Supplemental Material on Simple Superspaces William D. Linch iiië February, 2019 ë Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University. College Station, TX 77843 USA. Abstract This is a problem set on applications of 4D, N = 1 superspace to problems involving more supersymmetry, more dimensions, strings, and M-theory. ë [email protected] Contents 1 Four Dimensional, N = 1 \Simple" Superspace2 1.1 Flat Superspace D-Algebra............................2 1.2 Super-forms....................................4 1.3 Actions......................................6 1.4 Outlook......................................8 1 Four Dimensional, N = 1 \Simple" Superspace This is a problem set intended to speed up entry into research that relies on four- dimensional superspace. It should be considered to be supplementary material to proper treatments along the lines of those of the standard references. They are 1. Superspace by Gates, Grisaru, Roˇcek,and Siegel [1], henceforth GGRS. This is the field manual for superspace. It contains a \chapter 0" doing the whole book in 3D first where things are simpler, and then doing it all \for real" in 4D. As a bonus, it is on the arXiv as hep-th/0108200. 2. Ideas and Methods of Supersymmetry and Supergravity by Buchbinder and Kuzenko [2], henceforth B&K. This reference has been criticized as \Superspace rewritten so that people can understand it". So that makes it easier then right? Let's use it! It also has a few topics not treated in Superspace like free higher spin theories. Most importantly for us, it establishes the conventions we will be using, so its many explicit calculations and identities can be used without having to translate anything. 3. Supersymmetry and Supergravity by Wess and Bagger [3], henceforth W&B. This is the book of choice for most who want to use superspace for something and don't want to read 200 pages to get to the first Lagrangian. It also covers topics nicely that are buried somewhere deep in GGRS. It uses conventions that are very similar to B&K. A problem with it for us is that it does not provide all the tools needed for going beyond standard 4D, N = 1 superspace (e.g. it does not explain supergravity prepotentials). 1.1 Flat Superspace D-Algebra The workhorse for superspace is the spinorial covariant derivative (D ) = (D ; D¯ .) gen- α^ α . α erating infinitesimal translations in the odd directions coordinatized by (θα^) = (θα; θ¯α). They satisfy the defining rules fD ;D g = 0 ; fD ; D¯ .g = −2i@ ; fD¯ .; D¯ .g = 0 (1.1) α β α α a α β 2 where @ @ := (σm) . @ := (σm) . (1.2) a αα m αα @xm . Here σm for m = 0; 1; 2; 3 are the Pauli matrices, α; α = 1; 2 etc are the complex doublet indices of SL(2; C) = Spin(3; 1).1 Another invariant of this group is the antisymmetric symbols " = −" and " . = −" . .. They are both normalized so that " = 1. Together αβ βα αβ βα 12 they satisfy the all-important Fierz identity (σm) .η (σn) . = −2" " . (1.3) αα mn ββ αβ αβ (We are contracting bosonic coordinate indices with the Minkowski metric η = diag(−1; 1; 1; 1).) αβ βγ γ Since the "s are non-degenerate, we can define " so that "αβ" = δ , and similarly . α for "αβ. These are often called \spinor metrics" since they can be used to \raise and lower" spinor indices. Similarly to GR, if a spinor α is defined with the index down, we define α αβ the one with the index up as := " β. Unsimilarly to GR, this is not the same as βα βα αβ β" = " β = −" β! These signs and others will haunt your calculations. As our first example, we define the Pauli matrices with the indices up (~σm)αα := "αβ"αβ(σm) .. ββ We can now matrix-multiply to obtain the Clifford algebra multiplication rules m n n m β mn β (σ σ~ + σ σ~ )α = −2η δα . (~σmσn +σ ~nσm)α . = −2ηmnδα. (1.4) β β Problem 1. As another application of the spinor \metrics", note that (1.1) implies that DαDβ is antisymmetric. Show that DαDβDγ ≡ 0 as is the dotted version. Define . 2 α ¯ 2 ¯ . ¯ α D := D Dα D := DαD (1.5) and show that D D = 1 " D2 D¯ .D¯ . = − 1 " . .D¯ 2 (1.6) α β 2 αβ α β 2 αβ Problem 2. Using (1.1) show that . ¯ 2 ¯ α 2 ¯ . α [D ;Dα] = 4i@aD ; [D ; Dα] = −4i@aD (1.7a) . ¯ 2 ¯ . ¯ α 2 ¯ . α ¯ . fD ;Dαg = −2DαDαD ; fD ; Dαg = −2D DαDα (1.7b) . α ¯ 2 ¯ . 2 ¯ α D D Dα = DαD D (1.7c) 2 ¯ 2 a ¯ . 2 ¯ 2 α ¯ 2 [D ; D ] = −4i@ [Dα; Dα] ; fD ; D g = 2D D Dα + 162 (1.7d) D2D¯ 2D2 = 162D2 ; D¯ 2D2D¯ 2 = 162D¯ 2 (1.7e) 1 See §1.2.2 of B&K for more on this linear algebra. 3 Also of use is the silly observation that ( −2i@ − 2D¯ .D [D ; D¯ .] = a α α (1.8) α α ¯ . 2i@a + 2DαDα Congratulations! You are now in a position to understand any superfield representation of the 4D, N = 1 Poincar´esuperalgebra! Tune in next time to find out how! 1.2 Super-forms Scalar Multiplet In the lecture, we saw how to decompose a generic superfield into (anti- )chiral and linear parts by interpreting the second relation in (1.7d) as a statement about projection operators in superspace. The chiral scalar superfield satisfies the covariant con- straint ¯ . ¯ DαΦ(x; θ; θ) = 0 (1.9) by definition. By the superspace Poincar´elemma, we can solve this condition as Φ = D¯ 2' for an unconstrained complex scalar field ', although we will not usually need to do so explicitly. Since this field is chiral, the only covariant components of it were 1 2 φ(x) := Φj ; α(x) := DαΦj ;F (x) := − 4 D Φj (1.10) a statement we summarize schematically as α 2 Φ ∼ φ + θ α + θ F (1.11) Vector Multiplet Chiral superfields made a reappearance as the gauge parameter of the real scalar gauge superfield ¯ 1 ¯ ¯ . V = V ; δV = 2i Λ − Λ ; DαΛ = 0 (1.12) 2 Using the components Im(Λj), DαΛj, and D Λ, you should check that the only physical components are (again, schematically) a ¯ ¯2 α ¯2 2 V ∼ · · · + (θσ θ)Aa + θ θ λα + h.c. + θ θ D (1.13) In this gauge, called Wess-Zumino gauge, the elided terms are set to 0 algebraically. Problem 3. Define the component fields by 1 ¯ . 1 ¯ 2 1 2 ¯ 2 Aa(x) := 2 [Dα; Dα]V j ; λα(x) := − 4 D DαV j ;D(x) := 32 fD ; D gV j : (1.14) Show that the usual U(1) gauge parameter is the remaining part of Λ and that the \photino" λα and auxiliary field D are gauge-invariant. 4 Note that in this problem you have come across a result we saw in lecture that the superfield . 1 ¯ 2 ¯ . α ¯ . ¯ α Wα := − 4 D DαV , DαWα = 0 D Wα = DαW (1.15) is gauge-invariant. (In fact, the photino is the lowest component of this superfield.) Problem 4. Consider the Maxwell tensor Fab = @aAb − @bAa = −Fba. Contracting with Pauli matrices show that F = (σa) .(σb) .F = " . .f + " f¯. (1.16) ab αα ββ ab αβ αβ αβ αβ for some symmetric tensor fαβ = fβα. Find fαβ in terms of Fab. Define the Hodge dual ~ 1 cd Fab = 2 ab Fcd and show that F~ = i" . .f − i" f¯. (1.17) ab αβ αβ αβ αβ i Finally, show that fαβ = − 2 D(αWβ)j and compute the expression for this in terms of V . The Maxwell tensor satisfies the source-free Maxwell equations dF = 0 , F = dA or abcd @bFcd = 0 , Fab = 2@[aAb] (1.18) We will refer to these as Bianchi identities to distinguish them from the sourced Maxwell b equations (to which we will refer simply as \Maxwell equations") @ Fab = ja. In any case, we can solve the Bianchi identities on Fab in terms of the unconstrained potential Aa. In the super-analog, we solve the identities on the RHS of (1.15) for Wα for the unconstrained superfield V on the LHS. The latter is more basic than the vector potential in that it has a gauge symmetry that we used to get to Wess-Zumino gauge, that does not even exist for the component potential. For this reason the field V is called a prepotential for the vector multiplet. (Similarly the solution ' to the chiral constraint (1.9) is a prepotential.) Tensor Multiplet The chiral multiplet appeared as the gauge parameter for the vector multiplet. Similarly, we expect the vector multiplet to appear as the gauge parameter for a would-be 2-form multiplet, colloquially \tensor multiplet". ¯ . Problem 5. Show that a chiral spinor field Σα (i.e. DαΣα = 0) with gauge transformation 1 ¯ 2 ¯ δΣα = − 4 D DαU s.t. U = U (1.19) describes a gauge 2-form δBab = 2@[aΛb] by studying the gauge transformations of components and the Wess-Zumino gauge for them. Define the real superfield . 1 α ¯ . ¯ α H = 2i D Σα − DαΣ (1.20) 5 show that it is gauge invariant, and show that the Bianchi identity is D¯ 2H = 0 (i.e. it is a real, linear superfield). What is the component structure of this field strength? Use these last two results and the first identity in (1.7d) to show that the vector component, call it H~ a, ~ a ~ a 1 abcd is conserved, that is @aH = 0. Convince yourself that this implies H = 3! Hbcd for a closed 3-form Habc.