^{3}

We study a certain class of supersymmetric (SUSY) observables in three-dimensional

Perturbative series in quantum field theory (QFT) are usually divergent [

Borel resummation of

Resurgence theory has a long history in quantum mechanics and differential equations. There have been various applications in a variety of physical systems including quantum mechanics (QM) [^{1}

In general, it is much harder to study the resummation problem in QFT than in quantum mechanics since the Schrödinger equations are not available and we have to confront the saddle point analysis of path integrals “seriously.” According to the recent progress in understanding the resurgent structure of QM from the path integral viewpoint [

Find all critical points including complex saddles.

See which critical points contribute in terms of Lefschetz thimble decompositions.

Study perturbative expansions around contributing critical points.

We know that the first step is already technically hard in typical QFT and the second step is harder than the first step. Indeed, there are only a few known examples of physical quantities satisfying the following ideal conditions:

(1) physical quantities in

(2) quantities for which mathematically well-defined descriptions for their exact results are known;^{2}

(3) quantities with the non-trivial resurgent structure.

To the best of our knowledge, the only examples satisfying all three conditions are 2D pure Yang–Mills (YM) theory [^{3}^{4}

In this paper we propose an infinite number of examples satisfying all the above conditions, (1), (2), and (3). The examples are a certain class of supersymmetric observables in 3D ^{5}^{6}

Another motivation for this paper comes from mysterious results in the same setup previously found by one of the present authors [^{7}

Technically these results were obtained by rewriting the exact result and we did not have appropriate interpretations for them. To obtain more precise understanding of these results, we decompose the integration path of the Coulomb branch localization formula, Eq. (^{8}

We explicitly demonstrate the above arguments based on partition functions of a certain class of rank-1 3D

It has been shown [

We also discuss path integral interpretation of the non-perturbative contributions appearing in the resurgent transseries. Recently, one of the present authors has found complexified supersymmetric solutions in general 3D

This paper is organized as follows: In

In this section, we study the ^{9}^{10}^{11}

Applying the SUSY localization [^{12}

In

Let us take ^{13}

Note that this expression is similar to Borel resummation, Eq. (

Borel singularities (red crosses) are depicted for

Since the exact result is given by the integral along

The number of singularities in this region is

To show these results explicitly, first let us focus on

Note that it vanishes for

Note that this decomposition is well defined for almost all values of ^{14}

Comparing this with the above data, we identify the above parameters with

For

Instead, let us estimate the ambiguities of the perturbative and non-perturbative contributions by

Noting that

To sum up, the Borel ambiguity in the perturbative sector for ^{15}^{16}

We also note that the importance of the Borel singularities at the first and fourth quadrants on the perturbative Borel plane has been stressed in Ref. [

We also show the results for a generic number of flavors

We end this subsection with a comment on the uniqueness of the decomposition of the exact partition function into perturbative and non-perturbative parts in Eq. (

Here we decompose the Coulomb branch localization formula of Eq. (

We regard

Let us label the saddle points by

The Lefschetz thimble or the steepest descent contour

When

In general,

Let us analyze the structures of the Lefschetz thimbles in the present example. The saddle point equation in Eq. (

The saddle point equation, Eq. (

In the limit

As

Locations of critical points and their actions as functions of

We can easily compute the thimble flowing from

Note that

This always happens when we study the following type of integral:

By examining the limiting behavior of the critical points as

This fact has important implications for structures of (dual) thimbles. Since the actions at the poles are

Next we take into account a small-

From these actions, a necessary condition for a Stokes phenomenon is^{17}

Note that this condition is not satisfied by

Consequently, for finite

Now let us turn to the finite-^{18}

Thimble structure of the partition function of

Thimble structure of the partition function of

We first discuss the case with small

Close-up of

For

For

For larger

The red dotted line denotes

We have analyzed the cases for generic values of

Schematic expanded figures for

For

For

For

Therefore we can express the exact result in this regime as

At

The Lefschetz thimble decomposition of the exact result is well-defined at this point and the exact result is expressed as

For ^{19}

If we further increase

This shows that we have the decomposition

The thimble integral along

This decomposition is ambiguous at

Thus, at

Now we comment on the definition of the perturbative contribution. As we mentioned at the end of the previous subsection, the definition of the perturbative contribution based on the Borel resummation is just one possible definition. In our work, we define the perturbative part as the Borel resummation of the perturbative series and decompose the exact result into the perturbative and non-perturbative parts. We may be able to propose another feasible definition of the perturbative contribution: the thimble integral associated with the perturbative saddle

Finally, we mention a technical subtlety of

The difference of

So far we have discussed the Stokes phenomena and the resurgent structure by changing the real mass parameter while we have fixed the coupling ^{20}

Let us take complex

Repeating the argument of Ref. [

Borel singularities and the fan

^{21}

Note that the only differences from the

For

Therefore, the ambiguities are canceled and the whole transseries gives the unambiguous answer, which agrees with the exact result.

Let us decompose the exact result into Lefschetz thimble contributions. First, we discuss a small-

In the

In

Thimble structures for

For

For

For

As

The CS SQED has another description, which is connected to the original description by 3D mirror symmetry [

This is formally the same as the Coulomb branch localization formula for the

Let us perform thimble decomposition in this integral representation:

Note that the action becomes large for

Note that their roles in the transseries are unclear just from this information, in contrast to

For the other critical points it is hard to solve the flow equation analytically, as in the original theory.

In

Thimble structures in the mirror theory.

We next investigate the ^{22}

The partition function of the ^{23}

We again focus mainly on

This Borel transformation has simple poles at

As with the case of CS SQED, the exact result, Eq. (

The number of singularities in the region is

We first focus on

For

In these expressions of the full transseries expansion, each of the non-perturbative parts corresponds to the contribution with the action

In the case of general

As in the CS SQED cases, the transseries expression is apparently ambiguous for

It is clear that these ambiguities are canceled and we obtain the unambiguous result equivalent to the exact result.

The effective action of the present example with respect to

We consider the complexification

As in the

Note that while the third factor comes from the poles of the integrand, which we also had in the CS SQED case, the second factor comes from the zeros, which were absent in the

The zeros add qualitatively new features to the thimble structure because they can be end points of Lefschetz thimbles and thimbles may terminate at finite ^{24}

Now we present some samples of numerical results.

Thimble structures of the

Thimble structure of two-flavor

In the

So far we have analyzed the sphere partition functions of the

Let us consider general rank-1 ^{25}

The most important difference from the

We can extend the analyzes in Sects.

Note that the

Changing the integral contour as in

Noting that the poles start to come into the fourth quadrant when

As in the previous cases, this decomposition is apparently ambiguous for

Thus the ambiguities are canceled and we find the unambiguous answer consistent with the exact result.

We can also find the resurgent structure in the situation with fixed

Although this decomposition apparently has ambiguities for

We discuss thimble decomposition of the integral

The saddle point equation under this action is given by

We can analytically solve this equation in the weak-coupling limit as in the previous cases. For

As in the

So far we have considered only the partition function on a round sphere. In this subsection we discuss the extension of our argument to other observables.

Let us start with the Wilson loop

It is known that this operator preserves two supercharges if the contour

If we restrict ourselves to superconformal case, we can also compute the Bremsstrahlung function

As in the Wilson loop, the net effect is just insertion of the entire function and hence we basically arrive at the same conclusion as the Wilson loop. However, note that we cannot turn on real masses for this case since we are considering the superconformal case. In other words, we can formally turn on real masses at the level of the integral in Eq. (

We can also compute the two-point function of the

The derivatives by the real masses do not change the locations of singularities, while their degrees are changed. This difference, however, does not lead to a qualitative change of the resurgent structure and the thimble structures for weak coupling.

Let us consider the partition function on the squashed sphere

Note that the round-sphere case corresponds to

Borel singularities associated with each of the chiral multiplets become simple poles and are labeled by two integers.

The locations of the singularities depend on

Even for this case, we can still write the partition function as
^{26}

We can also put the supersymmetric Wilson loop on a squashed sphere constructed in Ref. [

For the superconformal case, we can also compute a two-point function of the normalized stress tensor at separate points, whose expression is determined by conformal symmetry as
^{27}

Although the derivative with respect to

In this section we discuss possible interpretations of the non-perturbative effects appearing in the transseries from the path integral viewpoint. It is technically obvious that the non-perturbative effects come from the Borel singularities, or equivalently the poles of the integrand of the Coulomb branch localization formula. In Ref. [

For example, in the

However, there are three subtleties in this interpretation. First, SUSY solutions are not necessarily saddle points on the curved space, contrary to the flat space. Indeed, the CSS constructed in Ref. [

Second, to verify our conjecture, we have to check the following two facts: (1) The Stokes phenomena regarding the non-perturbative contributions we have shown should be identified as jumps of intersection numbers between the original path integral contour and dual Lefschetz thimbles associated with the CSS. (2) The perturbative series in the non-perturbative sector of the transseries should agree with the perturbative series around the CSS. In particular, the perturbative series should terminate at the one-loop order. We may be able to check this statement in future work.

Third, to our knowledge, most analyses of SUSY localization in the literature have not performed serious saddle point analysis including complex saddles, and therefore there is a possibility that we are missing contributions from complex saddles. In particular, in the Coulomb branch localization formula for 3D

We summarize the results obtained in this paper as follows:

(1) We have expressed the exact results for the SUSY observables in 3D

(2) We have found that, when the real masses cross the special values, some of the Borel singularities are located on the real positive axis and come to contribute to the partition function as non-perturbative contributions. It leads to Stokes phenomena, where the perturbative Borel resummation becomes ambiguous. For example, in the

(3) We have shown that the relation between each of the thimble integrals in the thimble decomposition and each of the building blocks of the transseries do not necessarily have one-to-one correspondence for finite

(4) We have proposed path integral interpretations of the non-perturbative contributions appearing in the transseries. We interpret the non-perturbative effects as the complexified SUSY solutions constructed in Ref. [

(5) Based on our results, one may expect that, even if a perturbative series of a physical quantity is Borel summable along

We conclude this paper by discussing possible future studies. It is known that, in the Coulomb branch localization formula, picking up poles of the one-loop determinant gives rise to the Higgs branch representation of the partition function which includes a product of vortex and anti-vortex partition functions for some theories [

It is interesting to see whether the resurgent structures become simplified for higher SUSY theories such as 3D

We also comment on ref. [

Finally, although this paper has focused on weak-coupling expansions, it is also very interesting to study

We thank Muneto Nitta for his early collaboration and discussions. We are grateful to Sergei Gukov for useful comments on the draft in the first version on arXiv. Part of this work was completed during the workshop “Resurgent Asymptotics in Physics and Mathematics” at Kavli Institute for Theoretical Physics from October 2017. The authors are also grateful to the organizers and participants of “RIMS-iTHEMS International Workshop on Resurgence Theory” at RIKEN, Kobe. This work is supported by Ministry of Education, Culture, Sports, Science and Technology (MEXT)-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (Grant no. S1511006). This work is also supported in part by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (KAKENHI) grant numbers 18K03627 (T. F.), 16K17677 (T. M.), and 18H01217 (N. S.). This work is supported in part by the US Department of Energy Grant No. DE-FG02-03ER41260 (S. K.).

Open Access funding: SCOAP

In this appendix we write down supersymmetric actions in 3D

The 3D

If the gauge group includes

The 3D

In this appendix we compute the perturbative coefficients in the standard way, while we have derived the same results in the main text by Taylor-expanding the Borel transformations.

In terms of Euler number and the binomial theorem, we rewrite the hypermultiplet contribution as

Then the perturbative part of the partition function for

Using

For general

The only difference from SQED is the presence of

For general

In the thimble analysis, we first extend the real variable

The original integration contour

By choosing particular values of parameters, one might encounter the Stokes phenomenon, which is defined as

The complexified configuration space generally has not only critical points but also other objects such as singularities (sources) and zero-points (sinks), defined as

These points have the role of end-points of the thimbles.

^{1} See also reviews on the math side [

^{2} This does not necessarily mean that closed expressions for the exact results are explicitly known. For example, the localization method typically provides finite-dimensional integral representations for the exact results but we likely do not know how to perform the integrals analytically for gauge theories with multiple finite ranks. In this situation, we know the mathematically well-defined descriptions for the exact results but do not know their final closed expressions.

^{3} If we count so-called Cheshire cat resurgence [

^{4} When we have IR renormalons, this would not be true.

^{5} The partition function on

^{6} A real mass is given by a constant background of the flavor vector multiplet.

^{7} More precisely, the class of theory considered in Ref. [

^{8} Decomposition of localization formula by Lefschetz thimble has been considered in Ref. [

^{9} By “

^{10} The adjoint chiral multiplet with

^{11} In 3D

^{12} We are taking the radius of

^{13} Generalization to

^{14} We used

^{15} More precisely, at leading orders.

^{16} To avoid confusion, we note that the

^{17} We are assuming

^{18} Although the results could include small numerical errors, the main arguments in the following are not affected by the details.

^{19} Strictly speaking, we expect

^{20} More precisely, except for

^{21} We assume that

^{22} In 3D

^{23} In 3D

^{24} The fact that the number of the saddles around

^{25} We have rescaled

^{26} If

^{27} We take a normalization such that