, but it also becomes evident that such current exists when both the particle-hole and parity symmetries are explicitly broken. Since the current is induced thanks to helicity imbalance, we call it the Helical Magnetic Effect (HME). What is more interesting is that Yuta found the HME current becomes quite huge compared with the ordinary photocurrent in the infrared region. Therefore, we evaluate its magnitude based on the chiral kinetic theory and compute its frequency dependence. 

[1710.08923[hep-th]] Circle compactification and 't Hooft anomaly

 with Tatsuhiro Misumi & Norisuke Sakai

We develop a systematic procedure that derives an 't Hooft anomaly under circle compactifications (or, at finite temperatures) starting from that of the original uncompactified theory (i.e., at the zero temperature). Since an 't Hooft anomaly is the obstruction of gauging symmetries coming out of topology related to symmetry, that information is usually eliminated by thermal fluctuations at finite temperatures. We find that, under a certain twisted boundary condition, this difficulty does not appear and the phase structure of the circle-compactified theory obeys the constraint by 't Hooft anomaly matching. This applies to the ℤN-twisted ℂℙN-1​​ model and also to massless ℤN-QCD. We discuss the consistency of our computations with previous results and consider possible implications on the phase diagram of massless ℤN-QCD. Although a special limit or idealization is taken, we have a new rigorous result on the fundamental theory of strong interaction (QCD). 

This work is started through the discussion with Tatsuhiro about QCD with flavor-twisted boundary condition (Z(N)-QCD) at the Resurgence conference in Kobe, Japan. Indeed, my original motivation to study bifundamental QCD via anomaly is to understand ℤN-QCD from the viewpoint of recent developments on the anomaly, and this was a nice opportunity for me to understand the properties of the system very deeply.  Through the discussion on the resurgence of the ℤN-twisted ℂℙN-1​​ model with Sakai-san in the workshop at KITP, I got the idea of the general structure of anomaly under circle compactifications. 

[1706.03822[hep-lat]] Gradient flows without blow-up for Lefschetz thimbles

 with Hiromichi Nishimura & Jacobus J. M. Verbaarschot

The Lefschetz-thimble approach to the sign problem uses a gradient flow in the complexified field configurations to find appropriate stationary-phase cycles. We noticed that the gradient flow generically blows up in a finite time, so one must control its behavior very carefully in the numerical computation. Instead of paying attention to such blow-ups, we propose a new gradient flow that equally defines Lefschetz thimbles and does not suffer from blow-ups. We give its theoretical foundations, and also checked its properties numerically in simple examples. 

[1703.02040[cond-mat.mes-hall]]  Topological Properties of the Chiral Magnetic Effect in Multi-Weyl Semimetals

 with Yuta Kikuchi & Tomoya Hayata

Multi-Weyl semimetals are semimetals described by two-component fermions with non-linear dispersion, and they have the monopole charge greater than 1. We elucidate how their topological property appears in static and dynamic chiral magnetic effect. We also propose an experimental measurement to confirm the multiple monopole charge. 

This work was initiated by Yuta and Tomoya, and Yuta explained this work to me on the blackboard when I visited his office at Stony Brook. I realized that what they want to compute can be readily obtained and generalized by using differential forms. This mathematical simplification was quite helpful for us to realize that the trace part of dynamic chiral magnetic conductivity is always topologically determined, and the proposal for the experimental setup is based on this observation. 

[1609.06198[hep-th]]  Cheshire Cat resurgence, Self-resurgence and Quasi-Exactly Solvable Systems

 with Can Kozçaz, Tin Sulejmanpasic & Mithat Ünsal

There is a lot of attention to the asymptotic nature of the perturbation theory under the name of resurgence theory. In the case of quantum mechanics, the asymptotic nature is caused by the instanton–anti-instanton correlated event or the complex bion solution. In this study, we consider the impact of resurgence to the quasi-exact-solvability (QES), which is also known as the non-linear/multifold supersymmetry. In the QES system, the perturbation theory for first several states converges to give the exact result despite the fact that there exist instanton-like solutions (real bions) giving nonperturbative corrections. 

We showed that the complex bion describes a new nonperturbative contribution of QES systems, by looking at the asymptotic perturbative series after a tiny deformation of QES. We name this phenomenon as the Cheshire Cat Effect of Resurgence, which plays the key role to understand the absence of nonperturbative corrections in QES: Complex-bion contributions exactly cancel the nonperturbative correction from real bions. Furthermore, we extend this result to a “pseudo-QES” system that suffers from the nonperturbative correction to the convergent perturbative solution. 

We started this collaboration when we gathered at Harvard University in the last November. Mithat introduced relevant works to us there and brought my interest to the resurgence theory, which has a close relationship with the Lefschetz-thimble method of path integral. 

[1509.07146[hep-th]]  Lefschetz-thimble analysis of the sign problem in one-site fermion model

 with Yoshimasa Hidaka & Tomoya Hayata

Lefschetz-thimble approach is recently getting much attention, and its properties in practical applications need to be understood in more detail. In this paper, the one-site Hubbard model is studied using the Lefschetz-thimble integration method after the Hubbard-Stratonovich transformation. This toy model has a "quantum phase transition" as the chemical potential increases, and its partition function has the sign problem in the path integral expression. Therefore, this is a good playground of theoretical approaches to the sign problem. 

One day, Tomoya and Yoshimasa asked me whether this "phase transition" of the toy model can be nicely explained using the Lefschetz-thimble approach to the path integral. (Original question was about the Bose Hubbard model, but we solved the Fermi Hubbard model in this paper because of some inessential complexity, instead.) The computation showed that interference among Lefschetz thimbles is of great significance in order to explain such behaviors, and this fact surprised us a lot. This model has a remarkable similarity with the finite-density QCD, and we expect that a similar thing would happen also there. 

[1504.07268[hep-ph]]  From quantum to classical dynamics: The relativistic O(N) model in the framework of the real-time functional renormalization group

 with David Mesterházy & Jan H. Stockemer

By using nonperturbative RG to the Schwinger-Keldysh formalism of real-time dynamics, the model A dynamics and the microscopic theory are shown to be smoothly connected. We derive the RG equation within the local interaction approximation, and the nontrivial IR fixed point is discussed within this approximation. It elaborates how the dimensionally-reduced classical system emerges for static behaviors at the second-order phase transition, and allows us to address dynamical phenomena. We also discuss discrepancies between our result and the result obtained from Langevin dynamics, and it will tell a missing part of our approximation in order to complete our microscopic description of classical Langevin dynamics in future studies. 

David was a postdoc at University of Illinois at Chicago when I visited there, and we frequently talked about the related topic during my visit. David is quite familiar with the real-time formalism of quantum field theories and its nonperturbative RG, and he led this collaboration powerfully. 

[1412.2802[hep-th]] Structure of Lefschetz thimbles in simple fermionic systems

 with Takuya Kanazawa

Takuya is conversant with chiral symmetry, and he indicated me new possible connections of Lefschetz-thimble methods with other techniques used in random matrix models. Especially, the close relation between Lefschetz-thimble decomposition and Lee-Yang zeros is a big finding in this research. In this paper, we concentrate on simple models including fermionic degrees of freedom, and structure of Lefschetz thimbles are scrutinized carefully in order to demonstrate those new insights. In the part of QCD-like chiral symmetric models, techniques developed in the paper below are employed. 

[1406.2386[math-ph]] Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling

 with Takayuki Koike

In this paper, we discuss properties of real-time path integrals from the viewpoint of Picard–Lefschetz theory. Recently, this technique starts to gather much attention in the context of resurgence trans-series for perturbations in QFT, and also of the sign problem in numerical Monte Carlo simulation. I invited my mathematician friend, Takayuki, to start this collaborating work, and asked to tell me its basic mathematical structure. We applied it to the 'real-time' description of quantum tunneling in the double-well potential, and discuss how it can be useful and what is the problem to be solved for practical computations. 

[1402.0283[cond-mat.quant-gas]]  Multi-Regulator Functional Renormalization Group for Many-Fermion Systems

 with Tetsuo Hatsuda

Since we succeeded to describe the BCS theory and BEC of composite particles using fermionic FRG, we tried to combine those analyses to describe the whole region of the BCS-BEC crossover. This requires to control both the fermionic one-particle excitations and the bosonic composite-particle excitations without introducing the bosonic auxiliary fields. For this purpose, we constructed the fermionic functional renormalization group with multiple regulators, which we call Multi-Regulator FRG. The Nozieres–Schmitt-Rink theory turns out to be derived as a leading-order approximation in this theoretical formalism. We hope to apply it to strongly-coupled systems, such as nuclear matters, in future. 

[1310.5800[cond-mat.quant-gas]] Fermionic functional renormalization group approach to superfluid phase transition

 with Gergely Fejős & Tetsuo Hatsuda

This is the first paper including a main topic in my master's thesis. For an unbiased study of interacting fermions, superfluid phase transition is discussed within "fermionic" FRG. After reviewing its connection to the BCS theory, we reproduce the Gorkov & Melik-Barkhudarov correction and discuss effects of the fermion self-energy correction to the critical temperatures and chemical potentials.