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The Römelsberger Index, Berkooz Deconfinement, and Infinite Families of Seiberg Duals Matthew Sudano Kavli IPMU (MEXT-WPI, University of Tokyo-TODIAS) Based Largely on 1112.2996 Nagoya University, 2012 February 21

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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 Introduction

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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… Introduction 1 hep-th/9411149 2 hep-th/9812015

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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) Introduction hep-th/0510060 arXiv:0707.3702

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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 Introduction arXiv:0910.5944 arXiv:1107.5788arXiv:0801.4947

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Our index is an augmented Witten index Computed in radial quantization on The Römelsberger Prescription Fermion number operator R-symmetry current Cartan of Character of global group see also hep-th/0510251

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

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The Römelsberger Prescription

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

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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

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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…

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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…

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Elliptic Gamma Function First a bit of notation

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Elliptic Gamma Function Some properties of the gammas Math: Physics: massive states don’t contribute More explicitly, generic chiral has form

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Elliptic Gamma Function A less obviously physical case: Perhaps part of a bigger story… The part that I can make some sense of: example: r=0, charged under some sym arXiv:math/9907061

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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

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

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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

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

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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: The second conjecture may be wrong, but I don’t know an example with all anomalies OK except the pure R...

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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:”

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An Explicit Example Consider classic Seiberg duality

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An Explicit Example It could have been “Rains Duality” I would write

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An Explicit Example It could have been “Rains Duality” Following Spiridonov, Rains considered the equivalent integral measure+Adj stuff quarksanti-quarks arXiv:math/0309252

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An Explicit Example Proved the identity quarks anti-quarks different gauge group mesons RHS is index for

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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 No general algorithm (or statements about uniqueness or even finiteness of number of duals), but formalism appears algorithm friendly... arXiv:0912.4523

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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

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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

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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/9505067 hep-th/9603034

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BLySTer Deconfinement Berkooz deconfinement as refined by Luty, Schmaltz, Terning. The result:

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

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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

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

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

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An Application: CEHT Duality A chiral theory considered in a similar form long ago The unique symmetric renormalizable superpotential is

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

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An Application: CEHT Duality The magnetic (far right) theory:

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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

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An Application: CEHT Duality In terms of the index... assigning fugacities,

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An Application: CEHT Duality The (electric) index is

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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...

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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.

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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

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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.)

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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 Summary and Conclusions

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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?... More thoughts on future work

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Thank you for your attention.

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