FRAMED BPS QUIVERS AND LINE DEFECTS IN N=2 QFT

Transcription

1 FRAMED BPS QUIVERS AND LINE DEFECTS IN N=2 QFT Queen Mary, University of London Michele Cirafici CAMGSD & LARSyS based on

2 INTRODUCTION Lots of recent progress in the study of N=2 4d QFT BPS spectra, wall-crossing behavior, defects and exact results in general We will study line defects in N=2 QFTs Apply quiver technology to the study of framed BPS states (bound to the defect) New structures (cluster algebras) and insights (gluing and surgery rules)

3 OUTLINE BPS states Quiver Quantum Mechanics Line defects Framed quivers Applications

4 SUSY FIELD THEORIES AND THEIR BPS STATES

5 SEIBERG-WITTEN IN A FLASH We will consider 4d theories with N =2 Coulomb branch B : family of quantum vacua. At low energy: unbroken abelian gauge theory Seiberg-Witten: low energy Wilsonian action on B SuSy constraints solved by a family of Riemann surfaces u Lattice of charges: Central charge: u = H ( u ; Z) I Z (u) ' u h i, ji Measures mutual locality

6 BPS STATES Open problem: determine the stable BPS spectrum BPS (or short) reps: {R A, R B } =4(M Z ) AB BPS particles can form BPS bound states Single particle Hilbert space (normalizable wave functions) H u = M 2 H u, A good measure of the BPS degeneracies is the Witten index (vanishes on long reps) ( ; u) = 2 Tr H BP S,u (2J 3) 2 ( ) 2J 3

7 WALL-CROSSING The index jumps at walls of marginal stability [Gaiotto,Moore,Neitzke] [Kontsevich,Soibelman] [Denef] Example: the binding energy for a 2-particle bound state with charge = + 2 depends on u 2 B E bin = Z + 2 (u) Z (u) Z 2 (u) apple 0 The wall of marginal stability is when Z k Z 2 Crossing a wall of marginal stability the bound-state wave function is not anymore normalizable. The BPS state cannot be interpreted as a -particle state

8 WALL-CROSSING The moduli space of vacua is divided in chambers [Kontsevich,Soibelman] Universal wall-crossing formula [Joyce,Song]!Y K ( ;u) is a wall-crossing invariant To each central charge we associate a (BPS) ray Z (u) The chamber is specified by the ordering of the central charges: stability condition

9 QUIVER QUANTUM MECHANICS

10 BPS QUIVERS The info about the BPS spectrum can be packaged into a BPS quiver (with superpotential) Assume we have a positive basis of charges for : any BPS state is = X i n i i { i } [Alim,Cecotti,Cordova, Espahbody,Rastogi,Vafa] Each node represents an element of the basis { i } j The arrow from node to is the pairing (measure of mutual locality) i j h i, ji h i, ji i Given this quiver: what are the stable BPS states?

11 BPS QUIVERS Interaction of the elementary states: quiver quantum mechanics with 4 supercharges (and superpotential). The fields are quiver representations: Gauge group Y U(n i ) i2nodes B ij : V i! V j Its vacua correspond to 4d BPS particles. Mathematically: stable quiver representations (note that basis elements are by definition stable). Physically the notion of stability implies the choice of a chamber. Where do we get such a quiver? Stringy engineering

12 BPS QUIVERS AND TRIANGULATED CURVES To construct the BPS quiver we can use the String/M-theory engineering.theories of class S[g, C] energy dynamics of M M5 branes on are obtained as low R 3, C (today M=2) Lesson: questions on R 3, have an answer on C double cover of C One obtains a geometric description of BPS states by an ideal triangulation of quiver as follows C. To a triangulation we associate a BPS Edges Nodes [Gaiotto,Moore,Neitzke] [Alim,Cecotti,Cordova, Espahbody,Rastogi,Vafa] i j B 4 ij =+ B ij = X 4 B 4 ij Arrows

13 QUIVERS AND MUTATIONS Still a difficult problem! Idea: generate all the stable BPS states by Seiberg-like dualities A Seiberg duality gives a dual quiver which describes the same physics. On the quiver it acts as a mutation µ k,+ ( i )= This is just a change of basis. The idea is that if a charge can be a basis generator, then it is a stable BPS state After a few iteration the quiver will go back to itself and the full spectrum will have been generated k if i = k i +[h i, ki] + k if i 6= k [Alim,Cecotti,Cordova, Espahbody,Rastogi,Vafa] [Gaiotto,Moore,Neitzke] [Keller]

14 QUIVERS AND MUTATIONS For example in Argyres-Douglas A The order of the nodes at which we mutate is determined by the central charge (stability condition) Assume for example that arg Z (u) < arg Z 3 (u) < arg Z 2 (u)

15 QUIVERS AND MUTATIONS For example in Argyres-Douglas A

16 QUIVERS AND MUTATIONS For example in Argyres-Douglas A

17 QUIVERS AND MUTATIONS For example in Argyres-Douglas A

18 QUIVERS AND MUTATIONS For example in Argyres-Douglas A Spectrum {, 3, + 2, 2} + antiparticles Stability is obvious: these states appeared as basis elements in a Seiberg dual description

19 /3 SUMMARY In favourable circumstances the BPS spectrum of a susy field theory in a chamber is captured by a BPS quiver This quiver can be obtained from M-theory engineering (triangulation of C ) The spectrum can be computed with the mutation method by applying Seiberg-like dualities The spectrum in any chamber is then given by the wall crossing formula

20 LINE DEFECTS

21 LINE DEFECTS We will discuss line defects which are straight lines in R 3, These defects are /2-BPS and preserve a subalgebra labeled by as well as, and time translations 2 C SO(3) SU(2) R We think of a defect as a modification of the path integral, or of the Hilbert space. We call the new Hilbert space Hugely interesting structures: order parameters, OPE... H L

22 FRAMED BPS STATES The new Hilbert space is graded H BPS L = M H BPS L, New BPS bound M Re(Z (u)/ ) [Gaiotto,Moore,Neitzke] This defines framed BPS states, and their degeneracies (u, L, )=Tr H BP S u,l, ( ) 2J 3 Framed BPS states are BPS states bound to the defect They have their own wall-crossing formula

23 FRAMED BPS STATES Line defects are completely characterized by their framed spectrum and moduli space coordinates [Gaiotto,Moore,Neitzke] [Andriyash,Denef, Jafferis,Moore] hli = X (L, ) Y Framed wall crossing: add or remove a halo Y i! Y i ( + Y k ) h i, k i (L, ) changes so that hli is wall-crossing invariant!

24 FRAMED QUIVERS

25 FRAMED QUIVERS Idea: extend the quiver methods to study framed BPS states Model a line defect with an extra framing node To understand its origin, look back at M-theory engineering: M5 wrapping We can engineer a line defect with the boundary of a M2 brane ending on the M5. One leg of the boundary is on and the other on C. R 3, C other uses [Cordova,Neitzke] [Xie] [Chang,Diaconescu,Manschot,Moore,Soibleman] R 3, A line defect corresponds to a path on C

26 FRAMED QUIVERS A line defect is described by a lamination on C [Gaiotto,Moore,Neitzke] [Drukker,Morrison,Okuda] C This is just a collection of paths with additional conditions They can be closed They can be open. In this case each path carries a weight and the sum of the weights at a boundary segment must vanish R 3,

27 FRAMED QUIVERS We can identify each path on using a series of shear coordinates with respect to the triangulation C [Fomin,Thurston] L L L L b (T,L) =+ b (T,L) = b (T,L) =0 b (T,L) =0 Then we encode this information in the BPS quiver by adding a framing node connected to the nodes with b i (T,L) arrows. { i } there are exceptions! 2 3

28 LINE DEFECTS AND CLUSTER TRANSFORMATIONS Now we would like to generalize the mutation method to defects For framed quivers mutations generate new defects and we want to compute their framed BPS spectrum The natural objects which generalize quiver mutations are cluster algebras Encode the framed spectrum into the formal generating function L = X ( ; u) y

29 WHAT IS A CLUSTER ALGEBRA? Let s take a brief excursus [Fomin,Zelevinsky] [Fock,Goncharov] Cluster algebras are remarkable mathematical structures which pop up everywhere Given a quiver and a basis of charges associated with Q { i } each node, we introduce the formal variables each node (with ). y i y j = y i + j {y i }, one for Technically these are coefficients of a cluster algebra, or y- variables. No time to talk about X-variables!

30 WHAT IS A CLUSTER ALGEBRA? If we mutate the quiver the y-variables transform as mut k,+ y i = y i y [h i, k i] + k y i ( + y k ) h i, k i...which just looks like a complicated rational transformation. i = k i 6= k But: cluster algebras have periodicity properties. Anytime this happens we have a dilogarithm identity Deep connection with integrable systems, many applications in physics and math...

31 WHAT IS A CLUSTER ALGEBRA? For example for A2 Argyres-Douglas y (0) y (3) mut,+ ( y2 (0) mut,+ y (0) = y y2 (0) = y2 y2 (3) y (3) y2 (3) = = y +y2 +y y2 +y2 y y2 mut2,+ y () y2 () y () y2 () y (4) y (4) y2 (4) = = ( + y ) y2 y2 (4) = = ( mut2,+ y y2 y y2 +y2 y2 (2) y (2) y (2) y2 (2) +y2 +y y2 y y2 +y y2 = = y2 (5) y (5) mut,+ y (5) y2 (5) = = y2 y After few iterations the variables go back to themselves. The dilogarithm identity is: L(x) + L(y) = L x( y) + xy + L(xy) + L y( x) xy x= y + y y= L(x) = Li2 (x) + y2 ( + y ) + y2 + y y2 log x log( 2 x)

32 QUANTUM CLUSTER skip this slide! ALGEBRAS Cluster algebras have quantum cousins. Introduce formal variables {Y i } such that Y Y = q h i, j i Y i j i + j Quantum mutations is the quantum dilogarithm and mut q k = Ad (E(Y k )) k,+ where k,+ (Y i )=Y µk,+ ( i ) E(Y k ) Now a periodic sequence of quantum cluster transformations gives a quantum dilogarithm identity E(Y )E(Y 2 )=E(Y 2 )E(Y + 2 )E(Y 2 ) [Fock,Goncharov] [Keller] [Nagao]

33 PHYSICS OF CLUSTER TRANSFORMATIONS Physically a cluster (y-)transformation is the composition of a quiver mutation and a framed wall-crossing Therefore if we find a sequence of cluster transformations such that the sequence of quiver mutations acts trivially (eventually up to permutation), the result is just a sequence of framed wall-crossings The physical meaning of such a sequence of quiver mutation is a sequence of Seiberg dualities which generates the full BPS spectrum

34 LINE DEFECTS AND CLUSTER TRANSFORMATIONS Claim: consider a sequence of mutations which acts trivially on the BPS quiver (up to perm) but not on the framed BPS quiver mut 2,+ y 2 y y 2 (2) y (2) Then the mutated framed quiver corresponds to a new defect (because the framing corresponds to coordinates!) Its framed BPS spectrum is obtained by applying the operator mut km k,+

35 LINE DEFECTS AND CLUSTER TRANSFORMATIONS The reason is that the generating functions are BPS wall crossing invariant. But this can be seen as a condition on the new defect L old (y )=L new (y [i]) = L new (mut µo!n y ) The framed BPS spectrum of the new defect must be such that the above condition holds. To obtain we simply relabel the basis new (u, ) y [i]! y Equivalently since mutations are involutions, we write L new (y )=L old (mut µ o!n y )

36 FOR EXAMPLE Go back to mut 2,+ x[2] = +y+xy x y[2] = y+xy y L 5 = y The new defect must be such that L = X old (u, ) y = X 0 x y[2] L 5 = y new (u, x[2] = +y[2] + x[2]y[2] x[2] 0 ) y 0[i] Recipe: find that particular combination of mutated variables which leaves the spectrum invariant and then relabel them Easier: L 3 = x + y x + y = mut 2,+ y y x

37 variables associated with the nodes. More precisely we could associate to a lamination an element of the Leavitt path algebra LAQ constructed from the extended quiver Q. Now we would like to use the framed BPS quiver formalism to generate other line defects via cluster n the newmutations. line defect with the same mutation sequence and generate another andconsider so on. We find with three states a chamber y AND THEN... and 3 with h, 2 i = + and h 2, 3 i = +, and write the associated variables x = y 3, y = y 2 and z =c y. The BPS quiver is c L3 : : L5 O O xµz µyo µx y o z. (6.2), 2 =) (6.5) $ $ we called Lc ino Section 4.3 o o o whose framed Consider now the line operator spectrum is x y z x y and z Lc = z. Its framed BPS quiver is µx ) Lc7 c L 9 : =)...one can exploits these facts (6.6) systematically to x o y o z x o y o z compute framed BPS spectra zz µ z µy µ x x o Lc b b y o :. (6.3) z c Lon Lcfinally the nodes, starting from x, then y and We now mutate z dd µ z µy µ x y µx 0 ) =). (6.7) L = z $ Lc b Lc3 $ b B C o o o O O x o y z x y z C' y µz µy µx B, A L = mut mut mut L = y x,+ y,+ z,+ $ $ the new line defect act again on with the same mutation sequence and generate another z z o o o o to y the line zdefect xwe started with: y z we see t at the end we havex returned indeed line defect, and so on. We find efects come in cluster mutation orbits!c Of course all of these statements have and generate the new line operator L3 (we have chosen the labeling to uniformize with c al counterpart from the point of view of the triangulation associated with the Lc5 [22]). Its framed BPS spectrum its therefore L3c = mutx,+ muty,+ mutz,+ Lc = y + yz + z, L 3 : O O : µ z µy µ x. Applying algorithm to compute the framed BPS we see quiver. We can as can beour confirmed by looking at the corresponding pathspectra, in the extended =) (6.5) L5c = L7c = mutx,+ muty,+ mutz,+ L9c x =x+ +, y y L5c =47, x c L7 = x + x y + x y z, L9c = + +. xyz yz z mutx,+ muty,+ mutz,+ L3c = mutx,+ muty,+ mutz,+ c L = mutx,+ muty,+ mutz,+ µz µy µx =) x(6.8) o y o (6.9) Lc7 zz (6.0) x o y o (6.) µz µ y µx ts agree with Section 0.2 of [22] (we have chosen the same labeling=) for the line x o $$ z x o =) Lc x o µ z µy µ x $$ z y o (6.6) z Lc d d =) y o z c : : L9 µ z µy µ x z y o x o y o. (6.7) z We see that the end we have returned to the line defect we started with: indeed we see exemplify our methods, we now turn to a di erent chamber and a at di erent

38 " (2/3 + ) SUMMARY Methods based on BPS quivers can be extended to line defects Algebraically the presence of the defect is encoded in the framing of the quiver Quiver mutations are replaced by (quantum) cluster algebras Generate whole families of defects and compute their spectra (but not their full vevs)

39 GLUING AND SURGERY

40 GENERAL IDEAS The theories we are considering are based on a curve C Surgery or gluing of curves describes coupling or decoupling of physical theories [Gaiotto] [Cecotti,Vafa] Partial classification of N =2 theories It extends to defects! Namely from a framed quiver we can give general rules to consistently couple the defect in another theory

41 ELEMENTARY OPERATIONS The presence of a defect complicates things (combinatorially) Elementary cut and join rules on a surface boundary: adding a marked point adding a puncture adding a boundary component increasing the genus C with Using these rules one obtains a consistent framed quiver

42 ADDING A PUNCTURE For example, let s add a puncture to C Q L a b d Q L a b d Physically we can think of b b coupling a massive flavour Q L a d Q L a d If Q describes the original theory with a defect, how is Q L a b d Q L a b d it modified? The new quivers describe all Q L a b d Q L a b d the consistent couplings

43 AN APPLICATION

44 PUTTING ALL TOGETHER Using cluster mutations we can generate new defects from known ones Using gluing rules we can construct new framed quivers which describe defects in new theories Being clever we can also compute the framed BPS spectra Program: generate new QFT with line defects from a set of known QFT with known defects. Hopefully a first step towards a classification

45 PUTTING ALL TOGETHER generate Argyres-Douglas, again 8.3 new framed quivers: A4 in a simple example. We will now Finally we put all that we have learned to good Luse 5 take a line defect in the Argyres-Douglas superconformal theory of type A2 and use itµto yxuv generate new defects in the A4 theory. We are actually going to consider the most simple =) case available. We pick the line defect L5 y o xo u o v 4 A LA 6 A ( y o xo u o As an example we can consider a defect in A2 SCFT and use it to compute framed spectra in A4 SCFT 8.3 L5 Argyres-Douglas, again 4 B,LA 7 B Start with our A =and 8.2, we see that there exists=) =) L5 yy o in the appropriate chamber. From the rules ofo Section 8. o o 2 y yu x v a defect which we have already discussed in Sections 5.3 and 4.3. If we label y 2 = y and y = x, A4 then L5A2 = y. We can use the quiver gluing rules of [7, L 5 8] to obtain the A4 theory, whose BPS quiver we take to be Consistent coupling: y o xo zz y o u o xo v,, o u (8.4) 4 A LA 8 (8.3) µy x u v µ x y uy vo example. We will now Finally we put all that we have learned to good use in a ysimple 2 =) of type A2 and use it to =) take a line defect in the Argyres-Douglas superconformal theory which we have already discussed in Sections 5.3 and 4.3. If we label y = y and y = o o o generate new defects in the A4 theory. We are actually going to consider y the most x simple u 2 v x, A2 thendefect L5 = case available. We pick the line L5y. We can use the quiver gluing rules of [7, 8] to obtain the A4 theory, whose 4 BPS quiver we take to be LA 9\ L5 \ µy x u vx o µy x u v y o u o v, (8.4) 2, (8.3) v y o xo L5 4 y o Generate new defects! µy x u v =) µy x u v =) xo zz y o xo 4 u A o LA A 8 xo v µy x u v L7A4 = (8.7) with the property L5A4 = y. =) Indeed in this case the gluing rules are trivial, since the y o xo u o v y o xo u o v A4 lamination does not take part in the gluing. Equivalently, the relevant Leavitt path algebra B 4 LA involve 9 \ the L8 4 gluing. LA 0 = mutv u x y,+ L7A4 elements only node y, before and after the We can however start \ µy x u v 4 generating new line defects using quiver mutations. Since the theory (8.8) is complete, L9Awe = =) can pick a stability condition corresponding to the mutation sequence µ and start to y x u v y o xo u o v y o xo u o v A4 L0 = A4 L A A µy x u v =) y o (8.9) xo u o v mutv u x y,+ L6A4 mutv u x y,+ L8A4 mutv u x y,+ L9A4 u o 4 A LA = y A lamination does not take part in the gluing. Equivalently, the relevant Leavitt path algebra 4 The framed spectrum will now be given applying iteratively A LA 6 A elements only involve the node y, before and after the gluing. We can however start 4 µy x u v LA (8.6) 5 generating new line defects using quiver mutations. Since the theory is complete, we =) y A4 4 o o o o u vcan pick ya stability condition x ucorresponding v L(8.5) = mutvµuyxx uy,+ L5Astart =to + + y to the, mutation v and 6 sequence 4 o B LyA 7 u o ( ( and 8.2, we see that there exists generate new quivers: chamber. with in framed the appropriate Fromthe theproperty rules of Section L5A4 =8. y. Indeed in this case the gluing rules are trivial, since the a defect A A 4 LA 0 µ y x u v A4 (8.5) =) 5 y o L ( xo u o the operator mutv u x y,+ Compute their spectra! x x x = + +x u u u =u+ + v v =v = xyuv xuv uv v A4 A4 L = mutv u x y,+ L0 = y + xy + uxy + uvxy (8. (8. (8. (8. (8. (8. Remarkably starting from a defect in a known quantum field theory, we have obtained n The framed spectrum will now be given applying iteratively the operator mutv u x y,+

46 3/3 SUMMARY We have discussed a new approach to line defects using framed quivers and cluster algebras No need for the theory to admit a lagrangian description Framed quiver well behaved under gluing and surgery rules Exist techniques to compute the spectrum directly (via Leavitt path algebras, not included in this talk)

47 IN PROGRESS Extend to theories with no full CY) C (where you need the Classification of line defects? It would be nice to have a quiver description of surface defects Donaldson-Thomas theory of defects