Using results on topological band theory of phases of matter and discrete symmetries, we study topological properties of the band structure of physical systems involving spin and fermions. We apply this approach to study partial breaking in four-dimensional (4D) gauged supergravity in the rigid limit, and we describe the fermionic gapless mode in terms of the chiral anomaly. We also study the homologue of the usual spin–orbit coupling, , that opens the vanishing band gap for free fermions; we show that it is precisely given by the central extension of the supercurrent algebra in 4D spacetime. We also comment on the rigid limit of Andrianopoli et al. [Phys. Lett. B 744, 116 (2015)], and propose an interpretation of energy bands in terms of a chiral gapless isospin particle (iso-particle). Other features, such as discrete T-symmetry in the Fayet–Iliopoulos coupling space, the effect of quantum fluctuations, and the link with the Nielson–Ninomiya theorem, are also studied.
B10B11B12B311. Introduction
In the few years past there has been intensive interest in topological band theory in the Brillouin zone and in three-dimensional (3D) effective Chern–Simons field theories in connection with the phases of matter [1–9]. This interest has mainly concernd spin topological matter and spin topological gauge theories in lower spacetime dimensions; this is because of their particular properties in dealing with condensed matter systems like topological insulators and superconductors, and also for the role they play in the quantum Hall effect as well as in the study of boundary states and anomalies [10–17]. By looking to extend some special features of these studies to systems with spacetime spins beyond or 1, we fall into supergravity-like models where fermionic modes of higher spins such as relativistic spin are also known to play a basic role. In this paper we would like to explore some topological aspects of the band theory of systems having spins less than or equal to , and to look for a physical model where the topological properties obtained for spins can be extended to higher spins.
A priori, physical systems with spins may exist in spacetime dimensions where spin and particle fields have non-trivial gauge degrees of freedom; to get started, it is then natural to begin by fixing the full-spin content of the physical system we are interested in here, and also to define the Hamiltonian model or the field equations describing the full dynamics. To find a physical system with higher spins where such kinds of studies may be relevant, and also to identify the appropriate approach to use as the starting point, we give here two motivations: the first concerns the choice of a particular system having fermions with different spins, say two types of fermionic spins and , and the other regards the tools to use for approaching their band structure. First, by studying the constructions of Refs. [17–19], one comes to the conclusion that several topological condensed matter statements based on spin fermions may be approached by starting with the Dirac equation of relativistic theory. From this theory one may engineer effective Hamiltonians breaking explicitly the SO Lorentz symmetry by allowing non-linear dispersion relations due to underlying lattice geometries and interactions. It follows from this description that quantities like fermionic gapless/gapped modes, chiral ones, and edge states have interpretations in terms of massless/massive states, quasi-particles with exotic statistics, and anomalies whose explanation requires the use of topological notions such as manifold boundaries, left/right windings, the Berry connection, and the Nielson–Ninomiya theorem.
To look to extend results on spin topological matter to spin gravitinos, we then have to go beyond the Dirac equation, for instance by considering the Rarita–Schwinger equation of gravitinos, and try to mimic the analysis done for spin . Even though this is an interesting direction to take [20,21], we will not follow this path here because of the complicated supergravity interactions that make the field equations difficult to manage. Instead, we will rather use related equations given by extended supergravity Ward identities [22,23]. The use of these Ward identities has been motivated by the question of what kind of physical systems the specific properties of the gravitino band structure may serve. Recalling the role played by gravitinos in the spontaneous breaking of local supersymmetry, we immediately come to the point that gapless and gapped gravitinos can be applied to study the problem of partial supersymmetry breaking of -extended vector-like theories. Indeed, in the example of the effective gauged supergravity in 4D spacetime, one has, in addition to bosons (with ) and spin fermions, two gravitinos forming an isospin particle; that is to say, a doublet under the SU R-symmetry involving pairs of gapless gravitino modes. The breaking then requires a partial lifting of the degeneracies of mode doublets, which, as in the case of condensed matter with spin fermions, may be achieved by turning on a spin–orbit-like coupling [24]. The study of spin matter therefore offers a good opportunity to identify the iso-particle Hamiltonian including the homologue of that induces partial breaking of supersymmetry. This coupling will be denoted , where plays the role of the angular momentum and the isospin the role of the spin . In this regard, it interesting to recall that spontaneous partial breaking in supergravity may be done by the super-Higgs mechanism, in which, in supermultiplet language, a massive gravitino multiplet can be created by merging three multiplets: a massless gravitino eating a massless U multiplet and an chiral multiplet [25]. But here, the partial breaking will be done by the isospin–orbit coupling that opens the gap energy between the two gravitinos. In this study, we will show that the coupling is precisely given by the central anomaly of the supercurrent algebra in 4D spacetime [26,27].
The main purpose of this work is, then, to use results on topological band theory of fermionic matter and chiral anomalies as well as discrete symmetries to study partial breaking in gauged supergravity in four dimensions. The spacetime fields of our system are given by the field content of the standard supermultiplets; in particular, the field content of the gravity multiplet, vector multiplets, and matter multiplets. To perform this study we will use supergravity Ward identities in the rigid limit as considered in Ref. [28], and also study the partial breakings by using the topological approach along with the Nielson–Ninomiya theorem and the chiral anomaly. We also study the effect of quantum harmonic fluctuations in the Fayet–Iliopoulos (FI) coupling space, and show that the result of Ref. [28] is not affected by quantum corrections provided that a saturated condition holds.
The presentation is as follows: In Sect. 2, we describe some tools on partial breaking in the rigid limit of supergravity theory and present the basic equations to start with. We also give some useful comments. In Sect. 3, we derive the free Hamiltonian of the iso-particles in gauged supergravity, work out the isospin–orbit coupling that opens the zero gap between the two gravitino zero modes, and show how time reversing symmetry and (combined and parity ) can be implemented. In Sect. 4 we study gapless and gapped gravitinos in gauged supergravity, and describe the properties of partial supersymmetry breakings and their interpretation from the viewpoint of the Nielson–Ninomiya theorem and the chiral anomaly. We also discuss the effect of quantum fluctuations on partial breaking of supersymmetry. Section 5 is devoted to our conclusions and comments.
2. Rigid limit of <inline-formula><tex-math notation="LaTeX" id="ImEquation68"><![CDATA[$\boldsymbol{\mathcal{N}=2}$]]></tex-math></inline-formula> Ward identity: the U<inline-formula><tex-math notation="LaTeX" id="ImEquation69"><![CDATA[$(1)$]]></tex-math></inline-formula> model
Following Ref. [28], partial breaking of rigid and local extended supersymmetries is highly constrained; it can occur in a certain class of supersymmetric field theories provided one evades some no-go theorems [29–32]; see also Refs. [33–38]. In global 4D theories, this was first noticed in Refs. [26,39], and was explicitly realized in Refs. [40,41] for a model of a self-interacting vector multiplet in the presence of electric and magnetic FI terms. There, it was explicitly shown that the presence of electric and magnetic FI couplings is crucial to achieve partial breaking. The general conditions for partial supersymmetry breaking have been recently elucidated by L. Andrianopoli et al. in Ref. [28], where it was also shown that and should be non-aligned (). Their starting point for deriving the general conditions for partial supersymmetry breaking in the rigid limit^{1} was the reduced gauged supergravity Ward identity,
where the spin fermions and refer to the chiral and antichiral projections of the gauginos respectively. Here, the SO spacetime spin index of the fermions has been omitted for simplicity, while we have shown the other two indices, and . refers to the isospin representation of the SU symmetry of the supersymmetric algebra, since gauginos are iso-doublets under SU; this index is lowered and raised by the antisymmetric tensor and its inverse . The index designates the number of vector multiplets in the Coulomb branch of the gauged supergravity theory. Notice also that the quantity is a convention notation for the supersymmetric transformation of gauginos, which is given by , with the two fermions standing for the supersymmetric transformation parameters. In Eq. (2.1), the right-hand side is restricted to the pure Coulomb branch and so corresponds to the rigid limit of the following local identities:
The left-hand side of Eq. (2.1) contains two basic terms, namely the rigid limit of the scalar potential and an extra traceless constant matrix,
This Hermitian traceless matrix can be interpreted as an anomalous central extension in the supersymmetric current algebra [27,29,39]; it only affects the commutator of two supersymmetry transformations of the gauge field [29,41] and contains data on hidden gravity and matter sectors. Recall that the basic anticommutator of the supercurrent algebra is
where , , and are the time components of the supersymmetric current densities , , and , respectively. The time component densities in the current superalgebra, Eq. (2.4), are related to the , , and charges of the supersymmetric QFT in the usual manner. For example,
obeying ; the usual globally defined supersymmetric algebra with constrained to vanish.
By comparing Eq. (2.1) with the general form of the Ward identities, Eq. (2.2), we deduce that the term captures the contribution of the fermion shifts to the Ward identity coming from the rigid limit of the hidden gravity () and the matter () branches. For the simple example of an Abelian U gauge multiplet (), the anomaly iso-vector has been realized in terms of the electric and the magnetic FI coupling constant iso-vectors of the Coulomb branch of the effective U gauge theory as follows:^{2}
obeying the remarkable property and ; see Eq. (2.13). Moreover, partial breaking of supersymmetry takes place at [28,29]
This relation will be used later on when considering topological aspects of gapless fermions (Sect. 4.1) as well as harmonic fluctuations (Sect. 4.2), but before that let us add comments regarding Eqs. (2.6) and (2.7).
First, notice that in order to have a non-zero it is sufficient to take the following particular and simple choice,
satisfying and . This particular choice shows that a quadratic term of type
like the one appearing in Eq. (2.17) becomes necessary for the contribution of the -direction normal to the plane; that is to say:
This implication is obviously not usually true since, for non-orthogonal and , we have as long as . The trick will help us to detect the effect of , especially when studying quantum fluctuations around the supersymmetric ground states and .
Second, observe that by setting , , and , it follows that , and then the central extension matrix in Eq. (2.3) can be also expressed as
This way of expressing is interesting since, supported by the dimensional argument, it gives an idea of how to realize the factor in terms of the electrical and magnetic couplings of FI. Antisymmetry implies the natural factorization
which is nothing but the Andrianopoli et al. factorization of Eq. (2.6). We expect that this trick can also help to find the extension of the realization in Eq. (2.6) to higher supergravities, in particular to theory in the rigid limit where there is no matter branch; this generalization will not be considered here. Notice that for the simple choice of Eq. (2.8) we have the diagonal matrix
showing that, in the rest frame, we have , and then the current algebra of Eq. (2.4) splits into two copies with right-hand energy densities given by .
Third, a non-vanishing requires in general non-collinear and vectors, so the unit vectors , generate a two-dimensional plane with a normal vector given by . These three vectors together form a 3D vector basis of that we term the 3D iso-space:
By the terminology “3D iso-space” we intend to use its similarity with the usual Euclidean space of the classical mechanics of point-like particles to propose a physical interpretation of Eq. (2.6) by using the notion of an isospin particle; this will be done in Sect. 3.
Notice, moreover, the following features:
The relation in Eq. (2.6) concerns an effective U gauge theory with one gauge supermultiplet . The general expression of , extending Eq. (2.6), as well as the general form of the scalar potential energy associated with generic U effective gauge theories reduced to FI couplings, have been shown to be functions of the characteristic data of the special geometry of the scalar manifold. They are given by the factorizations
where are moment maps carrying quantum numbers of SU, is the metric of , and is a symmetric matrix of the form
encoding data on the scalar manifold of the theory; see Refs. [28,29] for more details. For the example of an Abelian U gauge model, is as in Eq. (2.6) while the scalar potential of Eq. (2.14) has the following remarkable quadratic shape:
with \gamma^{2}>0$]]> and assumed positive for later use. Notice that here should be viewed as the sum , with describing the coupling in the plane and in the normal directions; see also Eqs. (3.9) and (3.10).
By substituting Eq. (2.16) and into Eq. (2.14) we learn that the real parameters , , and in the above scalar potential indeed have a geometric interpretation in terms of the effective prepotential of the special geometry. For example, the parameters in Eq. (2.17) depend on both the real and imaginary parts of the second derivative of .
The scalar potential in Eq. (2.17) has a particular dependence on and ; it can be presented as a quadratic form , with and metric as follows:
with . This form will diagonalized later on for explicit calculations.
The above might be viewed as a special potential; a more general expression would involve more free parameters, such as
where is a number that depends on the vacuum expectation values (VEVs) of the scalar fields, and the parameters of the effective theory like masses and gauge coupling constants, and are two iso-vectors scaling in same manner as the FI constants, and , , and are dimensionless real 33 matrices— and are symmetric, but is a general matrix. These moduli may also characterize the scalar manifold of the effective supergravity and likely external fields as suspected from Table 1; see also Eq. (3.25), where of the is interpreted in terms of an external iso-magnetic field.
Comparison of classical particles and iso-particles.
Vectors in : electron
Iso-vectors in : gravitinos
particle
:
iso-particle
:
:
isotropy SO
:
R-symmetry SU
orbital moment
:
orbital moment
Hamiltonian
:
Hamiltonian
spin
:
isospin
gauge summetry U
:
gauge symmetry U
Finally, notice that by giving these somehow explicit details on the theory, we intend to use its simple properties to derive the iso-particle proposal and build the isospin–orbit coupling in supergravity mentioned in the introduction. We will also use these tools to study the isospin particle as well as hidden discrete symmetries that capture data on the topological phases of the right-hand side of the supersymmetry current algebra in Eq. (2.4).
The Andrianopoli et al. realization, Eq. (2.6), of the rigid anomaly iso-vector in effective supersymmetric gauge theory is interesting and is very suggestive; see Fig. 1 for an illustration. This is because of the wedge product that allows us to establish a correspondence between properties of the partial supersymmetry breaking and the electronic band theory with spin–orbit coupling turned on ().
A classical quasi-particle with angular momentum in FI coupling space parameters. The electric FI coupling is viewed as a position vector and the magnetic coupling as the momentum . In addition to , the quasi-particle also carries an intrinsic isospin charge as well as unit U charge due to the gauging of Abelian isometries of gauged supergravity. This image may be put in correspondence with an electron spinning around a nucleus.
Indeed, the axial vector , which we refer to below as the Andrianopoli et al. orbital vector, has the same form as the usual angular momentum vector,
of a 3D classical particle with coordinate position and momentum . By comparing the formula of Eq. (2.6) with the above , it follows that the FI electric coupling may be put in correspondence with the vector , and the magnetic with the vector . Hence, we have the schematic picture shown in Table 1 linking the physics of classical particles (electrons) with the physics of iso-particles of gauged supergravity (gravitinos and gauginos).
In the left column of Table 1, the Euclidian space is the usual 3D space with SO isotropy symmetry. In this real space live bosons and fermions; in particular, fermions with intrinsic properties like spin particles with symmetry
In the right column, the is an iso-space with isotropy symmetry SO given by the R-symmetry SU of the supersymmetric algebra. This is a global symmetry group that will be imagined here as a global isospin group SU characterizing the quasi-particle of Fig. 1. Thus, the homologue of the real space symmetry of Eq. (3.2) is given by
Matter in the iso-space is then given by quasi-particles carrying isospin charges under SU, in particular the isospin describing the two gravitinos and the pairs of gauginos of the Coulomb branch of the gauged supergravity. Recall that in this theory, the particle content belongs to three supermultiplets, namely the gravity , the vector , and the matter . The properties of the first two are summarized in Table 2.
Iso-space gravity and vector properties.
multiplets
Field content in spin
Spin
Isospin
graviton
:
2
2
0
Gravity
gravitinos
:
graviphoton
:
1
1
0
vector
:
1
1
0
Vector
gauginos
:
scalars
:
0
0
The field content includes the fermions (gravitinos and gauginos) with a non-trivial isospin charge. It also contains two spin gauge fields (graviphoton and Coulomb ) with
gauge transformations given by Abelian isometries of the scalar manifold of the supergravity theory. The fermionic fields carry a unit U charge, and interact with the gauge vector fields through the minimal coupling , where the covariant derivative with the electric/magnetic coupling ; see Refs. [31,32,40,41,44–46] for other features.
Moreover, in Table 1 we have an exotic variable playing the role of the real time of the left column of the table. This may be imagined in terms of an energy scale variable, and hence one is left with running complings and with
In what follows, we assume that the classical correspondence in Table 1 is also valid at quantum level, and study the energy band properties of the isospin particles (gravitinos and gauginos) of the gauged supergravity.
3.1. Deriving the free Hamiltonian of the iso-particle
Here, we use Table 1 to build the free Hamiltonian of the iso-particle and study its classical and quantum behaviors. We also comment on some interacting terms appearing in the scalar potential, Eq. (2.19).
3.1.1. Classical description
Using the proposal in Table 1, the free Hamiltonian of the classical iso-particle is given by the scalar potential of the supergravity theory. It is just the energy density of the supergravity theory,
Because this energy is quadratic in and as shown by the rigid limit of Ref. [28], then describes the free dynamics of a classical iso-particle in the 6D phase space parameterized by the FI coupling parameters. By using Eqs. (2.17) and (2.18), we have
where , , and the planar are three real parameters that have an interpretation in the special geometry of the scalar manifold of the effective theory. Here, they will be given an interpretation in terms of an effective mass and a frequency with relationships as in Eqs. (3.15) and (3.20). Notice the following useful features:
The above-described has the form of a classical harmonic oscillator energy , so one can take advantage of this feature to learn more about the properties of the iso-particle of the gauged supergravity.
The notation in Eq. (3.7) is to distinguish it from another contribution to be turned on later when switching on . By using the two types of vector products, a general quadratic term like has the typical form
showing that may come from two sources: (1) from a scalar product like , and/or (2) from the norm of the wedge product of the two vectors as follows:
In order to fix a freedom in the signs of , , and , we assume that the discriminant of the metric of Eq. (3.7) is positive definite,
0 .
\label{det}
\end{equation}]]>
As this discriminant is not sensitive to , we restrict and to both be positive; this constraint is also needed for to be bounded from below, which is an important condition for the quantization of coupling fluctuation.
The omission of the zero value in is because for , the Hamiltonian in Eq. (3.7) reduces to
ruling out the harmonic oscillations needed for quantum fluctuations; see Eq. (3.20). Nevertheless, the saturated limit also captures some interesting data; it will be discussed in Sect. 4.2.
With these features in mind, we are now in a position to deal with the Hamiltonian of Eq. (3.7). To do so, we perform a linear change of variables, , in order to put into the following normal form:
where now stands for “kinetic energy” and for the “potential energy.” The new , are related to the old , by
with , which diagonalizes the metric in Eq. (2.18). The resulting positive mass and (oscillation frequency) are functions of the , , and parameters; their explicit expressions are
Notice that, using the condition and the positivity of and , we have and then 0$]]>. Notice also the following properties:
The saturated value ; then, ( and .
Classically, the Hamiltonian in Eq. (3.13) is positive and bounded from below,
This vanishing lower value is important in the study of gauged supergravity in the rigid limit, since corresponds to an exact rigid supersymmetric phase. This property requires .
By restricting and to the particular choice in Eq. (2.8), the free Eq. (3.13) reduces to the Hamiltonian of a one-dimensional harmonic oscillator,
In what follows, we use this simple expression to study harmonic fluctuations of the FI couplings around the supersymmetric vacuum .
3.1.2. Quantum effect
The free iso-particle studied above is classical. However, like spin fermions in real 3D space the iso-particle also has intrinsic degrees of freedom, namely an isospin , as shown in Table 2, and a unit electric/magnetic charge given by Eq. (3.4).
Assuming the classical correspondence of Eq. (1) to also hold at the quantum level in the iso-space , it follows that the fluctuations of the FI couplings may also be governed by in same manner as for the usual Heisenberg uncertainty , which is expressed in terms of the usual phase space coordinates . If one accepts this assumption, then we cannot have exactly since , and so one expects supersymmetry in rigid limit to be broken by quantum effects since the ground state energy is now positive definite,
0 .
\end{equation}]]>
In what follows, we restrict our study to exhibiting this quantum behavior and to checking the breaking . We will return to study this feature in Sect. 4.2 when the isospin–orbit coupling is switched on. There, we will also give details of the condition for the partial breaking .
The quantum effect due to fluctuations of and around the supersymmetric ground state is induced by quantum isotropic oscillations with discrete energy and fundamental oscillation frequency
By using Eq. (3.15), we have
Observe that, because of the minus sign, this vanishes for those parameters , , and satisfying the degenerate condition , which has been ruled out by the constraint in Eq. (3.11). To illustrate the quantum effect for 0$]]>, we consider the particular choice of Eq. (2.8) bringing Eq. (3.13) to a one-dimensional quantum oscillator with Hamiltonian operator
This has a diagonal form , which by setting reads as usual like
with . The energy spectrum reduces to
with the frequency given by Eq. (3.20). The lowest energy value is given by ; it is non-zero for a non-vanishing frequency . Hence, the exact supersymmetry living at the classical vacuum gets completely broken by the quantum effect
0
\end{equation}]]>
We end this subsection by giving two brief comments on interactions. The first interacting potential energy has a linear expression in ,
and concerns the electric U gauge charge. This is a subgroup of the electric/magnetic U local symmetry of the gauged supergravity induced by gauging two Abelian isometries in the scalar manifold of the supergravity theory. The second interacting potential energy is given by the isospin–orbit coupling that we are particularly interested in here; it will be considered in detail in the next subsection.
Regarding Eq. (3.25), it is derived by taking the following two steps: First, start from the interaction energy of an electrically charged particle with momentum moving in the presence of an external magnetic field . Then, use the correspondence in Eq. (1) allowing us to imagine in terms of the FI magnetic vector and as an iso-vector . The obtained Eq. (3.25) describes just the term in Eq. (2.19), from which we learn that .
3.2. Isospin–orbit coupling
The proposal in Table 1 has been useful for the physical interpretation of the rigid Ward identity in terms of an iso-particle Hamiltonian with phase space coordinates , thanks to the Andrianoploli et al. formula giving the orbital momentum of this iso-particle, and thanks also to the structure of the scalar potential , which turns out to be nothing but the free Hamiltonian of Eq. (3.13). In this subsection, we derive the isospin–orbit coupling
where stands for the isospin vector and . For that purpose, recall that in Eq. (2.1) the rigid anomaly matrix appears in the form of a Hermitian traceless 22 matrix,
that reads in terms of the Pauli matrices and the Andrianopoli et al. orbital vector as follows:
This factorized form of teaches us that it can be imagined as describing the coupling of two things, namely the orbital isovector and the isospin vector
In what follows, we give two other different, but equivalent, ways to introduce . The first relies on comparing with the usual spin–orbit coupling of a particle with spin moving in real space with coordinate vector . The second way extends the approach of the previous section for deriving the free Hamiltonian (3.13) by including the isospin effect.
By comparing the effect of the spin–orbit coupling in electronic systems and the effect of in the partial breaking of supersymmetry, and by following Refs. [28,29], we learn that when the central extension matrix is turned off, i.e. , then supersymmetry is preserved (two gapless gravitinos). However, it can be partially broken when it is turned on, i.e. . This property can be viewed in terms of a non-zero gap energy E between the two fermionic iso-doublets, including the two charges of supersymmetry with the expression
This is exactly what happens for the case of two states of spin fermions in electronic condensed matter systems when the spin–orbit coupling is taken into account. This coupling is known to open the zero gap between the two states of free electrons. From this link with electronic properties, we deduce a correspondence between the central matrix of the supercurrent algebra and the Hamiltonian . This link reads explicitly like
where the isospin plays the role of the spin , and the Andrianopoli et al. vector the role of the angular momentum . Adding the isospin–orbit coupling term to the free Hamiltonian in Eq. (3.13) we get , which reads explicitly as
In matrix form, we have
with eigenvalues and eigenstates
The second way to introduce Eq. (3.26) is a purely algebraic approach. The key idea relies on thinking of the free energy density term of the two isospin states as
For each of the states we have used Eq. (3.13) to derive its free Hamiltonian, but this result is just the diagonal term of a general Hamiltonian matrix . The extension of to more interactions is then naturally given by the Ward identity, Eq. (2.1),
which is nothing but the right-hand side of the supersymmetric current algebra in Eq. (2.4), including the central matrix.
3.3. Discrete symmetries
From the rigid Ward identity of Andrianopoli et al., Eq. (2.1), we also learn that exact supersymmetry requires : no isospin–orbit coupling in our modeling. But this vanishing value is just the fixed point of the discrete symmetry acting on the anomaly iso-vector as follows:
To figure out the meaning of this discrete transformation we use Eq. (2.6), from which we learn that the minus sign can be generated in two ways, either by the change or by . To derive the physical interpretation of these two kinds of discrete symmetries, we use the analogy between the FI couplings and the classical phase coordinates . Promoting this correspondence to dynamical (running) couplings, say
it follows that the transformation in Eq. (3.37) corresponds, for example, to the usual time-reversing symmetry which maps the position and momentum respectively to and . On the side of the FI couplings, we then have the following action of the analog of iso-time ,
Notice that the usual space parity which maps the phase coordinates to allows us, by using the correspondence, to write
but this discrete transformation leaves invariant and so is not relevant for partial breaking. However, the combined transformation, which acts like
does affect the sign of . This combination can also be used to think about the transformation of Eq. (3.37). Actually, it corresponds to the second possibility of realizing from Eq. (2.6). Therefore, exact supersymmetry, which corresponds to , lives at the fixed point of the reversing time transformation of Eq. (3.37), or at the combined given by Eq. (3.41), or both.
4. Topological aspects and quantum effect
In this section we first study the topological behavior of gapless iso-particles of exact supersymmetry, as well as the gapless chiral ones that remain after partial breaking. We then study the effect of quantum fluctuations on partial supersymmetry breaking.
4.1. Chiral anomaly
Setting , we can turn the rigid Ward identities of Eq. (2.1) into the matrix equation
which is nothing but the Hamiltonian matrix of Eq. (3.32). Multiplying both sides of this 22 matrix relation by , describing the two states of the iso-particle, we end up with the eigenvalue equation whose two eigenvalues are given by ; the eigenstates associated with these are linear combinations of and , and read like with amplitudes and as follows:
The determinant that captures data on the singular points in the plane is given by the product of the eigenvalues , and reads as
It is a function of two real quantities, namely and , but here we will treat it as a parametric function of one variable like . The choice of the variable depends on the property we are interested in exhibiting; see Fig. 2. From the viewpoint of the scalar potential energy, the variable is given by , while is seen as a free parameter.
On the left, the discriminant as a function of and parameter . For there are two zeros, at ; one is visible in the rigid limit. At each zero, say , is a chiral gapless mode corresponding to a partially broken supersymmetric state. In the limit , the two chiral gapless modes at collide at the origin and form a gapless iso-doublet. On the right, as a function of . For non-zero the chiral gapless mode lives at each , merging for .
From the viewpoint of the vector we have the reverse picture: is the variable while stands for a free parameter. In the first image, has two zeros at , one positive, , that is visible in the global supersymmetry sector, and a hidden negative, . In the second picture, the discriminant has zeros at for positive and for negative . Let us express these two zeros in as with unit vector . The effective gap energy between the two energy density bands is given by
it vanishes for and then for . Because of the property , the zeros of are of two kinds: simple for 0$]]> and double for . At each simple zero lives a gapless fermionic mode (gravitino and gaugino) and a gapped one. For , with positive energy density , we have the conducting band, and for with negative we have the valence band. Notice that is conserved under^{3} the discrete change
Its two zeros are not fixed points of except for the origin; they are interchanged as shown by Eq. (4.5)—for instance, properties at may be deduced from those at .
Now let us approach from the viewpoint of the iso-space vector and consider the two-spheres and , with a surface normal to , surrounding respectively the zeros
The two-sphere is described by the vector , and by . These two-spheres should not be confused with the unit two-sphere
associated with the unit vectors of Eq. (2.13), but all three of , , live in the iso-space and are related to each other by continuous mappings like
Focusing, for instance, on , the continuity of shows that it has a winding describing the net number of times wraps the unit sphere ; the integer number just reflects the mathematical property . A similar thing can be said about thanks to the parity of Eq. (4.5), under which the gauge curvature of the underlying Berry connection is odd; see Eq. (4.11) below.
Moreover, each gapless state at the two zeros is anomalous in the sense that it has one gapless chiral mode and then violates the Nielson–Ninomiya theorem [17,42,43]. Recall that in theories that are free from chiral anomalies the usual Nielson–Ninomiya theorem [17,42] states that the sum of winding numbers around two-spheres surrounding the zeros where gapless modes live vanishes identically. Here, this statement reads explicitly as
where is a gauge curvature whose explicit expression will be given below. For positive , Eq. (4.3) has one zero given by an outgoing with positive sense in the normal direction; then, a two-sphere surrounding the point has a positive winding number
Here, the curvature is given by the following rank-2 antisymmetric tensor,
The Nielson–Ninomiya theorem is then violated due to the existence of one gapless chiral moving mode, and so the partially broken theory has a chiral anomaly: only one of the two supersymmetric charges , say the right, , is preserved; the left, , is broken. For the incoming we have the negative winding number
This negative value follows from the mapping , due to Eq. (4.5) and using Eq. (4.11). For the special case where the VEV of the scalar potential vanishes, , the discriminant of the matrix in Eq. (4.3) reduces to and its zero, , has a multiplicity 2. In this case, the Nielson–Ninomiya theorem reads as
At the fixed point of the transformation in Eq. (4.5), the two zeros collide at . Then, the two effective gravitino zero modes with opposite chiralities form a massless doublet (a massless iso-particle) and supersymmetry gets restored.
These results are summarized in Table 3.
Properties of zero modes of .
Zeros of
Multiplicity of zeros
Winding number
Conserved SUSY charges
1
+1
1
2
0
4.2. Quantum fluctuation
Here, we study quantum fluctuations in the FI couplings around the partial breaking vacuum and comment on their effect by using the special choice in Eq. (2.8). For that purpose we use to promote the matrix equation in Eq. (4.1) into an effective quantum eigenvalue matrix equation that we split into two eigenvalue equations:
In these relations we have , where the hatted and refer to the quantized operators associated with and expressed in terms of the phase space vectors and . For the particular FI coupling choice in Eq. (2.8) we have , and find, after repeating the steps between Eqs. (3.7) and (3.22), the two quantum 1D Hamiltonians
describing two oscillators with different frequencies . Their energies are given by , with
with the remarkable minus sign. Notice that imposing the constraint in Eq. (3.11) on both eigenstates, we have
leading to
and then to
For the case where one of the bounds of the constraints in Eq. (4.18) is saturated, for example if the upper bound of the squared deviation is saturated, we can fix one of the four parameters in terms of the three others like
By substituting back into Eq. (4.16), we end up with two energy spectrums. First, with
describing gapless iso-particles (gravitinos/gauginos) with , which corresponds to the ground state where partial breaking takes place, and second, with
0 , \label{m}
\end{equation}]]>
describing a gapped iso-particle. Thus, along with the gapless modes (, we have gapped states with harmonics . The energies are bounded as
0 ,
\end{equation}]]>
with the ground state energy corresponding to the classical , which is also the gap energy between the two polarizations of the iso-particle.
As a conclusion of this subsection, quantum fluctuations in the FI coupling space with do not destroy the partial breaking supersymmetry of the Andrianopoli et al. rigid limit; this property holds for the saturated condition of Eq. (4.20), otherwise quantum corrections also break the residual supersymmetry.
5. Conclusion
In this paper we have used results on topological band theory of usual matter to study partial breaking of gauged supergravity in the rigid limit. By using supergravity Ward identities and results from Refs. [28] and [17–19,31], we have derived a set of interesting conclusions on the band structure of gravitinos and gauginos in theory. Some of these conclusions have been obtained from the proposal in Table 1 and its quantum extension, and we rephrase them below:
(1) The interpretation of the Andrianopoli realization as an angular momentum vector of a quasi-particle with phase space coordinates allowed us to think of the two gravitinos and the two gauginos in terms of classical isospin particles (iso-particles) charged under U gauge symmetry. As a consequence of this observation, the scalar potential has been interpreted as the Hamiltonian, Eqs. (3.6) and (3.7), of a free iso-particle, and the central extension of the supercurrent algebra in Eq. (2.4) as describing the isospin–orbit coupling . This isospin–orbit interaction is the homologue of the usual spin–orbit coupling in electronic systems of condensed matter. The proposal in Table 1 also allowed us to derive two discrete symmetries, and , capturing data on partial breaking of supersymmetry; see Sect. 3.3 for details. Exact lives at the fixed point of these symmetries. In summary, we can say that the classical properties of the iso-particle are given by the supersymmetric current algebra in Eq. (2.3).
(2) By using the Nielson–Ninomiya theorem, we have studied the topological properties of the fermionic gapless states given by zeros of the discriminant in Eq. (4.3). The two bands of the rigid Ward operator are gapped except at isolated points in the phase space of the electric and magnetic coupling constants, where supersymmetry is partially broken and where there is a gapless chiral state with a chiral anomaly violating the Nielson–Ninomiya theorem. From the study of the properties of , it follows that the gap energy is given by and vanishes for ; that is, for a vanishing central extension in the supercurrent algebra. The zero modes of and their properties like windings and conserved supersymmetric charges are as reported in Table 3. At the particular point , the discriminant reduces to and has an SU singularity at the origin . There, the Nielson–Ninomiya theorem is trivially satisfied, as shown in Table 3, and supersymmetry is exact with compensating chiral anomalies.
(3) We have used the proposal in Table 1 to study the effect of quantum corrections induced by fluctuations of FI coupling constants (running couplings). We have found that the quantum effect in the iso-space of FI couplings may break supersymmetry completely except for the saturated bounds in Eq. (4.21), where half of the oscillating modes disappear.
Finally, we would like to add that this approach might be helpful to explore the picture in higher supergravities, in particular for ; progress in this direction will be reported in a future publication.
Funding
Open Access funding: SCOAP.
Footnotes
^{1}The rigid limit is implemented through a rescaling of the field contents of the theory and the spacetime supercoordinates by using the dimensionless parameter . For explicit details, see Refs. [29,41].
^{2}The exact expression found in Ref. [28] is . Here, the factor has been absorbed by scaling the FI couplings.
^{3}From the charged particle’s viewpoint, this mapping from valence- to conducting-like bands and vice versa may be imagined as a transformation combining time reversing and charge conjugation .
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