Fixed points and D-branes

Transcription

1 1 XVII Geometrical Seminar, Zlatibor, Serbia Grant MacEwan University, Edmonton, Canada September 3, joint work with Terry Gannon (University of Alberta) and Mark Walton (University of Lethbridge)

2 Outline Motivation Charges of WZW D-branes Fixed points and NIM-reps Summary and future work

3 Outline Motivation Charges of WZW D-branes Fixed points and NIM-reps Summary and future work

4 String theory In string theory, a particle is a finite curve of length approximately cm. Modern string theories contain both open (e.g. photon) and closed (e.g. graviton) strings. Topologically, a closed string is a circle S 1 and an open string is the interval [0, 1].

5 String worldsheet As a string evolves through time, it traces out a surface called a worldsheet. For example, an incoming closed string travelling from t = to t = 0 traces out a semi-infinite cylinder.

6 D-branes In 1989, higher dimensional objects ( membranes ) called D-branes (after Dirichlet) were introduced into string theory (Polchinski, Dai, Leigh, Hořava). Physically, D-branes are the membranes where the endpoints of open strings reside. In 1995, Polchinski proved that a consistent theory requires branes.

7 D-brane charges Branes are physical entities having tension and charge. During physical processes, these charges are conserved thus D-brane charges in string theory are analogous to electrical charges in particle physics. Unlike regular electrical charges, D-brane charges are usually preserved only modulo some integer M.

8 WZW models The Wess-Zumino-Witten models are rational conformal field theories (toy models for quantum field theories) that correspond to string theories on compact Lie groups (e.g. SU(n), the group of n n unitary matrices with determinant 1). Note: A conformal field theory is a two-dimensional quantum field theory whose symmetries include the conformal transformations. A rational CFT obeys an additional finiteness condition.

9 WZW models Most quantities in a WZW model have a natural interpretation in terms of the underlying affine Lie algebra g (1). For example, the primary fields are labelled by irreducible highest weight representations of g (1).

10 WZW models For example, the algebra associated to SU(n) is A (1) n 1, constructed from A n 1 = sl n (C) =: g as follows: Let L(g) = C[t ±1 ] g = { l Z a lt l } where only finitely many a l g are nonzero. Define the bracket [at l, bt m ] = [ab]t l+m on L(g). This makes L(g) into an infinite-dimensional Lie algebra.

11 WZW models For reasons of representation theory, centrally extend this algebra by the element C to obtain the algebra L(g) CC with bracket [C, x] = [x, C] = 0, and [at m, bt l ] = [ab]t m+l + mδ m+l,0 (a b)c. Extend once more by a derivation l 0 := t d dt to obtain g(1).

12 WZW models Note: Since C is a central element of g, any representation of g must send C to some scalar multiple k of the identity. Thus the representations come in levels k. For example, the set of irreducible highest weight representations for g (1) = A (1) n 1 is indexed by the set P+ k = {(λ 0 ;..., λ n 1 ) Z n 0 : λ λ n 1 = k}.

13 Outline Motivation Charges of WZW D-branes Fixed points and NIM-reps Summary and future work

14 Charge groups The charges of WZW D-branes form a discrete abelian group. These groups can be found through K-theory, or through a conformal field theory description. (Minasian-Moore, Witten, Kapustin, Bouwknegt-Mathai)

15 Charge groups For the simply connected groups (e.g. SU(n)), the charge groups are Z M for some integer M. The integer M has been found for all algebras and levels, as well as the charges. (Fredenhagen-Schomerus, Maldacena-Moore-Seiberg, Bouwknegt-Dawson-Ridout)

16 The problem Now suppose we have a non-simply connected Lie group. How can we find the actual charges themselves?

17 The problem In the simply connected case, the charges are determined uniquely by the charge equation. However, in the non-simply connected case, the charge equation leads to difficulties arising from fixed points of simple-currents. The simplest non-simply connected group is SO(3) = SU(2)/Z 2. The charge group is Z 2 Z 2 if 4 k and Z 4 if 4 k. Compare with the charge group Z k+2 for SU(2).

18 The tool A property of the S-matrix relating entries involving fixed points to entries of the S-matrix of the (smaller-rank) orbit Lie algebra ğ fixed point factorisation. Fixed point factorisation has been used successfully in the case of SU(n) to find D-brane charges for non-simply connected groups (Gaberdiel-Gannon).

19 Outline Motivation Charges of WZW D-branes Fixed points and NIM-reps Summary and future work

20 The S-matrix To each quantum field theory (e.g. a string theory) is associated an S-matrix. This matrix expresses amplitudes and thus is a fundamental ingredient of the theory. For the WZW models, the S-matrix can be given by e.g. the Kac-Peterson formula.

21 Simple-currents The S-matrix satisfies the inequality S λ0 S 00 > 0 with equality exactly when λ is a simple-current. Simple-currents correspond to permutations J of the vacuum 0 = (k; 0,..., 0) we refer to J also as a simple-current. In all cases except E (1) 8, level 2, simple-currents correspond to extended diagram automorphisms.

22 Affine Kac-Moody Coxeter-Dynkin diagrams x 1 1 x 2 1, 2 v 2 1, v 2 v v 1 1, 2 1, (1) A n 1 1 (1) B n x (1) C n 1 1 x x (1) D n (1) E (1) x E7 3 v v v , 4 1, , x (1) x x (1) (1) E8 F 4 G2 Figure: The Coxeter-Dynkin diagrams for the nontwisted affine Kac-Moody algebras

23 Fixed point factorisation Consider the S-matrix for the the affine algebra g k. If one of λ or µ is a fixed point of a simple-current, then S λµ can be written as a polynomial in terms of the S-matrix entries for the orbit Lie algebra ğ l. The rank of ğ l is smaller than the rank of g k (and l is at most k).

24 Simple-current modular invariants Let J be a simple-current for g k. The matrix M[J] λµ = ord(j) i=1 δ J i λ,µδ Z (Q J (λ) + ir J ) (when it corresponds to a modular invariant) corresponds to the WZW model with group G/ J. The number of D-branes is the trace of M[J].

25 Example For example, the D-series modular invariant for A (1) 1, corresponding to the order-2 simple current is D 4 = (1)

26 A (1) r fixed point factorisation Fix a level k, and let d (r + 1). The simple-current J d has order n/d. We have χ Λl (ϕ) = χ Λ ld/(r+1)( ϕ) d if l, where primes denote A(1) r+1 d 1 level kd/(r + 1) quantities. (Gannon-Walton) Note: χ λ (µ) := S λµ S 0µ.

27 C (1) r fixed point factorisation, r even For C (1) r (r even) at level k and the order-2 simple-current J: χ Λ2m (ϕ) = ( 1) m m l=0 χ Λ l ( ϕ) where primes denote A (2) 2( r ) level k quantities. 2

28 Fixed point factorisation algebras X (1) r, level k Simple-current FPF algebra Level A (1) r J d A (1) d 1 B (1) r J A (2) 2(r 1) C (1) r, r odd J C (1) r 1 2 C (1) r, r even J A (2) 2( r 2 ) k D (1) r J v C (1) r 2 D r (1), r odd J s C (1) r 3 2 D r (1), r even J s B (1) r 2 kd r+1 k k 2 k 2 k 4 k 2

29 Fixed point factorisation algebras X (1) r, level k Simple-current FPF algebra Level A (2) 2r 1 J C (1) r 1 D (2) r+1 D (2) r+1, r odd J A(2), r even J D(2) r k 2 2( r 1 2 ) 2 k k

30 The charge equation Let B be the set of all D-branes preserving the full affine symmetry g k (this corresponds to the charge conjugation modular invariant), and let q a be the charge of the D-brane a. dim(λ)q a = b N b λaq b (modulo an integer M), where λ P k +(g) and dim(λ) is the Weyl dimension of λ in g. The coefficient Nλa b gives the multiplicity of λ in the open string spectrum of an open string beginning on D-brane a and ending on D-brane b.

31 The NIM-rep The matrices N λ defined by (N λ ) ab = N b λa define a nonnegative integer matrix representation of the g k fusion ring. That is, for each λ P k +, the assignment λ N λ satisfies N λ N µ = ν N ν λµn ν where Nλµ ν formula are the fusion coefficients given by Verlinde s N ν λµ = κ S λκ S µκ S νκ S 0κ.

32 NIM-reps In general, the NIM-rep matrices are indexed by boundary states. For the WZW models, these are related to, but not generally equal to, the highest weight representations of g k (Gaberdiel, Gannon). They are given by ([ν], i), where [ν] = J ν and 1 i ord(ν). They satisfy a Verlinde-like formula N y λx = µ Ψ xµ S λµ Ψ yµ S 0µ where Ψ is a unitary matrix.

33 NIM-reps Calculating NIM-rep coefficients N ([κ],j) λ([ν],i) is easy when at least one of ν, κ is not a fixed point. In this case, they reduce to fusions for g k. When both indices are fixed points however, the NIM-reps are more difficult. Using fixed point factorisation, we find these also reduce to fusions, but this time, of both g k and ğ l.

34 NIM-reps of C r (1), r even Let J be the order-2 simple-current. Then N [κ] λ[ν] = Nλν κ + Nλν Jκ N ([κ],j) λ[ν] = N κ λν N (ψ,j) Λ n (ϕ,i) = { 1 2 ( N ψ Λ 2m ϕ + ) ( 1)i+j+m m l=0 Ñ ψ Λ l ϕ if n = 2m 1 N ψ 2 Λ nϕ if n = 2m + 1 where tildes indicate A (2) 2( r ) level k quantities. 2

35 NIM-reps These NIM-rep formulas have been found for all (non exceptional) affine algebras. (Gaberdiel, Gannon, Beltaos) With these, we can generalise the results for SU(n) to all WZW models.

36 Outline Motivation Charges of WZW D-branes Fixed points and NIM-reps Summary and future work

37 Summary To find actual D-brane charges, we need to use the conformal field theory description of D-branes (the charge equation). However, in the non-simply connected case the only way to handle these is to use fixed point factorisation.

38 Future work Compare the results with K-theory calculations Find more physical applications for fixed point factorisation Find a conceptual explanation for fixed point factorisation Explain the link between fixed point factorisation and the twining characters of Fuchs-Schellekens-Schweigert

39 Thank you Thank you for your attention.