text: what means that AH contains (or coincide with) the closure of the unstable manifold W U(O )7. 2 b) ε = 0.1463 d) ε = 0.1464 Figure 5: The location of manifolds of the saddle points sπ, sπ, sa and sr. Wu is the 1 2 1 1 a1 unstable manifold of the saddle point sa belonging to the attractor AH ; Ws and Ws are 1 1 1 2 the stable manifolds forming the boundary of the basin of attraction for this attractor. Ws is the stable manifold of sr belonging to the repeller RH , while Wu and Wu are r1 1 1 1 2 unstable manifolds forming the “basin of repulsion” for RH . 1 In the case under consideration the attractor AH belongs to the closure 1 of the unstable manifold W u of the saddle ﬁxed point sa. The basin of a1 1 attraction of AH is bounded from above by the stable manifold W s of the 1 1 symmetric saddle point sπ, and from below – by the stable manifold W s of 1 2 the saddle point sπ (see Fig. 5a). We also note that the manifolds W s and 2 1 W u intersect transversally. 1 When ε > ε , intersections between W s and W u , as well as W u and cris1 1 a1 1 W s appear (see Fig. 5b when ε = 0.1464) and, as a result, the attractor r1 AH collides with the upper boundary of its basin of attraction while the 1 repeller RH collides with the upper boundary of its “basin of repulsion”. 1 We note that after this collision the transversal intersection between W s and 1 W u remains and thus we can state here that the attractor AH intersects 1 1 with the repeller RH . But in this case such an intersection is not visible due 1 to the existing of homoclinic Henon-like attractor AH which attracts almost 2 all trajectories from the neighborhood of AH in forward time and RH which 1 2 attracts trajectories in backward time (here after the collision of AH and 1 RH the unstable manifold W u also intersects with the stable manifold W s 1 a1 2 of the symmetric saddle point sπ, see Fig. 6a, and this intersection gives a 2 transition mechanism from AH to AH ). 1 2 When ε = ε the unstable manifold W u of sa, belonging to the at- cris2 a2 2 tractor AH touches the stable manifold W s which forms the boundary of 2 2 the basin of attraction for AH . The same non-transversal heteroclinic tan- 2 gency appears between another pair of manifolds: W s and W u, one of which r2 2 7Recall that the prolongation of a set Q is a closed, invariant stable set consisting of all points attainable from Q by ε+-orbits for arbitrarily small ε [4]. Here ε+-orbits are ε-orbits for forward iteration of the map; ε > 0. 10 ##LTLine##

text: what

text: means

text: that

text: AH

text: contains

text: (or

text: coincide

text: with)

text: the

text: closure

text: of

text: the

text: unstable

text: manifold

text: W

text: U(O

text: b)

text: ε

text: d)

text: ε

text: Figure

text: The

text: location

text: of

text: manifolds

text: of

text: the

text: saddle

text: points

text: sπ,

text: sπ,

text: sa

text: and

text: sr.

text: Wu

text: is

text: the

text: unstable

text: manifold

text: of

text: the

text: saddle

text: point

text: sa

text: belonging

text: to

text: the

text: attractor

text: AH

text: Ws

text: and

text: Ws

text: are

text: the

text: stable

text: manifolds

text: forming

text: the

text: boundary

text: of

text: the

text: basin

text: of

text: attraction

text: for

text: this

text: attractor.

text: Ws

text: is

text: the

text: stable

text: manifold

text: of

text: sr

text: belonging

text: to

text: the

text: repeller

text: RH

text: while

text: Wu

text: and

text: Wu

text: are

text: unstable

text: manifolds

text: forming

text: the

text: “basin

text: of

text: repulsion”

text: for

text: RH

text: In

text: the

text: case

text: under

text: consideration

text: the

text: attractor

text: AH

text: belongs

text: to

text: the

text: closure

text: of

text: the

text: unstable

text: manifold

text: W

text: u

text: of

text: the

text: saddle

text: ﬁxed

text: point

text: sa.

text: The

text: basin

text: of

text: attraction

text: of

text: AH

text: is

text: bounded

text: from

text: above

text: by

text: the

text: stable

text: manifold

text: W

text: s

text: of

text: the

text: symmetric

text: saddle

text: point

text: sπ,

text: and

text: from

text: below

text: by

text: the

text: stable

text: manifold

text: W

text: s

text: of

text: the

text: saddle

text: point

text: sπ

text: (see

text: Fig.

text: We

text: also

text: note

text: that

text: the

text: manifolds

text: W

text: s

text: and

text: W

text: u

text: intersect

text: transversally.

text: When

text: ε

text: ε

text: intersections

text: between

text: W

text: s

text: and

text: W

text: u

text: as

text: well

text: as

text: W

text: u

text: and

text: W

text: s

text: appear

text: (see

text: Fig.

text: when

text: ε

text: and,

text: as

text: a

text: result,

text: the

text: attractor

text: AH

text: collides

text: with

text: the

text: upper

text: boundary

text: of

text: its

text: basin

text: of

text: attraction

text: while

text: the

text: repeller

text: RH

text: collides

text: with

text: the

text: upper

text: boundary

text: of

text: its

text: “basin

text: of

text: repulsion”.

text: We

text: note

text: that

text: after

text: this

text: collision

text: the

text: transversal

text: intersection

text: between

text: W

text: s

text: and

text: W

text: u

text: remains

text: and

text: thus

text: we

text: can

text: state

text: here

text: that

text: the

text: attractor

text: AH

text: intersects

text: with

text: the

text: repeller

text: RH

text: But

text: in

text: this

text: case

text: such

text: an

text: intersection

text: is

text: not

text: visible

text: due

text: to

text: the

text: existing

text: of

text: homoclinic

text: Henon-like

text: attractor

text: AH

text: which

text: attracts

text: almost

text: all

text: trajectories

text: from

text: the

text: neighborhood

text: of

text: AH

text: in

text: forward

text: time

text: and

text: RH

text: which

text: attracts

text: trajectories

text: in

text: backward

text: time

text: (here

text: after

text: the

text: collision

text: of

text: AH

text: and

text: RH

text: the

text: unstable

text: manifold

text: W

text: u

text: also

text: intersects

text: with

text: the

text: stable

text: manifold

text: W

text: s

text: of

text: the

text: symmetric

text: saddle

text: point

text: sπ,

text: see

text: Fig.

text: and

text: this

text: intersection

text: gives

text: a

text: transition

text: mechanism

text: from

text: AH

text: to

text: AH

text: When

text: ε

text: ε

text: the

text: unstable

text: manifold

text: W

text: u

text: of

text: sa,

text: belonging

text: to

text: the

text: at-

text: tractor

text: AH

text: touches

text: the

text: stable

text: manifold

text: W

text: s

text: which

text: forms

text: the

text: boundary

text: of

text: the

text: basin

text: of

text: attraction

text: for

text: AH

text: The

text: same

text: non-transversal

text: heteroclinic

text: tan-

text: gency

text: appears

text: between

text: another

text: pair

text: of

text: manifolds:

text: W

text: s

text: and

text: W

text: u,

text: one

text: of

text: which

text: that

text: the

text: prolongation

text: of

text: a

text: set

text: Q

text: is

text: a

text: closed,

text: invariant

text: stable

text: set

text: consisting

text: of

text: all

text: points

text: attainable

text: from

text: Q

text: by

text: ε+-orbits

text: for

text: arbitrarily

text: small

text: ε

text: Here

text: ε+-orbits

text: are

text: ε-orbits

text: for

text: forward

text: iteration

text: of

text: the

text: map;

text: ε

text: ##LTLine##


text: Interplay of coherent and dissipative dynamics in condensates of light 15 (cid:18) (cid:19) non-interacting interacting bosons in two dimensions [14, 15] π2 k T 2 ncr = (cid:126)B = 2640, (25) 3 Ω where the existence of two independent polarisations of light has been taken into account, and the temperature is T = 300 K. Note that the rise of the total photon number n around the transition point becomes smoothened with increasing the tot number of cavity levels L. This can be explained with the degeneracy d of the (cid:96)- (cid:96) th level, which becomes suﬃciently large for higher (cid:96) to contribute signiﬁcantly to n , tot as is seen in ﬁgure 3(a). The mode occupation n for a ﬁxed γ↑ is shown in a logarithmic representation in (cid:96) ﬁgure 4(a), along the triple-dashed vertical line of ﬁgure 3(a). The linear behaviour indicates that the modes are distributed according to Bose and Einstein via 1 n = − − . (26) (cid:96) exp[β(δ µ)] 1 (cid:96) Fitting a linear function to log (1 + 1/n ) yields the inverse temperature β and the (cid:96) chemical potential µ, due to equation (26). For a non-interacting BEC condensate, i.e., (cid:103) = 0, the temperature obtained by ﬁtting to a thermal cloud coincides with the β spectral temperature of the dye and the chemical potential is locked to the value δ [34]. 1 However, here (cid:103) is non-zero but small, which induces additional small corrections of the β occupation. As a consequence the eﬀective thermalisation temperature of the thermal cloud diﬀers generically from the spectral temperature of the dye. This behaviour depends on all system parameters, and especially a high pump parameter γ↑ can signiﬁcantly change this temperature. For a choice of parameters outside of some region the non-equilibrium properties of the model are dominating the tendency to thermalise, thus the distribution is, in general, not thermal any more. This happens even in the case (cid:103) = 0 [35]. Therefore, we restrict our further analysis only to the cases, where a thermal β cloud does exist and the corresponding temperature turns out to diﬀer maximally by 6 % from the dye spectral temperature. We have observed that a few of the lowest modes do not follow the linear dependency in ﬁgure 4(a), thus they are not considered in our ﬁtting procedure. Additionally, we drop some of the highest modes as well, as they carry the smallest occupation and can have, therefore, a larger relative error in our numerics. The resulting µ value from ﬁgure 4(a) is shown by a star symbol in ﬁgure 4(b). We repeat the procedure for diﬀerent values of the pumping rate γ↑ and obtain a linear equation of state according to ﬁgure 4(b). Note, that for the x-axis we now take n tot instead of γ↑, their connection is shown in ﬁgure 3(a). This ﬁnding is in contrast to the case (cid:103) = 0, where µ would be constant for all values of n [34]. β tot The slope ∂µ/∂n of the equation of state µ(n ) in ﬁgure 4(b) results, according tot tot to , from a dye-mediated eﬀective photon-photon interaction, which is measured by the dimensionless parameter g˜: 2π ∂µ g˜ = (cid:126) . (27) Ω ∂n tot ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Interplay

label: title

text: of

text: coherent

text: and

text: dissipative

text: dynamics

text: in

text: condensates

text: of

text: light

label: paragraph

label: equation

text: non-interacting

text: interacting

text: bosons

text: in

text: two

text: dimensions

text: π2

text: k

text: T

text: ncr

text: Ω

text: where

text: the

text: existence

text: of

text: two

text: independent

text: polarisations

text: of

text: light

text: has

text: been

text: taken

text: into

text: account,

text: and

text: the

text: temperature

text: is

text: T

text: K.

text: Note

text: that

text: the

text: rise

text: of

text: the

text: total

text: photon

text: number

text: n

text: around

text: the

text: transition

text: point

text: becomes

text: smoothened

text: with

text: increasing

text: the

text: tot

text: number

text: of

text: cavity

text: levels

text: L.

text: This

text: can

text: be

text: explained

text: with

text: the

text: degeneracy

text: d

text: of

text: the

text: th

text: level,

text: which

text: becomes

text: suﬃciently

text: large

text: for

text: higher

text: to

text: contribute

text: signiﬁcantly

text: to

text: n

text: tot

text: as

text: is

text: seen

text: in

text: ﬁgure

text: The

text: mode

text: occupation

text: n

text: for

text: a

text: ﬁxed

text: γ↑

text: is

text: shown

text: in

text: a

text: logarithmic

text: representation

text: in

text: ﬁgure

text: along

text: the

text: triple-dashed

text: vertical

text: line

text: of

text: ﬁgure

text: The

text: linear

text: behaviour

text: indicates

text: that

text: the

text: modes

text: are

text: distributed

text: according

text: to

text: Bose

text: and

text: Einstein

text: via

text: n

text: exp[β(δ

text: µ)]

text: Fitting

text: a

text: linear

text: function

text: to

text: log

text: yields

text: the

text: inverse

text: temperature

text: β

text: and

text: the

text: chemical

text: potential

text: µ,

text: due

text: to

text: equation

text: For

text: a

text: non-interacting

text: BEC

text: condensate,

text: i.e.,

text: the

text: temperature

text: obtained

text: by

text: ﬁtting

text: to

text: a

text: thermal

text: cloud

text: coincides

text: with

text: the

text: β

text: spectral

text: temperature

text: of

text: the

text: dye

text: and

text: the

text: chemical

text: potential

text: is

text: locked

text: to

text: the

text: value

text: δ

text: However,

text: here

text: is

text: non-zero

text: but

text: small,

text: which

text: induces

text: additional

text: small

text: corrections

text: of

text: the

text: β

text: occupation.

text: As

text: a

text: consequence

text: the

text: eﬀective

text: thermalisation

text: temperature

text: of

text: the

text: thermal

text: cloud

text: diﬀers

text: generically

text: from

text: the

text: spectral

text: temperature

text: of

text: the

text: dye.

text: This

text: behaviour

text: depends

text: on

text: all

text: system

text: parameters,

text: and

text: especially

text: a

text: high

text: pump

text: parameter

text: γ↑

text: can

text: signiﬁcantly

text: change

text: this

text: temperature.

text: For

text: a

text: choice

text: of

text: parameters

text: outside

text: of

text: some

text: region

text: the

text: non-equilibrium

text: properties

text: of

text: the

text: model

text: are

text: dominating

text: the

text: tendency

text: to

text: thermalise,

text: thus

text: the

text: distribution

text: is,

text: in

text: general,

text: not

text: thermal

text: any

text: more.

text: This

text: happens

text: even

text: in

text: the

text: case

text: Therefore,

text: we

text: restrict

text: our

text: further

text: analysis

text: only

text: to

text: the

text: cases,

text: where

text: a

text: thermal

text: β

text: cloud

text: does

text: exist

text: and

text: the

text: corresponding

text: temperature

text: turns

text: out

text: to

text: diﬀer

text: maximally

text: by

text: from

text: the

text: dye

text: spectral

text: temperature.

text: We

text: have

text: observed

text: that

text: a

text: few

text: of

text: the

text: lowest

text: modes

text: do

text: not

text: follow

text: the

text: linear

text: dependency

text: in

text: ﬁgure

text: thus

text: they

text: are

text: not

text: considered

text: in

text: our

text: ﬁtting

text: procedure.

text: Additionally,

text: we

text: drop

text: some

text: of

text: the

text: highest

text: modes

text: as

text: well,

text: as

text: they

text: carry

text: the

text: smallest

text: occupation

text: and

text: can

text: have,

text: therefore,

text: a

text: larger

text: relative

text: error

text: in

text: our

text: numerics.

text: The

text: resulting

text: µ

text: value

text: from

text: ﬁgure

text: is

text: shown

text: by

text: a

text: star

text: symbol

text: in

text: ﬁgure

text: We

text: repeat

text: the

text: procedure

text: for

text: diﬀerent

text: values

text: of

text: the

text: pumping

text: rate

text: γ↑

text: and

text: obtain

text: a

text: linear

text: equation

text: of

text: state

text: according

text: to

text: ﬁgure

text: Note,

text: that

text: for

text: the

text: x-axis

text: we

text: now

text: take

text: n

text: tot

text: instead

text: of

text: γ↑,

text: their

text: connection

text: is

text: shown

text: in

text: ﬁgure

text: This

text: ﬁnding

text: is

text: in

text: contrast

text: to

text: the

text: case

text: where

text: µ

text: would

text: be

text: constant

text: for

text: all

text: values

text: of

text: n

text: β

text: tot

text: The

text: slope

text: ∂µ/∂n

text: of

text: the

text: equation

text: of

text: state

text: µ(n

text: in

text: ﬁgure

text: results,

text: according

text: tot

text: tot

text: to

text: from

text: a

text: dye-mediated

text: eﬀective

text: photon-photon

text: interaction,

text: which

text: is

text: measured

text: by

text: the

text: dimensionless

text: parameter

text: g˜:

text: 2π

text: ∂µ

text: g˜

text: Ω

text: ∂n

text: tot

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Interplay of coherent an(cid:88)d d(cid:88)issip(cid:0)ative dynamics(cid:1)in condensates of light 5 M N † − V = (cid:103) a σ + a σ+ , (1c) A,C m j m j m=1 j=1 † where ω denote the cavity-mode frequencies and a (a ) the bosonic annihilation m m m (creation) operators of the cavity modes. The Hamiltonian H(j) describes the j-th ± A,R TLS including its reservoir, with σ and σz being its Pauli spin operators. Bosonic j j annihilation (creation) operators and frequencies of the reservoir oscillators are b † j,r (b ) and w , respectively, while λ are the appropriate interaction strengths. Since j,r r r the experimental spectra of the dye molecules are very broad, we are led to assume (cid:34) (cid:35) that the TLS-reservoir c(cid:88)oupling(cid:88)is strong. In order to treat it non-perturbatively to all † ˜ orders, we perform the polaron transformation H = UHU with N R λ † − U = exp σz r (b b ) , (2) j w j,r j,r r j=1 r=1 and ﬁnd (cid:88) (cid:88) (cid:88) M N N † ∆ H˜ = ω(cid:88)ma(cid:88)mam(cid:0)+ 2 σjz + V˜A,C +(cid:1) HR(j), (cid:88) (3a) m=1 j=1 j=1 M N R † − − † V˜ = (cid:103) a σ D + a σ+D+ , H(j) = w b b , (3b) A,C m j j m j j R r j,r j,r m=1 j=1 r=1 up to constant terms, wh(cid:20)ere (cid:21) ± ⊗ ±2λ † − D = R exp r (b b ) (4) j r=1 w j,r j,r r ˜ are the polaron displacement operators of the j-th TLS. In this way, V captures A,C the coupling of the TLS and the cavity modes which is dressed by the reservoir oscillators. In order to proceed further,(cid:104)we assum(cid:105)e that the oscillators in the polaron frame represent a bath in a thermal state at temperature T − ⊗ − ρ = Z 1 N exp βH(j) , (5) β β j=1 R where Z stands for the canonical partition function and β = 1/(k T ) [60]. We β B consider such initial conditions that the subsystem TLS-cavity is uncorrelated with the ⊗ bath in the polaron frame, i.e., ρ = ρ ρ . Since the coupling strength (cid:103) is total A,C β supposed to be weak, the bath inﬂuence can be incorporated by means of a master ˜ equation, i.e., by treating V as a perturbation up to the second order [61, 62]. The A,C ##LTLine## ##LTLine## ##LTLine##

text: Interplay

text: of

text: coherent

text: condensates

text: of

text: light

text: M

text: N

text: V

font: YQDAMK+DutchCalligraphic-Regular

text: a

text: σ

text: a

text: σ+

text: A,C

text: m

text: j

text: m

text: j

text: where

text: ω

text: denote

text: the

text: cavity-mode

text: frequencies

text: and

text: a

text: (a

text: the

text: bosonic

text: annihilation

text: m

text: m

text: m

text: (creation)

text: operators

text: of

text: the

text: cavity

text: modes.

text: The

text: Hamiltonian

text: H(j)

text: describes

text: the

text: j-th

text: A,R

text: TLS

text: including

text: its

text: reservoir,

text: with

text: σ

text: and

text: σz

text: being

text: its

text: Pauli

text: spin

text: operators.

text: Bosonic

text: j

text: j

text: annihilation

text: (creation)

text: operators

text: and

text: frequencies

text: of

text: the

text: reservoir

text: oscillators

text: are

text: b

text: j,r

text: (b

text: and

text: w

text: respectively,

text: while

text: λ

text: are

text: the

text: appropriate

text: interaction

text: strengths.

text: Since

text: j,r

text: r

text: r

text: the

text: experimental

text: spectra

text: of

text: the

text: dye

text: molecules

text: are

text: very

text: broad,

text: we

text: are

text: led

text: to

text: assume

text: that

text: the

text: TLS-reservoir

text: strong.

text: In

text: order

text: to

text: treat

text: it

text: non-perturbatively

text: to

text: all

text: orders,

text: we

text: perform

text: the

text: polaron

text: transformation

text: H

text: UHU

text: with

text: N

text: R

text: λ

text: U

text: exp

text: σz

text: r

text: (b

text: b

text: j

text: w

text: j,r

text: j,r

text: r

text: and

text: ﬁnd

text: M

text: N

text: N

text: H˜

text: σjz

text: V˜A,C

text: HR(j),

text: M

text: N

text: R

text: V˜

text: a

text: σ

text: D

text: a

text: σ+D+

text: H(j)

text: w

text: b

text: b

text: A,C

text: m

text: j

text: j

text: m

text: j

text: j

text: R

text: r

text: j,r

text: j,r

text: up

text: to

text: constant

text: terms,

text: ±2λ

text: D

text: R

text: exp

text: r

text: (b

text: b

text: j

text: w

text: j,r

text: j,r

text: r

text: are

text: the

text: polaron

text: displacement

text: operators

text: of

text: the

text: j-th

text: TLS.

text: In

text: this

text: way,

text: V

text: captures

text: A,C

text: the

text: coupling

text: of

text: the

text: TLS

text: and

text: the

text: cavity

text: modes

text: which

text: is

text: dressed

text: by

text: the

text: reservoir

text: oscillators.

text: In

text: order

text: to

text: proceed

text: that

text: the

text: oscillators

text: in

text: the

text: polaron

text: frame

text: represent

text: a

text: bath

text: in

text: a

text: thermal

text: state

text: at

text: temperature

text: T

text: ρ

text: Z

text: N

text: exp

text: βH(j)

text: β

text: β

text: R

text: where

text: Z

text: stands

text: for

text: the

text: canonical

text: partition

text: function

text: and

text: β

text: T

text: We

text: β

text: B

text: consider

text: such

text: initial

text: conditions

text: that

text: the

text: subsystem

text: TLS-cavity

text: is

text: uncorrelated

text: with

text: the

text: bath

text: in

text: the

text: polaron

text: frame,

text: i.e.,

text: ρ

text: ρ

text: ρ

text: Since

text: the

text: coupling

text: strength

text: is

text: total

text: A,C

text: β

text: supposed

text: to

text: be

text: weak,

text: the

text: bath

text: inﬂuence

text: can

text: be

text: incorporated

text: by

text: means

text: of

text: a

text: master

text: equation,

text: i.e.,

text: by

text: treating

text: V

text: as

text: a

text: perturbation

text: up

text: to

text: the

text: second

text: order

text: The

text: A,C

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Interplay of coherent and dissipative dynamics in condensates of light 6 ﬁrst order contributes to the coherent unitary evolution through the thermal-averaged (cid:88) (cid:88) (cid:0) (cid:1) term (cid:104) (cid:105) M N † −(cid:104) −(cid:105) (cid:104) (cid:105) V˜ = (cid:103) a σ D + a σ+ D+ , (6) A,C β m j j β m j j β m=1 j=1 (cid:104) (cid:105) ≡ where we introduced the notation(cid:20) X Tr[Xρ(cid:21)] for a bath expectation value. (cid:10) (cid:11) β β Using the result | | † − ∗ − α 2 βω exp [αb α b] = exp coth , (7) β 2 2 for a harmonic oscillat(cid:34)or of frequency ω in(cid:35)a thermal state, we ﬁnd for the bath (cid:88) expectation value of the polaron displacement operators (4) (cid:104) ±(cid:105) − R λ2 βw D = exp 2 r coth r . (8) j β w2 2 r=1 r Hence, one can naturally introduce a bath-dressed TLS-cavity coupling strength (cid:104) ±(cid:105) (cid:103) = (cid:103) D . Obviously, due to (8) we have 0 < (cid:103) /(cid:103) < 1, so that the inﬂuence of the β j β β (cid:88) (cid:88) (cid:0) (cid:1) bath in the ﬁrst order is to eﬀectively reduce the TLS-cavity interaction (cid:104) (cid:105) M N † − V˜ = (cid:103) a σ + a σ+ . (9) A,C β β m j m j m=1 j=1 At this point we note that the previous ﬁrst-order term was spuriously omitted by Kirton and Keeling [34, 35]. As we will demonstrate below, its inﬂuence in the photon BEC regime turns out to be negligible, so this regime can be described satisfactorily even if it is not taken into account. However, in the opposite laser-like regime such a term does play a major role. Anyhow, on formal grounds, it should be part of a proper treatment. We continue by applying the Born-Markov approximation as well as RWA, by tracing out the bath degrees of freedom and by taking into account the cavity losses along with the pumping and the decay of the TLS, similarly as [34, 35]. Incoherent pumping can be formally described as coupling each TLS to a bath of inverted harmonic oscillators [63]. With this we ﬁnd that the reduced density matrix ρ of the TLS-cavity A,C subsystem o(cid:34)beys the following master equation (cid:35) (cid:40) (cid:88) (cid:88) (cid:88)(cid:0) (cid:1) (cid:88) − M † M N † − − M κ (cid:41) ρ˙A,C = (cid:88)i (cid:16) δmamam + (cid:103)β (cid:17) (cid:88)amσ(cid:88)j +(cid:18)amσj+ , ρA,C 2 (cid:76)[a(cid:19)m] m=1 m=1 j=1 m=1 − N γ↑ γ↓ − M N γ+ γ † − + (cid:76)[σ+] + (cid:76)[σ ] + m (cid:76)[a σ+] + m (cid:76)[a σ ] ρ ,(10) 2 j 2 j 2 m j 2 m j A,C j=1 m=1 j=1 { † }− † where (cid:76)[X]ρ = X X, ρ 2XρX and we have moved into the frame rotating with − the frequency ∆, so that δ = ω ∆ stands for the detuning of the cavity mode from m m ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Interplay

text: of

text: coherent

text: and

text: dissipative

text: dynamics

text: in

text: condensates

text: of

text: light

text: ﬁrst

text: order

text: contributes

text: to

text: the

text: coherent

text: unitary

text: evolution

text: through

text: the

text: thermal-averaged

text: term

text: M

text: N

text: V˜

text: a

text: σ

text: D

text: a

text: σ+

text: D+

text: A,C

text: β

text: m

text: j

text: j

text: β

text: m

text: j

text: j

text: β

text: where

text: we

text: introduced

text: the

text: X

text: for

text: a

text: bath

text: expectation

text: value.

text: β

text: β

text: Using

text: the

text: result

text: α

text: βω

text: exp

text: [αb

text: α

text: b]

text: exp

text: coth

text: β

text: for

text: a

text: harmonic

text: of

text: frequency

text: ω

text: thermal

text: state,

text: we

text: ﬁnd

text: for

text: the

text: bath

text: expectation

text: value

text: of

text: the

text: polaron

text: displacement

text: operators

text: R

text: λ2

text: βw

text: D

text: exp

text: r

text: coth

text: r

text: j

text: β

text: r

text: Hence,

text: one

text: can

text: naturally

text: introduce

text: a

text: bath-dressed

text: TLS-cavity

text: coupling

text: strength

text: D

text: Obviously,

text: due

text: to

text: we

text: have

text: so

text: that

text: the

text: inﬂuence

text: of

text: the

text: β

text: j

text: β

text: β

text: bath

text: in

text: the

text: ﬁrst

text: order

text: is

text: to

text: eﬀectively

text: reduce

text: the

text: TLS-cavity

text: interaction

text: M

text: N

text: V˜

text: a

text: σ

text: a

text: σ+

text: A,C

text: β

text: β

text: m

text: j

text: m

text: j

text: At

text: this

text: point

text: we

text: note

text: that

text: the

text: previous

text: ﬁrst-order

text: term

text: was

text: spuriously

text: omitted

text: by

text: Kirton

text: and

text: Keeling

text: As

text: we

text: will

text: demonstrate

text: below,

text: its

text: inﬂuence

text: in

text: the

text: photon

text: BEC

text: regime

text: turns

text: out

text: to

text: be

text: negligible,

text: so

text: this

text: regime

text: can

text: be

text: described

text: satisfactorily

text: even

text: if

text: it

text: is

text: not

text: taken

text: into

text: account.

text: However,

text: in

text: the

text: opposite

text: laser-like

text: regime

text: such

text: a

text: term

text: does

text: play

text: a

text: major

text: role.

text: Anyhow,

text: on

text: formal

text: grounds,

text: it

text: should

text: be

text: part

text: of

text: a

text: proper

text: treatment.

text: We

text: continue

text: by

text: applying

text: the

text: Born-Markov

text: approximation

text: as

text: well

text: as

text: RWA,

text: by

text: tracing

text: out

text: the

text: bath

text: degrees

text: of

text: freedom

text: and

text: by

text: taking

text: into

text: account

text: the

text: cavity

text: losses

text: along

text: with

text: the

text: pumping

text: and

text: the

text: decay

text: of

text: the

text: TLS,

text: similarly

text: as

text: Incoherent

text: pumping

text: can

text: be

text: formally

text: described

text: as

text: coupling

text: each

text: TLS

text: to

text: a

text: bath

text: of

text: inverted

text: harmonic

text: oscillators

text: With

text: this

text: we

text: ﬁnd

text: that

text: the

text: reduced

text: density

text: matrix

text: ρ

text: of

text: the

text: TLS-cavity

text: A,C

text: subsystem

text: the

text: following

text: master

text: equation

text: M

text: M

text: N

text: M

text: κ

text: ρ˙A,C

text: δmamam

text: ρA,C

text: N

text: γ↑

text: γ↓

text: M

text: N

text: γ+

text: γ

text: m

text: σ+]

text: m

text: σ

text: ρ

text: j

text: j

text: m

text: j

text: m

text: j

text: A,C

text: where

text: X

text: X,

text: ρ

text: and

text: we

text: have

text: moved

text: into

text: the

text: frame

text: rotating

text: with

text: the

text: frequency

text: so

text: that

text: δ

text: ω

text: stands

text: for

text: the

text: detuning

text: of

text: the

text: cavity

text: mode

text: from

text: m

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Interplay of coherent and dissipative dynamics in condensates of light 10 − levels ω = ω + ((cid:96) 1)Ω with (cid:96) = 1, . . . , L, such that the energy level ω has the (cid:96) 1 (cid:96) degeneracy d = 2(cid:96), where the factor 2 comes from the two independent polarisations (cid:96) of light. The l(cid:88)owest frequency ω r(cid:88)epresents the cavity cutoﬀ. Since degenerate cavity 1 modes evolve in the same manner, we conclude M L † † f(a , a , . . .) = d f(a , a , . . .), (20) m m (cid:96) (cid:96) (cid:96) m=1 (cid:96)=1 for any arbitrary function f, where from each level ω we have chosen a † (cid:96) representative mode described by a and a . (cid:96) (cid:96) 2.3. Bath model (cid:80) In the following we analyse the stationary solutions of the equations (15)–(19) in the two regimes: a photon BEC and a laser-like regime. To this end, we specialise the − model by choosing the bath spectral density, deﬁned by J(w) = R λ2 δ(w w ), to r=1 r r have the convenient form η − J(w) = w3 exp ( w/w ) , (21) w2 c c where η represents a dimensionless parameter measuring the coupling strength of the TLS to the bath, and w is the cutoﬀ frequency of the bath. This allows us to obtain c for (8) and (1(cid:40)3) th(cid:34)e closed-form e(cid:35)x(cid:41)pressions (cid:10) (cid:11) (cid:48) ± (cid:40)− 2(cid:34)ψ ( 1 ) (cid:35)(cid:41) (cid:10)D = e(cid:11)xp 2η 1 βwc , (22a) j β (βw )2 c (cid:48) − (cid:48) − (cid:48) − − 1 ψ (1 iwct) + ψ (1+iwct) 2ψ ( 1 ) D (t)D+ = exp 4η 1 − + βwc βwc βwc ,(22b) j j β (1 iw t)2 (βw )2 c c (cid:48) where ψ(z) = Γ (z)/Γ(z) denotes a logarithmic derivative of the gamma function. In relation to the experiments, both parameters η and w characterise the impact of c the used solvent on the spectral properties of the dye molecules. Below we discuss ± in detail that they change the absorption and the emission spectra γ( δ) drastically. Furthermore, they turn out to have an impact upon the resulting bath-dressed coupling strength (cid:103) . β 3. Determination of realistic model parameters In order to apply our theory to the current experimental setups, we have to ﬁx the model parameters in the experimentally accessible regimes. One of the key ingredients for the thermalisation of photons are the spectral properties of the dye molecules [16,45]. (cid:18) (cid:19) Whereas the Einstein rate coeﬃcient for absorption B (ω) is usually measured, the 12 stimulated emission rate B (ω) is determined via the Kennard-Stepanov relation [68–70] 21 (cid:126) B (ω) ∼ − ω 21 exp , (23) B (ω) k T 12 B ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Interplay

text: of

text: coherent

text: and

text: dissipative

text: dynamics

text: in

text: condensates

text: of

text: light

text: levels

text: ω

text: ω

text: with

text: L,

text: such

text: that

text: the

text: energy

text: level

text: ω

text: has

text: the

text: degeneracy

text: d

text: where

text: the

text: factor

text: comes

text: from

text: the

text: two

text: independent

text: polarisations

text: of

text: light.

text: The

text: frequency

text: ω

text: the

text: cavity

text: cutoﬀ.

text: Since

text: degenerate

text: cavity

text: modes

text: evolve

text: in

text: the

text: same

text: manner,

text: we

text: conclude

text: M

text: L

text: f(a

text: a

text: d

text: f(a

text: a

text: m

text: m

text: for

text: any

text: arbitrary

text: function

text: f,

text: where

text: from

text: each

text: level

text: ω

text: we

text: have

text: chosen

text: a

text: representative

text: mode

text: described

text: by

text: a

text: and

text: a

text: Bath

text: model

text: In

text: the

text: following

text: we

text: analyse

text: the

text: stationary

text: solutions

text: of

text: the

text: equations

text: in

text: the

text: two

text: regimes:

text: a

text: photon

text: BEC

text: and

text: a

text: laser-like

text: regime.

text: To

text: this

text: end,

text: we

text: specialise

text: the

text: model

text: by

text: choosing

text: the

text: bath

text: spectral

text: density,

text: deﬁned

text: by

text: J(w)

text: R

text: λ2

text: δ(w

text: w

text: to

text: r

text: r

text: have

text: the

text: convenient

text: form

text: η

text: J(w)

text: exp

text: w/w

text: c

text: c

text: where

text: η

text: represents

text: a

text: dimensionless

text: parameter

text: measuring

text: the

text: coupling

text: strength

text: of

text: the

text: TLS

text: to

text: the

text: bath,

text: and

text: w

text: is

text: the

text: cutoﬀ

text: frequency

text: of

text: the

text: bath.

text: This

text: allows

text: us

text: to

text: obtain

text: c

text: for

text: and

text: closed-form

text: 2η

text: βwc

text: j

text: β

text: (βw

text: c

text: ψ

text: iwct)

text: ψ

text: 2ψ

text: D

text: (t)D+

text: exp

text: 4η

text: βwc

text: βwc

text: βwc

text: j

text: j

text: β

text: iw

text: (βw

text: c

text: c

text: where

text: ψ(z)

text: Γ

text: (z)/Γ(z)

text: denotes

text: a

text: logarithmic

text: derivative

text: of

text: the

text: gamma

text: function.

text: In

text: relation

text: to

text: the

text: experiments,

text: both

text: parameters

text: η

text: and

text: w

text: characterise

text: the

text: impact

text: of

text: c

text: the

text: used

text: solvent

text: on

text: the

text: spectral

text: properties

text: of

text: the

text: dye

text: molecules.

text: Below

text: we

text: discuss

text: in

text: detail

text: that

text: they

text: change

text: the

text: absorption

text: and

text: the

text: emission

text: spectra

text: γ(

text: δ)

text: drastically.

text: Furthermore,

text: they

text: turn

text: out

text: to

text: have

text: an

text: impact

text: upon

text: the

text: resulting

text: bath-dressed

text: coupling

text: strength

text: β

text: Determination

text: of

text: realistic

text: model

text: parameters

text: In

text: order

text: to

text: apply

text: our

text: theory

text: to

text: the

text: current

text: experimental

text: setups,

text: we

text: have

text: to

text: ﬁx

text: the

text: model

text: parameters

text: in

text: the

text: experimentally

text: accessible

text: regimes.

text: One

text: of

text: the

text: key

text: ingredients

text: for

text: the

text: thermalisation

text: of

text: photons

text: are

text: the

text: spectral

text: properties

text: of

text: the

text: dye

text: molecules

text: Whereas

text: the

text: Einstein

text: rate

text: coeﬃcient

text: for

text: absorption

text: B

text: (ω)

text: is

text: usually

text: measured,

text: the

text: stimulated

text: emission

text: rate

text: B

text: (ω)

text: is

text: determined

text: via

text: the

text: Kennard-Stepanov

text: relation

text: B

text: (ω)

text: ω

text: exp

text: B

text: (ω)

text: k

text: T

text: B

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: HIGHER TOPOLOGICAL HOCHSCHILD HOMOLOGY OF PERIODIC COMPLEX K-THEORY BRUNO STONEK Abstract. We describe the topological Hochschild homology of the periodic complex K-theory spectrum, THH(KU), as a commutative KU-algebra: it is equivalent to KU[K(Z, 3)] and to F(ΣKUQ), where F is the free commutative KU-algebra functor on (cid:39) ∨ a KU-module. Moreover, F(ΣKUQ) KU ΣKUQ, a square-zero extension. In order to prove these results, we ﬁrst establish that topological Hochschild homology commutes, as an algebra, with localization at an element. ⊗ Then, we prove that THHn(KU), the n-fold iteration of THH(KU), i.e. Tn KU, is equivalent to KU[G] where G is a certain product of integral Eilenberg-Mac Lane spaces, and to a free commutative KU-algebra on a rational KU-module. We prove that Sn ⊗ KU is equivalent to KU[K(Z, n + 2)] and to F(ΣnKUQ). We describe the topological André-Quillen homology of KU. 1. Introduction Topological Hochschild homology ( ) of structured ring spectra was introduced T HH by Bökstedt [Bök85] and Breen [Bre78]; for an introduction to the subject, see [DGM13, Chapter 4], [EKMM97, Chapter IX] and [Shi00]. It is the generalization to structured ring spectra of classical Hochschild homology ( ) of rings. HH One reason for its importance is its relation to algebraic -theory. If is a (discrete) K R → ring, then the trace map factors through the topological Hochschild K(R) HH(R) homology of the Eilenberg-Mac Lane ring spectrum of . Moreover, the trace map R → exists for any ring spectrum . Out of topological Hochschild homol- K(A) T HH(A) A ogy one can build topological cyclic homology, which has close ties to algebraic -theory: K see [DGM13]. We might thus see as a more easily approachable stepping stone on T HH the way to the more fundamental algebraic -theory. K In this paper we are interested with the topological Hochschild homology of , the KU periodic complex topological -theory spectrum. Previously, McClure and Staﬀeldt K (cid:39) ∨ [MS93, Theorem 8.1] showed that as spectra, where is the -adic T HH(L) L ΣLQ L p completion of the Adams summand of for a given odd prime . More recently, a lot KU p of eﬀort was devoted to describe , where is connective complex -theory T HH(ku) ku K [Aus05]: that case is markedly harder. It should also be noted that, rationally, K(KU) was determined in [AR12, Theorem 3.6]. Our ﬁrst expression for T HH(KU) as a commutative KU-algebra is obtained in The- orem 5.5: (cid:39) Z T HH(KU) KU[K( , 3)], Z ∧ Z where the underlying -module of is . The second one is KU KU[K( , 3)] KU K( , 3) + given in Theorem 5.10: there is a morphism of commutative KU-algebras → ˜ f : F (ΣKUQ) T HH(KU) 1

text: HIGHER

text: TOPOLOGICAL

text: HOCHSCHILD

text: HOMOLOGY

text: OF

text: PERIODIC

text: COMPLEX

text: K-THEORY

text: BRUNO

text: STONEK

text: Abstract.

text: We

label: abstract

text: describe

text: the

text: topological

text: Hochschild

text: homology

text: of

text: the

text: periodic

text: complex

text: K-theory

text: spectrum,

text: THH(KU),

text: as

text: a

text: commutative

text: KU-algebra:

text: it

text: is

text: equivalent

text: to

text: KU[K(Z,

text: and

text: to

text: F(ΣKUQ),

text: where

text: F

text: is

text: the

text: free

text: commutative

text: KU-algebra

text: functor

text: on

text: a

text: KU-module.

text: Moreover,

text: F(ΣKUQ)

text: KU

text: ΣKUQ,

text: a

text: square-zero

text: extension.

text: In

text: order

text: to

text: prove

text: these

text: results,

text: we

text: ﬁrst

text: establish

text: that

text: topological

text: Hochschild

text: homology

text: commutes,

text: as

text: an

text: algebra,

text: with

text: localization

text: at

text: an

text: element.

text: Then,

text: we

text: prove

text: that

text: THHn(KU),

text: the

text: n-fold

text: iteration

text: of

text: THH(KU),

text: i.e.

text: Tn

text: KU,

text: is

text: equivalent

text: to

text: KU[G]

text: where

text: G

text: is

text: a

text: certain

text: product

text: of

text: integral

text: Eilenberg-Mac

text: Lane

text: spaces,

text: and

text: to

text: a

text: free

text: commutative

text: KU-algebra

text: on

text: a

text: rational

text: KU-module.

text: We

text: prove

text: that

text: Sn

text: KU

text: is

text: equivalent

text: to

text: KU[K(Z,

text: n

text: and

text: to

text: F(ΣnKUQ).

text: We

text: describe

text: the

text: topological

text: André-Quillen

text: homology

text: of

text: KU.

text: Introduction

text: Topological

text: Hochschild

text: homology

text: of

text: structured

text: ring

text: spectra

text: was

text: introduced

text: T

text: HH

text: by

text: Bökstedt

text: and

text: Breen

text: for

text: an

text: introduction

text: to

text: the

text: subject,

text: see

text: Chapter

text: Chapter

text: IX]

text: and

text: It

text: is

text: the

text: generalization

text: to

text: structured

text: ring

text: spectra

text: of

text: classical

text: Hochschild

text: homology

text: of

text: rings.

text: HH

text: One

text: reason

text: for

text: its

text: importance

text: is

text: its

text: relation

text: to

text: algebraic

text: -theory.

text: If

text: is

text: a

text: (discrete)

text: K

text: R

text: ring,

text: then

text: the

text: trace

text: map

text: factors

text: through

text: the

text: topological

text: Hochschild

text: K(R)

text: HH(R)

text: homology

text: of

text: the

text: Eilenberg-Mac

text: Lane

text: ring

text: spectrum

text: of

text: Moreover,

text: the

text: trace

text: map

text: R

text: exists

text: for

text: any

text: ring

text: spectrum

text: Out

text: of

text: topological

text: Hochschild

text: homol-

text: K(A)

text: T

text: HH(A)

text: A

text: ogy

text: one

text: can

text: build

text: topological

text: cyclic

text: homology,

text: which

text: has

text: close

text: ties

text: to

text: algebraic

text: -theory:

text: K

text: see

text: We

text: might

text: thus

text: see

text: as

text: a

text: more

text: easily

text: approachable

text: stepping

text: stone

text: on

text: T

text: HH

text: the

text: way

text: to

text: the

text: more

text: fundamental

text: algebraic

text: -theory.

text: K

text: In

text: this

text: paper

text: we

text: are

text: interested

text: with

text: the

text: topological

text: Hochschild

text: homology

text: of

text: the

text: KU

text: periodic

text: complex

text: topological

text: -theory

text: spectrum.

text: Previously,

text: McClure

text: and

text: Staﬀeldt

text: K

text: Theorem

text: showed

text: that

text: as

text: spectra,

text: where

text: is

text: the

text: -adic

text: T

text: HH(L)

text: L

text: ΣLQ

text: L

text: p

text: completion

text: of

text: the

text: Adams

text: summand

text: of

text: for

text: a

text: given

text: odd

text: prime

text: More

text: recently,

text: a

text: lot

text: KU

text: p

text: of

text: eﬀort

text: was

text: devoted

text: to

text: describe

text: where

text: is

text: connective

text: complex

text: -theory

text: T

text: HH(ku)

text: ku

text: K

text: that

text: case

text: is

text: markedly

text: harder.

text: It

text: should

text: also

text: be

text: noted

text: that,

text: rationally,

text: K(KU)

text: was

text: determined

text: in

text: Theorem

text: Our

text: ﬁrst

text: expression

text: for

text: T

text: HH(KU)

text: as

text: a

text: commutative

text: KU-algebra

text: is

text: obtained

text: in

text: The-

text: orem

text: Z

text: T

text: HH(KU)

text: KU[K(

text: Z

text: Z

text: where

text: the

text: underlying

text: -module

text: of

text: is

text: The

text: second

text: one

text: is

text: KU

text: KU[K(

text: KU

text: K(

text: given

text: in

text: Theorem

text: there

text: is

text: a

text: morphism

text: of

text: commutative

text: KU-algebras

text: f

text: F

text: (ΣKUQ)

text: T

text: HH(KU)


text: 18 BRUNO STONEK ∧ − S → since the left adjoint fun∼ctor R(cid:95) : -M(cid:14) od R-Mod (cid:95)pre(cid:16)serves co(cid:14)limi(cid:17)ts. Therefore, ∧ ∧ FR(ΣRQ) = FR(R SQ1 ) = R FS(SQ1 ). We have FS(SQ1 ) = S ∨ SQ1 ∨ (SQ1 )∧n Σ (cid:39) S ∨ SQ1 ∨ (S1)∧n Σ n n Q ≥ ≥ n 2 n 2 Q ∧ − where we have used that the rationalization functor commutes with colimits and H (cid:14) that the rationalization of a smash product of ﬁnitely many factors is weakly equivalent to the smash product of the rationalizations. (cid:102) ∧ ≥ Now we claim that (S1) n Σ is contractible for n 2, and this ﬁnishes the proof. To n see(cid:102)this, let X be a based space and consider SP n(X) := X∧n/Σ and SP n(X) := X×n/Σ . − → n n There is a map f(cid:102): SP n 1(X) SP n(X) given by inserting a basepoint, and its coﬁber n ≥ is SP n(X). When X = S1, the map f is a homotopy equivalence for any n 2 [AGP02, 5.2.23], so that SP n(S1) is contractiblne for all n ≥ 2. (cid:3) Proof of Theorem 5.10. First, we work additively, and then we will determine the multi- plicative structure. ∞ (cid:39) S ∨ ∞ Recall that for any based space , there is a weak equivalence X Σ X Σ X (cid:39) ∨ + coming from the homotopy equivalence of spaces Σ(X ) S1 ΣX. Combining this + with Theorem 5.5 and Proposition 5.14, we obtain weak equivalences of -modules KU ∼ (cid:39) ∧ ∞ Z (cid:39) ∧ S ∨ ∞ Z T HH(KU) KU Σ K( , 3) KU ( Σ K( , 3)) = + (cid:39) ∨ ∧ ∞ Z (cid:39) ∨ KU (KU Σ K( , 3)) KU ΣKUQ. This splitting is compatible with the splitting (5.9) since both are just splitting oﬀ the →∼ unit, so we have a weak equivalence of -modules . KU ΣKUQ T HH(KU) In particular, we have an isomorphism ∼ Z ± ⊕ Q ± (5.18) π∗T HH(KU) = [t 1] Σ [t 1] Z ± of [t 1]-graded modules. We will now determine the multiplicative structure. Consider the map of -modules KU ∼ (5.19) f : ΣKUQ (cid:47)(cid:47) T HH(KU) (cid:47)(cid:47) T HH(KU). Q ± → In homotopy groups, this is an isomorphism Σ [t 1] π∗T HH(KU) followed by the inclusion into π∗T HH(KU). By the universal property satisﬁed by , we get that induces a map of commutative F f -algebras KU → ˜ f : F (ΣKUQ) T HH(KU). ◦ → Note that, by deﬁnition of , we have that is the trivial map, where f ε f ε : T HH(KU) is the augmentation. This implies that ˜ preserves the augmentation. KU f Now, ˜ is a weak equivalence. Indeed, after identifying with the split square- f F (ΣKUQ) ∨ zero extension (Proposition 5.16), ˜ amounts to the map of commutative KU ΣKUQ f ∨ → -algebras . But is a weak equivalence by KU (η, f) : KU ΣKUQ T HH(KU) (η, f) (cid:3) construction of . f ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BRUNO

text: STONEK

text: S

text: since

text: the

text: left

text: adjoint

text: fun∼ctor

text: od

text: R-Mod

text: Therefore,

text: FR(ΣRQ)

text: FR(R

text: R

text: We

text: have

text: S

text: )∧n

text: Σ

text: S

text: Σ

text: n

text: n

text: Q

text: n

text: n

text: Q

text: where

text: we

text: have

text: used

text: that

text: the

text: rationalization

text: functor

text: commutes

text: with

text: colimits

text: and

text: H

text: that

text: the

text: rationalization

text: of

text: a

text: smash

text: product

text: of

text: ﬁnitely

text: many

text: factors

text: is

text: weakly

text: equivalent

text: to

text: the

text: smash

text: product

text: of

text: the

text: rationalizations.

text: Now

text: we

text: claim

text: that

text: n

text: Σ

text: is

text: contractible

text: for

text: n

text: and

text: this

text: ﬁnishes

text: the

text: proof.

text: To

text: n

text: let

text: X

text: be

text: a

text: based

text: space

text: and

text: consider

text: SP

text: n(X)

text: X∧n/Σ

text: and

text: SP

text: n(X)

text: X×n/Σ

text: n

text: n

text: There

text: is

text: a

text: map

text: SP

text: n

text: SP

text: n(X)

text: given

text: by

text: inserting

text: a

text: basepoint,

text: and

text: its

text: coﬁber

text: n

text: is

text: SP

text: n(X).

text: When

text: X

text: the

text: map

text: f

text: is

text: a

text: homotopy

text: equivalence

text: for

text: any

text: n

text: so

text: that

text: SP

text: is

text: contractiblne

text: for

text: all

text: n

text: Proof

text: of

text: Theorem

text: First,

text: we

text: work

text: additively,

text: and

text: then

text: we

text: will

text: determine

text: the

text: multi-

text: plicative

text: structure.

text: S

text: Recall

text: that

text: for

text: any

text: based

text: space

text: there

text: is

text: a

text: weak

text: equivalence

text: X

text: Σ

text: X

text: Σ

text: X

text: coming

text: from

text: the

text: homotopy

text: equivalence

text: of

text: spaces

text: Σ(X

text: ΣX.

text: Combining

text: this

text: with

text: Theorem

text: and

text: Proposition

text: we

text: obtain

text: weak

text: equivalences

text: of

text: -modules

text: KU

text: Z

text: S

text: Z

text: T

text: HH(KU)

text: KU

text: Σ

text: K(

text: KU

text: Σ

text: K(

text: Z

text: KU

text: (KU

text: Σ

text: K(

text: KU

text: ΣKUQ.

text: This

text: splitting

text: is

text: compatible

text: with

text: the

text: splitting

text: since

text: both

text: are

text: just

text: splitting

text: oﬀ

text: the

text: unit,

text: so

text: we

text: have

text: a

text: weak

text: equivalence

text: of

text: -modules

text: KU

text: ΣKUQ

text: T

text: HH(KU)

text: In

text: particular,

text: we

text: have

text: an

text: isomorphism

text: Z

text: Q

text: π∗T

text: HH(KU)

text: [t

text: Σ

text: [t

text: Z

text: of

text: [t

text: modules.

text: We

text: will

text: now

text: determine

text: the

text: multiplicative

text: structure.

text: Consider

text: the

text: map

text: of

text: -modules

text: KU

text: f

text: ΣKUQ

text: T

text: HH(KU)

text: T

text: HH(KU).

text: Q

text: In

text: homotopy

text: groups,

text: this

text: is

text: an

text: isomorphism

text: Σ

text: [t

text: π∗T

text: HH(KU)

text: followed

text: by

text: the

text: inclusion

text: into

text: π∗T

text: HH(KU).

text: By

text: the

text: universal

text: property

text: satisﬁed

text: by

text: we

text: get

text: that

text: induces

text: a

text: map

text: of

text: commutative

text: F

text: f

text: -algebras

text: KU

text: f

text: F

text: (ΣKUQ)

text: T

text: HH(KU).

text: Note

text: that,

text: by

text: deﬁnition

text: of

text: we

text: have

text: that

text: is

text: the

text: trivial

text: map,

text: where

text: f

text: ε

text: f

text: ε

text: T

text: HH(KU)

text: is

text: the

text: augmentation.

text: This

text: implies

text: that

text: preserves

text: the

text: augmentation.

text: KU

text: f

text: Now,

text: is

text: a

text: weak

text: equivalence.

text: Indeed,

text: after

text: identifying

text: with

text: the

text: split

text: square-

text: f

text: F

text: (ΣKUQ)

text: zero

text: extension

text: (Proposition

text: amounts

text: to

text: the

text: map

text: of

text: commutative

text: KU

text: ΣKUQ

text: f

text: -algebras

text: But

text: is

text: a

text: weak

text: equivalence

text: by

text: KU

text: (η,

text: f)

text: KU

text: ΣKUQ

text: T

text: HH(KU)

text: (η,

text: f)

text: construction

text: of

text: f

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: HIGHER TOPOLOGICAL HOCHSCHILD HOMOLOGY OF PERIODIC COMPLEX K-THEORY 23 F the question, as [DT16] did for , of whether more generally is such that A = H KU ⊗ (cid:39) ⊗ (cid:39) p provided . More ambitiously, it would be interesting to X KU Y KU ΣX ΣY S ﬁnd conditions on any commutative -algebra that guarantee this property. A We ﬁrst need a generalization of Proposition 5.14:  ≥  Proposition 6.13. Let r 3. There are weak equivalences of KU-modules (cid:87)  if is odd ∧ Z (cid:39) ΣKUQ r , KU K( , r) if is even KUQ r . ≥ m 1 Proof. When is odd, the proof of Proposition 5.14 works just as well, and when is r r even it gives us ∧ Z (cid:39) ∧ Z (cid:87) KU K( , r) KUQ K( , r)Q. So let r be even. As note(cid:87)d in Section , K(Z, r)Q (cid:39) ΩSQr+1. Now we use the James (cid:39) ∧ splitting which says that, for X a connected pointed CW -complex, ΣΩΣX Σ ≥ X m. ∞ (cid:39) ∞ ∧ (cid:95) m 1 Therefore, Σ ΩΣX Σ ≥ X m. Rationalizing it and applying it to X = Sr, we m 1 obtain ∞ Z (cid:39) ∞ (cid:39) ∞ Σ K( , r)Q Σ ΩSQr+1 Σ SQrm. ≥ m 1 (cid:3) Since is even, Bott periodicity gives the result. r Corollary 6.14. The augmentation ideal T HHn(KU) is rational. Proof. The expression (6.7) gives, after splitting oﬀ the units of the spherical group rings, a weak equivalence of -modules KU (cid:39) ∧ S ∨ ∞ Z ∧ · · · ∧ S ∨ ∞ Z (6.15) T HHn(KU) KU ( Σ [K( , a + 2)]) ( Σ [K( , a − + 2)]). 1 2n 1 S S ∧ (cid:39) ∧ Observe that if is a rational -module and is any -module, then . T X T X T XQ Distributing the terms in the previous expression and applying Proposition 6.13 gives the (cid:3) result. From what we have just observed, we have that n is a non-unital commu- T HH (KU) Q ± tative KUQ-algebra. Thus, its homotopy groups are a non-unital commutative [t 1]- algebra, which we now aim to describe. Since commutes with rationalization, we get a weak equivalence T HH →∼ n n T HH (KU) T HH (KUQ) of non-unital commutative -algebras. We aim to describe KU ∼ n n T HH∗(KU) = T HH∗(KUQ). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: HIGHER

text: TOPOLOGICAL

text: HOCHSCHILD

text: HOMOLOGY

text: OF

text: PERIODIC

text: COMPLEX

text: K-THEORY

text: F

text: the

text: question,

text: as

text: did

text: for

text: of

text: whether

text: more

text: generally

text: is

text: such

text: that

text: A

text: H

text: KU

text: p

text: provided

text: More

text: ambitiously,

text: it

text: would

text: be

text: interesting

text: to

text: X

text: KU

text: Y

text: KU

text: ΣX

text: ΣY

text: S

text: ﬁnd

text: conditions

text: on

text: any

text: commutative

text: -algebra

text: that

text: guarantee

text: this

text: property.

text: A

text: We

text: ﬁrst

text: need

text: a

text: generalization

text: of

text: Proposition

text: Proposition

text: Let

text: r

text: There

text: are

text: weak

text: equivalences

text: of

text: KU-modules

text: if

text: is

text: odd

text: Z

text: ΣKUQ

text: r

text: KU

text: K(

text: r)

text: if

text: is

text: even

text: KUQ

text: r

text: m

text: Proof.

text: When

text: is

text: odd,

text: the

text: proof

text: of

text: Proposition

text: works

text: just

text: as

text: well,

text: and

text: when

text: is

text: r

text: r

text: even

text: it

text: gives

text: us

text: Z

text: Z

text: KU

text: K(

text: r)

text: KUQ

text: K(

text: r)Q.

text: So

text: let

text: r

text: be

text: even.

text: As

text: in

text: Section

text: K(Z,

text: r)Q

text: Now

text: we

text: use

text: the

text: James

text: splitting

text: which

text: says

text: that,

text: for

text: X

text: a

text: connected

text: pointed

text: CW

text: -complex,

text: ΣΩΣX

text: Σ

text: X

text: m.

text: m

text: Therefore,

text: Σ

text: ΩΣX

text: Σ

text: X

text: m.

text: Rationalizing

text: it

text: and

text: applying

text: it

text: to

text: X

text: Sr,

text: we

text: m

text: obtain

text: Z

text: Σ

text: K(

text: r)Q

text: Σ

text: Σ

text: SQrm.

text: m

text: Since

text: is

text: even,

text: Bott

text: periodicity

text: gives

text: the

text: result.

text: r

text: Corollary

text: The

text: augmentation

text: ideal

text: T

text: HHn(KU)

text: is

text: rational.

text: Proof.

text: The

text: expression

text: gives,

text: after

text: splitting

text: oﬀ

text: the

text: units

text: of

text: the

text: spherical

text: group

text: rings,

text: a

text: weak

text: equivalence

text: of

text: -modules

text: KU

text: S

text: Z

text: S

text: Z

text: T

text: HHn(KU)

text: KU

text: Σ

text: [K(

text: a

text: Σ

text: [K(

text: a

text: S

text: S

text: Observe

text: that

text: if

text: is

text: a

text: rational

text: -module

text: and

text: is

text: any

text: -module,

text: then

text: T

text: X

text: T

text: X

text: T

text: XQ

text: Distributing

text: the

text: terms

text: in

text: the

text: previous

text: expression

text: and

text: applying

text: Proposition

text: gives

text: the

text: result.

text: From

text: what

text: we

text: have

text: just

text: observed,

text: we

text: have

text: that

text: n

text: is

text: a

text: non-unital

text: commu-

text: T

text: HH

text: (KU)

text: Q

text: tative

text: KUQ-algebra.

text: Thus,

text: its

text: homotopy

text: groups

text: are

text: a

text: non-unital

text: commutative

text: [t

text: algebra,

text: which

text: we

text: now

text: aim

text: to

text: describe.

text: Since

text: commutes

text: with

text: rationalization,

text: we

text: get

text: a

text: weak

text: equivalence

text: T

text: HH

text: n

text: n

text: T

text: HH

text: (KU)

text: T

text: HH

text: (KUQ)

text: of

text: non-unital

text: commutative

text: -algebras.

text: We

text: aim

text: to

text: describe

text: KU

text: n

text: n

text: T

text: HH∗(KU)

text: T

text: HH∗(KUQ).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: HIGHER TOPOLOGICAL HOCHSCHILD HOMOLOGY OF PERIODIC COMPLEX K-THEORY 27 ⊗ ⊗ − the commutativity of (7.4). Note that ˜ is a weak equivalence because is a id f Y KU left Quillen functor, assuming is a -complex. Y CW We will now identify the members of the left column. Proposition 7.9. Let (X, x ) and (Y, y ) be based spaces, and let A be a commutative 0 0 -algebra. R (1) There is an isomorphism of commutative -algebras A ∼ ⊗ ⊗ ∧ ⊗ Y (X A) = (Y X) A A R + R ⊗ ⊗ where (resp. ) denotes the tensoring of - (resp. - ) over R CAlg A CAlg R A . Top Moreover, the isomorphism makes the following diagram in - commute. A CAlg → ∧ The morphism nY : X Y X was deﬁned in (7.7). X + ⊗ νY ⊗ ⊗ ⊗ (7.10) X A X RA (cid:47)(cid:47) Y (X A) R A R ∼ ⊗ = nYX id (cid:40)(cid:40) ∧ (cid:15)(cid:15) ⊗ (Y X) A + R → (2) Let be an -module. Let - - be the free commutative M A F : A Mod A CAlg algebra functor. There is an isomorphism ∼ ⊗ ∧ ∧ ∧ Y F (X M) = F (Y X M) A + making the following diagram commute. ∧ νY ∧ ⊗ ∧ F(X M) F (X M) (cid:47)(cid:47) Y F (X M) A ∼ ∧ = F(nYX id) (cid:41)(cid:41) ∧ (cid:15)(cid:15) ∧ F (Y X M) + ∧ In the expression for a based space we are using the standard tensoring Z M Z ∧ ∞ ∧ of - over , i.e. . A Mod Top∗ Z M = Σ Z M → Proof. (1) Let be a commutative -algebra with unit . Using the deﬁning B A ϕ : A B ⊗ − adjunction for , we get a homeomorphism Y A ∼ ⊗ ⊗ ⊗ (7.11) - - A CAlg(Y (X A), B) = Top(Y, A CAlg(X A, B)). A R R ⊗ → The morphisms of commutative -algebras are the morphisms of com- A X A B ⊗ → R mutative -algebras making the following diagram commute: R g : X A B R A νAX ϕ ⊗ (cid:123)(cid:123) (cid:32)(cid:32) X A (cid:47)(cid:47) B. R g ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: HIGHER

text: TOPOLOGICAL

text: HOCHSCHILD

text: HOMOLOGY

text: OF

text: PERIODIC

text: COMPLEX

text: K-THEORY

text: the

text: commutativity

text: of

text: Note

text: that

text: is

text: a

text: weak

text: equivalence

text: because

text: is

text: a

text: id

text: f

text: Y

text: KU

text: left

text: Quillen

text: functor,

text: assuming

text: is

text: a

text: -complex.

text: Y

text: CW

text: We

text: will

text: now

text: identify

text: the

text: members

text: of

text: the

text: left

text: column.

text: Proposition

text: Let

text: (X,

text: x

text: and

text: (Y,

text: y

text: be

text: based

text: spaces,

text: and

text: let

text: A

text: be

text: a

text: commutative

text: -algebra.

text: R

text: There

text: is

text: an

text: isomorphism

text: of

text: commutative

text: -algebras

text: A

text: Y

text: (X

text: A)

text: (Y

text: X)

text: A

text: A

text: R

text: R

text: where

text: (resp.

text: denotes

text: the

text: tensoring

text: of

text: (resp.

text: over

text: R

text: CAlg

text: A

text: CAlg

text: R

text: A

text: Top

text: Moreover,

text: the

text: isomorphism

text: makes

text: the

text: following

text: diagram

text: in

text: commute.

text: A

text: CAlg

text: The

text: morphism

text: nY

text: X

text: Y

text: X

text: was

text: deﬁned

text: in

text: X

text: νY

text: X

text: A

text: X

text: RA

text: Y

text: (X

text: A)

text: R

text: A

text: R

text: nYX

text: id

text: (Y

text: X)

text: A

text: R

text: Let

text: be

text: an

text: -module.

text: Let

text: be

text: the

text: free

text: commutative

text: M

text: A

text: F

text: A

text: Mod

text: A

text: CAlg

text: algebra

text: functor.

text: There

text: is

text: an

text: isomorphism

text: Y

text: F

text: (X

text: M)

text: F

text: (Y

text: X

text: M)

text: A

text: making

text: the

text: following

text: diagram

text: commute.

text: νY

text: F(X

text: M)

text: F

text: (X

text: M)

text: Y

text: F

text: (X

text: M)

text: A

text: F(nYX

text: id)

text: F

text: (Y

text: X

text: M)

text: In

text: the

text: expression

text: for

text: a

text: based

text: space

text: we

text: are

text: using

text: the

text: standard

text: tensoring

text: Z

text: M

text: Z

text: of

text: over

text: i.e.

text: A

text: Mod

text: Top∗

text: Z

text: M

text: Σ

text: Z

text: M

text: Proof.

text: Let

text: be

text: a

text: commutative

text: -algebra

text: with

text: unit

text: Using

text: the

text: deﬁning

text: B

text: A

text: ϕ

text: A

text: B

text: adjunction

text: for

text: we

text: get

text: a

text: homeomorphism

text: Y

text: A

text: A

text: CAlg(Y

text: (X

text: A),

text: B)

text: Top(Y,

text: A

text: CAlg(X

text: A,

text: B)).

text: A

text: R

text: R

text: The

text: morphisms

text: of

text: commutative

text: -algebras

text: are

text: the

text: morphisms

text: of

text: com-

text: A

text: X

text: A

text: B

text: R

text: mutative

text: -algebras

text: making

text: the

text: following

text: diagram

text: commute:

text: R

text: g

text: X

text: A

text: B

text: R

text: A

text: νAX

text: ϕ

text: X

text: A

text: B.

text: R

text: g

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: HIGHER TOPOLOGICAL HOCHSCHILD HOMOLOGY OF PERIODIC COMPLEX K-THEORY 9 ∞ → S Consider F = Σ : Top -Mod: it is a strong symmetric monoidal functor + S [EKMM97, II.1.2]. If is a topological commutative monoid, we denote by the G [G] S ∞ S -module together with the commutative -algebra structure induced by the monoid Σ G + structure of . G • Endow the cartesian category with the standard cosimplicial space and Top ∆ S ∞ t•op the symmetric monoidal category -Mod with the cosimplicial spectrum Σ ∆ . By + top Theorem 2.9 of [Sto18], these beget strong symmetric monoidal functors of geometric realization | − | → | − | S → S : sTop Top and : s -Mod -Mod. S If is a topological commutative monoid or a commutative -algebra, deﬁne A | | Bcy(A) := B•cy(A) . S It is a commutative -algebra. In the -module case, this object deﬁnes the topological A Hochschild homology of , denoted (which has good homotopical behavior when A T HH(A) S is a coﬁbrant commutative -algebra [EKMM97, IX.2.7]), and if is an -bimodule A M A | | → then B•cy(A, M) deﬁnes T HH(A, M). Note that if f : A B is a weak equivalence S → of coﬁbrant commutative -algebras, then is a weak equivalence. T HH(A) T HH(B) → ∧ First note that B•cy(A) B•cy(B) is a weak equivalence in each level. Indeed, f p+1 is a weak equivalence since is a weak equivalence and and are coﬁbrant. Then, since the f A B S simplicial cyclic bar construction gives a proper simplicial -module [EKMM97, IX.2.8], we can apply [EKMM97, X.2.4] to get the conclusion. | − | Since the functors are strong symmetric monoidal, by realizing the isomorphism (4.3) we obtain the following Proposition 4.4. Let G be a topological commutative monoid. There is an isomorphism S of commutative -algebras [G] ∼ T HH(S[G]) = (cid:47)(cid:47) S[Bcy(G)]. Versions of the previous proposition have appeared in [SVW00, Remark 4.4], in [HM97, Theorem 7.1] in the setting of functors with smash product, and in [DGM13, Example 4.2.2.7] in the setting of -spaces. Note that we take care to note that this isomorphism Γ S respects the commutative -algebra structures. [G] We will also need the following proposition, obtained by applying Lemma 4.1 to the ∞ → S strong symmetric monoidal functor Σ : Top -Mod. + ∈ Proposition 4.5. Consider G, H CMon(Top). There is an isomorphism of commu- S tative -algebras [G] ∼ S ∧ S = S × [G] [H] (cid:47)(cid:47) [G H] natural in . H ##LTLine## ##LTLine##

text: HIGHER

text: TOPOLOGICAL

text: HOCHSCHILD

text: HOMOLOGY

text: OF

text: PERIODIC

text: COMPLEX

text: K-THEORY

text: S

text: Consider

text: F

text: Σ

text: Top

text: -Mod:

text: it

text: is

text: a

text: strong

text: symmetric

text: monoidal

text: functor

text: S

text: If

text: is

text: a

text: topological

text: commutative

text: monoid,

text: we

text: denote

text: by

text: the

text: G

text: [G]

text: S

text: S

text: -module

text: together

text: with

text: the

text: commutative

text: -algebra

text: structure

text: induced

text: by

text: the

text: monoid

text: Σ

text: G

text: structure

text: of

text: G

text: Endow

text: the

text: cartesian

text: category

text: with

text: the

text: standard

text: cosimplicial

text: space

text: and

text: Top

text: S

text: t•op

text: the

text: symmetric

text: monoidal

text: category

text: -Mod

text: with

text: the

text: cosimplicial

text: spectrum

text: Σ

text: By

text: top

text: Theorem

text: of

text: these

text: beget

text: strong

text: symmetric

text: monoidal

text: functors

text: of

text: geometric

text: realization

text: S

text: S

text: sTop

text: Top

text: and

text: s

text: -Mod

text: -Mod.

text: S

text: If

text: is

text: a

text: topological

text: commutative

text: monoid

text: or

text: a

text: commutative

text: -algebra,

text: deﬁne

text: A

text: Bcy(A)

text: B•cy(A)

text: S

text: It

text: is

text: a

text: commutative

text: -algebra.

text: In

text: the

text: -module

text: case,

text: this

text: object

text: deﬁnes

text: the

text: topological

text: A

text: Hochschild

text: homology

text: of

text: denoted

text: (which

text: has

text: good

text: homotopical

text: behavior

text: when

text: A

text: T

text: HH(A)

text: S

text: is

text: a

text: coﬁbrant

text: commutative

text: -algebra

text: and

text: if

text: is

text: an

text: -bimodule

text: A

text: M

text: A

text: then

text: B•cy(A,

text: M)

text: deﬁnes

text: T

text: HH(A,

text: M).

text: Note

text: that

text: if

text: f

text: A

text: B

text: is

text: a

text: weak

text: equivalence

text: S

text: of

text: coﬁbrant

text: commutative

text: -algebras,

text: then

text: is

text: a

text: weak

text: equivalence.

text: T

text: HH(A)

text: T

text: HH(B)

text: First

text: note

text: that

text: B•cy(A)

text: B•cy(B)

text: is

text: a

text: weak

text: equivalence

text: in

text: each

text: level.

text: Indeed,

text: f

text: is

text: a

text: weak

text: equivalence

text: since

text: is

text: a

text: weak

text: equivalence

text: and

text: and

text: are

text: coﬁbrant.

text: Then,

text: since

text: the

text: f

text: A

text: B

text: S

text: simplicial

text: cyclic

text: bar

text: construction

text: gives

text: a

text: proper

text: simplicial

text: -module

text: we

text: can

text: apply

text: to

text: get

text: the

text: conclusion.

text: Since

text: the

text: functors

text: are

text: strong

text: symmetric

text: monoidal,

text: by

text: realizing

text: the

text: isomorphism

text: we

text: obtain

text: the

text: following

text: Proposition

text: Let

text: G

text: be

text: a

text: topological

text: commutative

text: monoid.

text: There

text: is

text: an

text: isomorphism

text: S

text: of

text: commutative

text: -algebras

text: [G]

text: T

text: HH(S[G])

text: S[Bcy(G)].

text: Versions

text: of

text: the

text: previous

text: proposition

text: have

text: appeared

text: in

text: Remark

text: in

text: Theorem

text: in

text: the

text: setting

text: of

text: functors

text: with

text: smash

text: product,

text: and

text: in

text: Example

text: in

text: the

text: setting

text: of

text: -spaces.

text: Note

text: that

text: we

text: take

text: care

text: to

text: note

text: that

text: this

text: isomorphism

text: Γ

text: S

text: respects

text: the

text: commutative

text: -algebra

text: structures.

text: [G]

text: We

text: will

text: also

text: need

text: the

text: following

text: proposition,

text: obtained

text: by

text: applying

text: Lemma

text: to

text: the

text: S

text: strong

text: symmetric

text: monoidal

text: functor

text: Σ

text: Top

text: -Mod.

text: Proposition

text: Consider

text: G,

text: H

text: CMon(Top).

text: There

text: is

text: an

text: isomorphism

text: of

text: commu-

text: S

text: tative

text: -algebras

text: [G]

text: S

text: S

text: S

text: [G]

text: [H]

text: [G

text: H]

text: natural

text: in

text: H

text: ##LTLine##

text: ##LTLine##


text: ∗ (Dated: May 30, 2020) We investigate the asymptotic safety conjecture for quantum gravity including curvature invariants beyond Ricci scalars. Our strategy is put to work for families of gravitational actions which depend on functions of the Ricci scalar, the Ricci tensor, and products thereof. Combining functional renormalisation with high order polynomial approximations and full numerical integration we derive the renormalisation group ﬂow for all couplings and analyse their ﬁxed points, scaling exponents, and the ﬁxed point eﬀective action as a function of the background Ricci curvature. The theory is characterised by three relevant couplings. Higher-dimensional couplings show near-Gaussian scaling with increasing canonical mass dimension. We ﬁnd that Ricci tensor invariants stabilise the UV ﬁxed point and lead to a rapid convergence of polynomial approximations. We apply our results to models for cosmology and establish that the gravitational ﬁxed point admits inﬂationary solutions. We also compare ﬁndings with those from f(R)-type theories in the same approximation and pin-point the key new eﬀects due to Ricci tensor interactions. Implications for the asymptotic safety conjecture of gravity are indicated.

text: (Dated:

text: May

label: date

text: We

text: investigate

text: the

text: asymptotic

text: safety

text: conjecture

text: for

text: quantum

text: gravity

text: including

text: curvature

text: invariants

text: beyond

text: Ricci

text: scalars.

text: Our

text: strategy

text: is

text: put

text: to

text: work

text: for

text: families

text: of

text: gravitational

text: actions

text: which

text: depend

text: on

text: functions

text: of

text: the

text: Ricci

text: scalar,

text: the

text: Ricci

text: tensor,

text: and

text: products

text: thereof.

text: Combining

text: functional

text: renormalisation

text: with

text: high

text: order

text: polynomial

text: approximations

text: and

text: full

text: numerical

text: integration

text: we

text: derive

text: the

text: renormalisation

text: group

text: ﬂow

text: for

text: all

text: couplings

text: and

text: analyse

text: their

text: ﬁxed

text: points,

text: scaling

text: exponents,

text: and

text: the

text: ﬁxed

text: point

text: eﬀective

text: action

text: as

text: a

text: function

text: of

text: the

text: background

text: Ricci

text: curvature.

text: The

text: theory

text: is

text: characterised

text: by

text: three

text: relevant

text: couplings.

text: Higher-dimensional

text: couplings

text: show

text: near-Gaussian

text: scaling

text: with

text: increasing

text: canonical

text: mass

text: dimension.

text: We

text: ﬁnd

text: that

text: Ricci

text: tensor

text: invariants

text: stabilise

text: the

text: UV

text: ﬁxed

text: point

text: and

text: lead

text: to

text: a

text: rapid

text: convergence

text: of

text: polynomial

text: approximations.

text: We

text: apply

text: our

text: results

text: to

text: models

text: for

text: cosmology

text: and

text: establish

text: that

text: the

text: gravitational

text: ﬁxed

text: point

text: admits

text: inﬂationary

text: solutions.

text: We

text: also

text: compare

text: ﬁndings

text: with

text: those

text: from

text: f(R)-type

text: theories

text: in

text: the

text: same

text: approximation

text: and

text: pin-point

text: the

text: key

text: new

text: eﬀects

text: due

text: to

text: Ricci

text: tensor

text: interactions.

text: Implications

text: for

text: the

text: asymptotic

text: safety

text: conjecture

text: of

text: gravity

text: are

text: indicated.


text: 14 vector ﬁeld. B. Wilsonian momentum cutoﬀ We will now discuss the speciﬁcs of the regulator term Rφiφj, which needs to be added to (3.1). k For convenience, we introduce the short-hand notation (cid:16) (cid:17) G = Γ(2) φiφj + Rφiφj (3.9) φiφj k k to denote the φ φ -component of the inverse regularised propagator. We will drop the φ φ i j i j indices when it is clear from the context. The coeﬃcients A (3.1) depend on the momentum scale m k through the couplings of the theory. For our purposes it is enough to determine the regulator for type I cutoﬀ. However, the process described here has a straightforward generalisation to the other types of cutoﬀs described in [30]. For type I cutoﬀs we deﬁne the Wilsonian regulator (see Sec. ) by demanding that the addition of the regulator term Rφiφj to the inverse propagator of k (cid:3) (cid:3) (cid:3) the corresponding ﬁeld leads to the replacement of the operator − by − + R (− ), where (cid:3) k R (− ) is a suitably chosen proﬁle function characterising the Wilsonian momentum cutoﬀ for k the ﬁeld modes. Using (3.9) together with (3.1) and the requirement on the regulator, the inverse regularised propagator takes the form (cid:88)p (cid:3) G = A (− + R )m (3.10) m k m=0 Solving (3.10) with (3.9) for R , we ﬁnd k (cid:88)p (cid:3) (cid:3) R = A [(− + R )m − (− )m] . (3.11) k m k m=1 Now that we have an explicit form for the regulator term we can proceed to determine its scale derivative, which is an essential component of the ﬂow equation (2.1). For type I cutoﬀ the operator (cid:3) − does not contain any couplings and thus it does not carry any RG scale-dependence. Therefore the ∂ derivative of the regulator reads t ∂ R = C ∂ R + C (3.12) t k 1 t k 2 where (cid:88)p (cid:3) − C = mA (− + R )m 1 (3.13) 1 m k m=1 (cid:88)p (cid:3) (cid:3) C = (∂ A ) [(− + R )m − (− )m] . (3.14) 2 t m k m=1 In this paper, we will use one and the same shape function R for all ﬁelds. This choice can be k omitted in more general settings.

text: vector

text: ﬁeld.

text: B.

label: section

text: Wilsonian

text: momentum

text: cutoﬀ

text: We

text: will

text: now

text: discuss

text: the

text: speciﬁcs

text: of

text: the

text: regulator

text: term

text: Rφiφj,

text: which

text: needs

text: to

text: be

text: added

text: to

text: k

text: For

text: convenience,

text: we

text: introduce

text: the

text: short-hand

text: notation

text: G

text: φiφj

text: Rφiφj

text: φiφj

text: k

text: k

text: to

text: denote

text: the

text: φ

text: φ

text: -component

text: of

text: the

text: inverse

text: regularised

text: propagator.

text: We

text: will

text: drop

text: the

text: φ

text: φ

text: i

text: j

text: i

text: j

text: indices

text: when

text: it

text: is

text: clear

text: from

text: the

text: context.

text: The

text: coeﬃcients

text: A

text: depend

text: on

text: the

text: momentum

text: scale

text: m

text: k

text: through

text: the

text: couplings

text: of

text: the

text: theory.

text: For

text: our

text: purposes

text: it

text: is

text: enough

text: to

text: determine

text: the

text: regulator

text: for

text: type

text: I

text: cutoﬀ.

text: However,

text: the

text: process

text: described

text: here

text: has

text: a

text: straightforward

text: generalisation

text: to

text: the

text: other

text: types

text: of

text: cutoﬀs

text: described

text: in

text: For

text: type

text: I

text: cutoﬀs

text: we

text: deﬁne

text: the

text: Wilsonian

text: regulator

text: (see

text: Sec.

text: by

text: demanding

text: that

text: the

text: addition

text: of

text: the

text: regulator

text: term

text: Rφiφj

text: to

text: the

text: inverse

text: propagator

text: of

text: k

text: the

text: corresponding

text: ﬁeld

text: leads

text: to

text: the

text: replacement

text: of

text: the

text: operator

text: by

text: R

text: where

text: k

text: R

text: is

text: a

text: suitably

text: chosen

text: proﬁle

text: function

text: characterising

text: the

text: Wilsonian

text: momentum

text: cutoﬀ

text: for

text: k

text: the

text: ﬁeld

text: modes.

text: Using

text: together

text: with

text: and

text: the

text: requirement

text: on

text: the

text: regulator,

text: the

text: inverse

text: regularised

text: propagator

text: takes

text: the

text: form

text: G

text: A

text: R

text: )m

text: m

text: k

text: Solving

text: with

text: for

text: R

text: we

text: ﬁnd

text: k

text: R

text: A

text: R

text: )m

text: )m]

text: k

text: m

text: k

text: Now

text: that

text: we

text: have

text: an

text: explicit

text: form

text: for

text: the

text: regulator

text: term

text: we

text: can

text: proceed

text: to

text: determine

text: its

text: scale

text: derivative,

text: which

text: is

text: an

text: essential

text: component

text: of

text: the

text: ﬂow

text: equation

text: For

text: type

text: I

text: cutoﬀ

text: the

text: operator

text: does

text: not

text: contain

text: any

text: couplings

text: and

text: thus

text: it

text: does

text: not

text: carry

text: any

text: RG

text: scale-dependence.

text: Therefore

text: the

text: derivative

text: of

text: the

text: regulator

text: reads

text: t

text: R

text: C

text: R

text: C

text: t

text: k

text: t

text: k

text: where

text: C

text: mA

text: R

text: )m

text: m

text: k

text: C

text: A

text: R

text: )m

text: )m]

text: t

text: m

text: k

text: In

text: this

text: paper,

text: we

text: will

text: use

text: one

text: and

text: the

text: same

text: shape

text: function

text: R

text: for

text: all

text: ﬁelds.

text: This

text: choice

text: can

text: be

text: k

text: omitted

text: in

text: more

text: general

text: settings.


text: 15 C. Diagonal pieces Now we have all the ingredients that we need in order to evaluate the trace of the ﬂow equation. For this purpose we split the RG ﬂow into two parts according to (3.12). Using (3.13), (3.14) the equation for the diagonal part reads 1 (cid:0) − (cid:1) 1 − 1 − Tr G 1∂ R = Tr(G 1 C ∂ R ) + Tr(G 1 C ) (3.15) t k 1 t k 2 2 2 2 where G is deﬁned in (3.10) in terms of the coeﬃcients A and the proﬁle function R . With m k the optimised cutoﬀ (2.45) and its scale derivative (2.46) we can evaluate the functionals (3.6) and (3.7) for the above expressions. We note that for a type III cutoﬀ there will be an extra term in (2.46) proportional to ∂ E. One observes that both the above integrals in (3.15) will have as a t (cid:82) ∞ common factor the step function θ(k2 − z) and therefore we can change the integral limits from (cid:82) k2 (cid:3) 0 to and replace the step functions by 1 everywhere. After substituting − = z, we then have 0 for the coeﬃcients C , C and G in (3.15) the following results, 1 2 (cid:88)p − C = mA k2m 2 1 m m=1 (cid:88)p C (z) = (∂ A ) (k2m − zm) (3.16) 2 t m m=1 (cid:88)p G = A k2m . m m=0 Both G and C have become independent of z for the optimised cutoﬀ. Now we turn our attention 1 to the evaluation of the integrals Q [W ] for positive n, (3.6). The integrand W (z) in (3.6) can n be split in two parts according to (3.15). Deﬁning W = (C ∂ R )/(2G) and W = C /(2G) we 1 1 t k 2 2 evaluate the corresponding integrals Ii = Q [W ] separately. Also these two integrals contain step n n i functions which are treated as above. Then we have (cid:80) k2n 1 p mA k2m I1 = (cid:80)m=1 m (3.17) n Γ(n) n p A k2m (cid:80)m=0 m I2 = 1 k2n 1 pm(cid:80)=1 mm+n∂tAmk2m . (3.18) n 2 Γ(n) n p A k2m m=0 m By summing the two contributions (3.17) and (3.18) we get an expression for the coeﬃcients Q of the heat kernel expansion for positive n, n Q = I1 + I2 , (n > 0) . (3.19) n n n For negative integer n, the z-dependence of W (z) in the Q-functionals (3.7) comes only from the ∂ Rφiφj term. It is then enough to compute the derivative of this term. In the previous sections we t k have disregarded the explicit form of the θ(k2−z)-functions, whose eﬀect, as overall factors, amounts to a change of integration limits. Here, we use the properties of the step function as introduced ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: C.

text: Diagonal

text: pieces

text: Now

text: we

text: have

text: all

text: the

text: ingredients

text: that

text: we

text: need

text: in

text: order

text: to

text: evaluate

text: the

text: trace

text: of

text: the

text: ﬂow

text: equation.

text: For

text: this

text: purpose

text: we

text: split

text: the

text: RG

text: ﬂow

text: into

text: two

text: parts

text: according

text: to

text: Using

text: the

text: equation

text: for

text: the

text: diagonal

text: part

text: reads

text: Tr

text: G

text: 1∂

text: R

text: Tr(G

text: C

text: R

text: Tr(G

text: C

text: t

text: k

text: t

text: k

text: where

text: G

text: is

text: deﬁned

text: in

text: in

text: terms

text: of

text: the

text: coeﬃcients

text: A

text: and

text: the

text: proﬁle

text: function

text: R

text: With

text: m

text: k

text: the

text: optimised

text: cutoﬀ

text: and

text: its

text: scale

text: derivative

text: we

text: can

text: evaluate

text: the

text: functionals

text: and

text: for

text: the

text: above

text: expressions.

text: We

text: note

text: that

text: for

text: a

text: type

text: III

text: cutoﬀ

text: there

text: will

text: be

text: an

text: extra

text: term

text: in

text: proportional

text: to

text: E.

text: One

text: observes

text: that

text: both

text: the

text: above

text: integrals

text: in

text: will

text: have

text: as

text: a

text: t

text: common

text: factor

text: the

text: step

text: function

text: z)

text: and

text: therefore

text: we

text: can

text: change

text: the

text: integral

text: limits

text: from

text: to

text: and

text: replace

text: the

text: step

text: functions

text: by

text: everywhere.

text: After

text: substituting

text: z,

text: we

text: then

text: have

text: for

text: the

text: coeﬃcients

text: C

text: C

text: and

text: G

text: in

text: the

text: following

text: results,

text: C

text: mA

text: m

text: C

text: (z)

text: A

text: zm)

text: t

text: m

text: G

text: A

text: m

text: Both

text: G

text: and

text: C

text: have

text: become

text: independent

text: of

text: z

text: for

text: the

text: optimised

text: cutoﬀ.

text: Now

text: we

text: turn

text: our

text: attention

text: to

text: the

text: evaluation

text: of

text: the

text: integrals

text: Q

text: [W

text: for

text: positive

text: n,

text: The

text: integrand

text: W

text: (z)

text: in

text: can

text: n

text: be

text: split

text: in

text: two

text: parts

text: according

text: to

text: Deﬁning

text: W

text: (C

text: R

text: and

text: W

text: C

text: we

text: t

text: k

text: evaluate

text: the

text: corresponding

text: integrals

text: Ii

text: Q

text: [W

text: separately.

text: Also

text: these

text: two

text: integrals

text: contain

text: step

text: n

text: n

text: i

text: functions

text: which

text: are

text: treated

text: as

text: above.

text: Then

text: we

text: have

text: p

text: mA

text: m

text: n

text: Γ(n)

text: n

text: p

text: A

text: m

text: n

text: Γ(n)

text: n

text: p

text: A

text: m

text: By

text: summing

text: the

text: two

text: contributions

text: and

text: we

text: get

text: an

text: expression

text: for

text: the

text: coeﬃcients

text: Q

text: of

text: the

text: heat

text: kernel

text: expansion

text: for

text: positive

text: n,

text: n

text: Q

text: (n

text: n

text: n

text: n

text: For

text: negative

text: integer

text: n,

text: the

text: z-dependence

text: of

text: W

text: (z)

text: in

text: the

text: Q-functionals

text: comes

text: only

text: from

text: the

text: Rφiφj

text: term.

text: It

text: is

text: then

text: enough

text: to

text: compute

text: the

text: derivative

text: of

text: this

text: term.

text: In

text: the

text: previous

text: sections

text: we

text: t

text: k

text: have

text: disregarded

text: the

text: explicit

text: form

text: of

text: the

text: whose

text: eﬀect,

text: as

text: overall

text: factors,

text: amounts

text: to

text: a

text: change

text: of

text: integration

text: limits.

text: Here,

text: we

text: use

text: the

text: properties

text: of

text: the

text: step

text: function

text: as

text: introduced

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##