University of Amsterdam
MSc Physics
GRAPPA
Master Thesis
Localization in supersymmetric
Chern-Simons theories
by
Vittorio Ricciardulli
10883398
August 2017
60 ECTS
Supervisor/Examiner:
Dr. Alejandra Castro Anich
Second Examiner:
Dr. Diego Hofman
Abstract
We provide a pedagogical introduction to U (1) R-symmetric N = 2 three-dimensional Chern-Simons-Yang-Mills-matter theories and to supersymmetric localization. First, we review the construction of the non-supersymmetric Chern-Simons action in flat d = 3 Euclidean space and discuss classical and quantum perturbative aspects of this theory. We then move on to introduce supersymmetric theories in flat space. In particular, we start by considering d = 4 N = 1 flat Lorentzian superspace and review the construction of R-symmetric actions for vector, chiral and anti-chiral multiplets. After Wick-rotating to Euclidean signature, we dimensionally reduce to d = 3 dimensions. The procedure results in N = 2 supersymmetric theories of the Yang-Mills type coupled to matter fields. Furthermore, we consider N = 2 d = 3 flat Euclidean superspace and derive a supersymmetric version of Chern-Simons theory. Then, we describe how to place these theories on S3 by coupling to a background
constituted by an off-shell new minimal supergravity multiplet in the rigid limit, in which the supergravity fields are frozen to appropriate classical configurations. Finally, we discuss the procedure of supersymmetric localization and localize our theories on the Coulomb branch of the moduli space where only the scalar in the vector multiplet acquires a VEV, showing that only a finite-dimensional subset of field space contributes to the path integral. We evaluate the partition function over this subset and obtain a result that is 1-loop exact, correctly reproducing known results in the literature. We conclude with a few remarks on the meaning and implication of those.
Contents
List of Tables and Figures iii
Note on Conventions v
Introduction 1
1 Chern-Simons Theory 5
1.1 Classical Chern-Simons theory . . . 6
1.1.1 Gauge invariance . . . 6
1.1.2 Equations of motion . . . 7
1.1.3 Topological nature of the theory and Hamiltonian . . . 9
1.2 Path integral formulation: perturbative approach . . . 10
1.2.1 Modified Faddeev-Popov gauge fixing . . . 12
1.2.2 1-loop contribution . . . 16
1.2.3 Gravitational Chern-Simons action and framing of the manifold . . . 18 2 Supersymmetry 21 2.1 Wess-Zumino model . . . 22 2.2 d = 4 N = 1 Lorentzian superspace . . . 26 2.2.1 Chiral superfield . . . 28 2.2.2 Vector superfield . . . 31 2.2.3 Invariant Lagrangians . . . 36 2.2.4 R-symmetry . . . 40
2.3 d = 4 N = 1 Euclidean Yang-Mills-matter theory . . . 41
2.3.1 Wick rotation . . . 42
2.4 Dimensional reduction: from d = 4 N = 1 to d = 3 N = 2 theories . . . 45
2.4.1 Spinor algebra . . . 46
2.4.2 Vector multiplet . . . 47
2.4.3 Matter multiplet . . . 48
2.4.4 d = 3 N = 2 Euclidean Yang-Mills-matter theory . . . 49
2.5 d = 3 N = 2 Euclidean superspace . . . 50
2.5.1 Components by projection . . . 50
2.5.3 An alternative derivation of Yang-Mills-matter theory . . . 56
2.5.4 Q-exact Lagrangians . . . 61
2.6 d = 3 N = 2 theories in curved space . . . 63
2.6.1 Supergravity . . . 64
2.6.2 Background: new minimal supergravity . . . 67
2.6.3 S3 . . . 69
2.6.4 Structure of the algebra . . . 70
2.6.5 Comments on the rigid supergravity approach . . . 72
3 Localization 75 3.1 Supersymmetry and fixed points . . . 77
3.2 Coulomb branch localization . . . 80
3.2.1 Gauge sector . . . 80
3.2.2 Matter sector . . . 87
3.3 Comments and conclusions . . . 90
3.3.1 Higgs branch localization . . . 90
3.3.2 Renormalization group flow . . . 90
A Characteristic classes 93 A.1 Invariant polynomials . . . 93
A.2 Chern classes and Chern characters . . . 99
A.3 Chern-Simons forms . . . 100
A.4 Pontryagin classes and framing of the manifold . . . 104
B Spinors 109 B.1 Clifford algebras . . . 110
B.1.1 Even dimension: d = 2m . . . 110
B.1.2 Odd dimension: d = 2m + 1 . . . 114
B.2 Useful spinor identities in (3, 1) spacetimes . . . 116
B.3 Useful spinor identities in (3, 0) spacetimes . . . 117
C Superconformal algebra 119 C.1 Conformal algebra . . . 119
C.2 d = 3 N = 2 superconformal algebra . . . 119
List of Tables
2.1 Dimensional reduction of vector multiplet V and matter multiplets Φ, ¯Φ. It is understood that the local dependence of the fields on the right is on the three-dimensional coordinates only. Barred fields are independent fields. . . . 45
2.2 Compendium of the number of off-shell and on-shell degrees of freedom for various massless fields in an arbitrary number of dimensions d. Here φ denotes a general scalar. For those, the on-shell number of d.o.f.s is 1 or 0 depending if they are auxiliary and can be integrated out, or not. . . 67
2.3 Summary of relevant quantum numbers for the fields in the vector and matter multiplets and for the SUSY parameters , ¯. ∆ is the scaling dimension and R the R-charge. . . 72
B.1 Possible spinor representations for spacetimes of dimensions and signatures (3, 1), (4, 0) and (3, 0). The arrow in the middle signifies that the last two columns follow from the content of the other two, respectively. The Dirac representation is, of course, always admitted, the Majorana and Weyl ones being, when present, a more fundamental way of writing the former. . . 109
List of Figures
Note on Conventions
Here we summarize general conventions and notation rules. Should any of them change, we will explicitly state so when needed. We use the following conventions for indices
• Greek lowercase letters from the start of the alphabet α, β, . . . to label the components of SU (2)-spinors. In Lorentzian theories, where two factors of SU (2) are present, we distinguish between the two by dotting ˙α, ˙β, . . . the indices of one factor. In Euclidean theories we use primes α0, β0, . . . .
• Greek lowercase letters from the middle of the alphabet µ, ν, . . . to label spacetime coordinates. When a different number of dimensions is present in the same context we use hatted indices ˆµ, ˆν, . . . or the capitalized italic version M, N, . . . to label coordinates of the higher dimensional manifold.
• Italic lowercase letters from the start of the alphabet a, b, c, . . . to label SO(3) structure group indices.
• Italic lowercase letters from the middle of the alphabet i, j, k, . . . to label Lie-algebra generators.
• Capitalized italic letters from the start of the alphabet A, B, C, . . . for mixed purposes. Summary of (possibly) ambiguous notation
• In superspace, the notations Dαor ˆDαare used to denote spinorial differential operators
acting on superfields. The notations ∇A or ¯∇A, with A = (α, µ), instead denote their
respective gauge covariant versions.
• In spacetime, we do not distinguish between metric and gauge covariant derivatives, but use the same symbol Dµ. In fact, its full expression — i.e. if it is gauge, or Lorentz
covariant, or both — can be inferred by the nature of the object it acts upon. For example, if it acts on a field X that is charged under some gauge symmetry, than one already knows that the derivative must carry a transformation dictated by the representation of the gauge group that the field sits in. If X is also living on a curved spacetime, than the derivative must also include a term that specifies the particular notion of parallel transport — i.e. affine or spin connection depending on the spin-statistics of X.
Introduction
The present work aims to provide a pedagogical introduction to graduate students in physics on (1) three-dimensional supersymmetric gauge theories coupled to matter fields in Euclidean space and (2) supersymmetric localization. Particular emphasis is given to the construction of this class of theories using the superspace formalism.
Chapter 1, introduces fundamental concepts in classical and quantum-perturbative as-pects of non-supersymmetric Chern-Simons theory at level k, described by the action
SCS = − ik 4π Z M trA ∧ dA + 2i 3A ∧ A ∧ A . (1)
The mathematical framework of characteristic classes represents the basis of such a theory and provides some intuition regarding the objects that one is dealing with and regarding their uses in the study of manifolds from a topological point of view. Therefore, following the exposition by Nakahara [1], this preparatory material is included in appendix A. Even though Chern-Simons theory is known to be exactly soluble quantum mechanically [2], we choose to discuss the perturbative regime. On the one hand, its exact treatment requires deeper knowledge about some areas of mathematics than it is generally expected at this level, whereas, on the other hand, analysis of the perturbative regime can make apparent fundamental aspects of the theory in a way that is directly accessible to physics students.
In chapter 2, we provide an introduction to supersymmetric1 and U (1) R-symmetric gauge theories. After a brief discussion using the usual quantum field theory framework, we move to the more natural arena of superspace and construct N = 1 theories in flat four-dimensional Lorentzian spacetime. The construction of flat, three-four-dimensional, Euclidean N = 2 theories is then derived by Wick rotation and dimensional reduction, which we carry out in detail. We then place such theories on S3 with radius r by coupling to off-shell
new minimal supergravity and freeze the supergravity fields to fixed values by sending the Planck mass MP → ∞. In this rigid limit, the supersymmetry algebra is enlarged to the
superconformal algebra (summarized in appendix C) and the geometry of the manifold and the supersymmetry blend via the superconformal Killing spinor equation
Dµ = −
i
2rγµ. (2)
1In view of the fermionic nature of supersymmetry, we saw fit to include a brief compendium of spinors
in four- and three-dimensional spacetimes in appendix B, where we deal with Clifford algebras and derive useful spinor identities.
The result is that the gauge sector is described by the following Chern-Simons term with level k SSCS = − ik 4π Z d3x√g tr εµνρ Aµ∂νAρ+ 2i 3AµAνAρ + 2λ¯λ − 2Dσ , (3)
and by a Yang-Mills term with gauge coupling g
SY M = 1 g2 Z d3x√g tr 1 4FµνF µν+ 1 2DµσD µσ + 1 2 D +σ r 2 − i¯λγµDµλ − i¯λ [σ, λ] − 1 2r ¯ λλ . (4) As for the matter sector, the dynamics of chiral and anti-chiral multiplets with the canonical R-charge q = 12 are described by
Sm = Z d3x√g DµφD¯ µφ − i ¯ψγµDµψ + ¯F F + i ¯φDφ + ¯φσ2φ + 3 4r2φφ − i ¯¯ ψσψ+ + i√2 ¯φλψ + i√2 ¯ψ¯λφ , (5)
We also allow for arbitrary R-charge assignments, in which case the action is the one in (2.6.33).
Finally, chapter 3 aims to introduce the tool of supersymmetric localization. We begin by reviewing a simple toy theory of one boson and two fermions to provide some insight into the general properties of the technique. We then apply the procedure to the Chern-Simons-matter path integral. Here we briefly summarize the procedure for the gauge sector and only quote the result for the matter sector. In a nutshell, localization tells us that the contribution to the partition function only comes from a finite subset of field space given by the intersection between the set of field configurations for which the bosonic part of (4) vanishes and those configurations that satisfy the equations
δλ = δ¯λ = 0. (6)
As the supersymmetry variations of the spinors λ and ¯λ are proportional to Fµν, σ and D,
these equations translate to the following constraints on the bosonic fields
D = −σ
r, Fµν = 0, Dµσ = ∂µσ = 0. (7)
Solutions to this set of equations represent the localization locus, that is the set of points in field space that give non-vanishing contribution to the partition function. On S3 with unit
radius, this equations are solved by
Aµ= 0, D = −σ0, (8)
where σ0 is a constant field configuration. We then compute the partition function by
that is exact at 1-loop order. One is then left with a classical contribution from the Chern-Simons action (3)
exp−ikπtr σ02 , (9)
and a 1-loop contribution from the Yang-Mills term (4) Y
α>0
2π−1sinh (πα · σ0)
2
, (10)
where α are root vectors. We then find that the partition function reduces to an integral over constant configurations σ0that take values in the Cartan subalgebra of the gauge group.
When matter fields are also present in a self-conjugate representation of the gauge group, the action (5) contributes a 1-loop factor only
Y
ρ
(2 cosh (πρ · σ0)) −1/2
, (11)
where ρ are the weight vectors of the representation.
As for the physics of the localization procedure, the previous results can be interpreted as computing the partition function of an infrared (IR) theory that sits at an IR fixed point along the RG flow. It is further interesting to notice that, in view of the fact that (3) is not only scale invariant, but also conformal, this fixed point can only be reached by deforming the theory with an operator that does not affect the IR. Incidentally, the Yang-Mills action (4) has exactly this role. The cartoon of figure1 aims to depict this argument.
R G flo w IR theory SSCS− g−2SY M SSCS
Chapter 1
Chern-Simons Theory
In this chapter we begin the study of Chern-Simons gauge field theory in three dimensions, which represents the basic ingredient for the theories we are going to study in later chapters. This gauge theory was first discussed in a comprehensive way by Witten [2].
Let M be a compact manifold with dim(M) = d = 2j −1, j ∈ Z+, and let G be a (gauge)
group that acts on points of the manifold. Then, the manifold is equipped with a principal G-bundle, whose local section A is a Lie algebra-valued 1-form — i.e. A ∈ Ω1(M, g) — the
connection
A = A(x)iµTidxµ, (1.0.1)
where Ti are the generators of the Lie algebra g of the gauge group, which satisfy
[Ti, Tj] = ifijkTk, (1.0.2)
where the factor of i is there because we take the generators to be Hermitian and fijk = −fjik are the structure constants that define the algebra. The corresponding curvature 2-form is
F = 1 2F i µν(x)Tidxµ∧ dxν = dA + iA ∧ A = = 1 2(∂µAν− ∂νAµ+ i [Aµ, Aν]) dx µ∧ dxν . (1.0.3)
We take A(x)iµ to be the fundamental bosonic field of our theory and define the action as the integral of the Chern-Simons form Q2j−1(A, F ) over the whole manifold
SCSd ≡ 2π Z
M
Q2j−1(A, F ) , (1.0.4)
where Q2j−1 is defined in (A.3.6) and we included a constant factor of 2π so that the overall
normalization is best suited to discuss the physics. In the rest of the thesis we restrict our attention to d = 3, since, at present, it is mainly in this dimensionality that the theory has found interesting applications in the field of high energy theoretical physics. Then, the action is given explicitly by
SCS = − 1 4π Z M trA ∧ dA + 2i 3A ∧ A ∧ A , (1.0.5)
or, in component notation SCS = − 1 8π Z M d3x µνρtrAµ ∂νAρ− ∂ρAν + 2i 3Aµ[Aν, Aρ] . (1.0.6)
For a non-Abelian gauge group the structure constants in (1.0.2) are non-zero and the cubic term does not vanish. It is this class of groups we will mostly consider in the rest of the thesis and the corresponding field theory is called non-Abelian Chern-Simons gauge theory. We will first study the classical theory in section1.1and then the perturbative regime of the quantum theory in section1.2.
1.1
Classical Chern-Simons theory
By classical we hereby mean that we wish to study the properties of the action alone of Chern-Simons theory, without using the path integral formulation. Then, we will discuss gauge invariance, the equations of motion and the Hamiltonian.
1.1.1
Gauge invariance
The gauge group, by acting on the field A, determines a symmetry on the bundle. This is an equivalence between different connections related by a gauge transformation given by the adjoint action of the group
A → AU = U AU−1− iU dU−1, (1.1.1)
where U ∈ G is an element of the gauge group with a local dependence on M, that is a map U : M → G. For those elements in G that are connected to the identity we can also define infinitesimal transformations: we take U (x) to be such a small element and expand it to first order around the identity as U (x) ≈ 1 + θ(x), U (x)−1 ≈ 1 − θ(x). Then, an infinitesimal transformation on the connection 1-form defines a covariant derivative
δUA ≡ AU− A = − (d + i[A, · ]) θ(x) ≡ −dAθ(x). (1.1.2)
Furthermore, the existence of this symmetry implies that, in a physical theory, each of the equivalent connections describes the same physics. This is a concept common to all gauge theories. We have shown in appendix A.3 that the change in the action under large gauge transformations (1.1.1) that are not connected to the identity is
δUSCS ≡ SCS AU − S(A) = = − i 4π Z M d tr A ∧ dU−1U − 1 12π Z M tr U dU−13 = ≈ − 1 12π Z M tr U dU−13, (1.1.3)
where we dropped the total derivative term since it vanishes if we assume suitable boundary conditions for the field A or, trivially, if the manifold does not have a boundary. As for the
remaining term, it does not vanish in general and, in principle, it can be computed explicitly, although only for a case by case study. In particular, it is instructive to consider the case in which M = S3 and G is a general simple Lie group with an SU (2) subgroup. Then, the computation is well-known and it is carried out in, for example, appendix A.15 of [3] and gives − 1 12π Z S3 tr U dU−13 = 2πn, n ∈ Z. (1.1.4)
This result has a nice topological interpretation: except for the constant factor of 2π, the result is an integer that specifies how many times the gauge group winds around S3, or, in other words, how many times the map U : S3 → G covers the three-sphere. This would
be formalized in homotopy theory by saying that the third homotopy group of the gauge group, π3(G), is isomorphic to the set of integers: π3(G) ∼= Z. As far as the general case is
concerned, on the other hand, in appendix A.3 we presented a heuristic argument based on characteristic classes and a reasoning by Witten and Donaldson. Both this latter argument and the particular example (1.1.4) lead to the following conclusions
1. any constant weight normalizing the action must be an integer,
2. classically the (pure) non-Abelian theory is not gauge invariant, but changes by an integer multiple of 2π.
Then, we see that, under such conditions, the non-invariance does not pose a problem for the quantum theory, since single-valuedness of the exponential of the action that goes in the path integral would still be preserved.
1.1.2
Equations of motion
As usual, to compute the e.o.m we set the infinitesimal variation of the action to zero
δSCS ∝ Z M trδ A ∧ dA + 2i 3A 3 = = 2 Z M trδA ∧ dA + iA ∧ A− Z M d tr δA ∧ A = = Z M d3x µνρtrδAµ ∂νAρ− ∂ρAν + i[Aν, Aρ] = = Z M d3x µνρδAiµFνρj tr TiTj ∝ Z M d3x µνρδAµiFνρi = 0, (1.1.5)
where in the first step we integrated by parts and used the cyclic property of the trace for Lie algebra-valued forms and in the last step we used that tr (TiTj) ∝ δij. Then, for arbitrary
variations of the fields δAµi, this implies the flatness condition
Fνρi = 0, (1.1.6)
and the general form of A that satisfies this constraint is
where we fixed the normalization in agreement with the conventions adopted earlier for the generators of the algebra. The classical solutions are then flat connections, which means that an A that satisfies this constraint gives rise to a notion of parallel transport on the bundle which only depends on the endpoints, but not on the particular path joining them. Let Γ be the set of curves γ(t) in M parametrized by t ∈ [0, 1] and consider the map g : Γ → G given by g (γ) = P exp −i Z γ A = P exp −i Z 1 0 dt Aµ˙xµ . (1.1.8)
Then, flatness implies that, for any two paths, γ(t) and γ0(t), starting and ending at the same points P0 and P1 and that can be continuously deformed one into the other (or, equivalently,
that are homotopic), we have
g(γ) = g(γ0). (1.1.9)
Indeed, let us consider an infinitesimal deformation of the curve δγ(t) that keeps the end-points fixed. This induces a deformation on the bundle that linearly affects the connection
δAµ(x) = −iδU ∂µU−1+ iU ∂µ U−1δU U−1 = i∂µδλ(x), (1.1.10)
where we used δU−1 = −U−1δU U−1 and δλ(x) is an infinitesimal, real function on the manifold that parametrizes the deformation as δU (x) = U (x)δλ(x). Then, we can inspect how the map g behaves under the variation
δ −i Z γ A = −i Z γ(1) γ(0) dxµδAµ= Z P1 P0 dxµ∂µδλ(x) = δλ(x) P1 P0 = 0, (1.1.11)
since the endpoints are kept fixed — that is δλ(x) = 0 at the endpoints. Then, δg = 0 and we can conclude that, for a flat connection, as long as there are no obstructions and we can continuously deform the path γ into γ0, g(γ) and g(γ0) map to the same element in G. Equivalently, this is to say that (1.1.8) depends on the homotopy class of the curve only.
In addition to this, flat connections are in one-to-one correspondence with group homo-morphisms f : π1(M) → G. This can be understood as follows. Let ΓC be the set of loops,
C(t), in M. The equivalence classes of loops that are homotopic to each other form the so-called fundamental group of the manifold, π1(M). Consider the map (1.1.8), but now
around a loop C(t) ∈ ΓC, which we denote by gC : ΓC → G. This represents the holonomy
of the connection A around the loop and it clearly gives the same answer for loops which are homotopic. This means that the holonomy map gC is actually a map from equivalence
classes of loops, [C] ∈ π1(M), to elements of the gauge group G. Then, we can write
gC : π1(M) → G, (1.1.12)
which is exactly the group homomorphism f that we just mentioned, since a product of holonomies gets mapped to a product of group elements in the following way
gC(C1◦ C2) = gC(C1) gC(C2), (1.1.13)
where ◦ denotes the product in the space of loops, defined via concatenation of loops, and on the RHS we have whatever kind of product there is in G. Of course, by flat connection
we really meant the whole equivalence class of flat connections that are related by a gauge transformation. If we now denote by Mf lat(M, G) the moduli space of the classical theory
— i.e. the set of solutions to the equations of motion (1.1.6) — we can put the previous consideration on a more formal footing by writing
Mf lat(M, G) ∼= Hom π1(M), G/G, (1.1.14)
where the RHS is the standard notation for continuous deformations of maps π1(M) → G,
modulo gauge transformations. Finally, this discussion essentially tells us that we can readily understand what flat connections the manifold M admits by only knowing the fundamental group of the manifold, π1(M). This feature will become useful when we discuss the quantum
theory in the next section.
1.1.3
Topological nature of the theory and Hamiltonian
Up to now, we have not commented upon a seemingly weird feature of the action of Chern-Simons theory: it does not depend on the metric. Theories which exhibit such a feature are called topological field theories of the Schwarz type and are indeed very interesting, since the fact that there is no metric in the action already tells us something on the dynamics of the theory. First of all, manifolds which have the same topology, but different geometries — i.e. different metrics — give rise to the same physical theory, the converse being true as well: the physical theory — i.e. the action — computes topological invariants of the manifold, mathematical quantities that are the same for topologically equivalent manifolds. Furthermore, it is interesting to notice that, although it is not a priori true that the transition to the quantum regime preserves topological invariance, observables in a sensible quantum theory, besides being gauge invariant, should have the additional property of being metric independent, if the topological nature of the theory is to be preserved. The second piece of information that we can directly acquire from a topological field theory of the Schwarz type is that each component of the energy-momentum tensor vanishes, since
Tµν ≡√1 g
δSCS
δgµν
= 0. (1.1.15)
This, in turn, immediately implies that the Hamiltonian of the theory vanishes as well
H = Z
d2x T00 = 0. (1.1.16)
As a last remark, we would like to comment a bit more on the implication of this. The vanishing of the Hamiltonian is a key ingredient to study the quantum theory in a non-perturbative regime, which has been shown by Witten to be completely soluble [2]. For the readers interested in seeing the full power of the theory from the topological point of view, we recommend to indeed study this latter work, together with [7], both of which treat the canonical quantization of Chern-Simons theory in great detail. In what follows, however, we do not delve into their description, as it would take us too far and since anyway those results wouldn’t be of much help to us in later chapters. The semiclassical regime, on the other
hand, is what interests us the most and it is this direction that we pursue in the next chapter. Hopefully, we will be able to construct a sensible quantum theory that in the perturbative regime still behaves as a topological field theory.
1.2
Path integral formulation: perturbative approach
We define the path integral as
Z = 1
Vol(G) Z
DA eikSCS(A), (1.2.1)
where the weight k — referred to as the level of Chern-Simons theory in the literature — is quantized to take integer values, as we discussed in the previous section, and the volume of the gauge group in front of the functional integral is a normalization that will become useful when we discuss the gauge fixing. Although the perturbative approach was first discussed in [2], in this section we prefer to review in detail the presentation by Mari˜no [8] and explicitly incorporate the work done in [9, 10,11].
In order to define a perturbative limit for Chern-Simons theory, let us first briefly review the ~ expansion in standard quantum field theory. Let us denote by S(φ) a generic action for a field φ whose scaling dimension we denote by λ and let the path integral be
Zφ= Z Dφ exp i ~ S (φ) . (1.2.2)
Let us now consider a classical field configuration, φ0, which solves
δS(φ) δφ φ=φ0 = 0. (1.2.3)
Then, for ~ → 0, we can evaluate (1.2.2) by saddle-point approximation: we treat the field as fluctuating around the extremum φ0
φ = φ0+ ~λ/2δφ, (1.2.4)
where the factor in front of δφ controls the relative size of the fluctuation, and we expand the action in powers of ~λ/2δφ
S(φ) = S(φ0) + ~λ/2 δS(φ) δφ φ=φ0 δφ + ~λ1 2!δφ δ2S(φ) δφ2 φ=φ0 δφ + . . . = ≡ S(φ0) + ~λ/2S(1)(δφ) + ~λS(2)(δφ) + O(~3λ/2). (1.2.5)
Here the first term is the classical contribution, S(1)(φ) obviously vanishes in view of (1.2.3) and the second term is the 1-loop contribution. Then, the path integral becomes
Zφ= Z Dφ exp i ~ S(φ0) + i~λ−1S(2)(δφ) + O(~(3λ−2)/2) . (1.2.6)
We can directly translate this reasoning to the case at hand if we restore ~ in the denominator of (1.2.1) Z = 1 Vol(G) Z DA exp ik ~ SCS(A) , (1.2.7)
Accordingly, we can evaluate the path integral with saddle-point approximation by expanding the action SCS(A) around classical configurations, which, as we saw in (1.1.6), are flat
connections labeled by homomorphisms π1(M) → G. Depending on the manifold M this
latter can be a continuous or discrete set. We assume it is a discrete and finite set and label the flat connections by the index (c) (not to be confused with the bracketed index in the saddle-point expansion of the action). Noticing now that the right scaling dimension for the field A is λ = 1, we decompose the field as
A = A(c)+ ~1/2B, (1.2.8)
and therefore the expansion for the action is
SCS(A) = SCS(A(c)) + ~ S(2)(B) + O(~3/2). (1.2.9)
Finally, by (1.2.6), we arrive at the following expression for the weak coupling limit of the Chern-Simons path integral
Z (M) =X c Z(c)(M) = =X c 1 Vol(G) Z DB exp ik ~ SCS(A(c)) + ikS(2)(B) + O(~1/2) = ≈X c exp ik ~SCS(A (c) ) · 1 Vol(G) Z DB expikS(2)(B) = ≡X c Zclassical(c) Z1−loop(c) (M) , (1.2.10)
where the terms O(~1/2) obviously vanish in the semiclassical limit and have been dropped
and the explicit dependence of the partition function on the manifold, Z = Z(M), will become clear once we get to the final result. Before we can compute the 1-loop term S(2)(B),
we need some definitions. First of all, we define covariant derivatives for the flat connections and their adjoint differentials (p is the degree of the form the differential acts upon)
dA(c) ≡ d + iA(c), ·, d
†
A(c) ≡ (−1)
3(1+p)+1∗ d
A(c)∗, (1.2.11)
which is a twisted de Rham differential, since flatness implies
iFA(c) ≡ d2A(c) = i dA(c)+ i A(c)∧ A(c) = 0. (1.2.12)
Then, we have the following twisted de Rham complex
0 dA(c)
−−−→ Ω0(M, g) dA(c)
−−−→ Ω1(M, g) dA(c)
−−−→ Ω2(M, g) dA(c)
which implies that the space of Lie algebra-valued p-forms is decomposed in orthogonal subsets. In particular, we are interested in 0- and 1-forms, for which
Ω0(M, g) = Ker dA(c)⊕ Im d
† A(c),
Ω1(M, g) = Ker d†A(c)⊕ Im dA(c),
(1.2.14)
as can be proved by considering the usual notion of inner product in the space of Lie algebra-valued forms. Going back to the evaluation of the quadratic term in the expansion of the action, instead of computing functional derivatives δ/δAiµ, we prefer to directly plug in the decomposition (1.2.8) in (1.0.5). Then, to get the 1-loop contribution we just have to look for terms linear in ~
SCS A(c)+ ~1/2B = SCS(A(c)) − ~ 4π Z M trB ∧ dB + i[A(c), B]− − 2 ~ 1/2 4π Z M trB ∧ dA(c)+ iA(c)∧ A(c) − − ~ 3/2 4π Z M tr 2i 3B 3 = = SCS(A(c)) − ~ 4π Z M trB ∧ dA(c)B + O ~3/2 = ≡ SCS(A(c)) + ~ S(B) + O ~3/2, (1.2.15)
where in the first step we integrated by parts and dropped a total derivative and in the second step we used (1.2.12). Mathematically, the first term, which we could already guess in (1.2.9), computes the Chern-Simons invariant of the flat connection, which is a topological invariant, whereas the factor multiplying ~ is the sought-for 1-loop contribution S(2)(B). In the path integral we then have
Z1−loop(c) = 1 Vol(G)
Z
DB eikS(B). (1.2.16)
To compute this functional integral we have to first fix a gauge to avoid overcounting the equivalent gauge connections in the path integral. Before getting to this, however, let us comment a bit more on the perturbative regime of Chern-Simons theory. In the literature, one seldom reasons in terms of ~, since the customary choice of natural units sets this latter to unity. Instead, the weak coupling limit is usually defined by considering the level k: following Witten [2], the saddle-point approximation is used in the limit k → ∞, but what he is considering there is really the combination ¯k ≡ k/~ for ~ → 0, this is why in the previous discussion we restored the correct powers of the reduced Planck’s constant. We also understand that when we talk about the level k being integer, what we really mean is that the action in the path integral is weighted by integer units of ~−1 and thus, keeping this in mind, we can safely go back to working with units in which ~ = 1.
1.2.1
Modified Faddeev-Popov gauge fixing
As usual, to compute the partition function in a gauge theory we want to integrate only over inequivalent gauge connections, that is over those connections which are not related by a
gauge transformation. Recall that the gauge group moves the connection along an orbit. The points on the orbit are then a collection of connections related by a gauge transformation or, equivalently, each orbit is an equivalence class of connections. By gauge fixing, one chooses a path in the set of gauge orbits that intersects each orbit only once. It is along this path that we wish to evaluate the path integral. The standard procedure is to choose a suitable gauge-fixing function g and compute the following Faddeev-Popov (FP) determinant
∆−1A(c)(B) =
Z
DU δ gA(c)(BU) , (1.2.17)
and then insert the identity
1 = ∆A(c)(B)
Z
DU δ gA(c)(BU) , (1.2.18)
in the path integral. We choose to work in covariant gauge gA(c)(B) = d
†
A(c)B = 0. (1.2.19)
In the case at hand, however, there is a subtlety that we need to discuss before computing the FP determinant: we assume that the connection is isolated and that it can be reducible. The former concept will become useful when we compute the 1-loop contribution later on. As for the latter, a reducible connection is one whose isotropy group — i.e. the subgroup of gauge transformations which leave the connection invariant — can be non-trivial, that is, it is neither an empty set, nor is it equal to the center of the group. It can be shown that the isotropy group, Hc, consists of constant gauge transformations that leave the connection
invariant
Hc= {φ = const. ∈ G | φA(c)φ−1 = A(c)}. (1.2.20)
A non-trivial isotropy group also means that
∅ 6= Lie(Hc) = H0(M, g) = Ker dA(c), (1.2.21)
which implies that the operator dA(c) has zero modes. Then, to evaluate the FP determinant
we have to take this into account in order to get a finite result. Furthermore, we notice that gA(c)(Bφ) = φ gA(c)(B)φ−1 = 0, φ ∈ Hc, (1.2.22)
which means that the gauge is not fixed completely by our gauge-fixing choice, hence there is a residual symmetry which we need to consider in the following computation. We are now ready to compute, at least formally, the determinant
∆−1A(c) = Z DU δ(gA(c)(BU)) = Z Hc⊕ (Lie Hc)⊥ DU δU det δg δU U =0 −1 = = Z Hc Dφ Z (Ker d A(c)) ⊥ DU0δU0 det d † A(c)dA −1 = = Z Hc Dφ det d † A(c)dA −1 (Ker d A(c)) ⊥ = = Vol (Hc) det d † A(c)dA −1 (Ker d A(c)) ⊥. (1.2.23)
There are a number of technicalities that needs to be explained. First of all, in the first step we used the well known property of the δ-function under a change of variables: we considered, for fixed B, gA(c)(BU) as a function of U and introduced the Jacobian determinant
upon changing δg → δU . Next, we split the integration over the group elements in a part along the isotropy group Hc, whose Lie algebra is Ker dA(c), and another along its
orthogonal complement. The delta function along this latter ensures that the evaluation of the determinant gets restricted to (Ker dA(c))⊥, thus avoiding the zero modes. Finally, in
view of (1.2.22), the final integral just gives a volume factor, since the integrand does not depend on the elements along Hc. Using, now, the well-known result of functional integration
theory that a path integral over Grassman fields gives a determinant (and not its inverse, as instead would a bosonic path integral), we can rewrite the FP determinant as
∆A(c)(B) = 1 Vol(Hc) Z (Ker d A(c)) ⊥ DC DC e−Sgh(C,C,B), (1.2.24)
where C and C are fermionic fields that take values in (Ker dA(c))⊥. As for the exponent,
from the formal manipulation carried out in (1.2.23), it is given by Sgh(C, C, B) = hC, d † A(c)dACi ≡ Z tr C ∧ ∗d† A(c)dAC = = hC, d†A(c)dA(c)Ci + ihC, d† A(c)[B, C] i = = hC, ∆0A(c)Ci + O(~λ/2), (1.2.25)
where the cubic interaction term between the ghosts and the fluctuation B is higher order in ~ and can be dropped in the present discussion. Here ∆0
A(c) is the Laplacian acting on
0-forms, not to be confused with the FP determinant. Using now that both the original 1-loop path integral and the FP determinant are gauge invariant, we can cast the gauge-fixed 1-loop contribution to the partition function as
Z1−loop(c) (M) = 1 Vol(G) Z DB Z DU δ(gA(c)(BU)) ∆A(c)(B) eikS(B)= = 1 Vol(G) Z DBU−1 Z DU δ(gA(c)(BU U −1 )) ∆A(c)(BU −1 ) eikS(BU −1) = = Z DB δ(gA(c)(B)) ∆A(c)(B) eikS(B)= = 1 Vol(Hc) Z Ω1(M,g) DB δ d†A(c)B Z (Ker d A(c)) ⊥ DC DC eikS(B)−Sgh(C,C,B), (1.2.26) where the integration over the gauge group,R DU , canceled against the factor of (Vol(G))−1. As for the δ-function constraint, we use that
B ∈ Ω1(M, g) = Ker d†A(c)⊕ Im dA(c), (1.2.27)
to write B as
where
θ ∈ (Ker dA(c))⊥, B0 ∈ Ker d
†
A(c). (1.2.29)
This change of variables results in a non-trivial Jacobian, both from the δ-function and from the integration measure. As for this latter, it can be computed by considering the norm in the space Ω1(M, g) of Lie algebra-valued 1-forms
||B||2 ≡ hB, Bi = hd A(c)θ, dA(c)θi + hB0, B0i + hdA(c)θ, B0i + hB0, dA(c)θi = = hθ, d†A(c)dA(c)θi + hB0, B0i = hθ, ∆0A(c)θi + ||B 0||2, (1.2.30) since hdA(c)θ, B0i = hθ, d † A(c)B 0i = 0. (1.2.31) From this we infer that the measure changes as
DB = (det ∆0
A(c))1/2Dθ DB
0
. (1.2.32)
Regarding the δ-function, on the other hand, it is sufficient to consider a generalization of the following identity
δ(ax) = 1
|a|δ(x), (1.2.33)
to matrices. Indeed, we have
δ d†A(c)B = δ ∆ 0 A(c)θ = det ∆ 0 A(c) −1 δ(θ), (1.2.34)
We stress that in both computations the determinants have their zero-modes removed, thus the Laplacian operator is positive-definite and its square root is well-defined. Putting all this together, we have
DB δ d†A(c)B = (det ∆
0 A(c))
−1/2Dθ DB0
δ(θ), (1.2.35)
where δ(θ) obviously sets θ = 0 when integrating over (Ker dA(c))⊥. Finally, the gauge-fixed
path integral is Z(c)(M) = e ikSCS(A(c)) Vol(Hc) det ∆0A(c) −1/2 Z Ker d† A(c) DB0 Z (Ker dA(c))⊥ DC DC eikS(B0)−Sgh(C,C,B0). (1.2.36) To summarize, the gauge fixing procedure has two main features: (1) it discards the zero-modes of the fluctuation B, that is those zero-modes along the kernel of the covariant derivative, making the functional determinant in (1.2.24) well-defined and positive definite; (2) it tells us that only the modes B0 of the fluctuation B contribute to the path integral, whereas θ does not. Therefore, in the following, we rename B0 → B and there should be no confusion. On the other hand, the FP procedure has a drawback: we were forced to pick a metric, as we can see from the action for the ghost fields (1.2.25), in which the Hodge star operator, and thus the metric, appears explicitly. Therefore, the gauge-fixed path integral manifestly breaks the topological invariance of the classical theory.
1.2.2
1-loop contribution
In the semiclassical limit k → ∞ we can evaluate the gauge-fixed path integral as a free theory, since the cubic interaction term between the ghosts C, C and the fluctuation B in (1.2.25) is higher order in ~. This is enough for the purposes of showing that one can still
obtain topological invariants of the three-dimensional manifold. On the other hand, if the aim was to develop a Feynman-like diagrammatic study of Chern-Simons theory, then the interaction term is of course very important and one should go deeper in perturbation theory to allow interaction terms.
From the previous discussion, we are left with two Gaussian functional integrals. The one over B gives
Z DB exp tr Z B ∧ ∗ −ik 4π ∗ dA(c) B ≡ Z Ker d† A(c) DB expnB,iQ 2 B o = = exp iπ 4 sign(Q) det Q 2π −1/2 Ker d† A(c) , (1.2.37)
where we inserted ∗∗ = (−1)p(d−p) = 1 for d = 3 and p = 2, the degree of d
A(c)B. Although
there are no zero modes, the operator appearing in the equation above is not positive-definite. We can instead consider the square, which is given by
det Q 2π 2 Ker d† A(c) = det k 4π2 2 ∆1A(c) Ker d† A(c) , (1.2.38)
where the operator
∆1A(c) Ker d† A(c) = d†A(c)dA(c), (1.2.39)
is the Laplacian on 1-forms, when restricted to Ker d†A(c), and is indeed positive-definite,
making its square root well-defined. Recall that the determinant is essentially a product of eigenvalues, which means that, for each eigenvalue in (1.2.37), we have a numerical factor of
k 4π2
−1/2
, (1.2.40)
and the regularized number of eigenvalues is formally given by the ζ-function ζ(A(c)) ≡ ζ−∗d
A(c)(0). (1.2.41)
It was shown in [13] that this quantity is closely related to the dimensions of the cohomology groups of the twisted de Rham complex (1.2.13) given by the covariant derivative. In fact, we have ζ(A(c)) = (−1)j+1 j X n=0 (−1)ndim Hn(M, g) = = dim H0(M, g) − dim H1(M, g) = = dim H0(M, g), (1.2.42)
where j = 1 for d = 2j − 1 = 3 and dim H1 = 0 for an isolated flat connection. Finally,
we need the sign of the operator Q. To this end, we consider the regularized difference of eigenvalues, which is formally given by the η-invariant evaluated at zero. We denote the absolute value of the positive and negative eigenvalues by λ±n, respectively. Then, the sign of Q is given by η(A(c)) ≡ η−∗d A(c)(0) = lims→0 X n λ+ −sn − λ− −sn = = η(0) + 4y π SCS(A (c)), (1.2.43)
where to get to the last line we used the Atiyah-Patodi-Singer (APS) theroem1, as pointed
out in [2, 8], and the number y appearing in the second term is the dual Coxeter number which is needed to have the correct normalization for the trace in SCS(A(c)). To clarify, we
wrote η(0) ≡ dG· η−∗d(0) ≡ dG· ηgrav, where dG = dim G. We will have more to say on this
term in a while. Therefore, for the integral over B we obtain Z Ker d† A(c) DB expnB,iQ 2 B o = k 4π2 −12dim H0 eiyS(A(c))+iπ4η(0) det d†A(c)dA(c) −1/4 . (1.2.44) As for the integral over the ghosts, we have
Z (Ker d A(c)) ⊥ DC DC exp−hC, ∆0 A(c)Ci = det ∆ 0 A(c) (Ker d A(c)) ⊥, (1.2.45)
which combines with the determinant coming from the Jacobian of the measure and the δ-function (1.2.35) of the gauge fixed path integral (1.2.36) to give
det ∆0A(c)
1/2
(Ker dA(c))⊥. (1.2.46)
Collecting these results, in the 1-loop contribution we have a ratio between the determinant of the Laplacian acting on 0-forms and that of the Laplacian acting on 1-forms that are restricted to Ker d†A(c). It was first noticed by Ray and Singer that such a ratio of determinants
is a good quantity to study topological properties of manifolds from an analytic point of view [12]. According to their definition, we have
q τR0 (M, A(c)) ≡ det ∆ 0 A(c) 1/2 det d†A(c)dA(c) 1/4 (1.2.47)
which is the square root of the so-called Ray-Singer torsion of the connection A(c). This is a topological invariant of the manifold when the connection is isolated and irreducible. For a reducible connection, however, it depends on the metric. Indeed, from [8], we have
τR0(M, A(c)) = (Vol(M))dimHc
τR(M, A(c)) (1.2.48)
1We point out that this is not the full expression of the theorem. Indeed, in the next section we will
introduce also the missing term, which we omitted here for simplicity. For an introduction to the APS theorem we refer the reader to chapter 12.8 of [1] and for a proof for physicists to the early work [15].
where the torsion, τR, on the RHS is now independent on the geometry of the manifold.
Recall now that in the path integral (1.2.36) the volume of the isotropy group appears in the denominator and that the isotropy group Hc is the set of constant 0-forms taking values in
a closed subset, hc, of the total Lie algebra g. Then, we can use the natural notion of inner
product that exists in hc to write this factor in a more explicit way. To this end, consider
an orthogonal basis of the isotropy group that is normalized to unity: the constant 0-form part of the elements is just 1, so that the basis is simply equal to the set of generators of the corresponding algebra, {ti}dim hi=1 c. Then, keeping in mind that dim Hc= dim hc, we have
Vol(Hc) ≡ dimHc Y i=1 ||ti|| = dimHc Y i=1 hti, tii1/2 = = Z M ∗1 dim(Hc)/2 × dimHc Y i=1 tr (titi)1/2=
= Vol(M)dim(Hc)/2Vol(hc),
(1.2.49)
which cancels against the volume of the manifold coming from the square root of the Ray-Singer torsion (1.2.48). Finally, putting together the classical and 1-loop contributions, the path integral reads
Z(c)(M) = 1 Vol(hc) k 4π2 −12dim H0
ei(k+y)SCS(A(c))+iπ4dG·ηgrav
q
τR(M, A(c)). (1.2.50)
A first important result is apparent: the Chern-Simons level k gets shifted by the dual Coxeter number y of the gauge group, representing a 1-loop renormalization of the coupling constant of the theory. It is indeed a known fact that in the full gauge-fixed theory with the cubic interaction vertex, which we discarded, one finds, by computing the relevant two- and three-point functions, that the level gets renormalized only at one-loop. In that framework, however, one also finds out that different regularization schemes seem to incorporate different UV corrections into the level k, but a discussion of this fact is beyond the scope of this thesis. Furthermore, it is apparent that topological invariance is lost in perturbation theory: the metric that we had to pick when writing the FP-determinant as a functional integral over ghosts (1.2.24) remains in the path integral in the phase involving the η-invariant of the operator − ∗ d: this is the gravitational anomaly. Luckily, it is a “good” anomaly, in the sense that it does not render the theory inconsistent (examples of this behaviour are gauge anomalies) and that it can be canceled by including an appropriate counterterm to the original Chern-Simons action. We now turn to this last effort.
1.2.3
Gravitational Chern-Simons action and framing of the
manifold
To restore topological invariance at the quantum level we can add a counterterm to cancel out the metric dependence in the gravitational eta invariant. The appropriate one is the
following gravitational Chern-Simons action (notice the different normalization) SCS(ω) ≡ − Z M Q3(ω, R) = 1 8π2 Z M tr ω ∧ d ω + 2 3ω 3 , (1.2.51)
where Q3(ω, R) is defined in (A.4.19), ω is the spin connection and R the corresponding
Riemannian curvature 2-form. The name “gravitational” is a historic one that comes from works by Witten [4, 5] — where he managed to discuss some aspects of (2 + 1)-dimensional gravity by building a gauge theory using the action above (in Lorentzian signature) — and a series of papers by Alvarez-Gaum´e and collaborators on anomalies in three-dimensions [14, 15]. The usefulness of this counterterm in the present discussion is due to the Atiyah-Patodi-Singer index theorem, according to which the combination
1
4ηgrav+ 1
12SCS(ω), (1.2.52)
does not depend on the metric on M and therefore is a topological invariant of the manifold. However, the gravitational action is plagued by an ambiguity similar to that of the Chern-Simons action: we need to specify a trivialization, or framing, of the tangent bundle to define the spin connection, so that to different trivializations can, in principle, correspond different gravitational actions. Luckily, in the last part of appendix A.4, we presented an argument according to which two actions corresponding to two different choices of framings only differ by a weight s0 ∈ 1
3Z and we can further spot that the factor of 1
3 is already included
in (1.2.52). Therefore, we finally understand that we can preserve topological invariance by replacing ηgravin (1.2.50) with the combination (1.2.52) and that at the same time we should
be careful and notice that the partition function is now “sensitive” to a change of framing as follows Z → exp 2πis ·dG 24 Z, (1.2.53) with s ∈ Z. (1.2.54)
Chapter 2
Supersymmetry
The aim of this chapter is to introduce fundamental concepts in supersymmetry (SUSY) that will be useful to understand the theories considered in chapter3. Spurred by theoretical interest, supersymmetry was initially introduced in physics to enlarge the symmetry of a field theory beyond Poincar´e and internal symmetries and it was later applied to the Standard Model, in an effort to solve some of its problems. However, although this symmetry is consistent with basic concepts of quantum field theory (for a proof of this see, for example, the first chapter of [17]), the most recent experiments in high energy particle physics have found no evidence that supersymmetry be realized in the real world, already during Run 1 of the LHC at√s = 8 TeV [16], but neither currently as the LHC is running at√s = 13 TeV and with higher luminosity. Still, given the major implications that such a symmetry would present for a fundamental theory of nature [18], the search for phenomenological realizations of supersymmetry continues. Be it or not an actual symmetry of spacetime, the theoretical interest remains.
Let us briefly consider the symmetry in d = 4 Minkowski spacetime and take the spinor generators Qα and Q
† ˙
α to be 2-component Weyl spinors1. The symmetry, which relates
bosonic and fermionic degrees of freedom, has the following algebraic structure [19] (σµν and
¯
σµν are the Lorentz generators in the spinor representation and are defined in (B.2.11) and (B.2.12)) {Qα, Q † ˙ α} = −2σ µ a ˙αPµ, (2.0.1) {Qα, Qβ} = {Q † ˙ α, Q † ˙ β} = 0, (2.0.2) Qα, Pµ = Q † ˙ α, Pµ = 0, (2.0.3) Qα, Mµν = i(σµν)αβQβ, (2.0.4) Q† ˙ α, M µν = −iε ˙ α ˙δ(¯σ µν)δ˙ ˙ βQ † ˙β, (2.0.5)
together with the usual commutation relations of the Poincar´e generators Pµ and Mµν. One
can also construct an extended version by allowing the theory to be symmetric under N different copies of the symmetry. We would then label the different supercharges by the indices A, B, . . . = 1, . . . , N as QαA and Q
† ˙
αB and modify the first two relations (2.0.1) and
(2.0.2) as {QαA, Q † ˙ αB} = −2δABσ µ α ˙αPµ, (2.0.6) {QαA, QβB} = ZABεαβ, (2.0.7)
and analogously for the complex conjugate supercharges. Here the matrix ZAB is
antisym-metric in its indices, so that for N = 1 we recover the right relation. As a remark, we point out that the amount of supersymmetry can not, in general, be arbitrarily imposed: being generated by spinor charges, it is constrained by the rules of existence of spinors in different spacetimes, as the spacetime symmetry group changes (or, equivalently, what changes is the Clifford algebra that determines the representations the spinors sit in2). For example, in
d = 4 Minkowski spacetime the Lorentz group SO(3, 1) ∼= SU (2) × SU (2) and the minimal spinors are either Majorana or complex Weyl, implying that the corresponding minimal su-persymmetry has an equivalent description in terms of a 4-component real Majorana spinor of supercharges or in terms of a 2-component complex Weyl spinor of supercharges (from the formulae above it is clear that we chose the Weyl “picture”). Therefore, we call this N = 1 SUSY. As for the d = 3 N = 2 Euclidean supersymmetric theories that we aim to study, they can be obtained by dimensional reduction of such a minimally supersymmetric theory, but in d = 4 Euclidean space.
This chapter is organized as follows. In section 2.1, we review a simple supersymmetric theory of scalars and spinors in d = 4 Minkowski spacetime by using the usual quantum field theory framework. In section 2.2, we recast it using the N = 1 superspace formulation and go on to derive a supersymmetric version of non-Abelian gauge theories. In the superspace arena, concepts like supercharges and supermultiplets will get an additional geometric mean-ing. Then, in section 2.3, we Wick rotate to get a Euclidean theory and in section 2.4 we dimensionally reduce it to obtain a supersymmetric theory in d = 3 space. As the symmetry group of three-dimensional space is smaller, the SUSY will get enhanced to N = 2. Section
2.5, on the other hand, is dedicated to deriving an action that cannot be obtained by di-mensional reduction of a higher-didi-mensional theory, thus we consider N = 2 superspace in d = 3 flat space. Finally, section 2.6 is dedicated to the last effort of placing the flat space theory on S3 by preserving the supersymmetry.
2.1
Wess-Zumino model
We would like to introduce the symmetry by studying a simple quantum field theory of massless scalars and spinors in four-dimensional Minkowski spacetime with signature ηµν =
diag(−, +, +, +). The model we present here is based on the first example of a supersym-metric theory, which appeared in 1974 [20]. The theory contains a complex scalar φ, a left-handed 2-component Weyl spinor ψα and a non-dynamical complex scalar F . The
rea-son we include this latter auxiliary field will become clear as we go through the example. We denote this collection of fields by X = (φ, ψ, F ), which we call chiral supermultiplet, whereas
2We included a compact review of Clifford algebras in d = 4 and d = 3 Minkowski and Euclidean
their conjugate fields constitute the anti-chiral one X† = (φ†, ψ†, F†). The simplest action that we can write down is
SW Z =
Z
d4x −∂µφ†∂µφ − iψ†σ¯µ∂µψ + F†F . (2.1.1)
The term supermultiplet has been coined by analogy with other symmetries: under super-symmetry the fields that belong to a supermultiplet are exchanged (only) among themselves, which is to say that supermultiplets are representations of the supersymmetry algebra. We take α to be a constant (infinitesimal) Weyl spinor and make the following ansatz for the
infinitesimal version of a symmetry that relates bosons and fermions
δφ =√2ψ, δφ† ≡ (δφ)† =√2ψ††, δψ = i√2σµ†∂µφ + √ 2F, δψ†≡ (δψ)†= −i√2σµ∂µφ†+ √ 2†F†, δF = i√2†σ¯µ∂µψ, δF†≡ (δF )† = −i √ 2∂µψ†σ¯µ, (2.1.2)
where the factor of √2 will become convenient later on and we suppressed the spinor indices for the moment. We check that the Lagrangian is invariant (the factors of√2 are not relevant here, thus we omit them)
δLφ= −∂µφ†∂µ(δφ) − ∂µ(δφ†)∂µφ =
= −∂µψ∂µφ†− †∂µψ†∂µφ,
δLψ = i∂µ(δψ†)¯σµψ − iψ†σ¯µ∂µ(δψ) − i∂µ δψ†σ¯µψ =
= ∂µψ∂µφ†+ †∂µψ†∂µφ − i†σ¯µ∂µψF + i∂µψ†σ¯µF†+
+ ∂µ iψ†σ¯µF − σν¯σµψ∂νφ†− ψ∂µφ†− †ψ†∂µφ ,
δLF = δF†F + F†δF =
= −i∂µψ†σ¯µF†+ i†σ¯µ∂µψF,
(2.1.3)
where for the variation δLψ we integrated by parts and, knowing that the partial derivatives
commute, we also used the identities (B.2.9) and (B.2.8). Indeed, up to the total derivative term ∂µKµin δLψ, the terms all cancel between themselves. However, we have to check that
the algebra closes as well. In particular, we consider the commutator of two infinitesimal transformations parametrized by 1 and 2. The one for the scalar φ is straightforward
[δ1, δ2] φ = √ 22(δ1ψ) − √ 21(δ2ψ) = = 2 (21− 12) F + 2i 2σµ†1 − 1σµ†2 ∂µφ = ≡ 2ivµ∂ µφ. (2.1.4) since 12 = 21. As for F [δ1, δ2] F = i √ 2†2σ¯µ∂µ(δ1ψ) − i √ 2†1σ¯µ∂µ(δ2ψ) = = −2i †1σ¯µ2− † 2σ¯ µ 1∂µF − 2 2†σ¯µσν†1− †1σ¯µσν†2∂µ∂νφ = = 2ivµ∂µF, (2.1.5)
where we used the fact that iσµ † j = −
†
jσ¯µi to rewrite the first term in terms of vµ and
(B.2.9) to cancel the second term in the second line. For the fermion the computation is a bit more involved
[δ1, δ2] ψ = √ 2δ1 iσµ†2∂µφ + 2F −√2δ2 iσµ†1∂µφ + 1F = = 2i (1∂µψ) σµ † 2+ 2i †1σ¯µ∂µψ 2− 2i (2∂µψ) σµ † 1− 2i †2σ¯µ∂µψ 1 = = 2ivµ∂µψ, (2.1.6) where to go from the second to the last line we used the Fierz identity (B.2.1) on the first and third terms, together with iσµ
† j = −
†
jσ¯µi, in order to cancel the other terms. Had we
not included the auxiliary scalar F , the previous commutator would have read
[δ1, δ2] ψ = √ 2δ1 iσµ†2∂µφ −√2δ2 iσµ†1∂µφ = = 2ivµ∂µψ − 2i †1σ¯µ∂µψ 2+ 2i †2σ¯µ∂µψ 1, (2.1.7)
and to cancel the last two terms we would have needed to use the equations of motion. It is therefore apparent why we included an auxiliary scalar whose SUSY transformation is proportional to the e.o.m of the spinor field: in this way, the algebra closes off-shell on any of the fields X into a spacetime translation (Pµ = (H, Pi)) parametrized by the vector vµ,
that is
[δ1, δ2] X = 2iv
µ∂
µX = −2vµ[Pµ, X] . (2.1.8)
The transformation rule for F looks like a trivial artifact and indeed it is, for this simple theory. As we study more complicated theories, however, the situation becomes more subtle and imposing off-shell closure of the algebra is going to give us interesting insights on the properties of the SUSY, not just an algebraic, ad hoc artifact. The purpose of F can also be seen at the level of the counting of degrees of freedom: off-shell, since no constraints have to be taken into account, the (real) bosonic d.o.f.s are four — two from φ and two from F — and the corresponding fermionic ones are four as well, since the spinor ψ has two complex components. When constructing general supersymmetric theories, this matching of bosonic and fermionic d.o.f.s between fields belonging to the same supermultiplet has to be satisfied off-shell.
We now want to inspect the quantum-mechanical structure of the symmetry by computing the charges that generate the infinitesimal variations. This will ultimately enable us to cast the commutation relations [δ1, δ1] X in terms of quantum-mechanical operators derived by
virtue of Noether’s theorem. We know, in fact, that to any continuous symmetry corresponds a conserved current. In the case at hand, the conserved current should carry a spinor index, besides a spacetime one, as it is fit for the current corresponding to a fermionic symmetry. We can use Noether’s theorem to compute the currents as (we now restore the factor of√2 in Kµ) Jµ+ J†µ† =X X δX δL δ(∂µX) − Kµ, (2.1.9)
from which we get the following supercurrents
Jαµ= −√2 (σνσ¯µψ)α∂νφ†, (2.1.10)
Jᆵ˙ = −√2 ψ†σ¯µσνα˙∂νφ, (2.1.11)
which are conserved on-shell. The corresponding supercharges (we assume that we already canonically quantized the theory) are
Qα = Z d3x Jα0 = −√2 Z d3x ψαπ + σiψ α∂iφ † = (2.1.12) = −√2 Z d3x (σνσ¯0ψ)α∂νφ†, Q†α˙ = Z d3x Jα†0˙ = −√2 Z d3x ψα†˙π†+ ψ†σiα˙∂iφ = (2.1.13) = −√2 Z d3x (ψ†σ¯0σν)α˙∂νφ,
where we have cast them in two equivalent forms, the first being more suited to compute the variation of the scalar field φ and the second that for the fermion ψ. We then define the quantum-mechanical operator
δS ≡ Q + Q††, (2.1.14)
that generates infinitesimal SUSY transformations via the commutator
[δS, X] = iδX, (2.1.15)
for all the fields X in the theory. We can check this assertion by using the equal-time commutation relations that arise from canonically quantizing the action (2.1.1)
[φ(x, t), π(x0, t)] = iδ3(x − x0), (2.1.16) ψα(x, t), ψ † ˙ α(x 0, t) = −σ0 α ˙αδ 3(x − x0 ). (2.1.17) We have Q + Q† †, φ = −√2 Z d3x ψ [π, φ] = i√2 ψ, (2.1.18) Q + Q† †, ψβ = α{Qα, ψβ} − n Q†α˙, ψβ o † ˙α= −nQ†α˙, ψβ o † ˙α = (2.1.19) =√2 Z d3x n ψ†˙ β, ψβ o ¯ σ0 ˙βασµα ˙α† ˙α∂µφ = = i√2 i σµ†β∂µφ, Q + Q† †, F = 0. (2.1.20)
It is not surprising that all we could derive was the on-shell version of the SUSY: the currents and the supercharges are valid on-shell, as equation (2.1.9) already implies use of the e.o.m.s. We will do better when we discuss superspace in the next section. However, for our purposes this form is already enough: using these relations, we can check that Qα
and Q†α˙, together with the charges for translation Pµ, satisfy the algebra (2.0.1). Consider,
in fact, the transformations δ1,2 generated by 1,2Q + Q
††
1,2, respectively. Then, in view of
(2.1.8) and (2.1.15), we have [δ1, δ2] X = − h 1Q + Q† † 1, h 2Q + Q† † 2, X ii +h2Q + Q† † 2, h 1Q + Q† † 1, X ii = = −hh1Q + Q† † 1, 2Q + Q† † 2 i , Xi= = −2vµ[Pµ, X] , (2.1.21)
where we used the Jacobi identity to get to the second line. Now, given that the parameter for translations is the vector vµ =
2σµ †
1− 1σµ †
2, the previous equation implies that
−2α1α†2˙ − α2α†1˙ σα ˙µαPµ = 2vµPµ= h 1Q + Q††1, 2Q + Q††2 i = =h1Q, Q† † 2 i +hQ††1, 2Q i = =α1† ˙2α− α 2 † ˙α 1 n Qα, Q † ˙ α o . (2.1.22)
which shows that we recover the correct form of the algebra also at the level of the quantum-mechanical operators Qα and Q
† ˙ α n Qα, Q † ˙ α o = −2σα ˙µαPµ, (2.1.23) {Qα, Qβ} = n Q†α˙, Q†˙ β o = 0. (2.1.24)
In this toy model the algebra has a simple form, but in a more complicated field theory, in which other symmetries are present (gauge symmetry, global symmetries, . . . ), the su-persymmetry algebra has to close with all of them, not only with the spacetime translation symmetry. We will get to see this in the field theories that we introduce at the end of this chapter. Before we get there, however, it is convenient to reconsider the model just pre-sented under the point of view of the superspace formalism, from which the construction of the symmetry, which here has been directly written down in (2.1.2), will get a deeper and more satisfying meaning.
2.2
d = 4 N = 1 Lorentzian superspace
In the present section, following [17, 22], we briefly introduce the superspace formalism. We focus on the physics and on the computational advantage it provides for supersymmetry.
First of all, a superspace, or supermanifold, in four dimensions, is a manifold equipped with the usual bosonic coordinates xµ, µ = 0, . . . , 3 — which we take to be Minkowski with the same signature as in the previous section — and a set of 4 Grassman-odd (fermionic) co-ordinates. In the present discussion we take these latter to be grouped into two 2-component Weyl spinors, θ and its complex conjugate θ†. Functions in superspace depend on all these coordinates, so that we can expand them in a power series in the fermionic coordinates,
which then truncates at quadratic order in these latter. The coefficients appearing order-by-order in θ, θ† are then functions of xµ only and represent the set of fields that make up the
corresponding supermultiplet on the bosonic manifold.
Let us then consider a complex function f (x, θ, θ†), to which we will refer to as scalar superfield. According to the previous discussion, its expansion has the general form
f (x, θ, θ†) = c(x) + θχ(x) + θ†ξ†(x) + θθ M (x) + θ†θ†N (x) + θσµθ†vµ(x) +
+ θθθ†η†(x) + θ†θ†θζ(x) + 1 2θθθ
†
θ†d(x), (2.2.1)
where all the component fields are complex and possible other scalar terms that may be allowed can be rewritten in this form by using spinor identities, so that they would be redundant. The power of this formalism is that infinitesimal transformations of the fields under supersymmetry are generated by differential operators that act on superfields, which therefore involve derivatives along both the bosonic and fermionic coordinates. The defining property of such operators is, of course, that they must satisfy anticommutation relations compatible with the SUSY algebra, that is they must “square” to the translation operator
ˆ
Pµ. In the following, hatted quantities refer to operators acting on superfields, whereas
non-hatted quantities are operators acting on the Hilbert space of the theory, such as the ones that we encountered in the previous section. Imposing Lorentz invariance, the appropriate form of the superspace operators is found to be
ˆ Qα = ∂ ∂θα − i σ µθ† α∂µ, (2.2.2) ˆ Q†α˙ = − ∂ ∂θ† ˙α + i (θσ µ) ˙ α∂µ. (2.2.3)
Indeed, using the usual rules for Grassman differentiation and (B.2.10), it is straightforward to check that they satisfy
n ˆQα, ˆQ† ˙ α o = 2iσα ˙µα∂µ≡ −2σµα ˙αPˆµ, (2.2.4) n ˆQα, ˆQβo = 0, n ˆQ†α˙, ˆQ†˙ β o = 0, (2.2.5)
where we defined ˆPµ= −i∂µ, the differential operator that generates spacetime translations
on superfields. An infinitesimal supersymmetry transformation on a superfield looks like
ˆ δSf ≡ αQˆα+ ˆQ † ˙ α † ˙αf = ∂ ∂θ + † ∂ ∂θ† + i θσ µ†− σµθ† ∂ µ f = ≈ fxµ+ i θσµ†− σµθ† , θ + , θ† + †− f (xµ, θ, θ† ), (2.2.6)
where we have expanded to first order in and † to get to the result. Then we understand that a supersymmetry transformation, at least to linear order in the parameters, can be regarded as a translation in superspace
xµ→ xµ+ i σµθ†
It is also useful to define another set of differential operators ˆ Dα = ∂ ∂θα + i σ µθ† α∂µ, (2.2.8) ˆ Dα†˙ = − ∂ ∂θ† ˙α − i (θσ µ) ˙ α∂µ, (2.2.9) which satisfy n ˆDα, ˆD† ˙ α o = 2σµα ˙αPˆµ, (2.2.10) n ˆDα, ˆDβo = 0, n ˆDα†˙, ˆD†˙ β o = 0. (2.2.11)
Furthermore, these are compatible with the supercharges (2.2.2) and (2.2.3), since n ˆQα, ˆDβo =n ˆQ†α˙, ˆDβ o =n ˆQα, ˆD†β˙ o =n ˆQ†α˙, ˆD†˙ β o = 0. (2.2.12)
We will then refer to Dα and D †
˙
α as supercovariant derivatives, because they are
compat-ible with supersymmetry: these anti-commutation relations imply that the supercovariant derivative of a superfield is again a superfield, in the sense of (2.2.6). Let us further notice that ˆ Qα = ˆDα− 2i σµθ† α∂µ, (2.2.13) ˆ Q†α˙ = ˆD†α˙ + 2i (θσµ)α˙∂µ. (2.2.14)
We would like to close this part with a remark on these differential operators. First of all, it is important to mention that other definitions of the ˆQs and ˆDs are possible and, in fact, there is no real agreement in the literature — of course, once a choice has been made, one must be consistent throughout. Let us clarify that it is not the form of the operators that differs, what does is just the placing of the minus signs and of the ‘i’s, as the former is constrained by the algebra (2.2.4). This, in turn, is intimately connected to the fact that the ˆQs, the ˆDs and ˆPµ generate a Lie (super )algebra and, accordingly, exponentiation
of a suitable linear combination of these operators gives elements of the corresponding Lie (super )group — the parameters for the ˆQs and the ˆDs are infinitesimal spinor parameters, so that the overall product is a bosonic quantity and the sum with the translation operator is well defined. From the point of view of the supergroup, the ˆQs would then represent the first order expansion of an element that acts on superfields by left -multiplication, whereas the ˆDs correspond to an element which acts by right -multiplication: this is what encodes their formal, not only conceptual, difference. Accordingly, we refer to (2.2.4) and (2.2.10) as left- and right-supersymmetry algebras, respectively. Incidentally, the fact that what we are considering is actually the first order expansion of a group element is the reason why a linear SUSY transformation is the translation (2.2.6) in superspace.
2.2.1
Chiral superfield
Let us consider superfields Φ and Φ†. The constraints ˆ
D†α˙Φ(x, θ, θ†) = 0, (2.2.15)
ˆ
