# the römelsberger index, berkooz deconfinement, and infinite families of seiberg duals matthew...

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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

• 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

• Seiberg duality1 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– Pouliot2 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

1hep-th/94111492hep-th/9812015

• 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

• 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:1107.5788 arXiv:0801.4947

• 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 groupsee also hep-th/0510251

The Römelsberger Prescription

• Only states annihilated by contribute

• Easy to find contributors in free case using

The Römelsberger Prescription

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

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

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 matterfield gives contributes

That function looks special…

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…

Elliptic Gamma Function

First a bit of notation

Elliptic Gamma Function

Some properties of the gammas

Math:

Physics: massive states don’t contribute

More explicitly,generic chiral has form

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

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

Elliptic Gamma FunctionFor example, one of the exponents is a cubic Casimir

Example: two-index symmetric of SU(3)

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

Total Ellipticity

The condition is equivalent to

All anomaly conditions, except R and R3

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...

An Explicit ExampleActually writing down the Römelsberger index for a given theory is easy. The vector/Haar parts are always the same. “Gauge theory measures:”

An Explicit Example

Consider classic Seiberg duality

An Explicit Example

It could have been “Rains Duality”

I would write

An Explicit Example

It could have been “Rains Duality”

Following Spiridonov, Rains considered the equivalent integral

measure+Adj stuff

quarks anti-quarks

arXiv:math/0309252

An Explicit Example

Proved the identity

quarksanti-quarksdifferent gauge groupmesons

RHS is index for

Comments/Questions

• Following Römelsberger’s prescription, conjecture, and perturbative evidence, Dolan and Osborn used Rains proof to demonstrate IE=IM.

• Doesn’t prove duality, but may be as good as we (can) do...

• Still many examples without IE=IM 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

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

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

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 mesonshep-th/9505067

hep-th/9603034

BLySTer Deconfinement

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

Index Deconfinement

It’s straightforward to write down the index on each side

where , , and .

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

Index Deconfinement

Consider just the index structure ( )

=

N+K

M

N+K-1

N+K

Index Deconfinement

There prove to be a consistent set of variables such that

M

N K

K

N-1

N K

=

An Application: CEHT Duality

A chiral theory considered in a similar form long ago

The unique symmetric renormalizable superpotential is

An Application: CEHT Duality

Summary of the procession of the duality

SU(N) SU(N)xSp(2M)

BLy

Ste

r

SU(Nf-N)xSp(2M)

Seib

erg

Sp s

-con

fSU(Nf-N)

An Application: CEHT Duality

The magnetic (far right) theory:

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

An Application: CEHT Duality

In terms of the index...

assigning fugacities,

An Application: CEHT Duality

The (electric) index is

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...

An Application: CEHT Duality

To dualize the SU group (apply An-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.

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

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.)

• Total ellipticity is a necessary condition for physical consistency, but possibly not a sufficient condition– Prove sufficiency; would imply TrR, TrR3 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

• 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

Thank you for your attention.