The Römelsberger Index, Berkooz Deconfinement, and Infinite Families of Seiberg Duals

Transcription

1 The Römelsberger Index, Berkooz Deconfinement, and Infinite Families of Seiberg Duals Matthew Sudano Kavli IPMU (MEXT-WPI, University of Tokyo-TODIAS) Based Largely on Nagoya University, 2012 February 21

2 Introduction SUSY theories have extraordinary renormalization properties Their dynamics, however, are remarkable primarily because they are tractable Exact results in SUSY theories have repeatedly substantiated long-held beliefs about strongly coupled non-susy theories In this sense, they have proved their value, independent of their role in Nature

3 Introduction Seiberg duality 1 is among the key advances in opening a window onto strong dynamics Relates distinct but IR-equivalent theories No derivation/proof (no doubt of validity) No algorithm for identifying duals Limited success in string theory/supergravity Pouliot 2 made an early effort in 4D by constructing a duality-invariant spectroscope of sorts that contained information about the chiral ring of a theory 1 hep-th/ hep-th/

4 Introduction Today I will discuss a refinement of this idea---the Römelsberger Index An augmented Witten Index Computed in radial quantization (on ) RG- (and therefore duality-) invariant For SCFT, counts protected operators Contains all information that can be learned from group theory (up to SC-commuting) hep-th/ arxiv:

5 Introduction Indices of almost all known examples of duality have been tested successfully Some identities have been proven using elliptic hypergeometric integral relations The existence of the precise necessary identities is probably not a coincidence Some stand as conjectures of integral identities New dualities have been proposed based on mathematical identities arxiv: arxiv: arxiv:

6 The Römelsberger Prescription Our index is an augmented Witten index Computed in radial quantization on Fermion number operator R-symmetry current Cartan of Character of global group see also hep-th/

7 The Römelsberger Prescription Only states annihilated by contribute Easy to find contributors in free case using

8 The Römelsberger Prescription

9 Römelsberger s Prescription Single-letter index for a matter superfield in a general theory follows from allowing other symmetries, including general R-charge Similarly, the vector superfield letter is

10 Römelsberger s Prescription Finally, we just have to compose the letters into words, which is taken care of by The Plethystic Exponential, and then project onto gauge invariant states by integrating againts the Haar measure

11 The Römelsberger Index In a line, the index for any 4D, N=1 theory is But we can do better. In terms of character monomials a matter field gives contributes That function looks special

12 Elliptic Gamma Function Superfields show up in the index as (products of) elliptic gammas So we should be able to extract physical meaning from the functions properties

13 Elliptic Gamma Function First a bit of notation

14 Elliptic Gamma Function Some properties of the gammas Math: Physics: massive states don t contribute More explicitly, generic chiral has form

15 Elliptic Gamma Function A less obviously physical case: Perhaps part of a bigger story arxiv:math/ The part that I can make some sense of: example: r=0, charged under some sym

16 Elliptic Gamma Function In terms of some products of thetas and using the basic theta quasi-periodicity relations This stuff starts to smell like physics

17 Elliptic Gamma Function For example, one of the exponents is a cubic Casimir Example: two-index symmetric of SU(3)

18 Total Ellipticity Recall that our gammas generally include Let such that it appears with only integral exponents Given two indices, take the ratio of the integrands; call this Rescale some fugacity by and divide by the original ; call this Total Ellipticity is invariance of under rescaling all by, including a rescaling

19 Total Ellipticity The condition is equivalent to All anomaly conditions, except R and R 3

20 Total Ellipticity I don t yet have a very clear understanding of the mathematical significance of total ellipticity It appears to be needed for there to be interesting (perhaps just provable?) relations; Spiridonov: C onjecture. The condition of total ellipticity for an elliptic hypergeometric term is necessary and sufficient for validity of the t Hooft anomaly matching conditions for dual superconformal fi eld theories whose superconformal indices are determined by this term. The second conjecture may be wrong, but I don t know an example with all anomalies OK except the pure R...

21 An Explicit Example Actually writing down the Römelsberger index for a given theory is easy. The vector/haar parts are always the same. Gauge theory measures:

22 An Explicit Example Consider classic Seiberg duality SU(N ) SU(N f ) 1 SU(N f ) 2 U(1) B U(1) R Φ N, W N E = 0 f Φ N N f SU(N f N) SU(N f ) 1 SU(N f ) 2 U(1) B U(1) R φ 1 N N f N φ 1 N N f N M ΦΦ 1 0 2(1 N N f ) N N f N N f, W M = φm φ

23 An Explicit Example It could have been Rains Duality SU(N ) SU(N f ) 1 SU(N f ) 2 U(1) B U(1) R Φ N N f I would write Φ N N f

24 An Explicit Example It could have been Rains Duality SU(N ) SU(N f ) 1 SU(N f ) 2 U(1) B U(1) R Φ N N f Φ N N f Following Spiridonov, Rains considered the equivalent integral quarks anti-quarks I (m ) A n (Z t 0,...t m+n+1 ;u 0,...u m+n+1 ;p,q) (4.1) := (p; p)n (q; q) n (n + 1)! 0 i n z i =Z 0 i n 0 r < m + n + 2 Γ(t r z i,u r /z i ;p,q) 0 i< j n Γ(z i /z j,z j /z i ;p,q) 1 i n dz i 2π 1z i. measure+adj stuff arxiv:math/

25 An Explicit Example Proved the identity Theorem 4.1. For otherwise genericparameters satisfying 0 r < m + n + 2 t r u r = (pq) m+1, I (m ) A n (Z... t i...;...u i...;p,q)= 0 r,s< m + n + 2 Γ(t r u s ;p,q)i (n) A m (Z... T 1 m + 1 /ti...;...u 1 m + 1 /ui...;p,q), where T = 0 r < m + n + 2 t r,u= 0 r < m + n + 2 u r. RHS is index for mesons different gauge group anti-quarks quarks SU(N f N) SU(N f ) 1 SU(N f ) 2 U(1) B U(1) R φ 1 N N f N φ 1 N N f N M ΦΦ 1 0 2(1 N N f ) N N f N N f, W M = φm φ

26 Comments/Questions Following Römelsberger s prescription, conjecture, and perturbative evidence, Dolan and Osborn used Rains proof to demonstrate I E =I M. Doesn t prove duality, but may be as good as we (can) do... Still many examples without I E =I M proof (e.g. Adj s) Is this related to Langland s duality? Many examples of putative new duals from math; Khmelnitsky studied some, most unsettled arxiv: No general algorithm (or statements about uniqueness or even finiteness of number of duals), but formalism appears algorithm friendly...

27 A Step Toward an Algorithm? The index accepts a finite and discrete set of data Rains s theorems give an equivalent index when the LHS index involves only fund s + anti-fund s For more-interesting representations, most physics examples yield (often somewhat unsatisfying) duals via the Berkooz deconfinement trick I will now prove the index version of deconfinement for the two-index anti-sym (also done for SU(N) adjoint) and apply it to a new example due to Craig, Essig, Hooke, Torroba

28 Berkooz Deconfinement Replace two-index anti-symmetric tensor, A, with confining gauge theory that outputs A as a meson Somewhat messy in initial iteration single confining field has constrained (D-term) moduli space accounted for in confined theory by including superpotential not immediately applicable to general A

29 BLySTer Deconfinement Berkooz deconfinement as refined by Luty, Schmaltz, Terning. The logic: Want one free anti-sym meson (no superpotential) Introduce confining Sp group Introduce two fundamentals to cure moduli space issue Introduce fields/interactions to cancel anomalies and give mass to unwanted mesons hep-th/ hep-th/

30 BLySTer Deconfinement Berkooz deconfinement as refined by Luty, Schmaltz, Terning. The result: Sp(N + K 4) G N SU(K ) U(1) U(1) R a 1 e r X 1 1 N e/k (2 N r)/k X 2 1 (N K )e/k [2(K 1) + (N K )r]/k X N e/k 2(K 2 + N r )/K A 1 2e 2r

31 Index Deconfinement It s straightforward to write down the index on each side where,, and.

32 Index Deconfinement It s also straightforward to check a few terms, but it isn t completely obvious how to prove the relation in general. The trick: Use the Sp s-confinement result Sp(2M ) SU(2M + 4) U(1) R Φ 1 M+2 M ΦΦ 2 M+2

33 Index Deconfinement Consider just the index structure ( ) N+K M N+K = N+K-1

34 Index Deconfinement There prove to be a consistent set of variables such that N K M = K N K N-1

35 An Application: CEHT Duality A chiral theory considered in a similar form long ago SU(N ) Sp(2N 2) SU(N + 3) U(1) U(1) R Q 1 e r Q 1 ẽ r P 1 1 (N + 3)e 2ẽ 4 (N + 3)r 2 r A 1 1 2ẽ 2 2 r The unique symmetric renormalizable superpotential is

36 An Application: CEHT Duality Summary of the procession of the duality BLySter Seiberg Sp s-conf SU(N) SU(N)xSp(2M) SU(N f -N)xSp(2M) SU(N f -N)

37 An Application: CEHT Duality The magnetic (far right) theory: SU(N + K 1) SU(K ) Sp(2N 2) SU(N + 3) U(1) U(1) R q 1 1 q 1 1 (K 4)e+ 2M ẽ N+K 1 (N + 3)e+ 3ẽ N+K 1 (K 4)r+ 2M (1 r) N+K 1 N + K + 2 (N + 3)r 3 r N+K 1 x e q e e X 2 2 r q r r X 2 A e q 2 2r q X e X 3 r X 3 M Q Q 1 1 e+ ẽ r+ r M Q P e+ e P r+ r P M QX e+ e X 2 r+ r X 2

38 An Application: CEHT Duality Some comments: The magnetic dual has a generally reasonable form: magnetic counterparts of fields, fundamental mesons, some extra singlets An odd feature: the gauge group is not fixed can have arbitrarily large rank A related odd feature: there is a fake global symmetry (of arbitrary rank) fixed by dynamical truncation of chiral ring Analysis relied on keeping track of the superpotential throughout duality steps

39 An Application: CEHT Duality In terms of the index... SU(N ) Sp(2N 2) SU(N + 3) U(1) U(1) R Q 1 e r Q 1 ẽ r P 1 1 (N + 3)e 2ẽ 4 (N + 3)r 2 r A 1 1 2ẽ 2 2 r assigning fugacities,

40 An Application: CEHT Duality The (electric) index is

41 An Application: CEHT Duality Directly applying our index deconfinement module gives the deconfined index Nothing to think about. Don t have to keep track of the superpotential. It s just an integral identity. The next step is similar...

42 An Application: CEHT Duality To dualize the SU group (apply A n -type integral transformation), one just has to fuse fundamentals (and anti-fundamentals) The consistency conditions look nasty in terms of natural variables, but this works as long as the theory is non-anomalous.

43 An Application: CEHT Duality So we can again blindly apply the integral transformation to get Integrating out heavy fields is automatic; you don t have to identify the heavy fields or apply their equations of motion

44 An Application: CEHT Duality A similar manipulation gives the final result Note: the index knows about the fake symmetry. The magnetic index appears to be a function of an fugacity, but it in fact cancels out. (Further note: the index doesn t appear to say anything interesting about true accidental symmetries.)

45 Summary and Conclusions Total ellipticity is a necessary condition for physical consistency, but possibly not a sufficient condition Prove sufficiency; would imply TrR, TrR 3 not indep. Prove insufficiency; would likely help find better condition Two-index anti-symmetric tensor and SU adjoint index deconfinement modules derived/proven Allows for efficient and systematic analysis of vast set of theories Two-index symmetric remains (is somewhat subtle) Novel example examined to demonstrate utility

46 More thoughts on future work Understand unproven index identities Many conjectured identities, including putative new dualities In this work, all followed from basic results Spiridonov believes adjoints are special Is it possible to use adjoint deconfinement for Kutasov duality? N=4? Connection to Langlands? Understand physical significance of modular transformations...if any Can we find a? Other RG-invariant data?...

47 Thank you for your attention.