^{3}

BPS vortex systems on closed Riemann surfaces with arbitrary genus are embedded into 2D supersymmetric Yang–Mills theory with matters. We turn on background

Vortices play an important role in many physical phenomena in diverse area of physics, and give vital information on non-perturbative dynamics of gauge theories in two dimensions. When the quartic coupling of the Higgs scalar field is given by the square of the gauge coupling, static forces between vortices cancel, leaving vortex position and orientation as moduli parameters of the solution. These vortices are called BPS vortices [

In recent years, the localization method in supersymmetric field theories [

The purpose of this paper is to characterize the BPS vortex equations on Riemann surfaces

If we consider localization of the path integral around the fixed point at non-zero values of the chiral multiplets, we find that the path integral gives an integral of the volume form over the moduli space. This is called the Higgs branch description of the volume of the moduli space. The Higgs branch description is useful to demonstrate the physical meaning of moduli space volume directly. However, it is difficult to evaluate explicitly in general since we need an explicit metric to construct the volume form as we have mentioned.

Using the same field theory, we can evaluate the vacuum expectation value of the operator in an alternative Coulomb branch description. The localization method is so powerful in the Coulomb branch description that the path integral will reduce to a simple contour integral. We can always perform this contour integral in principle. Since we are evaluating the same quantity in two different descriptions, this simple contour integral gives an alternative method of evaluating the moduli space volume. Although the relation between the contour integral (without knowing the metric explicitly) and the structure of the original moduli space is somewhat indirect, the field theory connects two different descriptions and explains why we can obtain the volume of the moduli space by the contour integral. To evaluate the path integral in this Coulomb branch description, we need to integrate over non-zero modes to find effective action for zero modes. Since we have inserted an operator, we also need to evaluate terms contributing to the correction to the operator, including fermionic terms. This point is also an improvement over our previous works [

We also consider a generating function of the moduli space volume. We can take a sum over vorticity

The organization of this paper is as follows. In Sect.

We first consider

We are interested in

In the flat space, the above theory can be embedded in a supersymmetric theory, and the BPS vortices preserve precisely half of the supersymmetry. This is the reason why BPS vortices have many nice features such as no static force between vortices, resulting in vast moduli space for multi-vortices.

Even in curved space-time such as

The Bogomolnyi completion of the energy of the vortices on the curved Riemann surface discussed in the previous section can be naturally embedded in 2D

The metric of the 4D space-time is

We first consider the vector multiplet part. The gauge fields in four dimensions reduce to gauge field

Following Refs. [

The Lagrangian density for the vector multiplet is given by

The action (

In order to preserve part of the supersymmetry on the curved space, we have to find a solution to Eq. (

By using explicit representation of the spinors and gamma matrices, we obtain the supersymmetry transformations in terms of components:

Now let us choose to keep a single supercharge for a rigid supersymmetry

Note here that the dependence of the

Furthermore, if we define

We find that the supercharge

Using this

To turn on the Fayet-Iliopoulos (FI) term and the

Next let us consider the chiral multiplet. We have the

For later convenience, we define

Similarly, for these complex conjugate fields, we have

Using this

This moment map consists of a part of the vortex equation.

To summarize, the total Lagrangian is written as a sum of the

The bosonic part of this Lagrangian reduces to the

So far, we have assigned the generic

Let us now consider a partition function for the supersymmetric theory that we have constructed in the previous section. We will see that the partition function is closely related to the volume of the vortex moduli space.

The partition function is defined by the following path integral over the configuration space of the whole fields

Then the partition function is given by a summation over the possible topological sector

We will see that this

First of all, we note that

After eliminating the auxiliary fields, the bosonic part of the

Using this coupling independence, we find that the path integral would be localized at solutions of a set of fixed (saddle) point equations as follows:

We examine the solutions to the above fixed point equations later, but we here focus on Eq. (

We refer to each kind of solution (

The moment map constraints (the BPS vortex equations) in Eq. (

Two different descriptions of the vortices on the Riemann surface. For the generic (small) size of the vortices (

Let us now elaborate on the possible significance of the Coulomb branch fixed points. From the exact solution of the ANO vortex (

If we take the large size limit of the vortices

The localization theorem says that the path integral is independent of the coupling ^{1}

Complementary descriptions of the Higgs and Coulomb branches with respect to the gauge coupling and FI parameter (vortex size)

Thus, we can use the extreme Coulomb branch description of the path integral instead of the Higgs branch. A similar complementarity of two descriptions between the Higgs and Coulomb branches through the FI parameters is also discussed in the quiver quantum mechanics [

In the following, we first discuss the Higgs branch description, but we will see that it is difficult to evaluate the path integral concretely in the Higgs branch. We will also see that the Coulomb branch description makes the evaluation of the path integral easy. This equivalence of two different descriptions is our key point of the calculation of the volume of the vortex.

Since our model has

Introducing the Faddeev–Popov (FP) ghosts

The BRST transformation acts on the fields as the gauge transformation with replacing the gauge transformation parameter by

Note that the BRST transformation is nilpotent

Once the gauge fixing function

The BRST symmetry of the above Lagrangian is apparent from the nilpotency of

Using

Since the total Lagrangian is written in the exact form of the nilpotent operator

Thus we can use the localization arguments again for the total gauge fixed Lagrangian. Hence we consider the localization for the

Now let us consider the 1-loop approximation of the

In the Higgs branch, we have fixed points of

Using this expansion (rescaling) of the fields, we also expand the rescaled total Lagrangian up to the quadratic order of the fluctuations,

Next let us consider the gauge fixing term. To find a suitable gauge fixing function, we pay attention to the terms proportional to

So if we adopt the gauge fixing function for the fluctuations by

Thus we obtain the gauge fixing and FP ghost Lagrangian for the above gauge fixing function:

Comparing the bosonic part of the Lagrangian (

For other fields, we now define sets of the bosonic and fermionic fields by

The 1-loop determinants of non-zero modes of the bosons and fermions cancel each other completely:

The bosonic zero modes are given by solutions to the linear equations:

On the other hand, the fermionic zero modes are given by the equations

As we discussed for the bosonic zero modes, we have seen

We need to integrate these bosonic and fermionic zero modes after integration of the non-zero modes, which gives the cancellation of the 1-loop determinant.

We have seen that the partition function of the fixed topological sector

In order to obtain a meaningful quantity from

^{2}

From the descent equation (

In order not to yield the extra contribution from the inserted operator, we need to modify

According to the localization theorem, the VEV of the

Since the solution to Eq. (

The bosonic part of this operator value at the Higgs branch fixed point gives just unity, but the fermionic part compensates all the fermionic zero modes since the exponent contains bi-linear terms of fermion pairs:

Since the operator

Let us now rewrite the above integral in the field configuration space in terms of the moduli parameters, which parametrize the BPS vortex solution. We denote the moduli parameters by complex coordinates

Thus we find that the path integral with the operator

Thanks to the localization theorem, we can also evaluate the above VEV of the operator in the other coupling region without changing the value of the path integral, and can reach even extreme Coulomb branch couplings

In this section, we consider the localization at the Coulomb branch, where the fields are expanded around the fixed point solution with non-vanishing

We will discuss the general non-Abelian case, but to see an essence of the Coulomb branch localization, we first explain the Abelian case.

In the Abelian theory, we denote the neutral scalar field by a lowercase letter

We now expand the bosonic fields in the vector multiplet around the classical solution (fixed points) by

For the fermionic fields, we expect that there are two 0-form zero modes and

Thus we also expand the fermionic fields in the vector multiplets around these zero modes as

In contrast to the Higgs branch evaluation, the fixed point solution of the bosonic field

This rescaling is always guaranteed by the invariance of the path integral measure

Using the above expansion and rescaling, we find that the Lagrangian becomes just quadratic order in the fluctuations:

The covariant derivatives

If the Lorentz gauge

Combining the quadratic part of the Lagrangian and the gauge fixing and FP terms, 1-loop determinants from the bosonic fields

On the other hand, the 1-loop determinant from the chiral multiplets is given by

To evaluate this determinant further, let us consider the eigenvalues of the Laplacians

So we find

On the other hand, for zero eigenvalue modes, there is no one-to-one correspondence between 0-forms and 1-forms. If we define the number of zero eigenvalue modes of the operator

So the 1-loop determinant reduces to

Next let us consider the contribution to the 1-loop determinant from

Thus we obtain

Now we can explicitly evaluate the vacuum expectation value of the operator

The integrand in Eq. (

On the other hand, in the Coulomb branch integration, the factor

Thus we can pick up the residues at

The choice of contours depending on the sign of

Let us give a concrete example. If we consider the vortices on the sphere (

The power of

By setting

In the case of

Finally we comment on the power of

The integral expression here does not depend on

In order for the moduli space to exist, at least the dimension should be equal to or greater than zero^{3}

Note here that there is a non-trivial lower bound for the vorticity

Now we generalize the above localization arguments to the non-Abelian case.

Let us consider the fixed point equation first. The fixed point equations for the non-Abelian theory are given by

Using the Cartan–Weyl bases (see Appendix A), the fixed point equations can be solved by

We now expand fields around the solution of the fixed point equations, i.e.,

Similarly, the fermions are expanded around the corresponding zero modes:

Other fields (auxiliary fields

Substituting the above expansion (

Introducing two component fermions by

At a generic value of

Because of the supersymmetry, we can expect essentially that the 1-loop determinants reduce to one by cancellation of bosons and fermions for the non-zero modes. We should, however, pay attention to the zero eigenvalue states of the operator

Actually, if we evaluate the 1-loop determinant from the off-diagonal components of the pseudo-zero modes, it reduces to

After integrating out all off-diagonal components of the fields, the argument of the localization for each Cartan part is almost parallel to the Abelian case in the previous subsection. Using the

Choosing the Lorentz gauge

By summing over the partition of the total vorticity

The power of

From the viewpoint of the BPS differential equations, it is difficult to find a topology (

The contour integral (

The condition is known as the Bradlow bound, which immediately follows from the BPS equations

A similar selection rule for the contour is also known as the Jeffrey–Kirwan residue formula [

The contour for the Jeffrey–Kirwan residue formula is chosen to get non-vanishing and vanishing residues if and only if

The Bradlow bound can be considered as a generalization of the Jeffrey–Kirwan residue formula for the effective FI parameter including the magnetic flux

For the non-Abelian theory, our integral formula (

This is also a generalization of the Jeffrey–Kirwan residue formula in the non-Abelian gauge theories.

So far, we have considered the volume of the moduli space under a fixed magnetic flux

Under this restriction of the summation over

Let us see some concrete examples. For the Abelian theory (

If we take the summation of

The integrand of the above contour integral has poles at zeros of the denominator, which are solutions of

The original degenerated pole at

The integral contour of the generating function and a split of the degenerated pole. After summing up the vorticity

There is no analytical solution of the transcendental equation (

If we assume

Because of the identity

Using these definitions and bounds, we can rewrite Eq. (

Next let us consider the non-Abelian case. Ignoring the Bradlow bound, we take the summation over the vorticity first, then we have

Again, if we denote a set of solutions to the transcendental equation

It is difficult to evaluate further the expression of the volume (

If we use the approximation (

Using this approximation, we find

Substituting this approximation into Eq. (

So we find that the volume of the moduli space of the non-Abelian local vortex becomes

This volume of the moduli space of the non-Abelian local vortex has been conjectured in Eq. (4.52) of Ref. [

In this paper, we derive an integral formula for the volume of the moduli space of the BPS vortex on the closed Riemann surface with the arbitrary genus. The BPS vortex system is embedded into

We firstly find that the path integral of the supersymmetric Yang–Mills theory in the Higgs branch gives directly the integral over the vortex moduli space. So the partition function of the supersymmetric Yang–Mills theory essentially gives the volume of the moduli space except for the integration of fermionic zero modes. Due to the fermionic zero modes, the partition function itself vanishes. We need to insert the appropriate operator in order to obtain the moduli space volume from the path integral. The inserted operator just compensates the fermionic zero modes and reduces to unity at the localization fixed point.

Secondly, in the Higgs branch description, we cannot perform the moduli space integral since the metric of the moduli space is not known in general. However, if we evaluate the same supersymmetric system in the Coulomb branch description by using the localization method, the path integral reduces to a simple finite-dimensional contour integral, which should give the volume of the vortex moduli space as discussed in the Higgs branch description. We also derive the exact 1-loop contribution to the gaugino mass including the higher genus case, which is needed to make the effective action supersymmetric.

The localization formula for the vortex moduli space captures the effect of the finite area of the Riemann surface, known as the Bradlow bound. The choice of the contours changes whether the area and vorticity satisfy the bound or not. This can be regarded as a kind of wall-crossing or Jeffrey–Kirwan residue formula where the choice of the contour depends on the flux in general.

We also discussed the generating function of the volume of the moduli space of the vortex. Under some assumptions, we can take the summation over the vorticity first. The summation modifies the contour integral whose poles and residues are given by the transcendental equations and are difficult to obtain analytically. However, this generating function can give a simple understanding of the reduction of the moduli space dimension in the case of the local vortices (

Our volume formula for the vortex moduli space on the Riemann surface suggests that there is a lower bound of vorticity (

In this paper, we consider only the case of the closed Riemann surface. If there are boundaries (punctures) of the Riemann surface, we should consider holonomies of the gauge fields around the boundaries. We expect that the partition function (the volume of the vortex moduli space) is a function of the boundary holonomies besides the vorticity and area. As known from Ref. [

Our system and evaluations can be extended to three dimensions, like

We would like to thank Toshiaki Fujimori, So Matsuura, Keisuke Ohashi, Yuya Sasai, and Yutaka Yoshida for useful discussions and comments. One of the authors (N.S.) thanks Takuya Okuda for useful discussion. This work was supported by JSPS KAKENHI(C) Grant Numbers JP26400256 and JP17K05422 (K.O.), by JSPS KAKENHI(B) Grant Number JP18H01217 (N.S.), and by the Ministry of Education, Culture, Sports, Science, and Technology(MEXT)-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (Grant No. S1511006) (N.S.).

Open Access funding: SCOAP

An

We use these notations in this paper.

To compute the 1-loop contributions to the fermion bi-linears, we need to consider the contribution from the propagators in the boson–fermion loop

We need to evaluate essentially

The heat kernel

The Laplacian

Thus we find

^{1}This gauge coupling

^{2}The two-form

^{3} If the dimension of the moduli space is zero, the moduli space becomes 0D isolated points.