Weyl Anomalies and D-brane Charges

Transcription

1 Weyl Anomalies and D-brane Charges Constantin Bachas 9th Crete Regional Meeting in String Theory Kolymbari, July

2 An elegant scientist and a very kind person whose memory lives also through this series of meetings

3 Introduction This talk is about an (analytic) extension of a beautiful paper by Gomis, Komargodski, Hsin, Schwimmer, Seiberg, Theisen ( ) These authors show that the geometry (Kähler potential) of CY moduli space M CY is related to Weyl anomalies, and that it can be computed from the sphere partition function. Gromov-Witten invariants This opens a new line of attack to a time-honoured problem, without relying on mirror symmetry. It also motivates a a striking conjecture: that M CY has zero Kähler class

4 With Daniel Plencner we generalized the Gomis et al anomaly to manifolds with boundary arxiv: This shows in particular that the hemisphere partition function computes the other piece of geometric data besides K(, ) : c ( ) The central charge, and the mass of CY D-branes M = c 2 e K

5 Few reminders and earlier work: CY moduli space factorizes locally: M c M tc h 2,1 complex structure (c,c) h 1,1 Kähler moduli (c,a) chiral twisted-chiral fields of N=(2,2) σ-model

6 Studied extensively for 30 years. Strong constraints from N =2 supersymmetry of of type-ii string theory compactified on CY3: Metric on complex-structure m.s. is classical but on Kähler m.s. it has instanton corrections Gromov-Witten invariants Assuming mirror symmetry, gives the latter from the former when mirror manifold and map is known. But usually it is not.

7 Recent progress came from calculations of partition functions of N=(2,2) GLSM using susy localization sphere hemisphere Benini, Cremonesi Doroud, Gomis, Le Floch, Lee Honda + Okuda Hori + Romo Sugishita + Terashima It was conjectured (and checked in examples) that: Jockers, Kumar, Lapan, Morrison, Romo ( ) (S 2 )= r r 0 c/3 e K(, ) (S 2 / 2 )= r r 0 c/6 c ( )

8 An argument based on tt* eqns was given by Gomis + Lee ( ) The proof of Gomis, Komargodski, Hsin, Schwimmer, Seiberg, Theisen is more elegant and powerful. It is based on the N=2 supersymmetric completion of a Weyl anomaly first discovered by Osborn 91 In the rest of this talk I will review this work of Gomis et al, then extend it to manifolds with boundary in 2D.

9 Bulk super-weyl anomaly Anomalies : non-conservation in correlation functions due to contact terms h@ µ j µ O 1 (p 1 ) O n (p n )i6=0 When r.h.s. is proportional to momenta: non-conservation only in presence of spacetime-dependent background µ j µ A e.g. µ j µ A = F ^ F G G axial charge violated by instantons

10 For chiral anomalies: background is gauge or gravitational For trace (Weyl) anomaly, can be exactly-marginal couplings: Osborn 91 Osborn, Petkou Bzowski, McFadden, Skenderis In 2D the 2-point function of marginal operators reads: amolodchikov metric anomaly ho I (z)ō J(w)i = g I J R 1 z w 4 = g I J log( z w 2 µ 2 ) 2 differential regularization of distribution

11 I Turn on space-dependent logµ z w I logµ ho I(z)Ō µ µ Jg I J where log z w (2) (z w) 9 NO anomaly for constant couplings. But supersymmetry relates it to a term that does not vanish µ I =0 Gomis et al ( ) This term can be removed by non-susy local counterterm; but SUSY gives it universal meaning

12 Technical details: N =(2, 2) SCFTs have U(1) V U(1) A R-symmetry. In computing the anomaly we choose to preserve the vector-like symmetry, so we must couple it to the N =2 supergravity in which this symmetry is gauged by a field V µ Closset + Cremonesi ( ) In superconformal gauge: g µ = e 2 µ, V µ = a Classically and a decouple, but in the quantum theory they dont due to the Weyl and axial anomalies.

13 Supersymmetry puts these fields in a twisted-chiral multiplet (y µ )=( + ia) w whose components are functions of y ± = x ± i ± ± The tc field obeys where D ± = D + = ± i ±, D ± + i ±. It is useful to also promote the marginal couplings to vevs of tc fields I = I (y ± )+, I = I (ȳ ± )+ Seiberg so as to make the susy of the anomaly manifest.

14 The bulk anomaly ia( ) := log V (M) is the susy invariant A closed := A (1) + A (2) = 1 4 M d 2 x d 4 h c 6 ( + ) ( + )K(, )i Gomis et al ( ) This obeys Wess-umino consistency A( 0 ) 0 A( ) =0 and can be integrated with the result: log V i 4 M d 2 x d 4 h c 6 ( + )Ki. super-liouville super-osborn

15 Expand in components: A (1) = c 12 M h d 2 x + a a + 1 i 2 ( w w + ww)+@µ b (1) µ + fermions, A (2) = 1 2 M h d 2 x (@ I µ J)@ JK 1 i 2 K (@µ a)k µ µ b (2) µ where K µ := i 2 (@ IK@ µ Ī K@ µ Ī) Kähler one-form (Cohomologically) non-trivial, real anomalies Variation of local invariant counterterm pgr (2) K(, )

16 The first term in A (2) is the scale anomaly in the 2-point function as follows from = log z 2 = (2) (z) JK = g I J contact term The non-vanishing term for constant couplings is the red one It could be removed by change of scheme in bosonic theory, but supersymmetry relates it to the non-trivial blue terms! Similar remarks for 4D Casimir energy Assel, Cassani, Di Pietro, Komargodski, Lorenzen, Martelli

17 Implications Integrating the anomaly for constant couplings gives S 2 K = 4 K =) E V (S 2 )= r r 0 c/3 e K(, ) so the 2-sphere free energy computes the Kähler potential on the SCFT2 moduli space (both chiral and twisted chiral) A puzzle E V (S 2 ) not invariant under Kähler-Weyl transformations K 0 (, ) =K(, )+H( )+ H( )

18 Resolution = The variation amounts to change of renormalization scheme: KWA (2) = 1 d 2 x d 4 ( + 4 )H + c.c. 1 4 M d 2 x M d + d ( D + D )H + M d 2 x (@ µ Y µ )+c.c. twisted F-term R = D + D = w ( ia)+ curvature superfield So local, susy and diffeo-invariant counterterm compensates the Kähler-Weyl (gauge) transformation!

19 An interesting conjecture Gomis et al ( ) If the moduli space had non-vanishing Kähler class one could S 2! M pick I (x) such that is non-trivial 2-cycle Then there would be no global renormalization scheme and no well-defined generating function Way out: Moduli space has Kähler class = 0 cf Morrison, Pleser in progress

20 Boundary anomaly Consider half space: x 1 apple 0 x 1 0 conformal boundary condition One-point functions of marginal operators: ho I (x)i = d I R 1 x 1 2 = d 2 1 [ ( x 1 ) log x 1 µ ]

21 Focus on B-type branes which are not obstructing Kahler deformations Take region of Kähler moduli space with no walls of marginal stability. The 1pt-function coefficients are related to a holomorphic boundary charge c ( ) 4d I = c I c I(K + log c ) Ooguri, Oz, Yin 96 For the mirror A-type branes c = Lag (3,0)

22 Argument: vacuum projection of boundary state vac ii := c 0i RR + X I c I Ii RR is flat section of the improved connection r C on moduli space structure constants of chiral ring

23 Our result: prove these relations from Weyl-Osborn anomaly, and show that hemisphere p.f. computes bnry charge + (D 2 )= r r 0 c/6 c ( ), (D 2 )= r r 0 c/6 c ( ). Under Kähler Weyl transformations c! c e F The boundary entropy is the scheme-independent combination g = c e K/2 = s + (D 2 ) (D 2 ) (S 2 ) D-brane mass

24 In string-theory compactifications, g and c I are the mass and RR charges of the 1/2 BPS D-brane states dyons in field-theory limits These are related to worldsheet anomalies!

25 Technical details: 3 steps in calculation: Take into account the divergence terms in A closed b (1) µ = 1 4 (@ µ ) 3 µ (@ µ a)a 3 4 µa b (2) µ = 1 4 (@ µ )K 1 µ K. Add `minimal boundary term needed for susy Extra boundary-superinvariant additions using formalism of boundary superspace

26 Consider the D-term : Reference boundary completion d 2 x M d 4 S = M d 2 x [S] top top component The type-b susy generator is D susy = (Q + + Q ) ( Q + + Q ) where Q ± + i ±, Q ± i ± The transformation of the D-term is a total derivative susy[s] top = d 4 D susy S = i d 4 ( + S S) + c.c. We want to write as the susy transformation of a boundary term.

27 Standard manipulations give: susy[s] top = susy(@ 1 [S] bnry )+@ 0 Y with [S] bnry = i 2 S + 1 S + 1 S ; so that I D (S) := d 2 x [S] top + dx 0 [S] bnry is our susy-invariant standard completion. For the case of interest, the integrated superfield is with S = 1 4 h c 6 ( + )Ki S

28 Boundary superspace Hori (hep-th/ ) x + = x, e i + = e i, e i + = e i Restrictions of bulk superfields, = + ia [w i@ 1 ( + ia)] Usual D-term and F-term integrals of bnry superfields are invariant Brunner + Hori (hep-th/ ) W-consistency, locality and parity covariance leads to ansatz for boundary-superinvariant contribution to anomaly: dx 0 [B] where B = i 8 h # c 12 ( 2 2 )+ G (, ) i G (, and reality condition G (, ) =[G (, )]?

29 Collecting everything: A open = M d 2 x [ S] top dx 0 ([ S] bnry +[ B] ) where S = 1 4 h c 6 ( + )Ki [S] bnry = i 2 S + 1 S + 1 S ; B = i 8 h # c 12 ( 2 2 )+ G (, ) i G (, central-charge anomaly Weyl-Osborn anomaly cf Polchinski; Solodukhin for higher D

30 Susy Ward identity: h L sugra L SCFT i =0 if = I =0 =) no terms propto I =) G (, ) =K(, ) + 2 log c ( ) Kähler-Weyl covariance (up to local counterterms) requires c := e h section of holomorphic line bundle K! K + H + H h! h H

31 final ingredient: susy hemisphere.... Seiberg, Festuccia A open n 1 4 d 2 x h ( ia)h + ( + ia) h i + i 4 dx 0 h wh w h io integrated anomaly subtracted so as to vanish for infinitesimal disk depends only the holomorphic boundary charge, plus the auxiliary field of the metric.

32 Killing-spinor equations imply w = 2i z( + ia + log )= 2i z( + ia + log + ), w = 2i z( ia + log + )= 2i z( ia + log ). where the unbroken superconformal symmetries are + = (z), = + ( z), + = (z), = + (z) Two solutions for hemisphere with B-type bnry condition: (+) : =1, + = z, = z, + =1, ( ): = z, + = 1, =1, + = z

33 Supersymmetric hemispheres with B-type bnry condition: = log(1 + z z) + constant, a =0 (+) : w = w = 2i 1+z z, ( ): w = w = 2i 1+z z which implies + (D 2, ) = 0 c ( ), (D 2, ) = 0 c ( ). qed

34 5. Summary + outlook Computed the super-weyl anomaly for N =(2, 2) models on a surface with boundary generalizing the result of Gomis, Komargodski, Hsin, Schwimmer, Seiberg, Theisen ( ) Not only the Kähler potential but also the brane charges & mass are given by an (`Osborn-type ) anomaly. They can be computed by localization of the hemisphere partition function

35 Argument easily extended to sphere partition function with moduli-changing interface 1 2 C I = e K( 1, 2) 2 log g I = K( 1, 1)+K( 2, 2) K( 1, 2) K( 2, 1) Calabi s diastasis function CB, Brunner, Douglas, Rastelli ( ) Extension to higher dimensions and other co-dimension defects [in progress with Daniel]

36 Many thanks to the organizers of this wonderful (series of) meeting