We also introduce some of the theories we will explore in the second half. In the second half of the dissertation we put our calculations from the first half to work, making non-trivial statements about strongly coupled field theories.
Supersymmetry Algebra
Multiplets
We will see in a moment that it can also be written as a D term, and so is a totalδ variation. We will review this action in more detail, together with the bosonic Chern-Simons action, to which it is closely related, in the next chapter.
Moduli Space
As we will see in some examples in the second part of the thesis, the exact module space in a theory can in many cases be determined precisely, especially in the case of extended supersymmetry. In the case of compact manifolds, which we look at later, there is no module space since constant backgrounds must be integrated over.
Superconformal Symmetry
We will return to this point when we discuss the superconformal index in Chapter 4. To construct explicit representations of the superconformal algebra on fields, we will again use the notationδ=αQα, for a fermionic parameter.
Extended Supersymmetry
In the previous chapter we discussed the class of theories we will be interested in and given their actions in flat Minkowskian space. However, as we will see in the next chapter, in many cases this quantity can be shown to be independent of β.
Conformal Theories on Conformally Flat Manifolds
- Chiral Multiplet
- Gauge Multiplet – Chern Simons Action
- Wilson Loops
- Yang-Mills Action
Later we will see that we can assign a more physical interpretation to some of the quantities we calculate. This is conformally invariant—in fact, it is topological, it is completely independent of the metric—and thus appears to be a viable option.
S 3 Actions
Since it is written as a sum of squares, we can immediately read the condition BPS:. Now we can immediately read off the two solutions of the conformal Killing equation: just take the components to be constant in this basis.
However, we can get around this problem if we are willing to expand our interpretation of the partition function a bit. Again, this depends on the discreteness of the H spectrum and is more subtle (and in many cases incorrect) in the presence of a continuous spectrum.
Example: SUSY Quantum Mechanics
We therefore find that the result is given by the 1-loop approximation around the saddle points of constantxµ and ψµ. One can generalize this in several directions to reproduce other well-known index theorems involving characteristic classes.
Localization on S 4
Chern-Simons Theory
To solve these equations explicitly, we will need to choose our manifold and spinor, which we will do in the next section. One then only needs to add this contribution over all zero modes, and we arrive at the leading approximation of the partition function in the largest limit.
S 3
In addition, we have taken into account the remaining Weyl symmetry by dividing by the order of the Weyl group, |W|. Note that the Lie algebra is spanned over a basisei of Cartan, in which= 1, .., r, up to rank r in the group, together with the root vectors Xα, where α runs over the roots of the Lie algebra.
Wilson Loop
We then see that the denominator of the 1-ring determinant cancels (up to a sign) against the Vandermonde determinant, and we are left with:. 4.22) This gives a great approximation to the partition function, which, by the above arguments, is actually correct for all. Tre2πσo (4.23) Thus, the only modification to the integral of the above matrix is the introduction of such a factor.
Matter
S 3
From the previous section we saw that the supersymmetric configurations for the gauge manifold on S3 only σ and D do not vanish. Let us first consider the case where the measure group is U(1), so thatσo is simply a real number.
General Result for Gauge Theory on S 3
Recall that chiral multiplets must come in pairs in conjugate representations that form hypermultiplets in order to provide the extended supersymmetry usually required for chiral multiplets to have an R-charge of 12.
Background Gauging of Global Symmetries
Finally, one can also connect to the U(1)J symmetry discussed in Chapter 2, which in flat space produces an FI termη. In the case of SQED above, we find that the most common distorted partition function, with U(1)A distortion andU(1)J distortionη, is:.
In the previous chapter, we wrote down the δ-exact actions for both gauge and material multiplets on S2 × S1, so we summarize them here. First, the classical contribution and the 1-loop determinant of the gauge-matter multiplets in each of these zero-mode backgrounds must be calculated.
Superconformal Index
This can be compared with results from the string theory side, calculated at weak coupling in the supergravity approximation,1 to provide a non-trivial test of the duality. Insertions of sources in gauge theory will correspond to insertions of vertex operators in string theory.
ABJM Theory
Supergravity Dual
In the supergravity approach, the field theory partition function is: This is anomalous, but can be regularized using holographic renormalization. Note that the free energy is in a sense supposed to count the degrees of freedom of the theory. see Chapter 7 for a more precise explanation).
Testing the Duality
Matrix Model at Large N
Matching the field theory and supergravity results provides a non-trivial test of the duality. The Higgs branch does not receive quantum corrections and can be parameterized by 2Nf−1 chiral multiples of the form M ∼QQ.˜.
General Case
Brane Construction
In the low-energy limit, where the N S5 branes are taken to zero separation, which corresponds to → ∞ in the field theory. As usual, we will be interested in the origin of moduli space, where the D3-branes coincide.
Proof of Duality in Matrix Model
That is, by exchanging the variables λa and τa, the contributions of the brane N S5 and D5 are exchanged and we obtain the partition function of the. Thus, we see that the effect of the two deformations is identical: they move the argument inside the cosh to Iα.
Including (1, 1) Branes
T-Transformation
But in this case the duality is already trivial at the field theory level. Thus, we can only look at a relatively small subset of the set of SL(2,Z) brane configurations and actions, although, as noted above, see [46] for a generalization.
ABJM Theory
However, we have written a special form for superconformal transformations of matter fields, and although this form is necessary in theories with extended supersymmetry, it is not the only one compatible with supersymmetry N= 2. Then, by the general arguments of Chapter 2, this must also be the scaling dimension of scalars in chiral multiplets .
General R Charge as Coupling to Supersymmetric Vector Multiplet
One can check whether these give rise to the correct anticommutators, especially with the R charge. But now we see that this is exactly the transformation law we would obtain if we coupled to a background vector multiplet with σ =on and Aµ =bµ, where the fields have charge ˆ∆.
Trial R-Charge
So to find the supersymmetric actions for this chiral manifold, one simply plugs these forms of the background manifolds into the action 3.6. Additionally, in both cases one can in principle consider the possibility that the R charge mixes with any U(1)J topological symmetries that may be present.
F-Maximization
With this in mind, let us consider Z(ca), the partition function, as a function of the trial R-symmetry. Since ca appears in the action as an (imaginary) real mass, taking a derivative with respect to toca calculates the one-point function of the corresponding flavor charge.
F-Theorem
An important difference is that, in three dimensions, there is the new possibility of adding a Chern-Simons term to the gauge set, and we will see that this plays an interesting role in duality. We will test these dualities by comparing partition functions deformed by real masses and FI terms as in theN= 2 case, using the results of the previous chapter and the extended supersymmetry case, which will turn out to be a special case.
Aharony Duality
Mapping of Partition Functions
We want to show that these two expressions are equal for all complex values of the deformations. The duality thus appears to be completely independent of the question of the superconformal R-charge.
Giveon-Kutasov Duality
This signals that new symmetries must arise in the IR, possibly from some of the fields being decoupled with which the R symmetry can mix. a Chern-Simons expression, one finds a constant phase which can be interpreted as being due to the fact that we compute a Chern-Simons partition function using a non-standard frame of S3, as discussed in [59] . One can repeat this argument for k hypermultiplets embedded in a U(Nc) theory, and one finds precisely the two partition functions in the Giveon Kutasov duals.
Extended Supersymmetry
Using this convention, we will often suppress the spinor indices unless this would cause unnecessary confusion. When we write an equality that is only numerically true in the base we are using (that is, the index structure is not the same on both sides), we will use the symbol .
Superconformal Algebra
Flat Space
We recognize the form ofvµ as the most general conformal killing vector in flat space. With this in mind, let us define the operators Qα, associating the transformations δ with the action esQ+cS, and the conformal generators Pµ, Mµν, D, and Kµ by assigning them toaµ, bµν, c, and dµ, respectively.
Gauge Multiplet
In this appendix we review some fundamental properties of some differential operators on conformal maps. From above, we see that we can express the relation in this new basis in terms of the new Lie brackets, and these are calculated directly: where ∂i is the alongei derivative.
Curvature
Spinor Covariant Derivative
One can see that killing spinors maps to Killing spinors under a conformal transformation provided we assign them dimension ∆ =−12, for then, in the new metric:. In other words, 0 itself is a killer spinor, and the corresponding 00 is only proportional to the Ricci.
Scalars
Killing Spinors
Gauge Case
Now we are looking for an action of the form: for some real functionsai, biµ. This is possible for an arbitrary spinor on the RHS, provided it does not vanish, as will always be the case.
Supersymmetric Chern-Simons Theory
Computation of [δ, δ]
If γ 6= 1, one finds a contribution to [δ1, δ2]λ proportional to [σ, λ∗], and the variation of the action under this transformation vanishes due to the cyclicity of the trace. Witten, "SL(2,Z) Action on Three-Dimensional Conformal Field Theories with Abelian Symmetry," In *Shifman, M .
