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Supercurrents Supercurrents Nathan Seiberg IAS 2011 Zohar Komargodski and NS arXiv:0904.1159, arXiv:1002.2228 Tom Banks and NS arXiv:1011.5120 Thomas T. Dumitrescu and NS arXiv:1106.0031 Summary I The supersymmetry algebra can have brane charges. I Depending on the brane charges, there are different supercurrent multiplets. I The nature of the multiplet is determined in the UV but is valid also in the IR. This leads to exact results about the renormalization group flow. I Different supermultiplets are associated with different off-shell supergravities. (They might be equivalent on shell.) I Understanding the multiplets leads to constraints on supergravity and string constructions. 1 The 4d N = 1 SUSY Algebra (imprecise) [Ferrara, Porrati; Gorsky, Shifman]: µ fQα_ ;Qαg = 2σαα_ (Pµ + Zµ) ; µν fQα;Qβg = σαβZµν : I Zµ is a string charge. I Z[µν] is a complex domain wall charge. I They are infinite { proportional to the volume. I They are not central. I They control the tension of BPS branes. I Algebraically Pµ and Zµ seem identical. But they are distinct... 2 The 4d N = 1 SUSY Current Algebra ν fQα_ ;Sαµg = 2σαα_ (Tνµ + Cνµ) + ··· ; νρ fQβ;Sαµg = σαβCνρµ : I Tµν and Sαµ are the energy momentum tensor and the supersymmetry current. I C[µν], C[µνρ] are conserved currents associated with strings and domain-walls. Their corresponding charges are Zµ, Z[µν] above. I Note that Tfµνg and C[µν] are distinct. 3 Properties of the Supercurrent Multiplet I Tµν is conserved and symmetric. It is subject to improvement (actually more general) 2 Tµν ! Tµν + @µ@ν − ηµν@ u : I Sαµ is conserved. It is subject to improvement β ν Sαµ ! Sαµ + (σµν)α @ ηβ : I We impose that Tµν is the highest spin operator in the multiplet. I We consider only well-defined (gauge invariant) local operators. 4 The S-Multiplet The most general supercurrent satisfying our requirements is the S-multiplet Sαα_ (real) α_ D Sαα_ = χα + Yα ; α α_ Dα_ χα = 0 ;D χα = Dα_ χ ; 2 DαYα + DβYα = 0 ; D Yα = 0 : Equivalently, α_ 2 D Sαα_ = D DαV + DαX; y Dα_ X = 0 ;V = V but V and X do not have to be well defined. 5 Components the S-Multiplet α_ D Sαα_ = χα + Yα α α_ ν Sµ = jµ + θ Sαµ + θα_ Sµ + (θσ θ)Tµν + ::: It includes 16+16 operators: I Energy momentum tensor Tµν (10 - 4 = 6) ρσ 2 I String current µνρσF in χα = D DαV (3) σ I A complex domain wall current µνρσ@ x in Yα = DαX (2) I A non-conserved R-current jµ (4) I A real scalar (1) I 16 fermionic operators 6 Improvements of the S-Multiplet The S-multiplet is not unique. The defining equation α_ D Sαα_ = χα + Yα is invariant under the transformation Sαα_ !Sαα_ + [Dα; Dα_ ]U; 3 2 χ ! χ + D D U; α α 2 α 1 2 Y !Y + D D U; α α 2 α with real U (well-defined up to an additive constant). This changes Sαµ, Tµν, the string and domain wall currents by improvement terms. 7 3 2 I If χα = 2 D DαU with a well defined U, we can set it to zero. This happens when there is no string charge (the string current can be improved to zero). Then, the S-multiplet is decomposed to a real (vector superfield) U and the Ferrara-Zumino multiplet (below). 1 2 I If Yα = DαX = 2 DαD U with a well defined U, we can set it to zero. This happens when there is no domain wall charge. Here the S-multiplet is decomposed to a real (vector superfield) U and the R-multiplet (below). I If both are true with the same U, we can set χα = Yα = 0. This happens when the theory is superconformal. Special Cases α_ D Sαα_ = χα + Yα In special cases the S-multiplet is decomposable: 8 1 2 I If Yα = DαX = 2 DαD U with a well defined U, we can set it to zero. This happens when there is no domain wall charge. Here the S-multiplet is decomposed to a real (vector superfield) U and the R-multiplet (below). I If both are true with the same U, we can set χα = Yα = 0. This happens when the theory is superconformal. Special Cases α_ D Sαα_ = χα + Yα In special cases the S-multiplet is decomposable: 3 2 I If χα = 2 D DαU with a well defined U, we can set it to zero. This happens when there is no string charge (the string current can be improved to zero). Then, the S-multiplet is decomposed to a real (vector superfield) U and the Ferrara-Zumino multiplet (below). 8 I If both are true with the same U, we can set χα = Yα = 0. This happens when the theory is superconformal. Special Cases α_ D Sαα_ = χα + Yα In special cases the S-multiplet is decomposable: 3 2 I If χα = 2 D DαU with a well defined U, we can set it to zero. This happens when there is no string charge (the string current can be improved to zero). Then, the S-multiplet is decomposed to a real (vector superfield) U and the Ferrara-Zumino multiplet (below). 1 2 I If Yα = DαX = 2 DαD U with a well defined U, we can set it to zero. This happens when there is no domain wall charge. Here the S-multiplet is decomposed to a real (vector superfield) U and the R-multiplet (below). 8 Special Cases α_ D Sαα_ = χα + Yα In special cases the S-multiplet is decomposable: 3 2 I If χα = 2 D DαU with a well defined U, we can set it to zero. This happens when there is no string charge (the string current can be improved to zero). Then, the S-multiplet is decomposed to a real (vector superfield) U and the Ferrara-Zumino multiplet (below). 1 2 I If Yα = DαX = 2 DαD U with a well defined U, we can set it to zero. This happens when there is no domain wall charge. Here the S-multiplet is decomposed to a real (vector superfield) U and the R-multiplet (below). I If both are true with the same U, we can set χα = Yα = 0. This happens when the theory is superconformal. 8 The Ferrara-Zumino (FZ) Multiplet 3 2 When χα = 2 D DαU we find the most familiar supercurrent { the FZ-multiplet α_ D Jαα_ = DαX; Dα_ X = 0 : I It contains 12+12 independent real operators: jµ (4), Tµν (6), x (2) and Sαµ (12). I It exists only if there are no string currents { it does not exist if there are FI-terms ζ or if the K¨ahlerform is not exact. Nontrivial ρσ Cµν ∼ ζµνρσF ρ i σ i Cµν ∼ iµνρσKii@ φ @ φ are obstructions to its existence. 9 The R-Multiplet 1 2 When Yα = DαX = 2 DαD U we find the R-multiplet α_ D Rαα_ = χα ; α α_ Dα_ χα = 0 ;D χα = Dα_ χ : I This multiplet includes 12+12 operators. Among them is a string current. But there is no domain wall current. I jµ = Rµ is a conserved R-current { the theory has a U(1)R symmetry. I This multiplet exists even when the K¨ahlerform is not exact or the theory has FI-terms. I Sαµ;T µν differ from those in the FZ-multiplet by improvement terms. 10 Example: Wess-Zumino Models Every theory has an S-multiplet. Example: a WZ theory i j Sαα_ = 2KijDαΦ Dα_ Φ ; 2 χα = D DαK; X = 4W: I All the operators are globally well defined. I Since X has to be well defined up to adding a constant, we can allow multi-valued W . 11 I If the theory has an R-symmetry, the S-multiplet can be improved to the R-multiplet (even when the K¨ahlerform is not exact) i j X i Rαα_ = 2KijDαΦ Dα_ Φ − [Dα; Dα_ ] RiΦ @iK; i ! 2 3 X χ = D D K − R Φi@ K ; α α 2 i i i Example: Wess-Zumino Models I If the K¨ahlerform is exact, there are no strings and the S-multiplet can be improved to the FZ-multiplet j 2 J = 2K D ΦiD Φ − [D ; D ]K; αα_ ij α α_ 3 α α_ 1 2 X = 4W − D K: 3 12 Example: Wess-Zumino Models I If the K¨ahlerform is exact, there are no strings and the S-multiplet can be improved to the FZ-multiplet j 2 J = 2K D ΦiD Φ − [D ; D ]K; αα_ ij α α_ 3 α α_ 1 2 X = 4W − D K: 3 I If the theory has an R-symmetry, the S-multiplet can be improved to the R-multiplet (even when the K¨ahlerform is not exact) i j X i Rαα_ = 2KijDαΦ Dα_ Φ − [Dα; Dα_ ] RiΦ @iK; i ! 2 3 X χ = D D K − R Φi@ K ; α α 2 i i i 12 I Consider a SUSY field theory with a U(1)R symmetry. It has an R-multiplet at every energy scale. Hence, there are no domain wall charges in the supersymmetry algebra and in particular, no BPS domain walls. Constraints on RG-Flow I Consider a SUSY field theory, which has an FZ-multiplet in the UV (e.g. a gauge theory without FI-terms). Hence, the FZ-multiplet exists at every energy scale. This constrains the low-energy theory: I No string charge in the SUSY algebra I No FI-terms, even for emergent gauge fields (previous arguments by [Shifman, Vainshtein; Dine; Weinberg]).
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