Twisted Hilbert Spaces of 3d Supersymmetric Gauge Theories

TL;DR

This paper explores the structure of supersymmetric ground states in 3d N=2 gauge theories on a line times a Riemann surface, revealing new insights into mirror symmetry through cohomological constructions.

Contribution

It introduces a cohomological framework for the space of ground states in twisted 3d supersymmetric theories, enabling new tests of mirror symmetry.

Findings

Constructed the space of ground states via cohomology of vortex moduli spaces.

Demonstrated dependence of ground states on deformation parameters like superpotentials and mass parameters.

Provided new checks of 3d abelian mirror symmetry.

Abstract

We study aspects of 3d $\mathcal{N}=2$ supersymmetric gauge theories on the product of a line and a Riemann surface. Performing a topological twist along the Riemann surface leads to an effective supersymmetric quantum mechanics on the line. We propose a construction of the space of supersymmetric ground states as a graded vector space in terms of a certain cohomology of generalized vortex moduli spaces on the Riemann surface. This exhibits a rich dependence on deformation parameters compatible with the topological twist, including superpotentials, real mass parameters, and background vector bundles associated to flavour symmetries. By matching spaces of supersymmetric ground states, we perform new checks of 3d abelian mirror symmetry.

Equations

$$\mathcal{H}=\underset{\mathfrak{m}\in \mathbb{Z}}{⨁}{\mathcal{H}}_{\mathfrak{m}}.$$

$${\mathcal{H}}_{\mathfrak{m}}=H_{{\bar{\partial }}_{m_f}+\delta }^{0,\bullet }({\mathcal{M}}_{\mathfrak{m}},{\mathcal{F}}_{\mathfrak{m}}),$$

$${\bar{\partial }}_{m_f}=e^{-h_f}\cdot \bar{\partial }\cdot e^{h_f},$$

$$\mathcal{I}$$

$$=\sum _{\mathfrak{m}\in \mathbb{Z}}{q}^{\mathfrak{m}}{\mathrm{Tr}}_{{\mathcal{H}}_{\mathfrak{m}}}(-1{)}^F{x}^{J_f},$$

$$\mathcal{H}=\underset{\mathfrak{m}=1-g}{\overset{\infty }{⨁}}{q}^{\mathfrak{m}}\underset{j=0}{\overset{\mathfrak{m}+g-1}{⨁}}{\wedge }^j({\mathbb{C}}^g),$$

$$\mathcal{I}=q^{1-g}(1-q{)}^{g-1},$$

$$\{Q,\bar{Q}\}$$

$$\{\bar{Q},\bar{Q}\}$$

$${\partial }_{m_f}Q$$

$${\partial }_{m_f}\bar{Q}$$

$${\partial }_{u}Q$$

$${\partial }_{\bar{u}}\bar{Q}$$

$$\mathcal{I}=\mathrm{Tr}(-1{)}^F{e}^{-2\pi \beta H}{e}^{-2\pi i\beta A_f\cdot J_f}.$$

$${\mathcal{I}}_{\alpha }$$

$$={\mathrm{Tr}}_{{\mathcal{H}}_{\alpha }}(-1{)}^F{x}^{J_f},$$

$$x=e^{-2\pi \beta (m_f+i{A}_f)}$$

$${\Omega }^{0,\bullet }(M)\otimes F$$

$$⟨\alpha ,\beta ⟩={\int }_M\bar{\alpha }\wedge \ast \beta .$$

$$h_f=m_f\cdot {\mu }_f.$$

$$Q$$

$$\bar{Q}$$

$${\bar{\partial }}_{m_f}^{†}$$

$${\bar{\partial }}_{m_f}$$

$$H_{{\bar{\partial }}_{m_f}+\delta }^{0,\bullet }(M,F).$$

$$\mathcal{H}=H_{\bar{\partial }+\delta }^{0,\bullet }(M,F).$$

$$Q$$

$$\bar{Q}$$

$$H$$

$$J_f$$

$${\mathfrak{c}}_{+}$$

$${\mathfrak{c}}_{-}$$

$$h_f=m_f|\varphi {|}^2.$$

$${\mathfrak{c}}_{+}:\varphi \cdot [{\varphi }^n]=[{\varphi }^{n+1}]$$

$${\mathfrak{c}}_{-}:\varphi \cdot [{\bar{\varphi }}^n]=[{\bar{\varphi }}^{n-1}],$$

Full text

1 Introduction

We study aspects of 3d $\mathcal{N}=2$ supersymmetric gauge theories on the product of a line and a Riemann surface $C$. We introduce a topological twist on the Riemann surface $C$, which preserves a pair of supercharges generating a supersymmetric quantum mechanics on the line with supermultiplets of the type obtained by the dimensional reduction of 2d $\mathcal{N}=(0,2)$ supermultiplets. The setup is illustrated in figure 1.

Following the philosophy of Bullimore:2016hdc , our ultimate goal is to find an effective supersymmetric quantum mechanics that captures correlation functions of the 3d $\mathcal{N}=2$ supersymmetric gauge theory compatible with the topological twist on $C$. As a first step, in this paper we consider the space of supersymmetric ground states of the supersymmetric quantum mechanics, which we call the twisted Hilbert space.

The supersymmetric ground states are charged under Fermion number and flavour symmetry. The twisted Hilbert space then transforms as $\mathbb{Z}_{2}$-graded or virtual representation of the flavour symmetry. Furthermore, it exhibits a rich dependence on supersymmetric deformation parameters obtained by coupling to background vectormultiplets for flavour symmetry in a manner compatible with the topological twist on $C$. In particular, for the flavour symmetry $G_{f}$ acting on chiral multiplets, we may turn on

A background $G_{f}$-vector bundle $E_{f}$ on $C$. 2. 2.

Real mass parameters $m_{f}$ valued in the Cartan subalgebra of $G_{f}$.

In supersymmetric gauge theories with abelian factors in the gauge group, we can also introduce deformation parameters associated to topological flavour symmetries. In particular, a real mass for a $U(1)$ topological symmetry corresponds to a real FI parameter.

In this paper, we will focus on supersymmetric gauge theories with $G=U(1)$. In this case, the supersymmetric twisted Hilbert space decomposes as a direct sum of contributions labelled by the quantized flux $\mathfrak{m}\in\mathbb{Z}$ on $C$,

[TABLE]

Supersymmetric ground states in $\mathcal{H}_{\mathfrak{m}}$ have charge $\mathfrak{m}$ under the topological flavour symmetry. Each summand is a finite-dimensional $\mathbb{Z}_{2}$-graded representation of the remaining flavour symmetry $G_{f}$ acting on the chiral multiplets. It is captured by an effective supersymmetric quantum mechanics whose target space is the moduli space $\mathcal{M}_{\mathfrak{m}}$ of solutions to a set of generalized vortex equations on $C$ with flux $\mathfrak{m}$ in the presence of the background vector bundle $E_{f}$.

The generalized vortex moduli spaces $\mathcal{M}_{\mathfrak{m}}$ are Kähler and may or may not be compact, depending on the matter content and flux $\mathfrak{m}\in\mathbb{Z}$. However, provided the moduli space $\mathcal{M}_{\mathfrak{m}}$ is smooth, we propose that the space of supersymmetric ground states can be understood in terms of an $L^{2}$-cohomology

[TABLE]

where $\mathcal{F}_{\mathfrak{m}}$ denotes a $\mathbb{Z}_{2}$-graded vector bundle that receives contributions from supersymmetric Chern-Simons terms, a background line bundle for the topological flavour symmetry on $C$, and the quantization of Fermion zero modes. The differential is a sum of a conjugated Dolbeault operator,

[TABLE]

where the real superpotential $h_{f}=m_{f}\cdot\mu_{f}$ is constructed from the moment map $\mu_{f}$ for the action of the flavour symmetry $G_{f}$ on $\mathcal{M}_{\mathfrak{m}}$, and an additional contribution $\delta$ from a 3d superpotential. This $L^{2}$-cohomology can depend intricately on the choice of background vector bundle $E_{f}$ and real mass parameters $m_{f}$.

An important consistency check for our proposal is to reproduce the supersymmetric twisted index on $S^{1}\times C$, which may be computed in the supersymmetric gauge theory using localization Benini:2015noa ; Benini:2016hjo ; Closset:2016arn . This should be recovered from the supersymmetric twisted Hilbert space by computing the graded trace,

[TABLE]

where $J_{T}$, $J_{f}$ are the generators of the topological and remaining $G_{f}$ flavour symmetry respectively, and the parameters $q$ and $x$ are the respective fugacities. Our proposal therefore provides a new interpretation of the supersymmetric twisted index as a generating function of equivariant Euler characters associated to the generalized vortex moduli spaces $\mathcal{M}_{\mathfrak{m}}$.

We emphasize, however, that the supersymmetric twisted Hilbert space exhibits information and structure that goes beyond the supersymmetric twisted index:

• •

There can be dramatic cancellations in computing the supersymmetric index via (4), particularly on Riemann surfaces of genus $g>0$.

• •

The supersymmetric twisted Hilbert space is sensitive to 3d superpotential deformations via the differential $\delta$, which removes pairs of supersymmetric ground states whose contribution to the supersymmetric twisted index cancel out.

• •

The supersymmetric twisted Hilbert space may jump across hyperplanes in the space of real mass parameters $m_{f}$ where there are non-compact massless degrees of freedom. On the other hand, the supersymmetric index is a meromorphic function with poles on these hyperplanes. The same remark applies to real FI parameters for topological flavour symmetries.

• •

The supersymmetric twisted Hilbert space depends on a choice of holomorphic vector bundle $E_{f}$ on $C$ for the flavour symmetry $G_{f}$, while the supersymmetric twisted index depends only on its Chern class. The same remark applies to background line bundles associated to topological flavour symmetries.

The supersymmetric twisted Hilbert space can therefore provide a more refined check of supersymmetric dualities such as 3d mirror symmetry Aharony:1997bx ; deBoer:1997kr ; deBoer:1996ck ; Intriligator:1996ex .

In order to illustrate some of these points, we provide a brief appetizer. Let us consider a supersymmetric $U(1)$ Chern-Simons theory at level $+\frac{1}{2}$ with a chiral multiplet of charge $+1$. In section 4.4, we will show that the supersymmetric twisted Hilbert space with real FI parameter $\zeta>0$ at infinite coupling is given by

[TABLE]

where $g>0$ is the genus of $C$ and the parameter $q$ keeps track of the grading by the topological flavour symmetry. Notice that there are non-vanishing contributions from an infinite number of fluxes, $\mathfrak{m}\geq 1-g$. On the other hand, the supersymmetric twisted index is a finite Laurent polynomial

[TABLE]

with the contributions from fluxes $\mathfrak{m}\geq 0$ cancelling out in the trace. Nevertheless, in section 5.1 we will demonstrate that equation (5) agrees with the supersymmetric twisted Hilbert space of a single chiral multiplet with a positive real mass parameter. Furthermore, we extend this agreement to include a holomorphic line bundle for the flavour symmetry. This constitutes a new check of the simplest 3d mirror symmetry.

Finally, let us mention some connections to related research. First, an important motivation for this work is to generate new realisations of the correspondence between 3d $\mathcal{N}=2$ theories and three-manifolds Dimofte:2011py ; Dimofte:2011ju ; Cecotti:2011iy and we note that computations of twisted Hilbert spaces have appeared in this context in Gukov:2017kmk ; Gukov:2016gkn . Secondly, it would be interesting to consider potential applications to supersymmetric quantum Hall systems on a Riemann surface Iengo:1993cs ; Klevtsov:2017aa ; Tong:2003vy ; Tong:2015xaa .

The paper is structured as follows. In section 2, we review aspects of $\mathcal{N}=(0,2)$ type supersymmetric quantum mechanics, emphasising how the space of supersymmetric ground states depends on various types of deformation parameters. In section 3, we consider the topological twist of 3d $\mathcal{N}=2$ theories on a Riemann surface $C$ and explain the general structure of the effective supersymmetric quantum mechanics. In section 4, we compute the supersymmetric twisted Hilbert space in a variety of examples, including theories of chiral multiplets with superpotentials and abelian supersymmetric gauge theories. In section 5, we perform new checks of 3d mirror symmetry using the results from section 4. Finally, in section 6 we conclude with a discussion of directions for further research.

2 Supersymmetric Quantum Mechanics

We review supersymmetric quantum mechanics with supermultiplets that arise from the dimensional reduction of $\mathcal{N}=(0,2)$ supermultiplets in two dimensions, emphasizing those aspects that will be important in applications to 3d $\mathcal{N}=2$ theories on a Riemann surface. For further background and examples of this class of supersymmetric quantum mechanics we refer the reader to Hori:2014tda ; Wong:2015qnf .

2.1 Setup

A supersymmetric quantum mechanics has odd generators $Q$ and $\bar{Q}$ that are adjoint with respect to a hermitian inner product on the Hilbert space. We suppose the supersymmetric quantum mechanics has flavour symmetry $G_{f}$ with conserved charges $J_{f}\in\mathfrak{t}_{f}^{*}$ and introduce an associated real mass parameter $m_{f}\in\mathfrak{t}_{f}$. Here $\mathfrak{t}_{f}$ is the Cartan subalgebra of $G_{f}$. The supersymmetry algebra is then

[TABLE]

where $H$ is the hamiltonian.

We define supersymmetric ground states as those annihilated by $H-m_{f}\cdot J_{f}$. A standard argument shows that the spectrum of $H-m_{f}\cdot J_{f}$ is non-negative and that supersymmetric ground states are equivalently annihilated by both of the supercharges. We assume that the spectrum is gapped, in which case the supersymmetric ground states have another equivalent description as the cohomology of either supercharge $Q$ or $\bar{Q}$.

The requirement that the spectrum is gapped may place constraints on the mass parameters $m_{f}$. We will denote the subspace of mass parameters where the spectrum is gapped by $\mathfrak{c}_{f}\subset\mathfrak{t}_{f}$. If $\mathfrak{c}_{f}\neq\mathfrak{t}_{f}$, this subspace typically consists of a union of chambers $\mathfrak{c}_{f}=\bigcup_{\alpha}\mathfrak{c}_{\alpha}\subset\mathfrak{t}_{f}$ cut out by hyperplanes where there are non-compact massless degrees of freedom.

In this paper, we will compute the space of supersymmetric ground states $\mathcal{H}$ as the cohomology of the supercharge $\bar{Q}$. Since $(-1)^{F}$ and $J_{f}$ commute with the supercharges, the space of supersymmetric ground states is graded by Fermion number and flavour symmetry. Alternatively, we can say that it is a $\mathbb{Z}_{2}$-graded or virtual representation of the flavour symmetry $G_{f}$.

It will be important to understand how the supersymmetric ground states change as the real mass parameters $m_{f}$ are varied. In the examples encountered below, the supercharges obey

[TABLE]

where $\mu_{f}\in\mathfrak{t}_{f}^{*}$ is a hermitian operator. This is an $A$-type deformation in the notation of Gaiotto:2016hvd . In particular, the operator $\partial_{m_{f}}+\mu_{f}$ commutes with $\bar{Q}$ and descends to a complex flat Berry connection on the sheaf of supersymmetric ground states over $\mathfrak{c}_{f}\subset\mathfrak{t}_{f}$. Put simply, while the wavefunctions of the supersymmetric ground states will depend explicitly on the real mass parameters, $\mathcal{H}$ remains constant as a graded vector space provided the spectrum remains gapped. Therefore, we associate a space of supersymmetric vacua $\mathcal{H}_{\alpha}$ to each chamber $\mathfrak{c}_{\alpha}$.

We will also encounter examples of $B$-type deformations of the supersymmetric quantum mechanics Gaiotto:2016hvd , where the supercharges depend holomorphically or anti-holomorphically on a set complex parameters $u$,

[TABLE]

In particular, the derivative $\partial_{\bar{u}}$ commutes with $\bar{Q}$ and provided the system is gapped will descend to a holomorphic Berry connection on the sheaf of supersymmetric ground states over the complex space parametrized by $u$.

The flavoured supersymmetric index is defined as a graded trace over the full Hilbert space of the supersymmetric quantum mechanics,

[TABLE]

In the euclidean path integral construction of the supersymmetric index, $\beta$ is the radius of the circle and $e^{-2\pi i\beta A_{f}\cdot J_{f}}$ is a background Wilson line for the flavour symmetry. A standard argument shows that only supersymmetric ground states contribute to the supersymmetric index and therefore in each chamber $\mathfrak{c}_{\alpha}$ we obtain an expression

[TABLE]

where

[TABLE]

is valued in the complexified maximal torus of the flavour symmetry $G_{f}$. The supersymmetric index can therefore be expressed as a graded trace over $\mathcal{H}_{\alpha}$.

The supersymmetric index $\mathcal{I}_{\alpha}$ computed in equation (11) will in general yield a different Laurent polynomial in $x$ in each chamber $\mathfrak{c}_{\alpha}$. However, they correspond to Laurent expansions of the same meromorphic function $\mathcal{I}(x)$ in the different chambers $\mathfrak{c}_{\alpha}$ under the identification (12). This meromorphic function then has poles on the hyperplanes separating these chambers. In the case $\mathfrak{c}_{f}=\mathfrak{t}_{f}$, the supersymmetric index is a finite Laurent polynomial in $x$.

Finally, the supersymmetric index is insensitive to $B$-type deformations.

2.2 Geometric Model

We now consider a general class of supersymmetric quantum mechanics of the above type that arise from supersymmetric sigma models. The construction of these supersymmetric quantum mechanics has much in common with the construction of 2d $\mathcal{N}=(0,2)$ supersymmetric sigma models Witten:1993yc .

We consider a supersymmetric sigma model specified by the following data:

• •

A complex manifold $M$ with hermitian metric.

• •

A $\mathbb{Z}$-graded hermitian vector bundle $F$.

• •

A holomorphic differential $\delta:F\to F$ of degree $+1$ obeying $\delta^{2}=0$.

The full Hilbert space of the supersymmetric quantum mechanics consists of smooth square-integrable sections of

[TABLE]

with respect to the hermitian inner product

[TABLE]

Here, $*$ denotes the Hodge star operator on $M$ and contraction along fiber directions using the hermitian metric on $F$ is understood.

The supersymmetric quantum mechanics has an $R$-symmetry transforming the supercharges $Q$, $\bar{Q}$ with charge $-1$, $+1$ respectively. Referring to the above geometric data, this R-symmetry can be identified with the sum of the form degree on the target space $M$ and the $\mathbb{Z}$-grading on the hermitian vector bundle $F$, modulo an additive constant.

In this paper, we will only keep track of the Fermion number $(-1)^{F}$. In particular, we regard $F$ as a $\mathbb{Z}_{2}$ graded vector bundle with decomposition $F=F_{e}\oplus F_{o}$ into even and odd components. The Fermion number $(-1)^{F}$ in the supersymmetric quantum mechanics is then given the sum of the form degree and the $\mathbb{Z}_{2}$-grading on $F$:

• •

$(-1)^{q}$ for a section of $\Omega^{0,q}(M)\otimes F_{e}$ and

• •

$(-1)^{q+1}$ for a section of $\Omega^{0,q}(M)\otimes F_{o}$   .

Let $G_{f}$ denote the group of isometries of $M$ that lift to an equivariant action on $F$ preserving its hermitian metric and commuting with the holomorphic differential $\delta$. This is the flavour symmetry of the supersymmetric quantum mechanics. At this point, we assume that $M$ is Kähler and there exists a corresponding real moment map $\mu_{f}\in\mathfrak{t}_{f}^{*}$. We may then introduce an $A$-type deformation of the supersymmetric quantum mechanics by real mass parameters $m_{f}\in\mathfrak{t}_{f}$, which can be understood as a real superpotential

[TABLE]

This superpotential is the moment map for the $U(1)_{m_{f}}\subset G_{f}$ isometry generated by the mass parameters $m_{f}$ witten-hol-morse .

Let $\mathfrak{c}_{f}\subset\mathfrak{t}_{f}$ denote the mass parameters where the fixed locus of the $U(1)_{m_{f}}$ isometry of $M$ is compact and the spectrum of the supersymmetric quantum mechanics is gapped. If $M$ is non-compact, this is a disjoint union of chambers $\mathfrak{c}_{f}=\bigcup_{\alpha}\mathfrak{c}_{\alpha}$ cut out by hyperplanes. If $M$ is compact, $\mathfrak{c}_{f}=\mathfrak{t}_{f}$.

The supercharges are identified with

[TABLE]

where

[TABLE]

and $\bar{\partial},\bar{\partial}^{\dagger}$ denote respectively the twisted Dolbeault operator acting on sections of (13) and its adjoint with respect to the hermitian inner product (14). Finally, $\delta^{\dagger}$ is the adjoint of holomorphic differential $\delta$ with respect to the hermitian metric on $F$. Note that the supercharge $\bar{Q}$ depends holomorphically on deformations of the holomorphic vector bundle $F$ and the differential $\delta$ : they are $B$-type deformations of the supersymmetric quantum mechanics.

These supercharges obey the supersymmetry algebra (7) where $H$ is a deformation of the Dolbeault Laplacian and $m_{f}\cdot J_{f}$ is the Lie derivative along the vector field generating $U(1)_{m_{f}}$.

Provided the mass parameters lie in $\mathfrak{c}_{f}\subset\mathfrak{t}_{f}$ and the spectrum of the supersymmetric quantum mechanics is gapped, the space of supersymmetric ground states can be identified with $L^{2}$-cohomology of the supercharge $\bar{Q}$, which we write schematically as

[TABLE]

Due to the exponential dependence of the supercharge on the mass parameter $m_{f}$ and the condition of square-normalizability, the computation of this cohomology for a non-compact target space $M$ will generally yield a different space of supersymmetric ground states $\mathcal{H}_{\alpha}$ in each chamber $\mathfrak{c}_{\alpha}$. In other words, the space of supersymmetric ground states may jump across hyperplanes in $\mathfrak{t}_{f}$ where there are massless non-compact degrees of freedom.

However, if $M$ is compact, the spectrum of the supersymmetric quantum mechanics is gapped for any $m_{f}$ and the space of supersymmetric vacua is constant on $\mathfrak{t}_{f}$. In this case, we can set $m_{f}=0$ and identify the space of supersymmetric ground states with the regular hypercohomology,

[TABLE]

Let us finally consider the supersymmetric index in this class of supersymmetric quantum mechanics. The supersymmetric index $\mathcal{I}_{\alpha}$ in each chamber computes the equivariant character of $\mathcal{H}_{\alpha}$ as a virtual representation of the flavour symmetry $G_{f}$. This index is independent of the differential and can be identified with an equivariant Euler character for $L^{2}$-cohomology classes of the conjugated Dolbeault operator $\bar{\partial}_{m_{f}}$ in equation (17). If $M$ is compact, the supersymmetric index $\mathcal{I}$ coincides with the regular equivariant Euler character $\chi(M,F)$.

2.3 Examples

2.3.1 Chiral Multiplets

Our first example is a single chiral multiplet $(\phi,\psi)$ with real a mass parameter $m_{f}$ for the $U(1)_{f}$ flavour symmetry. This model is a supersymmetric complex harmonic oscillator. In canonical quantization, the complex Fermion obeys $\{\psi,\bar{\psi}\}=1$ and the supercharges take the form

[TABLE]

The supercharges obey (7) with

[TABLE]

Note that while the supercharges and the combination $H-m_{f}J_{f}$ are unambiguous, $H$ and $J_{f}$ individually depend on a normal ordering constant $\kappa$, which can be understood as a supersymmetric Chern-Simons term for the $U(1)_{f}$ flavour symmetry Assel:2015nca ; Elitzur:1985xj .

The supersymmetric ground states wavefunctions are annihilated by both supercharges. The gapped region of parameter space consists of two chambers $\mathfrak{c}_{\pm}\subset\mathfrak{t}_{f}=\mathbb{R}$ corresponding to $m_{f}>0$ and $m_{f}<0$ respectively - see figure 2. Choosing the Fock vacuum annihilated by $\psi$, the normalizable ground state wavefunctions are

[TABLE]

where $n\geq 0$. Here we have defined a superpotential

[TABLE]

The supersymmetric ground state wavefunctions can also be viewed as harmonic representatives of $L^{2}$-cohomology classes for the supercharge $\bar{Q}$. We denote the associated cohomology classes by $[\,\phi^{n}\,]$ in the chamber $\mathfrak{c}_{+}$ and $[\,\bar{\phi}^{n}\,]$ in the chamber $\mathfrak{c}_{-}$. In the second chamber, it is important to remember the presence of the Fermion $\bar{\psi}$, which is suppressed in our notation. Since the operator $\phi$ commutes with the supercharge $\bar{Q}$, it has a well-defined action on these cohomology classes,

[TABLE]

which is compatible with the $U(1)_{f}$ flavour symmetry.

Although the supersymmetric ground state wavefunctions depend on $m_{f}$, as a vector space graded by $(-1)^{F}$ and $U(1)_{f}$, the space of supersymmetric ground states is constant in each chamber,

[TABLE]

Here we have introduced a formal parameter $x\in\mathbb{C}^{*}$ to keep track of the $U(1)_{f}$ charge measured by $J_{f}$. Note that there is a choice of Fermion number for the Fock vacuum, and we have assigned Fermion number zero to the Fock vacuum annihilated by $\psi$.

The supersymmetric index computed in each chamber is

[TABLE]

Recalling that in computing the supersymmetric index we identify $x=e^{-2\pi\beta(m_{f}+iA_{f})}$, this corresponds to the expanding the same meromorphic function

[TABLE]

in the appropriate regime, namely $|x|<1$ in $\mathfrak{c}_{+}$ and $|x|>1$ in $\mathfrak{c}_{-}$. This expression coincides with the 1-loop determinant for a chiral multiplet on a circle with a background supersymmetric Chern-Simons term for $U(1)_{f}$ at level $\kappa$.

This example can be extended to $N$ chiral multiplets $(\phi_{j},\psi_{j})$ with flavour symmetry $G_{f}=U(N)$ and mass parameters $m_{f}=(m_{1},\ldots,m_{N})\in\mathfrak{t}_{f}$. There are now massless degrees of freedom on all coordinate hyperplanes in $\mathfrak{t}_{f}=\mathbb{R}^{N}$. Removing these hyperplanes, the gapped region $\mathfrak{c}_{f}=\bigcup_{\alpha}\mathfrak{c}_{\alpha}$ consists of $2^{N}$ disjoint chambers

[TABLE]

labelled by a sign vector $\alpha=(\alpha_{1},\ldots,\alpha_{N})$.

The space of supersymmetric grounds states in each chamber is

[TABLE]

where

[TABLE]

We have chosen the same normal ordering constant $\kappa$ for each chiral multiplet to preserve the underlying $G_{f}=U(N)$ flavour symmetry. As expected, the result reproduces the expansion of the supersymmetric index

[TABLE]

in the appropriate regime $|x_{j}|^{\alpha_{j}}<1$.

This model can be understood as a supersymmetric sigma model to $M=\mathbb{C}^{N}$ with the standard flat Kähler metric, supplemented by a hermitian line bundle $F=K_{\mathbb{C}^{N}}^{\kappa+1/2}$. The flavour symmetry $G_{f}=U(N)$ corresponds to the isometries of $\mathbb{C}^{N}$. Introducing real mass parameters corresponds to a superpotential

[TABLE]

which is the moment map for the $U(1)_{m_{f}}$ isometry generated by $m_{f}\in\mathfrak{t}_{f}$. The chambers $\mathfrak{c}_{\alpha}$ correspond to values of the mass parameters where the fixed locus of $U(1)_{m_{f}}$ is compact, namely the origin of $\mathbb{C}^{N}$.

2.3.2 Fermi Multiplets

Let us now consider a single Fermi multiplet $(\eta,F)$ with a real mass parameter $m_{f}$ for the $U(1)_{f}$ flavour symmetry. In canonical quantization, the complex Fermion obeys $\{\eta,\bar{\eta}\}=1$ and

[TABLE]

The combination $H-m_{f}J_{f}=0$ is again unambiguous, whereas $H$ and $J_{f}$ individually depend on a normal ordering constant $\kappa$.

In canonical quantization, we can choose a Fock vacuum or reference state $|0\rangle$ annihilated by the Fermion $\eta$ and assign it Fermion number [math]. The supersymmetric ground states are then $|0\rangle$ and $\bar{\eta}|0\rangle$ with flavour charge $\kappa+\frac{1}{2}$ and $\kappa-\frac{1}{2}$ respectively, as measured by $J_{f}$. We therefore find

[TABLE]

in agreement with the supersymmetric index

[TABLE]

In the quantization of Fermi multiplets, there is a notational freedom to choose the Fock vacuum or reference state $|0\rangle$ to be annihilated by $\eta$ or $\bar{\eta}$. In more complicated examples below, we will use this freedom to choose the representation that is most convenient for enumerating the supersymmetric ground states. The Fermion number assigned to this Fock vacuum is, however, meaningful and sets the Fermion number grading of supersymmetric ground states. This corresponds to an overall sign in the supersymmetric index. The reader is forewarned that we will typically omit the reference state $|0\rangle$ from our notation.

2.3.3 Superpotentials

We now present a number of examples that couple chiral and Fermi multiplets with holomorphic superpotentials and will reappear in computations relevant for 3d $\mathcal{N}=2$ theories in section 4.

Let us first consider a chiral multiplet $(\phi,\psi)$ coupled to a Fermi multiplet $(\eta,F)$ with the following $J$-term superpotential

[TABLE]

where the complex mass parameter $u$ can be regarded as a vacuum expectation value for a background chiral multiplet. The model preserves a $G_{f}=U(1)$ flavour symmetry under which $(\phi,\psi)$ have charge $+1$ and $(\eta,F)$ have charge $-1$. We introduce a corresponding real mass parameter $m_{f}\neq 0$.

In canonical quantization, the supercharge

[TABLE]

is a sum of two contributions

[TABLE]

where $\bar{Q}^{(0)}$ is the contribution from the chiral multiplet and $\bar{Q}^{(1)}$ is the additional contribution from the Fermi multiplet with $J$-term superpotential. Note that in this model there is both a real $A$-type parameter $m_{f}$ and a complex $B$-type parameter $u$.

We assign Fermion number zero to the Fock vacuum annihilated by $\psi$ and $\eta$. First, the cohomology of $\bar{Q}^{(0)}$ consists of the supersymmetric ground states of the chiral multiplet tensored with those of the Fermi multiplet,

[TABLE]

If $u=0$, the computation ends here and there is an infinite number of supersymmetric ground states. Assuming $u\neq 0$, a short spectral sequence argument shows that the cohomology of the total supercharge $\bar{Q}$ is equivalent to the cohomology of $\bar{Q}^{(1)}$ acting on the states (39). This is computed as follows:

• •

$\mathfrak{c}_{+}$: $\bar{Q}^{(1)}$ removes pairs $[\,\phi^{n+1}\,]$ and $[\,\phi^{n}\,]\,\bar{\eta}$ with $n\geq 0$ leaving only $[\,1\,]$.

• •

$\mathfrak{c}_{-}$: $\bar{Q}^{(1)}$ removes pairs $[\,\bar{\phi}^{n}\,]$ and $[\,\bar{\phi}^{n+1}\,]\,\bar{\eta}$ with $n\geq 0$ leaving only $[\,1\,]\,\bar{\eta}$.

We therefore find that for $u\neq 0$ there is a unique supersymmetric ground state and, setting normal ordering constants $\kappa=0$, $\mathcal{H}_{\alpha}=\mathbb{C}$ in both chambers $\alpha=\pm$.

Let us compare this result with the supersymmetric index. This is computed by multiplying the contributions from a chiral multiplet of charge $+1$ and a Fermi multiplet of charge $-1$, with the result $\mathcal{I}=1$ for both $u=0$ and $u\neq 0$. In summary, the space of supersymmetric vacua is sensitive to the $J$-term superpotential whereas the supersymmetric index is not.

Let us now consider a second example with a pair of chiral multiplets $\phi_{1}$, $\phi_{2}$ coupled to a Fermi multiplet $\eta$ with superpotential $J(\phi)=\phi_{1}\phi_{2}$111As in our previous example, we could introduce a dimensionless complex parameter $u$ in the superpotential. Since the Hilbert space of supersymmetric vacua will not depend on this parameter provided $u\neq 1$, we set $u=1$ for convenience.. This preserves a $U(1)_{1}\times U(1)_{2}$ flavour symmetry whose charges can be chosen as follows,

[TABLE]

.

Introducing real mass parameters $m_{1},m_{2}$ for the flavour symmetry, there are four chambers $\mathfrak{c}_{\alpha}\subset\mathfrak{t}_{f}=\mathbb{R}^{2}$ labelled by a sign vector $\alpha=(\alpha_{1},\alpha_{2})$ - see figure 3. We again choose the Fock vacuum annihilated by $\psi_{1},\psi_{2}$ and $\eta$.

As in our previous example, the supercharge is a sum of contributions from the chiral multiplets and the superpotential. The cohomology of $\bar{Q}^{(0)}$ is again the tensor product of supersymmetric ground states for the chiral and Fermi multiplets. Let us first consider the chamber $\mathfrak{c}_{++}$, in which we compute the cohomology of $\bar{Q}^{(1)}=\phi_{1}\phi_{2}\eta$ acting on

[TABLE]

for $n_{1},n_{2}\geq 0$. The differential annihilates any $[\,\phi_{1}^{n_{1}}\phi_{2}^{n_{2}}\,]$ and sends the state $[\,\phi_{1}^{n_{1}}\phi_{2}^{n_{2}}\,]\bar{\eta}$ to $[\,\phi_{1}^{n_{1}+1}\phi_{2}^{n_{2}+1}\,]$. The remaining states in cohomology are

[TABLE]

and therefore

[TABLE]

where again we set $\kappa=0$.

In the chamber $\mathfrak{c}_{+-}$, we compute the cohomology of $\bar{Q}^{(1)}=\phi_{1}\phi_{2}\eta$ acting on

[TABLE]

with $n_{1},n_{2}\geq 0$. The differential annihilates any $[\,\phi_{1}^{n_{1}}\bar{\phi}_{2}^{n_{2}}\,]$ and sends the state $[\,\phi_{1}^{n_{1}}\bar{\phi}_{2}^{n_{2}}\,]\bar{\eta}$ to $[\,\phi_{1}^{n_{1}+1}\bar{\phi}_{2}^{n_{2}-1}\,]$ if it exists. The remaining states in cohomology are

[TABLE]

and therefore

[TABLE]

There are similar results on regions $\mathfrak{c}_{-+}$ and $\mathfrak{c}_{--}$.

This is consistent with expanding the supersymmetric index

[TABLE]

in the appropriate chambers. The supersymmetric index does not detect the presence of the $J$-term superpotential, except through the determination of the flavour symmetry and associated mass parameters. In particular, the contributions from states removed in pairs by $\bar{Q}^{(1)}$ cancel out in the supersymmetric index.

We consider one final example that will reappear in a three-dimensional problem in section 4.3. We introduce three chiral multiplets $\phi_{1},\phi_{2},\phi_{3}$ coupled to three Fermi multiplets $\eta_{1},\eta_{2},\eta_{3}$ with $J$-term superpotentials

[TABLE]

This model in fact arises from a supersymmetric quantum mechanics with four supercharges with chiral multiplets $\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$ and cubic superpotential $W=\Phi_{1}\Phi_{2}\Phi_{3}$. Here, we regard it as an $\mathcal{N}=(0,2)$ supersymmetric quantum mechanics with flavour symmetry

[TABLE]

.

Our notation and choice of charges is made with future applications in mind. It is straightforward to check that the supersymmetric index is $1$.

Introducing real mass parameters $m_{T}$ and $m_{A}$, there are six chambers $\mathfrak{c}_{\alpha}\subset\mathfrak{t}_{f}=\mathbb{R}^{2}$ labelled by sign vectors $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$. The sign vectors $(+++)$ and $(---)$ are not allowed as the mass parameters of all three chirals must sum to zero. Let us concentrate here on the chamber $(+-+)$ corresponding to mass parameters $0<m_{A}<m_{T}$. Following previous examples, we compute the cohomology of

[TABLE]

on states

[TABLE]

where $n_{1},n_{2},n_{3}\geq 0$ and $s_{1},s_{2},s_{3}=0,1$. We have chosen the Fock vacuum annihilated by $\psi_{1},\psi_{2},\psi_{3}$ and $\eta_{1},\eta_{2},\eta_{3}$ and assigned it Fermion number one.

This example is simple enough to compute representatives of cohomology classes directly. A more systematic method is to split the supercharge into three terms and apply the method spectral sequences to compute the cohomology of the total complex in steps. We summarize representatives of the remaining cohomology classes and their contribution to the supersymmetric index below,

[TABLE]

where $n\geq 0$. All of the contributions to the supersymmetric index cancel in pairs except for the final line, reproducing the expected result $\mathcal{I}=1$. A similar analysis can be performed in the remaining chambers.

A general model consists of $N$ chiral multiplets $(\phi_{j},\psi_{j})$ and $k$ Fermi multiplets $(\eta^{a},F^{a})$ coupled by holomorphic superpotentials $J_{a}(\phi)$ and $E^{a}(\phi)$. The Hilbert space of supersymmetric ground states is the cohomology of

[TABLE]

acting on the tensor product of supersymmetric ground states for the individual chiral and Fermi multiplets. This can be understood as a supersymmetric sigma model to $M=\mathbb{C}^{N}$ together with the $\mathbb{Z}_{2}$-graded hermitian vector bundle

[TABLE]

where $f$ denotes the odd rank-$k$ hermitian vector bundle on $\mathbb{C}^{N}$ with fibers spanned by the complex fermions $\eta^{a}$. The holomorphic differential $\delta=\bar{Q}^{(01)}$ is given by the sum of contraction with the holomorphic section $\eta^{a}J_{a}(\phi)$ of $f$ and the wedge product with the holomorphic section $\bar{\eta}_{a}E^{a}(\phi)$ of $f^{*}$.

2.3.4 Gauge Theory

A vectormultiplet in $\mathcal{N}=(0,2)$ supersymmetric quantum mechanics contains a gauge field $A_{\tau}$, a real scalar $\sigma$, and a real auxiliary field $D$, in addition to the complex fermions $\lambda$, $\tilde{\lambda}$. The real mass parameters introduced above can be regarded as coupling to a background vectormultiplet for the flavour symmetry $G_{f}$ and turning on a vacuum expectation value $m_{f}=\langle\sigma_{f}\rangle$ for the scalar component.

We now consider dynamical vectormultiplets for a gauge symmetry $G$. We focus on $G=U(1)$ and introduce $N$ chiral multiplets $(\phi_{j},\psi_{j})$ transforming with charge $Q_{j}$. We also introduce a real FI parameter $\zeta>0$ and a supersymmetric Wilson line of charge $q$. These contribute $\zeta D$ and $q(\sigma+iA_{\tau})$ respectively to the lagrangian. Global anomaly cancelation requires

[TABLE]

This model will arise in computations of the supersymmetric twisted Hilbert space of three-dimensional gauge theories with $G=U(1)$ on $C=\mathbb{CP}^{1}$ in sections 4.4-4.5

To compute the supersymmetric ground states, we introduce a supersymmetric sigma model onto configurations minimizing the euclidean action in the ‘geometric regime’. We first note that the auxiliary field can be eliminated by its equation of motion to give $D=e^{2}(\mu-\zeta)$, where $\mu=\sum_{j}Q_{j}|\phi_{j}|^{2}$ is the moment map for the $U(1)$ action on $\mathbb{C}^{N}$. The classical potential is then

[TABLE]

Assuming $\zeta>0$, the classical potential is minimized by configurations

[TABLE]

modulo constant $U(1)$ gauge transformations. At frequencies much smaller than $e^{2}\sqrt{\zeta}$, the system can be described by a supersymmetric sigma model to the Kähler quotient

[TABLE]

Since the dependence on $e^{2}$ and $\zeta$ is exact, provided the quotient $M$ is smooth we expect the supersymmetric sigma model to exactly capture the space of supersymmetric ground states in the supersymmetric gauge theory. Note that for the quotient $M$ to be smooth, we require

[TABLE]

such that $M$ is the total space of the line bundle $\mathcal{O}(-1)^{N-k}\to\mathbb{CP}^{k-1}$.

Let us specialize here to the compact case with $N$ chiral multiplets of charge $Q_{j}=+1$. The chiral multiplets transform in the fundamental representation of the $G_{f}=PSU(N)$ flavour symmetry. In this case, we find a supersymmetric sigma model to $M=\mathbb{CP}^{N-1}$. In addition, there is is a hermitian line bundle $F$ with the following contributions:

• •

A contribution $K_{\mathbb{CP}^{N-1}}^{1/2}=\mathcal{O}(-\frac{N}{2})$ from quantizing chiral multiplet fermions.

• •

A contribution $\mathcal{O}(q)$ from the supersymmetric Wilson line.

The anomaly cancellation condition (52) ensures that the combination $F=\mathcal{O}(q-\frac{N}{2})$ is well-defined.

We can further introduce real mass parameters for the chiral multiplets $m_{f}=(m_{1},\ldots,m_{N})$ with $\sum_{j}m_{j}=0$, which parametrize $\mathfrak{t}_{f}=\mathbb{R}^{N-1}$. In the supersymmetric sigma model, the real mass parameters introduce a real superpotential $h_{f}:\mathbb{CP}^{N-1}\to\mathbb{R}$ given by the moment map for the $U(1)_{m_{f}}$ isometry of $\mathbb{CP}^{N-1}$ generated by $m_{f}$.

Since the target of the supersymmetric sigma model is compact, the spectrum is always gapped and the space of supersymmetric ground states is constant over the whole parameter space $\mathfrak{t}_{f}=\mathbb{R}^{N-1}$. In particular, at $m_{f}=0$ the space of supersymmetric ground states can be identified with the Dolbeault cohomology

[TABLE]

We introducing complex parameters $(x_{1},\ldots,x_{N})$ with $\prod_{j}x_{j}=1$ parametrizing the complexified maximal torus of $G_{f}=PSU(N)$ to keep track of the grading by flavour symmetry. Then

[TABLE]

corresponding to symmetric powers of the fundamental and anti-fundamental representations of $G_{f}=PSU(N)$.

The supersymmetric index may be computed independently by localization in the supersymmetric gauge theory Hori:2014tda . This computation reproduces the characters of the above representations of $G_{f}=PSU(N)$,

[TABLE]

The contour $\Gamma$ surrounds the poles at $z=x_{j}^{-1}$ for all $j=1,\ldots,N$. The contour integral expression for the supersymmetric index coincides with the computation of the holomorphic Euler character $\chi(M,F)$ using the Hirzebruch-Riemann-Roch theorem.

3 Three-Dimensional Setup

We now consider 3d $\mathcal{N}=2$ theories with a topological twist on the product of a real line $\mathbb{R}$ and a compact connected Riemann surface $C$ of genus $g$. This setup falls into the general class of supersymmetric backgrounds introduced in Closset:2017zgf . The related supersymmetric background on $S^{1}\times C$ has been further studied in the context of the supersymmetric twisted index in Benini:2015noa ; Benini:2016hjo ; Closset:2016arn .

This setup preserves the same supersymmetry as an $\mathcal{N}=(0,2)$ supersymmetric quantum mechanics on $\mathbb{R}$ of the type considered in section 2. In this section, we will explain how the 3d supermultiplets decompose into those of the supersymmetric quantum mechanics and introduce an effective description that captures the twisted Hilbert space of supersymmetric ground states. The supersymmetric twisted Hilbert space is then computed explicitly in a number of examples in section 4.

3.1 Decomposing Supermultiplets

3.1.1 Chiral Multiplets

Let us first consider 3d chiral multiplets $\Phi$ of $R$-charge $r$ transforming in a unitary representation $R_{f}$ of a flavour symmetry $G_{f}$. We can introduce two deformation parameters associated to the flavour symmetry that are compatible with the topological twist on $C$:

• •

A holomorphic $G_{f}$ bundle $E_{f}$ on $C$ in the representation $R_{f}$.

• •

Real mass parameters $m_{f}\in\mathfrak{t}_{f}$.

As explained below, these deformation parameters can be understood as vacuum expectation values for a background vectormultiplet for $G_{f}$.

Performing the topological twist on $C$, the three-dimensional chiral multiplet decomposes into the following supermultiplets in supersymmetric quantum mechanics:

• •

A 1d $\mathcal{N}=(0,2)$ chiral multiplet $(\phi,\psi)$ valued in smooth sections of $E_{\Phi}$,

• •

A 1d $\mathcal{N}=(0,2)$ Fermi multiplet $(\eta,F)$ valued in smooth sections of $\Omega^{0,1}_{C}\otimes E_{\Phi}$,

where

[TABLE]

and $K_{C}$ is the canonical bundle on $C$. This may require a choice a spin structure on $C$, and different choices are related by tensoring $E_{f}$ with a flat line bundle on $C$. The holomorphic bundle $E_{\Phi}$ inherits a hermitian metric from that on the canonical bundle on $C$ and the hermitian metric on the vector space of the unitary representation $R_{f}$.

In addition to the standard kinetic term contributions in the supersymmetric quantum mechanics from these supermultiplets, there is an $E$-type superpotential for the Fermi multiplet,

[TABLE]

where $\bar{D}$ denotes the holomorphic structure on $E_{\Phi}$. Note that the $E$-term superpotential transforms in the same way as the Fermi multiplet $(\eta,F)$, as required for supersymmetry. The $E$-term contribution to the supercharge $\bar{Q}$ is proportional to

[TABLE]

where contraction using the hermitian metric on $E_{\Phi}$ is understood. The choice of holomorphic structure $\bar{D}$ on $E_{\Phi}$ is therefore a $B$-type parameter in the supersymmetric quantum mechanics.

The role of the $E$-type superpotential is to provide kinetic terms with derivatives along $C$ in the supersymmetric quantum mechanics. In particular, it contributes terms to the Lagrangian of the supersymmetric quantum mechanics proportional to

[TABLE]

where again contraction using the hermitian metric on $E_{\Phi}$ is understood.

Finally, real mass parameters generate the real superpotential

[TABLE]

where $\mu_{f}\in\mathfrak{t}^{*}_{f}$ is the moment map for the action of $G_{f}$ on the unitary representation $R_{f}$. In the presence of real mass parameters, the supercharges of the supersymmetric quantum mechanics are formally conjugated by the exponential factor $e^{h_{f}}$, as in section 2.2. This is an $A$-type deformation of the supersymmetric quantum mechanics.

3.1.2 3d Superpotentials

The above model can be deformed by a 3d superpotential $W(\Phi)$ preserving the $R$-symmetry used to perform the topological twist on $C$. A superpotential will place restrictions on the flavour symmetry $G_{f}$ and therefore the allowed background vector bundle $E_{f}$ and real mass parameters $m_{f}$.

The superpotential must have $R$-charge $+2$ and will therefore transform as a section of the canonical bundle on the curve $C$ in the twisted theory. It introduces a $J$-term superpotential to the supersymmetric quantum mechanics,

[TABLE]

which transforms as a smooth section of $\Omega_{C}^{1,0}\otimes\bar{E}_{\Phi}$. This generates an additional contribution to the supercharge $\bar{Q}$ proportional to

[TABLE]

where contraction using the hermitian metric on $E_{\Phi}$ is understood. The complex parameters in the 3d superpotential therefore become $B$-type deformation parameters in the supersymmetric quantum mechanics.

3.1.3 Gauge Theory

Now consider a 3d $\mathcal{N}=2$ vectormultiplet for a compact group $G$. The bosonic fields in the vectormultiplet consist of the connection $A_{\mu}$, a real scalar $\sigma$, and a real auxiliary field $D$. Performing the topological twist on $C$, the 3d $\mathcal{N}=2$ vectormultiplet decomposes into the following multiplets in the supersymmetric quantum mechanics:

• •

A 1d $\mathcal{N}=(0,2)$ vectormultiplet consisting of $A_{\tau}$, $\sigma$, and auxiliary field $D^{\mathrm{1d}}:=D+*\,F$ where $*$ is the Hodge operator on $C$.

• •

A 1d $\mathcal{N}=(0,2)$ chiral multiplet valued with complex scalar component given by the $(0,1)$-form component of the connection $\bar{D}$ on $C$.

In the supersymmetric quantum mechanics, the vectormultiplet is associated to the infinite-dimensional group $\mathcal{G}$ of gauge transformations $g:C\to G$.

Let us consider a dynamical vectormultiplet with gauge symmetry $G$ together with chiral multiplets $\Phi$ of $R$-charge $r$ transforming in a unitary representation $R$ of $G$. We suppose the chiral multiplets transform in a unitary representation $R_{f}$ of a residual flavour symmetry $G_{f}$.

We can again introduce deformation parameters $m_{f}$ and $E_{f}$ associated to the flavour symmetry $G_{f}$. In the supersymmetric quantum mechanics, the chiral multiplets decompose as above with

[TABLE]

where $E$, $E_{f}$ are the holomorphic vector bundles associated to the representations $R$, $R_{f}$. The parameters $m_{f}$, $E_{f}$ can in fact be understood as vacuum expectation values for a background vectormultiplet for the flavour symmetry $G_{f}$:

• •

The mass $m_{f}=\langle\sigma_{f}\rangle$ is a vacuum expectation for the 1d vectormultiplet.

• •

The holomorphic vector bundle $E_{f}$ is a vacuum expectation value for the 1d chiral multiplet $\bar{D}_{f}$.

Preserving both supercharges of the supersymmetric quantum mechanics requires $D_{f}^{\mathrm{1d}}=0$ and therefore we should also turn on a compensating auxiliary field $D_{f}=-*F_{f}$ given by the curvature of $E_{f}$.

If $G$ contains abelian factors, there is a topological flavour symmetry and we can turn on a vacuum expectation value for a background vectormultiplet for this symmetry. For example, if $G=U(1)$ there is a topological flavour symmetry $U(1)_{T}$ and we can introduce a real FI parameter $m_{T}=\zeta$ and holomorphic line bundle $L_{T}$. It is useful to note that the contribution to the Lagrangian of the supersymmetric quantum mechanics from a real FI parameter is

[TABLE]

where

[TABLE]

This can be regarded as an FI term in the supersymmetric quantum mechanics with parameter $\zeta_{1\mathrm{d}}=\mathrm{Vol_{C}}\zeta$ plus an additional contribution proportional to the flux.

Finally, for each abelian or simple factor in $G$, we can introduce a supersymmetric Chern-Simons term. For example, the contribution to the supersymmetric quantum mechanics from a $G=U(1)$ Chern-Simons term at level $k$ is

[TABLE]

In general, we can introduce various mixed Chern-Simons contributions between gauge, flavour and R-symmetries.

3.2 Effective Quantum Mechanics

In the previous section, we rephrased the topological twist of a 3d $\mathcal{N}=2$ theory on $\mathbb{R}\times C$ as an infinite dimensional supersymmetric quantum mechanics on $\mathbb{R}$. We now introduce an effective supersymmetric quantum mechanics that captures the space of supersymmetric ground states. We focus exclusively on regimes where there are only ‘Higgs branch’ vacua in the sense of Intriligator:2013lca . In this case, the effective supersymmetric quantum mechanics is a sigma model onto the moduli space of vortex like configurations that minimize the effective euclidean action, following the philosophy of Bershadsky:1995vm ; Kapustin:2006pk ; Kapustin:2006hi

3.2.1 Chiral Multiplets

Let us return to the model with chiral multiplets transforming a unitary representation $R_{f}$ of a flavour symmetry $G_{f}$ from section 3.1.1. The euclidean action is minimized by time-independent configurations that minimize of the potential of the supersymmetric quantum mechanics

[TABLE]

which is induced by the $E$-type superpotential term in equation (63). Such configurations therefore satisfy $\bar{D}\phi=0$. In addition, time-independent solutions of the Fermi multiplet equations of motion obey $\bar{D}\bar{\eta}=0$.

We therefore consider an effective supersymmetric quantum mechanics consisting of a finite number of fluctuations:

• •

Chiral multiplets $(\phi,\psi)$ valued in $H^{0}(E_{\Phi})$

• •

Fermi multiplets $(\eta,F)$ valued in $H^{1}(E_{\Phi})$ .

Let us define the number of chiral and Fermi multiplet fluctuations by

[TABLE]

The difference is determined by the Riemann-Roch theorem,

[TABLE]

and depends on the background holomorphic vector bundle only through $c_{1}(E_{\Phi})$ and $\mathrm{rk}(E_{\Phi})$. In contrast, the individual numbers of fluctuations may depend on the particular choice of holomorphic vector bundle with these parameters.

Introducing mass parameters $m_{f}\in\mathfrak{t}_{f}$, we can quantize the chiral and Fermi multiplet fluctuations as in section 2.3. We obtain a supersymmetric twisted Hilbert space $\mathcal{H}_{\alpha}$ in each chamber $\mathfrak{c}_{\alpha}\subset\mathfrak{t}_{f}$ where all of the fluctuations are massive. Furthermore, each $\mathcal{H}_{\alpha}$ will jump as the holomorphic vector bundle $E_{f}$ is varied whenever the numbers $n_{C}$ and $n_{F}$ change.

A supersymmetric Chern-Simons term for the flavour symmetry descends to a Chern-Simons term in the supersymmetric quantum mechanics and shifts the flavour grading of supersymmetric ground states. For example, for $G_{f}=U(1)$ the contribution (70) to the Lagrangian of the supersymmetric quantum mechanics is

[TABLE]

where $\mathfrak{m}_{f}=c_{1}(E_{f})$ is the flavour flux on $C$. This shifts the flavour conserved charge by $J_{f}\to J_{f}+k\mathfrak{m}_{f}$. This can be further supplemented by a superpotential $W(\Phi)$. In this case, we will assume that we can first quantize the fluctuations as above and then implement the $J$-term superpotential (65) arising from $W(\Phi)$ in the supersymmetric quantum mechanics of these fluctuations.

We construct explicitly the supersymmetric twisted Hilbert spaces $\mathcal{H}_{\alpha}$ for a single chiral multiplet with a supersymmetric Chern-Simons term for the $G_{f}=U(1)$ flavour symmetry in section 4.1. We then proceed examples involving superpotentials in sections 4.2 and 4.3.

3.2.2 Abelian Gauge Theories

We now return to supersymmetric gauge theory. At this point, we specialise to $G=U(1)$ with a supersymmetric Chern-Simons term at level $k$. We will introduce parameters $\zeta$, $L_{T}$ associated to the $U(1)_{T}$ topological symmetry and parameters $m_{f}$, $E_{f}$ associated to the flavour symmetry $G_{f}$ acting on chiral multiplets.

As above, we now consider configurations minimizing the euclidean effective action. We first set $m_{f}=0$ and then later turn the mass parameters back on in the effective supersymmetric sigma model description. Eliminating the auxiliary field in the vectormultiplet, the effective potential of the supersymmetric quantum mechanics is then given by

[TABLE]

where

[TABLE]

is a combination of the 1-loop quantum corrected parameters $\zeta_{eff}$, $k_{eff}$, which are piecewise constant functions of $\sigma$. In the absence of mass parameters $m_{f}$, we have $\zeta_{eff}=\zeta$. Finally, $\mu$ is the moment map for the action of $G=U(1)$ on the unitary representation $R$.

Configurations minimising the euclidean effective action are therefore solutions to

[TABLE]

modulo gauge transformations $g:C\to G$. In particular, the vectormultiplet scalar $\sigma$ is real and therefore must be a constant on $C$.

By an analysis similar to reference Intriligator:2013lca , the equations (77) admit a variety of different solutions depending on the supersymmetric Chern-Simons level $k$, the matter content and the value of $\zeta$