Research Statement

In the case that the target space of superstring theory is the Calabi-Yau manifold, we have a 2-dimensional N=2 superconformal field theory on the world sheet. The Landau-Ginzburg description of conformal field theory, that is, an infrared fixed point in a 2-dimensional supersymmetric theory (Landau-Ginzburg model) is believed to become superconformal [1]. In this research, I want to achieve a triumph of lattice field theory, and pave the way for the numerical investigation of the Landau-Ginzburg model. In particular, as an example of Landau-Ginzburg model, the authors in Refs. [2,3] simulated numerically the 2-dimensional N=2 massless Wess-Zumino model with a cubic superpotential, and then observed the infrared critical behavior.

In previous research, I established non-perturbative techniques for the Landau-Ginzburg model and verified, through numerical simulations, that its infrared fixed point corresponds to a superconformal field theory. Using the Kadoh-Suzuki formulation [4], which rigorously preserves supersymmetry, I applied numerical methods to various Landau-Ginzburg models belonging to the ADE classification and confirmed that the low-energy correlation functions correctly reproduce the expectations from the correspondence. Additionally, employing a finite-size scaling method considering the continuum limit, I performed precise measurements of correlation functions for the cubic potential, and provided an evidence for the restoration of locality.

As an example of simple target space, I considered the case of a torus given by x^3+y^3+z^3 and carried out numerical simulations under various deformations of the torus. Particularly, we can numerically verify the expectation that the central charge remains constant under the insertion of terms believed to represent a marginal operator xyz.

In lattice field theory, because of discretizing spacetime into a lattice,it has been a long posed problem to construct the energy-momentum tensor, a Noether current of spacetime symmetries. Recently, a method based on a kind of diffusion equation called gradient flow [1] has extensively developed. By deforming field configurations according to a diffusion equation in gradient flow, composite operators in terms of the flowed fields automatically become finite (renormalized). Similar problems arise for other symmetries such as supersymmetry that is incompatible with lattice spacetime structure.

Regarding this, the following research was studied:

1.  Although the Standard Model is a chiral gauge theory, we have not known any lattice formulation of chiral gauge theory except for the U(1) gauge theory. A new lattice formulation using gradient flow was proposed by Grabowska and Kaplan [2]. I showed that the proposed lattice formulation faces difficulties to possess desired phenomena like the fermion number anomaly.

2.  I discussed and exemplified the relationship between gradient flow and Wilsonian renormalization group. Particularly, I analyzed the Banks-Zaks fixed point for N_f-flavor QCD approximated at the two-loop level; the Wilson-Fisher fixed point for the large N limit of the 3-dimensional O(N) linear sigma model.

3.  I computed the chiral anomaly in a gravitational field from the chiral current and energy-momentum tensor constructed via gradient flow.

4. Furthermore, using gradient flow, I constructed a universal expression for the supercurrent, a Noether current of supersymmetry, for the 4-dimensional N=2 supersymmetric Yang-Mills theory.

Currently, I am carrying out lattice simulations for the case of the 3-dimensional O(N) sigma model and depicting a renormalization group flow diagram by using the gradient flow. Therefore I numerically and non-perturbatively demonstrate the existence of the Wilson-Fisher fixed point [3].

In general, perturbative expansions in field theory lead to divergent series, introducing uncertainties in theoretical predictions.The proliferation of Feynman diagrams and contributions from instantons result in uncertainties in both perturbative and non-perturbative realms, which are believed to cancel each other out in a phenomenon known as resurgence structure. We also have known the existence of renormalon uncertainties in the perturbation theory. Recent attention has focused on the conjecture that the resurgence structure compensates for the renormalon uncertainty due to the presence of bions, semi-classical objects, in spacetime compactified on S^1 [1]. Despite vigorous discussions on this conjecture, several negative results have been reported, leading to a somewhat chaotic situation.

Focusing on the uncertainty in perturbation theory in spacetime compactified on S^1, I studied the following issues:

1. Using large N approximation, I analyzed the renormalon analysis in a spacetime where one direction is compactified, and showed that the renormalon uncertainty does not match that induced by bions.

2. I calculated the vacuum energy of the supersymmetric CP^{N-1} model on RxS^1 up to next-to-leading order in 1/N expansion, and then demonstrated that renormalons itself cannot be discussed from this result. Especially this is quite different from the calculation of vacuum energy based on bions [2].

3. I discussed how the infrared structure of perturbation theory is altered under S^1 compactification, by revealing that the explosion in the number of Feynman diagrams possesses an uncertainty enhanced by N, corresponding to bion uncertainty. We specified the resurgence structure. This mechanism naturally explains the inconsistency observed in the vacuum energy calculations.

4. I showed that, for SU(N) gauge theories with adjoint fermions, renormalons do not exist for finite N.

Analysis of phase structures using generalized symmetry and the 't Hooft anomaly matching condition

In recent years, the concept of symmetry has been generalized, where the conventional 0-form symmetry associated with Noether's theorem is regarded as topological objects on a spacetime with a fixed time slice (in d dimensions). Furthermore, an extension of this symmetry allows for the discussion of topological objects extending (d-p) dimensions for p-form symmetries [1,2]. When such symmetries are gauged, leading to what is known as 't Hooft anomalies that quantum anomalies invariant under renormalization group flow, we can have nontrivial constraints on the phase structure in the strong coupling regime. This is expected to contribute to understanding the phase structure of QCD, and also many aspects not only of particle physics but also of condensed matter physics.

I analyzed the phase structure of linear quiver gauge theories based on the 't Hooft anomaly matching condition. In linear quiver gauge theories, (K+1) SU(N) gauge fields are coupled with K Weyl fermions such that a Weyl fermion \psi_i is in the fundamental representation of the i-th gauge group and the anti-fundamental representation of the (i+1)-th gauge group (where i=1, 2, ..., K). Particularly ``direction'' i constitutes a new dimension (dimensoinal deconstruction) [3]. It is interesting that this theory is quite similar to topological materials. I discussed scenarios that could occur in the strong coupling regime for K odd/even. A remarkable aspect of this analysis is that the gauge-invariant operator \psi_1\psi_2...\psi_K exhibits different types of condensation because it is either bosonic or fermionic if K is even or odd, respectively.

Research on generalized symmetry and topology in lattice regularized $U(1)$ gauge theory

Most of the recent developments in research on generalized symmetry, such as those mentioned above, have remained largely in the realm of continuous spacetime formalism. It have not yet reached fully regularized discussions. Intricate techniques, viewpoints, and mathematics are employed to discuss non-perturbative effects. Despite the common use of lattice description when introducing generalized symmetries, the difficulties arises from dealing with ``topology'' on the lattice. Actually, topology, an essential property, has been constructed in lattice SU(N) gauge theory in the 1980s [4]. An attempt to approach generalized symmetry from the lattice regularized perspective is quite nontrivial due to the complexity of this construction.

To address this, based on a simplified construction for the U(1) gauge theory [5], I initially demonstrated the fractionality of topological charges and mixed 't~Hooft anomalies by coupling Z_q 2-form gauge fields to the theory. It is however essential to note that introducing Z_q 2-form gauge fields is crucial for discussions on magnetic monopoles, but constructing monopoles is not possible on a lattice gauge configuration where topology is defined. We should assume the Witten effect happens. I have been developing methods to observe similar phenomena to the Witten effect on the lattice in 2-dimensional scalar field theories, awaiting further progress.

An other type of generalized symmetry is the higher-group symmetry, where symmetries are not direct products of gauge groups and cannot be gauged individually. An example is the anomaly cancellation structure known from superstring theory (the Green-Schwarz mechanism). Adding a Z_p 3-form symmetry to the theory restricts instanton numbers to pZ in the path integral without violating locality. In this case, however, only 1-from center symmetry cannot be gauged no longer, and then the structure becomes higher-group symmetry [6]. By applying this discussion to the U(1)/Z_q lattice gauge theory as mentioned above, we achieved the realization of higher-group symmetry in a more straightforward manner than in continuous theories. Particularly, while continuous theories are described by Z_q higher-form gauge fields as multiple U(1) gauge fields and their breaking via Higgs mechanism, lattice theories are described by them simply as integer lattice fields; it provides a clearer understanding of such structures in lattice gauge theory.