Supercurrents. Nathan Seiberg IAS

Transcription

1 Supercurrents Nathan Seiberg IAS 2011 Zohar Komargodski and NS arxiv: , arxiv: Tom Banks and NS arxiv: Thomas T. Dumitrescu and NS arxiv:

2 Summary The supersymmetry algebra can have brane charges. Depending on the brane charges, there are different supercurrent multiplets. 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. Different supermultiplets are associated with different off-shell supergravities. (They might be equivalent on shell.) Understanding the multiplets leads to constraints on supergravity and string constructions. 1

3 The 4d N = 1 SUSY Algebra (imprecise) [Ferrara, Porrati; Gorsky, Shifman]: {Q α, Q α } = 2σ µ α α (P µ + Z µ ), {Q α, Q β } = σ µν αβ Z µν. Z µ is a string charge. Z [µν] is a complex domain wall charge. They are infinite proportional to the volume. They are not central. They control the tension of BPS branes. Algebraically P µ and Z µ seem identical. But they are distinct... 2

4 The 4d N = 1 SUSY Current Algebra {Q α, S αµ } = 2σ ν α α (T νµ + C νµ ) +, {Q β, S αµ } = σ νρ αβ C νρµ. T µν and S αµ are the energy momentum tensor and the supersymmetry current. C [µν], C [µνρ] are conserved currents associated with strings and domain-walls. Their corresponding charges are Z µ, Z [µν] above. Note that T {µν} and C [µν] are distinct. 3

5 Properties of the Supercurrent Multiplet T µν is conserved and symmetric. It is subject to improvement (actually more general) T µν T µν + ( µ ν η µν 2) u. S αµ is conserved. It is subject to improvement S αµ S αµ + (σ µν ) α β ν η β. We impose that T µν is the highest spin operator in the multiplet. We consider only well-defined (gauge invariant) local operators. 4

6 The S-Multiplet The most general supercurrent satisfying our requirements is the S-multiplet S α α (real) D α S α α = χ α + Y α, D α χ α = 0, D α χ α = D α χ α, D α Y α + D β Y α = 0, D 2 Y α = 0. Equivalently, D α S α α = D 2 D α V + D α X, D α X = 0, V = V but V and X do not have to be well defined. 5

7 Components the S-Multiplet D α S α α = χ α + Y α S µ = j µ + θ α S αµ + θ α S α µ + (θσ ν θ)t µν +... It includes operators: Energy momentum tensor T µν (10-4 = 6) String current ɛ µνρσ F ρσ in χ α = D 2 D α V (3) A complex domain wall current ɛ µνρσ σ x in Y α = D α X (2) A non-conserved R-current j µ (4) A real scalar (1) 16 fermionic operators 6

8 Improvements of the S-Multiplet The S-multiplet is not unique. The defining equation D α S α α = χ α + Y α is invariant under the transformation S α α S α α + [D α, D α ]U, χ α χ α D2 D α U, Y α Y α D αd 2 U, with real U (well-defined up to an additive constant). This changes S αµ, T µν, the string and domain wall currents by improvement terms. 7

9 Special Cases D α S α α = χ α + Y α In special cases the S-multiplet is decomposable: 8

10 Special Cases D α S α α = χ α + Y α In special cases the S-multiplet is decomposable: If χ α = 3 2 D2 D α U 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

11 Special Cases D α S α α = χ α + Y α In special cases the S-multiplet is decomposable: If χ α = 3 2 D2 D α U 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). If Y α = D α X = 1 2 D αd 2 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

12 Special Cases D α S α α = χ α + Y α In special cases the S-multiplet is decomposable: If χ α = 3 2 D2 D α U 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). If Y α = D α X = 1 2 D αd 2 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). If both are true with the same U, we can set χ α = Y α = 0. This happens when the theory is superconformal. 8

13 The Ferrara-Zumino (FZ) Multiplet When χ α = 3 2 D2 D α U we find the most familiar supercurrent the FZ-multiplet D α J α α = D α X, D α X = 0. It contains independent real operators: j µ (4), T µν (6), x (2) and S αµ (12). It exists only if there are no string currents it does not exist if there are FI-terms ζ or if the Kähler form is not exact. Nontrivial C µν ζɛ µνρσ F ρσ C µν iɛ µνρσ K ii ρ φ i σ φ i are obstructions to its existence. 9

14 The R-Multiplet When Y α = D α X = 1 2 D αd 2 U we find the R-multiplet D α R α α = χ α, D α χ α = 0, D α χ α = D α χ α. This multiplet includes operators. Among them is a string current. But there is no domain wall current. j µ = R µ is a conserved R-current the theory has a U(1) R symmetry. This multiplet exists even when the Kähler form is not exact or the theory has FI-terms. S αµ, T µν differ from those in the FZ-multiplet by improvement terms. 10

15 Example: Wess-Zumino Models Every theory has an S-multiplet. Example: a WZ theory S α α = 2K ij D α Φ i D α Φ j, χ α = D 2 D α K, X = 4W. All the operators are globally well defined. Since X has to be well defined up to adding a constant, we can allow multi-valued W. 11

16 Example: Wess-Zumino Models If the Kähler form is exact, there are no strings and the S-multiplet can be improved to the FZ-multiplet J α α = 2K ij D α Φ i D α Φ j 2 3 [D α, D α ]K, X = 4W 1 3 D2 K. 12

17 Example: Wess-Zumino Models If the Kähler form is exact, there are no strings and the S-multiplet can be improved to the FZ-multiplet J α α = 2K ij D α Φ i D α Φ j 2 3 [D α, D α ]K, X = 4W 1 3 D2 K. If the theory has an R-symmetry, the S-multiplet can be improved to the R-multiplet (even when the Kähler form is not exact) R α α = 2K ij D α Φ i D α Φ j [D α, D α ] i ( ) χ α = D 2 D α K 3 R i Φ i i K, 2 i R i Φ i i K, 12

18 Constraints on RG-Flow 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: No string charge in the SUSY algebra No FI-terms, even for emergent gauge fields (previous arguments by [Shifman, Vainshtein; Dine; Weinberg]). The Kähler form of the target space is exact (previous argument by [Witten]). 13

19 Constraints on RG-Flow 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: No string charge in the SUSY algebra No FI-terms, even for emergent gauge fields (previous arguments by [Shifman, Vainshtein; Dine; Weinberg]). The Kähler form of the target space is exact (previous argument by [Witten]). 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. 13

20 The S-Multiplet in 3d The S-multiplet for N = 2 in 3d is given by D β S αβ = χ α + Y α, D α χ β = 1 2 Cε αβ, D α χ α = D α χ α, D α Y β + D β Y α = 0, D α Y α = C, where C is a complex constant. It leads to a new term in the SUSY current algebra: {Q α, S βµ } = 1 4 Cγ µαβ +. We interpret it as a space-filling brane current (not affected by improvements). This is consistent with dimensional reduction from 4d. 14

21 Application: Partial SUSY-Breaking If C 0, the vacuum preserves at most two of the four supercharges. SUSY can be partially broken from N = 2 to N = 1. It happens because of a deformation of the current algebra [Hughes, Polchinski]. This is fundamentally different from spontaneous breaking, where the current algebra is not modified. The nature of the multiplet and the value of C are determined in the UV. Hence, if C = 0 in the UV (e.g. in conventional SUSY gauge theories), there cannot be partial SUSY breaking. 15

22 Partial SUSY-Breaking Other places with the same phenomenon: In the 2d N = (0, 2) CP 1 model instantons generate nonzero C (earlier work by [Witten; Tan, Yagi]). N = (2, 2) in 2d Three different space-filling brane currents can break (2, 2) (1, 1), (2, 0), or (0, 2). Simple models realize these possibilities [Hughes, Polchinski; Losev, Shifman]. N = 2 N = 1 breaking in 4d [Hughes, Liu, Polchinski; Antoniadis, Partouche, Taylor; Ferrara, Girardello, Porrati]. 16

23 Constraints on Linearized SUGRA Linearized SUGRA is obtained by adding to the flat space Lagrangian L flat space + d 4 θh µ S µ + O(H 2 ) where S µ is the supercurrent and H µ is a superfield containing the deformation of the metric. 17

24 Constraints on Linearized SUGRA Linearized SUGRA is obtained by adding to the flat space Lagrangian L flat space + d 4 θh µ S µ + O(H 2 ) where S µ is the supercurrent and H µ is a superfield containing the deformation of the metric. Standard SUGRA ( old-minimal SUGRA ) uses the FZ-multiplet L =L flat space + h µν T µν + ψ µα S µα + ψ µ α S µ α + b µ j µ + Mx + Mx +... It exists only when the FZ-multiplet exists; i.e. when the Kähler form is exact [Witten and Bagger] and when there is no FI-term. 17

25 Constraints on Linearized SUGRA If the theory has a global U(1) R symmetry we can use new-minimal SUGRA, which is based on the R-multiplet. On shell it is equivalent to the old-minimal formalism. Even though the U(1) R symmetry of the matter theory is gauged, the resulting theory has a global U(1) R symmetry. 18

26 Constraints on Linearized SUGRA If the theory has a global U(1) R symmetry we can use new-minimal SUGRA, which is based on the R-multiplet. On shell it is equivalent to the old-minimal formalism. Even though the U(1) R symmetry of the matter theory is gauged, the resulting theory has a global U(1) R symmetry. We conclude that theories without an FZ-multiplet can be coupled to SUGRA only if they have a global U(1) R symmetry. This is consistent with earlier work of [Freedman; Barbieri, Ferrara, Nanopoulos, Stelle; Kallosh, Kofman, Linde, Van Proeyen]. 18

27 Constraints on Linearized SUGRA If the theory has a global U(1) R symmetry we can use new-minimal SUGRA, which is based on the R-multiplet. On shell it is equivalent to the old-minimal formalism. Even though the U(1) R symmetry of the matter theory is gauged, the resulting theory has a global U(1) R symmetry. We conclude that theories without an FZ-multiplet can be coupled to SUGRA only if they have a global U(1) R symmetry. This is consistent with earlier work of [Freedman; Barbieri, Ferrara, Nanopoulos, Stelle; Kallosh, Kofman, Linde, Van Proeyen]. Excluding gravity theories with global symmetries, such models are not acceptable. This constrains many supergravity and string constructions. 18

28 Constraints on Linearized SUGRA A rigid theory without an FZ-multiplet and without a U(1) R symmetry can be coupled to linearized SUGRA using the S-multiplet. 19

29 Constraints on Linearized SUGRA A rigid theory without an FZ-multiplet and without a U(1) R symmetry can be coupled to linearized SUGRA using the S-multiplet. The resulting SUGRA (16/16 SUGRA) has more degrees of freedom. One way to think about it is to add to the matter system another propagating chiral superfield such that it has an FZ-multiplet and then use the standard formalism [Siegel]. This is familiar from heterotic compactifications, where the additional propagating degrees of freedom are the dilaton, the dilatino and the two-form B. 19

30 Conclusions The supersymmetry current and the energy-momentum tensor are embedded in a supermultiplet. The S-multiplet is the most general supercurrent multiplet. It has components. It always exists. In special situations it is decomposable can be improved to a smaller multiplet. The most common supercurrent is the FZ-multiplet. It has components. It exists when there are no string charges in the SUSY algebra. This happens when the Kähler form is exact and there are no FI-terms. 20

31 Conclusions If the theory has a U(1) R symmetry it has an R-multiplet It has components. It does not admit domain wall charges in the algebra. 21

32 Conclusions If the theory has a U(1) R symmetry it has an R-multiplet It has components. It does not admit domain wall charges in the algebra. This discussion constrains the dynamics: If the UV theory has an FZ-multiplet, the low-energy theory has an exact Kähler form and no FI-terms it does not have string charges in the SUSY algebra. If the theory has a U(1)R symmetry, it has an R-multiplet and then it does not have charged domain walls. Space-filling brane currents give rise to partial SUSY breaking (fundamentally different from spontaneous breaking). If the corresponding current is not present in the UV, SUSY cannot be partially broken. 21

33 Conclusions This also constrains linearized supergravity Only theories with an FZ-multiplet can be coupled to old-minimal supergravity. Theories with nontrivial Kähler form or FI-terms can be coupled to new-minimal supergravity, but then they must have a continuous global U(1) R symmetry. More general theories can be coupled through their S-multiplet. But the resulting theory has more degrees of freedom. Constraints on SUGRA/string constructions. There conclusions are limited to linearized supergravity. Additional consistent possibilities exist in intrinsic gravitational theories with Planck size coupling constants [Witten and Bagger; NS]. 22